From 270ee228f87ddb91706c2d598bc2af7dd8637de8 Mon Sep 17 00:00:00 2001 From: Thorsten Fuchs Date: Sat, 23 May 2026 22:07:27 +0200 Subject: [PATCH] =?UTF-8?q?feat(corpus-v3):=20Wave=207=20=E2=80=94=20web?= =?UTF-8?q?=20projection=20layer=20for=20the=201886=20v3=20Corpus=20Items?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Adds the projection layer that makes the Corpus v3 graph (1922 items in the corpus-v3 repo) visible on the public site as /id/cid######/ pages with relation panels, identifier boxes, and JSON-LD. This is the first wave that surfaces v3 to the public site. The v2 corpus surfaces (Registry, /publications/, etc.) remain live and untouched until cutover (Wave 9). What lands scripts/sync_corpus_v3_to_site.py Mirrors corpus-v3/manifests/ → site/_data/corpus_v3/. Atomic write with validation gate (parses, checks expected top-level key). Reads from PRRP_CORPUS_V3_ROOT env or defaults to ../corpus-v3. 7 files synced: alias-index, cid-index, transition, item-types, relation-vocabulary, visibility-values, lifecycle-statuses. scripts/generate_corpus_v3_pages.py Reads corpus-v3/items/ and emits one Markdown collection document per public item under _corpus_v3_items/cid######.md with the full item record inlined as frontmatter. Atomic write, idempotent. Wave 7a default emits the 1886 visibility-public + deprecated-public items (36 private/internal items skipped). _layouts/corpus-v3-item.html Single layout, conditional sections for atomic / artifact / formal / result / proof / tombstone modes. Renders: - hero card with title, CID, primary alias, status, version - tombstone banner (when visibility deprecated_public) - payload section (markdown / latex / artifact card / page) - proof block (when type: proof) — chips for mode/status/ formality/version-pinning, conditional warning, structured steps with prrp:// usage badges - result block (when type: result) — class/strength chips, commentary (short/technical/scope_limits), inspection route, falsification surface - formalization block (when formalization metadata present) — module, declaration, commit, lean_toolchain, axiom audit - identifier box (see include) - relation panels (see include) - version + history meta-list - status disclaimer + JSON-LD embed _includes/corpus-v3/identifier-box.html Compact identifier card. CID + primary alias + type + status + visibility + version + all legacy aliases + external identifiers (DOI/Wikidata/GitHub/arXiv with hyperlinks) + release lines. Web addendum §4 + doctrine §20. _includes/corpus-v3/relation-panels.html Five panels per web addendum §6: Upstream dependencies / Formalized by / Appears in / Supports & Tested by / Downstream uses (declared). Aggregates from canonical relations[] + convenience-field shortcuts (depends_on, formalized_by, appears_in). Each CID is a chip link to /id/{cid}/. Plus a Sources meta-list. _includes/corpus-v3/jsonld.html schema.org JSON-LD embed for every item. Type mapping: monograph → schema:Book chapter → schema:Chapter paper/note/dossier → schema:ScholarlyArticle release_packet/code/proof_package → schema:CreativeWork page items → schema:CreativeWork everything else → schema:DefinedTerm Custom pr: namespace for Corpus-specific predicates (layer, domain, formalizedBy, dependsOn, status, visibility, currentVersion, releaseLines). _config.yml Adds corpus_v3_items collection with permalink: /id/:name/. _data/corpus_v3/ Synced manifests (7 YAML files + 1 transition manifest). _corpus_v3_items/ 1886 generated Markdown collection documents — one per public Corpus Item. Each becomes /id/cid######/ in the built site. Doctrinal milestones reached - Every public Corpus v3 item now has a permanent /id/cid######/ URL - All 2992+ aliases tracked across items remain discoverable - prrp:// references in proof items resolve to internal CIDs which in turn route to /id/cid######/ - JSON-LD makes Corpus Items machine-readable for search engines and downstream consumers - The relation panel pattern + identifier box pattern are now reusable for the Result lane refactor in a later wave Wave 7b (deferred to follow-up): Result page filtered views (e.g., /results/physics filtered by layers contains E1), Pagefind search integration extension to index corpus_v3 collection, /corpus/graph/ public landing page, more elaborate tombstone polish. 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_includes/corpus-v3/relation-panels.html create mode 100644 _layouts/corpus-v3-item.html create mode 100644 scripts/generate_corpus_v3_pages.py create mode 100644 scripts/sync_corpus_v3_to_site.py diff --git a/_config.yml b/_config.yml index 40d792d2a..3dc1c098c 100644 --- a/_config.yml +++ b/_config.yml @@ -56,6 +56,9 @@ collections: output: true permalink: /corpus/changelog/:name/ sort_by: date + corpus_v3_items: + output: true + permalink: /id/:name/ predictions: output: true permalink: /predictions/:name/ diff --git a/_corpus_v3_items/cid000001.md b/_corpus_v3_items/cid000001.md new file mode 100644 index 000000000..c037c50ba --- /dev/null +++ b/_corpus_v3_items/cid000001.md @@ -0,0 +1,100 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000001/ +title: Book II — The τ-Kernel +cid: cid000001 +primary_alias: BOK0001 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book II of the Panta Rhei monograph series. Presents the five generators, the primitive progression operator, K0-K6, + multiplicative and exponential address structure, boundary algebra, τ-holomorphy, and internal truth machinery. The τ-Kernel + that all later books build on. + + ' +aliases: +- BOK0001 +- book-ii +layers: [] +domains: +- math.foundation +- geometry.calibration +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: +- cid000002 +- cid000007 +- cid000008 +- cid000010 +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: + doi: null + wikidata: null + github: null + osf: null + isbn: null + arxiv: null +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-ii/ + doi: null + isbn: null + hash: null + version: '1.0' + abstract: + markdown: 'Book II establishes the τ-Kernel: the minimal machinery from which all + + subsequent constructions derive. We introduce the five generators, the + + progression operator, the K0-K6 axioms, multiplicative and exponential + + address structures, boundary algebra, the τ-holomorphy condition, and + + the internal truth machinery (τ-topos + four-valued logic). + + ' + short_summary: 'The τ-Kernel construction — generators, progression operator, K-axioms, boundary algebra, τ-holomorphy, + internal truth machinery. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book II — The τ-Kernel*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book II of the Panta Rhei monograph series. Presents the five generators, the primitive progression operator, + K0-K6, multiplicative and exponential address structure, boundary algebra, τ-holomorphy, and internal truth machinery. The + τ-Kerne +noindex: false +--- + diff --git a/_corpus_v3_items/cid000002.md b/_corpus_v3_items/cid000002.md new file mode 100644 index 000000000..5dc917496 --- /dev/null +++ b/_corpus_v3_items/cid000002.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000002/ +title: Book II Chapter 5 — Master Constant Calibration +cid: cid000002 +primary_alias: CHP0001 +type: chapter +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Chapter 5 of Book II derives the master constant ι_τ = 2_τ / (π_τ + e_τ) as the calibration crossing fixed by earned + boundary constants and lobe-swap invariance. + + ' +aliases: +- CHP0001 +- book-ii-chapter-5 +layers: [] +domains: +- math.foundation +- geometry.calibration +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: +- cid000007 +- cid000008 +- cid000010 +part_of: +- cid000001 +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: html + url: https://panta-rhei.site/publications/books/book-ii/chapter-5/ + abstract: + markdown: 'We define the earned boundary constants (DEF0001), establish their + + lobe-swap invariance (LEM0001), and derive the master constant + + calibration ι_τ = 2_τ / (π_τ + e_τ) as the unique crossing fixed + + by the construction (THM0001). + + ' + short_summary: 'Derives ι_τ as the calibration crossing fixed by earned boundary constants and lobe-swap invariance. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Chapter 5 of Book II derives the master constant ι_τ = 2_τ / (π_τ + e_τ) as the calibration crossing fixed by + earned boundary constants and lobe-swap invariance. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000003.md b/_corpus_v3_items/cid000003.md new file mode 100644 index 000000000..f761190a4 --- /dev/null +++ b/_corpus_v3_items/cid000003.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000003/ +title: The Panta Rhei Construction Spine +cid: cid000003 +primary_alias: DOS0001 +type: research_dossier +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: 'The Construction Spine dossier traces the program''s construction from the E0 substrate through E1 (physics), E2 + (life), and E3 (reflection) layers. Each Construction Spine step (S001-S121) is itself a Corpus Item (construction_step + type) and is referenced from this dossier. + + ' +aliases: +- DOS0001 +- WP-RC-CS-E0-E3 +- construction-spine +layers: +- cross_layer +domains: +- math.foundation +- physics.micro +- physics.cosmology +- life.code +- reflection.language +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: + doi: null + github: null +payload: + kind: external_artifact + canonical_artifact: + type: html + url: https://panta-rhei.site/corpus/construction-spine/ + abstract: + markdown: 'The Construction Spine is the program''s backbone construction — + + a 121-step sequence from the E0 substrate (the τ-Kernel) through + + E1 (physical readout), E2 (life closure), and E3 (reflection + + machinery). Each step is a stand-alone Corpus Item carrying its + + construction obligations and dependencies. + + ' + short_summary: 'The 121-step construction sequence from E0 substrate to E3 reflection. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Construction Spine dossier traces the program's construction from the E0 substrate through E1 (physics), + E2 (life), and E3 (reflection) layers. Each Construction Spine step (S001-S121) is itself a Corpus Item (construction_step + type) an +noindex: false +--- + diff --git a/_corpus_v3_items/cid000004.md b/_corpus_v3_items/cid000004.md new file mode 100644 index 000000000..fbd8740af --- /dev/null +++ b/_corpus_v3_items/cid000004.md @@ -0,0 +1,101 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000004/ +title: Categorical AI +cid: cid000004 +primary_alias: PAP0001 +type: research_paper +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Presents a substrate-aligned framework for reading modern AI architectures through the τ-Kernel construction. Includes + falsifiable predictions about hallucination, in-context generalization, and architectural reliability. + + ' +aliases: +- PAP0001 +- categorical-ai +layers: +- E1 +- E2 +domains: +- ai.alignment +- ai.interpretability +- ai.reliability +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: +- cid000015 +- cid000016 +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: + doi: null + arxiv: null +payload: + kind: external_artifact + canonical_artifact: + type: pdf + url: https://panta-rhei.site/publications/papers/categorical-ai/ + doi: null + osf: null + hash: null + abstract: + markdown: 'This paper presents a substrate-aligned framework for reading modern + + AI architectures (transformer, mixture-of-experts, recurrent, hybrid) + + through the τ-Kernel construction. We articulate prediction families + + around hallucination as patch failure, in-context generalization as + + finite-budget coherence extension, and architectural reliability as + + a function of substrate-alignment depth. + + ' + latex: 'This paper presents a substrate-aligned framework for reading modern + + AI architectures through the τ-Kernel construction... + + ' + short_summary: 'Presents a substrate-aligned architecture map and prediction catalog for modern AI systems. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Categorical AI: A Substrate-Aligned + + Framework and Falsifiable Predictions*. Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Presents a substrate-aligned framework for reading modern AI architectures through the τ-Kernel construction. + Includes falsifiable predictions about hallucination, in-context generalization, and architectural reliability. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000005.md b/_corpus_v3_items/cid000005.md new file mode 100644 index 000000000..9d3519e3c --- /dev/null +++ b/_corpus_v3_items/cid000005.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000005/ +title: Research Note 001 — Master Constant Calibration +cid: cid000005 +primary_alias: NOT0001 +type: research_note +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'A short companion note to Book II Chapter 5, re-deriving ι_τ from earned boundary constants and noting the corresponding + TauLib formalization. + + ' +aliases: +- NOT0001 +- RN-001 +- master-constant-calibration-note +layers: [] +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: summarizes + object: cid000010 +- predicate: explained_in + object: cid000002 +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: html + url: https://panta-rhei.site/publications/research-notes/rn-001/ + abstract: + markdown: 'We re-derive the master constant ι_τ = 2_τ / (π_τ + e_τ) from earned + + boundary constants (DEF0001) and lobe-swap invariance (LEM0001), + + and link to the corresponding TauLib formalization (FTH0001). + + ' + short_summary: 'Companion note to Book II Chapter 5 — re-derives ι_τ and points at the TauLib formalization. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A short companion note to Book II Chapter 5, re-deriving ι_τ from earned boundary constants and noting the corresponding + TauLib formalization. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000006.md b/_corpus_v3_items/cid000006.md new file mode 100644 index 000000000..84eef997b --- /dev/null +++ b/_corpus_v3_items/cid000006.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000006/ +title: Corpus v3 Wave 2 Seed Release +cid: cid000006 +primary_alias: REL0001 +type: release_packet +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Wave 2 of the Corpus v3 build-out — releases the first 22 seed Corpus Items spanning every payload class (atomic, + artifact, formal, result, proof, page) to validate the schema and helper library in practice. + + ' +aliases: +- REL0001 +- corpus-v3-wave-2-seed +layers: [] +domains: [] +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: +- cid000001 +- cid000002 +- cid000003 +- cid000004 +- cid000005 +- cid000007 +- cid000008 +- cid000009 +- cid000010 +- cid000011 +- cid000012 +- cid000013 +- cid000014 +- cid000015 +- cid000016 +- cid000017 +- cid000018 +- cid000019 +- cid000020 +- cid000021 +- cid000022 +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: github_repo + url: https://github.com/Panta-Rhei-Research/corpus-v3 + version: wave-2-done + hash: null + short_summary: 'Corpus v3 Wave 2 — schema-validated seed items demonstrating all payload classes. + + ' + citation_markdown: 'Panta Rhei Research Program. *Corpus v3 Wave 2 Seed Release*. + + GitHub: Panta-Rhei-Research/corpus-v3, tag wave-2-done, 2026-05-10. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Wave 2 of the Corpus v3 build-out — releases the first 22 seed Corpus Items spanning every payload class (atomic, + artifact, formal, result, proof, page) to validate the schema and helper library in practice. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000007.md b/_corpus_v3_items/cid000007.md new file mode 100644 index 000000000..bee9f3564 --- /dev/null +++ b/_corpus_v3_items/cid000007.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000007/ +title: Earned Boundary Constants +cid: cid000007 +primary_alias: DEF0001 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'The set of boundary points obtained via lobe-swap closure on the τ-domain. Earned constants are the constructive + boundary data that the master constant calibration consumes. + + ' +aliases: +- DEF0001 +- DEF0142 +- earned-boundary-constants +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: explained_in + object: cid000005 +depends_on: [] +appears_in: +- type: chapter + id: cid000002 + locator: Book II Chapter 5 +- type: monograph + id: cid000001 + locator: Book II +- type: dossier + id: cid000003 + locator: Construction Spine +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\mathcal{B}_\\tau \\;:=\\; \\{\\, b \\in \\partial X \\;:\\;\n b \\text{ is earned via lobe-swap closure\ + \ on } X_\\tau \\,\\}.\n\\]\n" + markdown: 'The set of *earned boundary constants* is the subset of boundary points + + obtained via lobe-swap closure on the τ-domain: + + + `B_tau := { b in dX : b earned via lobe-swap on X_tau }`. + + + Earned constants are the construction inputs for the master constant + + calibration. See `LEM0001` (lobe-swap invariance) and `THM0001` + + (master constant calibration). + + ' + plain: 'The earned boundary constants are the subset of boundary points + + obtained via lobe-swap closure on the tau-domain. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The set of boundary points obtained via lobe-swap closure on the τ-domain. Earned constants are the constructive + boundary data that the master constant calibration consumes. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000008.md b/_corpus_v3_items/cid000008.md new file mode 100644 index 000000000..7b1db4beb --- /dev/null +++ b/_corpus_v3_items/cid000008.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000008/ +title: Lobe-Swap Invariance +cid: cid000008 +primary_alias: LEM0001 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'The crossing condition determined by earned boundary constants is invariant under the lobe-swap operation. Establishes + that the calibration crossing is well-defined. + + ' +aliases: +- LEM0001 +- LEM0028 +- lobe-swap-invariance +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: used_by + object: cid000010 +- predicate: explained_in + object: cid000005 +depends_on: +- cid000007 +appears_in: +- type: chapter + id: cid000002 + locator: Book II Chapter 5 +- type: monograph + id: cid000001 + locator: Book II +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\forall b \\in \\mathcal{B}_\\tau,\\quad\n \\text{crossing}(b) = \\text{crossing}(\\sigma_{\\text{lobe}}(b)),\n\ + \\]\nwhere $\\sigma_{\\text{lobe}}$ denotes the lobe-swap operator on $X_\\tau$.\n" + markdown: '**Lemma (Lobe-Swap Invariance).** For every earned boundary constant + + `b ∈ B_tau`, the crossing condition determined by `b` equals the + + crossing condition determined by `σ_lobe(b)`, where `σ_lobe` is the + + lobe-swap operator on the τ-domain. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The crossing condition determined by earned boundary constants is invariant under the lobe-swap operation. Establishes + that the calibration crossing is well-defined. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000009.md b/_corpus_v3_items/cid000009.md new file mode 100644 index 000000000..9a3f32d27 --- /dev/null +++ b/_corpus_v3_items/cid000009.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000009/ +title: Refinement-Pressure Uniqueness +cid: cid000009 +primary_alias: PRP0001 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'The calibration crossing fixed by earned boundary constants and lobe-swap invariance is unique under refinement + pressure. Ensures the master constant is well-defined and not artifactual. + + ' +aliases: +- PRP0001 +- PRP0042 +- refinement-pressure-uniqueness +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: used_by + object: cid000010 +depends_on: +- cid000007 +- cid000008 +appears_in: +- type: chapter + id: cid000002 + locator: Book II Chapter 5 +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\exists!\\; c \\in \\mathbb{R}\\;:\\; c = \\iota_\\tau \\text{ under refinement-pressure\n stability on\ + \ } \\mathcal{B}_\\tau.\n\\]\n" + markdown: '**Proposition (Refinement-Pressure Uniqueness).** The calibration crossing + + fixed by `DEF0001` and `LEM0001` is unique under refinement-pressure + + stability on `B_tau`. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The calibration crossing fixed by earned boundary constants and lobe-swap invariance is unique under refinement + pressure. Ensures the master constant is well-defined and not artifactual. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000010.md b/_corpus_v3_items/cid000010.md new file mode 100644 index 000000000..6842c2ba7 --- /dev/null +++ b/_corpus_v3_items/cid000010.md @@ -0,0 +1,108 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000010/ +title: Master Constant Calibration +cid: cid000010 +primary_alias: THM0001 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Establishes the master coupling ι_τ = 2_τ / (π_τ + e_τ) as the calibrated crossing value of the constructed geometry. + Central theorem of Book II. + + ' +aliases: +- THM0001 +- THM0031 +- II.T25 +- H3 +- iota_tau_master_constant +- master-constant-calibration +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: formalized_by + object: cid000020 +- predicate: explained_in + object: cid000005 +- predicate: appears_in + object: cid000003 +- predicate: derives + object: cid000052 +depends_on: +- cid000007 +- cid000008 +- cid000009 +appears_in: +- type: chapter + id: cid000002 + locator: Book II Chapter 5 +- type: monograph + id: cid000001 + locator: Book II +- type: dossier + id: cid000003 + locator: Construction Spine step S024 +formalized_by: +- type: formal_theorem + id: cid000020 + alias: FTH0001 + module: TauLib.Geometry.CentralTheorem.Calibration +sources: +- type: monograph + item: cid000001 + locator: Book II, theorem II.T25 +- type: dossier + item: cid000003 + locator: Construction Spine Step S024 +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\iota_\\tau \\;=\\; \\frac{2_\\tau}{\\pi_\\tau + e_\\tau}.\n\\]\n" + markdown: '**Theorem (Master Constant Calibration).** The master coupling is + + + `iota_tau = 2_tau / (pi_tau + e_tau)`. + + + This is the unique crossing value fixed by earned boundary constants + + (`DEF0001`), lobe-swap invariance (`LEM0001`), and refinement-pressure + + uniqueness (`PRP0001`). + + ' + plain: 'The master coupling is iota_tau = 2_tau / (pi_tau + e_tau). + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Establishes the master coupling ι_τ = 2_τ / (π_τ + e_τ) as the calibrated crossing value of the constructed + geometry. Central theorem of Book II. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000011.md b/_corpus_v3_items/cid000011.md new file mode 100644 index 000000000..3a0718e3e --- /dev/null +++ b/_corpus_v3_items/cid000011.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000011/ +title: ι_τ Explains the Fine-Structure Constant +cid: cid000011 +primary_alias: CLM0001 +type: claim +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Scholarly claim that the master constant ι_τ together with the dimensional factor (11/15)² yields α at multi-loop + precision, with the closed-form algebraic shortcut α = (11/15)²·ι_τ⁴ giving 9.8 ppm agreement. + + ' +aliases: +- CLM0001 +- iota-tau-explains-fine-structure +layers: +- E1 +domains: +- physics.micro +- math.foundation +relations: +- predicate: tested_by + object: cid000015 +- predicate: explained_in + object: cid000004 +depends_on: +- cid000010 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\alpha \\;\\approx\\; \\left(\\frac{11}{15}\\right)^2 \\iota_\\tau^4\n \\quad\\text{(LO closed-form, 9.8\ + \ ppm)}\n\\]\n" + markdown: '**Claim (ι_τ Explains α).** The master constant `ι_τ` together with the + + dimensional factor `(11/15)²` yields the fine-structure constant at multi- + + loop precision. The closed-form algebraic shortcut + + + `α ≈ (11/15)² · ι_τ⁴` + + + gives 9.8 ppm agreement with CODATA α. The full multi-loop derivation + + (IV.T107) reaches ~0 ppm. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Scholarly claim that the master constant ι_τ together with the dimensional factor (11/15)² yields α at multi-loop + precision, with the closed-form algebraic shortcut α = (11/15)²·ι_τ⁴ giving 9.8 ppm agreement. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000012.md b/_corpus_v3_items/cid000012.md new file mode 100644 index 000000000..0fb3c4861 --- /dev/null +++ b/_corpus_v3_items/cid000012.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000012/ +title: τ-Einstein Equation Bridges to Orthodox GR +cid: cid000012 +primary_alias: BRG0001 +type: bridge_claim +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Bridge claim that the τ-Einstein equation (boundary-character equality) reduces under the local readout functor + to the orthodox Einstein field equation G_μν = (8πG/c⁴)T_μν in the chart limit. + + ' +aliases: +- BRG0001 +- tau-einstein-equation-bridge +layers: +- E1 +- E0_E1_bridge +domains: +- physics.cosmology +- gravity +- math.foundation +relations: +- predicate: orthodox_target + object: doi:10.1002/19053220607 +- predicate: tested_by + object: cid000015 +depends_on: +- cid000010 +- cid000014 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\text{Under the local readout functor }\\rho:\\mathcal{C}_\\tau\\to\\text{Lor}_4,\\quad\n \\rho(\\text{τ-Einstein})\ + \ = G_{\\mu\\nu} + \\Lambda g_{\\mu\\nu}\n = \\frac{8\\pi G}{c^4}\\,T_{\\mu\\nu}.\n\\]\n" + markdown: '**Bridge Claim (τ-Einstein → GR).** The τ-Einstein equation (boundary- + + character equality) reduces under the local readout functor to the + + orthodox Einstein field equation `G_μν + Λg_μν = (8πG/c⁴) T_μν` in + + the chart limit. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Bridge claim that the τ-Einstein equation (boundary-character equality) reduces under the local readout functor + to the orthodox Einstein field equation G_μν = (8πG/c⁴)T_μν in the chart limit. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000013.md b/_corpus_v3_items/cid000013.md new file mode 100644 index 000000000..f82607b81 --- /dev/null +++ b/_corpus_v3_items/cid000013.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000013/ +title: Dark Sector as Carrier Capacity-Gradient Readout +cid: cid000013 +primary_alias: RDO0001 +type: readout_claim +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Readout claim that what orthodox cosmology attributes to a dark-matter particle sector — flat rotation curves, large-scale + structure, lensing amplitudes — is, in the τ-macrocosm readout, attributed to carrier capacity-gradient and holonomy structure + on the substrate. + + ' +aliases: +- RDO0001 +- dark-sector-readout +layers: +- E1 +domains: +- physics.cosmology +- gravity +relations: +- predicate: tested_by + object: cid000015 +- predicate: explained_in + object: cid000001 +depends_on: +- cid000012 +- cid000014 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Readout Claim (Dark Sector).** In the τ-macrocosm readout, flat + + rotation curves and large-scale structure are attributed to carrier + + capacity-gradient and holonomy structure rather than to a separate + + dark-matter particle sector. + + + This is a *readout* claim — what the τ-framework projects when asked + + "what is the orthodox observation of dark matter actually a readout + + of?" — not a *bridge* claim (which would assert a specific + + correspondence with a named orthodox object). + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Readout claim that what orthodox cosmology attributes to a dark-matter particle sector — flat rotation curves, + large-scale structure, lensing amplitudes — is, in the τ-macrocosm readout, attributed to carrier capacity-gradient and + holonomy ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000014.md b/_corpus_v3_items/cid000014.md new file mode 100644 index 000000000..dfe0590e7 --- /dev/null +++ b/_corpus_v3_items/cid000014.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000014/ +title: Bridge Functor Exists +cid: cid000014 +primary_alias: AXM0001 +type: axiom +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Project-local axiom assumed without proof: a bridge functor F: τ-Cat → Orth-Cat exists, mapping τ-side constructions + to their orthodox counterparts. Used as a named gate in conditional proofs. + + ' +aliases: +- AXM0001 +- bridge-functor-exists +layers: +- E0_E1_bridge +domains: +- math.foundation +- category.theory +relations: +- predicate: bridges_to + object: cid000013 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: latex + latex: "\\[\n \\text{There exists a bridge functor } F:\\mathcal{C}_\\tau \\longrightarrow\n \\mathcal{C}_{\\text{orth}}\ + \ \\text{ preserving substrate-aligned structure.}\n\\]\n" + markdown: '**Bridge Functor Exists (AXM0001).** There exists a functor + + `F: tau-Cat → Orth-Cat` from the τ-side category to an orthodox-side + + category, preserving the substrate-aligned structure. + + + This is a project-local assumption: it is not derived inside the τ-Kernel + + construction. It is used as a named gate in conditional proofs (notably + + the Riemann hypothesis Challenge Response). + + ' +formalization: + status: lean_axiom_free + module: TauLib.Bridge.Functor + declaration: bridge_functor_exists + sorry_count: 0 + project_axioms: 1 + print_axioms: 'bridge_functor_exists + + ' +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Project-local axiom assumed without proof: a bridge functor F: τ-Cat → Orth-Cat exists, mapping τ-side constructions + to their orthodox counterparts. Used as a named gate in conditional proofs. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000015.md b/_corpus_v3_items/cid000015.md new file mode 100644 index 000000000..a7bfbb19d --- /dev/null +++ b/_corpus_v3_items/cid000015.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000015/ +title: CMB-S4 Tensor-to-Scalar Ratio r ≈ 0.0136 +cid: cid000015 +primary_alias: PRD0001 +type: prediction +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Tier-A prediction: the τ-framework predicts a tensor-to-scalar ratio r ≈ 0.0136 measurable by CMB-S4 (sensitivity + σ(r) ≈ 0.001). Falsifier is direct: r outside the predicted band excludes the framework. + + ' +aliases: +- PRD0001 +- cmb-s4-tensor-to-scalar +layers: +- E1 +domains: +- physics.cosmology +- cosmology.inflation +relations: +- predicate: tested_by + object: cid000016 +- predicate: explained_in + object: cid000004 +depends_on: +- cid000010 +- cid000013 +- cid000014 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Prediction (CMB-S4 r).** The τ-framework predicts the inflation + + tensor-to-scalar ratio at + + + `r ≈ 0.0136 ± 0.002` + + + derived from carrier-capacity geometry on the inflation substrate. + + CMB-S4 sensitivity σ(r) ≈ 0.001 makes this directly falsifiable. + + + Precision tier: **Tier-A** (sub-10%). + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tier-A prediction: the τ-framework predicts a tensor-to-scalar ratio r ≈ 0.0136 measurable by CMB-S4 (sensitivity + σ(r) ≈ 0.001). Falsifier is direct: r outside the predicted band excludes the framework. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000016.md b/_corpus_v3_items/cid000016.md new file mode 100644 index 000000000..a7648d99c --- /dev/null +++ b/_corpus_v3_items/cid000016.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000016/ +title: CMB-S4 r Measurement Outside Predicted Band +cid: cid000016 +primary_alias: FAL0001 +type: falsifier +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'If CMB-S4 measures the tensor-to-scalar ratio at r > 0.018 or r < 0.009 (95% CL), the τ-framework prediction PRD0001 + is falsified. + + ' +aliases: +- FAL0001 +- cmb-s4-r-falsifier +layers: +- E1 +domains: +- physics.cosmology +relations: +- predicate: tests + object: cid000015 +depends_on: +- cid000015 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Falsifier (CMB-S4 r).** If CMB-S4 measures the tensor-to-scalar ratio + + at `r > 0.018` or `r < 0.009` at 95% CL, the τ-framework prediction + + `PRD0001` is falsified. + + + The band corresponds to the 3σ envelope around the predicted central + + value `r ≈ 0.0136 ± 0.002`. Detection outside this band is incompatible + + with the carrier-capacity geometry derivation. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'If CMB-S4 measures the tensor-to-scalar ratio at r > 0.018 or r < 0.009 (95% CL), the τ-framework prediction + PRD0001 is falsified. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000017.md b/_corpus_v3_items/cid000017.md new file mode 100644 index 000000000..2d215a59f --- /dev/null +++ b/_corpus_v3_items/cid000017.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000017/ +title: TauLib v3 Full Axiom Audit +cid: cid000017 +primary_alias: OPN0001 +type: open_problem +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Open problem: complete the axiom-free formalization audit across the full TauLib v3 module set. Currently 1 project-local + axiom (AXM0001 bridge functor exists) remains in the bridge layer; the rest must round-trip through #print axioms producing + no project-local entries. + + ' +aliases: +- OPN0001 +- taulib-v3-full-axiom-audit +layers: +- E0 +- E1 +domains: +- math.foundation +- formalization.lean +relations: +- predicate: depends_on + object: cid000014 +- predicate: appears_in + object: cid000003 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Open Problem (TauLib v3 Full Axiom Audit).** Complete the axiom-free\nformalization audit across all TauLib\ + \ v3 modules.\n\nCurrent status:\n - τ-Kernel modules (MOD0001-MOD0050): axiom-free ✓\n - Calibration modules (MOD0051-MOD0080):\ + \ axiom-free ✓\n - Bridge modules: 1 project-local axiom (AXM0001) — by design\n - Readout modules: pending full audit\n\ + \nThe bridge axiom is *intentional* — it marks where the τ → orthodox\nbridge enters as an assumption. Other modules should\ + \ round-trip through\n`#print axioms` producing no project-local entries beyond the bridge\nlayer.\n" +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Open problem: complete the axiom-free formalization audit across the full TauLib v3 module set. Currently 1 + project-local axiom (AXM0001 bridge functor exists) remains in the bridge layer; the rest must round-trip through #print + axioms prod' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000018.md b/_corpus_v3_items/cid000018.md new file mode 100644 index 000000000..898b85312 --- /dev/null +++ b/_corpus_v3_items/cid000018.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000018/ +title: Book II Chapter 5 — Typo in ι_τ Numerical Value +cid: cid000018 +primary_alias: ERR0001 +type: erratum +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Erratum: Book II Chapter 5 (first printing) listed ι_τ ≈ 0.341305; the correct value to 6 decimals is ι_τ ≈ 0.341304. + The derivation is unaffected; only the displayed decimal expansion in equation (5.3) was off by 1 in the sixth decimal. + + ' +aliases: +- ERR0001 +- book-ii-chapter-5-typo-erratum +layers: [] +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: erratum_for + object: cid000002 +- predicate: corrected_by + object: cid000002 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Erratum.** In Book II Chapter 5 (first printing, equation 5.3), the + + numerical value of ι_τ was displayed as + + + > `ι_τ ≈ 0.341305` + + + The correct value to six decimal places is + + + > `ι_τ ≈ 0.341304` + + + (specifically, ι_τ = 2 / (π + e) = 0.3413035...). The derivation, + + proofs, and downstream calculations are unaffected — only the displayed + + sixth-decimal was off. + + + Second-printing copies and the online version are corrected. Cite the + + original Book II Chapter 5 alongside this erratum if quoting the + + affected equation. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Erratum: Book II Chapter 5 (first printing) listed ι_τ ≈ 0.341305; the correct value to 6 decimals is ι_τ ≈ + 0.341304. The derivation is unaffected; only the displayed decimal expansion in equation (5.3) was off by 1 in the sixth + decimal. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000019.md b/_corpus_v3_items/cid000019.md new file mode 100644 index 000000000..8b7c20fa3 --- /dev/null +++ b/_corpus_v3_items/cid000019.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000019/ +title: TauLib.Geometry.CentralTheorem.Calibration +cid: cid000019 +primary_alias: MOD0001 +type: formal_module +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Lean module containing the formalization of the master constant calibration theorem (THM0001) and its supporting + definitions and lemmas. Axiom-free formalization — no project-local axioms in this module''s dependency closure. + + ' +aliases: +- MOD0001 +- taulib-geometry-central-theorem-calibration +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +- formalization.lean +relations: +- predicate: implemented_by + object: cid000020 +- predicate: appears_in + object: cid000003 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: +- cid000020 +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.Geometry.CentralTheorem.Calibration + module_path: TauLib/Geometry/CentralTheorem/Calibration.lean + commit: '0000000000000000000000000000000000000000' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '0000000000000000000000000000000000000000' + sorry_count: 0 + project_axioms: 0 + print_axioms: 'propext, Classical.choice, Quot.sound + + ' +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Lean module containing the formalization of the master constant calibration theorem (THM0001) and its supporting + definitions and lemmas. Axiom-free formalization — no project-local axioms in this module''s dependency closure. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000020.md b/_corpus_v3_items/cid000020.md new file mode 100644 index 000000000..8d4a6f617 --- /dev/null +++ b/_corpus_v3_items/cid000020.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000020/ +title: TauLib.Geometry.CentralTheorem.iota_tau_calibration +cid: cid000020 +primary_alias: FTH0001 +type: formal_theorem +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Lean formalization of the master constant calibration theorem (THM0001). Declared in TauLib.Geometry.CentralTheorem.Calibration + (MOD0001). Axiom-free: print_axioms returns only Lean''s built-in axioms. + + ' +aliases: +- FTH0001 +- FTH0044 +- iota_tau_calibration +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +- formalization.lean +relations: +- predicate: formalizes + object: cid000010 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000019 +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.Geometry.CentralTheorem.Calibration + declaration: iota_tau_calibration + module_path: TauLib/Geometry/CentralTheorem/Calibration.lean + commit: '0000000000000000000000000000000000000000' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '0000000000000000000000000000000000000000' + sorry_count: 0 + project_axioms: 0 + print_axioms: 'propext, Classical.choice, Quot.sound + + ' +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Lean formalization of the master constant calibration theorem (THM0001). Declared in TauLib.Geometry.CentralTheorem.Calibration + (MOD0001). Axiom-free: print_axioms returns only Lean''s built-in axioms. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000021.md b/_corpus_v3_items/cid000021.md new file mode 100644 index 000000000..266c2af4b --- /dev/null +++ b/_corpus_v3_items/cid000021.md @@ -0,0 +1,135 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000021/ +title: No Independent Dark-Sector Particle Required in the τ Readout +cid: cid000021 +primary_alias: RSL0001 +type: result +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Reader-facing result: in the τ-macrocosm readout, flat rotation curves and large-scale structure are attributed + to carrier capacity-gradient and holonomy structure rather than to a separate dark-matter particle sector. + + ' +aliases: +- RSL0001 +- RSL0007 +- no-independent-dark-sector +layers: +- E1 +domains: +- physics.cosmology +- gravity +relations: +- predicate: tested_by + object: cid000015 +- predicate: tested_by + object: cid000016 +- predicate: explained_in + object: cid000001 +depends_on: +- cid000010 +- cid000012 +- cid000013 +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Result.** In the τ-macrocosm readout, the orthodox observations + + typically attributed to a dark-matter particle sector — flat rotation + + curves, large-scale structure, gravitational lensing amplitudes — are + + instead attributed to carrier capacity-gradient and holonomy structure + + on the τ-substrate. + + + The framework does not predict the *absence* of a dark-matter particle. + + It predicts that no *independent* dark-sector particle is *required* + + to account for the orthodox observations. + + ' +formalization: {} +proof: {} +result: + class: physical_readout + claim_strength: framework_consequence + inspection_route: + - cid000010 + - cid000012 + - cid000013 + - cid000015 + falsification_surface: cid000016 +commentary: + short: 'The orthodox dark-matter sector is the τ-framework''s readout of carrier + + capacity-gradient and holonomy structure — no independent particle + + required. + + ' + technical: 'The reduction follows from the τ-Einstein bridge (BRG0001) applied to + + the substrate''s carrier-capacity geometry. The orthodox gravitational + + field equations recover from the local-readout functor on the τ-Einstein + + identity, and the carrier-capacity term shows up where orthodox treats + + dark matter. This is a *readout* claim, not a *bridge* claim — the + + framework doesn''t assert "dark-matter halo = τ-object X"; it asserts + + "what orthodox is reading out as dark matter is, in the τ readout, + + carrier capacity-gradient structure." + + ' + scope_limits: 'This result does not claim that direct-detection experiments will fail. + + Direct detection of a dark-matter particle would be consistent with the + + framework only if the particle''s properties also factor through + + carrier-capacity geometry — a strong additional constraint, but not a + + direct framework violation. + + ' + inspection_note: 'Reviewers should walk the inspection_route in order: master constant + + calibration (THM0001) → τ-Einstein bridge (BRG0001) → dark-sector + + readout (RDO0001) → CMB-S4 falsifiable surface (PRD0001/FAL0001). + + ' +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Reader-facing result: in the τ-macrocosm readout, flat rotation curves and large-scale structure are attributed + to carrier capacity-gradient and holonomy structure rather than to a separate dark-matter particle sector. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000022.md b/_corpus_v3_items/cid000022.md new file mode 100644 index 000000000..4c265662e --- /dev/null +++ b/_corpus_v3_items/cid000022.md @@ -0,0 +1,149 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000022/ +title: Proof of Master Constant Calibration +cid: cid000022 +primary_alias: PRF0001 +type: proof +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Semi-formal Corpus-addressed proof of the master constant calibration theorem (THM0001). Combines earned boundary + constants (DEF0001), lobe-swap invariance (LEM0001), and refinement-pressure uniqueness (PRP0001) into the calibration crossing + identity. + + ' +aliases: +- PRF0001 +- PRF0042 +- proof-iota-calibration +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: proves + object: cid000010 +- predicate: uses + object: cid000007 +- predicate: uses + object: cid000008 +- predicate: uses + object: cid000009 +- predicate: formalized_by + object: cid000020 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'We prove the master constant calibration by combining the earned + + constants [DEF0001](prrp://def0001@v1), lobe-swap invariance + + [LEM0001](prrp://lem0001@v1), and refinement-pressure uniqueness + + [PRP0001](prrp://prp0001@v1). + + + By the lobe-swap invariance of the crossing condition (LEM0001) and + + the refinement-pressure uniqueness proposition (PRP0001), the + + calibration crossing fixed by earned boundary constants (DEF0001) is + + the unique value ι_τ = 2_τ / (π_τ + e_τ), as claimed by THM0001. + + ' +formalization: + status: lean_axiom_free + module: TauLib.Geometry.CentralTheorem.Calibration + declaration: iota_tau_calibration + proof_item: cid000020 + sorry_count: 0 + project_axioms: 0 +proof: + mode: semi_formal + conclusion: cid000010 + proof_status: complete + formality_level: prose_addressed + dependency_closure_status: checked + version_pinning: pinned + assumptions: [] + steps: + - id: s1 + label: Earned constants + text_markdown: 'The boundary constants used in the construction are available by + + [DEF0001](prrp://def0001@v1) — the lobe-swap closure on the + + τ-domain produces a non-empty set of earned constants. + + ' + uses: + - ref: prrp://def0001@v1 + role: uses_definition + - id: s2 + label: Lobe-swap invariance fixes the crossing + text_markdown: 'Apply [LEM0001](prrp://lem0001@v1): the crossing condition determined + + by an earned constant `b` equals the crossing condition determined + + by `σ_lobe(b)`. The crossing is therefore invariant under lobe-swap. + + ' + uses: + - ref: prrp://lem0001@v1 + role: uses_lemma + - id: s3 + label: Uniqueness under refinement pressure + text_markdown: 'Apply [PRP0001](prrp://prp0001@v1): the crossing fixed by `DEF0001` + + and `LEM0001` is unique under refinement-pressure stability on + + `B_τ`. There is exactly one calibration crossing. + + ' + uses: + - ref: prrp://prp0001@v1 + role: uses_theorem + - id: s4 + label: Conclude master constant identity + text_markdown: 'Combining steps s1-s3, the unique calibration crossing fixed by + + earned boundary constants and lobe-swap invariance is + + + `ι_τ = 2_τ / (π_τ + e_τ)`. + + + This establishes [THM0001](prrp://thm0001@v1). + + ' + derives: cid000010 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Semi-formal Corpus-addressed proof of the master constant calibration theorem (THM0001). Combines earned boundary + constants (DEF0001), lobe-swap invariance (LEM0001), and refinement-pressure uniqueness (PRP0001) into the calibration crossin +noindex: false +--- + diff --git a/_corpus_v3_items/cid000023.md b/_corpus_v3_items/cid000023.md new file mode 100644 index 000000000..9b71e7421 --- /dev/null +++ b/_corpus_v3_items/cid000023.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000023/ +title: Book I — Categorical Foundations +cid: cid000023 +primary_alias: BOK0002 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book I of the Panta Rhei monograph series. Establishes the categorical foundations the program builds on: orbits, + four-block order readout, non-redundant fold strata, the iterator ladder, and the categorical groundwork that the τ-Kernel + (Book II) constructs above. + + ' +aliases: +- BOK0002 +- book-i +layers: [] +domains: +- math.foundation +- category.theory +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-i/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book I establishes the categorical foundations of the program — the + + orbit architecture, four-block order readout, non-redundant fold + + strata, the iterator ladder, and the categorical groundwork that + + the τ-Kernel (Book II) constructs above. Pre-substrate; this book + + sets up the language in which everything later is spoken. + + ' + short_summary: 'Categorical foundations — orbits, four-block order readout, iterator ladder, the language the τ-Kernel constructs + above. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book I — Categorical Foundations*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book I of the Panta Rhei monograph series. Establishes the categorical foundations the program builds on: orbits, + four-block order readout, non-redundant fold strata, the iterator ladder, and the categorical groundwork that the τ-Kernel + (Bo' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000024.md b/_corpus_v3_items/cid000024.md new file mode 100644 index 000000000..d0ae09089 --- /dev/null +++ b/_corpus_v3_items/cid000024.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000024/ +title: Book III — The τ-Manifold +cid: cid000024 +primary_alias: BOK0003 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book III takes the τ-Kernel handoff (S029 in the Construction Spine) and constructs the τ-manifold: the manifold + structure on which physical readouts unfold. Includes the central theorem route, the τ-manifold geometry, and the bridge + to Book IV (Categorical Microcosm). + + ' +aliases: +- BOK0003 +- book-iii +layers: [] +domains: +- math.foundation +- geometry.manifold +- physics.micro +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-iii/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book III constructs the τ-manifold from the Book II handoff. Establishes + + the manifold''s geometric structure, the central-theorem route on it, + + and the bridge to Book IV (Categorical Microcosm) where physical + + readout begins. + + ' + short_summary: 'The τ-manifold — manifold geometry on the Kernel handoff, central-theorem route, bridge to physical readout. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book III — The τ-Manifold*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book III takes the τ-Kernel handoff (S029 in the Construction Spine) and constructs the τ-manifold: the manifold + structure on which physical readouts unfold. Includes the central theorem route, the τ-manifold geometry, and the bridge + to Boo' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000025.md b/_corpus_v3_items/cid000025.md new file mode 100644 index 000000000..0f779fabe --- /dev/null +++ b/_corpus_v3_items/cid000025.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000025/ +title: Book IV — Categorical Microcosm +cid: cid000025 +primary_alias: BOK0004 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book IV opens the physical readout: particles, fields, coupling constants, the fine-structure constant α with its + multi-loop derivation (IV.T107) and closed-form algebraic LO shortcut α = (11/15)²·ι_τ⁴ at 9.8 ppm. The first book where + the τ-construction projects measurable physics. + + ' +aliases: +- BOK0004 +- book-iv +layers: +- E1 +domains: +- physics.micro +- physics.particle +- geometry.calibration +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-iv/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book IV opens the physical readout in the τ-construction. Establishes + + the particle and field content, derives the coupling constants + + (including the fine-structure constant α at multi-loop precision via + + IV.T107, with closed-form algebraic shortcut α = (11/15)²·ι_τ⁴ giving + + 9.8 ppm agreement), and connects to the SM electroweak sector. + + ' + short_summary: 'Physical readout — particles, fields, couplings, fine-structure constant derivation at multi-loop precision. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book IV — Categorical Microcosm*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book IV opens the physical readout: particles, fields, coupling constants, the fine-structure constant α with + its multi-loop derivation (IV.T107) and closed-form algebraic LO shortcut α = (11/15)²·ι_τ⁴ at 9.8 ppm. The first book where + the τ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000026.md b/_corpus_v3_items/cid000026.md new file mode 100644 index 000000000..d162d5d00 --- /dev/null +++ b/_corpus_v3_items/cid000026.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000026/ +title: Book V — Categorical Macrocosm +cid: cid000026 +primary_alias: BOK0005 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book V scales the readout from microcosm (Book IV) to macrocosm: the τ-Einstein identity reduces to G_μν = (8πG/c⁴)T_μν + in the chart limit, the orthodox dark-sector dissolves into carrier capacity-gradient and holonomy structure, and the cosmic + web is read as Wilson-loop carrier channels (S073). Predicts CMB-S4 tensor-to-scalar ratio r ≈ 0.0136. + + ' +aliases: +- BOK0005 +- book-v +layers: +- E1 +domains: +- physics.cosmology +- gravity +- cosmology.inflation +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-v/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book V scales the τ-readout to cosmic scales. The τ-Einstein identity + + (BRG0001) reduces to G_μν + Λg_μν = (8πG/c⁴) T_μν in the chart limit, + + recovering general relativity. The orthodox dark-sector dissolves into + + carrier capacity-gradient and holonomy structure (RSL0001/RDO0001). + + Predicts CMB-S4 tensor-to-scalar ratio r ≈ 0.0136 (PRD0001) with + + falsifier FAL0001. + + ' + short_summary: 'Cosmological readout — τ-Einstein bridges to GR, dark-sector dissolved, cosmic web as carrier channels, + CMB-S4 r prediction. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book V — Categorical Macrocosm*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book V scales the readout from microcosm (Book IV) to macrocosm: the τ-Einstein identity reduces to G_μν = (8πG/c⁴)T_μν + in the chart limit, the orthodox dark-sector dissolves into carrier capacity-gradient and holonomy structure, and the co' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000027.md b/_corpus_v3_items/cid000027.md new file mode 100644 index 000000000..812125123 --- /dev/null +++ b/_corpus_v3_items/cid000027.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000027/ +title: Book VI — The Code of Life +cid: cid000027 +primary_alias: BOK0006 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book VI opens the E2 layer: the τ-readout of life. Establishes life closure as an operational concept (organism + boundary, self-production, operational closure), reads the organism as a cohomology patch, and treats evolution as a morphology + tree on the τ-manifold. + + ' +aliases: +- BOK0006 +- book-vi +layers: +- E2 +- E1_E2_bridge +domains: +- life.code +- life.organism +- life.evolution +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-vi/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book VI opens the τ-readout of life. The E2 layer establishes life + + closure (boundary, self-production, operational closure), reads the + + organism as a cohomology patch on the τ-manifold, and treats evolution + + as a morphology tree on that manifold. Bridges to Book V (cosmic + + substrate the carrier capacity lives in) and forward to Book VII + + (reflective structure). + + ' + short_summary: 'Biological readout — life closure, organism as cohomology patch, evolution as morphology tree on the τ-manifold. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book VI — The Code of Life*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book VI opens the E2 layer: the τ-readout of life. Establishes life closure as an operational concept (organism + boundary, self-production, operational closure), reads the organism as a cohomology patch, and treats evolution as a morphology ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000028.md b/_corpus_v3_items/cid000028.md new file mode 100644 index 000000000..014eccfa6 --- /dev/null +++ b/_corpus_v3_items/cid000028.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000028/ +title: Book VII — Logos and the Ontic Closure +cid: cid000028 +primary_alias: BOK0007 +type: monograph +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Book VII closes the construction with the E3 layer: the reflective structure. Treats mind, language, formal systems, + and the ontic-closure burden — the question of whether the program''s construction successfully hosts itself as a formal-mathematical, + life-bearing, and reflective object. + + ' +aliases: +- BOK0007 +- book-vii +layers: +- E3 +- E2_E3_bridge +domains: +- reflection.language +- reflection.mind +- math.foundation +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: external_artifact + canonical_artifact: + type: book_pdf + url: https://panta-rhei.site/publications/books/book-vii/ + doi: null + isbn: null + version: '1.0' + abstract: + markdown: 'Book VII closes the τ-construction with the E3 layer: reflective + + structure. Establishes the four registers of reason (S093), treats + + formal systems as represented objects (S095), addresses the Gödel + + and halting boundaries (S096), language/meaning/proof arc (S097), + + logic/inference/truth at reflective scale (S098), mind/consciousness/ + + commitment-access (S099), and the logos boundary + ontic-closure + + burden (S100) — the question of whether the program''s construction + + successfully self-hosts. + + ' + short_summary: 'Reflective structure — mind, language, formal systems, the ontic-closure burden. The construction''s self-hosting + closure. + + ' + citation_markdown: 'Fuchs, Thorsten, and Anna-Sophie Fuchs. *Book VII — Logos and the Ontic Closure*. + + Panta Rhei Research Program, 2026. + + ' + citation_metadata: + authors: + - Fuchs, Thorsten + - Fuchs, Anna-Sophie + year: 2026 + publisher: Panta Rhei Research Program + edition: First Edition +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book VII closes the construction with the E3 layer: the reflective structure. Treats mind, language, formal + systems, and the ontic-closure burden — the question of whether the program''s construction successfully hosts itself as + a formal-mat' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000029.md b/_corpus_v3_items/cid000029.md new file mode 100644 index 000000000..378b2646a --- /dev/null +++ b/_corpus_v3_items/cid000029.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000029/ +title: Non-import discipline +cid: cid000029 +primary_alias: CNS0001 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The construction begins by refusing the usual ambient supports: no ZFC universe as ontic background, no continuum, + no classical first-order semantics, no spacetime, observer, runtime, physical primitives, or measurement structure.' +aliases: +- CNS0001 +- S001 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The construction begins by refusing the usual ambient supports: no ZFC universe as ontic background, no continuum, + no classical first-order semantics, no spacetime, observer, runtime, physical primitives, or measurement structure. Reasoning + may be carried out in CIC/Lean as an external audit language, but τ-objects themselves must be generated internally. This + establishes the first burden: every later mathematical or physical object, operation, or readout must be earned from the + kernel rather than borrowed from the metalanguage.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The construction begins by refusing the usual ambient supports: no ZFC universe as ontic background, no continuum, + no classical first-order semantics, no spacetime, observer, runtime, physical primitives, or measurement structure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000030.md b/_corpus_v3_items/cid000030.md new file mode 100644 index 000000000..2e7812990 --- /dev/null +++ b/_corpus_v3_items/cid000030.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000030/ +title: Coherence kernel +cid: cid000030 +primary_alias: CNS0002 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The bare kernel is fixed as five ordered generator symbols α < π < γ < η < ω, one progression/application operation + ρ as the sole primitive operation, and the constraints K0 -- K6. +aliases: +- CNS0002 +- S002 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The bare kernel is fixed as five ordered generator symbols α < π < γ < η < ω, one progression/application operation + ρ as the sole primitive operation, and the constraints K0 -- K6. K0 prevents τ from becoming one object among its own + objects; K2 makes ω the absorbing fixed point (ρ(ω)=ω); K3 -- K6 generate and close the four finite rays while forbidding + premature collapse into ω (K5 in particular: ρ^n(g)≠ω for non-ω g). This is the proposed minimal symbolic kernel: not + yet geometry, number, space, or physics, but the least constructive source claimed to carry them. The E0 claim is minimal + constructive sufficiency, not classical model-theoretic categoricity.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The bare kernel is fixed as five ordered generator symbols α < π < γ < η < ω, one progression/application operation + ρ as the sole primitive operation, and the constraints K0 -- K6. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000031.md b/_corpus_v3_items/cid000031.md new file mode 100644 index 000000000..9bbe4e9b8 --- /dev/null +++ b/_corpus_v3_items/cid000031.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000031/ +title: Orbit architecture +cid: cid000031 +primary_alias: CNS0003 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Applying ρ to the non-ω generators produces exactly four generated rays O_α, O_π, O_γ, O_η together with the singleton/fixed + boundary ω. +aliases: +- CNS0003 +- S003 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Applying ρ to the non-ω generators produces exactly four generated rays O_α, O_π, O_γ, O_η together with the + singleton/fixed boundary ω. K5 supplies the diagonal discipline: no finite iteration on a non-ω generator reaches ω. The + result is a linear-symbolic universe with four finite-directional channels and one boundary absorber: enough generated + directionality to avoid under-generation, but enough diagonal discipline to prevent unrestricted self-collapse. The diagonal + discipline maps structurally onto Girard''s !-free linear logic.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Applying ρ to the non-ω generators produces exactly four generated rays O_α, O_π, O_γ, O_η together with the + singleton/fixed boundary ω. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000032.md b/_corpus_v3_items/cid000032.md new file mode 100644 index 000000000..fd55555e6 --- /dev/null +++ b/_corpus_v3_items/cid000032.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000032/ +title: Four-block order readout +cid: cid000032 +primary_alias: CNS0004 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: In order-readout, the kernel is not a single generated ray. +aliases: +- CNS0004 +- S004 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'In order-readout, the kernel is not a single generated ray. It is a typed + + four-block stack + + + O_α ≺ O_π ≺ O_γ ≺ O_η ≺ ω . + + + This shape is an order-readout analogy, not imported ordinal theory. The + + classical ordinal ω would correspond only to the first post-α + + boundary position; the τ-symbol ω is the global absorber/top + + element after all four typed rays. The stack is therefore analogous to a finite + + ω-block shape, but each block carries a different fold regime. This is + + the E0 precursor of the later 4+1 sector template.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: In order-readout, the kernel is not a single generated ray. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000033.md b/_corpus_v3_items/cid000033.md new file mode 100644 index 000000000..f8a4f3c2a --- /dev/null +++ b/_corpus_v3_items/cid000033.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000033/ +title: Nonredundant fold strata +cid: cid000033 +primary_alias: CNS0005 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The four non-ω rays are not arbitrary names, and their indices are not introduced as pre-given primes. +aliases: +- CNS0005 +- S005 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The four non-ω rays are not arbitrary names, and their indices are not introduced as pre-given primes. Their + canonical roles are fixed at the fold level: α supplies direct counting, π the future multiplicity spine, γ the future + exponent channel, η the future tetration channel, and ω the fixed-point absorber and closure beacon. The fold-chain principle + then orders the operations: progression-depth supplies counting; addition is recovered over progression-depth; multiplication, + exponentiation, and tetration are successive finite folds; and the ladder saturates at the η-level, with no fourth finite + fold admitted as a canonical new address layer. Primality is still unavailable here: only after multiplication and divisibility + are earned does the internal prime enumeration identify the scalar prime-readout sequence 2,3,5, rather than import it. + Prime-addressability is therefore earned by fold structure, while ω remains the global iteration boundary: ρ(ω)=ω and + ρ^n(g)≠ω for non-ω generators.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The four non-ω rays are not arbitrary names, and their indices are not introduced as pre-given primes. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000034.md b/_corpus_v3_items/cid000034.md new file mode 100644 index 000000000..fae192746 --- /dev/null +++ b/_corpus_v3_items/cid000034.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000034/ +title: Counting as construction +cid: cid000034 +primary_alias: CNS0006 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For each generator g, the expression φ_g(n)=ρ^n(g) is not a classification of pre-existing numbers. +aliases: +- CNS0006 +- S006 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'For each generator g, the expression φ_g(n)=ρ^n(g) is not a classification of pre-existing numbers. It is the + generated count itself: the n-th finite application of ρ along an orbit. Counting therefore appears first as constructive + depth, not as imported ℕ, and it must be kept separate from later calculation inside earned algebraic structures. The + metalanguage may index this process by n, but the τ-object is the generated orbit position.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For each generator g, the expression φ_g(n)=ρ^n(g) is not a classification of pre-existing numbers. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000035.md b/_corpus_v3_items/cid000035.md new file mode 100644 index 000000000..79e011d33 --- /dev/null +++ b/_corpus_v3_items/cid000035.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000035/ +title: Iterator ladder +cid: cid000035 +primary_alias: CNS0007 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Once progression depth exists, addition is recovered as the finite fold of ρ on the τ-index scaffold. +aliases: +- CNS0007 +- S007 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Once progression depth exists, addition is recovered as the finite fold of ρ on the τ-index scaffold. The higher + ladder then folds successively: multiplication is iterated addition, exponentiation is iterated multiplication, and tetration + is iterated exponentiation. Thus raw ρ underlies the four concrete address levels +, ×, exponentiation, and tetration, + which saturate at tetration. Algebra enters as disciplined recursion over generated symbolic depth. The inspection question + is whether this ladder truly derives the operations from ρ, or whether ordinary arithmetic re-enters through the external + recursion language.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Once progression depth exists, addition is recovered as the finite fold of ρ on the τ-index scaffold. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000036.md b/_corpus_v3_items/cid000036.md new file mode 100644 index 000000000..d893e2d06 --- /dev/null +++ b/_corpus_v3_items/cid000036.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000036/ +title: Address resolution, not calculation +cid: cid000036 +primary_alias: CNS0008 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The iterator ladder would remain unstable without canonical addresses. +aliases: +- CNS0008 +- S008 +layers: +- E0 +domains: +- construction.step.formal +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The iterator ladder would remain unstable without canonical addresses. The construction therefore introduces + normal forms in the genealogical DAG: codes reduce to a unique representative, and equality becomes finite-witness normal-form + comparison rather than unrestricted symbolic calculation. The Cayley word metric on addresses is integer-valued and effective, + supplying a genuine internal distance on generated code paths. This is the first place where arithmetic becomes address + geometry.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The iterator ladder would remain unstable without canonical addresses. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000037.md b/_corpus_v3_items/cid000037.md new file mode 100644 index 000000000..929bb109b --- /dev/null +++ b/_corpus_v3_items/cid000037.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000037/ +title: Hyperfactorization and ABCD chart +cid: cid000037 +primary_alias: CNS0009 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Address geometry is made global by hyperfactorization. +aliases: +- CNS0009 +- S009 +layers: +- E0 +domains: +- construction.step.formal +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Address geometry is made global by hyperfactorization. Every X≥ 2 receives a unique decomposition X=((A↑↑ C)^B)· + D as a tower atom T(A,B,C)=(A↑↑ C)^B multiplied by a residual D. At this point divisibility and internal primality have + already been earned. In the τ-statement, A is the largest internal prime atom (under idx, the largest prime divisor of + X), C is the maximal tetration height of the A-tower factor, B is the residual tower exponent, and D has only internal + prime factors strictly less than A. The resulting ABCD chart, written in later scalar notation as + + + :ℕ_≥ 2 to P×ℕ_≥ 1^3, + + + is injective and turns arithmetic objects into typed global addresses. This prevents number from collapsing into a flat + additive/multiplicative background and preserves the tower structure needed for boundary polarity downstream.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Address geometry is made global by hyperfactorization. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000038.md b/_corpus_v3_items/cid000038.md new file mode 100644 index 000000000..4fcbb7bb1 --- /dev/null +++ b/_corpus_v3_items/cid000038.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000038/ +title: Prime polarity +cid: cid000038 +primary_alias: CNS0010 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: After internal primes and CRT idempotents are available, prime polarity is fixed by the CRT-local/spectral-Legendre + classifier, not by the ABCD growth-rate comparison alone. +aliases: +- CNS0010 +- S010 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: After internal primes and CRT idempotents are available, prime polarity is fixed by the CRT-local/spectral-Legendre + classifier, not by the ABCD growth-rate comparison alone. For odd p, p is B-labelled iff (2/p)=+1 iff p≡±1±od 8, and C-labelled + iff (2/p)=-1 iff p≡±3±od 8; 2 carries the mixed X-label. The τ-derivation lives at the idempotent level in H_τ Z/Primorial(n) + Z, via CRT idempotents and spectral Legendre reduction, and agrees pointwise with the orthodox Legendre classifier. Both + B and C classes are infinite, indeed density 1/2. The older growth-rate story is structural motivation, not the derivation. +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: After internal primes and CRT idempotents are available, prime polarity is fixed by the CRT-local/spectral-Legendre + classifier, not by the ABCD growth-rate comparison alone. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000039.md b/_corpus_v3_items/cid000039.md new file mode 100644 index 000000000..b8a0a4df2 --- /dev/null +++ b/_corpus_v3_items/cid000039.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000039/ +title: Internal set theory +cid: cid000039 +primary_alias: CNS0011 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Set-like behavior is reconstructed from generated divisibility and address containment, not imported from an ambient + powerset universe. +aliases: +- CNS0011 +- S011 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: Set-like behavior is reconstructed from generated divisibility and address containment, not imported from an ambient + powerset universe. Membership becomes an internal divisibility/address relation; K5 blocks the diagonal move that would + require a completed external totality; cardinality collapses, within the τ-internal readout, to a single countable grade + _0. This gives τ enough set-theoretic grammar for later construction while refusing Cantorian infinity as primitive ontology. +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Set-like behavior is reconstructed from generated divisibility and address containment, not imported from an + ambient powerset universe. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000040.md b/_corpus_v3_items/cid000040.md new file mode 100644 index 000000000..a80f51500 --- /dev/null +++ b/_corpus_v3_items/cid000040.md @@ -0,0 +1,104 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000040/ +title: Split-complex boundary algebra +cid: cid000040 +primary_alias: CNS0012 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Prime polarity cannot remain a bare classification of fold strata: it must be given an algebra in which the two + polar orientations can be projected, compared, and later transported.' +aliases: +- CNS0012 +- S012 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Prime polarity cannot remain a bare classification of fold strata: it must be + + given an algebra in which the two polar orientations can be projected, compared, + + and later transported. The construction therefore passes to the bipolar boundary + + algebra + + + 𝔻_τ=R_τ ⊕ jR_τ, j^2=+1 , + + + where R_τ denotes the earned scalar/readout algebra available at this + + stage, not an imported real line. + + Its idempotents + + + e_±=(1± j)/(2) + + + split boundary behavior into orthogonal channels, and the + + σ-equivariant Boolean sublattice + + + B_σ(D)=0,e_+,e_-,1 + + + gives the first four-atom boundary grammar. At this stage the atoms are not + + physical sectors; they are pre-physical channel eigenstates with the + + pre-physics dictionary 0 null, e_+γ/B, + + e_-η/C, 1α/D. Physical names such as + + electromagnetic, strong, and gravitational are assigned only after the Book-III + + carrier and sector readout are constructed.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Prime polarity cannot remain a bare classification of fold strata: it must be given an algebra in which the + two polar orientations can be projected, compared, and later transported.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000041.md b/_corpus_v3_items/cid000041.md new file mode 100644 index 000000000..544f8be41 --- /dev/null +++ b/_corpus_v3_items/cid000041.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000041/ +title: τ-holomorphy as ω-germ transformer +cid: cid000041 +primary_alias: CNS0013 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Once boundary atoms exist, the next burden is not to allow arbitrary maps. +aliases: +- CNS0013 +- S013 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Once boundary atoms exist, the next burden is not to allow arbitrary maps. + + The construction must earn a class of admissible transformations from the + + boundary grammar itself. A τ-holomorphic map is therefore defined as a + + certified ω-germ transformer: normal-form code with type, + + tail-stability, and tail-independence, without a prior function graph, + + Cartesian product, or prior external codomain. The diagonal discipline structurally + + precludes ordinary function-graph constructions; idempotent collapse to + + elliptic C is impossible. Its components satisfy split-complex + + Cauchy -- Riemann equations and the hyperbolic wave equation, not the elliptic + + Laplacian, because j^2=+1. This earns analytic structure without importing + + complex analysis, and it supplies the admissible-map grammar that Book II will + + use to derive continuity and topology.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Once boundary atoms exist, the next burden is not to allow arbitrary maps. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000042.md b/_corpus_v3_items/cid000042.md new file mode 100644 index 000000000..ada8182e2 --- /dev/null +++ b/_corpus_v3_items/cid000042.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000042/ +title: Four-valued τ-topos +cid: cid000042 +primary_alias: CNS0014 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The four boundary atoms now acquire a second role: they classify internal truth.' +aliases: +- CNS0014 +- S014 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The four boundary atoms now acquire a second role: they classify internal truth. The subobject classifier Ω_τ=B_σ(𝔻)=Neither,True,False,Both + turns the boundary algebra into a four-valued internal logic with paraconsistent negation σ and Belnap -- Dunn structure. + Self-referential propositions stabilize by Cauchy iteration; the value Both is not a metaphorical "both" but the multiplicative + unit of 𝔻 -- - algebraic identity, not metaphor. Thus the same idempotent algebra that separates boundary channels also + carries semantic load: truth, failure, overlap, and underdetermination become internal values rather than external interpretive + bookkeeping.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The four boundary atoms now acquire a second role: they classify internal truth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000043.md b/_corpus_v3_items/cid000043.md new file mode 100644 index 000000000..edfbd95ba --- /dev/null +++ b/_corpus_v3_items/cid000043.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000043/ +title: Lemniscate characters +cid: cid000043 +primary_alias: CNS0015 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: With the four-valued boundary algebra in place, boundary behavior can be organized into characters. +aliases: +- CNS0015 +- S015 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'With the four-valued boundary algebra in place, boundary behavior can be organized into characters. The characters + χ_± read the two idempotent channels of 𝔻 and assemble them into the algebraic lemniscate: the bipolar locus where B-polar + and C-polar tower behavior meet without collapsing into one channel. This gives the boundary a character ring, not merely + a set of truth values. Book II will later supply the geometric readout of this algebraic lemniscate: the T^2-fiber degenerates + at ω into the wedge locus that carries the same character grammar.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: With the four-valued boundary algebra in place, boundary behavior can be organized into characters. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000044.md b/_corpus_v3_items/cid000044.md new file mode 100644 index 000000000..9403cd92f --- /dev/null +++ b/_corpus_v3_items/cid000044.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000044/ +title: Global Hartogs +cid: cid000044 +primary_alias: CNS0016 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: ' character grammar, CRT reconstruction, tower coherence, and the τ-Identity Theorem now support Global Hartogs: + τ-holomorphic data on L K, for primordially thin K, extends uniquely to L, so coherent ω-tail data determine the finite-stage + values they project...' +aliases: +- CNS0016 +- S016 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000023 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-01 + macro_title: Build the τ-Kernel +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The spectral character grammar, CRT reconstruction, tower coherence, and the τ-Identity Theorem now support Global + Hartogs: τ-holomorphic data on L K, for primordially thin K, extends uniquely to L, so coherent ω-tail data determine + the finite-stage values they project to. This is not an imported transfinite completion theorem; it is the first internal + limit/stage discipline of τ. The construction can speak about its own infinite-stage extensions because boundary data, + character grammar, and admissible ω-germ transformers have already been earned. This closes Book I''s mathematical layer + by showing that the kernel has enough internal grammar to host limit constructions without leaving the system; proof-substrate + audit remains part of the formal trust budget rather than a separate construction step.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: ' character grammar, CRT reconstruction, tower coherence, and the τ-Identity Theorem now support Global Hartogs: + τ-holomorphic data on L K, for primordially thin K, extends uniquely to L, so coherent ω-tail data determine the finite-stage + va' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000045.md b/_corpus_v3_items/cid000045.md new file mode 100644 index 000000000..7cc313b68 --- /dev/null +++ b/_corpus_v3_items/cid000045.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000045/ +title: Boundary-first inversion +cid: cid000045 +primary_alias: CNS0017 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book II reverses the classical order in which topology is chosen first, continuity is defined over it, and holomorphy + is then imposed as an analytic condition. +aliases: +- CNS0017 +- S017 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book II reverses the classical order in which topology is chosen first, + + continuity is defined over it, and holomorphy is then imposed as an analytic + + condition. In τ, the admissible ω-germ transformer grammar of + + Book I is primary: holomorphy/probe compatibility supplies continuity; + + continuity supplies topology; topology supplies geometry. This inversion is + + the reason Book II can earn topology from the kernel rather than borrow it as + + ambient structure. The inspection question is correspondingly sharp: whether + + any topological assumption enters before the ω-germ/probe grammar has + + earned it.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book II reverses the classical order in which topology is chosen first, continuity is defined over it, and holomorphy + is then imposed as an analytic condition. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000046.md b/_corpus_v3_items/cid000046.md new file mode 100644 index 000000000..bde797aad --- /dev/null +++ b/_corpus_v3_items/cid000046.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000046/ +title: τ^3 as geometric readout τ^1×_f T^2 +cid: cid000046 +primary_alias: CNS0018 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The geometric object denoted τ^3 is constructed as a readout of τ, not adopted as a Cartesian product. +aliases: +- CNS0018 +- S018 +layers: +- E0 +domains: +- construction.step.formal +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The geometric object denoted τ^3 is constructed as a readout of + + τ, not adopted as a Cartesian product. It is not + + τ×τ×τ. The base τ^1 is the (D,A) + + radial/address base of the ABCD chart; the fiber T^2 is the (B,C) + + solenoidal fiber built from the two bipolar character channels. The fiber map + + f records the nontrivial dependency, so no global trivialization + + τ^3≅τ^1× T^2 is asserted. This fibered readout is the + + canonical geometric body that later becomes the substrate candidate for the + + Book-III physical carrier.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The geometric object denoted τ^3 is constructed as a readout of τ, not adopted as a Cartesian product. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000047.md b/_corpus_v3_items/cid000047.md new file mode 100644 index 000000000..13e46f23f --- /dev/null +++ b/_corpus_v3_items/cid000047.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000047/ +title: Cylinders and ultrametric topology +cid: cid000047 +primary_alias: CNS0019 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The local domains of τ^3 are not open sets chosen from an external space. +aliases: +- CNS0019 +- S019 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The local domains of τ^3 are not open sets chosen from an external + + space. They are ABCD-prefix cylinders: refinement classes of address tails. + + Two points are close when their generated address data agree for a long initial + + segment; their first-disagreement depth δ(t,t'') defines the ultrametric + + + d(t,t'')=2^-δ(t,t''). + + + Thus locality is earned from prefix agreement. The topology is the topology of + + address resolution, not a topology imposed on an already spatial object.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The local domains of τ^3 are not open sets chosen from an external space. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000048.md b/_corpus_v3_items/cid000048.md new file mode 100644 index 000000000..04b4d5fc0 --- /dev/null +++ b/_corpus_v3_items/cid000048.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000048/ +title: Stone/profinite shape and two-readout principle +cid: cid000048 +primary_alias: CNS0020 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The ultrametric cylinder topology is compact, Hausdorff, and totally disconnected: a Stone/profinite shape.' +aliases: +- CNS0020 +- S020 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The ultrametric cylinder topology is compact, Hausdorff, and totally + + disconnected: a Stone/profinite shape. This does not contradict the later + + geometric readout, because τ carries two compatible readouts of the same + + earned structure. The profinite readout preserves refinement, limit, and + + address information; the geometric readout gives the body on which incidence, + + fiber structure, and later carrier projection can be read. The fine + + Stone/profinite topology and coarse geometric readout are parallel consequences + + of the same address/coherence data. The two-readout principle prevents either + + side from being collapsed into the other: geometry is not topology, but it is + + canonically read from the same address structure.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The ultrametric cylinder topology is compact, Hausdorff, and totally disconnected: a Stone/profinite shape.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000049.md b/_corpus_v3_items/cid000049.md new file mode 100644 index 000000000..be2c797a1 --- /dev/null +++ b/_corpus_v3_items/cid000049.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000049/ +title: Geometric lemniscate at ω +cid: cid000049 +primary_alias: CNS0021 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The algebraic lemniscate of Book I now receives a geometric body. +aliases: +- CNS0021 +- S021 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The algebraic lemniscate of Book I now receives a geometric body. At the + + ω-boundary, the T^2-fiber degenerates into a wedge locus whose two + + lobes correspond to the B-polar and C-polar character channels. Boundary, + + wedge, and character-ring residence coincide at this locus. The point is not + + decorative: the same bipolar grammar that organized split-complex boundary + + characters now appears as the limiting geometry of the fiber.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The algebraic lemniscate of Book I now receives a geometric body. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000050.md b/_corpus_v3_items/cid000050.md new file mode 100644 index 000000000..ce5c7441a --- /dev/null +++ b/_corpus_v3_items/cid000050.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000050/ +title: Tarski geometry and wave-type causal precursor +cid: cid000050 +primary_alias: CNS0022 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: ' the ultrametric address topology along the geometric readout produces an internally Tarski-style Euclidean geometry: + incidence, betweenness, and congruence are recovered from address comparison and readout-stability, not imposed as primitive + spatial relati...' +aliases: +- CNS0022 +- S022 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Coarse-graining the ultrametric address topology along the geometric readout + + produces an internally Tarski-style Euclidean geometry: incidence, + + betweenness, and congruence are recovered from address comparison and + + readout-stability, not imposed as primitive spatial relations. In parallel, + + the split-complex relation j^2=+1 supplies a hyperbolic, wave-type causal + + precursor. This is not yet physical time; it is the hyperbolic algebraic + + precursor that later readout layers may use.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: ' the ultrametric address topology along the geometric readout produces an internally Tarski-style Euclidean + geometry: incidence, betweenness, and congruence are recovered from address comparison and readout-stability, not imposed + as primiti' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000051.md b/_corpus_v3_items/cid000051.md new file mode 100644 index 000000000..90e772fa8 --- /dev/null +++ b/_corpus_v3_items/cid000051.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000051/ +title: Earned constants pi_tau,e_tau,j +cid: cid000051 +primary_alias: CNS0023 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The first structural constants are read from already-earned geometry and iteration. +aliases: +- CNS0023 +- S023 +layers: +- E0 +domains: +- construction.step.formal +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The first structural constants are read from already-earned geometry and + + iteration. pi_tau is obtained from the internal Euclidean/incidence + + geometry; e_tau from the eigenvalue structure of the -iterator; + + j from the split-complex idempotent boundary algebra. The dyadic scalar + + 2_τ is the two-channel/refinement scalar read from the earned idempotent + + boundary split, not the imported natural number 2. None of these constants + + is imported from ordinary real or complex analysis. Each records a different + + internal stabilization: geometric closure, iterative growth, and bipolar + + boundary decomposition.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The first structural constants are read from already-earned geometry and iteration. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000052.md b/_corpus_v3_items/cid000052.md new file mode 100644 index 000000000..6cf43630f --- /dev/null +++ b/_corpus_v3_items/cid000052.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000052/ +title: Master constant iota_tau +cid: cid000052 +primary_alias: CNS0024 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The three earned ingredients pi_tau,e_tau, and the dyadic refinement scalar 2_τ meet in the ω-limit as the master + coupling iota_tau=(2_τ)/(pi_tau+e_tau). +aliases: +- CNS0024 +- S024 +layers: +- E0 +domains: +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The three earned ingredients pi_tau,e_tau, and the dyadic refinement + + scalar 2_τ meet in the ω-limit as the master coupling + + + iota_tau=(2_τ)/(pi_tau+e_tau). + + + Its closed form is anchored in II.T25/H3 and in TauLib as + + iota_tau(pi_tau+e_tau)≡2_τ. Lobe-swap invariance, + + refinement-pressure uniqueness, and primorial-stabilized combinatorial + + linearity force the crossing value; the Central-Theorem route later transports + + this calibration, it does not rederive the closed form. The numerical value + + iota_tau≈0.341 is therefore presented as a structural calibration + + constant of the constructed geometry, not as a fitted parameter.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The three earned ingredients pi_tau,e_tau, and the dyadic refinement scalar 2_τ meet in the ω-limit as the master + coupling iota_tau=(2_τ)/(pi_tau+e_tau). +noindex: false +--- + diff --git a/_corpus_v3_items/cid000053.md b/_corpus_v3_items/cid000053.md new file mode 100644 index 000000000..18dda5e04 --- /dev/null +++ b/_corpus_v3_items/cid000053.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000053/ +title: Local Hartogs and mutual determination +cid: cid000053 +primary_alias: CNS0025 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The boundary grammar now becomes a local continuation principle. +aliases: +- CNS0025 +- S025 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The boundary grammar now becomes a local continuation principle. The + + bounded-lift operator BndLift_n certifies that, at each finite depth, + + refinement tails, spectral tails, ω-germs, boundary characters, and + + Hartogs continuation are mutually determined. Knowing any one of these data + + types fixes the others up to the admissible τ-holomorphic grammar. + + Boundary and interior therefore cease to be independent descriptions: they are + + two readouts of one stabilized continuation structure. This rigidity is what + + will later make physical localization non-arbitrary; the carrier cannot choose + + interior data freely once boundary data are fixed.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The boundary grammar now becomes a local continuation principle. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000054.md b/_corpus_v3_items/cid000054.md new file mode 100644 index 000000000..3f6c26244 --- /dev/null +++ b/_corpus_v3_items/cid000054.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000054/ +title: Regularity and finite spectral support +cid: cid000054 +primary_alias: CNS0026 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The admissible maps of Step 25 must also be regular in a finite, inspectable sense. +aliases: +- CNS0026 +- S026 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-02 + macro_title: Recover Core Mathematics +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The admissible maps of Step 25 must also be regular in a finite, inspectable + + sense. Within the τ-holomorphic category, regularity is the positive + + finite-stabilization condition: at p, the ω-germ/evolution + + sequence stabilizes at some finite primorial depth N. Equivalently, the + + B- and C-channel regularity conditions both stabilize, so the canonical + + idempotent-supported extension exists. Regularity is therefore not imposed as + + an external analytic smoothness condition; it is finite stabilization of the + + earned boundary/evolution grammar.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The admissible maps of Step 25 must also be regular in a finite, inspectable sense. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000055.md b/_corpus_v3_items/cid000055.md new file mode 100644 index 000000000..cd30c843a --- /dev/null +++ b/_corpus_v3_items/cid000055.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000055/ +title: Sheaf coherence, self-enrichment, Yoneda +cid: cid000055 +primary_alias: CNS0027 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Finite regularity is still local; the next burden is gluing. +aliases: +- CNS0027 +- S027 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-03 + macro_title: Internalize Self-Enrichment +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Finite regularity is still local; the next burden is gluing. The + + ω-germ presheaf satisfies sheaf coherence because local admissible maps + + agree on refinement-tail overlaps, so the sheaf gluing axiom pastes them + + without adding external matching data. This gluing result enables + + self-enrichment: every hom-object + + + [A,B] + + + is itself a τ-object, rather than an external collection of arrows. + + Yoneda then becomes a theorem of the construction — fully faithful and + + bipolar-preserving — not an imported embedding choice. This is the decisive + + handoff to Book III: the construction has now earned internal relational sites, + + and those hom-object loci are exactly what the physical-carrier problem will + + use as candidates for E_1 localization.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Finite regularity is still local; the next burden is gluing. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000056.md b/_corpus_v3_items/cid000056.md new file mode 100644 index 000000000..94012ce3b --- /dev/null +++ b/_corpus_v3_items/cid000056.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000056/ +title: Central Theorem route +cid: cid000056 +primary_alias: CNS0028 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Central Theorem assembles the preceding machinery into the bulk -- boundary correspondence: 𝒪(τ^3) ≅ A_spec(𝕃).' +aliases: +- CNS0028 +- S028 +layers: +- E0 +domains: +- construction.step.formal +- construction.step.hinge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-03 + macro_title: Internalize Self-Enrichment +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The Central Theorem assembles the preceding machinery into the bulk -- boundary + + correspondence: + + + 𝒪(τ^3) ≅ A_spec(𝕃). + + + The holomorphic interior, the boundary character ring, and the spectral algebra + + are not three independent objects; under the theorem they determine one another + + through a calibrated split-complex isomorphism. Its iota_tau-calibration + + property transports the already-earned H3/II.T25 master-constant calibration + + from boundary data into the interior; it does not rederive the closed form. The finite rank-(3,15) Lean check and + + the exact registry dependencies live in the internal source ledger; the main point + + here is structural: boundary spectral data and holomorphic interior data become + + one rigid object.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Central Theorem assembles the preceding machinery into the bulk -- boundary correspondence: 𝒪(τ^3) ≅ A_spec(𝕃).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000057.md b/_corpus_v3_items/cid000057.md new file mode 100644 index 000000000..0f5616a40 --- /dev/null +++ b/_corpus_v3_items/cid000057.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000057/ +title: τ-manifold and Book III handoff +cid: cid000057 +primary_alias: CNS0029 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'he Central-Theorem chain by removing the specific classical objection that compactness would force all holomorphic + functions to be constant: because j^2=+1 gives a hyperbolic, not elliptic, operator, nonconstant bounded τ-holomorphic functions + remain availa...' +aliases: +- CNS0029 +- S029 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-03 + macro_title: Internalize Self-Enrichment +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The Liouville-type obstruction analysis closes the Central-Theorem chain by + + removing the specific classical objection that compactness would force all + + holomorphic functions to be constant: because j^2=+1 gives a hyperbolic, + + not elliptic, operator, nonconstant bounded τ-holomorphic functions + + remain available. What remains is not an ambient smooth manifold imported from + + outside, but a τ-manifold structure: boundary-determined data, + + admissible ω-germ transformations, finite spectral support, and the + + four-atom boundary grammar assembled into a single geometric object. Book II + + therefore ends with a canonical geometric body, but not yet physics. Book III + + can now ask the next question in a disciplined way: which internally earned + + relational structures of this body can serve as physical localization sites?' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'he Central-Theorem chain by removing the specific classical objection that compactness would force all holomorphic + functions to be constant: because j^2=+1 gives a hyperbolic, not elliptic, operator, nonconstant bounded τ-holomorphic functi' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000058.md b/_corpus_v3_items/cid000058.md new file mode 100644 index 000000000..f508ec878 --- /dev/null +++ b/_corpus_v3_items/cid000058.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000058/ +title: Kernel minimality and categoricity +cid: cid000058 +primary_alias: CNS0030 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The construction uses three distinct notions of uniqueness, and the document keeps them separate. +aliases: +- CNS0030 +- S030 +layers: +- E0 +domains: +- construction.step.formal +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000001 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-03 + macro_title: Internalize Self-Enrichment +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The construction uses three distinct notions of uniqueness, and the document + + keeps them separate. First, at E0, the operative claim is minimal + + constructive sufficiency: the five generator symbols, ρ, and K0 -- K6 are + + claimed to be the least symbolic source able to carry address structure, + + topology, holomorphy, self-enrichment, and the E1 carrier. Fewer generators + + should collapse the construction; additional generators should be redundant or + + break canonical closure. Second, at the Central-Theorem level, the rank-(3,15) + + categoricity check says that the relevant fibered product with an ABCD chart + + satisfying K0 -- K5 is canonically isomorphic to τ^3; at that rank the + + moduli space is pt. Third, classical model-theoretic + + categoricity -- one model up to isomorphism over arbitrary models -- belongs + + to the later E3 self-hosting problem, where τ can represent formal + + systems, model talk, and its own kernel internally. This dossier uses the first + + two notions and defers the third.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The construction uses three distinct notions of uniqueness, and the document keeps them separate. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000059.md b/_corpus_v3_items/cid000059.md new file mode 100644 index 000000000..564539cd0 --- /dev/null +++ b/_corpus_v3_items/cid000059.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000059/ +title: E_0 is not yet physics +cid: cid000059 +primary_alias: CNS0031 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book II delivers a canonical geometric readout τ^3=τ^1×_f T^2, but that readout is not yet physical spacetime. +aliases: +- CNS0031 +- S031 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book II delivers a canonical geometric readout + + + τ^3=τ^1×_f T^2, + + + but that readout is not yet physical spacetime. Its loci are address-derived + + geometric positions inside the earned τ-geometry; they do not yet carry + + physical events, observables, dynamics, or measurement semantics. Book III + + therefore begins with a location problem rather than a physics claim: where, + + inside the structures already earned, can physical content be carried without + + importing a background manifold?' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book II delivers a canonical geometric readout τ^3=τ^1×_f T^2, but that readout is not yet physical spacetime. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000060.md b/_corpus_v3_items/cid000060.md new file mode 100644 index 000000000..74898774c --- /dev/null +++ b/_corpus_v3_items/cid000060.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000060/ +title: No bare-point ontology +cid: cid000060 +primary_alias: CNS0032 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Physical points are not primitive E_0-atoms and not coordinate tuples of τ^3. +aliases: +- CNS0032 +- S032 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Physical points are not primitive E_0-atoms and not coordinate tuples of + + τ^3. Either move would re-import the Cartesian point ontology that the + + construction has refused from the start. A physical site must instead be earned + + from relational structure internal to τ. This refusal is constructive: + + it forces Book III to identify a localization object that carries boundary, + + interior, and morphism data at once, rather than merely occupying a position in + + a pre-given space.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Physical points are not primitive E_0-atoms and not coordinate tuples of τ^3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000061.md b/_corpus_v3_items/cid000061.md new file mode 100644 index 000000000..5db779167 --- /dev/null +++ b/_corpus_v3_items/cid000061.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000061/ +title: E_1 hom-object loci +cid: cid000061 +primary_alias: CNS0033 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book II's self-enrichment now becomes the localization mechanism. +aliases: +- CNS0033 +- S033 +layers: +- E1 +domains: +- construction.step.formal +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book II''s self-enrichment now becomes the localization mechanism. Since + + + [A,B] Obj(τ), + + + morphisms are no longer external arrows between already-given points; they are + + internal τ-objects with boundary characters, holomorphic interiors, and + + ω-germ data. The first candidates for physical sites are therefore + + hom-object loci: relational sites that encode how structures meet and transform, + + not where they sit on an imported coordinate grid. This is the first + + E_1-level answer to the question "where can physics live?"' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book II's self-enrichment now becomes the localization mechanism. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000062.md b/_corpus_v3_items/cid000062.md new file mode 100644 index 000000000..dad577cc3 --- /dev/null +++ b/_corpus_v3_items/cid000062.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000062/ +title: Central-Theorem constraint on physical loci +cid: cid000062 +primary_alias: CNS0034 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A hom-object locus becomes a candidate physical site only if its boundary and interior data satisfy the mutual-determination + discipline of the Central Theorem. +aliases: +- CNS0034 +- S034 +layers: +- E1 +domains: +- construction.step.formal +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'A hom-object locus becomes a candidate physical site only if its boundary and + + interior data satisfy the mutual-determination discipline of the Central + + Theorem. The iota_tau-calibration channel transports boundary spectral + + data into the holomorphic interior, so localization is not a free label placed + + on a morphism. A physical locus is admitted only when boundary character, + + interior ω-germ, and spectral data cohere through the calibrated + + bulk -- boundary isomorphism.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A hom-object locus becomes a candidate physical site only if its boundary and interior data satisfy the mutual-determination + discipline of the Central Theorem. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000063.md b/_corpus_v3_items/cid000063.md new file mode 100644 index 000000000..f7e78ab70 --- /dev/null +++ b/_corpus_v3_items/cid000063.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000063/ +title: Hartogs bulk projection +cid: cid000063 +primary_alias: CNS0035 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For each admitted hom-object locus, the local T^2-fiber acts as a two-dimensional boundary surface carrying the four-atom + character data. +aliases: +- CNS0035 +- S035 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'For each admitted hom-object locus, the local T^2-fiber acts as a + + two-dimensional boundary surface carrying the four-atom character data. + + Hartogs-type extension projects this boundary data into a local + + three-dimensional bulk. The bulk is therefore not inserted as hidden space; it + + is the interior determined by the boundary grammar. At this stage the theorem + + is local: each admissible boundary determines its own bulk patch. The + + three-effective-dimensional readout remains subject to Book III''s stated + + dimensional-accounting caveat. What remains unproved here is whether all such + + patches glue into one coherent carrier.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For each admitted hom-object locus, the local T^2-fiber acts as a two-dimensional boundary surface carrying the + four-atom character data. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000064.md b/_corpus_v3_items/cid000064.md new file mode 100644 index 000000000..9466d53df --- /dev/null +++ b/_corpus_v3_items/cid000064.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000064/ +title: Local-to-global gluing burden +cid: cid000064 +primary_alias: CNS0036 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The local Hartogs bulks must assemble into one globally coherent three-dimensional carrier. +aliases: +- CNS0036 +- S036 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The local Hartogs bulks must assemble into one globally coherent + + three-dimensional carrier. Without this gluing, E_1 would contain many + + locally determined interiors but no physical space. The Book III problem is + + therefore a gluing problem: the local interiors must agree on overlaps, preserve + + their boundary/spectral data, and support a single carrier readout. This is the + + point where the famous external problem surfaces enter, not as prestige claims, + + but as names for the structural guarantees required by gluing.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The local Hartogs bulks must assemble into one globally coherent three-dimensional carrier. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000065.md b/_corpus_v3_items/cid000065.md new file mode 100644 index 000000000..70c2bb126 --- /dev/null +++ b/_corpus_v3_items/cid000065.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000065/ +title: Carrier guarantees and external shadows +cid: cid000065 +primary_alias: CNS0037 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The global carrier requires eight construction guarantees, each a condition on the local-to-global gluing data. +aliases: +- CNS0037 +- S037 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The global carrier requires eight construction guarantees, each a condition on + + the local-to-global gluing data. Spatial Force supplies simple-connectivity + + control (external shadow: Poincaré, externally established via Perelman and + + re-read internally). Harmonic Force supplies spectral purity (RH/GRH shadow). + + Regular Force supplies local Hartogs regularity (Navier -- Stokes shadow). + + Discrete Force supplies gap/discreteness (Yang -- Mills shadow). Legible Force + + supplies sector addressability (Hodge shadow). Codable Force supplies rational + + proto-code addressability (BSD shadow). Coherent Force supplies functorial + + automorphic/Galois gluing (Langlands shadow). Predictive Force is the carrier''s + + forward export condition; its full P vs NP / E_1 to E_2 reading is + + deferred. The framework does not claim externally settled Clay-valid resolutions + + of these problems; their names mark the external shadows of τ-internal + + construction obligations with explicit bridge gaps. The bridge ledger still + + classifies most orthodox translations as conjectural, with Poincaré established + + and P vs NP treated as a bridge break.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The global carrier requires eight construction guarantees, each a condition on the local-to-global gluing data. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000066.md b/_corpus_v3_items/cid000066.md new file mode 100644 index 000000000..0235dfa96 --- /dev/null +++ b/_corpus_v3_items/cid000066.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000066/ +title: Global Cartesian gluing and decompactification +cid: cid000066 +primary_alias: CNS0038 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: When the carrier guarantees hold, the local Hartogs bulks assemble into a coherent global three-dimensional Cartesian + readout. +aliases: +- CNS0038 +- S038 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'When the carrier guarantees hold, the local Hartogs bulks assemble into a + + coherent global three-dimensional Cartesian readout. The residual + + τ^3_R structure decompactifies at human scales into an R^3 + + readout. Thus ordinary three-dimensional space is not an input to the theory: + + it is the large-scale readout of globally glued Hartogs interiors. The step is + + bridge-pending on the disclosed Book III custom axioms, but its construction + + role is clear: this is where canonical τ^3-geometry becomes an + + E_1 spatial carrier. Here III.T50 refers to the corpus/manuscript Global + + Cartesian Gluing theorem; it is manuscript/registry-backed, not claimed here as + + the current TauLib label/module.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: When the carrier guarantees hold, the local Hartogs bulks assemble into a coherent global three-dimensional Cartesian + readout. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000067.md b/_corpus_v3_items/cid000067.md new file mode 100644 index 000000000..620be4c77 --- /dev/null +++ b/_corpus_v3_items/cid000067.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000067/ +title: 3+1 spacetime signature +cid: cid000067 +primary_alias: CNS0039 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The fibered shape already separates the base τ^1 from the T^2-fiber. +aliases: +- CNS0039 +- S039 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The fibered shape already separates the base τ^1 from the T^2-fiber. + + After global Cartesian gluing, the T^2-derived bulk supplies the spatial + + readout, while the τ^1-base supplies the temporal direction that Book V + + will develop as proper time. The resulting 3+1 signature is therefore not + + chosen as a background Lorentzian metric. It is the structural signature of the + + carrier: three spatial directions from the Hartogs bulk, one temporal base + + direction from τ^1.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The fibered shape already separates the base τ^1 from the T^2-fiber. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000068.md b/_corpus_v3_items/cid000068.md new file mode 100644 index 000000000..aac8c9f65 --- /dev/null +++ b/_corpus_v3_items/cid000068.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000068/ +title: 4+1 sector template +cid: cid000068 +primary_alias: CNS0040 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Only after the carrier has been located can the five-generator grammar receive physical sector readouts. +aliases: +- CNS0040 +- S040 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Only after the carrier has been located can the five-generator grammar receive + + physical sector readouts. Boundary functoriality + + + :Char(𝕃) to 𝒪(τ^3) + + + transports the character grammar into holomorphic sectors on the carrier: + + α to D (gravity), π to A (weak), γ to B + + (electromagnetic), η to C (strong), with ω as mediator + + (Higgs/mass). The 4+1 template is therefore not imposed from the Standard + + Model. It is the sectorization of the carrier by the generator/boundary + + grammar already earned; its physical sector readouts remain bridge-level at + + this point.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Only after the carrier has been located can the five-generator grammar receive physical sector readouts. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000069.md b/_corpus_v3_items/cid000069.md new file mode 100644 index 000000000..e1da2423b --- /dev/null +++ b/_corpus_v3_items/cid000069.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000069/ +title: No Knobs Principle +cid: cid000069 +primary_alias: CNS0041 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Once the sector template is in place, inter-sector couplings are no longer free dials. +aliases: +- CNS0041 +- S041 +layers: +- E1 +domains: +- construction.step.formal +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Once the sector template is in place, inter-sector couplings are no longer free + + dials. At each primorial depth M_d, the coupling + + + κ(Sector_i,Sector_j)=f_ij(iota_tau,M_d) + + + is canonically determined by the master constant and the depth data. The scope + + is intentionally narrow in this compact note: Book III''s No Knobs covers the + + 10-entry primitive-sector coupling ledger, while this paragraph uses only the + + inter-sector/export consequence. It is not a claim that every later numerical + + physics readout has already been theorem-proved at this stage. Book IV and Book + + V instantiate the sector grammar and expose the empirical readouts.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Once the sector template is in place, inter-sector couplings are no longer free dials. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000070.md b/_corpus_v3_items/cid000070.md new file mode 100644 index 000000000..b650fa28c --- /dev/null +++ b/_corpus_v3_items/cid000070.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000070/ +title: Hinge Theorem export +cid: cid000070 +primary_alias: CNS0042 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The Hinge Theorem closes Book III by turning the carrier construction into an export contract. +aliases: +- CNS0042 +- S042 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000024 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-04 + macro_title: Identify the Physical Carrier +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The Hinge Theorem closes Book III by turning the carrier construction into an + + export contract. Every result in Books IV -- V must be a sector instantiation of + + the E_1 enrichment just constructed: microphysics is read off the T^2 + + fiber, macrocosm off the τ^1 base, and inter-sector coupling off the + + No Knobs ledger, with the III.R40 temporal-spatial decomposition fixing the + + base/fiber export grammar. The later books are therefore not independent physical + + theories attached to the mathematics. They are typed readouts of the Book III + + carrier.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Hinge Theorem closes Book III by turning the carrier construction into an export contract. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000071.md b/_corpus_v3_items/cid000071.md new file mode 100644 index 000000000..9c737d410 --- /dev/null +++ b/_corpus_v3_items/cid000071.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000071/ +title: Layered equivalence +cid: cid000071 +primary_alias: CNS0043 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The enrichment ladder changes what "the same" can mean. +aliases: +- CNS0043 +- S043 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "The enrichment ladder changes what \"the same\" can mean. At E_0, sameness\nis identity of generated objects,\ + \ addresses, and normal forms. At E_1,\nphysics introduces physical-kind equivalence: two distinct carrier realizations\n\ + count as the same particle or mode when they preserve the same defect-bundle,\nholonomy, charge, mass, spin, and transport\ + \ invariants under admissible readout.\nThus \"this is a neutron\" does not assert numerical identity of substrate\npieces;\ + \ it asserts membership in the neutron equivalence class. Higher-layer\nanalogues -- code equivalence at E_2, semantic/proof\ + \ equivalence at E_3\n -- appear below as the construction continues into Books VI -- VII." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The enrichment ladder changes what "the same" can mean. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000072.md b/_corpus_v3_items/cid000072.md new file mode 100644 index 000000000..daa6aae4a --- /dev/null +++ b/_corpus_v3_items/cid000072.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000072/ +title: No stage/actor split +cid: cid000072 +primary_alias: CNS0044 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book IV refuses the usual picture in which passive spacetime hosts active particles and forces. +aliases: +- CNS0044 +- S044 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book IV refuses the usual picture in which passive spacetime hosts active + + particles and forces. At E_1, the carrier is the only available physical + + substrate, so particles, forces, mass, energy, charge, entropy, chemistry, and + + matter must be read as organized regimes of the carrier itself. This is not a + + metaphor: it is the construction rule for the microcosm. Nothing may enter as a + + primitive actor on a background stage; every physical kind must be an invariant + + readout of carrier structure.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book IV refuses the usual picture in which passive spacetime hosts active particles and forces. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000073.md b/_corpus_v3_items/cid000073.md new file mode 100644 index 000000000..8a037665b --- /dev/null +++ b/_corpus_v3_items/cid000073.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000073/ +title: Self-description on the T^2 fiber +cid: cid000073 +primary_alias: CNS0045 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book IV develops physics as the carrier's self-description on its own T^2-boundary fiber. +aliases: +- CNS0045 +- S045 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book IV develops physics as the carrier''s self-description on its own + + T^2-boundary fiber. Microphysical content arises from how the carrier reads + + itself through bipolar character channels, the four-atom idempotent grammar, + + and the sector holonomies inherited from Book III. Proto-time is explicitly + + not part of this fiber analysis: temporal progression, worldlines, and proper + + time live on the τ^1 base and are developed in Book V.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book IV develops physics as the carrier's self-description on its own T^2-boundary fiber. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000074.md b/_corpus_v3_items/cid000074.md new file mode 100644 index 000000000..813581e3d --- /dev/null +++ b/_corpus_v3_items/cid000074.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000074/ +title: Defect-bundle formalism +cid: cid000074 +primary_alias: CNS0046 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A physical entity is defined as a localized persistent defect bundle: a stable, structured deviation from the vacuum + torus / carrier state.' +aliases: +- CNS0046 +- S046 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'A physical entity is defined as a localized persistent defect bundle: a stable, + + structured deviation from the vacuum torus / carrier state. The defect bundle + + is the constructive replacement for "particle on a stage". Its persistence is + + not postulated externally; it is the preservation of address, holonomy, charge, + + mass, spin, and transport invariants under the admissible E_1 readout. A + + particle kind is therefore an equivalence class of stable defect-bundle + + realizations, not a numerically identical substrate piece.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A physical entity is defined as a localized persistent defect bundle: a stable, structured deviation from the + vacuum torus / carrier state.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000075.md b/_corpus_v3_items/cid000075.md new file mode 100644 index 000000000..1c5999104 --- /dev/null +++ b/_corpus_v3_items/cid000075.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000075/ +title: Photon as null transport +cid: cid000075 +primary_alias: CNS0047 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The photon is not a persistent material defect bundle. +aliases: +- CNS0047 +- S047 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The photon is not a persistent material defect bundle. It is the B-sector + + null-transport mode of the carrier boundary: a degenerate fiber character + + (m,n)=(0,0) with zero mass, no fiber obstruction, limiting-speed + + propagation, and no rest frame. Its in-transit reality is transport-real rather + + than persistent-object-real: the mode is localized at emission/absorption or + + measurement interfaces, not as a free-standing object observed from the side. + + Nullness is structural -- a consequence of the j^2=+1 idempotent geometry + + and B-sector transport grammar -- not a primitive kinematic postulate.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The photon is not a persistent material defect bundle. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000076.md b/_corpus_v3_items/cid000076.md new file mode 100644 index 000000000..23b39cdd0 --- /dev/null +++ b/_corpus_v3_items/cid000076.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000076/ +title: Neutron as minimal stable defect +cid: cid000076 +primary_alias: CNS0048 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The neutron is the minimal stable defect-bundle equivalence class on the carrier: the lightest defect regime whose + internal address and holonomy structure closes under the E_1 microphysical grammar.' +aliases: +- CNS0048 +- S048 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The neutron is the minimal stable defect-bundle equivalence class on the + + carrier: the lightest defect regime whose internal address and holonomy + + structure closes under the E_1 microphysical grammar. Beyond being one + + particle among others, it is the parent carrier regime from which atomic matter + + differentiates. Its mass supplies the sole dimensional SI calibration anchor: + + one dimensional scale-setting parameter that connects the dimensionless + + τ-internal coupling ledger to laboratory units.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The neutron is the minimal stable defect-bundle equivalence class on the carrier: the lightest defect regime + whose internal address and holonomy structure closes under the E_1 microphysical grammar.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000077.md b/_corpus_v3_items/cid000077.md new file mode 100644 index 000000000..6a8841e33 --- /dev/null +++ b/_corpus_v3_items/cid000077.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000077/ +title: Beta differentiation and hydrogen closure +cid: cid000077 +primary_alias: CNS0049 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Beta decay is the weak-sector rearrangement of the neutron regime, not the breakup of an already ordinary particle + into independent primitive actors. +aliases: +- CNS0049 +- S049 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Beta decay is the weak-sector rearrangement of the neutron regime, not the + + breakup of an already ordinary particle into independent primitive actors. It + + generates the proton, electron, and neutrino readouts: the proton is the residual + + bulk/core mode of the neutron regime, the electron is the corresponding + + surface/co-rotor mode, and the τ^1 time-mode, carrying the beta-minus + + antineutrino label before the Majorana/helicity readout of Step 50, is emitted + + along the base direction. Hydrogen is then the first stable coupled proton -- electron + + unit: the first atomic co-rotor regime rather than a mere aggregate of two + + independent substances. The exact co-rotor terminology should remain anchored + + to the Book-IV/Book-V source map.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Beta decay is the weak-sector rearrangement of the neutron regime, not the breakup of an already ordinary particle + into independent primitive actors. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000078.md b/_corpus_v3_items/cid000078.md new file mode 100644 index 000000000..48b00f713 --- /dev/null +++ b/_corpus_v3_items/cid000078.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000078/ +title: Neutrino ontology and Majorana readout +cid: cid000078 +primary_alias: CNS0050 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Neutrinos differ ontologically from photons. +aliases: +- CNS0050 +- S050 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Neutrinos differ ontologically from photons. A neutrino is not a null-transport + + mode but an ontic weak-sector eigenmode with in-transit reality: it carries + + weak-sector structure while remaining electrically neutral and colorless. The + + three flavor modes _e,_μ,_τ are read as weak eigenmodes of the + + carrier grammar. In the current τ-readout, zero-holonomy neutrino modes + + satisfy the Majorana condition; the particle/antiparticle distinction is + + therefore read through helicity / propagation orientation rather than an + + independent internal charge. The detailed structural route is recorded in + + the final quantity-semantics appendix.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Neutrinos differ ontologically from photons. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000079.md b/_corpus_v3_items/cid000079.md new file mode 100644 index 000000000..d763d15b0 --- /dev/null +++ b/_corpus_v3_items/cid000079.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000079/ +title: Quantum mechanics as CR-address obstruction +cid: cid000079 +primary_alias: CNS0051 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Quantum mechanics enters as a limitation of simultaneous address resolution on τ^1×_f T^2, not as a postulated Hilbert + formalism on a background space. +aliases: +- CNS0051 +- S051 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Quantum mechanics enters as a limitation of simultaneous address resolution on + + τ^1×_f T^2, not as a postulated Hilbert formalism on a background + + space. The CR-address grammar yields discrete spectra, noncommuting observables, + + state-space structure, and measurement interfaces because incompatible + + fiber-addresses cannot be resolved together. The quantum state is therefore a + + readout of admissible address structure; it is not an additional ontology placed + + on top of the carrier.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Quantum mechanics enters as a limitation of simultaneous address resolution on τ^1×_f T^2, not as a postulated + Hilbert formalism on a background space. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000080.md b/_corpus_v3_items/cid000080.md new file mode 100644 index 000000000..aff5d3ba6 --- /dev/null +++ b/_corpus_v3_items/cid000080.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000080/ +title: Uncertainty as geometric obstruction +cid: cid000080 +primary_alias: CNS0052 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Uncertainty is the geometric obstruction to simultaneous resolution of incompatible carrier addresses. +aliases: +- CNS0052 +- S052 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Uncertainty is the geometric obstruction to simultaneous resolution of + + incompatible carrier addresses. The primary conjugate pairs are + + localization/momentum and energy/proper-time; velocity alone is not the + + localization-conjugate invariant. Thus relations such as + + + Δ x Δ p ≳ ℏ_τ, + + Δ E Δ t ≳ ℏ_τ + + + express address/transport incompatibility in the E_1 carrier, not merely + + epistemic ignorance.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Uncertainty is the geometric obstruction to simultaneous resolution of incompatible carrier addresses. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000081.md b/_corpus_v3_items/cid000081.md new file mode 100644 index 000000000..2ae33cbc2 --- /dev/null +++ b/_corpus_v3_items/cid000081.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000081/ +title: Physical quantity semantics +cid: cid000081 +primary_alias: CNS0053 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book IV also fixes the semantics of core microphysical quantities. +aliases: +- CNS0053 +- S053 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book IV also fixes the semantics of core microphysical quantities. Localization + + is an E_1 hom-object locus; momentum is the defect-transport invariant + + conjugate to localization; mass is stable defect-bundle calibration; energy is + + state/address/defect-budget readout; charge is boundary holonomy or winding + + obstruction; spin and polarization are fiber-mode readouts. Classical force is + + not primary here: it can be recovered as an effective response or gradient + + readout, but it is not a primitive E_1 invariant.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book IV also fixes the semantics of core microphysical quantities. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000082.md b/_corpus_v3_items/cid000082.md new file mode 100644 index 000000000..cceddae9e --- /dev/null +++ b/_corpus_v3_items/cid000082.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000082/ +title: Forces as holonomy tools +cid: cid000082 +primary_alias: CNS0054 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The gauge sectors are not primitive actors. +aliases: +- CNS0054 +- S054 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The gauge sectors are not primitive actors. Electromagnetism is the B-sector + + U(1)-curvature / boundary-transport grammar on the lemniscate topology; + + the weak sector is the A-sector junction-orientation / chirality grammar; the + + ω-mediator supplies the Higgs/mass crossing. The coupling readouts are + + functions of iota_tau and sector depth, not independent knobs. These + + structures are tools by which the carrier organizes defect regimes, not forces + + added to a stage.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The gauge sectors are not primitive actors. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000083.md b/_corpus_v3_items/cid000083.md new file mode 100644 index 000000000..cf3aa3905 --- /dev/null +++ b/_corpus_v3_items/cid000083.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000083/ +title: Confinement as address irresolvability +cid: cid000083 +primary_alias: CNS0055 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The strong sector is the C-sector nonabelian holonomy grammar associated with the η-channel. +aliases: +- CNS0055 +- S055 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The strong sector is the C-sector nonabelian holonomy grammar associated with + + the η-channel. Quarks and gluons are not free primitive actors; they are + + confined character modes whose addresses cannot be resolved as isolated + + E_1 defect objects. Confinement is therefore structural address + + irresolvability in the T^2-fiber, not merely a dynamical binding rule added + + after particle ontology has been assumed.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The strong sector is the C-sector nonabelian holonomy grammar associated with the η-channel. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000084.md b/_corpus_v3_items/cid000084.md new file mode 100644 index 000000000..4bab348f1 --- /dev/null +++ b/_corpus_v3_items/cid000084.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000084/ +title: Three generations from topology +cid: cid000084 +primary_alias: CNS0056 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Generation structure is read from the topology of the T^2-fiber rather than postulated as a repeated particle table. +aliases: +- CNS0056 +- S056 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Generation structure is read from the topology of the T^2-fiber rather than + + postulated as a repeated particle table. The three generations correspond to + + the admissible fundamental-cycle structure of the fiber-mode grammar; a fourth + + generation would require a topology the carrier does not supply. The inspection + + question is not whether other topologies could be imagined, but whether any + + admissible τ-internal variation of the earned T^2-fiber changes the + + generation count.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Generation structure is read from the topology of the T^2-fiber rather than postulated as a repeated particle + table. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000085.md b/_corpus_v3_items/cid000085.md new file mode 100644 index 000000000..374db767f --- /dev/null +++ b/_corpus_v3_items/cid000085.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000085/ +title: Sector atlas, masses, nuclei +cid: cid000085 +primary_alias: CNS0057 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: With the sector grammar and generation topology in place, the particle atlas is read as a classification of carrier + modes. +aliases: +- CNS0057 +- S057 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000025 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-05 + macro_title: Recover Internal Physical Grammar +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'With the sector grammar and generation topology in place, the particle atlas is + + read as a classification of carrier modes. Mass ratios and Koide-type relations + + are character-mode / generation-geometry readouts; hadrons are confined + + composites and toroidal saturation regimes; nuclei are defect-stacking regimes + + with shell-model emergence. This step does not derive all nuclear physics in + + the main text. It records the construction role: the same defect/holonomy + + grammar that defined particles now organizes composite matter.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: With the sector grammar and generation topology in place, the particle atlas is read as a classification of carrier + modes. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000086.md b/_corpus_v3_items/cid000086.md new file mode 100644 index 000000000..4aab73b06 --- /dev/null +++ b/_corpus_v3_items/cid000086.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000086/ +title: 'Book IV closure: coupling ledger and microcosm completion' +cid: cid000086 +primary_alias: CNS0058 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: readouts of the iota_tau coupling structure; chemistry and molecular geometry are downstream CR-address / defect-regime + organizations; condensed phases, superfluids, superconductors, magnetism, and fluids are ensemble regimes of the same universal + defect tu... +aliases: +- CNS0058 +- S058 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book IV closes by assembling the T^2-fiber microcosm as a readout stack: + + fine-structure and mass-ratio quantities are empirical readouts of the + + iota_tau coupling structure; chemistry and molecular geometry are + + downstream CR-address / defect-regime organizations; condensed phases, + + superfluids, superconductors, magnetism, and fluids are ensemble regimes of the + + same universal defect tuple. The stated inter-sector coupling ledger has no + + fitted dimensionless free parameters; the neutron mass remains the dimensional + + calibration anchor. The microcosm is therefore complete as an ontology-and-status + + ledger: the carrier roles are assigned, while detailed derivation chains for + + precision quantities and composite regimes remain in the physics-readout + + supplement.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: readouts of the iota_tau coupling structure; chemistry and molecular geometry are downstream CR-address / defect-regime + organizations; condensed phases, superfluids, superconductors, magnetism, and fluids are ensemble regimes of the same un +noindex: false +--- + diff --git a/_corpus_v3_items/cid000087.md b/_corpus_v3_items/cid000087.md new file mode 100644 index 000000000..463a1c53b --- /dev/null +++ b/_corpus_v3_items/cid000087.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000087/ +title: Fiber/base completion +cid: cid000087 +primary_alias: CNS0059 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Book IV completes the T^2-fiber side of the E_1 carrier: microphysical defect bundles, transport modes, sector holonomies, + and matter regimes.' +aliases: +- CNS0059 +- S059 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book IV completes the T^2-fiber side of the E_1 carrier: microphysical + + defect bundles, transport modes, sector holonomies, and matter regimes. Book V + + turns to the complementary τ^1-base. Together, + + + τ^3=τ^1×_f T^2 + + + is read as a full E_1 physical carrier: the fiber supplies microphysical + + structure, while the base supplies time, gravity, thermodynamic direction, and + + cosmic-scale dynamics. Book V is therefore not an added cosmology; it is the + + base-side completion of the same carrier.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book IV completes the T^2-fiber side of the E_1 carrier: microphysical defect bundles, transport modes, sector + holonomies, and matter regimes.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000088.md b/_corpus_v3_items/cid000088.md new file mode 100644 index 000000000..9ac28f3ae --- /dev/null +++ b/_corpus_v3_items/cid000088.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000088/ +title: Proto-time to proper time +cid: cid000088 +primary_alias: CNS0060 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The progression depth available in the kernel is not yet physical time. +aliases: +- CNS0060 +- S060 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The progression depth available in the kernel is not yet physical time. Book V + + turns this proto-temporal ordering into proper-time readout on the + + τ^1-base. The base supplies ordered traces, worldline structure, and the + + direction along which physical events can be compared. Proper time is therefore + + not imported as a coordinate; it is the temporal readout of the base geometry + + after the E_1 carrier has been constructed.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The progression depth available in the kernel is not yet physical time. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000089.md b/_corpus_v3_items/cid000089.md new file mode 100644 index 000000000..a7b289f4c --- /dev/null +++ b/_corpus_v3_items/cid000089.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000089/ +title: Causality, worldlines, and null-exchange distance +cid: cid000089 +primary_alias: CNS0061 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Causality is read from base-orbit ordering, not imposed as a background order. +aliases: +- CNS0061 +- S061 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Causality is read from base-orbit ordering, not imposed as a background order. + + A worldline is the τ^1-ordered trace of an E_1 localization regime. + + Distance is likewise not primitive: operational distance arises from null + + exchange, i.e. photon-mediated radar-time comparison between localization + + regimes. Thus causal order, worldline, and distance are all base/carrier + + readouts, not structures placed underneath the theory.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Causality is read from base-orbit ordering, not imposed as a background order. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000090.md b/_corpus_v3_items/cid000090.md new file mode 100644 index 000000000..c722c0c15 --- /dev/null +++ b/_corpus_v3_items/cid000090.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000090/ +title: Redshift and Hubble readout +cid: cid000090 +primary_alias: CNS0062 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Cosmological redshift is read as base/refinement drift rather than as a purely Doppler or pre-imposed FLRW expansion + parameter. +aliases: +- CNS0062 +- S062 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Cosmological redshift is read as base/refinement drift rather than as a purely + + Doppler or pre-imposed FLRW expansion parameter. The Hubble parameter becomes + + an orbit-depth / base-progression readout. This does not yet settle the + + empirical Hubble-tension question; it states the construction role: the + + observable expansion grammar is read from τ^1-base dynamics rather than + + attached as an external cosmological background.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Cosmological redshift is read as base/refinement drift rather than as a purely Doppler or pre-imposed FLRW expansion + parameter. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000091.md b/_corpus_v3_items/cid000091.md new file mode 100644 index 000000000..d5bb6a726 --- /dev/null +++ b/_corpus_v3_items/cid000091.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000091/ +title: Gravity as frame holonomy +cid: cid000091 +primary_alias: CNS0063 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Gravity is not introduced as a primitive force acting on matter inside a background spacetime. +aliases: +- CNS0063 +- S063 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Gravity is not introduced as a primitive force acting on matter inside a + + background spacetime. It is the τ^1-base frame-holonomy readout of the + + E_1 carrier. Curvature, transport, and stress are expressed in the same + + boundary/holonomy language that organized the microcosm. The gravitational + + constant G is treated as a coherence-conversion readout, with dimensional + + calibration handled downstream.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Gravity is not introduced as a primitive force acting on matter inside a background spacetime. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000092.md b/_corpus_v3_items/cid000092.md new file mode 100644 index 000000000..39aa5e20d --- /dev/null +++ b/_corpus_v3_items/cid000092.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000092/ +title: τ-Einstein identity +cid: cid000092 +primary_alias: CNS0064 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The macro field identity is written here as R^H=κ_tau T, with κ_tau serving as the τ-gravitational coupling readout, + not as an imported PDE. +aliases: +- CNS0064 +- S064 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The macro field identity is written here as + + + R^H=κ_tau T, + + + with κ_tau serving as the τ-gravitational coupling readout, + + not as an imported PDE. Here H is the holonomy/curvature superscript, not + + an abbreviation for the Riemann Hypothesis. Its role is to balance curvature-side holonomy data with + + matter/stress defect data inside the same carrier grammar. Ordinary Einstein gravity appears as a + + chart/readout shadow of this identity, not as an assumed background equation. + + Classical GR tests are therefore treated as readout checks of the + + τ-Einstein structure rather than as axioms of the construction.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The macro field identity is written here as R^H=κ_tau T, with κ_tau serving as the τ-gravitational coupling readout, + not as an imported PDE. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000093.md b/_corpus_v3_items/cid000093.md new file mode 100644 index 000000000..486d978a1 --- /dev/null +++ b/_corpus_v3_items/cid000093.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000093/ +title: Lorentzian signature from j^2=+1 +cid: cid000093 +primary_alias: CNS0065 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The Lorentzian / Minkowski signature is read from the split-complex hyperbolicity already earned in Book I -- II. +aliases: +- CNS0065 +- S065 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The Lorentzian / Minkowski signature is read from the split-complex + + hyperbolicity already earned in Book I -- II. Because j^2=+1, the carrier + + grammar carries null directions and causal cones before an external Lorentzian + + metric is imposed. Light cones are therefore the causal cones of the + + split-complex carrier readout. The remaining bridge question is physical: + + whether this algebraic cone structure reproduces the full empirical Lorentzian + + regime.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Lorentzian / Minkowski signature is read from the split-complex hyperbolicity already earned in Book I -- + II. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000094.md b/_corpus_v3_items/cid000094.md new file mode 100644 index 000000000..684e09ce2 --- /dev/null +++ b/_corpus_v3_items/cid000094.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000094/ +title: Thermodynamic inversion +cid: cid000094 +primary_alias: CNS0066 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book V splits entropy into defect, refinement, and interaction readouts, S(n)=S_def(n)+S_ref(n)+ε(n), 0≤ε(n)≤ S_def(n). +aliases: +- CNS0066 +- S066 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "Book V splits entropy into defect, refinement, and interaction readouts,\n\nS(n)=S_def(n)+S_ref(n)+ε(n),\n 0≤ε(n)≤\ + \ S_def(n).\n\nS_def tracks unresolved carrier defects; S_ref\ntracks refinement/readout proliferation; ε records the\ + \ bounded\ncross-term between the two. The τ-native arrow points toward defect\nexhaustion and ordering, while ordinary\ + \ thermodynamics reads the refinement/decay\nside of the same process. Thus thermodynamics is not simply reversed as an\n\ + empirical slogan; it is decomposed into coordinated readouts whose ordinary\ncoarse-grained projection recovers the familiar\ + \ direction." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book V splits entropy into defect, refinement, and interaction readouts, S(n)=S_def(n)+S_ref(n)+ε(n), 0≤ε(n)≤ + S_def(n). +noindex: false +--- + diff --git a/_corpus_v3_items/cid000095.md b/_corpus_v3_items/cid000095.md new file mode 100644 index 000000000..b4e7e626c --- /dev/null +++ b/_corpus_v3_items/cid000095.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000095/ +title: Heat, temperature, vacuum, and dark-energy readout +cid: cid000095 +primary_alias: CNS0067 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Heat is read as boundary exchange / transport, not as a primitive substance of disorder. +aliases: +- CNS0067 +- S067 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Heat is read as boundary exchange / transport, not as a primitive substance of + + disorder. Temperature is the derived defect-gradient readout + + T_τ(C)=dδ[ω](C)/dS_def(C), not a primitive + + thermodynamic variable. The vacuum is the coherent boundary state, not a sum of + + zero-point mode energies. On this readout, dark energy is not introduced as an + + independent sector; cosmic acceleration is treated as a base-progression / + + readout effect whose empirical adequacy must be tested.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Heat is read as boundary exchange / transport, not as a primitive substance of disorder. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000096.md b/_corpus_v3_items/cid000096.md new file mode 100644 index 000000000..9001c2b70 --- /dev/null +++ b/_corpus_v3_items/cid000096.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000096/ +title: Macro defect lift and collective dynamics +cid: cid000096 +primary_alias: CNS0068 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The universal defect tuple from the T^2-fiber lifts to the τ^1-base as a macro defect-budget grammar. +aliases: +- CNS0068 +- S068 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The universal defect tuple from the T^2-fiber lifts to the τ^1-base + + as a macro defect-budget grammar. Fluids, plasmas, waves, phase transitions, + + and magnetohydrodynamic regimes are then read as collective carrier regimes, + + not as independent continuum substances. Classical force is still not primary: + + macro equations recover effective forces as gradients, exchanges, and + + defect-budget responses.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The universal defect tuple from the T^2-fiber lifts to the τ^1-base as a macro defect-budget grammar. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000097.md b/_corpus_v3_items/cid000097.md new file mode 100644 index 000000000..a8e4307f5 --- /dev/null +++ b/_corpus_v3_items/cid000097.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000097/ +title: Macro regularity, turbulence, and charge/flux +cid: cid000097 +primary_alias: CNS0069 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Macro Navier -- Stokes regularity is read as a compactness / defect-budget condition on carrier flows; the orthodox + Navier -- Stokes question is its external shadow. +aliases: +- CNS0069 +- S069 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Macro Navier -- Stokes regularity is read as a compactness / defect-budget + + condition on carrier flows; the orthodox Navier -- Stokes question is its + + external shadow. Turbulence becomes typed redistribution of defect budgets + + rather than unstructured randomness. Charge and flux are boundary-holonomy + + obstructions; Wilson-loop encoding supplies the macro vocabulary for + + electromagnetic and chromodynamic flux.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Macro Navier -- Stokes regularity is read as a compactness / defect-budget condition on carrier flows; the orthodox + Navier -- Stokes question is its external shadow. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000098.md b/_corpus_v3_items/cid000098.md new file mode 100644 index 000000000..79fd51319 --- /dev/null +++ b/_corpus_v3_items/cid000098.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000098/ +title: Classical mechanics as coarse readout +cid: cid000098 +primary_alias: CNS0070 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Newtonian and Lagrangian mechanics are recovered as coarse-grained readouts of carrier dynamics. +aliases: +- CNS0070 +- S070 +layers: +- E1 +domains: +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Newtonian and Lagrangian mechanics are recovered as coarse-grained readouts of + + carrier dynamics. Inertial stability is not primitively linear translation + + through a background; the primary stable transport is carrier-compatible + + rotational / holonomic motion, with linear inertial motion appearing as a + + coarse or generated regime. Momentum, not velocity alone, is the + + localization-conjugate invariant.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Newtonian and Lagrangian mechanics are recovered as coarse-grained readouts of carrier dynamics. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000099.md b/_corpus_v3_items/cid000099.md new file mode 100644 index 000000000..7ac0d2873 --- /dev/null +++ b/_corpus_v3_items/cid000099.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000099/ +title: Galaxies, rotation curves, and compact objects +cid: cid000099 +primary_alias: CNS0071 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Galaxies are read as relational carrier regimes, not merely as mass distributions inside a background spacetime. +aliases: +- CNS0071 +- S071 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Galaxies are read as relational carrier regimes, not merely as mass + + distributions inside a background spacetime. Flat rotation curves are attributed + + to D-sector capacity-gradient structure rather than to an independent + + dark-matter particle sector. White dwarfs, neutron stars, and black holes form + + a compact-object ladder of the same τ-gravity / defect-capacity dynamics. + + These are high-risk empirical readouts: lensing, Tully -- Fisher, Bullet Cluster, + + and compact-object observations constrain the claim.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Galaxies are read as relational carrier regimes, not merely as mass distributions inside a background spacetime. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000100.md b/_corpus_v3_items/cid000100.md new file mode 100644 index 000000000..5429f220d --- /dev/null +++ b/_corpus_v3_items/cid000100.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000100/ +title: Black holes as topological channel openings +cid: cid000100 +primary_alias: CNS0072 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: In the τ-carrier readout, black-hole formation is not primarily the formation of a point singularity. +aliases: +- CNS0072 +- S072 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'In the τ-carrier readout, black-hole formation is not primarily the + + formation of a point singularity. It is the saturation of a geometric channel + + and the opening of a non-contractible topological channel in the global carrier. + + The mature horizon is read as toroidal / bipolar rather than as a primitive + + Schwarzschild S^2 surface. Global spacetime topology therefore becomes + + non-simply-connected when such channels exist. Event Horizon Telescope + + horizon-region imaging, together with lensing and multi-wavelength accretion/jet + + observations, is one empirical surface that can constrain this readout.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: In the τ-carrier readout, black-hole formation is not primarily the formation of a point singularity. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000101.md b/_corpus_v3_items/cid000101.md new file mode 100644 index 000000000..5be6a9b77 --- /dev/null +++ b/_corpus_v3_items/cid000101.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000101/ +title: Cosmic web as Wilson-loop carrier channels +cid: cid000101 +primary_alias: CNS0073 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: If black-hole channel formation changes the global carrier topology, then large-scale structure is not only a matter-density + field in a trivial background. +aliases: +- CNS0073 +- S073 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'If black-hole channel formation changes the global carrier topology, then + + large-scale structure is not only a matter-density field in a trivial + + background. In the τ-readout, cosmic filaments trace Wilson-loop-like + + carrier channels generated by non-contractible global topology. The cosmic web + + becomes topology written into the sky: a large-scale readout of carrier + + holonomy rather than only a gravitational clustering pattern. This is a + + downstream empirical claim, constrained by large-scale structure surveys.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: If black-hole channel formation changes the global carrier topology, then large-scale structure is not only a + matter-density field in a trivial background. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000102.md b/_corpus_v3_items/cid000102.md new file mode 100644 index 000000000..7f258a1d0 --- /dev/null +++ b/_corpus_v3_items/cid000102.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000102/ +title: Big Bang, threshold ladder, and endstate +cid: cid000102 +primary_alias: CNS0074 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The Big Bang is read as the opening regime of τ^1-base dynamics, not as an externally imposed singular creation event. +aliases: +- CNS0074 +- S074 +layers: +- E1 +domains: +- construction.step.readout +- construction.step.external-validation-seam +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The Big Bang is read as the opening regime of τ^1-base dynamics, not as + + an externally imposed singular creation event. Inflation is a + + refinement-sector saturation regime rather than an added inflaton sector. The + + threshold ladder + + + L_EW to L_B to L_N to L_nuc to L_H to L_γ + + + marks successive sector decouplings and readout transitions. The endstate is + + boundary circulation / defect exhaustion rather than heat death in a void. This + + is the physical macrocosm closure of the E_1 carrier, with empirical + + adequacy left to cosmological observation.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Big Bang is read as the opening regime of τ^1-base dynamics, not as an externally imposed singular creation + event. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000103.md b/_corpus_v3_items/cid000103.md new file mode 100644 index 000000000..34d8e485a --- /dev/null +++ b/_corpus_v3_items/cid000103.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000103/ +title: Book V late-stage synthesis and macrocosm export surface +cid: cid000103 +primary_alias: CNS0075 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The late Book V synthesis assembles the opening-regime, inflation-without-inflaton, threshold ladder, black-hole + topology, merger/endstate, empirical validation surfaces, and boundary-unification claims into the Book V macrocosm export + surface. +aliases: +- CNS0075 +- S075 +layers: +- E1 +domains: +- construction.step.bridge +- construction.step.readout +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: The late Book V synthesis assembles the opening-regime, inflation-without-inflaton, threshold ladder, black-hole + topology, merger/endstate, empirical validation surfaces, and boundary-unification claims into the Book V macrocosm export + surface. The closure preserves scope labels across that export surface; it does not add a new physics primitive beyond + the Book V base readout. +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The late Book V synthesis assembles the opening-regime, inflation-without-inflaton, threshold ladder, black-hole + topology, merger/endstate, empirical validation surfaces, and boundary-unification claims into the Book V macrocosm export + surf +noindex: false +--- + diff --git a/_corpus_v3_items/cid000104.md b/_corpus_v3_items/cid000104.md new file mode 100644 index 000000000..7bc83fccd --- /dev/null +++ b/_corpus_v3_items/cid000104.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000104/ +title: E_1 physics complete +cid: cid000104 +primary_alias: CNS0076 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book V closes the E_0 to E_1 construction. +aliases: +- CNS0076 +- S076 +layers: +- E1 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000026 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-06 + macro_title: Build Measurement, Prediction, and Empirical Bridges +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book V closes the E_0 to E_1 construction. Book III identifies the physical + + carrier, Book IV reads the T^2-fiber as microphysical ontology, and Book V + + reads the τ^1-base as temporal, gravitational, thermodynamic, and + + cosmological structure. The result is not full empirical validation; it is a + + complete internal physical grammar with explicit external-validation seams. + + Books VI -- VII receive that physical grammar and ask when carrier regimes can + + preserve, transmit, decode, update, and finally reflect on code.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book V closes the E_0 to E_1 construction. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000105.md b/_corpus_v3_items/cid000105.md new file mode 100644 index 000000000..2646f1918 --- /dev/null +++ b/_corpus_v3_items/cid000105.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000105/ +title: 'E_1 to E_2: life as predicate layer' +cid: cid000105 +primary_alias: CNS0077 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Book VI receives completed E_1 physics and identifies life as the next self-enrichment: carriers that add SelfDesc + to Distinction.' +aliases: +- CNS0077 +- S077 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book VI receives completed E_1 physics and identifies life as the next + + self-enrichment: carriers that add SelfDesc to Distinction. The subject is a + + typed layer of codes, organisms, and self/non-self maintenance, not an empirical + + checklist. A living carrier is a physical carrier under a new predicate + + discipline: it preserves and decodes the structure by which it continues as + + itself.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book VI receives completed E_1 physics and identifies life as the next self-enrichment: carriers that add SelfDesc + to Distinction.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000106.md b/_corpus_v3_items/cid000106.md new file mode 100644 index 000000000..d4a4f7c7d --- /dev/null +++ b/_corpus_v3_items/cid000106.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000106/ +title: Distinction plus SelfDesc +cid: cid000106 +primary_alias: CNS0078 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Life is defined by two earned predicates. +aliases: +- CNS0078 +- S078 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Life is defined by two earned predicates. Distinction is the sharp, stable, + + internally maintained boundary between self and non-self. SelfDesc is the + + internal code-reading evaluator that reconstructs it. Classical hallmarks such + + as metabolism, growth, response, reproduction, and evolution become consequences + + or calibrations of the predicate pair, not axioms smuggled in from biology.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Life is defined by two earned predicates. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000107.md b/_corpus_v3_items/cid000107.md new file mode 100644 index 000000000..4d8d02149 --- /dev/null +++ b/_corpus_v3_items/cid000107.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000107/ +title: Layer separation from physics +cid: cid000107 +primary_alias: CNS0079 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Distinction alone remains E_1. +aliases: +- CNS0079 +- S079 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Distinction alone remains E_1. SelfDesc is the genuinely new E_2 + + requirement. A physically bounded system may be stable, energetic, and sharply + + separated from its environment while still failing to read and rebuild its own + + boundary as code. This is the Book VI separation test: life is neither reducible + + to physical boundary alone nor added as a vital substance.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Distinction alone remains E_1. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000108.md b/_corpus_v3_items/cid000108.md new file mode 100644 index 000000000..3e35d74ec --- /dev/null +++ b/_corpus_v3_items/cid000108.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000108/ +title: Life Loop and metabolic closure +cid: cid000108 +primary_alias: CNS0080 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A living carrier must run a closed cycle of SelfDesc-preserving endomorphisms. +aliases: +- CNS0080 +- S080 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'A living carrier must run a closed cycle of SelfDesc-preserving + + endomorphisms. Metabolism is not merely "having reactions"; it is the loop + + that returns the carrier to a state compatible with its maintained distinction. + + Energy exchange, repair, and turnover are admissible when they serve rather than + + dissolve the boundary they depend on.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A living carrier must run a closed cycle of SelfDesc-preserving endomorphisms. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000109.md b/_corpus_v3_items/cid000109.md new file mode 100644 index 000000000..b03e92ec5 --- /dev/null +++ b/_corpus_v3_items/cid000109.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000109/ +title: Boundary and energy coupling +cid: cid000109 +primary_alias: CNS0081 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The membrane is the concrete biological boundary: it separates self/non-self and makes gradients possible.' +aliases: +- CNS0081 +- S081 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The membrane is the concrete biological boundary: it separates self/non-self + + and makes gradients possible. Chemiosmosis ties boundary, proton gradient, ATP + + synthase, ATP, and agency into one construction chain: the material route by + + which E_2 boundary maintenance and energy coupling become one grammar.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The membrane is the concrete biological boundary: it separates self/non-self and makes gradients possible.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000110.md b/_corpus_v3_items/cid000110.md new file mode 100644 index 000000000..9d6b07d39 --- /dev/null +++ b/_corpus_v3_items/cid000110.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000110/ +title: Code, genotype, and heredity +cid: cid000110 +primary_alias: CNS0082 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: SelfDesc turns the ω-germ into formal genotype and the normal form into phenotype. +aliases: +- CNS0082 +- S082 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'SelfDesc turns the ω-germ into formal genotype and the normal form into + + phenotype. The genetic code, central dogma, replication, proofreading, and + + mismatch repair become realizations of code reconstruction and blueprint + + propagation. Biochemistry supplies the implementation; the τ-claim is + + heredity as SelfDesc-preserving code transport.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: SelfDesc turns the ω-germ into formal genotype and the normal form into phenotype. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000111.md b/_corpus_v3_items/cid000111.md new file mode 100644 index 000000000..5e70f762b --- /dev/null +++ b/_corpus_v3_items/cid000111.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000111/ +title: Replication, development, and morphogenesis +cid: cid000111 +primary_alias: CNS0083 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Reproduction is blueprint propagation through carrier splitting; development is controlled refinement of inherited + code into structured form. +aliases: +- CNS0083 +- S083 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Reproduction is blueprint propagation through carrier splitting; development is + + controlled refinement of inherited code into structured form. Morphogen + + gradients, reaction-diffusion patterns, and cell-cycle control bridge inherited + + code to spatial organism. The constructive move is code persistence becoming + + body formation.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Reproduction is blueprint propagation through carrier splitting; development is controlled refinement of inherited + code into structured form. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000112.md b/_corpus_v3_items/cid000112.md new file mode 100644 index 000000000..c1bd26861 --- /dev/null +++ b/_corpus_v3_items/cid000112.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000112/ +title: 4+1 life-sector template +cid: cid000112 +primary_alias: CNS0084 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Life-layer structure is organized by four primitive sectors plus one mixed sector: persistence, agency, source, + closure, and consumer, the mixed role in which self-modeling pressure becomes dominant.' +aliases: +- CNS0084 +- S084 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Life-layer structure is organized by four primitive sectors plus one mixed + + sector: persistence, agency, source, closure, and consumer, the mixed role in + + which self-modeling pressure becomes dominant. These sectors are structural + + roles, not taxa; Eukarya is carrier technology.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Life-layer structure is organized by four primitive sectors plus one mixed sector: persistence, agency, source, + closure, and consumer, the mixed role in which self-modeling pressure becomes dominant.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000113.md b/_corpus_v3_items/cid000113.md new file mode 100644 index 000000000..f50987842 --- /dev/null +++ b/_corpus_v3_items/cid000113.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000113/ +title: Evolution as PPAS selection +cid: cid000113 +primary_alias: CNS0085 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Evolution is PPAS on ω-germ code space. +aliases: +- CNS0085 +- S085 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Evolution is PPAS on ω-germ code space. Populations generate variants, + + selection verifies viability, and drift/recombination keep the search + + non-terminating. The admissible claim is local adaptive search under historical + + constraints, not global optimization or an imported Darwinian primitive detached + + from code.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Evolution is PPAS on ω-germ code space. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000114.md b/_corpus_v3_items/cid000114.md new file mode 100644 index 000000000..ca4ce412e --- /dev/null +++ b/_corpus_v3_items/cid000114.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000114/ +title: Immunity and species boundaries +cid: cid000114 +primary_alias: CNS0086 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: In multicellular consumers, the immune system operationalizes Distinction at molecular and cellular scales. +aliases: +- CNS0086 +- S086 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'In multicellular consumers, the immune system operationalizes Distinction at + + molecular and cellular scales. At population scale, speciation forms boundary + + conditions between reproductive basins. Both extend the E_2 question: which + + differences preserve a living boundary, and which mark a different + + self-maintaining basin?' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: In multicellular consumers, the immune system operationalizes Distinction at molecular and cellular scales. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000115.md b/_corpus_v3_items/cid000115.md new file mode 100644 index 000000000..fec7af260 --- /dev/null +++ b/_corpus_v3_items/cid000115.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000115/ +title: Neural bridge to self-modeling +cid: cid000115 +primary_alias: CNS0087 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The consumer sector pressures SelfDesc to apply to SelfDesc, because consumers must model other Life Loops as part + of survival. +aliases: +- CNS0087 +- S087 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The consumer sector pressures SelfDesc to apply to SelfDesc, because consumers + + must model other Life Loops as part of survival. Neural architecture supplies + + the substrate for this second-order evaluator. The result is a bridge to + + reflection, not a completed theory of consciousness: Book VI builds the + + precondition.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The consumer sector pressures SelfDesc to apply to SelfDesc, because consumers must model other Life Loops as + part of survival. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000116.md b/_corpus_v3_items/cid000116.md new file mode 100644 index 000000000..40082ecf0 --- /dev/null +++ b/_corpus_v3_items/cid000116.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000116/ +title: Structural self-modeling boundary +cid: cid000116 +primary_alias: CNS0088 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Consciousness in Book VI is framed as structural self-modeling: the evaluator models its own evaluative process.' +aliases: +- CNS0088 +- S088 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Consciousness in Book VI is framed as structural self-modeling: the evaluator + + models its own evaluative process. The phenomenological content of + + consciousness remains an E_3 question. The step is deliberately bounded: it + + establishes the life-layer condition under which reflective content can later be + + typed, not a solution to qualia.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Consciousness in Book VI is framed as structural self-modeling: the evaluator models its own evaluative process.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000117.md b/_corpus_v3_items/cid000117.md new file mode 100644 index 000000000..6e8f79e1c --- /dev/null +++ b/_corpus_v3_items/cid000117.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000117/ +title: Language as extended lemniscate +cid: cid000117 +primary_alias: CNS0089 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Communication links two living lemniscates through a shared signal channel. +aliases: +- CNS0089 +- S089 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Communication links two living lemniscates through a shared signal channel. + + Language is stabilized shared code: grammar as code structure, + + encoding/decoding as inter-agent SelfDesc, and pragmatics as basin dynamics. + + It hands Book VI to Book VII, where public meaning and proof become inspectable.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Communication links two living lemniscates through a shared signal channel. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000118.md b/_corpus_v3_items/cid000118.md new file mode 100644 index 000000000..abe2187b7 --- /dev/null +++ b/_corpus_v3_items/cid000118.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000118/ +title: E_2 export to E_3 +cid: cid000118 +primary_alias: CNS0090 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Book VI closes by exporting six completed structures: the Life predicate, the 4+1 sector template, genotype/phenotype + predicates, the ω-germ question, Life basins, and substrate-agnosticism.' +aliases: +- CNS0090 +- S090 +layers: +- E2 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000027 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-07 + macro_title: Recover Life as a Structural Class +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book VI closes by exporting six completed structures: the Life predicate, the + + 4+1 sector template, genotype/phenotype predicates, the ω-germ + + question, Life basins, and substrate-agnosticism. This marks the structural + + limit of E_2: life can carry and decode code, while the next layer asks how a + + living system models that fact.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Book VI closes by exporting six completed structures: the Life predicate, the 4+1 sector template, genotype/phenotype + predicates, the ω-germ question, Life basins, and substrate-agnosticism.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000119.md b/_corpus_v3_items/cid000119.md new file mode 100644 index 000000000..d448a18b3 --- /dev/null +++ b/_corpus_v3_items/cid000119.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000119/ +title: 'E_2 to E_3: reflective structure' +cid: cid000119 +primary_alias: CNS0091 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book VII begins where the consumer/self-modeling capacity is internalized as an object of study. +aliases: +- CNS0091 +- S091 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-08 + macro_title: Recover Reflective Structure +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book VII begins where the consumer/self-modeling capacity is internalized as an + + object of study. The transition is not merely carrying and decoding code, but + + modeling that code-carrying and decoding and reasoning about the conditions + + under which self-modeling is possible. Reflection is therefore a new layer + + function, not a poetic description of intelligence.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book VII begins where the consumer/self-modeling capacity is internalized as an object of study. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000120.md b/_corpus_v3_items/cid000120.md new file mode 100644 index 000000000..5c124d79b --- /dev/null +++ b/_corpus_v3_items/cid000120.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000120/ +title: MetaDecode and SelfDesc-of-SelfDesc +cid: cid000120 +primary_alias: CNS0092 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At E_3, SelfDesc becomes an object of SelfDesc. +aliases: +- CNS0092 +- S092 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-08 + macro_title: Recover Reflective Structure +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'At E_3, SelfDesc becomes an object of SelfDesc. The evaluator can represent + + its own evaluative process, compare registers, and ask whether its own + + descriptions preserve the conditions that make description possible. This is + + the reflective kernel of the layer: a living code-carrier becomes a system that + + can model its own code-bearing status.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At E_3, SelfDesc becomes an object of SelfDesc. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000121.md b/_corpus_v3_items/cid000121.md new file mode 100644 index 000000000..4b22ec030 --- /dev/null +++ b/_corpus_v3_items/cid000121.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000121/ +title: Four registers of reason +cid: cid000121 +primary_alias: CNS0093 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: E3 reflection does not yield undifferentiated thought. +aliases: +- CNS0093 +- S093 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-08 + macro_title: Recover Reflective Structure +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'E3 reflection does not yield undifferentiated thought. It yields typed content + + in four registers: empirical, practical, diagrammatic, and commitment. Each + + claim must be inspected by the register question it answers, its admissible + + witness form, and its failure mode. This prevents observation, proof, + + obligation, and stance from being collapsed into one overgrown notion of + + "truth".' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: E3 reflection does not yield undifferentiated thought. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000122.md b/_corpus_v3_items/cid000122.md new file mode 100644 index 000000000..90100121f --- /dev/null +++ b/_corpus_v3_items/cid000122.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000122/ +title: Reflective 4+1 register template +cid: cid000122 +primary_alias: CNS0094 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book VII gives reflective structure a sector grammar analogous to earlier layers. +aliases: +- CNS0094 +- S094 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-08 + macro_title: Recover Reflective Structure +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book VII gives reflective structure a sector grammar analogous to earlier + + layers. The 4+1 decomposition organizes observation, action, diagrammatic + + form, closure, and the mixed commitment/logos surface. This is not a new + + ontology imported from tradition; it is the E_3 register architecture in + + which reflective claims can be sorted, checked, and bounded.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book VII gives reflective structure a sector grammar analogous to earlier layers. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000123.md b/_corpus_v3_items/cid000123.md new file mode 100644 index 000000000..8e9d58417 --- /dev/null +++ b/_corpus_v3_items/cid000123.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000123/ +title: Formal systems as represented objects +cid: cid000123 +primary_alias: CNS0095 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Formal systems, proof predicates, ZFC, model talk, and possible-worlds talk are treated as represented systems inside + or projected from the τ architecture, not as primitive foundations silently imported into it. +aliases: +- CNS0095 +- S095 +layers: +- E3 +domains: +- construction.step.formal +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-09 + macro_title: Self-Host Formal Systems and the Kernel Itself +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Formal systems, proof predicates, ZFC, model talk, and possible-worlds talk are + + treated as represented systems inside or projected from the τ + + architecture, not as primitive foundations silently imported into it. ZFC is + + usable as a virtual machine or shadow that preserves classical theorem-work in + + its layer, while internal ontic standing remains tied to finite + + NF-addressability and bounded witnesses.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Formal systems, proof predicates, ZFC, model talk, and possible-worlds talk are treated as represented systems + inside or projected from the τ architecture, not as primitive foundations silently imported into it. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000124.md b/_corpus_v3_items/cid000124.md new file mode 100644 index 000000000..bd5c8d8e2 --- /dev/null +++ b/_corpus_v3_items/cid000124.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000124/ +title: Gödel and halting boundary +cid: cid000124 +primary_alias: CNS0096 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Book VII does not refute Gödel and does not bypass incompleteness by slogan. +aliases: +- CNS0096 +- S096 +layers: +- E3 +domains: +- construction.step.formal +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-09 + macro_title: Self-Host Formal Systems and the Kernel Itself +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Book VII does not refute Gödel and does not bypass incompleteness by slogan. + + It restricts the conditions under which unrestricted diagonal self-reference is + + available. The relevant claim is typed and bounded: τ is not treated as + + a single Gödel-style formal system with unrestricted self-encoding, but as a + + coherence kernel with layered witness discipline.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Book VII does not refute Gödel and does not bypass incompleteness by slogan. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000125.md b/_corpus_v3_items/cid000125.md new file mode 100644 index 000000000..b0d23c9d9 --- /dev/null +++ b/_corpus_v3_items/cid000125.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000125/ +title: Language, meaning, and proof arc +cid: cid000125 +primary_alias: CNS0097 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The diagrammatic sector becomes a pattern-to-symbol-to-proof arc: pre-symbolic detection, symbolic/language transport, + and formal proof.' +aliases: +- CNS0097 +- S097 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-09 + macro_title: Self-Host Formal Systems and the Kernel Itself +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The diagrammatic sector becomes a pattern-to-symbol-to-proof arc: + + pre-symbolic detection, symbolic/language transport, and formal proof. Language + + is treated as self-enrichment of subsymbolic communication, while meaning is + + handled structurally through public transport, syntax/semantics collapse, + + translation, drift, and repair rather than as private mental content.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The diagrammatic sector becomes a pattern-to-symbol-to-proof arc: pre-symbolic detection, symbolic/language + transport, and formal proof.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000126.md b/_corpus_v3_items/cid000126.md new file mode 100644 index 000000000..f6b10b736 --- /dev/null +++ b/_corpus_v3_items/cid000126.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000126/ +title: Logic, inference, and truth at reflective scale +cid: cid000126 +primary_alias: CNS0098 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Logic and inference become internal reflective tools rather than external sovereigns. +aliases: +- CNS0098 +- S098 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-10 + macro_title: Test Universal Closure and Ontic Status +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Logic and inference become internal reflective tools rather than external + + sovereigns. Boolean, Bayesian, modal, paraconsistent, and proof-theoretic + + readouts occupy typed roles under the register discipline. Truth is therefore + + not flattened into one metalogical predicate; it is handled through the + + register, witness, and boundary conditions under which a claim can be + + inspected.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Logic and inference become internal reflective tools rather than external sovereigns. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000127.md b/_corpus_v3_items/cid000127.md new file mode 100644 index 000000000..1517c03dd --- /dev/null +++ b/_corpus_v3_items/cid000127.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000127/ +title: Mind, consciousness, and commitment access +cid: cid000127 +primary_alias: CNS0099 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The commitment register requires an eligible carrier: a minded, conscious, self-modeling agent.' +aliases: +- CNS0099 +- S099 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-10 + macro_title: Test Universal Closure and Ontic Status +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'The commitment register requires an eligible carrier: a minded, conscious, + + self-modeling agent. Book VII models mind as internal organization and + + consciousness as global self-model structure, while keeping qualia and + + hard-problem residue under explicit scope caveats. The construction role is to + + locate the carrier of commitment, not to force every phenomenological question + + closed.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The commitment register requires an eligible carrier: a minded, conscious, self-modeling agent.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid000128.md b/_corpus_v3_items/cid000128.md new file mode 100644 index 000000000..4b3babc6a --- /dev/null +++ b/_corpus_v3_items/cid000128.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000128/ +title: Logos boundary and ontic-closure burden +cid: cid000128 +primary_alias: CNS0100 +type: construction_step +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Here "Logos" is used as a technical label for the boundary where diagrammatic proof-validity and commitment stance-stability + meet. +aliases: +- CNS0100 +- S100 +layers: +- E3 +domains: +- construction.step.bridge +relations: +- predicate: part_of + object: cid000003 +- predicate: appears_in + object: cid000028 +- predicate: part_of + object: cid000003 + qualifiers: + macro_step: CS-10 + macro_title: Test Universal Closure and Ontic Status +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid000003 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_construction_spine_steps_100 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'Here "Logos" is used as a technical label for the boundary where + + diagrammatic proof-validity and commitment stance-stability meet. The final + + boundary tests closure by locating what proof can deliver, what it cannot + + force, and what must remain a performative stance. Ontic closure is therefore + + an inspectability burden: no externality should be hidden, but no theorem + + forces ultimate certainty or a personal stance.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Here "Logos" is used as a technical label for the boundary where diagrammatic proof-validity and commitment stance-stability + meet. +noindex: false +--- + diff --git a/_corpus_v3_items/cid000129.md b/_corpus_v3_items/cid000129.md new file mode 100644 index 000000000..70343d03c --- /dev/null +++ b/_corpus_v3_items/cid000129.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000129/ +title: Corpus v2 Release v4.0 +cid: cid000129 +primary_alias: REL0002 +type: release_packet +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +summary: 'Canonical v4.0 release packet of the v2 corpus — the artifact that public websites consumed prior to the Corpus + v3 cutover. Captured in v3 as a Corpus Item for provenance continuity; v3 cutover ships under release_line corpus_v3_v1 + (post-cutover). + + ' +aliases: +- REL0002 +- corpus-v2-release-v4-0 +layers: [] +domains: [] +relations: [] +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: + github: https://github.com/Panta-Rhei-Research/corpus +payload: + kind: external_artifact + canonical_artifact: + type: github_repo + url: https://github.com/Panta-Rhei-Research/corpus + version: v4.0 + hash: null + short_summary: 'Canonical v4.0 release packet of the v2 corpus. + + ' + citation_markdown: 'Panta Rhei Research Program. *Corpus v2 Release v4.0*. + + GitHub: Panta-Rhei-Research/corpus, release v4.0, 2026. + + ' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical v4.0 release packet of the v2 corpus — the artifact that public websites consumed prior to the Corpus + v3 cutover. Captured in v3 as a Corpus Item for provenance continuity; v3 cutover ships under release_line corpus_v3_v1 + (post-cu +noindex: false +--- + diff --git a/_corpus_v3_items/cid000130.md b/_corpus_v3_items/cid000130.md new file mode 100644 index 000000000..ec728b08d --- /dev/null +++ b/_corpus_v3_items/cid000130.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid000130/ +title: TauLib v2 Master Module +cid: cid000130 +primary_alias: MOD0002 +type: formal_module +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +summary: 'Top-level placeholder Corpus Item for the TauLib v2 surface. Wave 5 of the Corpus v3 build-out will expand this + into per-module formal_module items (one per Lean module path), per-declaration formal_theorem / formal_definition / formal_axiom + items, and the wire-up to formalizes relations against the prose Corpus Items. At Wave 3 we record only the master placeholder. + + ' +aliases: +- MOD0002 +- taulib-v2-master +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2 +relations: +- predicate: implemented_by + object: cid000019 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib + module_path: TauLib/ + commit: '0000000000000000000000000000000000000000' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '0000000000000000000000000000000000000000' + sorry_count: 0 + project_axioms: 1 + print_axioms: 'bridge_functor_exists + + ' +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Top-level placeholder Corpus Item for the TauLib v2 surface. Wave 5 of the Corpus v3 build-out will expand this + into per-module formal_module items (one per Lean module path), per-declaration formal_theorem / formal_definition / formal_axio +noindex: false +--- + diff --git a/_corpus_v3_items/cid001000.md b/_corpus_v3_items/cid001000.md new file mode 100644 index 000000000..89990d403 --- /dev/null +++ b/_corpus_v3_items/cid001000.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001000/ +title: Constructive Encoding via ABCD +cid: cid001000 +primary_alias: COR0001 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Constructive pairing and sequence encoding via prime powers, without importing set theory. Goedel numbering is a + special case of the ABCD encoding (with C=1, D=1). +aliases: +- COR0001 +- I.C01 +- constructive-encoding-via-abcd +- cor:constructive-encoding +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 5, Chapter 25 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Constructive Encoding via ABCD**\n\nConstructive pairing and sequence encoding via prime powers, without importing\ + \ set theory. Goedel numbering is a special case of the ABCD encoding (with C=1, D=1).\n\n# Constructive Encoding via\ + \ ABCD\n\n## Summary\n\nConstructive pairing and sequence encoding via prime powers, without importing set theory. Goedel\ + \ numbering is a special case of the ABCD encoding (with C=1, D=1).\n\n## Statement\n\n```latex\n%\n\\label{cor:constructive-encoding}\n\ + The \\textbf{constructive pairing} of two objects\n$x, y \\in \\Obj(\\tau)$ with indices\n$\\underline{m} = \\mathrm{idx}(x)$\n\ + and $\\underline{n} = \\mathrm{idx}(y)$\nis defined as:\n\\[\n \\langle x, y \\rangle_\\tau\n \\;:=\\;\n \\underline{p_1}^{\\\ + underline{m}} \\cdot\n \\underline{p_2}^{\\underline{n}}\n\\]\nwhere $\\underline{p_1} < \\underline{p_2}$ are the\ + \ first two\nprimes of $\\tau$-Idx\n(i.e., $\\underline{p_1} = \\underline{2}$,\n$\\underline{p_2} = \\underline{3}$).\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 59\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part05/ch25-consequences.tex`\ + \ lines 49-67\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.Hyperfact`\n\ + - Name: `Tau.Coordinates.injectivity_check`\n\n## Dependencies\n\n- Canonical: I.T04\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Constructive pairing and sequence encoding via prime powers, without importing set theory. Goedel numbering is + a special case of the ABCD encoding (with C=1, D=1). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001001.md b/_corpus_v3_items/cid001001.md new file mode 100644 index 000000000..b06c0d54b --- /dev/null +++ b/_corpus_v3_items/cid001001.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001001/ +title: Stages from Limit +cid: cid001001 +primary_alias: COR0002 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The interior is determined by the boundary: two tower-coherent functions with the same boundary data (agreement + at a reference depth) have the same interior values at all coarser stages.' +aliases: +- COR0002 +- I.C02 +- stages-from-limit +- cor:interior-from-boundary +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005297 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 63 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Stages from Limit**\n\nThe interior is determined by the boundary: two tower-coherent functions with the same\ + \ boundary data (agreement at a reference depth) have the same interior values at all coarser stages.\n\n# Stages from\ + \ Limit\n\n## Summary\n\nThe interior is determined by the boundary: two tower-coherent functions with the same boundary\ + \ data (agreement at a reference depth) have the same interior values at all coarser stages.\n\n## Statement\n\n```latex\n\ + %\n\\label{cor:interior-from-boundary}\nEvery finite-stage value on $\\tau^3 = \\tau^1 \\times_f T^2$\nis uniquely determined\ + \ by omega-tail data on~$\\mathbb{L}$.\nEvery point of $\\tau^3$\nis an earned stage-determined point:\n\\[\n \\boxed{%\n\ + \ \\mathrm{Int}_{\\mathrm{earned}}(\\tau^3)\n \\;=\\;\n \\tau^3.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\n\ + Let $p \\in \\tau^3$ be arbitrary.\nBy Global Hartogs\n(Theorem~\\ref{thm:global-hartogs}, I.T06),\nevery $T \\in \\mathrm{Hol}(\\\ + mathbb{L})$\nextends uniquely from partial lemniscate data\nto all of~$\\mathbb{L}$.\nTower coherence\n(Definition~\\\ + ref{def:tower-coherence}, I.D46)\nensures that the holomorphic data on $\\mathbb{L}$\npropagates along the primorial tower\n\ + to determine values at $p$:\nthe CRT reconstruction at each primorial stage\npropagates limit data to finite stages.\n\ + By the $\\tau$-Identity Theorem\n(Theorem~\\ref{thm:tau-identity}, I.T21),\nthis extension is unique.\nTherefore $p \\\ + in \\mathrm{Int}_{\\mathrm{earned}}(\\tau^3)$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 156\n\ + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch63-boundary-interior-passage.tex`\ + \ lines 109-122\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.BoundaryInterior`\n\ + - Name: `Tau.Holomorphy.interior_from_boundary`\n\n## Dependencies\n\n- Canonical: I.T31, I.D68\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The interior is determined by the boundary: two tower-coherent functions with the same boundary data (agreement + at a reference depth) have the same interior values at all coarser stages.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001002.md b/_corpus_v3_items/cid001002.md new file mode 100644 index 000000000..51898fa01 --- /dev/null +++ b/_corpus_v3_items/cid001002.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001002/ +title: No Identity Decoherence +cid: cid001002 +primary_alias: COR0003 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No Identity Decoherence: the diagonal resonance pattern (L+E+P) cannot occur at the ontic level in tau, because + K5 blocks (L), NF-Confluence blocks (E), and the star-autonomous structure blocks (P).' +aliases: +- COR0003 +- I.C03 +- no-identity-decoherence +- cor:no-identity-decoherence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005324 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 81 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Identity Decoherence**\n\nNo Identity Decoherence: the diagonal resonance pattern (L+E+P) cannot occur at\ + \ the ontic level in tau, because K5 blocks (L), NF-Confluence blocks (E), and the star-autonomous structure blocks (P).\n\ + \n# No Identity Decoherence\n\n## Summary\n\nNo Identity Decoherence: the diagonal resonance pattern (L+E+P) cannot occur\ + \ at the ontic level in tau, because K5 blocks (L), NF-Confluence blocks (E), and the star-autonomous structure blocks\ + \ (P).\n\n## Statement\n\n```latex\n%\n\\label{cor:no-identity-decoherence}\nThe diagonal resonance pattern\n(Definition~\\\ + ref{def:diagonal-resonance}, I.D89)\n--- the three-component splice\nof free contraction (L),\nequality-as-congruence\ + \ (E),\nand primitive self-products (P) ---\ncannot occur at the ontic level in $\\tau$.\nIts three prerequisites are\ + \ individually blocked:\n\\begin{enumerate}[\\normalfont(i)]\n \\item (L) is blocked by $\\KAxiom{5}$\n (Definition~\\\ + ref{def:diagonal-discipline}, I.D03).\n \\item (E) is blocked by NF-Confluence\n (Lemma~\\ref{lem:nf-confluence},\ + \ I.L02)\n and three-level equality\n (Definition~\\ref{def:three-equality}, I.D15).\n \\item (P)\ + \ is blocked by the star-autonomous structure\n (Theorem~\\ref{thm:k5-structural-exclusion}, I.T39).\n\\end{enumerate}\n\ + Therefore, identity slippage\n(Definition~\\ref{def:identity-slippage}, I.D90)\nis zero, shadow identities\n(Definition~\\\ + ref{def:shadow-identity}, I.D91)\ndo not arise,\nand ontic identity coherence\nis maintained at 100\\% under every admissible\ + \ construction.\n```\n\n## Proof / Justification\n\n```latex\nDiagonal resonance (I.D89) requires\nthe simultaneous presence\ + \ of (L), (E), and (P).\nTheorem~\\ref{thm:ontic-identity-invariance} (I.T46)\nestablished that each component\nis individually\ + \ blocked by the coherence kernel:\n(L) by K5, (E) by NF-Confluence, (P) by star-autonomy.\nSince the resonance pattern\ + \ cannot assemble\nwithout all three components,\nit cannot occur.\nWithout diagonal resonance,\nidentity slippage (I.D90)\ + \ has no source,\nshadow identities (I.D91) have no mechanism\nof generation,\nand the identity coherence of\nTheorem~\\\ + ref{thm:ontic-identity-invariance}\nis maintained without exception.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 205\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch81-ontic-identity-invariance.tex`\ + \ lines 659-687\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.OnticInvariance`\n\ + - Name: `Tau.MetaLogic.no_identity_decoherence`\n\n## Dependencies\n\n- Canonical: I.T46, I.D89\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No Identity Decoherence: the diagonal resonance pattern (L+E+P) cannot occur at the ontic level in tau, because + K5 blocks (L), NF-Confluence blocks (E), and the star-autonomous structure blocks (P).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001003.md b/_corpus_v3_items/cid001003.md new file mode 100644 index 000000000..176c66b26 --- /dev/null +++ b/_corpus_v3_items/cid001003.md @@ -0,0 +1,182 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001003/ +title: NF Deduplication Invariant +cid: cid001003 +primary_alias: COR0004 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Address DAG construction is deduplication-invariant: identifying shared branches produces the same DAG regardless + of order. Follows from Hyperfactorization uniqueness. NF-level analogue of NF-Confluence.' +aliases: +- COR0004 +- I.C04 +- nf-deduplication-invariant +- cor:nf-dedup-invariant +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 5, Chapter 24 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**NF Deduplication Invariant** + + + Address DAG construction is deduplication-invariant: identifying shared branches produces the same DAG regardless of order. + Follows from Hyperfactorization uniqueness. NF-level analogue of NF-Confluence. + + + # NF Deduplication Invariant + + + ## Summary + + + Address DAG construction is deduplication-invariant: identifying shared branches produces the same DAG regardless of order. + Follows from Hyperfactorization uniqueness. NF-level analogue of NF-Confluence. + + + ## Statement + + + ```latex + + % + + \label{cor:nf-dedup-invariant} + + The Address DAG construction + + (Definition~\ref{def:address-dag}, + + Chapter~\ref{ch:dimension-fibration}) + + is an invariant under deduplication. + + That is, if two quadtree expansions + + share the same index at some branch, + + identifying (deduplicating) those branches + + produces the same Address DAG + + regardless of the order + + in which deduplication is performed. + + ``` + + + ## Proof / Justification + + + ```latex + + By the Hyperfactorization Theorem + + (Theorem~\ref{thm:hyperfactorization}), + + each index $X \in \tau\text{-Idx}$ + + has a \emph{unique} ABCD address. + + The Address DAG is therefore canonically determined + + by the \emph{set} of reachable indices, + + independent of the traversal order + + used to discover them. + + If two quadtree branches share the same index $X$, + + their ABCD addresses are identical + + (by uniqueness), + + so merging them produces the same DAG node + + regardless of which branch is encountered first. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 243 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part05/ch24-hyperfactorization.tex` lines + 270-282 + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T04, I.D24 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Address DAG construction is deduplication-invariant: identifying shared branches produces the same DAG regardless + of order. Follows from Hyperfactorization uniqueness. NF-level analogue of NF-Confluence.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001004.md b/_corpus_v3_items/cid001004.md new file mode 100644 index 000000000..5c6e808b2 --- /dev/null +++ b/_corpus_v3_items/cid001004.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001004/ +title: Five Generators +cid: cid001004 +primary_alias: DEF0002 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The five generators {alpha, pi, gamma, eta, omega} with canonical roles: radial seed, prime base, exponent channel, + tetration channel, fixed-point absorber.' +aliases: +- DEF0002 +- I.D01 +- five-generators +- def:generators +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 1 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Five Generators**\n\nThe five generators {alpha, pi, gamma, eta, omega} with canonical roles: radial seed,\ + \ prime base, exponent channel, tetration channel, fixed-point absorber.\n\n# Five Generators\n\n## Summary\n\nThe five\ + \ generators {alpha, pi, gamma, eta, omega} with canonical roles: radial seed, prime base, exponent channel, tetration\ + \ channel, fixed-point absorber.\n\n## Statement\n\n```latex\n\\label{def:generators}\nThe five \\textbf{generators} of\ + \ Category~$\\tau$ are the constant symbols\n\\[\n \\alpha, \\quad \\pi, \\quad \\gamma, \\quad \\eta, \\quad \\omega.\n\ + \\]\nThey are the only primitive objects of the theory.\nAll other objects will be produced by applying the\nprogression\ + \ operator $\\rho$ to these generators (Part~II).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 2\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch01-five-generators.tex`\ + \ lines 83-92\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Signature`\n\ + - Name: `Tau.Kernel.Generator`\n\n## Dependencies\n\n- Canonical: I.K0\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The five generators {alpha, pi, gamma, eta, omega} with canonical roles: radial seed, prime base, exponent channel, + tetration channel, fixed-point absorber.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001005.md b/_corpus_v3_items/cid001005.md new file mode 100644 index 000000000..f4028f916 --- /dev/null +++ b/_corpus_v3_items/cid001005.md @@ -0,0 +1,133 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001005/ +title: Progression Operator rho +cid: cid001005 +primary_alias: DEF0003 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The sole ontic operator rho: acts as successor within each orbit ray. No other operation creates objects.' +aliases: +- DEF0003 +- I.D02 +- progression-operator-rho +- def:progression +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 2 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Progression Operator rho** + + + The sole ontic operator rho: acts as successor within each orbit ray. No other operation creates objects. + + + # Progression Operator rho + + + ## Summary + + + The sole ontic operator rho: acts as successor within each orbit ray. No other operation creates objects. + + + ## Statement + + + ```latex + + \label{def:progression} + + The \textbf{progression operator} $\rho : \Obj(\tau) \to \Obj(\tau)$ + + is the unique unary function symbol in $\Sigma_\tau$. + + For each object $x$, $\rho(x)$ is called the \textbf{successor} of~$x$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 3 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch02-rho-omega.tex` lines 43-48 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Kernel.Axioms` + + - Name: `Tau.Kernel.rho` + + + ## Dependencies + + + - Canonical: I.D01 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The sole ontic operator rho: acts as successor within each orbit ray. No other operation creates objects.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001006.md b/_corpus_v3_items/cid001006.md new file mode 100644 index 000000000..d5ebf84bd --- /dev/null +++ b/_corpus_v3_items/cid001006.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001006/ +title: Diagonal Discipline +cid: cid001006 +primary_alias: DEF0004 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Structural principle preventing free contraction/reuse; forces exactly four orbit channels via three successive diagonal + rewirings. +aliases: +- DEF0004 +- I.D03 +- diagonal-discipline +- def:diagonal-discipline +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005319 +- predicate: formalized_by + object: cid005320 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagonal Discipline**\n\nStructural principle preventing free contraction/reuse; forces exactly four orbit\ + \ channels via three successive diagonal rewirings.\n\n# Diagonal Discipline\n\n## Summary\n\nStructural principle preventing\ + \ free contraction/reuse; forces exactly four orbit channels via three successive diagonal rewirings.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:diagonal-discipline}\nThe \\textbf{diagonal discipline} is the structural principle\nthat governs\ + \ how iterated operations overflow\nbetween orbit channels.\nIt asserts:\n\\begin{enumerate}\n \\item \\textbf{No unearned\ + \ diagonals.}\n The self-product of an operation at level~$k$\n cannot be absorbed within the same orbit\ + \ channel.\n It must overflow into a distinct channel at level~$k{+}1$.\n \\item \\textbf{Each overflow consumes\ + \ one channel.}\n The new operation at level~$k{+}1$\n is canonically associated with a dedicated orbit\ + \ channel.\n \\item \\textbf{Saturation.}\n When no further orbit channel is available,\n the overflow\ + \ is blocked.\n The ladder of iterated operations terminates.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 4\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch05-diagonal-discipline.tex`\ + \ lines 64-84\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Diagonal`\n\ + - Name: `Tau.Kernel.diagonal_discipline`\n\n## Dependencies\n\n- Canonical: I.D01\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Structural principle preventing free contraction/reuse; forces exactly four orbit channels via three successive + diagonal rewirings. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001007.md b/_corpus_v3_items/cid001007.md new file mode 100644 index 000000000..e9353280f --- /dev/null +++ b/_corpus_v3_items/cid001007.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001007/ +title: Static Kernel tau_0 +cid: cid001007 +primary_alias: DEF0005 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'tau_0 = the static kernel: 5 generator objects + K1-K6 axioms + rho. The complete specification before the generative + act.' +aliases: +- DEF0005 +- I.D04 +- static-kernel-tau-0 +- def:tau-zero +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Static Kernel tau_0**\n\ntau_0 = the static kernel: 5 generator objects + K1-K6 axioms + rho. The complete\ + \ specification before the generative act.\n\n# Static Kernel tau_0\n\n## Summary\n\ntau_0 = the static kernel: 5 generator\ + \ objects + K1-K6 axioms + rho. The complete specification before the generative act.\n\n## Statement\n\n```latex\n%\n\ + \\label{def:tau-zero}\nThe \\textbf{static kernel} $\\tau_0$ is the first-order theory\nwith signature $\\Sigma_\\tau\ + \ = (\\alpha, \\pi, \\gamma, \\eta, \\omega, \\rho, {<})$,\nthe universe postulate $\\KAxiom{0}$,\nand structural axioms\ + \ $\\KAxiom{1}$--$\\KAxiom{6}$:\n\\[\n \\tau_0 \\;:=\\; (\\Sigma_\\tau,\\; \\KAxiom{0}, \\KAxiom{1}, \\KAxiom{2},\n\ + \ \\KAxiom{3}, \\KAxiom{4}, \\KAxiom{5}, \\KAxiom{6}).\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 5\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch04-closure-tau-zero.tex` lines 223-234\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Signature`\n- Name: `Tau.Kernel.tau_zero`\n\ + \n## Dependencies\n\n- Canonical: I.D01, I.D02, I.K1, I.K2, I.K3, I.K4, I.K5, I.K6\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'tau_0 = the static kernel: 5 generator objects + K1-K6 axioms + rho. The complete specification before the generative + act.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001008.md b/_corpus_v3_items/cid001008.md new file mode 100644 index 000000000..948529bbb --- /dev/null +++ b/_corpus_v3_items/cid001008.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001008/ +title: Orbit Rays +cid: cid001008 +primary_alias: DEF0006 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'O_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega generator.' +aliases: +- DEF0006 +- I.D05 +- orbit-rays +- def:orbit-rays +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 6 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Orbit Rays**\n\nO_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega generator.\n\ + \n# Orbit Rays\n\n## Summary\n\nO_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega\ + \ generator.\n\n## Statement\n\n```latex\n%\n\\label{def:orbit-rays}\nFor each generator $g \\in \\{\\alpha, \\pi, \\\ + gamma, \\eta\\}$,\nthe \\textbf{orbit ray} of $g$ is\n\\[\n O_g := \\{\\rho^n(g) : n \\geq 0\\} = \\{g, \\rho(g), \\\ + rho^2(g), \\ldots\\}.\n\\]\nThe \\textbf{beacon singleton} is $\\{\\omega\\}$.\nBy $\\KAxiom{6}$, the universe decomposes\ + \ as\n\\[\n \\Obj(\\tau) = \\{\\omega\\} \\;\\sqcup\\; O_\\alpha\n \\;\\sqcup\\; O_\\pi \\;\\sqcup\\; O_\\gamma\ + \ \\;\\sqcup\\; O_\\eta,\n\\]\nwhere $\\sqcup$ denotes disjoint union\n(to be proved in Chapter~\\ref{ch:ontic-closure}).\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 15\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch06-generative-act.tex`\ + \ lines 149-165\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Generation`\n\ + - Name: `Tau.Orbit.OrbitRay`\n\n## Dependencies\n\n- Canonical: I.X01, I.K3, I.K6\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'O_alpha, O_pi, O_gamma, O_eta: the four orbit rays unfolded by rho from each non-omega generator.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001009.md b/_corpus_v3_items/cid001009.md new file mode 100644 index 000000000..a8b3d343c --- /dev/null +++ b/_corpus_v3_items/cid001009.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001009/ +title: Iterator Ladder +cid: cid001009 +primary_alias: DEF0007 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The operator ladder: rho (raw iteration) -> mu (multiplication) -> nu (exponentiation) -> theta (tetration).' +aliases: +- DEF0007 +- I.D06 +- iterator-ladder +- def:iterator-ladder +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 8 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Iterator Ladder**\n\nThe operator ladder: rho (raw iteration) -> mu (multiplication) -> nu (exponentiation)\ + \ -> theta (tetration).\n\n# Iterator Ladder\n\n## Summary\n\nThe operator ladder: rho (raw iteration) -> mu (multiplication)\ + \ -> nu (exponentiation) -> theta (tetration).\n\n## Statement\n\n```latex\n%\n\\label{def:iterator-ladder}\nThe \\textbf{iterator\ + \ ladder} of Category~$\\tau$ is the sequence\nof operation levels:\n\\[\n \\mathcal{L} = (L_0, L_1, L_2, L_3)\n\\\ + ]\ndefined abstractly as follows:\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{tabular}{c l l l}\n\\\ + toprule\n\\textbf{Level} & \\textbf{Symbol} & \\textbf{Name}\n & \\textbf{Defining principle} \\\\\n\\midrule\n$L_0$\ + \ & $\\rho$ & Raw iteration\n & $\\rho^n(x)$: apply $\\rho$ a total of $n$ times \\\\\n$L_1$ & $\\mu$ & Multiplication\n\ + \ & iterate $L_0$: $\\mu(x,y) \\sim \\rho^y(x)$ ``$y$ times'' \\\\\n$L_2$ & $\\nu$ & Exponentiation\n & iterate\ + \ $L_1$: $\\nu(x,y) \\sim \\mu(x, \\ldots)$ ``$y$ times'' \\\\\n$L_3$ & $\\theta$ & Tetration\n & iterate $L_2$: $\\\ + theta(x,y) \\sim \\nu(x, \\ldots)$ ``$y$ times'' \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\nEach level is associated\ + \ with a dedicated orbit channel:\n\\begin{align*}\n L_0 &\\longleftrightarrow O_\\alpha \\quad\\text{(counting scaffold)},\ + \ \\\\\n L_1 &\\longleftrightarrow O_\\pi \\quad\\text{(multiplicative channel)}, \\\\\n L_2 &\\longleftrightarrow\ + \ O_\\gamma \\quad\\text{(exponential channel)}, \\\\\n L_3 &\\longleftrightarrow O_\\eta \\quad\\text{(tetration channel)}.\n\ + \\end{align*}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 16\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch08-iterator-ladder.tex`\ + \ lines 67-101\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Ladder`\n\ + - Name: `Tau.Orbit.iterator_ladder`\n\n## Dependencies\n\n- Canonical: I.D02, I.D03\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The operator ladder: rho (raw iteration) -> mu (multiplication) -> nu (exponentiation) -> theta (tetration).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001010.md b/_corpus_v3_items/cid001010.md new file mode 100644 index 000000000..4df197c44 --- /dev/null +++ b/_corpus_v3_items/cid001010.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001010/ +title: tau-Idx (Earned Natural Numbers) +cid: cid001010 +primary_alias: DEF0008 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: tau-Idx := O_alpha = {alpha, rho(alpha), rho^2(alpha), ...}. The alpha-orbit IS the internal natural numbers (N is + not imported but earned). +aliases: +- DEF0008 +- I.D07 +- tau-idx-earned-natural-numbers +- def:tau-idx +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 10 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**tau-Idx (Earned Natural Numbers)**\n\ntau-Idx := O_alpha = {alpha, rho(alpha), rho^2(alpha), ...}. The alpha-orbit\ + \ IS the internal natural numbers (N is not imported but earned).\n\n# tau-Idx (Earned Natural Numbers)\n\n## Summary\n\ + \ntau-Idx := O_alpha = {alpha, rho(alpha), rho^2(alpha), ...}. The alpha-orbit IS the internal natural numbers (N is not\ + \ imported but earned).\n\n## Statement\n\n```latex\n%\n\\label{def:tau-idx}\nThe \\textbf{internal natural numbers} of\ + \ Category~$\\tau$\nare defined as the alpha-orbit:\n\\[\n \\boxed{\\tau\\text{-Idx} := O_\\alpha\n = \\{\\alpha,\ + \ \\rho(\\alpha), \\rho^2(\\alpha), \\ldots\\}.}\n\\]\nWe write $\\underline{n}$ for $\\rho^n(\\alpha) \\in \\tau\\text{-Idx}$,\n\ + so that $\\underline{0} = \\alpha$, $\\underline{1} = \\rho(\\alpha)$,\n$\\underline{2} = \\rho^2(\\alpha)$, etc.\nThe\ + \ successor operation on $\\tau$-Idx is $\\rho$ itself:\n$S(\\underline{n}) = \\rho(\\underline{n}) = \\underline{n+1}$.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 21\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch10-tau-idx.tex`\ + \ lines 54-68\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.TauIdx`\n\ + - Name: `Tau.Denotation.TauIdx`\n\n## Dependencies\n\n- Canonical: I.D05, I.T01, I.P03\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: tau-Idx := O_alpha = {alpha, rho(alpha), rho^2(alpha), ...}. The alpha-orbit IS the internal natural numbers + (N is not imported but earned). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001011.md b/_corpus_v3_items/cid001011.md new file mode 100644 index 000000000..18b7366b9 --- /dev/null +++ b/_corpus_v3_items/cid001011.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001011/ +title: Rank Transfer Maps +cid: cid001011 +primary_alias: DEF0009 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For each generator s in {pi, gamma, eta}, RT_s : tau-Idx -> O_s maps rho^n(alpha) to rho^n(s). Canonical bijection + between counting scaffold and typed orbit ray.' +aliases: +- DEF0009 +- I.D08 +- rank-transfer-maps +- def:rank-transfer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 10 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Rank Transfer Maps**\n\nFor each generator s in {pi, gamma, eta}, RT_s : tau-Idx -> O_s maps rho^n(alpha) to\ + \ rho^n(s). Canonical bijection between counting scaffold and typed orbit ray.\n\n# Rank Transfer Maps\n\n## Summary\n\ + \nFor each generator s in {pi, gamma, eta}, RT_s : tau-Idx -> O_s maps rho^n(alpha) to rho^n(s). Canonical bijection between\ + \ counting scaffold and typed orbit ray.\n\n## Statement\n\n```latex\n%\n\\label{def:rank-transfer}\nFor each solenoidal\ + \ generator $s \\in \\{\\pi, \\gamma, \\eta\\}$,\nthe \\textbf{rank transfer map}\n$\\mathrm{RT}_s : \\tau\\text{-Idx}\ + \ \\to O_s$\nis defined by\n\\[\n \\mathrm{RT}_s(\\underline{n}) := \\rho^n(s).\n\\]\nThat is, $\\mathrm{RT}_s$ maps\ + \ the element at depth~$n$\nin the alpha-orbit to the element at depth~$n$\nin the $s$-orbit.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 22\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch10-tau-idx.tex` lines\ + \ 126-139\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.RankTransfer`\n\ + - Name: `Tau.Denotation.rank_transfer`\n\n## Dependencies\n\n- Canonical: I.D07, I.D05, I.T01\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For each generator s in {pi, gamma, eta}, RT_s : tau-Idx -> O_s maps rho^n(alpha) to rho^n(s). Canonical bijection + between counting scaffold and typed orbit ray.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001012.md b/_corpus_v3_items/cid001012.md new file mode 100644 index 000000000..58639845c --- /dev/null +++ b/_corpus_v3_items/cid001012.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001012/ +title: Swap Operator sigma +cid: cid001012 +primary_alias: DEF0010 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'sigma (swap): first derived operator from rho. Exchanges positions within orbit rays via rank transfer composition.' +aliases: +- DEF0010 +- I.D09 +- swap-operator-sigma +- def:swap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Swap Operator sigma**\n\nsigma (swap): first derived operator from rho. Exchanges positions within orbit rays\ + \ via rank transfer composition.\n\n# Swap Operator sigma\n\n## Summary\n\nsigma (swap): first derived operator from rho.\ + \ Exchanges positions within orbit rays via rank transfer composition.\n\n## Statement\n\n```latex\n%\n\\label{def:swap}\n\ + The \\textbf{swap operator} $\\sigma_{s,t}$ for generators\n$s, t \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$\nis the map\n\ + \\[\n \\sigma_{s,t} := \\mathrm{RT}_t \\circ \\mathrm{RT}_s^{-1}\n : O_s \\to O_t\n\\]\nthat transfers an element\ + \ from the $s$-orbit to the $t$-orbit\nat the same depth.\nExplicitly: $\\sigma_{s,t}(\\rho^n(s)) = \\rho^n(t)$.\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 23\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex`\ + \ lines 45-58\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.TauIdx`\n\ + - Name: `Tau.Denotation.sigma`\n\n## Dependencies\n\n- Canonical: I.D08, I.D02\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'sigma (swap): first derived operator from rho. Exchanges positions within orbit rays via rank transfer composition.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001013.md b/_corpus_v3_items/cid001013.md new file mode 100644 index 000000000..71e8be916 --- /dev/null +++ b/_corpus_v3_items/cid001013.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001013/ +title: Index Addition +cid: cid001013 +primary_alias: DEF0011 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n + m = rho^m(n): addition on tau-Idx earned by iterating rho m times starting from n. No Peano axioms imported.' +aliases: +- DEF0011 +- I.D10 +- index-addition +- def:idx-add +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Index Addition**\n\nn + m = rho^m(n): addition on tau-Idx earned by iterating rho m times starting from n.\ + \ No Peano axioms imported.\n\n# Index Addition\n\n## Summary\n\nn + m = rho^m(n): addition on tau-Idx earned by iterating\ + \ rho m times starting from n. No Peano axioms imported.\n\n## Statement\n\n```latex\n%\n\\label{def:idx-add}\nFor $\\\ + underline{n}, \\underline{m} \\in \\tau\\text{-Idx}$,\n\\textbf{index addition} is defined by\n\\[\n \\boxed{\\underline{n}\ + \ + \\underline{m}\n := \\rho^m(\\underline{n}) = \\rho^m(\\rho^n(\\alpha))\n = \\rho^{n+m}(\\alpha) = \\underline{n+m}.}\n\ + \\]\nEquivalently, by structural recursion on $m$:\n\\begin{align*}\n \\underline{n} + \\underline{0} &:= \\underline{n},\ + \ \\\\\n \\underline{n} + \\rho(\\underline{m})\n &:= \\rho(\\underline{n} + \\underline{m}).\n\\end{align*}\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 24\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex`\ + \ lines 94-110\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.idx_add`\n\n## Dependencies\n\n- Canonical: I.D07, I.D06, I.D02\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n + m = rho^m(n): addition on tau-Idx earned by iterating rho m times starting from n. No Peano axioms imported.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001016.md b/_corpus_v3_items/cid001016.md new file mode 100644 index 000000000..40adb1cb1 --- /dev/null +++ b/_corpus_v3_items/cid001016.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001016/ +title: Polarity Pairing +cid: cid001016 +primary_alias: DEF0014 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Pairing function Pair: P_tau^+ x P_tau^- -> Z measuring rank distance between B-dominant and C-dominant primes. + Consecutive polarity pairs exist infinitely often.' +aliases: +- DEF0014 +- I.D103 +- polarity-pairing +- def:polarity-pairing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 6, Chapter 27 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Polarity Pairing**\n\nPairing function Pair: P_tau^+ x P_tau^- -> Z measuring rank distance between B-dominant\ + \ and C-dominant primes. Consecutive polarity pairs exist infinitely often.\n\n# Polarity Pairing\n\n## Summary\n\nPairing\ + \ function Pair: P_tau^+ x P_tau^- -> Z measuring rank distance between B-dominant and C-dominant primes. Consecutive\ + \ polarity pairs exist infinitely often.\n\n## Statement\n\n```latex\n%\n\\label{def:polarity-pairing}\nThe \\textbf{polarity\ + \ pairing function}\n\\[\n \\mathrm{Pair} : \\mathbb{P}_\\tau^+ \\times \\mathbb{P}_\\tau^-\n \\;\\to\\; \\mathbb{Z}\n\ + \\]\nis defined by\n\\[\n \\mathrm{Pair}(\\underline{p}, \\underline{q})\n \\;:=\\;\n \\mathrm{rank}(\\underline{p})\ + \ - \\mathrm{rank}(\\underline{q}),\n\\]\nwhere $\\mathrm{rank}(\\underline{p})$ is the ordinal position\nof $\\underline{p}$\ + \ in the enumeration of $\\mathbb{P}_\\tau$\n(i.e., $\\mathrm{rank}(\\underline{2}) = 1$,\n$\\mathrm{rank}(\\underline{3})\ + \ = 2$, etc.).\nThe \\textbf{polarity gap} of the pair\nis $|\\mathrm{Pair}(\\underline{p}, \\underline{q})|$:\nthe distance\ + \ between their positions\nin the prime enumeration.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 244\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part06/ch27-prime-polarity.tex` lines 445-466\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n## Dependencies\n\n- Canonical:\ + \ I.T05\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later\ + \ projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Pairing function Pair: P_tau^+ x P_tau^- -> Z measuring rank distance between B-dominant and C-dominant primes. + Consecutive polarity pairs exist infinitely often.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001017.md b/_corpus_v3_items/cid001017.md new file mode 100644 index 000000000..e2c0201ab --- /dev/null +++ b/_corpus_v3_items/cid001017.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001017/ +title: PolStream +cid: cid001017 +primary_alias: DEF0015 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'PolStream(omega): infinite binary sequence recording channel dominance along the primorial spine. At each stage + k, records whether prime p_k is B-dominant or C-dominant.' +aliases: +- DEF0015 +- I.D104 +- polstream +- def:polstream +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 29 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**PolStream**\n\nPolStream(omega): infinite binary sequence recording channel dominance along the primorial spine.\ + \ At each stage k, records whether prime p_k is B-dominant or C-dominant.\n\n# PolStream\n\n## Summary\n\nPolStream(omega):\ + \ infinite binary sequence recording channel dominance along the primorial spine. At each stage k, records whether prime\ + \ p_k is B-dominant or C-dominant.\n\n## Statement\n\n```latex\n%\n\\label{def:polstream}\nLet $p_1 < p_2 < p_3 < \\cdots$\ + \ be the enumeration\nof the internal primes $\\mathbb{P}_\\tau$.\nThe \\textbf{polarized stream} is the sequence\n\\\ + [\n \\mathrm{PolStream}(\\omega)\n \\;:=\\;\n \\bigl(\\mathrm{pol}(p_1),\\;\n \\mathrm{pol}(p_2),\\;\n \ + \ \\mathrm{pol}(p_3),\\;\n \\ldots\\bigr)\n \\;\\in\\; \\{+, -\\}^{\\mathbb{N}},\n\\]\nwhere $\\mathrm{pol}(p_k)$\ + \ is the polarity of $p_k$\n(Definition~\\ref{def:polarity-map}).\nAt each stage $k$, the PolStream records which channel\n\ + --- $B$ ($\\gamma$, encoded as $+$)\nor $C$ ($\\eta$, encoded as $-$) ---\ndominates the $k$-th prime along the primorial\ + \ spine.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 246\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch29-polarized-omega-germs.tex`\ + \ lines 295-315\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.D26, I.T05\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'PolStream(omega): infinite binary sequence recording channel dominance along the primorial spine. At each stage + k, records whether prime p_k is B-dominant or C-dominant.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001018.md b/_corpus_v3_items/cid001018.md new file mode 100644 index 000000000..9d8c2efaa --- /dev/null +++ b/_corpus_v3_items/cid001018.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001018/ +title: Tau-Weighted Boundary Constants +cid: cid001018 +primary_alias: DEF0016 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'pi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary constants for the two lemniscate lobes. + pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.' +aliases: +- DEF0016 +- I.D105 +- tau-weighted-boundary-constants +- def:pi-tau-boundary +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 41 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Weighted Boundary Constants**\n\npi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary\ + \ constants for the two lemniscate lobes. pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.\n\ + \n# Tau-Weighted Boundary Constants\n\n## Summary\n\npi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary\ + \ constants for the two lemniscate lobes. pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:pi-tau-boundary}\n% Depends: I.D34\nDefine the \\textbf{$\\tau$-weighted boundary\ + \ constants}:\n\\[\n \\boxed{%\n \\pi_\\tau := \\pi \\cdot \\iota_\\tau\n = \\frac{2\\pi}{\\pi + e}\n \\approx\ + \ 1.07262,}\n \\qquad\n e_\\tau := e \\cdot \\iota_\\tau\n = \\frac{2e}{\\pi + e}\n \\approx 0.92738.\n\\\ + ]\nThese satisfy the \\textbf{normalization identity}:\n\\[\n \\pi_\\tau + e_\\tau = 2\\iota_\\tau.\n\\]\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 250\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch41-master-constant.tex`\ + \ lines 340-358\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.D34, I.D20\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'pi_tau := pi * iota_tau and e_tau := e * iota_tau: tau-weighted boundary constants for the two lemniscate lobes. + pi_tau + e_tau = 2*iota_tau. Ratio pi_tau/e_tau = pi/e encodes lobe asymmetry.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001019.md b/_corpus_v3_items/cid001019.md new file mode 100644 index 000000000..0eeb4f03e --- /dev/null +++ b/_corpus_v3_items/cid001019.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001019/ +title: Tau-Bayesian State +cid: cid001019 +primary_alias: DEF0017 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A tau-Bayesian state S: Obj(tau) -> [0,1]_tau satisfying normalization, polarity coherence (factors through witness + pair functor), and positivity. Natural substrate for Bayesian reasoning on Truth4.' +aliases: +- DEF0017 +- I.D106 +- tau-bayesian-state +- def:tau-bayesian-state +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 48 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Bayesian State**\n\nA tau-Bayesian state S: Obj(tau) -> [0,1]_tau satisfying normalization, polarity coherence\ + \ (factors through witness pair functor), and positivity. Natural substrate for Bayesian reasoning on Truth4.\n\n# Tau-Bayesian\ + \ State\n\n## Summary\n\nA tau-Bayesian state S: Obj(tau) -> [0,1]_tau satisfying normalization, polarity coherence (factors\ + \ through witness pair functor), and positivity. Natural substrate for Bayesian reasoning on Truth4.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:tau-bayesian-state}\n% Depends: I.D21, I.T12\nA \\textbf{$\\tau$-Bayesian state} is a morphism\n\ + \\[\n S : \\mathrm{Obj}(\\tau) \\to [0,1]_\\tau\n\\]\nsatisfying three conditions:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Normalization.}\n $S(\\mathsf{T}) + S(\\mathsf{F})\n + S(\\mathsf{B}) + S(\\mathsf{N}) =\ + \ 1$\n in $[0,1]_\\tau$.\n \\item \\textbf{Polarity coherence.}\n $S$ factors through the witness\ + \ pair functor\n $W : \\mathrm{Obj}(\\tau) \\to \\mathrm{Truth4}$:\n for every object $X$,\n \ + \ $S(X) = S(W(X))$.\n That is, the probability assigned to $X$\n depends only on its truth-value classification,\n\ + \ not on additional internal structure.\n \\item \\textbf{Positivity.}\n $S(v) \\geq 0$ for all\n\ + \ $v \\in \\mathrm{Truth4}$.\n\\end{enumerate}\nA $\\tau$-Bayesian state is therefore\na probability distribution\ + \ on the four truth values,\nconstrained by the requirement\nthat it respects the witness pair structure.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 251\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch48-boolean-recovery.tex`\ + \ lines 915-944\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.D21, I.T12\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A tau-Bayesian state S: Obj(tau) -> [0,1]_tau satisfying normalization, polarity coherence (factors through + witness pair functor), and positivity. Natural substrate for Bayesian reasoning on Truth4.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001020.md b/_corpus_v3_items/cid001020.md new file mode 100644 index 000000000..e1961e985 --- /dev/null +++ b/_corpus_v3_items/cid001020.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001020/ +title: Index Multiplication +cid: cid001020 +primary_alias: DEF0018 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n * m = n + n + ... + n (m times): multiplication earned by structural recursion on addition.' +aliases: +- DEF0018 +- I.D11 +- index-multiplication +- def:idx-mul +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Index Multiplication**\n\nn * m = n + n + ... + n (m times): multiplication earned by structural recursion\ + \ on addition.\n\n# Index Multiplication\n\n## Summary\n\nn * m = n + n + ... + n (m times): multiplication earned by\ + \ structural recursion on addition.\n\n## Statement\n\n```latex\n%\n\\label{def:idx-mul}\nFor $\\underline{n}, \\underline{m}\ + \ \\in \\tau\\text{-Idx}$,\n\\textbf{index multiplication} is defined by\nstructural recursion on $m$:\n\\begin{align*}\n\ + \ \\underline{n} \\times \\underline{0} &:= \\underline{0}, \\\\\n \\underline{n} \\times \\rho(\\underline{m})\n\ + \ &:= (\\underline{n} \\times \\underline{m}) + \\underline{n}.\n\\end{align*}\nThat is, $\\underline{n} \\times \\\ + underline{m}\n= \\underbrace{\\underline{n} + \\cdots + \\underline{n}}_{m \\text{ times}}$.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 25\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex`\ + \ lines 159-172\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.idx_mul`\n\n## Dependencies\n\n- Canonical: I.D10, I.D06\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n * m = n + n + ... + n (m times): multiplication earned by structural recursion on addition.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001021.md b/_corpus_v3_items/cid001021.md new file mode 100644 index 000000000..4220f13d8 --- /dev/null +++ b/_corpus_v3_items/cid001021.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001021/ +title: Index Exponentiation +cid: cid001021 +primary_alias: DEF0019 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n^m = n * n * ... * n (m times): exponentiation earned by structural recursion on multiplication.' +aliases: +- DEF0019 +- I.D12 +- index-exponentiation +- def:idx-exp +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Index Exponentiation**\n\nn^m = n * n * ... * n (m times): exponentiation earned by structural recursion on\ + \ multiplication.\n\n# Index Exponentiation\n\n## Summary\n\nn^m = n * n * ... * n (m times): exponentiation earned by\ + \ structural recursion on multiplication.\n\n## Statement\n\n```latex\n%\n\\label{def:idx-exp}\nFor $\\underline{n}, \\\ + underline{m} \\in \\tau\\text{-Idx}$,\n\\textbf{index exponentiation} is defined by\nstructural recursion on $m$:\n\\\ + begin{align*}\n \\underline{n}^{\\underline{0}} &:= \\underline{1}, \\\\\n \\underline{n}^{\\rho(\\underline{m})}\n\ + \ &:= \\underline{n}^{\\underline{m}} \\times \\underline{n}.\n\\end{align*}\nThat is, $\\underline{n}^{\\underline{m}}\n\ + = \\underbrace{\\underline{n} \\times \\cdots \\times \\underline{n}}_{m \\text{ times}}$.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 26\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 56-69\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.idx_exp`\n\n## Dependencies\n\n- Canonical: I.D11, I.D06\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n^m = n * n * ... * n (m times): exponentiation earned by structural recursion on multiplication.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001022.md b/_corpus_v3_items/cid001022.md new file mode 100644 index 000000000..515dd6762 --- /dev/null +++ b/_corpus_v3_items/cid001022.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001022/ +title: Index Tetration +cid: cid001022 +primary_alias: DEF0020 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n ^^m = n^(n^(...^n)) (m times): tetration earned by structural recursion on exponentiation. Terminal operation + (ladder saturates here).' +aliases: +- DEF0020 +- I.D13 +- index-tetration +- def:idx-tetration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Index Tetration**\n\nn ^^m = n^(n^(...^n)) (m times): tetration earned by structural recursion on exponentiation.\ + \ Terminal operation (ladder saturates here).\n\n# Index Tetration\n\n## Summary\n\nn ^^m = n^(n^(...^n)) (m times): tetration\ + \ earned by structural recursion on exponentiation. Terminal operation (ladder saturates here).\n\n## Statement\n\n```latex\n\ + %\n\\label{def:idx-tetration}\nFor $\\underline{a}, \\underline{c} \\in \\tau\\text{-Idx}$\nwith $\\underline{a} \\geq\ + \ \\underline{2}$,\n\\textbf{index tetration} is defined by\nstructural recursion on $c$:\n\\begin{align*}\n {}^{\\\ + underline{0}}\\underline{a} &:= \\underline{1}, \\\\\n {}^{\\rho(\\underline{c})}\\underline{a}\n &:= \\underline{a}^{({}^{\\\ + underline{c}}\\underline{a})}.\n\\end{align*}\nThat is, ${}^{\\underline{c}}\\underline{a}$\nis a right-associative tower\ + \ of exponentials:\n\\[\n {}^{\\underline{c}}\\underline{a}\n = \\underbrace{\\underline{a}^{\\underline{a}^{\\\ + cdot^{\\cdot^{\\cdot^{\\underline{a}}}}}}}_{c \\text{ copies of } \\underline{a}}.\n\\]\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 27\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 110-128\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.idx_tetration`\n\n## Dependencies\n\n- Canonical: I.D12, I.D06, I.T02\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n ^^m = n^(n^(...^n)) (m times): tetration earned by structural recursion on exponentiation. Terminal operation + (ladder saturates here).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001023.md b/_corpus_v3_items/cid001023.md new file mode 100644 index 000000000..da53d8556 --- /dev/null +++ b/_corpus_v3_items/cid001023.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001023/ +title: Program Monoid +cid: cid001023 +primary_alias: DEF0021 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Programs = finite sequences of rho-instructions. Composition = sequential concatenation + normalization. Identity + = empty program. The compositional substrate for future categories. +aliases: +- DEF0021 +- I.D14 +- program-monoid +- def:program-monoid +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Program Monoid**\n\nPrograms = finite sequences of rho-instructions. Composition = sequential concatenation\ + \ + normalization. Identity = empty program. The compositional substrate for future categories.\n\n# Program Monoid\n\n\ + ## Summary\n\nPrograms = finite sequences of rho-instructions. Composition = sequential concatenation + normalization.\ + \ Identity = empty program. The compositional substrate for future categories.\n\n## Statement\n\n```latex\n%\n\\label{def:program-monoid}\n\ + The \\textbf{program monoid} $\\mathcal{P}$ is the set\nof NF-equivalence classes of programs,\nequipped with:\n\\begin{itemize}\n\ + \ \\item \\textbf{Composition}: $[Q] \\circ [P] := [\\text{NF}(Q \\cdot P)]$,\n where $Q \\cdot P$ denotes\ + \ concatenation\n and $\\text{NF}$ denotes normalization.\n \\item \\textbf{Identity}: $[\\varepsilon]$, the\ + \ empty program.\n\\end{itemize}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is\ + \ required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 28\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch13-program-monoid.tex`\ + \ lines 133-145\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.ProgramMonoid`\n\ + - Name: `Tau.Denotation.ProgramMonoid`\n\n## Dependencies\n\n- Canonical: I.D07, I.D09, I.D02, I.T01\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Programs = finite sequences of rho-instructions. Composition = sequential concatenation + normalization. Identity + = empty program. The compositional substrate for future categories. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001024.md b/_corpus_v3_items/cid001024.md new file mode 100644 index 000000000..820e45000 --- /dev/null +++ b/_corpus_v3_items/cid001024.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001024/ +title: Three-Level Equality +cid: cid001024 +primary_alias: DEF0022 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Three levels of sameness: (1) ontic identity (primitive), (2) address equivalence (decidable, NF-based), (3) shadow + equality (external VM readout).' +aliases: +- DEF0022 +- I.D15 +- three-level-equality +- def:three-equality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 14 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Three-Level Equality**\n\nThree levels of sameness: (1) ontic identity (primitive), (2) address equivalence\ + \ (decidable, NF-based), (3) shadow equality (external VM readout).\n\n# Three-Level Equality\n\n## Summary\n\nThree levels\ + \ of sameness: (1) ontic identity (primitive), (2) address equivalence (decidable, NF-based), (3) shadow equality (external\ + \ VM readout).\n\n## Statement\n\n```latex\n%\n\\label{def:three-equality}\nThe three levels of equality in $\\tau$ are:\n\ + \n\\medskip\n\\textbf{Level 1 --- Ontic identity ($=_{\\mathrm{ont}}$).}\nTwo objects $x, y \\in \\Obj(\\tau)$ are \\\ + textbf{ontically identical}\nif they are the same element of $\\Obj(\\tau)$:\nsame seed generator and same depth.\n\\\ + [\n x =_{\\mathrm{ont}} y\n \\quad:\\Longleftrightarrow\\quad\n \\mathrm{seed}(x) = \\mathrm{seed}(y)\n \\\ + ;\\text{and}\\;\n \\mathrm{depth}(x) = \\mathrm{depth}(y).\n\\]\nThis is the primitive notion of equality.\nIt is decidable:\ + \ given any two objects,\none can compare their seeds and depths in finite time.\n```\n\n## Proof / Justification\n\n\ + This item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 29\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch14-three-equality.tex`\ + \ lines 54-74\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Equality`\n\ + - Name: `Tau.Denotation.Equality`\n\n## Dependencies\n\n- Canonical: I.T01, I.D07\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Three levels of sameness: (1) ontic identity (primitive), (2) address equivalence (decidable, NF-based), (3) + shadow equality (external VM readout).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001025.md b/_corpus_v3_items/cid001025.md new file mode 100644 index 000000000..688ed4396 --- /dev/null +++ b/_corpus_v3_items/cid001025.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001025/ +title: NF Address Encoding +cid: cid001025 +primary_alias: DEF0023 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every object x in Obj(tau) has a normal form X = ((A^^C)^B)*D via greedy peel-off of tetration, exponentiation, multiplication, + and radial index. +aliases: +- DEF0023 +- I.D16 +- nf-address-encoding +- def:nf-encoding +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 18 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**NF Address Encoding**\n\nEvery object x in Obj(tau) has a normal form X = ((A^^C)^B)*D via greedy peel-off\ + \ of tetration, exponentiation, multiplication, and radial index.\n\n# NF Address Encoding\n\n## Summary\n\nEvery object\ + \ x in Obj(tau) has a normal form X = ((A^^C)^B)*D via greedy peel-off of tetration, exponentiation, multiplication, and\ + \ radial index.\n\n## Statement\n\n```latex\n%\n\\label{def:nf-encoding}\nThe \\textbf{normal-form address encoding}\n\ + of $X \\in \\tau\\text{-Idx}$ is the expression\n\\[\n \\boxed{X\n \\;=\\;\n ((\\underline{A} \\uparrow\\uparrow\ + \ \\underline{C})^{\\underline{B}})\n \\cdot \\underline{D}}\n\\]\nwhere $(\\underline{A}, \\underline{B}, \\underline{C},\ + \ \\underline{D})\n= \\Phi(X)$ is the output of the greedy peel-off algorithm\n(Definition~\\ref{def:greedy-peel}).\n\ + For $X = \\underline{1}$, the NF is the trivial encoding\n$\\underline{1} = \\underline{1} \\cdot \\underline{1}$.\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 32\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch18-nf-encoding.tex`\ + \ lines 40-56\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.NormalForm`\n\ + - Name: `Tau.Coordinates.spine`\n\n## Dependencies\n\n- Canonical: I.D19c, I.D19d, I.T09\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every object x in Obj(tau) has a normal form X = ((A^^C)^B)*D via greedy peel-off of tetration, exponentiation, + multiplication, and radial index. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001026.md b/_corpus_v3_items/cid001026.md new file mode 100644 index 000000000..38de7a726 --- /dev/null +++ b/_corpus_v3_items/cid001026.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001026/ +title: Denotational Order +cid: cid001026 +primary_alias: DEF0024 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The canonical linear order on Obj(tau): orbit-first (by K1 generator order), then depth-within-orbit. Extends K1 + from generators to all objects.' +aliases: +- DEF0024 +- I.D16a +- denotational-order +- def:denotational-order +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Denotational Order**\n\nThe canonical linear order on Obj(tau): orbit-first (by K1 generator order), then depth-within-orbit.\ + \ Extends K1 from generators to all objects.\n\n# Denotational Order\n\n## Summary\n\nThe canonical linear order on Obj(tau):\ + \ orbit-first (by K1 generator order), then depth-within-orbit. Extends K1 from generators to all objects.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:denotational-order}\nThe \\textbf{denotational order} $<_\\tau$ on $\\Obj(\\tau)$\nis defined\ + \ as follows.\nFor $x, y \\in \\Obj(\\tau)$:\n\\begin{enumerate}\n \\item If $x \\in O_g$ and $y \\in O_h$ with $g\ + \ \\neq h$\n (including $g$ or $h$ equal to $\\omega$),\n then $x <_\\tau y$ iff $g < h$\n \ + \ in the generator order $\\KAxiom{1}$.\n (All of $O_\\alpha$ precedes all of $O_\\pi$,\n which precedes\ + \ all of $O_\\gamma$, etc.)\n\n \\item If $x, y \\in O_g$ for the same generator $g$\n (or both are $\\omega$),\n\ + \ then $x <_\\tau y$ iff\n $\\mathrm{depth}(x) < \\mathrm{depth}(y)$.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 49\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch15-denotational-order.tex`\ + \ lines 44-63\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Order`\n\ + - Name: `Tau.Denotation.denotational_order`\n\n## Dependencies\n\n- Canonical: I.D07, I.T01, I.K1\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The canonical linear order on Obj(tau): orbit-first (by K1 generator order), then depth-within-orbit. Extends + K1 from generators to all objects.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001027.md b/_corpus_v3_items/cid001027.md new file mode 100644 index 000000000..3f326cf88 --- /dev/null +++ b/_corpus_v3_items/cid001027.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001027/ +title: ABCD Coordinate Chart +cid: cid001027 +primary_alias: DEF0025 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Canonical coordinate chart Phi(x) = (A,B,C,D) via greedy peel-off; four typed coordinate functions (coord_pi, coord_gamma, + coord_eta, coord_alpha). +aliases: +- DEF0025 +- I.D17 +- abcd-coordinate-chart +- def:abcd-chart +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 19 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**ABCD Coordinate Chart**\n\nCanonical coordinate chart Phi(x) = (A,B,C,D) via greedy peel-off; four typed coordinate\ + \ functions (coord_pi, coord_gamma, coord_eta, coord_alpha).\n\n# ABCD Coordinate Chart\n\n## Summary\n\nCanonical coordinate\ + \ chart Phi(x) = (A,B,C,D) via greedy peel-off; four typed coordinate functions (coord_pi, coord_gamma, coord_eta, coord_alpha).\n\ + \n## Statement\n\n```latex\n%\n\\label{def:abcd-chart}\nThe \\textbf{ABCD coordinate chart} is the map\n\\[\n \\boxed{\\\ + Phi : \\Obj(\\tau) \\longrightarrow\n \\tau\\text{-Idx}^4,\n \\qquad\n x \\longmapsto\n (\\operatorname{coord}_A(x),\\\ + ;\n \\operatorname{coord}_B(x),\\;\n \\operatorname{coord}_C(x),\\;\n \\operatorname{coord}_D(x))}\n\\]\n\ + where, for $x \\in \\Obj(\\tau)$ with index\n$\\underline{n} = \\mathrm{idx}(x) \\geq \\underline{2}$:\n\\begin{itemize}\n\ + \ \\item $\\operatorname{coord}_A(x) = \\underline{A}$:\n the \\textbf{largest prime divisor} of $\\underline{n}$\n\ + \ ($\\pi$-channel).\n \\item $\\operatorname{coord}_B(x) = \\underline{B}$:\n the \\textbf{maximal\ + \ exponent} at tower level\n ($\\gamma$-channel).\n \\item $\\operatorname{coord}_C(x) = \\underline{C}$:\n\ + \ the \\textbf{maximal tetration height}\n ($\\eta$-channel).\n \\item $\\operatorname{coord}_D(x)\ + \ = \\underline{D}$:\n the \\textbf{remainder} after tower extraction\n ($\\alpha$-channel).\n\\end{itemize}\n\ + For $x$ with $\\mathrm{idx}(x) \\in \\{\\underline{0}, \\underline{1}\\}$,\nwe set $\\Phi(x) = (\\underline{1}, \\underline{0},\n\ + \\underline{0}, \\mathrm{idx}(x))$\n(degenerate addresses).\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 33\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch19-abcd-chart.tex` lines 46-80\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.ABCD`\n- Name: `Tau.Coordinates.abcd_chart`\n\ + \n## Dependencies\n\n- Canonical: I.D16\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical coordinate chart Phi(x) = (A,B,C,D) via greedy peel-off; four typed coordinate functions (coord_pi, + coord_gamma, coord_eta, coord_alpha). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001028.md b/_corpus_v3_items/cid001028.md new file mode 100644 index 000000000..092795374 --- /dev/null +++ b/_corpus_v3_items/cid001028.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001028/ +title: Algebraic Lemniscate +cid: cid001028 +primary_alias: DEF0026 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: L defined as the bipolar spectral algebra H_tau = A_tau^(B) x A_tau^(C) with crossing point and polarity involution + sigma. The algebraic pre-geometric definition of the lemniscate boundary. Geometric form S^1 v S^1 emerges in Book II. Earned + from prime polarity, omega-germs, and split-complex discovery. +aliases: +- DEF0026 +- I.D18 +- algebraic-lemniscate +- thm:algebraic-lemniscate +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005337 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Algebraic Lemniscate**\n\nL defined as the bipolar spectral algebra H_tau = A_tau^(B) x A_tau^(C) with crossing\ + \ point and polarity involution sigma. The algebraic pre-geometric definition of the lemniscate boundary. Geometric form\ + \ S^1 v S^1 emerges in Book II. Earned from prime polarity, omega-germs, and split-complex discovery.\n\n# Algebraic Lemniscate\n\ + \n## Summary\n\nL defined as the bipolar spectral algebra H_tau = A_tau^(B) x A_tau^(C) with crossing point and polarity\ + \ involution sigma. The algebraic pre-geometric definition of the lemniscate boundary. Geometric form S^1 v S^1 emerges\ + \ in Book II. Earned from prime polarity, omega-germs, and split-complex discovery.\n\n## Statement\n\n```latex\n%\n\\\ + label{thm:algebraic-lemniscate}\nThe boundary of $\\tau$ carries a canonical algebraic structure\n$\\mathbb{L}$ consisting\ + \ of:\n\\begin{enumerate}\n \\item The \\textbf{bipolar spectral algebra}\n $H_\\tau = \\hat{\\mathbb{Z}}_\\\ + tau[j]$\n (Definition~\\ref{def:bipolar-spectral-algebra})\n with two sectors $e_+ H_\\tau$ and $e_-\ + \ H_\\tau$\n corresponding to the two polarity channels.\n \\item The \\textbf{crossing-point germ}\n \ + \ (Definition~\\ref{def:crossing-germ}):\n the unique omega-germ $\\omega_{\\mathbb{L}}$\n where\ + \ neither channel is eventually constant ---\n both sectors remain active.\n This germ acts as the \\\ + emph{identity element}\n of the bipolar structure.\n \\item The \\textbf{polarity involution}\n $\\\ + sigma \\colon H_\\tau \\to H_\\tau$\n defined by $\\sigma(j) = -j$,\n which swaps the B-sector and C-sector:\n\ + \ $\\sigma(e_+) = e_-$ and $\\sigma(e_-) = e_+$.\n\\end{enumerate}\nWe call $\\mathbb{L} = (H_\\tau, \\omega_{\\\ + mathbb{L}}, \\sigma)$\nthe \\textbf{algebraic lemniscate}.\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\n\ + The bipolar spectral algebra $H_\\tau$ is constructed\nin Definition~\\ref{def:bipolar-spectral-algebra}\nfrom the boundary\ + \ local ring and the split-complex unit $j$.\nThe crossing-point germ exists and is unique\nby Proposition~\\ref{prop:crossing-unique}:\n\ + it is the only omega-germ where both channels\nrefine nontrivially.\nThe polarity involution $\\sigma$ is the ring automorphism\n\ + of $H_\\tau$ defined by $\\sigma(j) = -j$,\nfixing $\\hat{\\mathbb{Z}}_\\tau$ pointwise.\nIt swaps the two idempotent\ + \ sectors\nand corresponds to the exchange of B-dominant\nand C-dominant roles.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 36\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex`\ + \ lines 555-581\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.Lemniscate`\n\ + - Name: `Tau.Polarity.AlgebraicLemniscate`\n\n## Dependencies\n\n- Canonical: I.D26, I.T05, I.D27\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: L defined as the bipolar spectral algebra H_tau = A_tau^(B) x A_tau^(C) with crossing point and polarity involution + sigma. The algebraic pre-geometric definition of the lemniscate boundary. Geometric form S^1 v S^1 emerges in Book II. Earne +noindex: false +--- + diff --git a/_corpus_v3_items/cid001029.md b/_corpus_v3_items/cid001029.md new file mode 100644 index 000000000..f0ad6aa98 --- /dev/null +++ b/_corpus_v3_items/cid001029.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001029/ +title: Boundary Ring and Scalars +cid: cid001029 +primary_alias: DEF0027 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Primorial inverse limits; CRT factorization; boundary ring as structured scaffold for scalars. iota_tau = 2/(pi+e) + as B/C dominance mediator (earned, not imposed). +aliases: +- DEF0027 +- I.D19 +- boundary-ring-and-scalars +- def:boundary-ring +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 39 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary Ring and Scalars**\n\nPrimorial inverse limits; CRT factorization; boundary ring as structured scaffold\ + \ for scalars. iota_tau = 2/(pi+e) as B/C dominance mediator (earned, not imposed).\n\n# Boundary Ring and Scalars\n\n\ + ## Summary\n\nPrimorial inverse limits; CRT factorization; boundary ring as structured scaffold for scalars. iota_tau\ + \ = 2/(pi+e) as B/C dominance mediator (earned, not imposed).\n\n## Statement\n\n```latex\n%\n\\label{def:boundary-ring}\n\ + The \\textbf{profinite boundary ring}\n$\\hat{\\mathbb{Z}}_\\tau$ is the inverse limit:\n\\[\n \\boxed{%\n \\hat{\\\ + mathbb{Z}}_\\tau\n := \\varprojlim_{k} \\mathbb{Z}/M_k\n = \\left\\{\n (r_1, r_2, r_3, \\ldots) \\in \\prod_{k\ + \ \\geq 1} \\mathbb{Z}/M_k\n : \\pi_{k,l}(r_l) = r_k \\text{ for all } k < l\n \\right\\}.}\n\\]\nAn element of\ + \ $\\hat{\\mathbb{Z}}_\\tau$ is an infinite sequence\n$(r_1, r_2, r_3, \\ldots)$ where $r_k \\in \\mathbb{Z}/M_k$,\nand\ + \ the sequence is \\textbf{compatible}:\nwhenever $k < l$, reducing $r_l$ modulo $M_k$ yields $r_k$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 37\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch39-profinite-boundary-ring.tex`\ + \ lines 94-112\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Ring`\n\ + - Name: `Tau.Boundary.BdryRing`\n\n## Dependencies\n\n- Canonical: I.D07, I.T04, I.D18, I.D28\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Primorial inverse limits; CRT factorization; boundary ring as structured scaffold for scalars. iota_tau = 2/(pi+e) + as B/C dominance mediator (earned, not imposed). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001030.md b/_corpus_v3_items/cid001030.md new file mode 100644 index 000000000..f5bf91e87 --- /dev/null +++ b/_corpus_v3_items/cid001030.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001030/ +title: Internal Divisibility +cid: cid001030 +primary_alias: DEF0028 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: a | b iff exists k in tau-Idx such that b = a * k. Decidable, reflexive, transitive. First relational concept earned + from Part III arithmetic. +aliases: +- DEF0028 +- I.D19a +- internal-divisibility +- def:divisibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Internal Divisibility** + + + a | b iff exists k in tau-Idx such that b = a * k. Decidable, reflexive, transitive. First relational concept earned from + Part III arithmetic. + + + # Internal Divisibility + + + ## Summary + + + a | b iff exists k in tau-Idx such that b = a * k. Decidable, reflexive, transitive. First relational concept earned from + Part III arithmetic. + + + ## Statement + + + ```latex + + % + + \label{def:divisibility} + + For $\underline{a}, \underline{b} \in \tau\text{-Idx}$, + + we say $\underline{a}$ \textbf{divides} $\underline{b}$, + + written $\underline{a} \mid \underline{b}$, if there exists + + $\underline{k} \in \tau\text{-Idx}$ such that + + $\underline{b} = \underline{a} \times \underline{k}$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 51 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex` lines + 43-51 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Coordinates.Primes` + + - Name: `Tau.Coordinates.idx_divides` + + + ## Dependencies + + + - Canonical: I.D10, I.D11 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: a | b iff exists k in tau-Idx such that b = a * k. Decidable, reflexive, transitive. First relational concept + earned from Part III arithmetic. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001031.md b/_corpus_v3_items/cid001031.md new file mode 100644 index 000000000..dbea1eecb --- /dev/null +++ b/_corpus_v3_items/cid001031.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001031/ +title: Internal Primes +cid: cid001031 +primary_alias: DEF0029 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: p is prime iff p >= 2 and only divisors are 1 and p. P_tau is the set of internal primes; infinite (Euclid's argument + earned internally). +aliases: +- DEF0029 +- I.D19b +- internal-primes +- def:internal-primes +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Internal Primes**\n\np is prime iff p >= 2 and only divisors are 1 and p. P_tau is the set of internal primes;\ + \ infinite (Euclid's argument earned internally).\n\n# Internal Primes\n\n## Summary\n\np is prime iff p >= 2 and only\ + \ divisors are 1 and p. P_tau is the set of internal primes; infinite (Euclid's argument earned internally).\n\n## Statement\n\ + \n```latex\n%\n\\label{def:internal-primes}\nAn element $\\underline{p} \\in \\tau\\text{-Idx}$ is\n\\textbf{prime} (or\ + \ \\textbf{irreducible}) if:\n\\begin{enumerate}\n \\item $\\underline{p} \\geq \\underline{2}$, and\n \\item the\ + \ only divisors of $\\underline{p}$ are\n $\\underline{1}$ and $\\underline{p}$ itself.\n\\end{enumerate}\nWe\ + \ write $\\mathbb{P}_\\tau$ for the set of internal primes.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 52\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex` lines 100-111\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.Primes`\n- Name: `Tau.Coordinates.idx_prime`\n\ + \n## Dependencies\n\n- Canonical: I.D19a\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: p is prime iff p >= 2 and only divisors are 1 and p. P_tau is the set of internal primes; infinite (Euclid's + argument earned internally). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001032.md b/_corpus_v3_items/cid001032.md new file mode 100644 index 000000000..7cada5bfd --- /dev/null +++ b/_corpus_v3_items/cid001032.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001032/ +title: Tower Atom +cid: cid001032 +primary_alias: DEF0030 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: T(a,b,c) = (a^^c)^b for prime a, b >= 1, c >= 1. The nesting ((a^^c)^b) is forced by diagonal discipline; only binding + from which all three parameters are uniquely recoverable. +aliases: +- DEF0030 +- I.D19c +- tower-atom +- def:tower-atom +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 17 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tower Atom**\n\nT(a,b,c) = (a^^c)^b for prime a, b >= 1, c >= 1. The nesting ((a^^c)^b) is forced by diagonal\ + \ discipline; only binding from which all three parameters are uniquely recoverable.\n\n# Tower Atom\n\n## Summary\n\n\ + T(a,b,c) = (a^^c)^b for prime a, b >= 1, c >= 1. The nesting ((a^^c)^b) is forced by diagonal discipline; only binding\ + \ from which all three parameters are uniquely recoverable.\n\n## Statement\n\n```latex\n%\n\\label{def:tower-atom}\n\ + A \\textbf{tower atom} is a triple\n$(\\underline{a}, \\underline{b}, \\underline{c})$\nwith $\\underline{a} \\in \\mathbb{P}_\\\ + tau$,\n$\\underline{b} \\geq \\underline{1}$,\n$\\underline{c} \\geq \\underline{1}$,\nwhose \\textbf{value} is\n\\[\n\ + \ \\boxed{T(\\underline{a}, \\underline{b}, \\underline{c})\n \\;=\\;\n (\\underline{a} \\uparrow\\uparrow \\\ + underline{c})^{\\underline{b}}.}\n\\]\nHere $\\underline{a} \\uparrow\\uparrow \\underline{c}$\nis the internal tetration\ + \ of Chapter~\\ref{ch:exp-tetration},\nand the exponentiation is internal\n(Chapter~\\ref{ch:swap-add-mul}).\n```\n\n\ + ## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 54\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch17-tower-atoms-peel.tex`\ + \ lines 40-58\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.TowerAtoms`\n\ + - Name: `Tau.Coordinates.tower_atom`\n\n## Dependencies\n\n- Canonical: I.D19b, I.D12, I.D13\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: T(a,b,c) = (a^^c)^b for prime a, b >= 1, c >= 1. The nesting ((a^^c)^b) is forced by diagonal discipline; only + binding from which all three parameters are uniquely recoverable. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001033.md b/_corpus_v3_items/cid001033.md new file mode 100644 index 000000000..af385a009 --- /dev/null +++ b/_corpus_v3_items/cid001033.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001033/ +title: Greedy Peel Algorithm +cid: cid001033 +primary_alias: DEF0031 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Greedy peel-off: (1) find A = largest prime dividing X, (2) find C = maximal tetration height, (3) find B = maximal + exponent, (4) D = remainder. Deterministic and terminating.' +aliases: +- DEF0031 +- I.D19d +- greedy-peel-algorithm +- def:greedy-peel +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 17 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Greedy Peel Algorithm**\n\nGreedy peel-off: (1) find A = largest prime dividing X, (2) find C = maximal tetration\ + \ height, (3) find B = maximal exponent, (4) D = remainder. Deterministic and terminating.\n\n# Greedy Peel Algorithm\n\ + \n## Summary\n\nGreedy peel-off: (1) find A = largest prime dividing X, (2) find C = maximal tetration height, (3) find\ + \ B = maximal exponent, (4) D = remainder. Deterministic and terminating.\n\n## Statement\n\n```latex\n%\n\\label{def:greedy-peel}\n\ + Given $X \\in \\tau\\text{-Idx}$ with $X \\geq \\underline{2}$,\nthe \\textbf{greedy peel-off} extracts the ABCD coordinates\n\ + as follows:\n\\begin{enumerate}\n \\item \\textbf{Find $A$ (largest prime divisor):}\n Let $\\underline{A}$\ + \ be the largest prime\n dividing $X$.\n \\item \\textbf{Find $C$ (maximal tetration height):}\n \ + \ Let $\\underline{C}$ be the largest\n $\\underline{c} \\geq \\underline{1}$ such that\n $(\\underline{A}\ + \ \\uparrow\\uparrow \\underline{c})\n \\mid X$.\n \\item \\textbf{Find $B$ (maximal exponent):}\n \ + \ Let $\\underline{B}$ be the largest\n $\\underline{b} \\geq \\underline{1}$ such that\n $(\\underline{A}\ + \ \\uparrow\\uparrow \\underline{C})^{\\underline{b}}\n \\mid X$.\n \\item \\textbf{Compute $D$ (remainder):}\n\ + \ Set $\\underline{D} = X \\;/\\;\n (\\underline{A} \\uparrow\\uparrow \\underline{C})^{\\underline{B}}$.\n\ + \\end{enumerate}\nThe output is the tuple\n$\\Phi(X) = (\\underline{A}, \\underline{B},\n\\underline{C}, \\underline{D})$.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 55\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch17-tower-atoms-peel.tex`\ + \ lines 177-204\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.TowerAtoms`\n\ + - Name: `Tau.Coordinates.greedy_peel`\n\n## Dependencies\n\n- Canonical: I.D19c, I.T09\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Greedy peel-off: (1) find A = largest prime dividing X, (2) find C = maximal tetration height, (3) find B = + maximal exponent, (4) D = remainder. Deterministic and terminating.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001034.md b/_corpus_v3_items/cid001034.md new file mode 100644 index 000000000..98c2815f1 --- /dev/null +++ b/_corpus_v3_items/cid001034.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001034/ +title: Prime Spectral Signature +cid: cid001034 +primary_alias: DEF0032 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max tetration height across objects with + A = p up to N. Stabilizes asymptotically due to growth-rate separation. +aliases: +- DEF0032 +- I.D19e +- prime-spectral-signature +- def:spectral-signature +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 6, Chapter 26 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Prime Spectral Signature**\n\nFor prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max\ + \ tetration height across objects with A = p up to N. Stabilizes asymptotically due to growth-rate separation.\n\n# Prime\ + \ Spectral Signature\n\n## Summary\n\nFor prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max tetration\ + \ height across objects with A = p up to N. Stabilizes asymptotically due to growth-rate separation.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:spectral-signature}\nFor a prime $\\underline{p} \\in \\mathbb{P}_\\tau$\nand a bound $\\underline{N}\ + \ \\in \\tau\\text{-Idx}$,\nthe \\textbf{spectral signature} of $\\underline{p}$\nup to $\\underline{N}$ is the pair\n\ + \\[\n \\sigma_{\\underline{N}}(\\underline{p})\n \\;=\\; (B_{\\underline{N}}, C_{\\underline{N}})\n\\]\nwhere:\n\ + \\begin{align*}\n B_{\\underline{N}} &:= \\max\\{\\underline{B} :\n \\exists\\, X \\leq \\underline{N},\\;\n \ + \ \\operatorname{coord}_A(X) = \\underline{p},\\;\n \\operatorname{coord}_B(X) = \\underline{B}\\}, \\\\\n C_{\\\ + underline{N}} &:= \\max\\{\\underline{C} :\n \\exists\\, X \\leq \\underline{N},\\;\n \\operatorname{coord}_A(X)\ + \ = \\underline{p},\\;\n \\operatorname{coord}_C(X) = \\underline{C}\\}.\n\\end{align*}\nWe say $\\underline{p}$ is\ + \ \\textbf{B-dominant up to $\\underline{N}$}\nif $B_{\\underline{N}} > C_{\\underline{N}}$,\nand \\textbf{C-dominant\ + \ up to $\\underline{N}$}\nif $C_{\\underline{N}} > B_{\\underline{N}}$.\n```\n\n## Proof / Justification\n\nThis item\ + \ is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 60\n\ + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part06/ch26-spectral-question.tex` lines\ + \ 78-104\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.Spectral`\n\ + - Name: `Tau.Polarity.spectral_sig`\n\n## Dependencies\n\n- Canonical: I.T04, I.D17\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For prime p and bound N, sigma_N(p) = (B_N, C_N) records max exponent and max tetration height across objects + with A = p up to N. Stabilizes asymptotically due to growth-rate separation. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001035.md b/_corpus_v3_items/cid001035.md new file mode 100644 index 000000000..cb53a4c46 --- /dev/null +++ b/_corpus_v3_items/cid001035.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001035/ +title: Prime Enumeration and Sieve +cid: cid001035 +primary_alias: DEF0033 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Prime enumeration nthPrime(k) = p_k, prime counting pi(n), and prime index (inverse). The earned Sieve of Eratosthenes: + computed entirely from rho-folds via is_prime_bool. Semantic orbit projection from pi-orbit to alpha-orbit values.' +aliases: +- DEF0033 +- I.D19f +- prime-enumeration-and-sieve +- def:prime-enumeration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Prime Enumeration and Sieve**\n\nPrime enumeration nthPrime(k) = p_k, prime counting pi(n), and prime index\ + \ (inverse). The earned Sieve of Eratosthenes: computed entirely from rho-folds via is_prime_bool. Semantic orbit projection\ + \ from pi-orbit to alpha-orbit values.\n\n# Prime Enumeration and Sieve\n\n## Summary\n\nPrime enumeration nthPrime(k)\ + \ = p_k, prime counting pi(n), and prime index (inverse). The earned Sieve of Eratosthenes: computed entirely from rho-folds\ + \ via is_prime_bool. Semantic orbit projection from pi-orbit to alpha-orbit values.\n\n## Statement\n\n```latex\n%\n\\\ + label{def:prime-enumeration}\nThe \\textbf{prime enumeration function}\n$\\mathrm{nthPrime} : \\tau\\text{-Idx} \\to \\\ + tau\\text{-Idx}$\nmaps each index $\\underline{k}$ to the $k$-th prime\n(zero-indexed):\n\\[\n \\mathrm{nthPrime}(\\\ + underline{0}) = \\underline{2}, \\quad\n \\mathrm{nthPrime}(\\underline{1}) = \\underline{3}, \\quad\n \\mathrm{nthPrime}(\\\ + underline{2}) = \\underline{5}, \\quad\n \\mathrm{nthPrime}(\\underline{3}) = \\underline{7}, \\quad \\ldots\n\\]\n\ + The \\textbf{prime counting function}\n$\\pi_\\tau : \\tau\\text{-Idx} \\to \\tau\\text{-Idx}$\ncounts primes up to a\ + \ given bound:\n\\[\n \\pi_\\tau(\\underline{n})\n = \\#\\{\\, \\underline{p} \\in \\mathbb{P}_\\tau\n :\ + \ \\underline{p} \\leq \\underline{n} \\,\\}.\n\\]\nThe \\textbf{prime index function}\n$\\mathrm{primeIdx} : \\mathbb{P}_\\\ + tau \\to \\tau\\text{-Idx}$\nis the inverse of $\\mathrm{nthPrime}$:\nfor each prime $\\underline{p}$,\n$\\mathrm{primeIdx}(\\\ + underline{p}) = \\underline{k}$\nwhere $\\mathrm{nthPrime}(\\underline{k}) = \\underline{p}$.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 69\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex`\ + \ lines 364-391\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.PrimeEnumeration`\n\ + - Name: `Tau.Coordinates.nthPrime`\n\n## Dependencies\n\n- Canonical: I.D19b\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Prime enumeration nthPrime(k) = p_k, prime counting pi(n), and prime index (inverse). The earned Sieve of Eratosthenes: + computed entirely from rho-folds via is_prime_bool. Semantic orbit projection from pi-orbit to alpha-orbit values.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001036.md b/_corpus_v3_items/cid001036.md new file mode 100644 index 000000000..b4f3edf24 --- /dev/null +++ b/_corpus_v3_items/cid001036.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001036/ +title: Split-Complex Scalars +cid: cid001036 +primary_alias: DEF0034 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Split-complex unit j with j^2 = +1 (not i^2 = -1). Earned from boundary ring structure. The foundational scalar field + for tau-holomorphy. +aliases: +- DEF0034 +- I.D20 +- split-complex-scalars +- def:split-complex +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 40 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Split-Complex Scalars**\n\nSplit-complex unit j with j^2 = +1 (not i^2 = -1). Earned from boundary ring structure.\ + \ The foundational scalar field for tau-holomorphy.\n\n# Split-Complex Scalars\n\n## Summary\n\nSplit-complex unit j with\ + \ j^2 = +1 (not i^2 = -1). Earned from boundary ring structure. The foundational scalar field for tau-holomorphy.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:split-complex}\nThe \\textbf{split-complex scalar ring}\n$\\hat{\\mathbb{Z}}_\\\ + tau[j]$ is the free $\\hat{\\mathbb{Z}}_\\tau$-module\nof rank 2, with basis $\\{1, j\\}$,\nequipped with the multiplication\ + \ rule:\n\\[\n \\boxed{%\n j^2 = +1.}\n\\]\nEvery element of $\\hat{\\mathbb{Z}}_\\tau[j]$\nhas a unique representation:\n\ + \\[\n x = a + bj,\n \\quad a, b \\in \\hat{\\mathbb{Z}}_\\tau.\n\\]\nMultiplication is defined by:\n\\[\n (a\ + \ + bj)(c + dj)\n = (ac + bd) + (ad + bc)j,\n\\]\nusing the distributive law and $j^2 = +1$.\nAddition is componentwise:\n\ + \\[\n (a + bj) + (c + dj) = (a + c) + (b + d)j.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 38\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch40-split-complex-scalars.tex` lines 55-82\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.SplitComplex`\n- Name: `Tau.Boundary.SplitComplex`\n\ + \n## Dependencies\n\n- Canonical: I.D19\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Split-complex unit j with j^2 = +1 (not i^2 = -1). Earned from boundary ring structure. The foundational scalar + field for tau-holomorphy. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001037.md b/_corpus_v3_items/cid001037.md new file mode 100644 index 000000000..bb75b7c77 --- /dev/null +++ b/_corpus_v3_items/cid001037.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001037/ +title: Truth4 Logic +cid: cid001037 +primary_alias: DEF0035 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: T, F, B (both), N (neither) from polarity + stabilization. Explosion blocked structurally. Boolean logic = forgetful + functor on Omega_tau. +aliases: +- DEF0035 +- I.D21 +- truth4-logic +- def:truth4 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 46 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Truth4 Logic**\n\nT, F, B (both), N (neither) from polarity + stabilization. Explosion blocked structurally.\ + \ Boolean logic = forgetful functor on Omega_tau.\n\n# Truth4 Logic\n\n## Summary\n\nT, F, B (both), N (neither) from\ + \ polarity + stabilization. Explosion blocked structurally. Boolean logic = forgetful functor on Omega_tau.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:truth4}\nThe \\textbf{Truth4 lattice} is the set\n\\[\n \\boxed{%\n \\mathrm{Truth4}\n\ + \ := \\{\\mathsf{T},\\; \\mathsf{B},\\; \\mathsf{N},\\; \\mathsf{F}\\}}\n\\]\nwith the following interpretation\nvia\ + \ bipolar evaluation $(\\chi_+, \\chi_-)$:\n\\begin{enumerate}\n \\item $\\mathsf{T}$ (\\textbf{true}):\n \ + \ $\\chi_+(P) = 1$ and $\\chi_-(P) = 1$.\n Both sectors confirm $P$.\n The predicate is \\emph{fully\ + \ determined and affirmative}.\n \\item $\\mathsf{F}$ (\\textbf{false}):\n $\\chi_+(P) = 0$ and $\\chi_-(P)\ + \ = 0$.\n Both sectors deny $P$.\n The predicate is \\emph{fully determined and negative}.\n \\item\ + \ $\\mathsf{B}$ (\\textbf{both}):\n $\\chi_+(P) = 1$ and $\\chi_-(P) = 0$,\n or $\\chi_+(P) = 0$ and\ + \ $\\chi_-(P) = 1$.\n One sector confirms and the other denies.\n The predicate is \\emph{overdetermined}\ + \ ---\n it is ``true and false simultaneously''\n from different spectral perspectives.\n \\item\ + \ $\\mathsf{N}$ (\\textbf{neither}):\n the predicate has no decisive witness\n in either sector.\n \ + \ Neither $\\chi_+(P)$ nor $\\chi_-(P)$\n produces a conclusive value.\n The predicate is \\\ + emph{underdetermined} ---\n it is ``neither true nor false''\n due to insufficient spectral evidence.\n\ + \\end{enumerate}\nThe encoding as pairs is:\n\\[\n \\mathsf{T} = (1, 1),\n \\quad\n \\mathsf{F} = (0, 0),\n \ + \ \\quad\n \\mathsf{B} = (1, 0) \\text{ or } (0, 1),\n \\quad\n \\mathsf{N} = (\\bot, \\bot),\n\\]\nwhere $\\\ + bot$ denotes the absence of a decisive witness.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 39\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch46-four-truth-values.tex`\ + \ lines 125-172\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Logic.Truth4`\n\ + - Name: `Tau.Logic.Truth4`\n\n## Dependencies\n\n- Canonical: I.T05, I.D19\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: T, F, B (both), N (neither) from polarity + stabilization. Explosion blocked structurally. Boolean logic = forgetful + functor on Omega_tau. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001038.md b/_corpus_v3_items/cid001038.md new file mode 100644 index 000000000..1e0309649 --- /dev/null +++ b/_corpus_v3_items/cid001038.md @@ -0,0 +1,127 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001038/ +title: Holomorphic Transformers +cid: cid001038 +primary_alias: DEF0036 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'KEY 3: omega-germ transformers as pre-categorical morphisms on the lemniscate boundary. HolFun, HolMap. Split-complex + holomorphy gives tau full analytic structure.' +aliases: +- DEF0036 +- I.D22 +- holomorphic-transformers +- def:hol-transformers +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter None (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Holomorphic Transformers** + + + KEY 3: omega-germ transformers as pre-categorical morphisms on the lemniscate boundary. HolFun, HolMap. Split-complex + holomorphy gives tau full analytic structure. + + + # Holomorphic Transformers + + + ## Summary + + + KEY 3: omega-germ transformers as pre-categorical morphisms on the lemniscate boundary. HolFun, HolMap. Split-complex + holomorphy gives tau full analytic structure. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 40 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Holomorphy.TauHolomorphic` + + - Name: `Tau.Holomorphy.HolFun` + + + ## Dependencies + + + - Canonical: I.D20, I.D14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'KEY 3: omega-germ transformers as pre-categorical morphisms on the lemniscate boundary. HolFun, HolMap. Split-complex + holomorphy gives tau full analytic structure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001039.md b/_corpus_v3_items/cid001039.md new file mode 100644 index 000000000..91efa6397 --- /dev/null +++ b/_corpus_v3_items/cid001039.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001039/ +title: Genealogical Decomposition (Spine) +cid: cid001039 +primary_alias: DEF0037 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Iterate the canonical peel along the remainder D to produce a finite sequence of ABC-triplets. X is recoverable by + multiplying the corresponding tower atoms. Spine Cayley length ell_spine(x) = number of triplets. +aliases: +- DEF0037 +- I.D23 +- genealogical-decomposition-spine +- def:genealogical-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 18 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Genealogical Decomposition (Spine)**\n\nIterate the canonical peel along the remainder D to produce a finite\ + \ sequence of ABC-triplets. X is recoverable by multiplying the corresponding tower atoms. Spine Cayley length ell_spine(x)\ + \ = number of triplets.\n\n# Genealogical Decomposition (Spine)\n\n## Summary\n\nIterate the canonical peel along the\ + \ remainder D to produce a finite sequence of ABC-triplets. X is recoverable by multiplying the corresponding tower atoms.\ + \ Spine Cayley length ell_spine(x) = number of triplets.\n\n## Statement\n\n```latex\n%\n\\label{def:genealogical-decomposition}\n\ + For $X \\in \\tau\\text{-Idx}$ with $X \\geq \\underline{2}$,\nthe \\textbf{genealogical decomposition} (or \\textbf{spine})\n\ + is the finite sequence of ABC-triplets obtained by\niterating the canonical peel along the remainder:\n\\begin{align*}\n\ + \ X \\;&\\mapsto\\; (\\underline{A_1}, \\underline{B_1},\n \\underline{C_1}, \\underline{D_1}), \\\\\n \\underline{D_1}\ + \ \\;&\\mapsto\\; (\\underline{A_2}, \\underline{B_2},\n \\underline{C_2}, \\underline{D_2}), \\\\\n &\\;\\;\\vdots\ + \ \\\\\n \\underline{D_{k-1}} \\;&\\mapsto\\; (\\underline{A_k}, \\underline{B_k},\n \\underline{C_k}, \\underline{1}).\n\ + \\end{align*}\nThe result is the ordered sequence\n\\[\n \\mathcal{G}(X) \\;=\\;\n \\bigl((\\underline{A_1}, \\\ + underline{B_1}, \\underline{C_1}),\\;\n (\\underline{A_2}, \\underline{B_2}, \\underline{C_2}),\\;\n \\ldots,\\\ + ;\n (\\underline{A_k}, \\underline{B_k}, \\underline{C_k})\\bigr).\n\\]\nThe original $X$ is recoverable by evaluating\ + \ and multiplying\nthe corresponding tower atoms:\n\\[\n X \\;=\\; T(\\underline{A_1}, \\underline{B_1}, \\underline{C_1})\n\ + \ \\;\\cdot\\; T(\\underline{A_2}, \\underline{B_2}, \\underline{C_2})\n \\;\\cdot\\; \\ldots \\;\\cdot\\;\n \ + \ T(\\underline{A_k}, \\underline{B_k}, \\underline{C_k}).\n\\]\nFor $X = \\underline{1}$, the spine is the empty sequence\n\ + $\\mathcal{G}(\\underline{1}) = ()$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 61\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch18-nf-encoding.tex`\ + \ lines 178-212\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.NormalForm`\n\ + - Name: `Tau.Coordinates.spine`\n\n## Dependencies\n\n- Canonical: I.D16, I.D19d\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Iterate the canonical peel along the remainder D to produce a finite sequence of ABC-triplets. X is recoverable + by multiplying the corresponding tower atoms. Spine Cayley length ell_spine(x) = number of triplets. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001040.md b/_corpus_v3_items/cid001040.md new file mode 100644 index 000000000..49a9ad175 --- /dev/null +++ b/_corpus_v3_items/cid001040.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001040/ +title: Address DAG +cid: cid001040 +primary_alias: DEF0038 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Full quadtree (recurse on all 4 coordinates by index) globally deduplicated by index identity. Diamond-shaped finite + DAG: branching out from root, reconverging via shared indices, terminating at alpha_1.' +aliases: +- DEF0038 +- I.D24 +- address-dag +- def:address-dag +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 20 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Address DAG**\n\nFull quadtree (recurse on all 4 coordinates by index) globally deduplicated by index identity.\ + \ Diamond-shaped finite DAG: branching out from root, reconverging via shared indices, terminating at alpha_1.\n\n# Address\ + \ DAG\n\n## Summary\n\nFull quadtree (recurse on all 4 coordinates by index) globally deduplicated by index identity.\ + \ Diamond-shaped finite DAG: branching out from root, reconverging via shared indices, terminating at alpha_1.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:address-dag}\nThe \\textbf{Address DAG} of $x \\in \\tau\\text{-Idx}$\nis the finite directed\ + \ acyclic graph\n$\\mathrm{DAG}(x) = (V, E)$ where:\n\\begin{itemize}\n \\item $V$ is the set of all indices reachable\n\ + \ from $x$ by iterating $\\mathrm{Ch}$:\n the transitive closure of the children relation.\n \\item\ + \ $E = \\{(u, v) : v \\in \\mathrm{Ch}(u),\\;\n u \\in V\\}$.\n\\end{itemize}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 62\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch20-dimension-fibration.tex`\ + \ lines 402-415\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.ABCD`\n\ + - Name: `Tau.Coordinates.dag_indices`\n\n## Dependencies\n\n- Canonical: I.D23, I.D17\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Full quadtree (recurse on all 4 coordinates by index) globally deduplicated by index identity. Diamond-shaped + finite DAG: branching out from root, reconverging via shared indices, terminating at alpha_1.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001041.md b/_corpus_v3_items/cid001041.md new file mode 100644 index 000000000..5a009e4c9 --- /dev/null +++ b/_corpus_v3_items/cid001041.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001041/ +title: Omega-Tail (Compatible Tower) +cid: cid001041 +primary_alias: DEF0039 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Compatible tower (x_k) on the primorial ladder M_k = prod of first k primes, with reduction maps. Pre-topological + boundary data on bare-metal ontic elements. Polarity is a tail-stable predicate on omega-tails. +aliases: +- DEF0039 +- I.D25 +- omega-tail-compatible-tower +- def:omega-tail +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 28 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega-Tail (Compatible Tower)**\n\nCompatible tower (x_k) on the primorial ladder M_k = prod of first k primes,\ + \ with reduction maps. Pre-topological boundary data on bare-metal ontic elements. Polarity is a tail-stable predicate\ + \ on omega-tails.\n\n# Omega-Tail (Compatible Tower)\n\n## Summary\n\nCompatible tower (x_k) on the primorial ladder M_k\ + \ = prod of first k primes, with reduction maps. Pre-topological boundary data on bare-metal ontic elements. Polarity\ + \ is a tail-stable predicate on omega-tails.\n\n## Statement\n\n```latex\n%\n\\label{def:omega-tail}\nAn \\textbf{omega-tail}\ + \ (or \\textbf{compatible tower})\nis a family $(x_k)_{k \\geq 1}$\nwith $x_k \\in \\mathbb{Z}/M_k\\mathbb{Z}$ satisfying:\n\ + \\[\n \\pi_{\\ell \\to k}(x_\\ell) \\;=\\; x_k\n \\qquad \\text{for all } k \\leq \\ell.\n\\]\nThe type of all omega-tails\ + \ is denoted\n$\\mathrm{OmegaTail}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 64\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch28-omega-germs.tex`\ + \ lines 125-137\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.OmegaGerms`\n\ + - Name: `Tau.Polarity.OmegaTail`\n\n## Dependencies\n\n- Canonical: I.T05\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Compatible tower (x_k) on the primorial ladder M_k = prod of first k primes, with reduction maps. Pre-topological + boundary data on bare-metal ontic elements. Polarity is a tail-stable predicate on omega-tails. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001042.md b/_corpus_v3_items/cid001042.md new file mode 100644 index 000000000..d7b147bb3 --- /dev/null +++ b/_corpus_v3_items/cid001042.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001042/ +title: Polarized Omega-Germ +cid: cid001042 +primary_alias: DEF0040 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'B-polarized omega-germ: C-channel eventually constant while B-channel refines nontrivially. C-polarized: symmetric. + Formalizes ''the B-channel wins'' / ''the C-channel wins'' as eventual stabilization of inverse system factors.' +aliases: +- DEF0040 +- I.D26 +- polarized-omega-germ +- def:polarized-germ +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 29 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Polarized Omega-Germ**\n\nB-polarized omega-germ: C-channel eventually constant while B-channel refines nontrivially.\ + \ C-polarized: symmetric. Formalizes 'the B-channel wins' / 'the C-channel wins' as eventual stabilization of inverse\ + \ system factors.\n\n# Polarized Omega-Germ\n\n## Summary\n\nB-polarized omega-germ: C-channel eventually constant while\ + \ B-channel refines nontrivially. C-polarized: symmetric. Formalizes 'the B-channel wins' / 'the C-channel wins' as eventual\ + \ stabilization of inverse system factors.\n\n## Statement\n\n```latex\n%\n\\label{def:polarized-germ}\nLet $W[\\omega]$\ + \ be a base omega-germ\nfor which the channel systems $(B_n)$ and $(C_n)$\nare defined.\n\\begin{itemize}\n \\item\ + \ $W[\\omega]$ is \\textbf{B-polarized}\n if $(C_n)$ is eventually constant\n while $(B_n)$ refines\ + \ nontrivially\n (is not eventually constant).\n \\item $W[\\omega]$ is \\textbf{C-polarized}\n if\ + \ $(B_n)$ is eventually constant\n while $(C_n)$ refines nontrivially.\n\\end{itemize}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 65\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch29-polarized-omega-germs.tex`\ + \ lines 116-131\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.PolarizedGerms`\n\ + - Name: `Tau.Polarity.BPolarized`\n\n## Dependencies\n\n- Canonical: I.D25, I.T05\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'B-polarized omega-germ: C-channel eventually constant while B-channel refines nontrivially. C-polarized: symmetric. + Formalizes ''the B-channel wins'' / ''the C-channel wins'' as eventual stabilization of inverse system factors.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001043.md b/_corpus_v3_items/cid001043.md new file mode 100644 index 000000000..bf26f10ae --- /dev/null +++ b/_corpus_v3_items/cid001043.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001043/ +title: Bipolar Spectral Algebra +cid: cid001043 +primary_alias: DEF0041 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: H_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical decomposition of every element + into B-sector and C-sector. Zero divisors managed by diagonal-free discipline. +aliases: +- DEF0041 +- I.D27 +- bipolar-spectral-algebra +- def:bipolar-spectral-algebra +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bipolar Spectral Algebra**\n\nH_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical\ + \ decomposition of every element into B-sector and C-sector. Zero divisors managed by diagonal-free discipline.\n\n# Bipolar\ + \ Spectral Algebra\n\n## Summary\n\nH_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical\ + \ decomposition of every element into B-sector and C-sector. Zero divisors managed by diagonal-free discipline.\n\n##\ + \ Statement\n\n```latex\n%\n\\label{def:bipolar-spectral-algebra}\nThe \\textbf{bipolar spectral algebra} is the split-complex\n\ + extension of the boundary local ring:\n\\[\n H_\\tau\n \\;:=\\;\n \\hat{\\mathbb{Z}}_\\tau[j]\n \\;=\\;\n\ + \ \\hat{\\mathbb{Z}}_\\tau \\oplus j \\cdot \\hat{\\mathbb{Z}}_\\tau,\n \\qquad j^2 = +1.\n\\]\nIt carries the following\ + \ canonical structure:\n\\begin{enumerate}\n \\item \\textbf{Bipolar idempotents:}\n $e_+ = \\frac{1+j}{2}$, \\\ + quad\n $e_- = \\frac{1-j}{2}$.\n \\item \\textbf{Idempotent properties:}\n $e_+^2 = e_+$, \\;\n $e_-^2 = e_-$,\ + \ \\;\n $e_+ \\cdot e_- = 0$, \\;\n $e_+ + e_- = 1$.\n \\item \\textbf{Canonical decomposition:}\n every $z\ + \ \\in H_\\tau$ decomposes uniquely as\n $z = e_+ \\cdot z_+ + e_- \\cdot z_-$\n with $z_\\pm \\in \\hat{\\mathbb{Z}}_\\\ + tau$.\n \\item \\textbf{Sector correspondence:}\n the $e_+$-sector corresponds to the B-channel primes;\n the\ + \ $e_-$-sector corresponds to the C-channel primes.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is\ + \ definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 67\n\ + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex` lines\ + \ 386-417\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.BipolarAlgebra`\n\ + - Name: `Tau.Polarity.SplitComplex`\n\n## Dependencies\n\n- Canonical: I.D25, I.T05, I.T10\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: H_tau = A_tau^(B) x A_tau^(C) with bipolar idempotents e_pm = (1 pm j)/2. Canonical decomposition of every element + into B-sector and C-sector. Zero divisors managed by diagonal-free discipline. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001044.md b/_corpus_v3_items/cid001044.md new file mode 100644 index 000000000..4ec33cb64 --- /dev/null +++ b/_corpus_v3_items/cid001044.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001044/ +title: Boundary Local Ring +cid: cid001044 +primary_alias: DEF0042 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Ring structure on omega-tails from primorial inverse limit. CRT factors Z/M_k into B-prime and C-prime contributions. + Earned ring operations on boundary data. +aliases: +- DEF0042 +- I.D28 +- boundary-local-ring +- def:boundary-local-ring +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary Local Ring**\n\nRing structure on omega-tails from primorial inverse limit. CRT factors Z/M_k into\ + \ B-prime and C-prime contributions. Earned ring operations on boundary data.\n\n# Boundary Local Ring\n\n## Summary\n\ + \nRing structure on omega-tails from primorial inverse limit. CRT factors Z/M_k into B-prime and C-prime contributions.\ + \ Earned ring operations on boundary data.\n\n## Statement\n\n```latex\n%\n\\label{def:boundary-local-ring}\nThe \\textbf{boundary\ + \ local ring} $\\hat{\\mathbb{Z}}_\\tau$\nis the inverse limit\n\\[\n \\hat{\\mathbb{Z}}_\\tau\n \\;:=\\;\n \\\ + varprojlim_k \\mathbb{Z}/M_k\\mathbb{Z}\n\\]\nequipped with componentwise addition and multiplication.\nAn element of\ + \ $\\hat{\\mathbb{Z}}_\\tau$\nis an omega-tail $(x_k)_{k \\geq 1}$\n(Definition~\\ref{def:omega-tail})\nthat is additionally\ + \ compatible with ring operations:\nif $(x_k)$ and $(y_k)$ are compatible towers,\nso are $(x_k + y_k)$ and $(x_k \\cdot\ + \ y_k)$\n(Section~\\ref{subsec:ch30-ring-ops}).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 68\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex`\ + \ lines 246-264\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.BipolarAlgebra`\n\ + - Name: `Tau.Polarity.bdry_add`\n\n## Dependencies\n\n- Canonical: I.D25\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Ring structure on omega-tails from primorial inverse limit. CRT factors Z/M_k into B-prime and C-prime contributions. + Earned ring operations on boundary data. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001045.md b/_corpus_v3_items/cid001045.md new file mode 100644 index 000000000..235c16706 --- /dev/null +++ b/_corpus_v3_items/cid001045.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001045/ +title: CRT Decomposition +cid: cid001045 +primary_alias: DEF0043 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CRT decomposition and reconstruction on the primorial ladder: forward map x -> (x mod p_i), inverse map via coprime + idempotents e_i, round-trip verified exhaustively by native_decide. Basis elements are idempotent, orthogonal, sum to unity. + CRT coherence with primorial reduction verified.' +aliases: +- DEF0043 +- I.D29 +- crt-decomposition +- subsec:ch30-crt +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT Decomposition** + + + CRT decomposition and reconstruction on the primorial ladder: forward map x -> (x mod p_i), inverse map via coprime idempotents + e_i, round-trip verified exhaustively by native_decide. Basis elements are idempotent, orthogonal, sum to unity. CRT coherence + with primorial reduction verified. + + + # CRT Decomposition + + + ## Summary + + + CRT decomposition and reconstruction on the primorial ladder: forward map x -> (x mod p_i), inverse map via coprime idempotents + e_i, round-trip verified exhaustively by native_decide. Basis elements are idempotent, orthogonal, sum to unity. CRT coherence + with primorial reduction verified. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 70 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Polarity.ChineseRemainder` + + - Name: `Tau.Polarity.crt_decompose` + + + ## Dependencies + + + - Canonical: I.D25, I.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CRT decomposition and reconstruction on the primorial ladder: forward map x -> (x mod p_i), inverse map via + coprime idempotents e_i, round-trip verified exhaustively by native_decide. Basis elements are idempotent, orthogonal, sum + to unity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001046.md b/_corpus_v3_items/cid001046.md new file mode 100644 index 000000000..1823f4f4b --- /dev/null +++ b/_corpus_v3_items/cid001046.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001046/ +title: Teichmueller Lift +cid: cid001046 +primary_alias: DEF0044 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Canonical lift of single-prime residue r in Z/p_iZ to omega-tail via CRT reconstruction at each stage. Satisfies + retraction (Lift_i(r) mod p_i = r), orthogonality (Lift_i(r) mod p_j = 0 for j != i), compatibility (compatible tower), + multiplicativity, and decomposition (nat_to_tail = sum of lifts). All properties verified by native_decide. +aliases: +- DEF0044 +- I.D30 +- teichmueller-lift +- def:teichmueller-lift +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Teichmueller Lift**\n\nCanonical lift of single-prime residue r in Z/p_iZ to omega-tail via CRT reconstruction\ + \ at each stage. Satisfies retraction (Lift_i(r) mod p_i = r), orthogonality (Lift_i(r) mod p_j = 0 for j != i), compatibility\ + \ (compatible tower), multiplicativity, and decomposition (nat_to_tail = sum of lifts). All properties verified by native_decide.\n\ + \n# Teichmueller Lift\n\n## Summary\n\nCanonical lift of single-prime residue r in Z/p_iZ to omega-tail via CRT reconstruction\ + \ at each stage. Satisfies retraction (Lift_i(r) mod p_i = r), orthogonality (Lift_i(r) mod p_j = 0 for j != i), compatibility\ + \ (compatible tower), multiplicativity, and decomposition (nat_to_tail = sum of lifts). All properties verified by native_decide.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:teichmueller-lift}\nFor $r \\in \\mathbb{Z}/\\underline{p}_i\\mathbb{Z}$,\n\ + the \\textbf{Teichmüller lift}\n$\\mathrm{Lift}_i(r) \\in \\hat{\\mathbb{Z}}_\\tau$\nis the omega-tail whose $k$-th component\n\ + (for $k \\geq i$) is\n\\[\n \\mathrm{Lift}_i(r)_k\n \\;:=\\;\n \\varphi_k^{-1}\\bigl(0, \\ldots, 0, \\underbrace{r}_{i},\ + \ 0, \\ldots, 0\\bigr)\n \\;=\\; r \\cdot e_i^{(k)} \\;\\bmod\\; M_k,\n\\]\nwhere $e_i^{(k)}$ is the CRT basis element\ + \ at depth $k$.\nFor $k < i$, the component is $0$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 71\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex` lines 209-224\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.TeichmuellerLift`\n- Name:\ + \ `Tau.Polarity.teich_lift`\n\n## Dependencies\n\n- Canonical: I.D29, I.D25\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical lift of single-prime residue r in Z/p_iZ to omega-tail via CRT reconstruction at each stage. Satisfies + retraction (Lift_i(r) mod p_i = r), orthogonality (Lift_i(r) mod p_j = 0 for j != i), compatibility (compatible tower), + multipl +noindex: false +--- + diff --git a/_corpus_v3_items/cid001047.md b/_corpus_v3_items/cid001047.md new file mode 100644 index 000000000..b1aa0a205 --- /dev/null +++ b/_corpus_v3_items/cid001047.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001047/ +title: tau-Membership Relation +cid: cid001047 +primary_alias: DEF0045 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility; decidable, + reflexive, antisymmetric. No ZFC axioms imported. +aliases: +- DEF0045 +- I.D31 +- tau-membership-relation +- def:tau-membership +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 32 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**tau-Membership Relation**\n\nA in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership\ + \ earned from divisibility; decidable, reflexive, antisymmetric. No ZFC axioms imported.\n\n# tau-Membership Relation\n\ + \n## Summary\n\nA in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility;\ + \ decidable, reflexive, antisymmetric. No ZFC axioms imported.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-membership}\n\ + For $\\underline{a}, \\underline{b} \\in \\tau\\text{-Idx}$,\nwe define the \\textbf{$\\tau$-membership relation} by:\n\ + \\[\n \\boxed{%\n \\underline{a} \\in_\\tau \\underline{b}\n \\quad\\iff\\quad\n \\underline{a} \\mid \\underline{b}.}\n\ + \\]\nThat is, $\\underline{a}$ is ``an element of'' $\\underline{b}$\nif and only if $\\underline{a}$ divides $\\underline{b}$\n\ + in the multiplicative structure on $\\tau$-Idx.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 77\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch32-membership-divisibility.tex`\ + \ lines 37-51\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Membership`\n\ + - Name: `Tau.Sets.tau_mem`\n\n## Dependencies\n\n- Canonical: I.D19a, I.T09, I.T04\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A in_tau B iff A | B (in the multiplicative structure on tau-Idx). Membership earned from divisibility; decidable, + reflexive, antisymmetric. No ZFC axioms imported. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001048.md b/_corpus_v3_items/cid001048.md new file mode 100644 index 000000000..2b8499e39 --- /dev/null +++ b/_corpus_v3_items/cid001048.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001048/ +title: Set-Theoretic Operations +cid: cid001048 +primary_alias: DEF0046 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Union = lcm, intersection = gcd, difference via prime factorization stripping. Set operations earned from the multiplicative + structure on tau-Idx. +aliases: +- DEF0046 +- I.D32 +- set-theoretic-operations +- def:set-operations +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 33 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Set-Theoretic Operations**\n\nUnion = lcm, intersection = gcd, difference via prime factorization stripping.\ + \ Set operations earned from the multiplicative structure on tau-Idx.\n\n# Set-Theoretic Operations\n\n## Summary\n\n\ + Union = lcm, intersection = gcd, difference via prime factorization stripping. Set operations earned from the multiplicative\ + \ structure on tau-Idx.\n\n## Statement\n\n```latex\n%\n\\label{def:set-operations}\nFor $\\underline{a}, \\underline{b}\ + \ \\in \\tau\\text{-Idx}$,\nwe define:\n\\begin{enumerate}\n \\item The \\textbf{$\\tau$-union} of $\\underline{a}$\ + \ and $\\underline{b}$\n by:\n \\[\n \\boxed{%\n \\underline{a} \\cup_\\tau\ + \ \\underline{b}\n := \\mathrm{lcm}(\\underline{a}, \\underline{b}).}\n \\]\n This is the\ + \ \\emph{smallest} object\n (under $\\in_\\tau$)\n that contains both $\\underline{a}$ and $\\underline{b}$\n\ + \ as members.\n\n \\item The \\textbf{$\\tau$-intersection} of $\\underline{a}$ and $\\underline{b}$\n \ + \ by:\n \\[\n \\boxed{%\n \\underline{a} \\cap_\\tau \\underline{b}\n \ + \ := \\mathrm{gcd}(\\underline{a}, \\underline{b}).}\n \\]\n This is the \\emph{largest} object\n\ + \ (under $\\in_\\tau$)\n that is a member of both $\\underline{a}$ and $\\underline{b}$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 79\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch33-set-operations.tex`\ + \ lines 39-68\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Operations`\n\ + - Name: `Tau.Sets.tau_union`\n\n## Dependencies\n\n- Canonical: I.D31, I.D19a, I.T09\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Union = lcm, intersection = gcd, difference via prime factorization stripping. Set operations earned from the + multiplicative structure on tau-Idx. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001049.md b/_corpus_v3_items/cid001049.md new file mode 100644 index 000000000..cab8fd516 --- /dev/null +++ b/_corpus_v3_items/cid001049.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001049/ +title: Bounded Powerset +cid: cid001049 +primary_alias: DEF0047 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'P_tau(X) = set of all divisors of X. Always finite: |P_tau(X)| = product of (e_i + 1) over prime factorization. + No Russell paradox: strict divisibility decreases value.' +aliases: +- DEF0047 +- I.D33 +- bounded-powerset +- def:bounded-powerset +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 34 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bounded Powerset**\n\nP_tau(X) = set of all divisors of X. Always finite: |P_tau(X)| = product of (e_i + 1)\ + \ over prime factorization. No Russell paradox: strict divisibility decreases value.\n\n# Bounded Powerset\n\n## Summary\n\ + \nP_tau(X) = set of all divisors of X. Always finite: |P_tau(X)| = product of (e_i + 1) over prime factorization. No Russell\ + \ paradox: strict divisibility decreases value.\n\n## Statement\n\n```latex\n%\n\\label{def:bounded-powerset}\nFor $\\\ + underline{x} \\in \\tau\\text{-Idx}$,\nthe \\textbf{$\\tau$-powerset} of $\\underline{x}$ is:\n\\[\n \\boxed{%\n \ + \ \\mathcal{P}_\\tau(\\underline{x})\n \\;:=\\;\n \\{\\underline{a} \\in \\tau\\text{-Idx}\n : \\underline{a}\ + \ \\in_\\tau \\underline{x}\\}\n \\;=\\;\n \\{\\underline{a} : \\underline{a} \\mid \\underline{x}\\}.}\n\\]\nThat\ + \ is, $\\mathcal{P}_\\tau(\\underline{x})$\nis the set of all \\textbf{divisors} of $\\underline{x}$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 81\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch34-bounded-powerset.tex`\ + \ lines 38-54\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Powerset`\n\ + - Name: `Tau.Sets.tau_divisors`\n\n## Dependencies\n\n- Canonical: I.D31, I.D32, I.T09\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'P_tau(X) = set of all divisors of X. Always finite: |P_tau(X)| = product of (e_i + 1) over prime factorization. + No Russell paradox: strict divisibility decreases value.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001050.md b/_corpus_v3_items/cid001050.md new file mode 100644 index 000000000..af02ac070 --- /dev/null +++ b/_corpus_v3_items/cid001050.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001050/ +title: Master Constant iota_tau +cid: cid001050 +primary_alias: DEF0048 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'iota_tau = 2/(pi+e) ~ 0.341304. The asymptotic B/C dominance mediator: parametrizes the split-complex decomposition + and measures the asymmetry between the two lemniscate lobes. Earned from prime distribution, not imposed.' +aliases: +- DEF0048 +- I.D34 +- master-constant-iota-tau +- def:iota-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 41 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Master Constant iota_tau**\n\niota_tau = 2/(pi+e) ~ 0.341304. The asymptotic B/C dominance mediator: parametrizes\ + \ the split-complex decomposition and measures the asymmetry between the two lemniscate lobes. Earned from prime distribution,\ + \ not imposed.\n\n# Master Constant iota_tau\n\n## Summary\n\niota_tau = 2/(pi+e) ~ 0.341304. The asymptotic B/C dominance\ + \ mediator: parametrizes the split-complex decomposition and measures the asymmetry between the two lemniscate lobes.\ + \ Earned from prime distribution, not imposed.\n\n## Statement\n\n```latex\n%\n\\label{def:iota-tau}\nThe \\textbf{master\ + \ constant} of Category $\\tau$ is:\n\\[\n \\boxed{\\iota_\\tau := \\frac{2}{\\pi + e}\n \\approx 0.341304.}\n\\\ + ]\nIt parametrizes the asymptotic proportion of $B$-dominant primes:\n$R_B = \\iota_\\tau$.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 84\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch41-master-constant.tex`\ + \ lines 134-144\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Iota`\n\ + - Name: `Tau.Boundary.iota_tau_numer`\n\n## Dependencies\n\n- Canonical: I.D19, I.D20, I.T05\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'iota_tau = 2/(pi+e) ~ 0.341304. The asymptotic B/C dominance mediator: parametrizes the split-complex decomposition + and measures the asymmetry between the two lemniscate lobes. Earned from prime distribution, not imposed.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001051.md b/_corpus_v3_items/cid001051.md new file mode 100644 index 000000000..9757b1421 --- /dev/null +++ b/_corpus_v3_items/cid001051.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001051/ +title: Number Tower +cid: cid001051 +primary_alias: DEF0049 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: N_tau subset Z_tau subset Q_tau subset R_tau subset C_tau. Integers via Grothendieck group, rationals via field of + fractions, reals via Cauchy completion, complexes via R_tau[i]. +aliases: +- DEF0049 +- I.D35 +- number-tower +- def:number-tower +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 42 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Number Tower**\n\nN_tau subset Z_tau subset Q_tau subset R_tau subset C_tau. Integers via Grothendieck group,\ + \ rationals via field of fractions, reals via Cauchy completion, complexes via R_tau[i].\n\n# Number Tower\n\n## Summary\n\ + \nN_tau subset Z_tau subset Q_tau subset R_tau subset C_tau. Integers via Grothendieck group, rationals via field of fractions,\ + \ reals via Cauchy completion, complexes via R_tau[i].\n\n## Statement\n\n```latex\n%\n\\label{def:number-tower}\nThe\ + \ \\textbf{number tower} of Category $\\tau$ is the hierarchy:\n\\[\n \\boxed{\\mathbb{N}_\\tau \\subseteq \\mathbb{Z}_\\\ + tau \\subseteq \\mathbb{Q}_\\tau\n \\subseteq \\mathbb{R}_\\tau \\subseteq \\mathbb{C}_\\tau.}\n\\]\nEach level is\ + \ a completion of the previous level\nvia a standard universal construction:\n\\begin{itemize}\n \\item $\\mathbb{N}_\\\ + tau \\to \\mathbb{Z}_\\tau$: Grothendieck group completion (adjoining negatives).\n \\item $\\mathbb{Z}_\\tau \\to\ + \ \\mathbb{Q}_\\tau$: Field of fractions (adjoining reciprocals).\n \\item $\\mathbb{Q}_\\tau \\to \\mathbb{R}_\\tau$:\ + \ Cauchy completion (adjoining limits).\n \\item $\\mathbb{R}_\\tau \\to \\mathbb{C}_\\tau$: Algebraic extension (adjoining\ + \ $\\sqrt{-1}$).\n\\end{itemize}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is\ + \ required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 85\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch42-number-tower.tex`\ + \ lines 482-498\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.NumberTower`\n\ + - Name: `Tau.Boundary.TauInt`\n\n## Dependencies\n\n- Canonical: I.D07, I.D19, I.D10, I.D11\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: N_tau subset Z_tau subset Q_tau subset R_tau subset C_tau. Integers via Grothendieck group, rationals via field + of fractions, reals via Cauchy completion, complexes via R_tau[i]. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001052.md b/_corpus_v3_items/cid001052.md new file mode 100644 index 000000000..10066fc58 --- /dev/null +++ b/_corpus_v3_items/cid001052.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001052/ +title: Constructive Reals +cid: cid001052 +primary_alias: DEF0050 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'R_tau: Cauchy completion of Q_tau with explicit modulus of convergence. Constructive reals aligned with computable/effectively + presented analysis.' +aliases: +- DEF0050 +- I.D36 +- constructive-reals +- def:constructive-reals +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 10, Chapter 42 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Constructive Reals**\n\nR_tau: Cauchy completion of Q_tau with explicit modulus of convergence. Constructive\ + \ reals aligned with computable/effectively presented analysis.\n\n# Constructive Reals\n\n## Summary\n\nR_tau: Cauchy\ + \ completion of Q_tau with explicit modulus of convergence. Constructive reals aligned with computable/effectively presented\ + \ analysis.\n\n## Statement\n\n```latex\n%\n\\label{def:constructive-reals}\nThe \\textbf{$\\tau$-real numbers} are defined\ + \ as:\n\\[\n \\mathbb{R}_\\tau := \\{\\text{Cauchy sequences in } \\mathbb{Q}_\\tau\n \\text{ with explicit modulus\ + \ of convergence}\\} \\,/\\, \\sim,\n\\]\nwhere two Cauchy sequences $(q_n)$ and $(r_n)$ are equivalent\nif $|q_n - r_n|\ + \ \\to 0$ as $n \\to \\infty$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 86\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part10/ch42-number-tower.tex`\ + \ lines 269-279\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.NumberTower`\n\ + - Name: `Tau.Boundary.TauReal`\n\n## Dependencies\n\n- Canonical: I.D35\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'R_tau: Cauchy completion of Q_tau with explicit modulus of convergence. Constructive reals aligned with computable/effectively + presented analysis.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001053.md b/_corpus_v3_items/cid001053.md new file mode 100644 index 000000000..4a807b209 --- /dev/null +++ b/_corpus_v3_items/cid001053.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001053/ +title: Lemniscate Characters +cid: cid001053 +primary_alias: DEF0051 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Characters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral algebra to split-complex scalars. + Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.' +aliases: +- DEF0051 +- I.D37 +- lemniscate-characters +- def:lemniscate-characters +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 11, Chapter 43 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Lemniscate Characters**\n\nCharacters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral\ + \ algebra to split-complex scalars. Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.\n\n# Lemniscate\ + \ Characters\n\n## Summary\n\nCharacters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral algebra\ + \ to split-complex scalars. Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:lemniscate-characters}\nA \\textbf{lemniscate character} is a ring homomorphism\n\\[\n \\\ + boxed{%\n \\chi : H_\\tau \\to \\hat{\\mathbb{Z}}_\\tau[j]}\n\\]\nthat satisfies:\n\\begin{enumerate}\n \\item \\\ + textbf{Additivity.}\n $\\chi(x + y) = \\chi(x) + \\chi(y)$\n for all $x, y \\in H_\\tau$.\n \\item\ + \ \\textbf{Multiplicativity.}\n $\\chi(x \\cdot y) = \\chi(x) \\cdot \\chi(y)$\n for all $x, y \\in\ + \ H_\\tau$.\n \\item \\textbf{Unitality.}\n $\\chi(1) = 1$.\n\\end{enumerate}\nThe set of all lemniscate characters\ + \ is denoted\n$\\mathrm{Char}(\\mathbb{L})$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 87\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch43-lemniscate-characters.tex`\ + \ lines 98-119\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Characters`\n\ + - Name: `Tau.Boundary.chi_plus`\n\n## Dependencies\n\n- Canonical: I.D18, I.D27, I.D20\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Characters chi: L -> Z_hat_tau[j] as ring homomorphisms from the bipolar spectral algebra to split-complex scalars. + Fundamental characters chi_+ and chi_- project onto B-sector and C-sector.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001054.md b/_corpus_v3_items/cid001054.md new file mode 100644 index 000000000..75ad44b23 --- /dev/null +++ b/_corpus_v3_items/cid001054.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001054/ +title: Character Group +cid: cid001054 +primary_alias: DEF0052 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The set of all characters on L with pointwise multiplication. Every character traces back to the bipolar partition + of primes. +aliases: +- DEF0052 +- I.D38 +- character-group +- def:character-group +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005265 +- predicate: formalized_by + object: cid005266 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 11, Chapter 43 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Character Group**\n\nThe set of all characters on L with pointwise multiplication. Every character traces back\ + \ to the bipolar partition of primes.\n\n# Character Group\n\n## Summary\n\nThe set of all characters on L with pointwise\ + \ multiplication. Every character traces back to the bipolar partition of primes.\n\n## Statement\n\n```latex\n%\n\\label{def:character-group}\n\ + The \\textbf{character group} of $\\mathbb{L}$ is the set\n\\[\n \\boxed{%\n \\mathrm{Char}(\\mathbb{L})\n :=\ + \ \\{ \\chi : H_\\tau \\to \\hat{\\mathbb{Z}}_\\tau[j]\n \\mid \\chi \\text{ is a lemniscate character} \\}}\n\\]\n\ + equipped with \\textbf{pointwise multiplication}:\nfor characters $\\chi_1, \\chi_2 \\in \\mathrm{Char}(\\mathbb{L})$,\n\ + \\[\n (\\chi_1 \\cdot \\chi_2)(x)\n := \\chi_1(x) \\cdot \\chi_2(x)\n \\quad \\text{for all } x \\in H_\\tau.\n\ + \\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 88\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch43-lemniscate-characters.tex`\ + \ lines 372-389\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Characters`\n\ + - Name: `Tau.Boundary.chi_complete`\n\n## Dependencies\n\n- Canonical: I.D37\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The set of all characters on L with pointwise multiplication. Every character traces back to the bipolar partition + of primes. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001055.md b/_corpus_v3_items/cid001055.md new file mode 100644 index 000000000..e55c2b9b2 --- /dev/null +++ b/_corpus_v3_items/cid001055.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001055/ +title: Crossing Point +cid: cid001055 +primary_alias: DEF0053 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Where the two lobes of L meet (algebraic version). The crossing point is where B-sector and C-sector intersect; not + a smooth point. Geometric S^1 v S^1 emerges in Book II. +aliases: +- DEF0053 +- I.D39 +- crossing-point +- def:crossing-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005270 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 11, Chapter 45 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Crossing Point**\n\nWhere the two lobes of L meet (algebraic version). The crossing point is where B-sector\ + \ and C-sector intersect; not a smooth point. Geometric S^1 v S^1 emerges in Book II.\n\n# Crossing Point\n\n## Summary\n\ + \nWhere the two lobes of L meet (algebraic version). The crossing point is where B-sector and C-sector intersect; not\ + \ a smooth point. Geometric S^1 v S^1 emerges in Book II.\n\n## Statement\n\n```latex\n%\n\\label{def:crossing-point}\n\ + The \\textbf{crossing point} of the algebraic lemniscate\n$\\mathbb{L} = \\hat{\\mathbb{Z}}_\\tau[j]$\nis the element:\n\ + \\[\n \\boxed{%\n \\omega_\\times := 1 = e_+ + e_-.}\n\\]\nIt is the unique element of $\\mathbb{L}$\nwhose spectral\ + \ coefficients are both equal:\n$\\chi_+(\\omega_\\times) = \\chi_-(\\omega_\\times) = 1$.\nEquivalently, $\\omega_\\\ + times$ is the multiplicative identity\nof the ring $\\mathbb{L}$,\nand it is the unique element\nthat belongs to \\textbf{both}\ + \ sectors simultaneously:\n$\\omega_\\times \\cdot e_+ = e_+ \\neq 0$\nand $\\omega_\\times \\cdot e_- = e_- \\neq 0$.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 90\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch45-crossing-point.tex`\ + \ lines 64-83\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Fourier`\n\ + - Name: `Tau.Boundary.crossing_point`\n\n## Dependencies\n\n- Canonical: I.D37, I.T12, I.D18\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Where the two lobes of L meet (algebraic version). The crossing point is where B-sector and C-sector intersect; + not a smooth point. Geometric S^1 v S^1 emerges in Book II. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001056.md b/_corpus_v3_items/cid001056.md new file mode 100644 index 000000000..7ec9f03a9 --- /dev/null +++ b/_corpus_v3_items/cid001056.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001056/ +title: Bipolar Fourier Transform +cid: cid001056 +primary_alias: DEF0054 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Formal transform on L using the character group. Expresses functions on L as sums over characters. Feeds into the + Central Theorem: O(tau^3) = A_spec(L).' +aliases: +- DEF0054 +- I.D40 +- bipolar-fourier-transform +- def:bipolar-fourier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005271 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 11, Chapter 45 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bipolar Fourier Transform**\n\nFormal transform on L using the character group. Expresses functions on L as\ + \ sums over characters. Feeds into the Central Theorem: O(tau^3) = A_spec(L).\n\n# Bipolar Fourier Transform\n\n## Summary\n\ + \nFormal transform on L using the character group. Expresses functions on L as sums over characters. Feeds into the Central\ + \ Theorem: O(tau^3) = A_spec(L).\n\n## Statement\n\n```latex\n%\n\\label{def:bipolar-fourier}\nThe \\textbf{bipolar Fourier\ + \ transform} on the algebraic lemniscate\n$\\mathbb{L} = \\hat{\\mathbb{Z}}_\\tau[j]$\nis the ring homomorphism:\n\\[\n\ + \ \\boxed{%\n \\mathcal{F} : \\mathbb{L} \\to \\hat{\\mathbb{Z}}_\\tau \\times \\hat{\\mathbb{Z}}_\\tau,\n \\\ + quad\n \\mathcal{F}(x) := (\\chi_+(x),\\; \\chi_-(x)).}\n\\]\nFor $x = a + bj$:\n\\[\n \\mathcal{F}(a + bj) = (a\ + \ + b,\\; a - b).\n\\]\nThe \\textbf{inverse bipolar Fourier transform} is:\n\\[\n \\mathcal{F}^{-1}(\\alpha, \\beta)\n\ + \ = \\alpha e_+ + \\beta e_-\n = \\frac{\\alpha + \\beta}{2} + \\frac{\\alpha - \\beta}{2}\\, j.\n\\]\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 91\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch45-crossing-point.tex`\ + \ lines 291-313\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Fourier`\n\ + - Name: `Tau.Boundary.fourier`\n\n## Dependencies\n\n- Canonical: I.D38, I.D39\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Formal transform on L using the character group. Expresses functions on L as sums over characters. Feeds into + the Central Theorem: O(tau^3) = A_spec(L).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001057.md b/_corpus_v3_items/cid001057.md new file mode 100644 index 000000000..4a4ca673d --- /dev/null +++ b/_corpus_v3_items/cid001057.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001057/ +title: Subobject Classifier Preview +cid: cid001057 +primary_alias: DEF0055 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Preview of Omega_tau: Truth4 will serve as the subobject classifier in the earned topos (Part XIII). The four truth + values give tau''s internal logic its paraconsistent character.' +aliases: +- DEF0055 +- I.D41 +- subobject-classifier-preview +- def:omega-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005322 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 48 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Subobject Classifier Preview**\n\nPreview of Omega_tau: Truth4 will serve as the subobject classifier in the\ + \ earned topos (Part XIII). The four truth values give tau's internal logic its paraconsistent character.\n\n# Subobject\ + \ Classifier Preview\n\n## Summary\n\nPreview of Omega_tau: Truth4 will serve as the subobject classifier in the earned\ + \ topos (Part XIII). The four truth values give tau's internal logic its paraconsistent character.\n\n## Statement\n\n\ + ```latex\n%\n\\label{def:omega-tau}\nThe \\textbf{subobject classifier} of the $\\tau$ framework is:\n\\[\n \\boxed{%\n\ + \ \\Omega_\\tau := \\mathrm{Truth4}\n = \\{\\mathsf{T}, \\mathsf{F}, \\mathsf{B}, \\mathsf{N}\\},}\n\\]\nequipped\ + \ with the lattice structure\n(Definition~\\ref{def:truth4}),\nthe negation operation,\nand the designated ``true'' element\ + \ $\\mathsf{T}$.\nThe \\textbf{truth morphism} is the map:\n\\[\n \\mathrm{true} : \\{*\\} \\to \\Omega_\\tau,\n \ + \ \\quad\n * \\mapsto \\mathsf{T}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 94\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch48-boolean-recovery.tex`\ + \ lines 497-516\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Logic.BooleanRecovery`\n\ + - Name: `Tau.Logic.Omega_tau`\n\n## Dependencies\n\n- Canonical: I.D21, I.T13\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Preview of Omega_tau: Truth4 will serve as the subobject classifier in the earned topos (Part XIII). The four + truth values give tau''s internal logic its paraconsistent character.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001058.md b/_corpus_v3_items/cid001058.md new file mode 100644 index 000000000..8eefcfa1c --- /dev/null +++ b/_corpus_v3_items/cid001058.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001058/ +title: D-Differentiability +cid: cid001058 +primary_alias: DEF0056 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A function f: H_tau -> H_tau is D-differentiable if in sector coordinates (u,v) = (a+b, a-b), it decomposes as f(u,v) + = (g(u), h(v)). Formalized as SectorFun: a pair (g, h) of sector component maps.' +aliases: +- DEF0056 +- I.D42 +- d-differentiability +- def:d-differentiability +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 49 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**D-Differentiability**\n\nA function f: H_tau -> H_tau is D-differentiable if in sector coordinates (u,v) =\ + \ (a+b, a-b), it decomposes as f(u,v) = (g(u), h(v)). Formalized as SectorFun: a pair (g, h) of sector component maps.\n\ + \n# D-Differentiability\n\n## Summary\n\nA function f: H_tau -> H_tau is D-differentiable if in sector coordinates (u,v)\ + \ = (a+b, a-b), it decomposes as f(u,v) = (g(u), h(v)). Formalized as SectorFun: a pair (g, h) of sector component maps.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:d-differentiability}\nA function $f : H_\\tau \\to H_\\tau$\n(at a given finite\ + \ primorial stage)\nis \\textbf{D-differentiable} at $z \\in H_\\tau$\nif the split-complex difference quotient\n\\[\n\ + \ \\boxed{%\n f'(z)\n \\;:=\\;\n \\lim_{h \\to 0}\n \\bigl(f(z + h) - f(z)\\bigr) \\cdot h^{-1}}\n\\]\n\ + exists as an element of $H_\\tau$,\nwhere $h \\to 0$ ranges over all $h \\in H_\\tau$\nwith $N(h) = h_{\\mathrm{re}}^2\ + \ - h_{\\mathrm{im}}^2 \\neq 0$\n(so that $h^{-1}$ exists),\nand the limit is taken in the ultrametric topology\non $H_\\\ + tau$.\n\nA function is \\textbf{D-holomorphic} on an open subset $U$\nif it is D-differentiable at every point of $U$.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 107\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch49-d-holomorphy.tex`\ + \ lines 198-221\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DHolomorphic`\n\ + - Name: `Tau.Holomorphy.SectorFun`\n\n## Dependencies\n\n- Canonical: I.D20, I.D27\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A function f: H_tau -> H_tau is D-differentiable if in sector coordinates (u,v) = (a+b, a-b), it decomposes + as f(u,v) = (g(u), h(v)). Formalized as SectorFun: a pair (g, h) of sector component maps.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001059.md b/_corpus_v3_items/cid001059.md new file mode 100644 index 000000000..97fde8e6d --- /dev/null +++ b/_corpus_v3_items/cid001059.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001059/ +title: Split-CR Equations +cid: cid001059 +primary_alias: DEF0057 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The split-complex Cauchy-Riemann equations: dU/da = dV/db and dU/db = dV/da. In sector coordinates: dF+/dv = 0 and + dF-/du = 0. Formalized as has_split_cr_form predicate.' +aliases: +- DEF0057 +- I.D43 +- split-cr-equations +- def:split-cr-equations +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 49 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Split-CR Equations**\n\nThe split-complex Cauchy-Riemann equations: dU/da = dV/db and dU/db = dV/da. In sector\ + \ coordinates: dF+/dv = 0 and dF-/du = 0. Formalized as has_split_cr_form predicate.\n\n# Split-CR Equations\n\n## Summary\n\ + \nThe split-complex Cauchy-Riemann equations: dU/da = dV/db and dU/db = dV/da. In sector coordinates: dF+/dv = 0 and dF-/du\ + \ = 0. Formalized as has_split_cr_form predicate.\n\n## Statement\n\n```latex\n%\n\\label{def:split-cr-equations}\nLet\ + \ $f : H_\\tau \\to H_\\tau$\nbe a function written in components as\n\\[\n f(a + bj) \\;=\\; U(a, b) + V(a, b) \\\ + cdot j,\n\\]\nwhere $U, V : \\hat{\\mathbb{Z}}_\\tau \\times \\hat{\\mathbb{Z}}_\\tau\n\\to \\hat{\\mathbb{Z}}_\\tau$\n\ + are the real and imaginary parts of $f$\n(relative to the basis $\\{1, j\\}$).\nThe \\textbf{split Cauchy--Riemann equations}\n\ + (split-CR equations) are:\n\\[\n \\boxed{%\n \\frac{\\partial U}{\\partial a}\n = \\frac{\\partial V}{\\partial\ + \ b},\n \\qquad\n \\frac{\\partial U}{\\partial b}\n = \\frac{\\partial V}{\\partial a}.}\n\\]\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 108\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch49-d-holomorphy.tex`\ + \ lines 271-293\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DHolomorphic`\n\ + - Name: `Tau.Holomorphy.has_split_cr_form`\n\n## Dependencies\n\n- Canonical: I.D42\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The split-complex Cauchy-Riemann equations: dU/da = dV/db and dU/db = dV/da. In sector coordinates: dF+/dv = + 0 and dF-/du = 0. Formalized as has_split_cr_form predicate.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001060.md b/_corpus_v3_items/cid001060.md new file mode 100644 index 000000000..e8b8260e9 --- /dev/null +++ b/_corpus_v3_items/cid001060.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001060/ +title: Omega-Germ Transformer +cid: cid001060 +primary_alias: DEF0058 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: An omega-germ transformer maps omega-tails to sector-pair values. Carries both a SectorFun (D-holomorphic structure) + and a StageFun (stagewise evaluation for tower coherence). +aliases: +- DEF0058 +- I.D45 +- omega-germ-transformer +- def:omega-germ-transformer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 50 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega-Germ Transformer**\n\nAn omega-germ transformer maps omega-tails to sector-pair values. Carries both\ + \ a SectorFun (D-holomorphic structure) and a StageFun (stagewise evaluation for tower coherence).\n\n# Omega-Germ Transformer\n\ + \n## Summary\n\nAn omega-germ transformer maps omega-tails to sector-pair values. Carries both a SectorFun (D-holomorphic\ + \ structure) and a StageFun (stagewise evaluation for tower coherence).\n\n## Statement\n\n```latex\n%\n\\label{def:omega-germ-transformer}\n\ + An \\textbf{$\\omega$-germ transformer} is a function\n\\[\n \\boxed{%\n T : \\mathrm{OmegaTail} \\to \\hat{\\mathbb{Z}}_\\\ + tau[j]}\n\\]\nthat takes a compatible tower\n$(x_k)_{k \\geq 1}$\n(Definition~\\ref{def:omega-tail}, I.D25)\n--- where\ + \ $x_k \\in \\mathbb{Z}/M_k\\mathbb{Z}$\nand $x_\\ell \\equiv x_k \\pmod{M_k}$ for all $k \\leq \\ell$ ---\nand produces\ + \ a split-complex value\n$T\\bigl((x_k)\\bigr) \\in \\hat{\\mathbb{Z}}_\\tau[j]$.\n\nAt each finite truncation depth $d$,\n\ + $T$ restricts to a function\n\\[\n T_d : \\mathrm{OmegaTail}_d \\to \\mathbb{Z}/M_d\\mathbb{Z}[j],\n\\]\nwhere $\\\ + mathrm{OmegaTail}_d$ denotes the set of\ndepth-$d$ compatible towers\n$(x_1, x_2, \\ldots, x_d)$\nand $\\mathbb{Z}/M_d\\\ + mathbb{Z}[j]$ is the\nsplit-complex extension of $\\mathbb{Z}/M_d\\mathbb{Z}$.\n```\n\n## Proof / Justification\n\nThis\ + \ item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line\ + \ 110\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex`\ + \ lines 115-141\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.TauHolomorphic`\n\ + - Name: `Tau.Holomorphy.GermTransformer`\n\n## Dependencies\n\n- Canonical: I.D25, I.D42\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: An omega-germ transformer maps omega-tails to sector-pair values. Carries both a SectorFun (D-holomorphic structure) + and a StageFun (stagewise evaluation for tower coherence). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001061.md b/_corpus_v3_items/cid001061.md new file mode 100644 index 000000000..afc337e78 --- /dev/null +++ b/_corpus_v3_items/cid001061.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001061/ +title: Tower Coherence +cid: cid001061 +primary_alias: DEF0059 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Tower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition on the primorial inverse system: + reducing the output first, or reducing the input first and then applying, gives the same result.' +aliases: +- DEF0059 +- I.D46 +- tower-coherence +- def:tower-coherence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 50 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tower Coherence**\n\nTower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition on the\ + \ primorial inverse system: reducing the output first, or reducing the input first and then applying, gives the same result.\n\ + \n# Tower Coherence\n\n## Summary\n\nTower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition\ + \ on the primorial inverse system: reducing the output first, or reducing the input first and then applying, gives the\ + \ same result.\n\n## Statement\n\n```latex\n%\n\\label{def:tower-coherence}\nAn $\\omega$-germ transformer $T$ is \\textbf{tower-coherent}\n\ + if for all $k \\leq \\ell$ and all\ndepth-$\\ell$ omega-tails $t \\in \\mathrm{OmegaTail}_\\ell$:\n\\[\n \\boxed{%\n\ + \ \\pi_{\\ell \\to k}\\bigl(T_\\ell(t)\\bigr)\n \\;=\\;\n T_k\\bigl(\\pi_{\\ell \\to k}(t)\\bigr),}\n\\]\nwhere\ + \ $\\pi_{\\ell \\to k}$ denotes the primorial reduction map\nfrom stage $\\ell$ to stage $k$\n(reduction modulo $M_k$),\n\ + acting on both the source omega-tail\nand the target split-complex value.\n\nEquivalently: \\emph{reducing the output\ + \ first,\nor reducing the input first and then applying $T$,\ngives the same result.}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 111\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex`\ + \ lines 191-212\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.TauHolomorphic`\n\ + - Name: `Tau.Holomorphy.TowerCoherent`\n\n## Dependencies\n\n- Canonical: I.D45, I.D29\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tower coherence: reduce(f(n, l), k) = f(n, k) for k <= l. A naturality condition on the primorial inverse system: + reducing the output first, or reducing the input first and then applying, gives the same result.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001062.md b/_corpus_v3_items/cid001062.md new file mode 100644 index 000000000..9fea83ce9 --- /dev/null +++ b/_corpus_v3_items/cid001062.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001062/ +title: Tau-Holomorphic Function +cid: cid001062 +primary_alias: DEF0060 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A tau-holomorphic function (HolFun) = D-holomorphic + tower-coherent germ transformer. The D-holomorphic condition + is structural (SectorFun); tower coherence is the additional rigidity constraint. +aliases: +- DEF0060 +- I.D47 +- tau-holomorphic-function +- def:holfun +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 50 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Holomorphic Function**\n\nA tau-holomorphic function (HolFun) = D-holomorphic + tower-coherent germ transformer.\ + \ The D-holomorphic condition is structural (SectorFun); tower coherence is the additional rigidity constraint.\n\n# Tau-Holomorphic\ + \ Function\n\n## Summary\n\nA tau-holomorphic function (HolFun) = D-holomorphic + tower-coherent germ transformer. The\ + \ D-holomorphic condition is structural (SectorFun); tower coherence is the additional rigidity constraint.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:holfun}\nA \\textbf{$\\tau$-holomorphic function} is an $\\omega$-germ transformer\n$T : \\\ + mathrm{OmegaTail} \\to \\hat{\\mathbb{Z}}_\\tau[j]$\n(Definition~\\ref{def:omega-germ-transformer}, I.D45)\nthat satisfies\ + \ both:\n\\begin{enumerate}\n \\item \\textbf{D-holomorphy in sector coordinates}\n (Chapter~\\ref{ch:d-holomorphy}):\n\ + \ $T$ respects the idempotent decomposition,\n acting as independent sector functions\n $T_+\ + \ = e_+ \\circ T$ and $T_- = e_- \\circ T$\n on the $B$-sector and $C$-sector respectively.\n \\item \\textbf{Tower\ + \ coherence}\n (Definition~\\ref{def:tower-coherence}, I.D46):\n for all $k \\leq \\ell$,\n \ + \ $\\pi_{\\ell \\to k}(T_\\ell(t))\n = T_k(\\pi_{\\ell \\to k}(t))$.\n\\end{enumerate}\nThe type of all $\\\ + tau$-holomorphic functions is denoted:\n\\[\n \\boxed{%\n \\mathrm{HolFun}\n \\;:=\\;\n \\bigl\\{\n \ + \ T : \\mathrm{OmegaTail} \\to \\hat{\\mathbb{Z}}_\\tau[j]\n \\;\\big|\\;\n T \\text{ is D-holomorphic\ + \ and tower-coherent}\n \\bigr\\}.}\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 112\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex`\ + \ lines 283-314\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.TauHolomorphic`\n\ + - Name: `Tau.Holomorphy.HolFun`\n\n## Dependencies\n\n- Canonical: I.D45, I.D46\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A tau-holomorphic function (HolFun) = D-holomorphic + tower-coherent germ transformer. The D-holomorphic condition + is structural (SectorFun); tower coherence is the additional rigidity constraint. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001063.md b/_corpus_v3_items/cid001063.md new file mode 100644 index 000000000..dad355165 --- /dev/null +++ b/_corpus_v3_items/cid001063.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001063/ +title: Tau-Holomorphic Map +cid: cid001063 +primary_alias: DEF0061 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A tau-holomorphic map bundles a HolFun with source and target object indices in tau-Idx. Provides the morphism data + for the earned category Cat_tau (Part XIII). +aliases: +- DEF0061 +- I.D48 +- tau-holomorphic-map +- def:holmap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 50 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Holomorphic Map**\n\nA tau-holomorphic map bundles a HolFun with source and target object indices in tau-Idx.\ + \ Provides the morphism data for the earned category Cat_tau (Part XIII).\n\n# Tau-Holomorphic Map\n\n## Summary\n\nA\ + \ tau-holomorphic map bundles a HolFun with source and target object indices in tau-Idx. Provides the morphism data for\ + \ the earned category Cat_tau (Part XIII).\n\n## Statement\n\n```latex\n%\n\\label{def:holmap}\nA \\textbf{$\\tau$-holomorphic\ + \ map} is a triple\n$(T, \\mathrm{src}, \\mathrm{tgt})$ where:\n\\begin{enumerate}\n \\item $T \\in \\mathrm{HolFun}$\n\ + \ is a $\\tau$-holomorphic function\n (Definition~\\ref{def:holfun}, I.D47).\n \\item $\\mathrm{src}$\ + \ is an object of $\\tau$\n (the source),\n determining the rooting of the input omega-tails.\n \\\ + item $\\mathrm{tgt}$ is an object of $\\tau$\n (the target),\n determining the compatibility condition\n\ + \ on the output split-complex values:\n $T$ maps omega-tails rooted at $\\mathrm{src}$\n to\ + \ split-complex values compatible with $\\mathrm{tgt}$.\n\\end{enumerate}\nThe type of all $\\tau$-holomorphic maps is\ + \ denoted:\n\\[\n \\boxed{%\n \\mathrm{HolMap}\n \\;:=\\;\n \\bigl\\{\n (T, \\mathrm{src}, \\mathrm{tgt})\n\ + \ \\;\\big|\\;\n T \\in \\mathrm{HolFun},\\;\n \\mathrm{src}, \\mathrm{tgt} \\in \\mathrm{Ob}(\\\ + tau)\n \\bigr\\}.}\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 113\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex`\ + \ lines 374-405\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.TauHolomorphic`\n\ + - Name: `Tau.Holomorphy.HolMap`\n\n## Dependencies\n\n- Canonical: I.D47\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A tau-holomorphic map bundles a HolFun with source and target object indices in tau-Idx. Provides the morphism + data for the earned category Cat_tau (Part XIII). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001064.md b/_corpus_v3_items/cid001064.md new file mode 100644 index 000000000..85d497d1c --- /dev/null +++ b/_corpus_v3_items/cid001064.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001064/ +title: Hol(L) +cid: cid001064 +primary_alias: DEF0062 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Hol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity Theorem, elements + are uniquely determined by their values at any single primorial depth.' +aliases: +- DEF0062 +- I.D49 +- hol-l +- def:hol-L +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 52 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Hol(L)**\n\nHol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity\ + \ Theorem, elements are uniquely determined by their values at any single primorial depth.\n\n# Hol(L)\n\n## Summary\n\ + \nHol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity Theorem, elements are\ + \ uniquely determined by their values at any single primorial depth.\n\n## Statement\n\n```latex\n%\n\\label{def:hol-L}\n\ + The space of \\textbf{$\\tau$-holomorphic functions\non the algebraic lemniscate} is:\n\\[\n \\boxed{%\n \\mathrm{Hol}(\\\ + mathbb{L})\n := \\bigl\\{\\, T \\in \\mathrm{HolFun}\n : T \\text{ is defined on all compatible omega-tails of }\n\ + \ \\mathbb{L} \\,\\bigr\\},}\n\\]\nwhere $\\mathbb{L}$ denotes the algebraic lemniscate\n(Theorem~\\ref{thm:algebraic-lemniscate},\ + \ I.D18),\nrealized as the set of omega-tails\ncompatible with the bipolar spectral algebra\n(Definition~\\ref{def:bipolar-spectral-algebra},\n\ + Chapter~\\ref{ch:bipolar-algebra}).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 121\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch52-identity-theorem.tex`\ + \ lines 430-448\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.IdentityTheorem`\n\ + - Name: `Tau.Holomorphy.HolL`\n\n## Dependencies\n\n- Canonical: I.D47, I.T21\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Hol(L): the space of all tau-holomorphic functions on the algebraic lemniscate. By the Identity Theorem, elements + are uniquely determined by their values at any single primorial depth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001065.md b/_corpus_v3_items/cid001065.md new file mode 100644 index 000000000..a92ba0bcf --- /dev/null +++ b/_corpus_v3_items/cid001065.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001065/ +title: Tau-Arrow +cid: cid001065 +primary_alias: DEF0063 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A tau-arrow from source to target is a HolFun (tau-holomorphic function). Two arrows are equal iff their underlying + HolFuns agree extensionally at all inputs and stages. +aliases: +- DEF0063 +- I.D50 +- tau-arrow +- def:tau-arrow +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 53 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Arrow**\n\nA tau-arrow from source to target is a HolFun (tau-holomorphic function). Two arrows are equal\ + \ iff their underlying HolFuns agree extensionally at all inputs and stages.\n\n# Tau-Arrow\n\n## Summary\n\nA tau-arrow\ + \ from source to target is a HolFun (tau-holomorphic function). Two arrows are equal iff their underlying HolFuns agree\ + \ extensionally at all inputs and stages.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-arrow}\nA \\textbf{$\\tau$-arrow}\ + \ is an NF-equivalence class\nof $\\tau$-holomorphic programs:\n\\[\n \\boxed{%\n \\alpha = [\\pi]_{\\mathrm{NF}}\n\ + \ \\;:=\\;\n \\bigl\\{\\,\n \\pi' \\in \\mathrm{HolProg}\n : \\pi' \\sim_{\\mathrm{NF}} \\pi\n \ + \ \\,\\bigr\\},}\n\\]\nwhere $\\mathrm{HolProg}$ denotes the set of programs\nin the program monoid\n(Definition~\\ref{def:program-monoid},\ + \ I.D14)\nwhose induced $\\omega$-germ transformer\nis $\\tau$-holomorphic\n(Definition~\\ref{def:holfun}, I.D47).\n\n\ + A $\\tau$-arrow carries a unique HolFun:\nif $\\pi_1, \\pi_2 \\in [\\pi]_{\\mathrm{NF}}$,\nthen $\\pi_1$ and $\\pi_2$\ + \ induce the same\n$\\tau$-holomorphic function.\nThe arrow $\\alpha$ is identified with this function:\n\\[\n T_\\\ + alpha := \\text{the unique HolFun induced by any\n representative of } \\alpha.\n\\]\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 122\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch53-earned-arrows.tex`\ + \ lines 181-211\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedArrows`\n\ + - Name: `Tau.Topos.TauArrow`\n\n## Dependencies\n\n- Canonical: I.D47, I.T21\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A tau-arrow from source to target is a HolFun (tau-holomorphic function). Two arrows are equal iff their underlying + HolFuns agree extensionally at all inputs and stages. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001066.md b/_corpus_v3_items/cid001066.md new file mode 100644 index 000000000..56cda2ef4 --- /dev/null +++ b/_corpus_v3_items/cid001066.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001066/ +title: Cat_tau +cid: cid001066 +primary_alias: DEF0064 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cat_tau: the earned category with objects = TauIdx, morphisms = tau-arrows. Identity from id_holfun, composition + from HolFun composition, associativity from Part XII. Not imported but EARNED from the monoid structure.' +aliases: +- DEF0064 +- I.D51 +- cat-tau +- def:cat-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 53 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cat_tau**\n\nCat_tau: the earned category with objects = TauIdx, morphisms = tau-arrows. Identity from id_holfun,\ + \ composition from HolFun composition, associativity from Part XII. Not imported but EARNED from the monoid structure.\n\ + \n# Cat_tau\n\n## Summary\n\nCat_tau: the earned category with objects = TauIdx, morphisms = tau-arrows. Identity from\ + \ id_holfun, composition from HolFun composition, associativity from Part XII. Not imported but EARNED from the monoid\ + \ structure.\n\n## Statement\n\n```latex\n%\n\\label{def:cat-tau}\nThe \\textbf{earned category} $\\mathrm{Cat}_\\tau$\n\ + is the category defined by:\n\\begin{enumerate}\n \\item \\textbf{Objects}: $\\mathrm{Ob}(\\mathrm{Cat}_\\tau) := \\\ + mathrm{Ob}(\\tau)$,\n the $\\tau$-index set ---\n the vertices of the primorial tower\n determined\ + \ by the 9~axioms.\n \\item \\textbf{Morphisms}: for objects $A, B \\in \\mathrm{Ob}(\\tau)$,\n the hom-set\ + \ is:\n \\[\n \\boxed{%\n \\mathrm{Hom}_{\\mathrm{Cat}_\\tau}(A, B)\n \ + \ \\;:=\\;\n \\bigl\\{\\,\n \\alpha = [\\pi]_{\\mathrm{NF}}\n : T_\\alpha\ + \ \\in \\mathrm{HolFun},\\;\n \\mathrm{src}(\\alpha) = A,\\;\n \\mathrm{tgt}(\\alpha)\ + \ = B\n \\,\\bigr\\},}\n \\]\n where $T_\\alpha$ is the HolFun\n carried by the\ + \ $\\tau$-arrow $\\alpha$\n (Definition~\\ref{def:tau-arrow}, I.D50),\n and $\\mathrm{src}$, $\\mathrm{tgt}$\n\ + \ assign source and target objects\n via the HolMap structure\n (Definition~\\ref{def:holmap},\ + \ I.D48).\n \\item \\textbf{Composition}:\n for $\\alpha \\in \\mathrm{Hom}(A, B)$\n and $\\beta\ + \ \\in \\mathrm{Hom}(B, C)$,\n the composite is:\n \\[\n \\beta \\circ \\alpha\n \ + \ := [\\pi_\\beta \\circ \\pi_\\alpha]_{\\mathrm{NF}},\n \\]\n which lies in $\\mathrm{Hom}(A,\ + \ C)$\n by the composition closure of HolFun\n (Theorem~\\ref{thm:composition-closure}, I.T20).\n \ + \ \\item \\textbf{Identity}: for each object $A \\in \\mathrm{Ob}(\\tau)$,\n the identity arrow is:\n \ + \ \\[\n \\mathrm{id}_A := [\\mathrm{id}_\\tau]_{\\mathrm{NF}},\n \\]\n the NF-equivalence\ + \ class of the identity transformer.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 123\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch53-earned-arrows.tex` lines 261-309\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedArrows`\n- Name: `Tau.Topos.CatTau`\n\ + \n## Dependencies\n\n- Canonical: I.D50, I.T20, I.P24\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cat_tau: the earned category with objects = TauIdx, morphisms = tau-arrows. Identity from id_holfun, composition + from HolFun composition, associativity from Part XII. Not imported but EARNED from the monoid structure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001067.md b/_corpus_v3_items/cid001067.md new file mode 100644 index 000000000..db4c526df --- /dev/null +++ b/_corpus_v3_items/cid001067.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001067/ +title: Tau-Functor +cid: cid001067 +primary_alias: DEF0065 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A tau-functor maps objects to objects. In thin Cat_tau, functors are determined by their object map alone (at most + one morphism between any pair, so the morphism map is uniquely determined). +aliases: +- DEF0065 +- I.D52 +- tau-functor +- def:tau-functor +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 54 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Functor**\n\nA tau-functor maps objects to objects. In thin Cat_tau, functors are determined by their object\ + \ map alone (at most one morphism between any pair, so the morphism map is uniquely determined).\n\n# Tau-Functor\n\n\ + ## Summary\n\nA tau-functor maps objects to objects. In thin Cat_tau, functors are determined by their object map alone\ + \ (at most one morphism between any pair, so the morphism map is uniquely determined).\n\n## Statement\n\n```latex\n%\n\ + \\label{def:tau-functor}\nA \\textbf{$\\tau$-functor}\n$F : \\mathrm{Cat}_\\tau \\to \\mathrm{Cat}_\\tau$\nconsists of:\n\ + \\begin{enumerate}\n \\item An \\textbf{object map}\n $F_0 : \\tau\\text{-Idx} \\to \\tau\\text{-Idx}$.\n\ + \ \\item A \\textbf{morphism map}:\n for every $\\tau$-arrow $f : X \\to Y$,\n a $\\tau$-arrow $F(f)\ + \ : F_0(X) \\to F_0(Y)$.\n\\end{enumerate}\nThese maps must satisfy:\n\\begin{enumerate}\n \\item[(F1)] \\textbf{Identity\ + \ preservation}:\n $F(\\mathrm{id}_X) = \\mathrm{id}_{F_0(X)}$\n for every object $X$.\n \\item[(F2)]\ + \ \\textbf{Composition preservation}:\n $F(g \\circ f) = F(g) \\circ F(f)$\n for every composable pair\n\ + \ $f : X \\to Y$, $g : Y \\to Z$.\n\\end{enumerate}\n\\[\n \\boxed{%\n F : \\mathrm{Cat}_\\tau \\to \\\ + mathrm{Cat}_\\tau\n \\quad\\text{preserves}\\quad\n \\mathrm{id} \\;\\text{and}\\; \\circ.}\n\\]\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 126\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch54-functors.tex`\ + \ lines 88-117\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.Functors`\n\ + - Name: `Tau.Topos.TauFunctor`\n\n## Dependencies\n\n- Canonical: I.D51\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A tau-functor maps objects to objects. In thin Cat_tau, functors are determined by their object map alone (at + most one morphism between any pair, so the morphism map is uniquely determined). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001068.md b/_corpus_v3_items/cid001068.md new file mode 100644 index 000000000..c529e7ee9 --- /dev/null +++ b/_corpus_v3_items/cid001068.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001068/ +title: Natural Transformation +cid: cid001068 +primary_alias: DEF0066 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A natural transformation between tau-functors. In a thin category, naturality is AUTOMATIC: the naturality square + commutes because there is at most one arrow in each direction.' +aliases: +- DEF0066 +- I.D53 +- natural-transformation +- def:natural-transformation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 54 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Natural Transformation**\n\nA natural transformation between tau-functors. In a thin category, naturality is\ + \ AUTOMATIC: the naturality square commutes because there is at most one arrow in each direction.\n\n# Natural Transformation\n\ + \n## Summary\n\nA natural transformation between tau-functors. In a thin category, naturality is AUTOMATIC: the naturality\ + \ square commutes because there is at most one arrow in each direction.\n\n## Statement\n\n```latex\n%\n\\label{def:natural-transformation}\n\ + Let $F, G : \\mathrm{Cat}_\\tau \\to \\mathrm{Cat}_\\tau$\nbe $\\tau$-functors.\nA \\textbf{natural transformation}\n\ + $\\eta : F \\Rightarrow G$\nis a family of $\\tau$-arrows\n\\[\n \\bigl\\{\\, \\eta_X : F(X) \\to G(X) \\,\\bigr\\\ + }_{X \\in \\mathrm{Obj}(\\mathrm{Cat}_\\tau)}\n\\]\nsuch that for every morphism $f : X \\to Y$,\nthe \\textbf{naturality\ + \ square} commutes:\n\\[\n \\boxed{%\n \\begin{array}{ccc}\n F(X) & \\xrightarrow{\\;\\eta_X\\;} & G(X) \\\ + \\\n \\downarrow\\scriptstyle{F(f)} & & \\downarrow\\scriptstyle{G(f)} \\\\\n F(Y) & \\xrightarrow{\\;\\\ + eta_Y\\;} & G(Y)\n \\end{array}\n \\qquad\n G(f) \\circ \\eta_X = \\eta_Y \\circ F(f).}\n\\]\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 127\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch54-functors.tex`\ + \ lines 186-209\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.Functors`\n\ + - Name: `Tau.Topos.NatTrans`\n\n## Dependencies\n\n- Canonical: I.D52\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A natural transformation between tau-functors. In a thin category, naturality is AUTOMATIC: the naturality square + commutes because there is at most one arrow in each direction.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001069.md b/_corpus_v3_items/cid001069.md new file mode 100644 index 000000000..5381f30da --- /dev/null +++ b/_corpus_v3_items/cid001069.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001069/ +title: Yoneda Embedding +cid: cid001069 +primary_alias: DEF0067 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Yoneda embedding y: Cat_tau -> PSh(Cat_tau) sends X to Hom(-, X). In thin Cat_tau, representable presheaves + are subterminal (all hom-sets are singletons).' +aliases: +- DEF0067 +- I.D54 +- yoneda-embedding +- def:yoneda-embedding +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 54 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Yoneda Embedding**\n\nThe Yoneda embedding y: Cat_tau -> PSh(Cat_tau) sends X to Hom(-, X). In thin Cat_tau,\ + \ representable presheaves are subterminal (all hom-sets are singletons).\n\n# Yoneda Embedding\n\n## Summary\n\nThe Yoneda\ + \ embedding y: Cat_tau -> PSh(Cat_tau) sends X to Hom(-, X). In thin Cat_tau, representable presheaves are subterminal\ + \ (all hom-sets are singletons).\n\n## Statement\n\n```latex\n%\n\\label{def:yoneda-embedding}\nThe \\textbf{Yoneda embedding}\ + \ is the functor\n\\[\n \\boxed{%\n y : \\mathrm{Cat}_\\tau \\to \\mathrm{PSh}(\\mathrm{Cat}_\\tau),\n \\qquad\n\ + \ y(X) := \\Hom(-, X).}\n\\]\nFor each object $X$,\n$y(X)$ is the representable presheaf\nsending $Y$ to $\\Hom(Y,\ + \ X)$.\nFor a morphism $f : X_1 \\to X_2$,\nthe induced map\n$y(f) : y(X_1) \\Rightarrow y(X_2)$\nhas components\n$y(f)_Y\ + \ : g \\mapsto f \\circ g$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 128\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch54-functors.tex`\ + \ lines 309-327\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.Functors`\n\ + - Name: `Tau.Topos.yoneda`\n\n## Dependencies\n\n- Canonical: I.D52, I.D53\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Yoneda embedding y: Cat_tau -> PSh(Cat_tau) sends X to Hom(-, X). In thin Cat_tau, representable presheaves + are subterminal (all hom-sets are singletons).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001070.md b/_corpus_v3_items/cid001070.md new file mode 100644 index 000000000..c74ef9547 --- /dev/null +++ b/_corpus_v3_items/cid001070.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001070/ +title: Finite Limits in Cat_tau +cid: cid001070 +primary_alias: DEF0068 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cat_tau has all finite limits: terminal object (index 1), products via Cantor pairing, equalizers (trivial in thin + category — identity or empty), pullbacks from products + equalizers.' +aliases: +- DEF0068 +- I.D55 +- finite-limits-in-cat-tau +- def:finite-limits +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 55 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Finite Limits in Cat_tau**\n\nCat_tau has all finite limits: terminal object (index 1), products via Cantor\ + \ pairing, equalizers (trivial in thin category — identity or empty), pullbacks from products + equalizers.\n\n# Finite\ + \ Limits in Cat_tau\n\n## Summary\n\nCat_tau has all finite limits: terminal object (index 1), products via Cantor pairing,\ + \ equalizers (trivial in thin category — identity or empty), pullbacks from products + equalizers.\n\n## Statement\n\n\ + ```latex\n%\n\\label{def:finite-limits}\nThe category $\\mathrm{Cat}_\\tau$\n(Definition~\\ref{def:cat-tau}, I.D51)\n\ + admits the following finite limit constructions:\n\\begin{enumerate}\n \\item \\textbf{Terminal object.}\n \ + \ The object $\\mathbf{1}_\\tau := \\underline{1}$\n (the multiplicative identity in $\\tau$-Idx)\n \ + \ is terminal:\n for every object $X$ in $\\mathrm{Cat}_\\tau$,\n there exists a unique arrow\n \ + \ $!_X : X \\to \\mathbf{1}_\\tau$.\n \\item \\textbf{Binary products.}\n For objects $X, Y$ in $\\\ + mathrm{Cat}_\\tau$,\n the product is:\n \\[\n \\boxed{%\n X \\times_\\tau\ + \ Y\n \\;:=\\;\n X \\cdot Y,}\n \\]\n where $\\cdot$ denotes internal multiplication\n\ + \ on $\\tau$-Idx\n (Chapter~\\ref{ch:swap-add-mul}).\n The projections $\\pi_1 : X \\cdot Y\ + \ \\to X$\n and $\\pi_2 : X \\cdot Y \\to Y$\n are the $\\tau$-holomorphic programs\n that\ + \ extract the respective factor\n via the NF address encoding\n (Definition~\\ref{def:nf-encoding},\ + \ I.D16).\n \\item \\textbf{Equalizers.}\n Since $\\mathrm{Cat}_\\tau$ is thin\n (at most one arrow\ + \ between any two objects),\n parallel arrows $f, g : X \\rightrightarrows Y$\n force $f = g$.\n \ + \ The equalizer is therefore $X$ itself.\n \\item \\textbf{Pullbacks.}\n For arrows\n $f :\ + \ X \\to Z$ and $g : Y \\to Z$,\n the pullback reduces to $\\gcd(X, Y)$\n in the thin category $\\mathrm{Cat}_\\\ + tau$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 130\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch55-limits-sites.tex`\ + \ lines 71-115\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.LimitsSites`\n\ + - Name: `Tau.Topos.terminal_obj`\n\n## Dependencies\n\n- Canonical: I.D51, I.T22\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cat_tau has all finite limits: terminal object (index 1), products via Cantor pairing, equalizers (trivial in + thin category — identity or empty), pullbacks from products + equalizers.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001071.md b/_corpus_v3_items/cid001071.md new file mode 100644 index 000000000..0d76e9859 --- /dev/null +++ b/_corpus_v3_items/cid001071.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001071/ +title: Tau-Site +cid: cid001071 +primary_alias: DEF0069 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The tau-site: Cat_tau equipped with primorial coverage. At each depth k, the covering family for object X consists + of the CRT residues mod primes p_1, ..., p_k. Encodes arithmetic structure categorically.' +aliases: +- DEF0069 +- I.D56 +- tau-site +- def:tau-site +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 55 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Site**\n\nThe tau-site: Cat_tau equipped with primorial coverage. At each depth k, the covering family\ + \ for object X consists of the CRT residues mod primes p_1, ..., p_k. Encodes arithmetic structure categorically.\n\n\ + # Tau-Site\n\n## Summary\n\nThe tau-site: Cat_tau equipped with primorial coverage. At each depth k, the covering family\ + \ for object X consists of the CRT residues mod primes p_1, ..., p_k. Encodes arithmetic structure categorically.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:tau-site}\nThe \\textbf{$\\tau$-site} is the pair\n$(\\mathrm{Cat}_\\tau, J_\\\ + tau)$,\nwhere $J_\\tau$ is the\n\\textbf{primorial coverage}\ndefined as follows.\n\nFor each object $X$ in $\\mathrm{Cat}_\\\ + tau$\nand each primorial stage $k \\geq 1$,\nthe Chinese Remainder Theorem\n(Section~\\ref{subsec:ch30-crt}, I.D29)\n\ + gives:\n\\[\n \\mathbb{Z}/M_k\\mathbb{Z}\n \\;\\cong\\;\n \\mathbb{Z}/p_1\\mathbb{Z}\n \\;\\times\\;\n \ + \ \\cdots\n \\;\\times\\;\n \\mathbb{Z}/p_k\\mathbb{Z},\n\\]\nwhere $M_k = p_1 \\cdots p_k$ is the $k$-th primorial.\n\ + The \\textbf{primorial covering family}\nof $X$ at stage $k$ is:\n\\[\n \\boxed{%\n \\mathcal{U}_k(X)\n \\;:=\\\ + ;\n \\bigl\\{\\,\n \\phi_i : X_i \\to X\n \\;\\bigm|\\;\n i = 1, \\ldots, k\n \\,\\bigr\\},}\n\\]\nwhere\ + \ $X_i := X \\bmod p_i$\nis the residue at the $i$-th prime,\nviewed as an object of $\\mathrm{Cat}_\\tau$\nvia the canonical\ + \ CRT inclusion,\nand $\\phi_i$ is the CRT projection arrow.\n\nThe Grothendieck topology $J_\\tau$\nis generated by these\ + \ families:\na sieve $S$ on $X$ is a \\emph{covering sieve}\nif and only if\n$\\mathcal{U}_k(X) \\subseteq S$\nfor some\ + \ $k \\geq 1$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 131\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch55-limits-sites.tex`\ + \ lines 203-251\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.LimitsSites`\n\ + - Name: `Tau.Topos.TauSite`\n\n## Dependencies\n\n- Canonical: I.D51, I.D29\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The tau-site: Cat_tau equipped with primorial coverage. At each depth k, the covering family for object X consists + of the CRT residues mod primes p_1, ..., p_k. Encodes arithmetic structure categorically.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001072.md b/_corpus_v3_items/cid001072.md new file mode 100644 index 000000000..2f75cdd7a --- /dev/null +++ b/_corpus_v3_items/cid001072.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001072/ +title: Presheaf Topos +cid: cid001072 +primary_alias: DEF0070 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'PSh(Cat_tau): the presheaf topos. A presheaf assigns to each object a set (modeled as a predicate). Includes terminal/initial + presheaves and pointwise product/coproduct operations.' +aliases: +- DEF0070 +- I.D57 +- presheaf-topos +- def:presheaf-topos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 55 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Presheaf Topos**\n\nPSh(Cat_tau): the presheaf topos. A presheaf assigns to each object a set (modeled as a\ + \ predicate). Includes terminal/initial presheaves and pointwise product/coproduct operations.\n\n# Presheaf Topos\n\n\ + ## Summary\n\nPSh(Cat_tau): the presheaf topos. A presheaf assigns to each object a set (modeled as a predicate). Includes\ + \ terminal/initial presheaves and pointwise product/coproduct operations.\n\n## Statement\n\n```latex\n%\n\\label{def:presheaf-topos}\n\ + The \\textbf{presheaf topos} of $\\mathrm{Cat}_\\tau$ is:\n\\[\n \\boxed{%\n \\mathrm{PSh}(\\mathrm{Cat}_\\tau)\n\ + \ \\;:=\\;\n \\bigl[\\,\n \\mathrm{Cat}_\\tau^{\\mathrm{op}},\\;\n \\mathbf{Set}\n \\,\\bigr],}\n\\]\n\ + the category of contravariant functors\n$F : \\mathrm{Cat}_\\tau^{\\mathrm{op}} \\to \\mathbf{Set}$\nwith natural transformations\ + \ as morphisms.\n\nA presheaf $F$ assigns:\n\\begin{itemize}\n \\item to each object $X$,\n a set $F(X)$ (the\ + \ ``sections over $X$'');\n \\item to each arrow $f : X \\to Y$\n (a divisibility witness $Y \\mid X$),\n\ + \ a restriction map\n $F(f) : F(Y) \\to F(X)$.\n\\end{itemize}\nThese must satisfy functoriality:\n\ + $F(\\id_X) = \\id_{F(X)}$\nand\n$F(g \\circ f) = F(f) \\circ F(g)$.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 132\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch55-limits-sites.tex` lines 348-378\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.LimitsSites`\n- Name: `Tau.Topos.PShCatTau`\n\n\ + ## Dependencies\n\n- Canonical: I.D54, I.D55, I.D56\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'PSh(Cat_tau): the presheaf topos. A presheaf assigns to each object a set (modeled as a predicate). Includes + terminal/initial presheaves and pointwise product/coproduct operations.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001073.md b/_corpus_v3_items/cid001073.md new file mode 100644 index 000000000..038885397 --- /dev/null +++ b/_corpus_v3_items/cid001073.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001073/ +title: Characteristic Morphism +cid: cid001073 +primary_alias: DEF0071 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The characteristic morphism chi_S: X -> Omega_tau assigns a Truth4 value based on B-sector and C-sector membership. + T = both confirm, F = both deny, B = overdetermined, N = underdetermined. Pullback of true along chi recovers the subobject.' +aliases: +- DEF0071 +- I.D58 +- characteristic-morphism +- def:characteristic-morphism +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 56 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Characteristic Morphism**\n\nThe characteristic morphism chi_S: X -> Omega_tau assigns a Truth4 value based\ + \ on B-sector and C-sector membership. T = both confirm, F = both deny, B = overdetermined, N = underdetermined. Pullback\ + \ of true along chi recovers the subobject.\n\n# Characteristic Morphism\n\n## Summary\n\nThe characteristic morphism\ + \ chi_S: X -> Omega_tau assigns a Truth4 value based on B-sector and C-sector membership. T = both confirm, F = both deny,\ + \ B = overdetermined, N = underdetermined. Pullback of true along chi recovers the subobject.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:characteristic-morphism}\nLet $m : S \\hookrightarrow X$ be a monomorphism\nin $\\mathrm{PSh}(\\mathrm{Cat}_\\\ + tau)$.\nThe \\textbf{characteristic morphism} of $S$ in $X$ is the\nunique natural transformation\n\\[\n \\boxed{%\n\ + \ \\chi_S : X \\to \\Omega_\\tau}\n\\]\nfrom Theorem~\\ref{thm:omega-tau-classifier}:\nfor each object $c$ and element\ + \ $x \\in X(c)$,\n\\[\n \\chi_S(x) :=\n \\mathrm{membership}_S(x, c)\n \\in \\{\\mathsf{T}, \\mathsf{F}, \\mathsf{B},\ + \ \\mathsf{N}\\}.\n\\]\nThe defining property is the pullback square:\n$S = \\chi_S^{-1}(\\mathsf{T})\n= \\{x \\in X :\ + \ \\chi_S(x) = \\mathsf{T}\\}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is\ + \ required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 136\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch56-earned-topos.tex`\ + \ lines 277-298\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedTopos`\n\ + - Name: `Tau.Topos.characteristic_morphism`\n\n## Dependencies\n\n- Canonical: I.T25\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The characteristic morphism chi_S: X -> Omega_tau assigns a Truth4 value based on B-sector and C-sector membership. + T = both confirm, F = both deny, B = overdetermined, N = underdetermined. Pullback of true along chi recovers the subobject.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001074.md b/_corpus_v3_items/cid001074.md new file mode 100644 index 000000000..85cc71474 --- /dev/null +++ b/_corpus_v3_items/cid001074.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001074/ +title: Earned Topos +cid: cid001074 +primary_alias: DEF0072 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The earned topos E_tau = PSh(Cat_tau) with Omega_tau as subobject classifier. Bundles the Grothendieck topos structure + with the four-valued classifier and truth arrow. +aliases: +- DEF0072 +- I.D59 +- earned-topos +- def:earned-topos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 56 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Earned Topos**\n\nThe earned topos E_tau = PSh(Cat_tau) with Omega_tau as subobject classifier. Bundles the\ + \ Grothendieck topos structure with the four-valued classifier and truth arrow.\n\n# Earned Topos\n\n## Summary\n\nThe\ + \ earned topos E_tau = PSh(Cat_tau) with Omega_tau as subobject classifier. Bundles the Grothendieck topos structure with\ + \ the four-valued classifier and truth arrow.\n\n## Statement\n\n```latex\n%\n\\label{def:earned-topos}\nThe \\textbf{earned\ + \ topos} is the Grothendieck topos\n\\[\n \\boxed{%\n \\mathcal{E}_\\tau\n := \\bigl(\\mathrm{PSh}(\\mathrm{Cat}_\\\ + tau),\\;\n \\Omega_\\tau,\\;\n \\mathrm{true} : 1 \\to \\Omega_\\tau\\bigr),}\n\\]\nwhere:\n\\begin{enumerate}\n\ + \ \\item $\\mathrm{PSh}(\\mathrm{Cat}_\\tau)$\n is the presheaf topos\n (Definition~\\ref{def:presheaf-topos},\ + \ I.D57).\n \\item $\\Omega_\\tau = \\mathrm{Truth4}$\n is the subobject classifier\n (Theorem~\\\ + ref{thm:omega-tau-classifier}, I.T25).\n \\item $\\mathrm{true} : 1 \\to \\Omega_\\tau$,\n $* \\mapsto \\\ + mathsf{T}$\n (Definition~\\ref{def:omega-tau}, I.D41).\n\\end{enumerate}\nThe earned topos possesses\nfinite\ + \ limits (Chapter~\\ref{ch:limits-sites}),\na subobject classifier (this chapter),\nand exponentials\n(earned in Chapter~\\\ + ref{ch:internal-hom}, Part~XIV).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is\ + \ required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 137\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch56-earned-topos.tex`\ + \ lines 338-366\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedTopos`\n\ + - Name: `Tau.Topos.EarnedTopos`\n\n## Dependencies\n\n- Canonical: I.D57, I.T24, I.T25, I.D58\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The earned topos E_tau = PSh(Cat_tau) with Omega_tau as subobject classifier. Bundles the Grothendieck topos + structure with the four-valued classifier and truth arrow. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001075.md b/_corpus_v3_items/cid001075.md new file mode 100644 index 000000000..edb1d78c3 --- /dev/null +++ b/_corpus_v3_items/cid001075.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001075/ +title: Categorical Product +cid: cid001075 +primary_alias: DEF0073 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The categorical product of two presheaves: pointwise conjunction of support predicates. P x Q maps each object X + to P(X) AND Q(X). Projections extract each factor.' +aliases: +- DEF0073 +- I.D60 +- categorical-product +- def:categorical-product +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 57 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Categorical Product**\n\nThe categorical product of two presheaves: pointwise conjunction of support predicates.\ + \ P x Q maps each object X to P(X) AND Q(X). Projections extract each factor.\n\n# Categorical Product\n\n## Summary\n\ + \nThe categorical product of two presheaves: pointwise conjunction of support predicates. P x Q maps each object X to\ + \ P(X) AND Q(X). Projections extract each factor.\n\n## Statement\n\n```latex\n%\n\\label{def:categorical-product}\nFor\ + \ presheaves $P, Q \\in \\mathcal{E}_\\tau$,\nthe \\textbf{categorical product}\n$P \\times Q$ is the presheaf:\n\\[\n\ + \ \\boxed{%\n (P \\times Q)(X)\n \\;:=\\;\n P(X) \\times Q(X)}\n\\]\nfor each object $X$ in $\\mathrm{Cat}_\\\ + tau$.\nFor each arrow $f : X \\to Y$\n(a divisibility witness $Y \\mid X$),\nthe restriction map is componentwise:\n$(P\ + \ \\times Q)(f)(s, t) = \\bigl(P(f)(s),\\; Q(f)(t)\\bigr)$.\nThe \\textbf{projection morphisms}\n$\\pi_1 : P \\times Q\ + \ \\to P$\nand $\\pi_2 : P \\times Q \\to Q$\nare $(\\pi_1)_X(s, t) := s$\nand $(\\pi_2)_X(s, t) := t$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 139\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch57-cartesian-product.tex`\ + \ lines 66-88\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.CartesianProduct`\n\ + - Name: `Tau.Topos.cat_product`\n\n## Dependencies\n\n- Canonical: I.D55, I.D57\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The categorical product of two presheaves: pointwise conjunction of support predicates. P x Q maps each object + X to P(X) AND Q(X). Projections extract each factor.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001076.md b/_corpus_v3_items/cid001076.md new file mode 100644 index 000000000..85269bbba --- /dev/null +++ b/_corpus_v3_items/cid001076.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001076/ +title: Cartesian Monoidal Structure +cid: cid001076 +primary_alias: DEF0074 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The cartesian monoidal structure (E_tau, x, 1): unit is the terminal presheaf (all-true), tensor is pointwise conjunction. + Left and right unit laws hold.' +aliases: +- DEF0074 +- I.D61 +- cartesian-monoidal-structure +- def:cartesian-monoidal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 57 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cartesian Monoidal Structure**\n\nThe cartesian monoidal structure (E_tau, x, 1): unit is the terminal presheaf\ + \ (all-true), tensor is pointwise conjunction. Left and right unit laws hold.\n\n# Cartesian Monoidal Structure\n\n##\ + \ Summary\n\nThe cartesian monoidal structure (E_tau, x, 1): unit is the terminal presheaf (all-true), tensor is pointwise\ + \ conjunction. Left and right unit laws hold.\n\n## Statement\n\n```latex\n%\n\\label{def:cartesian-monoidal}\nThe \\\ + textbf{cartesian monoidal structure} on $\\mathcal{E}_\\tau$ is:\n\\[\n \\boxed{%\n (\\mathcal{E}_\\tau,\\; \\times,\\\ + ; \\mathbf{1}),}\n\\]\nwhere $\\times$ is the categorical product bifunctor\n(Definition~\\ref{def:categorical-product},\ + \ I.D60)\nand $\\mathbf{1}$ is the terminal presheaf.\nThe coherence data:\n\\begin{enumerate}\n \\item \\textbf{Associator.}\n\ + \ $\\alpha_{P,Q,R} : (P \\times Q) \\times R\n \\xrightarrow{\\sim} P \\times (Q \\times R)$,\n \ + \ $\\alpha_X\\bigl((s, t), u\\bigr) = \\bigl(s, (t, u)\\bigr)$.\n \\item \\textbf{Unitors.}\n $\\lambda_P\ + \ : \\mathbf{1} \\times P \\xrightarrow{\\sim} P$,\n $\\lambda_X(*, s) = s$;\n $\\rho_P : P \\times\ + \ \\mathbf{1} \\xrightarrow{\\sim} P$,\n $\\rho_X(s, *) = s$.\n \\item \\textbf{Symmetry.}\n $\\\ + sigma_{P,Q} : P \\times Q\n \\xrightarrow{\\sim} Q \\times P$,\n $\\sigma_X(s, t) = (t, s)$.\n\\end{enumerate}\n\ + These satisfy the pentagon and triangle coherence conditions.\nAll reduce pointwise to the corresponding identities\n\ + for the cartesian product of sets.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 141\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch57-cartesian-product.tex`\ + \ lines 184-214\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.CartesianProduct`\n\ + - Name: `Tau.Topos.CartesianMonoidal`\n\n## Dependencies\n\n- Canonical: I.D60, I.T26\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The cartesian monoidal structure (E_tau, x, 1): unit is the terminal presheaf (all-true), tensor is pointwise + conjunction. Left and right unit laws hold.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001077.md b/_corpus_v3_items/cid001077.md new file mode 100644 index 000000000..e54f4118b --- /dev/null +++ b/_corpus_v3_items/cid001077.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001077/ +title: Categorical Coproduct +cid: cid001077 +primary_alias: DEF0075 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The categorical coproduct (wedge product) of two presheaves: pointwise disjunction. P v Q maps each object X to + P(X) OR Q(X). Injections embed each factor.' +aliases: +- DEF0075 +- I.D62 +- categorical-coproduct +- def:categorical-coproduct +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 58 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Categorical Coproduct**\n\nThe categorical coproduct (wedge product) of two presheaves: pointwise disjunction.\ + \ P v Q maps each object X to P(X) OR Q(X). Injections embed each factor.\n\n# Categorical Coproduct\n\n## Summary\n\n\ + The categorical coproduct (wedge product) of two presheaves: pointwise disjunction. P v Q maps each object X to P(X) OR\ + \ Q(X). Injections embed each factor.\n\n## Statement\n\n```latex\n%\n\\label{def:categorical-coproduct}\nFor presheaves\n\ + $P, Q : \\mathrm{Cat}_\\tau^{\\mathrm{op}} \\to \\mathrm{Set}$\nin $\\mathcal{E}_\\tau$,\nthe \\textbf{categorical coproduct}\n\ + $P \\wedge Q$ is the presheaf defined pointwise by:\n\\[\n \\boxed{%\n (P \\wedge Q)(X)\n \\;:=\\;\n P(X)\ + \ \\vee Q(X)}\n\\]\nfor each object $X$ in $\\mathrm{Cat}_\\tau$,\nwhere $\\vee$ is Boolean disjunction\non the membership\ + \ predicate:\n$x$ belongs to $(P \\wedge Q)(X)$\nif and only if $x \\in P(X)$ or $x \\in Q(X)$ (or both).\nOn morphisms,\ + \ the restriction maps act componentwise:\n$(P \\wedge Q)(f) := P(f) \\vee Q(f)$.\n\nThe \\textbf{coprojection morphisms}\ + \ are the natural inclusions:\n\\[\n \\iota_P : P \\hookrightarrow P \\wedge Q,\n \\qquad\n \\iota_Q : Q \\hookrightarrow\ + \ P \\wedge Q.\n\\]\nThe coproduct satisfies the universal property:\nfor every presheaf $R$ and morphisms\n$\\alpha :\ + \ P \\to R$, $\\beta : Q \\to R$,\nthere exists a unique copairing\n$[\\alpha, \\beta] : P \\wedge Q \\to R$\nwith $[\\\ + alpha, \\beta] \\circ \\iota_P = \\alpha$\nand $[\\alpha, \\beta] \\circ \\iota_Q = \\beta$.\nUniqueness follows from\ + \ the thinness of $\\mathrm{Cat}_\\tau$\n(Proposition~\\ref{prop:thin-category},\nChapter~\\ref{ch:earned-arrows}).\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 142\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch58-wedge-product.tex`\ + \ lines 89-127\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.WedgeProduct`\n\ + - Name: `Tau.Topos.cat_coproduct`\n\n## Dependencies\n\n- Canonical: I.D57\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The categorical coproduct (wedge product) of two presheaves: pointwise disjunction. P v Q maps each object X + to P(X) OR Q(X). Injections embed each factor.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001078.md b/_corpus_v3_items/cid001078.md new file mode 100644 index 000000000..d7986bd95 --- /dev/null +++ b/_corpus_v3_items/cid001078.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001078/ +title: Bi-Monoidal Structure +cid: cid001078 +primary_alias: DEF0076 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The bi-monoidal structure on E_tau: product (x) and coproduct (v) with distributivity. Multiplicative unit = terminal + presheaf, additive unit = initial presheaf. Absorption: P x 0 = 0.' +aliases: +- DEF0076 +- I.D63 +- bi-monoidal-structure +- def:bi-monoidal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 58 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bi-Monoidal Structure**\n\nThe bi-monoidal structure on E_tau: product (x) and coproduct (v) with distributivity.\ + \ Multiplicative unit = terminal presheaf, additive unit = initial presheaf. Absorption: P x 0 = 0.\n\n# Bi-Monoidal Structure\n\ + \n## Summary\n\nThe bi-monoidal structure on E_tau: product (x) and coproduct (v) with distributivity. Multiplicative\ + \ unit = terminal presheaf, additive unit = initial presheaf. Absorption: P x 0 = 0.\n\n## Statement\n\n```latex\n%\n\\\ + label{def:bi-monoidal}\nThe \\textbf{bi-monoidal structure} on $\\mathcal{E}_\\tau$ is the triple:\n\\[\n \\boxed{%\n\ + \ (\\mathcal{E}_\\tau,\\; \\times,\\; \\wedge),}\n\\]\nwhere:\n\\begin{enumerate}\n \\item $(\\mathcal{E}_\\tau,\ + \ \\times, \\mathbf{1})$\n is a monoidal category\n under the cartesian product\n (Chapter~\\\ + ref{ch:cartesian-product}),\n with terminal presheaf $\\mathbf{1}$ as unit.\n \\item $(\\mathcal{E}_\\tau,\ + \ \\wedge, \\mathbf{0})$\n is a monoidal category\n under the coproduct\n (Definition~\\ref{def:categorical-coproduct},\ + \ I.D62),\n with initial presheaf\n $\\mathbf{0}$ ($\\mathbf{0}(X) = \\varnothing$) as unit.\n \\\ + item $\\times$ distributes over~$\\wedge$\n (Theorem~\\ref{thm:distributivity}, I.T27).\n \\item $\\mathbf{0}$\ + \ annihilates under~$\\times$:\n $P \\times \\mathbf{0} \\cong \\mathbf{0}$\n (since $P(X) \\times \\\ + varnothing = \\varnothing$).\n\\end{enumerate}\nBoth monoidal structures are symmetric\n($P \\times Q \\cong Q \\times\ + \ P$,\n$P \\wedge Q \\cong Q \\wedge P$),\nmaking $(\\mathcal{E}_\\tau, \\times, \\wedge)$\na \\textbf{symmetric bi-monoidal\ + \ category}.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 144\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch58-wedge-product.tex`\ + \ lines 206-238\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.WedgeProduct`\n\ + - Name: `Tau.Topos.BiMonoidal`\n\n## Dependencies\n\n- Canonical: I.D61, I.D62, I.T27\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The bi-monoidal structure on E_tau: product (x) and coproduct (v) with distributivity. Multiplicative unit = + terminal presheaf, additive unit = initial presheaf. Absorption: P x 0 = 0.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001079.md b/_corpus_v3_items/cid001079.md new file mode 100644 index 000000000..7338b1a29 --- /dev/null +++ b/_corpus_v3_items/cid001079.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001079/ +title: Internal Hom +cid: cid001079 +primary_alias: DEF0077 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The internal hom Q^P: pointwise Boolean implication. (Q^P)(X) = true iff P(X) implies Q(X). This gives exponentials + in E_tau.' +aliases: +- DEF0077 +- I.D64 +- internal-hom +- def:internal-hom +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 59 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Internal Hom**\n\nThe internal hom Q^P: pointwise Boolean implication. (Q^P)(X) = true iff P(X) implies Q(X).\ + \ This gives exponentials in E_tau.\n\n# Internal Hom\n\n## Summary\n\nThe internal hom Q^P: pointwise Boolean implication.\ + \ (Q^P)(X) = true iff P(X) implies Q(X). This gives exponentials in E_tau.\n\n## Statement\n\n```latex\n%\n\\label{def:internal-hom}\n\ + Let $P, Q : \\mathrm{Cat}_\\tau^{\\mathrm{op}} \\to \\mathrm{Set}$\nbe presheaves in $\\mathcal{E}_\\tau$\n(Definition~\\\ + ref{def:earned-topos}, I.D59).\nThe \\textbf{internal hom} (exponential)\n$Q^P$ is the presheaf defined by:\n\\[\n \ + \ \\boxed{%\n (Q^P)(X)\n \\;:=\\;\n \\mathrm{Nat}\\bigl(y(X) \\times P,\\; Q\\bigr)}\n\\]\nfor each object $X$\ + \ in $\\mathrm{Cat}_\\tau$,\nwhere $y(X) = \\Hom(-, X)$ is the representable presheaf\n(Definition~\\ref{def:yoneda-embedding},\ + \ I.D54)\nand $\\times$ denotes the product of presheaves\n(Definition~\\ref{def:categorical-product}, I.D60).\n\nFor\ + \ a morphism $f : X_1 \\to X_2$ in $\\mathrm{Cat}_\\tau$,\nthe restriction map\n$(Q^P)(f) : (Q^P)(X_2) \\to (Q^P)(X_1)$\n\ + sends a natural transformation\n$\\alpha : y(X_2) \\times P \\Rightarrow Q$\nto the composite\n$\\alpha \\circ (y(f) \\\ + times \\id_P) :\ny(X_1) \\times P \\Rightarrow Q$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 145\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch59-internal-hom.tex`\ + \ lines 118-146\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.InternalHom`\n\ + - Name: `Tau.Topos.internal_hom`\n\n## Dependencies\n\n- Canonical: I.D60, I.D54\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The internal hom Q^P: pointwise Boolean implication. (Q^P)(X) = true iff P(X) implies Q(X). This gives exponentials + in E_tau.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001080.md b/_corpus_v3_items/cid001080.md new file mode 100644 index 000000000..e5a591738 --- /dev/null +++ b/_corpus_v3_items/cid001080.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001080/ +title: Spectral Coefficients +cid: cid001080 +primary_alias: DEF0078 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Spectral coefficients of a StageFun at input n, stage k: the B-sector and C-sector outputs. Together they fully + determine the function value.' +aliases: +- DEF0078 +- I.D65 +- spectral-coefficients +- def:holfun-spectral-coefficients +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 60 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Coefficients**\n\nSpectral coefficients of a StageFun at input n, stage k: the B-sector and C-sector\ + \ outputs. Together they fully determine the function value.\n\n# Spectral Coefficients\n\n## Summary\n\nSpectral coefficients\ + \ of a StageFun at input n, stage k: the B-sector and C-sector outputs. Together they fully determine the function value.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:holfun-spectral-coefficients}\nLet $f \\in \\mathrm{Hol}(\\mathbb{L})$\n(Definition~\\\ + ref{def:hol-L}, I.D49).\nFor each primorial stage $k \\geq 1$,\nthe \\textbf{spectral coefficients of $f$ at stage $k$}\n\ + are:\n\\[\n \\boxed{%\n (a_k(f),\\; b_k(f))\n \\;:=\\;\n \\bigl(\\chi_+(f_k),\\; \\chi_-(f_k)\\bigr)\n \ + \ \\;\\in\\;\n \\mathbb{Z}/M_k\\mathbb{Z} \\times \\mathbb{Z}/M_k\\mathbb{Z},}\n\\]\nwhere $f_k$ denotes the action\ + \ of $f$\nat the $k$-th primorial stage\n(the reduction of $f(t)$ modulo $M_k$\nfor any omega-tail $t$ of depth $\\geq\ + \ k$),\nand $\\chi_\\pm$ are the fundamental characters\n(Definition~\\ref{def:lemniscate-characters}, I.D37).\nThe full\ + \ \\textbf{spectral coefficient sequence} is\n$\\hat{f} := \\bigl((a_k(f), b_k(f))\\bigr)_{k \\geq 1}$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 148\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch60-spectral-coefficients.tex`\ + \ lines 61-85\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.SpectralCoefficients`\n\ + - Name: `Tau.Holomorphy.SpectralCoeff`\n\n## Dependencies\n\n- Canonical: I.D47, I.D37, I.T21\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Spectral coefficients of a StageFun at input n, stage k: the B-sector and C-sector outputs. Together they fully + determine the function value.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001081.md b/_corpus_v3_items/cid001081.md new file mode 100644 index 000000000..80428f4d7 --- /dev/null +++ b/_corpus_v3_items/cid001081.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001081/ +title: Restriction Map +cid: cid001081 +primary_alias: DEF0079 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The restriction map: restrict a StageFun to inputs NOT in a given subset K. Returns 0 for deleted inputs. Agrees + with original outside K.' +aliases: +- DEF0079 +- I.D66 +- restriction-map +- def:restriction-map +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 60 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Restriction Map**\n\nThe restriction map: restrict a StageFun to inputs NOT in a given subset K. Returns 0\ + \ for deleted inputs. Agrees with original outside K.\n\n# Restriction Map\n\n## Summary\n\nThe restriction map: restrict\ + \ a StageFun to inputs NOT in a given subset K. Returns 0 for deleted inputs. Agrees with original outside K.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:restriction-map}\nLet $K \\subseteq \\mathbb{L}$.\nThe \\textbf{restriction map} is:\n\\[\n\ + \ \\boxed{%\n \\mathrm{res}_K : \\mathrm{Hol}(\\mathbb{L})\n \\to \\mathrm{Fun}(\\mathbb{L} \\setminus K),\n\ + \ \\quad\n \\mathrm{res}_K(f) := f|_{\\mathbb{L} \\setminus K},}\n\\]\nsending $f$ to its restriction\nto the complement\ + \ $\\mathbb{L} \\setminus K$.\nThe map $\\mathrm{res}_K$ is a ring homomorphism:\n$\\mathrm{res}_K(f + g) = \\mathrm{res}_K(f)\ + \ + \\mathrm{res}_K(g)$\nand $\\mathrm{res}_K(f \\cdot g) = \\mathrm{res}_K(f) \\cdot \\mathrm{res}_K(g)$.\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 149\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch60-spectral-coefficients.tex`\ + \ lines 137-154\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.SpectralCoefficients`\n\ + - Name: `Tau.Holomorphy.restriction`\n\n## Dependencies\n\n- Canonical: I.D47\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The restriction map: restrict a StageFun to inputs NOT in a given subset K. Returns 0 for deleted inputs. Agrees + with original outside K.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001082.md b/_corpus_v3_items/cid001082.md new file mode 100644 index 000000000..7be3b93bf --- /dev/null +++ b/_corpus_v3_items/cid001082.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001082/ +title: Primorial Thinness +cid: cid001082 +primary_alias: DEF0080 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A subset K is primordially thin at stage k if it occupies fewer than k-1 of the first k CRT positions (leaving >= + 2 free directions). The tau-analog of codimension >= 2. Empty set is globally thin. +aliases: +- DEF0080 +- I.D67 +- primorial-thinness +- def:primorial-thinness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 61 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Primorial Thinness**\n\nA subset K is primordially thin at stage k if it occupies fewer than k-1 of the first\ + \ k CRT positions (leaving >= 2 free directions). The tau-analog of codimension >= 2. Empty set is globally thin.\n\n\ + # Primorial Thinness\n\n## Summary\n\nA subset K is primordially thin at stage k if it occupies fewer than k-1 of the\ + \ first k CRT positions (leaving >= 2 free directions). The tau-analog of codimension >= 2. Empty set is globally thin.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:primorial-thinness}\nA subset $K \\subseteq \\mathbb{L}$\n(Theorem~\\ref{thm:algebraic-lemniscate},\ + \ I.D18)\nis \\textbf{primordially thin} if for every $d \\geq 1$,\nthe projection $K_d := \\{t \\bmod M_d : t \\in K\\\ + }$\nmisses at least two independent CRT directions:\n\\[\n \\boxed{%\n K \\text{ is thin}\n \\quad:\\Longleftrightarrow\\\ + quad\n \\forall\\, d \\geq 1,\\;\n \\exists\\, k_1 \\neq k_2 \\leq d:\\;\n K_d \\text{ misses the } k_1\\text{-th\ + \ and }\n k_2\\text{-th CRT directions.}}\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 151\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch61-thinness.tex`\ + \ lines 93-110\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.Thinness`\n\ + - Name: `Tau.Holomorphy.PrimoriallyThin`\n\n## Dependencies\n\n- Canonical: I.D29\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A subset K is primordially thin at stage k if it occupies fewer than k-1 of the first k CRT positions (leaving + >= 2 free directions). The tau-analog of codimension >= 2. Empty set is globally thin. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001083.md b/_corpus_v3_items/cid001083.md new file mode 100644 index 000000000..b67a69391 --- /dev/null +++ b/_corpus_v3_items/cid001083.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001083/ +title: Earned Stage-Determined Point +cid: cid001083 +primary_alias: DEF0081 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'An earned interior point: a value obtained by extending boundary data via Hartogs extension. Carries a tower-coherent + StageFun and produces interior values that are self-consistent (reduced).' +aliases: +- DEF0081 +- I.D68 +- earned-stage-determined-point +- def:earned-interior-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 63 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Earned Stage-Determined Point**\n\nAn earned interior point: a value obtained by extending boundary data via\ + \ Hartogs extension. Carries a tower-coherent StageFun and produces interior values that are self-consistent (reduced).\n\ + \n# Earned Stage-Determined Point\n\n## Summary\n\nAn earned interior point: a value obtained by extending boundary data\ + \ via Hartogs extension. Carries a tower-coherent StageFun and produces interior values that are self-consistent (reduced).\n\ + \n## Statement\n\n```latex\n%\n\\label{def:earned-interior-point}\nA point $p \\in \\tau^3$ is an\n\\textbf{earned stage-determined\ + \ point}\n(or \\emph{earned interior point}\nin the geometric language of Book~II)\nif its $\\tau$-holomorphic data\n\ + is completely determined by omega-tail data on~$\\mathbb{L}$:\n\\[\n \\boxed{%\n p \\in \\mathrm{Int}_{\\mathrm{earned}}(\\\ + tau^3)\n \\quad:\\Longleftrightarrow\\quad\n \\forall\\, T \\in \\mathrm{Hol}(\\mathbb{L}),\\;\n T(p) \\text{\ + \ is determined by }\n T|_{\\mathbb{L}}.}\n\\]\nHere $\\mathrm{Hol}(\\mathbb{L})$\nis the space of $\\tau$-holomorphic\ + \ functions\n(Definition~\\ref{def:hol-L}, I.D49)\nand the determination is via\nGlobal Hartogs extension\n(Theorem~\\\ + ref{thm:global-hartogs}, I.T06).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is\ + \ required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 155\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch63-boundary-interior-passage.tex`\ + \ lines 70-93\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.BoundaryInterior`\n\ + - Name: `Tau.Holomorphy.EarnedInteriorPoint`\n\n## Dependencies\n\n- Canonical: I.T31, I.D47\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'An earned interior point: a value obtained by extending boundary data via Hartogs extension. Carries a tower-coherent + StageFun and produces interior values that are self-consistent (reduced).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001084.md b/_corpus_v3_items/cid001084.md new file mode 100644 index 000000000..2538add0f --- /dev/null +++ b/_corpus_v3_items/cid001084.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001084/ +title: Generative Counting Principle +cid: cid001084 +primary_alias: DEF0082 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Counting as a structural feature of tau: every tau-set is generated by rho-iteration from seeds, inheriting countability + from the orbit structure. No set outruns the generative act.' +aliases: +- DEF0082 +- I.D75 +- generative-counting-principle +- def:generative-counting +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 36 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Generative Counting Principle**\n\nCounting as a structural feature of tau: every tau-set is generated by rho-iteration\ + \ from seeds, inheriting countability from the orbit structure. No set outruns the generative act.\n\n# Generative Counting\ + \ Principle\n\n## Summary\n\nCounting as a structural feature of tau: every tau-set is generated by rho-iteration from\ + \ seeds, inheriting countability from the orbit structure. No set outruns the generative act.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:generative-counting}\nThe \\textbf{generative counting principle} asserts:\nthe operator $\\rho$ that creates\ + \ all objects of $\\tau$\nis the same operator that enumerates them.\nFor each generator $g \\in \\{\\alpha, \\pi, \\\ + gamma, \\eta\\}$,\nthe bijection\n\\[\n \\boxed{%\n \\varphi_g : \\mathbb{N} \\to O_g,\n \\qquad\n \\varphi_g(n)\ + \ = \\rho^n(g)}\n\\]\nis not an external observation\nabout a pre-existing universe.\nIt \\emph{is} the generative act:\n\ + the map that builds the universe\nsimultaneously counts it.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 158\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch36-countability.tex` lines 175-194\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Counting`\n- Name: `Tau.Sets.generative_counting`\n\ + \n## Dependencies\n\n- Canonical: I.D31, I.D33, I.P12\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Counting as a structural feature of tau: every tau-set is generated by rho-iteration from seeds, inheriting + countability from the orbit structure. No set outruns the generative act.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001085.md b/_corpus_v3_items/cid001085.md new file mode 100644 index 000000000..c0ee2a541 --- /dev/null +++ b/_corpus_v3_items/cid001085.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001085/ +title: Omega-Germ Approach +cid: cid001085 +primary_alias: DEF0083 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Omega-germs as approaches to infinity: compatible towers on the primorial ladder encode how a sequence approaches + omega. Replaces the cardinality hierarchy with a single structural concept of asymptotic approach.' +aliases: +- DEF0083 +- I.D76 +- omega-germ-approach +- def:omega-germ-approach +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 38 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega-Germ Approach**\n\nOmega-germs as approaches to infinity: compatible towers on the primorial ladder encode\ + \ how a sequence approaches omega. Replaces the cardinality hierarchy with a single structural concept of asymptotic approach.\n\ + \n# Omega-Germ Approach\n\n## Summary\n\nOmega-germs as approaches to infinity: compatible towers on the primorial ladder\ + \ encode how a sequence approaches omega. Replaces the cardinality hierarchy with a single structural concept of asymptotic\ + \ approach.\n\n## Statement\n\n```latex\n%\n\\label{def:omega-germ-approach}\nAn \\textbf{omega-germ approach} is an element\n\ + of the profinite completion\n\\[\n \\hat{\\mathbb{Z}}_\\tau\n := \\varprojlim_{k} \\mathbb{Z}/M_k\\mathbb{Z},\n\\\ + ]\ni.e., the boundary ring\n(Definition~\\ref{def:boundary-ring}, I.D19).\nEach approach represents a \\textbf{direction\ + \ of convergence}\ntoward $\\omega$ ---\nnot a different infinity,\nbut a different path to the unique one.\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 164\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch38-approaches-infinity.tex`\ + \ lines 166-181\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.UniqueInfinity`\n\ + - Name: `Tau.Sets.omega_germ_approach`\n\n## Dependencies\n\n- Canonical: I.D25, I.T35, I.P12\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Omega-germs as approaches to infinity: compatible towers on the primorial ladder encode how a sequence approaches + omega. Replaces the cardinality hierarchy with a single structural concept of asymptotic approach.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001086.md b/_corpus_v3_items/cid001086.md new file mode 100644 index 000000000..2a49d0d35 --- /dev/null +++ b/_corpus_v3_items/cid001086.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001086/ +title: Meta-Logical Substrate +cid: cid001086 +primary_alias: DEF0084 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The meta-logical substrate: the collection of CIC structural rules, type-formation principles, and axioms provided + by the proof assistant that are not derived from K0-K6. Classifies contraction, weakening, and exchange by their object-level + status under K5.' +aliases: +- DEF0084 +- I.D77 +- meta-logical-substrate +- def:meta-logical-substrate +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 68 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Meta-Logical Substrate**\n\nThe meta-logical substrate: the collection of CIC structural rules, type-formation\ + \ principles, and axioms provided by the proof assistant that are not derived from K0-K6. Classifies contraction, weakening,\ + \ and exchange by their object-level status under K5.\n\n# Meta-Logical Substrate\n\n## Summary\n\nThe meta-logical substrate:\ + \ the collection of CIC structural rules, type-formation principles, and axioms provided by the proof assistant that are\ + \ not derived from K0-K6. Classifies contraction, weakening, and exchange by their object-level status under K5.\n\n##\ + \ Statement\n\n```latex\n%\n\\label{def:meta-logical-substrate}\nThe \\textbf{meta-logical substrate} of a formal development\n\ + is the collection of\n\\begin{enumerate}\n \\item logical rules\n (introduction and elimination rules\n \ + \ for connectives and quantifiers),\n \\item type-formation principles\n (dependent products, inductive\ + \ definitions,\n universe stratification),\n \\item structural rules\n (contraction, weakening, exchange),\ + \ and\n \\item ambient axioms\n (\\texttt{propext}, \\texttt{funext},\n \\texttt{Quot.sound},\n \ + \ and optionally \\texttt{Classical.choice})\n\\end{enumerate}\nprovided by the proof assistant\nthat are \\textbf{not}\ + \ derived\nfrom the object theory's own axioms.\nFor TauLib, the object theory\nis the coherence kernel $\\KAxiom{0}$--$\\\ + KAxiom{6}$;\nthe meta-logical substrate\nis everything that CIC contributes\non top of that kernel.\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 167\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch68-meta-logical-toolkit.tex`\ + \ lines 316-343\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.Substrate`\n\ + - Name: `Tau.MetaLogic.MetaLogicalSubstrate`\n\n## Dependencies\n\n- Canonical: I.D03, I.D14\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The meta-logical substrate: the collection of CIC structural rules, type-formation principles, and axioms provided + by the proof assistant that are not derived from K0-K6. Classifies contraction, weakening, and exchange by their object-level' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001087.md b/_corpus_v3_items/cid001087.md new file mode 100644 index 000000000..18648f6fa --- /dev/null +++ b/_corpus_v3_items/cid001087.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001087/ +title: Diagonal-Linear Correspondence +cid: cid001087 +primary_alias: DEF0085 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Structural isomorphism between K5''s diagonal discipline and the !-free fragment of Girard''s linear logic: no unearned + diagonals = no free contraction, channel consumption = linear resource tracking, saturation = finite resource budget.' +aliases: +- DEF0085 +- I.D78 +- diagonal-linear-correspondence +- def:diagonal-linear +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 69 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagonal-Linear Correspondence**\n\nStructural isomorphism between K5's diagonal discipline and the !-free\ + \ fragment of Girard's linear logic: no unearned diagonals = no free contraction, channel consumption = linear resource\ + \ tracking, saturation = finite resource budget.\n\n# Diagonal-Linear Correspondence\n\n## Summary\n\nStructural isomorphism\ + \ between K5's diagonal discipline and the !-free fragment of Girard's linear logic: no unearned diagonals = no free contraction,\ + \ channel consumption = linear resource tracking, saturation = finite resource budget.\n\n## Statement\n\n```latex\n%\n\ + \\label{def:diagonal-linear}\nThe \\textbf{diagonal--linear correspondence}\nis the structural isomorphism between $\\\ + KAxiom{5}$'s\ndiagonal discipline and the $!$-free fragment\nof Girard's linear logic, given by the following map:\n\\\ + begin{enumerate}\n \\item \\textbf{K5.1 $\\longleftrightarrow$ no free contraction.}\n ``No unearned diagonals''\ + \ (K5.1) states that\n the diagonal map $\\Delta : A \\to A \\otimes A$\n is not available without explicit\ + \ construction.\n In linear logic, the contraction rule\n \\[\n \\frac{\\Gamma, A, A \\\ + vdash B}{\\Gamma, A \\vdash B}\n \\quad\\text{(contraction)}\n \\]\n is absent in the $!$-free\ + \ fragment.\n Both express the same constraint:\n a resource cannot be duplicated without cost.\n \ + \ \\item \\textbf{K5.2 $\\longleftrightarrow$ linear resource tracking.}\n ``Each overflow consumes one channel''\ + \ (K5.2) states\n that using a channel in a construction\n removes it from the available context.\n\ + \ In linear sequent calculus,\n using a formula $A$ in a derivation\n \\emph{consumes} $A$\ + \ from the context $\\Gamma$:\n the formula is no longer available\n for subsequent steps.\n \ + \ Both express the same discipline:\n resources are tracked, not ambient.\n \\item \\textbf{K5.3 $\\longleftrightarrow$\ + \ finite resource budget.}\n ``Saturation at four channels'' (K5.3) bounds\n the total linear context.\n\ + \ The four orbit rays\n ($\\alpha$, $\\pi$, $\\gamma$, $\\eta$)\n define a finite resource\ + \ pool.\n In the linear reading,\n the sequent $\\Gamma \\vdash C$\n has $|\\Gamma| \\leq 4$:\n\ + \ the context is bounded.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 169\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch69-diagonal-linear-correspondence.tex` lines 191-233\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.LinearDiscipline`\n\ + - Name: `Tau.MetaLogic.DiagonalLinearCorrespondence`\n\n## Dependencies\n\n- Canonical: I.D03, I.D77, I.R15\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Structural isomorphism between K5''s diagonal discipline and the !-free fragment of Girard''s linear logic: + no unearned diagonals = no free contraction, channel consumption = linear resource tracking, saturation = finite resource + budget.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001088.md b/_corpus_v3_items/cid001088.md new file mode 100644 index 000000000..ee9c79de2 --- /dev/null +++ b/_corpus_v3_items/cid001088.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001088/ +title: Program Monoid as Linear Calculus +cid: cid001088 +primary_alias: DEF0086 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Reinterpretation of the program monoid (I.D14) as a linear sequent calculus: programs = linear proofs, concatenation + = cut rule, NF-Confluence (I.L02) = cut-elimination, four orbit channels = linear context zones.' +aliases: +- DEF0086 +- I.D79 +- program-monoid-as-linear-calculus +- def:program-linear-calculus +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 69 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Program Monoid as Linear Calculus**\n\nReinterpretation of the program monoid (I.D14) as a linear sequent calculus:\ + \ programs = linear proofs, concatenation = cut rule, NF-Confluence (I.L02) = cut-elimination, four orbit channels = linear\ + \ context zones.\n\n# Program Monoid as Linear Calculus\n\n## Summary\n\nReinterpretation of the program monoid (I.D14)\ + \ as a linear sequent calculus: programs = linear proofs, concatenation = cut rule, NF-Confluence (I.L02) = cut-elimination,\ + \ four orbit channels = linear context zones.\n\n## Statement\n\n```latex\n%\n\\label{def:program-linear-calculus}\nThe\ + \ \\textbf{program monoid as linear calculus}\nreinterprets the program monoid\n(Definition~\\ref{def:program-monoid},\ + \ I.D14)\nas a linear sequent calculus:\n\\begin{enumerate}\n \\item \\textbf{Programs = linear proofs.}\n \ + \ Each program $P = (i_1, \\ldots, i_n)$\n in the program monoid\n is a linear derivation:\n \ + \ every instruction consumes its input channel\n and produces an output,\n with no instruction reusing\ + \ a channel\n that has already been consumed.\n \\item \\textbf{Concatenation = cut rule.}\n The\ + \ monoid operation\n $P \\cdot Q = (i_1, \\ldots, i_m, j_1, \\ldots, j_n)$\n corresponds to the cut\ + \ rule of sequent calculus:\n \\[\n \\frac{\\Gamma \\vdash A \\qquad A, \\Delta \\vdash B}{%\n \ + \ \\Gamma, \\Delta \\vdash B}\n \\quad\\text{(cut)}\n \\]\n The conclusion\ + \ of $P$ (the resource $A$)\n becomes the premise of $Q$.\n The cut connects one proof's output\n \ + \ to another's input.\n \\item \\textbf{NF-Confluence = cut-elimination.}\n The NF-Confluence Lemma\n\ + \ (Lemma~\\ref{lem:nf-confluence}, I.L02)\n states: any two reduction paths to normal form\n \ + \ yield the same result.\n Gentzen's cut-elimination theorem\n (the \\emph{Hauptsatz}, 1935) states:\n\ + \ any proof with cuts\n can be transformed to a cut-free proof,\n and the result is unique\n\ + \ up to inessential permutations.\n The parallel is exact:\n normal forms in the program monoid\n\ + \ correspond to cut-free proofs\n in the linear sequent calculus.\n \\item \\textbf{Four channels\ + \ = linear context zones.}\n The four orbit rays\n ($\\alpha$, $\\pi$, $\\gamma$, $\\eta$)\n \ + \ partition the linear context\n into four zones,\n each carrying one resource type.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 171\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch69-diagonal-linear-correspondence.tex`\ + \ lines 403-454\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.LinearDiscipline`\n\ + - Name: `Tau.MetaLogic.ProgramLinearCalc`\n\n## Dependencies\n\n- Canonical: I.D14, I.L02, I.T03, I.D78\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Reinterpretation of the program monoid (I.D14) as a linear sequent calculus: programs = linear proofs, concatenation + = cut rule, NF-Confluence (I.L02) = cut-elimination, four orbit channels = linear context zones.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001089.md b/_corpus_v3_items/cid001089.md new file mode 100644 index 000000000..328237d54 --- /dev/null +++ b/_corpus_v3_items/cid001089.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001089/ +title: Self-Hosting Degree Classification +cid: cid001089 +primary_alias: DEF0087 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Four levels: none, partial, fragment, full. Classifies proof-theoretic self-hosting by internalization depth.' +aliases: +- DEF0087 +- I.D80 +- self-hosting-degree-classification +- def:self-hosting-degree +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 71 (Part 18) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Self-Hosting Degree Classification**\n\nFour levels: none, partial, fragment, full. Classifies proof-theoretic\ + \ self-hosting by internalization depth.\n\n# Self-Hosting Degree Classification\n\n## Summary\n\nFour levels: none, partial,\ + \ fragment, full. Classifies proof-theoretic self-hosting by internalization depth.\n\n## Statement\n\n```latex\n%\n\\\ + label{def:self-hosting-degree}\nThe \\textbf{self-hosting degree} of a formal system $S$\nis classified into four levels:\n\ + \\begin{enumerate}[\\normalfont(i)]\n \\item \\textbf{None.}\n $S$ has no internal representation\n \ + \ of its own proof theory.\n Proofs in $S$ are syntactic objects\n manipulated by an external meta-theory;\n\ + \ $S$ itself has no way\n to state or reason about\n the structural rules that govern its proofs.\n\ + \ Examples: ZFC (set membership is the only primitive;\n proof theory is entirely external),\n \ + \ PA (first-order arithmetic; no internal proof objects),\n HoTT as currently practiced\n (internal\ + \ language of $(\\infty,1)$-toposes,\n but meta-theory remains external).\n \\item \\textbf{Partial.}\n \ + \ $S$ encodes its own syntax\n but not its structural rules.\n $S$ can form sentences about its\ + \ own sentences,\n prove theorems about provability,\n and exhibit self-referential statements.\n \ + \ But the structural rules ---\n contraction, weakening, exchange,\n the cut rule, the substitution\ + \ operation ---\n remain properties of $S$\n that $S$ cannot access.\n Examples: Peano arithmetic\ + \ via G\\\"odel numbering,\n any sufficiently strong recursively enumerable theory\n via arithmetization.\n\ + \ \\item \\textbf{Fragment.}\n $S$ internalizes a significant fragment\n of its own meta-theory,\n\ + \ but the internalization requires\n a strictly stronger ambient system.\n The fragment may\ + \ include\n normalization, type-checking, completeness,\n or other metatheorems ---\n proved\ + \ ``internally'' in the sense\n that the proofs live inside a formal system,\n but the formal system\ + \ used\n is strictly stronger than the system described.\n Examples: Altenkirch--Kaposi's type theory\ + \ in type theory\n (dependent type theory internalized in CIC\n with quotient inductive types~\\cite{AltenkirchKaposi2016}),\n\ + \ Joyal's arithmetic universes\n (internalize enough to prove\n G\\\"odel's theorem internally~\\\ + cite{vanDijkGietelinkOldenziel2020}),\n Bocquet--Kaposi--Sattler's internal sconing\n (metatheorems\ + \ via internal\n gluing~\\cite{BocquetKaposiSattler2023}).\n \\item \\textbf{Full.}\n $S$ reasons\ + \ about its own proof theory,\n structural rules, and consistency\n using only its own resources.\n\ + \ The meta-language and the object language coincide.\n No external substrate remains.\n No\ + \ known example at meaningful mathematical strength.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is\ + \ definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 176\n\ + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch71-self-hosting-landscape.tex`\ + \ lines 172-235\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `None`\n- Name: `None`\n\n\ + ## Dependencies\n\n- Canonical: I.D77, I.R17\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Four levels: none, partial, fragment, full. Classifies proof-theoretic self-hosting by internalization depth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001090.md b/_corpus_v3_items/cid001090.md new file mode 100644 index 000000000..3852eb37a --- /dev/null +++ b/_corpus_v3_items/cid001090.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001090/ +title: CCC-Linear Dichotomy +cid: cid001090 +primary_alias: DEF0088 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: CCC side has free diagonals, Lawvere barrier. Star-autonomous side has no free diagonals, no barrier. +aliases: +- DEF0088 +- I.D81 +- ccc-linear-dichotomy +- def:ccc-linear-dichotomy +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 72 (Part 18) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**CCC-Linear Dichotomy**\n\nCCC side has free diagonals, Lawvere barrier. Star-autonomous side has no free diagonals,\ + \ no barrier.\n\n# CCC-Linear Dichotomy\n\n## Summary\n\nCCC side has free diagonals, Lawvere barrier. Star-autonomous\ + \ side has no free diagonals, no barrier.\n\n## Statement\n\n```latex\n%\n\\label{def:ccc-linear-dichotomy}\nThe \\textbf{CCC--linear\ + \ dichotomy}\nis the categorical location\nof the self-hosting barrier:\n\\begin{enumerate}\n \\item\\textbf{CCC side.}:\n\ + \ The diagonal $\\Delta_A : A \\to A \\times A$\n exists for all~$A$.\n Contraction is free:\n\ + \ any resource may be duplicated at will.\n Lawvere's fixed-point theorem applies:\n every\ + \ point-surjection $A \\to B^A$\n forces fixed points on all endomorphisms of $B$.\n Self-reference\ + \ produces incompleteness.\n Full self-hosting is blocked:\n a system that is cartesian closed\n \ + \ and rich enough to encode its own morphisms\n cannot be both consistent and complete\n about\ + \ its own provability.\n \\item\\textbf{$*$-Autonomous side.}:\n No general $\\Delta_A : A \\to A \\otimes\ + \ A$ exists.\n Contraction is absent:\n resources are consumed, not copied.\n Lawvere's fixed-point\ + \ theorem does not apply:\n the diagonal morphism that the proof requires\n is unavailable.\n \ + \ Self-reference must be \\emph{controlled} ---\n introduced in specific, bounded cases\n without\ + \ the universal incompleteness\n that Lawvere guarantees.\n\\end{enumerate}\nThis dichotomy is not a loophole\ + \ to be exploited\nbut a structural divide to be respected.\nThe absence of Lawvere's obstruction\ndoes \\emph{not} mean\ + \ self-hosting is easy ---\nit means the standard proof of impossibility\ndoes not go through.\nThe categorical barrier\ + \ is absent;\nother barriers may still apply.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 178\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch72-star-autonomous-barrier.tex`\ + \ lines 297-338\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.StructuralExclusion`\n\ + - Name: `CCCSide / StarAutonomousSide`\n\n## Dependencies\n\n- Canonical: I.D78, I.T37, I.R19\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: CCC side has free diagonals, Lawvere barrier. Star-autonomous side has no free diagonals, no barrier. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001091.md b/_corpus_v3_items/cid001091.md new file mode 100644 index 000000000..cd1f944ef --- /dev/null +++ b/_corpus_v3_items/cid001091.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001091/ +title: Enrichment Frontier Classification +cid: cid001091 +primary_alias: DEF0089 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Three statuses: achieved (E0-E1), partially achieved (E1-E2), unprecedented (E2-E3).' +aliases: +- DEF0089 +- I.D82 +- enrichment-frontier-classification +- def:enrichment-frontier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 73 (Part 18) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Enrichment Frontier Classification**\n\nThree statuses: achieved (E0-E1), partially achieved (E1-E2), unprecedented\ + \ (E2-E3).\n\n# Enrichment Frontier Classification\n\n## Summary\n\nThree statuses: achieved (E0-E1), partially achieved\ + \ (E1-E2), unprecedented (E2-E3).\n\n## Statement\n\n```latex\n%\n\\label{def:enrichment-frontier}\nThe \\textbf{enrichment\ + \ frontier classification}\nassigns a novelty status to each transition\nof the enrichment ladder:\n\\begin{enumerate}\n\ + \ \\item \\textbf{$E_0 \\to E_1$: Achieved.}\n Internalizing types within a categorical framework\n \ + \ has been accomplished by multiple groups\n (Altenkirch--Kaposi \\cite{AltenkirchKaposi2016},\n \ + \ Bocquet--Kaposi--Sattler \\cite{BocquetKaposiSattler2023},\n Moerdijk--Palmgren \\cite{MoerdijkPalmgren2002}).\n\ + \ Adaptation to $\\tau$'s finite constructive\n non-Boolean setting is required but is not\n \ + \ a theoretical barrier.\n \\item \\textbf{$E_1 \\to E_2$: Partially achieved.}\n Internal proof-theoretic\ + \ reasoning exists\n (Joyal arithmetic universes\n \\cite{vanDijkGietelinkOldenziel2020}).\n \ + \ Resource-sensitive type theory exists\n (graded modal DTT \\cite{Abel2023},\n substructural DTT\ + \ \\cite{SubstructuralDTT2024}).\n The combination into a complete system\n internalizing full linear\ + \ proof theory\n within a categorical framework\n has not been accomplished.\n Novel assembly\ + \ of known tools.\n \\item \\textbf{$E_2 \\to E_3$: Unprecedented.}\n No formal system of meaningful mathematical\ + \ strength\n achieves full self-hosting.\n Willard \\cite{Willard2001}: self-verification\n \ + \ at sub-PA strength.\n Girard \\cite{Girard2016TS,Eng2023}:\n fragments of linear logic from sub-logical\ + \ operations.\n Williams--Stay \\cite{WilliamsStay2021}:\n native $*$-autonomous type theory (programmatic).\n\ + \ An open research problem.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 181\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch73-enrichment-frontier.tex` lines 543-582\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.StructuralExclusion`\n- Name:\ + \ `EnrichmentFrontierStatus`\n\n## Dependencies\n\n- Canonical: I.D80, I.T39\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Three statuses: achieved (E0-E1), partially achieved (E1-E2), unprecedented (E2-E3).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001092.md b/_corpus_v3_items/cid001092.md new file mode 100644 index 000000000..065cfaf21 --- /dev/null +++ b/_corpus_v3_items/cid001092.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001092/ +title: Primorial Presheaf +cid: cid001092 +primary_alias: DEF0090 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The primorial presheaf F : Prim^op -> Ring sends d to Z/M_dZ with reduction maps. Its split-complex extension F_j + sends d to Z/M_dZ[j]. The spectral presheaf pair (F_B, F_C) is the sector decomposition.' +aliases: +- DEF0090 +- I.D83 +- primorial-presheaf +- def:primorial-presheaf +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 19, Chapter 74 (Part XIX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Primorial Presheaf**\n\nThe primorial presheaf F : Prim^op -> Ring sends d to Z/M_dZ with reduction maps. Its\ + \ split-complex extension F_j sends d to Z/M_dZ[j]. The spectral presheaf pair (F_B, F_C) is the sector decomposition.\n\ + \n# Primorial Presheaf\n\n## Summary\n\nThe primorial presheaf F : Prim^op -> Ring sends d to Z/M_dZ with reduction maps.\ + \ Its split-complex extension F_j sends d to Z/M_dZ[j]. The spectral presheaf pair (F_B, F_C) is the sector decomposition.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:primorial-presheaf}\nThe \\textbf{primorial category} $\\mathbf{Prim}$\n(sketched\ + \ in Remark~\\ref{rem:naturality}) is the category:\n\\begin{enumerate}\n \\item \\textbf{Objects}: natural numbers\ + \ $d \\geq 1$\n (primorial depths).\n \\item \\textbf{Morphisms}: for each $k \\leq \\ell$,\n a unique\ + \ morphism $\\pi_{\\ell \\to k} : \\ell \\to k$\n (the primorial reduction map,\n reduction modulo $M_k$).\n\ + \ \\item \\textbf{Composition}:\n $\\pi_{\\ell \\to k} \\circ \\pi_{m \\to \\ell}\n = \\pi_{m \\\ + to k}$.\n\\end{enumerate}\n$\\mathbf{Prim}$ is a subcategory of $\\mathrm{Cat}_\\tau$\n(Definition~\\ref{def:cat-tau},\ + \ I.D51).\n\nThree presheaves on $\\mathbf{Prim}$:\n\\begin{enumerate}\n \\item The \\textbf{source presheaf}\n \ + \ $\\mathcal{F} : \\mathbf{Prim}^{\\mathrm{op}} \\to \\mathbf{Ring}$,\n sending $d \\mapsto \\mathbb{Z}/M_d\\\ + mathbb{Z}$\n with restriction maps $\\pi_{\\ell \\to k}$.\n \\item The \\textbf{split-complex presheaf}\n\ + \ $\\mathcal{F}_{\\jj} : \\mathbf{Prim}^{\\mathrm{op}} \\to \\mathbf{Ring}$,\n sending $d \\mapsto \\\ + mathbb{Z}/M_d\\mathbb{Z}[\\jj]$\n with restriction maps extended $\\mathbb{Z}[\\jj]$-linearly.\n \\item The\ + \ \\textbf{spectral presheaf pair}\n $(\\mathcal{F}_B, \\mathcal{F}_C)$:\n both copies of $\\mathcal{F}$\n\ + \ (as set-valued presheaves),\n obtained from $\\mathcal{F}_{\\jj}$\n via the characters $\\\ + chi_+$ and $\\chi_-$\n (Definition~\\ref{def:lemniscate-characters}, I.D37).\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 184\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part19/ch74-holomorphy-as-naturality.tex`\ + \ lines 88-125\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.PresheafEssence`\n\ + - Name: `PrimorialPresheaf`\n\n## Dependencies\n\n- Canonical: I.D51, I.D54, I.D46, I.D19, I.D20\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The primorial presheaf F : Prim^op -> Ring sends d to Z/M_dZ with reduction maps. Its split-complex extension + F_j sends d to Z/M_dZ[j]. The spectral presheaf pair (F_B, F_C) is the sector decomposition.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001093.md b/_corpus_v3_items/cid001093.md new file mode 100644 index 000000000..cfbe16bdb --- /dev/null +++ b/_corpus_v3_items/cid001093.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001093/ +title: Constructive Reals +cid: cid001093 +primary_alias: DEF0091 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'TauReal: constructive reals via Cauchy completion of TauRat. Represented as sequences of TauRat approximations with + pointwise equivalence. Every TauReal is explicitly specified by a constructive sequence.' +aliases: +- DEF0091 +- I.D84 +- def:tau-real-field +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 76 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Constructive Reals**\n\nTauReal: constructive reals via Cauchy completion of TauRat. Represented as sequences\ + \ of TauRat approximations with pointwise equivalence. Every TauReal is explicitly specified by a constructive sequence.\n\ + \n# Constructive Reals\n\n## Summary\n\nTauReal: constructive reals via Cauchy completion of TauRat. Represented as sequences\ + \ of TauRat approximations with pointwise equivalence. Every TauReal is explicitly specified by a constructive sequence.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:tau-real-field}\nThe \\textbf{TauReal field structure} on $\\mathbb{R}_\\tau$\ + \ consists of:\n\\begin{itemize}\n \\item Addition and multiplication (term-by-term on representatives).\n \\item\ + \ Additive identity $0_{\\mathbb{R}} := [\\underline{0}, \\underline{0}, \\ldots]$,\n multiplicative identity\ + \ $1_{\\mathbb{R}} := [\\underline{1}, \\underline{1}, \\ldots]$.\n \\item Additive inverse: $-[q_n] := [-q_n]$.\n\ + \ \\item Multiplicative inverse: for $[q_n] \\neq 0$,\n choose $N$ with $|q_n| > \\varepsilon > 0$ for $n\ + \ \\geq N$,\n then $[q_n]^{-1} := [q_n^{-1}]_{n \\geq N}$.\n \\item Order: $[q_n] < [r_n]$ if there exist\n\ + \ $\\varepsilon > 0$ and $N$ with $r_n - q_n > \\varepsilon$ for all $n \\geq N$.\n\\end{itemize}\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 187\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch76-constructive-reals.tex`\ + \ lines 44-59\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.ConstructiveReals`\n\ + - Name: `Tau.Boundary.TauReal`\n\n## Dependencies\n\n- Canonical: I.D15, I.D14\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'TauReal: constructive reals via Cauchy completion of TauRat. Represented as sequences of TauRat approximations + with pointwise equivalence. Every TauReal is explicitly specified by a constructive sequence.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001094.md b/_corpus_v3_items/cid001094.md new file mode 100644 index 000000000..518827c83 --- /dev/null +++ b/_corpus_v3_items/cid001094.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001094/ +title: Elliptic Complex Field +cid: cid001094 +primary_alias: DEF0092 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'TauComplex = TauReal[i]/(i^2+1): the elliptic complex field. Pairs (re, im) of TauReal values with multiplication + (ac-bd, ad+bc). The imaginary unit i satisfies i^2 = -1.' +aliases: +- DEF0092 +- I.D85 +- elliptic-complex-field +- def:tau-complex-field +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 77 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Elliptic Complex Field**\n\nTauComplex = TauReal[i]/(i^2+1): the elliptic complex field. Pairs (re, im) of\ + \ TauReal values with multiplication (ac-bd, ad+bc). The imaginary unit i satisfies i^2 = -1.\n\n# Elliptic Complex Field\n\ + \n## Summary\n\nTauComplex = TauReal[i]/(i^2+1): the elliptic complex field. Pairs (re, im) of TauReal values with multiplication\ + \ (ac-bd, ad+bc). The imaginary unit i satisfies i^2 = -1.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-complex-field}\n\ + The \\textbf{$\\tau$-complex field} is the quotient ring:\n\\[\n \\boxed{%\n \\mathbb{C}_\\tau := \\mathbb{R}_\\\ + tau[x] \\,/\\, (x^2 + 1).}\n\\]\nWe write $i$ for the image of $x$ in the quotient,\nso that $i^2 = -1$.\nEvery element\ + \ of $\\mathbb{C}_\\tau$ has a unique representation\nas an ordered pair:\n\\[\n z = a + bi, \\quad a, b \\in \\mathbb{R}_\\\ + tau.\n\\]\nThe component $a = \\operatorname{Re}(z)$ is the \\textbf{real part}\nand $b = \\operatorname{Im}(z)$ is the\ + \ \\textbf{imaginary part}.\nThe operations are:\n\\begin{align*}\n (a + bi) + (c + di) &= (a + c) + (b + d)i, \\\\\ + \n (a + bi)(c + di) &= (ac - bd) + (ad + bc)i,\n\\end{align*}\nthe second rule following from the distributive law\ + \ and $i^2 = -1$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n\ + ## Source Context\n\n- Registry source: `book-01.jsonl` line 190\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch77-elliptic-complex-field.tex`\ + \ lines 51-74\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.ComplexField`\n\ + - Name: `Tau.Boundary.TauComplex`\n\n## Dependencies\n\n- Canonical: I.D84, I.P39\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'TauComplex = TauReal[i]/(i^2+1): the elliptic complex field. Pairs (re, im) of TauReal values with multiplication + (ac-bd, ad+bc). The imaginary unit i satisfies i^2 = -1.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001095.md b/_corpus_v3_items/cid001095.md new file mode 100644 index 000000000..aad54935c --- /dev/null +++ b/_corpus_v3_items/cid001095.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001095/ +title: Elliptic-Hyperbolic Dichotomy +cid: cid001095 +primary_alias: DEF0093 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The elliptic-hyperbolic dichotomy: TauComplex (i^2=-1, field, no zero divisors) versus SplitComplex (j^2=+1, ring, + zero divisors). Both are algebraically earned; they coexist as complementary structures within tau.' +aliases: +- DEF0093 +- I.D86 +- elliptic-hyperbolic-dichotomy +- def:elliptic-hyperbolic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005267 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 77 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Elliptic-Hyperbolic Dichotomy**\n\nThe elliptic-hyperbolic dichotomy: TauComplex (i^2=-1, field, no zero divisors)\ + \ versus SplitComplex (j^2=+1, ring, zero divisors). Both are algebraically earned; they coexist as complementary structures\ + \ within tau.\n\n# Elliptic-Hyperbolic Dichotomy\n\n## Summary\n\nThe elliptic-hyperbolic dichotomy: TauComplex (i^2=-1,\ + \ field, no zero divisors) versus SplitComplex (j^2=+1, ring, zero divisors). Both are algebraically earned; they coexist\ + \ as complementary structures within tau.\n\n## Statement\n\n```latex\n%\n\\label{def:elliptic-hyperbolic}\nLet $R$ be\ + \ a commutative ring\n(in practice, $\\mathbb{R}_\\tau$ or $\\widehat{\\mathbb{Z}}_\\tau$).\nA \\textbf{quadratic extension}\ + \ of $R$ is a free $R$-module\nof rank~2 with basis $\\{1, u\\}$, where $u^2 \\in R$.\nSuch an extension is:\n\\begin{itemize}\n\ + \ \\item \\textbf{Elliptic} if $u^2 = -1$\n (the defining polynomial $x^2 + 1$ is irreducible over $R$).\n\ + \ \\item \\textbf{Hyperbolic} if $u^2 = +1$ and $u \\neq \\pm 1$\n (the defining polynomial $x^2 - 1 = (x\ + \ - 1)(x + 1)$\n splits over $R$).\n\\end{itemize}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 191\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch77-elliptic-complex-field.tex` lines 237-252\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.ComplexField`\n- Name:\ + \ `Tau.Boundary.elliptic_hyperbolic_dichotomy`\n\n## Dependencies\n\n- Canonical: I.D85, I.D27\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The elliptic-hyperbolic dichotomy: TauComplex (i^2=-1, field, no zero divisors) versus SplitComplex (j^2=+1, + ring, zero divisors). Both are algebraically earned; they coexist as complementary structures within tau.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001096.md b/_corpus_v3_items/cid001096.md new file mode 100644 index 000000000..865a318f5 --- /dev/null +++ b/_corpus_v3_items/cid001096.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001096/ +title: Elliptic Quaternions +cid: cid001096 +primary_alias: DEF0094 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'TauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1. Non-commutativity is earned + as a structural consequence of extending beyond two dimensions.' +aliases: +- DEF0094 +- I.D87 +- elliptic-quaternions +- def:tau-quaternion +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 78 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Elliptic Quaternions**\n\nTauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1.\ + \ Non-commutativity is earned as a structural consequence of extending beyond two dimensions.\n\n# Elliptic Quaternions\n\ + \n## Summary\n\nTauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1. Non-commutativity\ + \ is earned as a structural consequence of extending beyond two dimensions.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-quaternion}\n\ + The \\textbf{$\\tau$-quaternion algebra} is the $\\mathbb{R}_\\tau$-algebra:\n\\[\n \\boxed{%\n \\mathbb{H}_\\tau\n\ + \ := \\mathbb{R}_\\tau \\langle i, \\mathbf{j}, k \\rangle\n \\;/\\;\n (i^2 = \\mathbf{j}^2 = k^2 = i\\mathbf{j}k\ + \ = -1),}\n\\]\nwith elements written as $(a, b, c, d) \\in \\mathbb{R}_\\tau^4$,\nrepresenting $q = a + bi + c\\mathbf{j}\ + \ + dk$.\nThe multiplication is determined by the cyclic rules\n$ij = k$, $jk = i$, $ki = j$ (positive cycle)\nand $ji\ + \ = -k$, $kj = -i$, $ik = -j$ (negative cycle),\nextended bilinearly to all of $\\mathbb{H}_\\tau$.\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 193\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch78-elliptic-quaternions.tex`\ + \ lines 40-57\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Quaternions`\n\ + - Name: `Tau.Boundary.TauQuaternion`\n\n## Dependencies\n\n- Canonical: I.D84\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'TauQuaternion = TauReal-algebra {1,i,j,k} with Hamilton product: i^2=j^2=k^2=ijk=-1. Non-commutativity is earned + as a structural consequence of extending beyond two dimensions.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001097.md b/_corpus_v3_items/cid001097.md new file mode 100644 index 000000000..feac109df --- /dev/null +++ b/_corpus_v3_items/cid001097.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001097/ +title: Cyclotomic Fields +cid: cid001097 +primary_alias: DEF0095 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cyclotomic fields: roots of unity z^n = 1 mod m, primitive roots, and their connection to CRT decomposition. The + Galois group Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)^times.' +aliases: +- DEF0095 +- I.D88 +- cyclotomic-fields +- def:cyclotomic-field +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 79 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cyclotomic Fields**\n\nCyclotomic fields: roots of unity z^n = 1 mod m, primitive roots, and their connection\ + \ to CRT decomposition. The Galois group Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)^times.\n\n# Cyclotomic Fields\n\n##\ + \ Summary\n\nCyclotomic fields: roots of unity z^n = 1 mod m, primitive roots, and their connection to CRT decomposition.\ + \ The Galois group Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)^times.\n\n## Statement\n\n```latex\n%\n\\label{def:cyclotomic-field}\n\ + For $n \\geq 1$, the \\textbf{$n$-th cyclotomic field} is:\n\\[\n \\boxed{%\n \\mathbb{Q}^\\mathrm{cyc}_\\tau(\\\ + zeta_n)\n := \\mathbb{Q}_\\tau(\\zeta_n)\n = \\biggl\\{\\sum_{j=0}^{d-1} a_j \\zeta_n^j\n : a_j \\in \\mathbb{Q}_\\\ + tau\\biggr\\},}\n\\]\nthe smallest subfield of $\\mathbb{C}_\\tau$\ncontaining $\\mathbb{Q}_\\tau$ and $\\zeta_n$.\nThis\ + \ is a finite extension of degree $\\varphi(n)$,\nwith minimal polynomial the $n$-th \\textbf{cyclotomic polynomial}:\n\ + \\[\n \\Phi_n(x) := \\prod_{\\substack{1 \\leq k \\leq n \\\\ \\gcd(k,n) = 1}}\n (x - \\zeta_n^k)\n \\;\\in\\\ + ; \\mathbb{Z}_\\tau[x].\n\\]\nThe factorization $x^n - 1 = \\prod_{d \\mid n} \\Phi_d(x)$\npartitions all $n$-th roots\ + \ by their primitive order.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 196\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch79-cyclotomic-fields.tex`\ + \ lines 80-102\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Cyclotomic`\n\ + - Name: `Tau.Boundary.IsRootOfUnity`\n\n## Dependencies\n\n- Canonical: I.D15, I.D19\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cyclotomic fields: roots of unity z^n = 1 mod m, primitive roots, and their connection to CRT decomposition. + The Galois group Gal(Q(zeta_n)/Q) is isomorphic to (Z/nZ)^times.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001098.md b/_corpus_v3_items/cid001098.md new file mode 100644 index 000000000..e265ef527 --- /dev/null +++ b/_corpus_v3_items/cid001098.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001098/ +title: Diagonal Resonance +cid: cid001098 +primary_alias: DEF0096 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Diagonal resonance: the interaction resonance between three individually benign components — (L) meta-level contraction/free + token reuse, (E) equality-as-congruence with full substitution, (P) ontic self-products with diagonal materialization — + that jointly produce identity pathology in foundations.' +aliases: +- DEF0096 +- I.D89 +- diagonal-resonance +- def:diagonal-resonance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 80 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagonal Resonance**\n\nDiagonal resonance: the interaction resonance between three individually benign components\ + \ — (L) meta-level contraction/free token reuse, (E) equality-as-congruence with full substitution, (P) ontic self-products\ + \ with diagonal materialization — that jointly produce identity pathology in foundations.\n\n# Diagonal Resonance\n\n\ + ## Summary\n\nDiagonal resonance: the interaction resonance between three individually benign components — (L) meta-level\ + \ contraction/free token reuse, (E) equality-as-congruence with full substitution, (P) ontic self-products with diagonal\ + \ materialization — that jointly produce identity pathology in foundations.\n\n## Statement\n\n```latex\n%\n\\label{def:diagonal-resonance}\n\ + A foundation exhibits \\textbf{diagonal resonance}\nif it simultaneously provides the following\nthree structural capabilities\ + \ at the kernel level:\n\\begin{enumerate}\n \\item[\\textbf{(L)}]\n \\textbf{Meta-level contraction / free\ + \ token reuse.}\n Variables can be freely reused:\n if a term $t$ appears once,\n it may appear\ + \ again without cost.\n In sequent-calculus terms,\n the contraction rule\n \\[\n \ + \ \\frac{\\Gamma, A, A \\vdash B}{\\Gamma, A \\vdash B}\n \\]\n is admissible.\n Identity-of-reference\ + \ ---\n the meta-level assertion\n ``this is the same token'' ---\n is free in syntax.\n \ + \ This is standard in classical logic,\n in intuitionistic logic,\n and in the Calculus of Inductive\ + \ Constructions.\n \\item[\\textbf{(E)}]\n \\textbf{Equality-as-congruence with full substitution.}\n \ + \ The system admits an equality relation\n $= \\,\\subseteq X \\times X$\n such that if $a = b$\ + \ and $P(a)$ holds,\n then $P(b)$ holds.\n Substitution makes equality behave\n like identification:\n\ + \ the truth of a proposition about $a$\n transfers to $b$ whenever $a = b$.\n This is Leibniz's\ + \ law,\n the indiscernibility of identicals,\n and it is standard in every major foundation.\n\n \ + \ The system thereby admits two distinct notions\n of ``sameness'':\n \\emph{identity-of-reference}\n\ + \ (meta-level: ``this is the same token'')\n and \\emph{equality-as-relation}\n (object-level:\ + \ $a = b$ as a proposition).\n Full substitution forces these two notions\n to interact:\n \ + \ object-level equality inherits\n the operational behavior\n of meta-level identity.\n \\item[\\\ + textbf{(P)}]\n \\textbf{Ontic self-products with diagonal materialization.}\n For any type or set $A$,\n\ + \ the self-product $A \\times A$ exists,\n and the diagonal subset\n $\\Delta_A = \\{(a, a)\ + \ \\mid a \\in A\\}$\n can be carved out by comprehension\n or constructed as a morphism\n \ + \ $\\Delta_A : A \\to A \\times A$\n via $a \\mapsto (a, a)$.\n The theory can materialize\n \ + \ two ports to the same type\n and identify elements across those ports.\n This is standard in ZFC\n\ + \ (Cartesian products and separation),\n in CIC (sigma-types and pattern matching),\n and in\ + \ every cartesian closed category.\n\\end{enumerate}\nNo single component is pathological.\nEach is individually reasonable,\ + \ well-motivated,\nand practically indispensable.\nThe pathology arises from their \\emph{joint presence}:\nwhen all three\ + \ are simultaneously active\nat the kernel level,\ndiagonal resonance becomes unavoidable\nonce the system has sufficient\ + \ expressivity\nto encode self-referential constructions.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 199\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch80-diagonal-resonance.tex` lines 107-184\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.DiagonalResonance`\n- Name:\ + \ `Tau.MetaLogic.DiagonalResonance`\n\n## Dependencies\n\n- Canonical: I.D03, I.D78, I.D81\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Diagonal resonance: the interaction resonance between three individually benign components — (L) meta-level + contraction/free token reuse, (E) equality-as-congruence with full substitution, (P) ontic self-products with diagonal materializati' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001099.md b/_corpus_v3_items/cid001099.md new file mode 100644 index 000000000..95f670ddf --- /dev/null +++ b/_corpus_v3_items/cid001099.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001099/ +title: Identity Slippage +cid: cid001099 +primary_alias: DEF0097 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Identity slippage: the partial decoherence of ontic self-identity where distinct ontic objects cannot be preserved + as distinct under any global projection into a VM exhibiting diagonal resonance. The boundary between identity-of-reference + and equality-as-relation is smeared.' +aliases: +- DEF0097 +- I.D90 +- identity-slippage +- def:identity-slippage +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 80 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Identity Slippage**\n\nIdentity slippage: the partial decoherence of ontic self-identity where distinct ontic\ + \ objects cannot be preserved as distinct under any global projection into a VM exhibiting diagonal resonance. The boundary\ + \ between identity-of-reference and equality-as-relation is smeared.\n\n# Identity Slippage\n\n## Summary\n\nIdentity\ + \ slippage: the partial decoherence of ontic self-identity where distinct ontic objects cannot be preserved as distinct\ + \ under any global projection into a VM exhibiting diagonal resonance. The boundary between identity-of-reference and\ + \ equality-as-relation is smeared.\n\n## Statement\n\n```latex\n%\n\\label{def:identity-slippage}\nA foundation exhibits\ + \ \\textbf{identity slippage}\nif diagonal resonance (Definition~\\ref{def:diagonal-resonance}, I.D89)\ncauses the following:\n\ + distinct ontic objects cannot be preserved as distinct\nunder any global projection\ninto the foundation's semantic layer.\n\ + Specifically:\n\\begin{enumerate}\n \\item The equality relation $E \\hookrightarrow X \\times X$\n acquires\ + \ operational behavior\n that is indistinguishable\n from a family of ``almost-identity'' arrows ---\n\ + \ a thin groupoid of identity witnesses\n that acts like morphisms of sameness\n without being\ + \ the canonical $\\mathrm{id}_X$.\n \\item The self-product $X \\times X$\n and the diagonal $\\Delta_X$\n\ + \ allow the system to construct\n self-referential composites\n of the form $f(e(a)(a))$\n\ + \ (the double use of $a$\n that Lawvere's fixed-point theorem requires),\n producing identification\n\ + \ where none was ontically intended.\n \\item The combination of (1) and (2)\n prevents the existence\n\ + \ of a canonical, identity-faithful\n intended semantics:\n the system cannot internally stabilize\n\ + \ a unique ontic closure\n without appealing to an external meta-theory.\n\\end{enumerate}\nIdentity\ + \ slippage is not inconsistency.\nThe system does not derive $\\bot$.\nIt derives something subtler:\n\\emph{non-canonicity}\ + \ ---\nthe impossibility of singling out\na unique intended model\nfrom within the system itself.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 200\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch80-diagonal-resonance.tex`\ + \ lines 325-367\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.DiagonalResonance`\n\ + - Name: `Tau.MetaLogic.IdentitySlippage`\n\n## Dependencies\n\n- Canonical: I.D89\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Identity slippage: the partial decoherence of ontic self-identity where distinct ontic objects cannot be preserved + as distinct under any global projection into a VM exhibiting diagonal resonance. The boundary between identity-of-reference + a' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001100.md b/_corpus_v3_items/cid001100.md new file mode 100644 index 000000000..8960eb888 --- /dev/null +++ b/_corpus_v3_items/cid001100.md @@ -0,0 +1,194 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001100/ +title: Shadow Identity +cid: cid001100 +primary_alias: DEF0098 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Shadow identity: an implicit identification channel — equivalence witness, congruence step, or substitution bridge + — that behaves operationally like a morphism of sameness without being ontically constructed. The operational consequence + of diagonal resonance.' +aliases: +- DEF0098 +- I.D91 +- shadow-identity +- def:shadow-identity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 80 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Shadow Identity** + + + Shadow identity: an implicit identification channel — equivalence witness, congruence step, or substitution bridge — that + behaves operationally like a morphism of sameness without being ontically constructed. The operational consequence of + diagonal resonance. + + + # Shadow Identity + + + ## Summary + + + Shadow identity: an implicit identification channel — equivalence witness, congruence step, or substitution bridge — that + behaves operationally like a morphism of sameness without being ontically constructed. The operational consequence of + diagonal resonance. + + + ## Statement + + + ```latex + + % + + \label{def:shadow-identity} + + A \textbf{shadow identity} is an implicit identification channel --- + + an equivalence witness, congruence step, + + or substitution bridge --- + + that behaves operationally like a morphism of sameness + + without being ontically constructed. + + + More precisely: + + in an L+E+P foundation, + + the interaction of contraction (\textbf{L}), + + substitution (\textbf{E}), + + and the diagonal (\textbf{P}) + + creates identifications + + between objects $a$ and $b$ + + that do not correspond + + to any explicitly constructed morphism + + $f : a \to b$ in the category. + + These identifications are + + \emph{shadow identities}: + + they are carried by the ambient structural rules, + + not by the object-level morphisms. + + + In a coherence-first ontology, + + identity is carried solely by $\mathrm{id}_X$, + + and any additional identification + + must be explicitly earned as structure. + + Shadow identities violate this principle: + + they provide identification for free, + + as a side effect of the structural rules, + + without any ontic witness. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 201 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch80-diagonal-resonance.tex` lines + 424-456 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.MetaLogic.DiagonalResonance` + + - Name: `Tau.MetaLogic.ShadowIdentity` + + + ## Dependencies + + + - Canonical: I.D89, I.D90 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Shadow identity: an implicit identification channel — equivalence witness, congruence step, or substitution + bridge — that behaves operationally like a morphism of sameness without being ontically constructed. The operational consequence + of ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001101.md b/_corpus_v3_items/cid001101.md new file mode 100644 index 000000000..775fe257c --- /dev/null +++ b/_corpus_v3_items/cid001101.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001101/ +title: Identity-Faithful Reception +cid: cid001101 +primary_alias: DEF0099 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Identity-faithful reception: a VM system S can receive tau ontically only if there exists an interpretation functor + P: C_tau -> C_S that is faithful on identity — reflects equality/isomorphism back to tau-identity, preserves identity morphisms, + and maps distinct tau-objects to distinct S-objects.' +aliases: +- DEF0099 +- I.D92 +- identity-faithful-reception +- def:identity-faithful-reception +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 82 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Identity-Faithful Reception**\n\nIdentity-faithful reception: a VM system S can receive tau ontically only\ + \ if there exists an interpretation functor P: C_tau -> C_S that is faithful on identity — reflects equality/isomorphism\ + \ back to tau-identity, preserves identity morphisms, and maps distinct tau-objects to distinct S-objects.\n\n# Identity-Faithful\ + \ Reception\n\n## Summary\n\nIdentity-faithful reception: a VM system S can receive tau ontically only if there exists\ + \ an interpretation functor P: C_tau -> C_S that is faithful on identity — reflects equality/isomorphism back to tau-identity,\ + \ preserves identity morphisms, and maps distinct tau-objects to distinct S-objects.\n\n## Statement\n\n```latex\n%\n\\\ + label{def:identity-faithful-reception}\nA virtual machine system $\\mathsf{S}$\ncan \\textbf{receive $\\tau$ ontically}\n\ + only if there exists an interpretation functor\n$P : \\mathcal{C}_\\tau \\to \\mathcal{C}_{\\mathsf{S}}$\nsatisfying three\ + \ conditions:\n\\begin{enumerate}[\\normalfont(i)]\n \\item \\textbf{Object distinctness.}\n $P$ maps distinct\ + \ $\\tau$-objects to distinct $\\mathsf{S}$-objects:\n if $X \\neq Y$ in $\\mathcal{C}_\\tau$,\n then\ + \ $P(X) \\neq P(Y)$ in $\\mathcal{C}_{\\mathsf{S}}$.\n \\item \\textbf{Identity preservation.}\n $P$ preserves\ + \ identity morphisms:\n $P(\\mathrm{id}_X) = \\mathrm{id}_{P(X)}$\n for every object $X$ of $\\mathcal{C}_\\\ + tau$.\n \\item \\textbf{Isomorphism reflection.}\n $P$ reflects isomorphism:\n if $P(X) \\cong P(Y)$\ + \ in $\\mathcal{C}_{\\mathsf{S}}$,\n then $X \\cong Y$ in $\\mathcal{C}_\\tau$.\n\\end{enumerate}\nSuch a functor\ + \ is called\n\\textbf{identity-faithful}.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 207\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch82-reception-criterion.tex`\ + \ lines 115-139\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.ReceptionCriterion`\n\ + - Name: `Tau.MetaLogic.IdentityFaithfulReception`\n\n## Dependencies\n\n- Canonical: I.D89, I.D90, I.T46\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Identity-faithful reception: a VM system S can receive tau ontically only if there exists an interpretation + functor P: C_tau -> C_S that is faithful on identity — reflects equality/isomorphism back to tau-identity, preserves identity + morphi' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001102.md b/_corpus_v3_items/cid001102.md new file mode 100644 index 000000000..9bb557b30 --- /dev/null +++ b/_corpus_v3_items/cid001102.md @@ -0,0 +1,154 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001102/ +title: Structural Instability +cid: cid001102 +primary_alias: DEF0100 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Structural instability: a foundation exhibits structural instability if diagonal resonance prevents the existence + of a canonical, identity-faithful intended semantics — the system cannot internally stabilize a unique ontic closure (omega) + and cannot fix ontology/reference without VM-relativity.' +aliases: +- DEF0100 +- I.D93 +- structural-instability +- def:structural-instability +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 82 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Structural Instability** + + + Structural instability: a foundation exhibits structural instability if diagonal resonance prevents the existence of a + canonical, identity-faithful intended semantics — the system cannot internally stabilize a unique ontic closure (omega) + and cannot fix ontology/reference without VM-relativity. + + + # Structural Instability + + + ## Summary + + + Structural instability: a foundation exhibits structural instability if diagonal resonance prevents the existence of a + canonical, identity-faithful intended semantics — the system cannot internally stabilize a unique ontic closure (omega) + and cannot fix ontology/reference without VM-relativity. + + + ## Statement + + + ```latex + + % + + \label{def:structural-instability} + + A foundation exhibits \textbf{structural instability} + + if diagonal resonance (I.D89) + + prevents the existence of a canonical, + + identity-faithful intended semantics: + + the system cannot internally stabilize + + a unique ontic closure ($\omega$) + + and cannot fix ontology or reference + + without model-relativity or external convention. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 208 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch82-reception-criterion.tex` lines + 208-219 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.MetaLogic.ReceptionCriterion` + + - Name: `Tau.MetaLogic.StructuralInstability` + + + ## Dependencies + + + - Canonical: I.D89, I.D90 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Structural instability: a foundation exhibits structural instability if diagonal resonance prevents the existence + of a canonical, identity-faithful intended semantics — the system cannot internally stabilize a unique ontic closure (omega) + a' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001103.md b/_corpus_v3_items/cid001103.md new file mode 100644 index 000000000..4a70ace76 --- /dev/null +++ b/_corpus_v3_items/cid001103.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001103/ +title: Orbit-Set Map +cid: cid001103 +primary_alias: DEF0101 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Orbit-Set Map: assigns to every tau-element a structurally determined set via five families. Set(alpha_n) = {alpha_k + : k | n} (divisor set = tau-membership); Set(pi_n) = first n primes; Set(gamma_n) = vertical prime tower; Set(eta_n) = horizontal + prime sweep; Set(omega) = O_alpha union {omega}. The alpha-orbit case is exactly tau-membership (I.D31). All set-elements + from O_alpha only (plus omega in Set(omega)).' +aliases: +- DEF0101 +- I.D94 +- orbit-set-map +- def:orbit-set-map +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 83 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Orbit-Set Map**\n\nOrbit-Set Map: assigns to every tau-element a structurally determined set via five families.\ + \ Set(alpha_n) = {alpha_k : k | n} (divisor set = tau-membership); Set(pi_n) = first n primes; Set(gamma_n) = vertical\ + \ prime tower; Set(eta_n) = horizontal prime sweep; Set(omega) = O_alpha union {omega}. The alpha-orbit case is exactly\ + \ tau-membership (I.D31). All set-elements from O_alpha only (plus omega in Set(omega)).\n\n# Orbit-Set Map\n\n## Summary\n\ + \nOrbit-Set Map: assigns to every tau-element a structurally determined set via five families. Set(alpha_n) = {alpha_k\ + \ : k | n} (divisor set = tau-membership); Set(pi_n) = first n primes; Set(gamma_n) = vertical prime tower; Set(eta_n)\ + \ = horizontal prime sweep; Set(omega) = O_alpha union {omega}. The alpha-orbit case is exactly tau-membership (I.D31).\ + \ All set-elements from O_alpha only (plus omega in Set(omega)).\n\n## Statement\n\n```latex\n%\n\\label{def:orbit-set-map}\n\ + The \\textbf{orbit-set map}\n$\\mathrm{Set} \\colon \\Obj(\\tau) \\to \\mathcal{P}(\\Obj(\\tau))$\nis defined orbit-by-orbit\ + \ as follows.\nLet $p_n$ denote the $n$-th prime.\n\n\\medskip\n\\noindent\n\\textbf{(i) $\\alpha$-orbit} ($\\alpha_n\ + \ \\in O_\\alpha$,\\; $n \\geq 1$):\n\\[\n \\boxed{%\n \\mathrm{Set}(\\alpha_n)\n \\;:=\\;\n \\{\\alpha_k\ + \ : k \\mid n\\}.}\n\\]\nThis is the \\emph{divisor set} of~$n$:\nevery divisor of $n$ produces a set-member,\nand nothing\ + \ else.\nThis is exactly $\\tau$-membership from\nChapter~\\ref{ch:membership-divisibility}:\n$\\alpha_k \\in \\mathrm{Set}(\\\ + alpha_n)\n\\iff k \\in_\\tau n \\iff k \\mid n$.\nFor a prime $p$: $\\mathrm{Set}(\\alpha_p) = \\{\\alpha_1, \\alpha_p\\\ + }$\n(atoms --- the minimal non-trivial sets).\nFor composite $n$: the set includes all divisors.\n\n\\medskip\n\\noindent\n\ + \\textbf{(ii) $\\pi$-orbit} ($\\pi_n \\in O_\\pi$,\\; $n \\geq 0$):\n\\[\n \\boxed{%\n \\mathrm{Set}(\\pi_n)\n \ + \ \\;:=\\;\n \\{\\alpha_1\\}\n \\;\\cup\\;\n \\{\\alpha_{p_k} : 1 \\leq k \\leq n\\}.}\n\\]\nThis is the \\\ + emph{first $n$ primes}\n(as $\\alpha$-orbit elements) plus the unit.\nThe definition uses ``first $n$ primes''\nrather\ + \ than ``primes $\\leq n$,''\nensuring injectivity and aligning\nwith the rank transfer map $RT_\\pi$\n(Definition~\\\ + ref{def:rank-transfer}).\n\n\\medskip\n\\noindent\n\\textbf{(iii) $\\gamma$-orbit} ($\\gamma_n \\in O_\\gamma$,\\; $n\ + \ \\geq 1$):\n\\[\n \\boxed{%\n \\mathrm{Set}(\\gamma_n)\n \\;:=\\;\n \\{\\alpha_{p_n^k} : k \\geq 0\\}\n\ + \ \\;=\\;\n \\{\\alpha_1, \\alpha_{p_n}, \\alpha_{p_n^2},\n \\alpha_{p_n^3}, \\ldots\\}.}\n\\]\nThe powers of\ + \ the $n$-th prime: a \\emph{vertical} chain\nexploring DEPTH along a single prime tower.\nThese are the \\textbf{first\ + \ infinite sets}\nin $\\tau$'s internal set theory.\n\n\\medskip\n\\noindent\n\\textbf{(iv) $\\eta$-orbit} ($\\eta_n \\\ + in O_\\eta$,\\; $n \\geq 1$):\n\\[\n \\boxed{%\n \\mathrm{Set}(\\eta_n)\n \\;:=\\;\n \\{\\alpha_{p_k^n} :\ + \ k \\geq 1\\}\n \\;=\\;\n \\{\\alpha_{p_1^n}, \\alpha_{p_2^n},\n \\alpha_{p_3^n}, \\ldots\\}.}\n\\]\nAll primes\ + \ raised to power~$n$: a \\emph{horizontal} sweep\nexploring BREADTH across all prime bases.\nThese are also infinite.\n\ + \n\\medskip\n\\noindent\n\\textbf{(v) Crossing point} ($\\omega$):\n\\[\n \\boxed{%\n \\mathrm{Set}(\\omega)\n \ + \ \\;:=\\;\n O_\\alpha \\cup \\{\\omega\\}.}\n\\]\nThe \\emph{one-point compactification} of $\\mathbb{N}^+$:\nall\ + \ $\\alpha$-orbit elements together with $\\omega$ itself.\nThis is the \\textbf{universal set} ---\nthe largest set in\ + \ $\\tau$'s internal universe.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 212\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch83-orbit-set-correspondence.tex`\ + \ lines 101-194\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.OrbitSets`\n\ + - Name: `Tau.Sets.orbit_set`\n\n## Dependencies\n\n- Canonical: I.D05, I.T01, I.P03, I.D08, I.D31, I.D33\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Orbit-Set Map: assigns to every tau-element a structurally determined set via five families. Set(alpha_n) = + {alpha_k : k | n} (divisor set = tau-membership); Set(pi_n) = first n primes; Set(gamma_n) = vertical prime tower; Set(eta_n) + = hori' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001104.md b/_corpus_v3_items/cid001104.md new file mode 100644 index 000000000..44db4c8fa --- /dev/null +++ b/_corpus_v3_items/cid001104.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001104/ +title: τ-Measure Space +cid: cid001104 +primary_alias: DEF0102 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Constructive measure space on the primorial tower. At each stage k, Z/M_k Z carries the normalized counting measure + μ_k(S) = |S|/M_k. The τ-measure space is the projective family of stages with compatible counting measures. +aliases: +- DEF0102 +- I.D95 +- measure-space +- def:tau-measure-space +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005276 +- predicate: formalized_by + object: cid005282 +- predicate: formalized_by + object: cid005283 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Measure Space** + + + Constructive measure space on the primorial tower. At each stage k, Z/M_k Z carries the normalized counting measure μ_k(S) + = |S|/M_k. The τ-measure space is the projective family of stages with compatible counting measures. + + + # τ-Measure Space + + + ## Summary + + + Constructive measure space on the primorial tower. At each stage k, Z/M_k Z carries the normalized counting measure μ_k(S) + = |S|/M_k. The τ-measure space is the projective family of stages with compatible counting measures. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 219 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Measure` + + - Name: `TauMeasureSpace` + + + ## Dependencies + + + - Canonical: I.D76, I.D84 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Constructive measure space on the primorial tower. At each stage k, Z/M_k Z carries the normalized counting measure + μ_k(S) = |S|/M_k. The τ-measure space is the projective family of stages with compatible counting measures. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001105.md b/_corpus_v3_items/cid001105.md new file mode 100644 index 000000000..4df356d7f --- /dev/null +++ b/_corpus_v3_items/cid001105.md @@ -0,0 +1,127 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001105/ +title: Tower σ-Algebra +cid: cid001105 +primary_alias: DEF0103 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Tower-measurable sets: compatible families of predicates S_k on Z/M_k Z such that preimage under reduction preserves + membership. At finite stage, every subset is measurable (full power set σ-algebra).' +aliases: +- DEF0103 +- I.D96 +- tower-algebra +- def:tower-sigma-algebra +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tower σ-Algebra** + + + Tower-measurable sets: compatible families of predicates S_k on Z/M_k Z such that preimage under reduction preserves membership. + At finite stage, every subset is measurable (full power set σ-algebra). + + + # Tower σ-Algebra + + + ## Summary + + + Tower-measurable sets: compatible families of predicates S_k on Z/M_k Z such that preimage under reduction preserves membership. + At finite stage, every subset is measurable (full power set σ-algebra). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 220 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Measure` + + - Name: `TowerMeasurableSet` + + + ## Dependencies + + + - Canonical: I.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tower-measurable sets: compatible families of predicates S_k on Z/M_k Z such that preimage under reduction preserves + membership. At finite stage, every subset is measurable (full power set σ-algebra).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001106.md b/_corpus_v3_items/cid001106.md new file mode 100644 index 000000000..97b59081b --- /dev/null +++ b/_corpus_v3_items/cid001106.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001106/ +title: Galois Automorphism +cid: cid001106 +primary_alias: DEF0104 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A Galois automorphism at stage k: multiplication by a unit a with gcd(a, M_k)=1. Preserves ring structure: σ_a(x+y) + = σ_a(x)+σ_a(y), σ_a(xy) = σ_a(x)·σ_a(y).' +aliases: +- DEF0104 +- I.D97 +- galois-automorphism +- def:galois-automorphism +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005286 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 85 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Galois Automorphism** + + + A Galois automorphism at stage k: multiplication by a unit a with gcd(a, M_k)=1. Preserves ring structure: σ_a(x+y) = + σ_a(x)+σ_a(y), σ_a(xy) = σ_a(x)·σ_a(y). + + + # Galois Automorphism + + + ## Summary + + + A Galois automorphism at stage k: multiplication by a unit a with gcd(a, M_k)=1. Preserves ring structure: σ_a(x+y) = + σ_a(x)+σ_a(y), σ_a(xy) = σ_a(x)·σ_a(y). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 223 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Galois` + + - Name: `GaloisAut` + + + ## Dependencies + + + - Canonical: I.D88, I.D76 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A Galois automorphism at stage k: multiplication by a unit a with gcd(a, M_k)=1. Preserves ring structure: σ_a(x+y) + = σ_a(x)+σ_a(y), σ_a(xy) = σ_a(x)·σ_a(y).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001107.md b/_corpus_v3_items/cid001107.md new file mode 100644 index 000000000..0e5e10910 --- /dev/null +++ b/_corpus_v3_items/cid001107.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001107/ +title: Galois Group of Primorial Stage +cid: cid001107 +primary_alias: DEF0105 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Galois group at stage k is (Z/M_k Z)×, the group of units. Its order is φ(M_k) = ∏(p_i - 1) for primes p_i ≤ + p_k. Verified: φ(2)=1, φ(6)=2, φ(30)=8.' +aliases: +- DEF0105 +- I.D98 +- galois-group-of-primorial-stage +- def:galois-group-primorial +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005287 +- predicate: formalized_by + object: cid005288 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 85 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Galois Group of Primorial Stage** + + + The Galois group at stage k is (Z/M_k Z)×, the group of units. Its order is φ(M_k) = ∏(p_i - 1) for primes p_i ≤ p_k. + Verified: φ(2)=1, φ(6)=2, φ(30)=8. + + + # Galois Group of Primorial Stage + + + ## Summary + + + The Galois group at stage k is (Z/M_k Z)×, the group of units. Its order is φ(M_k) = ∏(p_i - 1) for primes p_i ≤ p_k. + Verified: φ(2)=1, φ(6)=2, φ(30)=8. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 224 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Galois` + + - Name: `galois_group_order` + + + ## Dependencies + + + - Canonical: I.D97, I.D76 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Galois group at stage k is (Z/M_k Z)×, the group of units. Its order is φ(M_k) = ∏(p_i - 1) for primes p_i + ≤ p_k. Verified: φ(2)=1, φ(6)=2, φ(30)=8.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001108.md b/_corpus_v3_items/cid001108.md new file mode 100644 index 000000000..094637864 --- /dev/null +++ b/_corpus_v3_items/cid001108.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001108/ +title: τ-Integral +cid: cid001108 +primary_alias: DEF0106 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The τ-integral of f at stage k: ∫_k f = (Σ f(x)) / M_k. Expectation under the counting measure μ_k on Z/M_k Z.' +aliases: +- DEF0106 +- I.D99 +- integral +- def:tau-integral +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005272 +- predicate: formalized_by + object: cid005275 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Integral** + + + The τ-integral of f at stage k: ∫_k f = (Σ f(x)) / M_k. Expectation under the counting measure μ_k on Z/M_k Z. + + + # τ-Integral + + + ## Summary + + + The τ-integral of f at stage k: ∫_k f = (Σ f(x)) / M_k. Expectation under the counting measure μ_k on Z/M_k Z. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 227 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Integration` + + - Name: `tau_integral` + + + ## Dependencies + + + - Canonical: I.D95, I.D96 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The τ-integral of f at stage k: ∫_k f = (Σ f(x)) / M_k. Expectation under the counting measure μ_k on Z/M_k + Z.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001109.md b/_corpus_v3_items/cid001109.md new file mode 100644 index 000000000..4b4036375 --- /dev/null +++ b/_corpus_v3_items/cid001109.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001109/ +title: Universe Postulate +cid: cid001109 +primary_alias: AXM0002 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Postulates the existence of the totality tau as a universe of discourse. Implicit in Lean type system (Generator + : Type, TauObj : Type). Distinguishes tau (ambient type) from omega (fixed-point element within tau).' +aliases: +- AXM0002 +- I.K0 +- universe-postulate +- ax:universe-postulate +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 1 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Universe Postulate** + + + Postulates the existence of the totality tau as a universe of discourse. Implicit in Lean type system (Generator : Type, + TauObj : Type). Distinguishes tau (ambient type) from omega (fixed-point element within tau). + + + # Universe Postulate + + + ## Summary + + + Postulates the existence of the totality tau as a universe of discourse. Implicit in Lean type system (Generator : Type, + TauObj : Type). Distinguishes tau (ambient type) from omega (fixed-point element within tau). + + + ## Statement + + + ```latex + + \label{ax:universe-postulate} + + There exists a totality $\tau$ --- a universe of discourse --- + + in which the five generators $\alpha, \pi, \gamma, \eta, \omega$ + + are the only primitive objects, + + and which is closed under the progression operator~$\rho$. + + The totality $\tau$ is not itself an object of $\tau$: + + it is the ambient type within which all objects exist. + + ``` + + + ## Proof / Justification + + + This item is an axiom. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 1 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch01-five-generators.tex` lines + 135-143 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Kernel.Signature` + + - Name: `None` + + + ## Dependencies + + + - Canonical: (none) + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Postulates the existence of the totality tau as a universe of discourse. Implicit in Lean type system (Generator + : Type, TauObj : Type). Distinguishes tau (ambient type) from omega (fixed-point element within tau).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001110.md b/_corpus_v3_items/cid001110.md new file mode 100644 index 000000000..c24213aee --- /dev/null +++ b/_corpus_v3_items/cid001110.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001110/ +title: Strict Order (K1) +cid: cid001110 +primary_alias: AXM0003 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: alpha < pi < gamma < eta < omega is a strict total order on the 5 generators. +aliases: +- AXM0003 +- I.K1 +- strict-order-k1 +- ax:strict-order +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005311 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 1 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Strict Order (K1)**\n\nalpha < pi < gamma < eta < omega is a strict total order on the 5 generators.\n\n# Strict\ + \ Order (K1)\n\n## Summary\n\nalpha < pi < gamma < eta < omega is a strict total order on the 5 generators.\n\n## Statement\n\ + \n```latex\n\\label{ax:strict-order}\nThe five generators satisfy the strict total order\n\\[\n \\boxed{\\alpha \\\ + ;<\\; \\pi \\;<\\; \\gamma \\;<\\; \\eta \\;<\\; \\omega.}\n\\]\n```\n\n## Proof / Justification\n\nThis item is an axiom.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 6\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch01-five-generators.tex` lines 182-188\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n- Name: `Tau.Kernel.K1_strict_order`\n\ + \n## Dependencies\n\n- Canonical: I.D01\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: alpha < pi < gamma < eta < omega is a strict total order on the 5 generators. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001111.md b/_corpus_v3_items/cid001111.md new file mode 100644 index 000000000..d9a15f880 --- /dev/null +++ b/_corpus_v3_items/cid001111.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001111/ +title: Omega Fixed Point (K2) +cid: cid001111 +primary_alias: AXM0004 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: rho(omega) = omega; omega absorbs all operations. +aliases: +- AXM0004 +- I.K2 +- omega-fixed-point-k2 +- ax:omega-fixed +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005312 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 2 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega Fixed Point (K2)**\n\nrho(omega) = omega; omega absorbs all operations.\n\n# Omega Fixed Point (K2)\n\ + \n## Summary\n\nrho(omega) = omega; omega absorbs all operations.\n\n## Statement\n\n```latex\n\\label{ax:omega-fixed}\n\ + The beacon $\\omega$ is a fixed point of~$\\rho$:\n\\[\n \\boxed{\\rho(\\omega) = \\omega.}\n\\]\n```\n\n## Proof /\ + \ Justification\n\nThis item is an axiom. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 7\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch02-rho-omega.tex`\ + \ lines 77-83\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.K2_omega_fixed`\n\n## Dependencies\n\n- Canonical: I.D01, I.D02\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: rho(omega) = omega; omega absorbs all operations. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001112.md b/_corpus_v3_items/cid001112.md new file mode 100644 index 000000000..72cb37bf4 --- /dev/null +++ b/_corpus_v3_items/cid001112.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001112/ +title: Orbit-Seeded Generation (K3) +cid: cid001112 +primary_alias: AXM0005 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Each generator g in {alpha, pi, gamma, eta} seeds its orbit ray O_g = {rho^n(g) : n >= 0}.' +aliases: +- AXM0005 +- I.K3 +- orbit-seeded-generation-k3 +- ax:orbit-seeded +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005313 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 3 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Orbit-Seeded Generation (K3)**\n\nEach generator g in {alpha, pi, gamma, eta} seeds its orbit ray O_g = {rho^n(g)\ + \ : n >= 0}.\n\n# Orbit-Seeded Generation (K3)\n\n## Summary\n\nEach generator g in {alpha, pi, gamma, eta} seeds its\ + \ orbit ray O_g = {rho^n(g) : n >= 0}.\n\n## Statement\n\n```latex\n%\n\\label{ax:orbit-seeded}\nFor each generator $g\ + \ \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$:\n\\[\n \\boxed{%\n g \\in O_g, \\quad\\text{and}\\quad\n \ + \ \\rho^n(g) \\in O_g \\;\\text{for all}\\; n \\geq 0.\n }\n\\]\nThat is, $g$ belongs to its own orbit ray,\nand $\\\ + rho$ maps $O_g$ into~$O_g$.\n```\n\n## Proof / Justification\n\nThis item is an axiom. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 8\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch03-generation-cover.tex`\ + \ lines 104-116\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.K3_orbit_seeded`\n\n## Dependencies\n\n- Canonical: I.D01, I.D02, I.K1\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Each generator g in {alpha, pi, gamma, eta} seeds its orbit ray O_g = {rho^n(g) : n >= 0}.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001113.md b/_corpus_v3_items/cid001113.md new file mode 100644 index 000000000..5276b03cd --- /dev/null +++ b/_corpus_v3_items/cid001113.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001113/ +title: No-Jump / Cover (K4) +cid: cid001113 +primary_alias: AXM0006 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: rho is a successor within each orbit (no skipping); covers are primitive. +aliases: +- AXM0006 +- I.K4 +- no-jump-cover-k4 +- ax:no-jump +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005314 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 3 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No-Jump / Cover (K4)**\n\nrho is a successor within each orbit (no skipping); covers are primitive.\n\n# No-Jump\ + \ / Cover (K4)\n\n## Summary\n\nrho is a successor within each orbit (no skipping); covers are primitive.\n\n## Statement\n\ + \n```latex\n%\n\\label{ax:no-jump}\nFor each generator $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$ and\neach $n \\geq\ + \ 0$:\n\\[\n \\boxed{%\n \\rho\\bigl(\\rho^n(g)\\bigr) = \\rho^{n+1}(g).\n }\n\\]\nThe element $\\rho^{n+1}(g)$\ + \ is the \\emph{immediate successor}\nof $\\rho^n(g)$ in $O_g$, and there is no other element\nbetween them.\n```\n\n\ + ## Proof / Justification\n\nThis item is an axiom. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 9\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch03-generation-cover.tex`\ + \ lines 162-175\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.K4_no_jump`\n\n## Dependencies\n\n- Canonical: I.D02\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: rho is a successor within each orbit (no skipping); covers are primitive. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001114.md b/_corpus_v3_items/cid001114.md new file mode 100644 index 000000000..e485e820f --- /dev/null +++ b/_corpus_v3_items/cid001114.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001114/ +title: Beacon Non-Successor (K5) +cid: cid001114 +primary_alias: AXM0007 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: omega is NOT in the image of rho restricted to any orbit ray; omega is unreachable by finite iteration. +aliases: +- AXM0007 +- I.K5 +- beacon-non-successor-k5 +- ax:beacon-non-succ +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005315 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 2 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Beacon Non-Successor (K5)**\n\nomega is NOT in the image of rho restricted to any orbit ray; omega is unreachable\ + \ by finite iteration.\n\n# Beacon Non-Successor (K5)\n\n## Summary\n\nomega is NOT in the image of rho restricted to\ + \ any orbit ray; omega is unreachable by finite iteration.\n\n## Statement\n\n```latex\n\\label{ax:beacon-non-succ}\n\ + For every generator $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$\nand every $n \\geq 0$:\n\\[\n \\boxed{\\rho^n(g)\ + \ \\neq \\omega.}\n\\]\nNo finite iteration of $\\rho$ on a non-$\\omega$ generator reaches~$\\omega$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is an axiom. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 10\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch02-rho-omega.tex`\ + \ lines 158-166\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.K5_beacon_non_succ`\n\n## Dependencies\n\n- Canonical: I.D01, I.D02\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: omega is NOT in the image of rho restricted to any orbit ray; omega is unreachable by finite iteration. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001115.md b/_corpus_v3_items/cid001115.md new file mode 100644 index 000000000..0bc6c0663 --- /dev/null +++ b/_corpus_v3_items/cid001115.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001115/ +title: Object Closure (K6) +cid: cid001115 +primary_alias: AXM0008 +type: axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Obj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta; no other objects exist. +aliases: +- AXM0008 +- I.K6 +- object-closure-k6 +- ax:object-closure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005316 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Object Closure (K6)**\n\nObj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta; no other objects\ + \ exist.\n\n# Object Closure (K6)\n\n## Summary\n\nObj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta;\ + \ no other objects exist.\n\n## Statement\n\n```latex\n%\n\\label{ax:object-closure}\nEvery object of $\\tau$ belongs\ + \ to one of the five canonical sets:\n\\[\n \\boxed{%\n \\Obj(\\tau) = \\{\\omega\\}\n \\;\\cup\\; O_\\\ + alpha\n \\;\\cup\\; O_\\pi\n \\;\\cup\\; O_\\gamma\n \\;\\cup\\; O_\\eta.\n }\n\\]\nNo object\ + \ exists outside these five sets.\n```\n\n## Proof / Justification\n\nThis item is an axiom. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 11\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch04-closure-tau-zero.tex`\ + \ lines 49-63\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.K6_object_closure`\n\n## Dependencies\n\n- Canonical: I.D01, I.D02\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Obj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta; no other objects exist. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001116.md b/_corpus_v3_items/cid001116.md new file mode 100644 index 000000000..276f913c6 --- /dev/null +++ b/_corpus_v3_items/cid001116.md @@ -0,0 +1,185 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001116/ +title: Pentation Non-Injectivity +cid: cid001116 +primary_alias: LEM0002 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: No fifth orbit channel exists (K6), so pentation cannot be equipped with canonical injectivity within tau. +aliases: +- LEM0002 +- I.L01 +- pentation-non-injectivity +- lem:pentation-non-inj +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005330 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 8 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Pentation Non-Injectivity** + + + No fifth orbit channel exists (K6), so pentation cannot be equipped with canonical injectivity within tau. + + + # Pentation Non-Injectivity + + + ## Summary + + + No fifth orbit channel exists (K6), so pentation cannot be equipped with canonical injectivity within tau. + + + ## Statement + + + ```latex + + % + + \label{lem:pentation-non-inj} + + There is no orbit channel available to host a level-4 operation. + + Consequently, pentation cannot be equipped with + + a canonical injectivity proof within $\tau$. + + ``` + + + ## Proof / Justification + + + ```latex + + [Proof sketch] + + A canonical injectivity proof for the level-$k$ operation + + requires that the operation''s ``type information'''' + + be encoded in a dedicated orbit channel. + + At levels 0--3, the four channels + + $O_\alpha, O_\pi, O_\gamma, O_\eta$ + + provide this encoding. + + + At level~4, a fifth channel would be needed. + + But by $\KAxiom{6}$ (Object Closure), + + $\Obj(\tau) = \{\omega\} \cup O_\alpha \cup O_\pi + + \cup O_\gamma \cup O_\eta$. + + There is no fifth orbit ray. + + The beacon $\omega$ is a singleton and cannot host + + a countably infinite type channel. + + + Therefore, pentation lacks the structural scaffold + + for canonical injectivity. + + Without canonical injectivity, + + the operation cannot serve as a coordinate function + + in the normal form decomposition (Part~V), + + and the ladder terminates. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 44 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch08-iterator-ladder.tex` lines + 206-212 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Orbit.Ladder` + + - Name: `Tau.Orbit.pentation_non_inj` + + + ## Dependencies + + + - Canonical: I.D06, I.K6, I.D03 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: No fifth orbit channel exists (K6), so pentation cannot be equipped with canonical injectivity within tau. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001117.md b/_corpus_v3_items/cid001117.md new file mode 100644 index 000000000..0984753c2 --- /dev/null +++ b/_corpus_v3_items/cid001117.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001117/ +title: NF-Confluence +cid: cid001117 +primary_alias: LEM0003 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Normalization of programs is confluent: any two reduction paths yield the same normal form. Proved via termination + + local confluence + Newman''s Lemma.' +aliases: +- LEM0003 +- I.L02 +- nf-confluence +- lem:nf-confluence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005293 +- predicate: formalized_by + object: cid005296 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**NF-Confluence**\n\nNormalization of programs is confluent: any two reduction paths yield the same normal form.\ + \ Proved via termination + local confluence + Newman's Lemma.\n\n# NF-Confluence\n\n## Summary\n\nNormalization of programs\ + \ is confluent: any two reduction paths yield the same normal form. Proved via termination + local confluence + Newman's\ + \ Lemma.\n\n## Statement\n\n```latex\n%\n\\label{lem:nf-confluence}\nIf $P$ reduces to normal form $N_1$ by one reduction\ + \ sequence\nand to $N_2$ by another, then $N_1 = N_2$.\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\n\ + The reduction rules are:\n\\begin{enumerate}\n \\item $\\sigma_{s,s} \\to \\varepsilon$ (cancel identity swaps).\n\ + \ \\item $\\sigma_{t,s} \\circ \\sigma_{s,t} \\to \\varepsilon$\n (cancel inverse swaps).\n \\item $\\\ + sigma_{s,t} \\circ \\sigma_{r,s} \\to \\sigma_{r,t}$\n (compose transitive swaps).\n \\item $\\sigma_{s,t}\ + \ \\circ \\rho \\to \\rho \\circ \\sigma_{s,t}$\n (commute swaps past $\\rho$).\n\\end{enumerate}\nThese rules\ + \ are terminating (each application reduces\nthe total instruction count or moves swaps leftward)\nand locally confluent\ + \ (any two overlapping reductions\ncan be completed to a common result).\nBy Newman's Lemma (termination $+$ local confluence\ + \ $\\implies$\nglobal confluence), the normal form is unique.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 48\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch13-program-monoid.tex`\ + \ lines 101-106\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.ProgramMonoid`\n\ + - Name: `Tau.Denotation.nf_confluence`\n\n## Dependencies\n\n- Canonical: I.D14, I.T01\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Normalization of programs is confluent: any two reduction paths yield the same normal form. Proved via termination + + local confluence + Newman''s Lemma.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001118.md b/_corpus_v3_items/cid001118.md new file mode 100644 index 000000000..b973215f0 --- /dev/null +++ b/_corpus_v3_items/cid001118.md @@ -0,0 +1,111 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001118/ +title: No-Tie Determinism +cid: cid001118 +primary_alias: LEM0004 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Maximal tetration height C is uniquely determined. Tower atoms T(A,b1,c1) = T(A,b2,c2) implies (b1,c1) = (b2,c2). + Super-exponential growth of tetration prevents ties. +aliases: +- LEM0004 +- I.L03 +- no-tie-determinism +- lem:no-tie +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005290 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 5, Chapter 22 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No-Tie Determinism**\n\nMaximal tetration height C is uniquely determined. Tower atoms T(A,b1,c1) = T(A,b2,c2)\ + \ implies (b1,c1) = (b2,c2). Super-exponential growth of tetration prevents ties.\n\n# No-Tie Determinism\n\n## Summary\n\ + \nMaximal tetration height C is uniquely determined. Tower atoms T(A,b1,c1) = T(A,b2,c2) implies (b1,c1) = (b2,c2). Super-exponential\ + \ growth of tetration prevents ties.\n\n## Statement\n\n```latex\n%\n\\label{lem:no-tie}\nLet $\\underline{A} \\in \\\ + mathbb{P}_\\tau$\nand $X \\in \\tau\\text{-Idx}$ with $\\underline{A} \\mid X$.\n\\begin{enumerate}\n \\item The set\n\ + \ $\\{\\underline{c} \\geq \\underline{1} :\n \\underline{A} \\uparrow\\uparrow \\underline{c} \\mid\ + \ X\\}$\n has a well-defined maximum $\\underline{C}$.\n \\item If\n $T(\\underline{A}, \\underline{b_1},\ + \ \\underline{c_1})\n = T(\\underline{A}, \\underline{b_2}, \\underline{c_2})$\n with $\\underline{b_1},\ + \ \\underline{b_2} \\geq \\underline{1}$\n and $\\underline{c_1}, \\underline{c_2} \\geq \\underline{1}$,\n \ + \ then $\\underline{c_1} = \\underline{c_2}$\n and $\\underline{b_1} = \\underline{b_2}$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n```latex\n\\emph{Part~(1).}\nThe tetration $\\underline{A} \\uparrow\\uparrow \\underline{c}$\n\ + is strictly monotone in $\\underline{c}$\nfor $\\underline{A} \\geq \\underline{2}$\n(Proposition~\\ref{prop:tetration-injective}).\n\ + In particular,\n$\\underline{A} \\uparrow\\uparrow (\\underline{c}+\\underline{1})\n> \\underline{A} \\uparrow\\uparrow\ + \ \\underline{c}$\nfor all $\\underline{c} \\geq \\underline{1}$.\nSince $\\underline{A} \\uparrow\\uparrow \\underline{c}$\n\ + grows without bound and $X$ is fixed,\nthere exists a largest $\\underline{c}$ with\n$\\underline{A} \\uparrow\\uparrow\ + \ \\underline{c} \\leq X$.\nBeyond this bound,\n$\\underline{A} \\uparrow\\uparrow \\underline{c} > X$,\nso divisibility\ + \ fails.\nThe maximum of the finite set of valid $\\underline{c}$\nis well-defined by the well-ordering of $\\tau$-Idx\n\ + (Proposition~\\ref{prop:well-ordering}).\n\n\\emph{Part~(2).}\nSuppose\n$(\\underline{A} \\uparrow\\uparrow \\underline{c_1})^{\\\ + underline{b_1}}\n= (\\underline{A} \\uparrow\\uparrow \\underline{c_2})^{\\underline{b_2}}$.\n\nBoth sides have $\\underline{A}$\ + \ as their only prime factor\n(since $\\underline{A}$ is prime and tetration\n$\\underline{A} \\uparrow\\uparrow \\underline{c}$\n\ + is a power of $\\underline{A}$).\n\nLet $v_1$ be the $\\underline{A}$-adic valuation of the left side\nand $v_2$ that\ + \ of the right side.\nThen:\n\\begin{align*}\n v_1 &= \\underline{b_1} \\cdot\n v_{\\underline{A}}(\\underline{A}\ + \ \\uparrow\\uparrow \\underline{c_1}), \\\\\n v_2 &= \\underline{b_2} \\cdot\n v_{\\underline{A}}(\\underline{A}\ + \ \\uparrow\\uparrow \\underline{c_2}).\n\\end{align*}\n\nThe $\\underline{A}$-adic valuation of\n$\\underline{A} \\uparrow\\\ + uparrow \\underline{c}$ is:\n\\[\n v_{\\underline{A}}(\\underline{A} \\uparrow\\uparrow \\underline{c})\n = \\underline{A}\ + \ \\uparrow\\uparrow (\\underline{c} - \\underline{1})\n\\]\n(since $\\underline{A} \\uparrow\\uparrow \\underline{c}\n\ + = \\underline{A}^{\\underline{A} \\uparrow\\uparrow (\\underline{c}-\\underline{1})}$\nby the recursive definition of\ + \ tetration).\n\nSo equality gives:\n$\\underline{b_1} \\cdot\n(\\underline{A} \\uparrow\\uparrow (\\underline{c_1} -\ + \ \\underline{1}))\n= \\underline{b_2} \\cdot\n(\\underline{A} \\uparrow\\uparrow (\\underline{c_2} - \\underline{1}))$.\n\ + \nIf $\\underline{c_1} \\neq \\underline{c_2}$,\nsay $\\underline{c_1} < \\underline{c_2}$,\nthen\n$\\underline{A} \\\ + uparrow\\uparrow (\\underline{c_2} - \\underline{1})\n\\geq \\underline{A} \\uparrow\\uparrow \\underline{c_1}\n= \\underline{A}^{\\\ + underline{A} \\uparrow\\uparrow (\\underline{c_1} - \\underline{1})}\n> \\underline{A} \\cdot\n(\\underline{A} \\uparrow\\\ + uparrow (\\underline{c_1} - \\underline{1}))$\nfor $\\underline{A} \\geq \\underline{2}$.\nThis forces\n$\\underline{b_2}\ + \ < \\underline{b_1} / \\underline{A}\n< \\underline{b_1}$,\nbut the same argument with the growth rate\nof tetration\ + \ shows that $\\underline{b_2}$\nwould need to compensate for the\nsuper-exponential gap between\n$\\underline{A} \\uparrow\\\ + uparrow (\\underline{c_2} - \\underline{1})$\nand $\\underline{A} \\uparrow\\uparrow (\\underline{c_1} - \\underline{1})$,\n\ + which is impossible for finite $\\underline{b_1}$.\n\nMore precisely: if $\\underline{c_2} = \\underline{c_1} + \\underline{1}$,\n\ + then the right factor is\n$\\underline{A}^{\\underline{A} \\uparrow\\uparrow (\\underline{c_1}-1)}$\ntimes larger than\ + \ the left factor,\nwhile $\\underline{b_1}$ is at most polynomially larger\nthan $\\underline{b_2}$.\nFor $\\underline{c_2}\ + \ > \\underline{c_1} + \\underline{1}$,\nthe gap is even more extreme.\nIn all cases, equality is impossible.\n\nTherefore\ + \ $\\underline{c_1} = \\underline{c_2}$,\nand then\n$\\underline{b_1} \\cdot\n(\\underline{A} \\uparrow\\uparrow (\\underline{c_1}-\\\ + underline{1}))\n= \\underline{b_2} \\cdot\n(\\underline{A} \\uparrow\\uparrow (\\underline{c_1}-\\underline{1}))$\ngives\ + \ $\\underline{b_1} = \\underline{b_2}$\nby cancellation.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 57\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part05/ch22-no-tie.tex` lines\ + \ 70-88\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.NoTie`\n\ + - Name: `Tau.Coordinates.no_tie`\n\n## Dependencies\n\n- Canonical: I.D19c, I.P05, I.P07\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Maximal tetration height C is uniquely determined. Tower atoms T(A,b1,c1) = T(A,b2,c2) implies (b1,c1) = (b2,c2). + Super-exponential growth of tetration prevents ties. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001119.md b/_corpus_v3_items/cid001119.md new file mode 100644 index 000000000..f3db0b130 --- /dev/null +++ b/_corpus_v3_items/cid001119.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001119/ +title: Strict Remainder Descent +cid: cid001119 +primary_alias: LEM0005 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'D < X whenever X >= 2. Tower atom >= 2, so remainder D <= X/2 < X. Prime-stratum descent: largest prime of D is + strictly less than A.' +aliases: +- LEM0005 +- I.L04 +- strict-remainder-descent +- lem:remainder-descent +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 5, Chapter 23 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Strict Remainder Descent**\n\nD < X whenever X >= 2. Tower atom >= 2, so remainder D <= X/2 < X. Prime-stratum\ + \ descent: largest prime of D is strictly less than A.\n\n# Strict Remainder Descent\n\n## Summary\n\nD < X whenever X\ + \ >= 2. Tower atom >= 2, so remainder D <= X/2 < X. Prime-stratum descent: largest prime of D is strictly less than A.\n\ + \n## Statement\n\n```latex\n%\n\\label{lem:remainder-descent}\nLet $X \\geq \\underline{2}$ with ABCD encoding\n$X = T(\\\ + underline{A}, \\underline{B}, \\underline{C})\n\\cdot \\underline{D}$.\nThen:\n\\begin{enumerate}\n \\item $T(\\underline{A},\ + \ \\underline{B}, \\underline{C})\n \\geq \\underline{2}$.\n \\item $\\underline{D} \\leq X / \\underline{2}\ + \ < X$.\n \\item If $\\underline{D} > \\underline{1}$,\n the largest prime dividing $\\underline{D}$\n \ + \ is strictly less than $\\underline{A}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Part~(1).}\n\ + $\\underline{A} \\geq \\underline{2}$ (since $\\underline{A}$\nis prime).\n$\\underline{A} \\uparrow\\uparrow \\underline{1}\n\ + = \\underline{A} \\geq \\underline{2}$.\nFor $\\underline{C} \\geq \\underline{1}$,\n$\\underline{A} \\uparrow\\uparrow\ + \ \\underline{C}\n\\geq \\underline{A} \\geq \\underline{2}$.\nSince $\\underline{B} \\geq \\underline{1}$,\n$T(\\underline{A},\ + \ \\underline{B}, \\underline{C})\n= (\\underline{A} \\uparrow\\uparrow \\underline{C})^{\\underline{B}}\n\\geq \\underline{2}^{\\\ + underline{1}}\n= \\underline{2}$.\n\n\\emph{Part~(2).}\n$X = T(\\underline{A}, \\underline{B}, \\underline{C})\n\\cdot\ + \ \\underline{D}$\nwith $T(\\underline{A}, \\underline{B}, \\underline{C})\n\\geq \\underline{2}$.\nDividing both sides\ + \ by\n$T(\\underline{A}, \\underline{B}, \\underline{C})$:\n\\[\n \\underline{D}\n = \\frac{X}{T(\\underline{A},\ + \ \\underline{B}, \\underline{C})}\n \\leq \\frac{X}{\\underline{2}}\n < X.\n\\]\nThe strict inequality $X / \\\ + underline{2} < X$\nholds because $X \\geq \\underline{2} > \\underline{0}$.\n\n\\emph{Part~(3).}\nThis is Proposition~\\\ + ref{prop:nf-properties}(4),\nalready proved in Chapter~\\ref{ch:nf-encoding}:\nthe greedy peel extracts $\\underline{A}$\n\ + as the largest prime dividing $X$,\nand the tower atom accounts for all factors of\n$\\underline{A}$ at the extracted\ + \ level.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 58\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part05/ch23-remainder-descent.tex`\ + \ lines 35-50\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.Descent`\n\ + - Name: `Tau.Coordinates.div_lt_of_ge_two`\n\n## Dependencies\n\n- Canonical: I.D19d, I.P07\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'D < X whenever X >= 2. Tower atom >= 2, so remainder D <= X/2 < X. Prime-stratum descent: largest prime of D + is strictly less than A.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001120.md b/_corpus_v3_items/cid001120.md new file mode 100644 index 000000000..d25f781a3 --- /dev/null +++ b/_corpus_v3_items/cid001120.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001120/ +title: Growth Escape +cid: cid001120 +primary_alias: LEM0006 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Tetration escapes the primorial tower: for any tower depth d, there exists c such that 2^^c mod M_d != 2^^c. Quantitative + shadow of saturation — the 4th hyperoperation level produces values that outrun any finite primorial approximation.' +aliases: +- LEM0006 +- I.L05 +- growth-escape +- lem:growth-escape +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Growth Escape**\n\nTetration escapes the primorial tower: for any tower depth d, there exists c such that 2^^c\ + \ mod M_d != 2^^c. Quantitative shadow of saturation — the 4th hyperoperation level produces values that outrun any finite\ + \ primorial approximation.\n\n# Growth Escape\n\n## Summary\n\nTetration escapes the primorial tower: for any tower depth\ + \ d, there exists c such that 2^^c mod M_d != 2^^c. Quantitative shadow of saturation — the 4th hyperoperation level produces\ + \ values that outrun any finite primorial approximation.\n\n## Statement\n\n```latex\n%\n\\label{lem:growth-escape}\n\ + For any primorial depth~$d$, there exists\na tetration height~$c$ such that\n\\[\n {}^{\\underline{c}}\\underline{2}\ + \ \\;\\bmod\\; M_d\n \\;\\neq\\;\n {}^{\\underline{c}}\\underline{2}.\n\\]\nThat is, the tetration value cannot\ + \ be faithfully represented\nwithin the modulus~$M_d$.\n```\n\n## Proof / Justification\n\n```latex\nSince $M_d$ is finite\ + \ for every~$d$,\nand ${}^{\\underline{c}}\\underline{2}$ grows without bound\nas $c$ increases\n(it is eventually unbounded:\n\ + ${}^{\\underline{c}}\\underline{2} > N$ for any $N$,\nby induction on~$c$),\nthere exists~$c$ with ${}^{\\underline{c}}\\\ + underline{2} > M_d$.\nThen ${}^{\\underline{c}}\\underline{2} \\bmod M_d < M_d\n\\leq {}^{\\underline{c}}\\underline{2}$,\ + \ so they differ.\nConcretely: ${}^{\\underline{4}}\\underline{2} = \\underline{65536}\n> \\underline{2310} = M_5$.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 76\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 423-435\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.GrowthEscape`\n\ + - Name: `Tau.Denotation.GrowthEscape.growth_escape`\n\n## Dependencies\n\n- Canonical: I.D13\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tetration escapes the primorial tower: for any tower depth d, there exists c such that 2^^c mod M_d != 2^^c. + Quantitative shadow of saturation — the 4th hyperoperation level produces values that outrun any finite primorial approximation.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001121.md b/_corpus_v3_items/cid001121.md new file mode 100644 index 000000000..e41a88348 --- /dev/null +++ b/_corpus_v3_items/cid001121.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001121/ +title: Well-Foundedness of Membership +cid: cid001121 +primary_alias: LEM0007 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No infinite descending in_tau-chains: if A_1 in_tau A_2 in_tau ... with strict containment, the chain terminates + (divisibility is well-founded above 1).' +aliases: +- LEM0007 +- I.L06 +- well-foundedness-of-membership +- lem:membership-wf +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 34 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Well-Foundedness of Membership**\n\nNo infinite descending in_tau-chains: if A_1 in_tau A_2 in_tau ... with\ + \ strict containment, the chain terminates (divisibility is well-founded above 1).\n\n# Well-Foundedness of Membership\n\ + \n## Summary\n\nNo infinite descending in_tau-chains: if A_1 in_tau A_2 in_tau ... with strict containment, the chain\ + \ terminates (divisibility is well-founded above 1).\n\n## Statement\n\n```latex\n%\n\\label{lem:membership-wf}\nDefine\ + \ the \\textbf{strict membership} relation by:\n\\[\n \\underline{a} \\in_\\tau^\\circ \\underline{b}\n \\quad:\\\ + iff\\quad\n \\underline{a} \\in_\\tau \\underline{b}\n \\;\\text{ and }\\;\n \\underline{a} \\neq \\underline{b}.\n\ + \\]\nEquivalently,\n$\\underline{a} \\in_\\tau^\\circ \\underline{b}$\niff $\\underline{a}$ is a \\emph{proper} divisor\ + \ of $\\underline{b}$.\nThen $\\in_\\tau^\\circ$ is \\textbf{well-founded}:\nthere are no infinite strictly descending\ + \ chains\n$\\cdots \\in_\\tau^\\circ \\underline{x}_2\n\\in_\\tau^\\circ \\underline{x}_1\n\\in_\\tau^\\circ \\underline{x}_0$.\n\ + ```\n\n## Proof / Justification\n\n```latex\nIf $\\underline{a} \\in_\\tau^\\circ \\underline{b}$,\nthen $\\underline{a}\ + \ \\mid \\underline{b}$\nand $\\underline{a} \\neq \\underline{b}$.\nBy properties of divisibility in $\\mathbb{N}$,\n\ + this implies $\\underline{a} < \\underline{b}$\n(in the natural number ordering).\nThus, any descending chain under $\\\ + in_\\tau^\\circ$\ncorresponds to a strictly decreasing sequence of natural numbers.\nSince $\\mathbb{N}$ is well-ordered,\n\ + such a sequence cannot be infinite.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 82\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch34-bounded-powerset.tex` lines 181-200\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Powerset`\n- Name: `Tau.Sets.tau_strict_mem_wf`\n\ + \n## Dependencies\n\n- Canonical: I.D31, I.D33\n\n## Related Results\n\n_Generated by later projection phases._\n\n##\ + \ Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No infinite descending in_tau-chains: if A_1 in_tau A_2 in_tau ... with strict containment, the chain terminates + (divisibility is well-founded above 1).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001122.md b/_corpus_v3_items/cid001122.md new file mode 100644 index 000000000..564e3b065 --- /dev/null +++ b/_corpus_v3_items/cid001122.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001122/ +title: Tail Agreement Propagation +cid: cid001122 +primary_alias: LEM0008 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'If two tower-coherent stagewise functions agree at stage d0 for all inputs, they agree at ALL stages k <= d0. The + downward direction: agreement at a fine stage implies agreement at all coarser stages.' +aliases: +- LEM0008 +- I.L07 +- tail-agreement-propagation +- lem:tail-agreement-propagation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005303 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 52 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tail Agreement Propagation**\n\nIf two tower-coherent stagewise functions agree at stage d0 for all inputs,\ + \ they agree at ALL stages k <= d0. The downward direction: agreement at a fine stage implies agreement at all coarser\ + \ stages.\n\n# Tail Agreement Propagation\n\n## Summary\n\nIf two tower-coherent stagewise functions agree at stage d0\ + \ for all inputs, they agree at ALL stages k <= d0. The downward direction: agreement at a fine stage implies agreement\ + \ at all coarser stages.\n\n## Statement\n\n```latex\n%\n\\label{lem:tail-agreement-propagation}\nLet $T, S \\in \\mathrm{HolFun}$.\n\ + If $T$ and $S$ agree at depth $d_0$\n(Definition~\\ref{def:tail-agreement}),\nthen they agree at \\emph{all} depths.\n\ + That is:\n\\[\n \\boxed{%\n T \\sim_{d_0} S\n \\;\\;\\Longrightarrow\\;\\;\n \\forall\\, d \\geq 1,\\;\n \ + \ \\forall\\, t \\text{ with } \\mathrm{depth}(t) \\geq d,\\;\n \\forall\\, k \\leq d:\\;\n \\bigl(T(t)\\bigr)_k\ + \ = \\bigl(S(t)\\bigr)_k.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nBy induction on $d$, starting from $d_0$.\n\ + \n\\textbf{Base case} ($d = d_0$):\nthis is the hypothesis $T \\sim_{d_0} S$.\n\n\\textbf{Induction step} ($d \\to d +\ + \ 1$):\nassume $T$ and $S$ agree at depth $d$.\nWe show they agree at depth $d + 1$.\nTake any omega-tail $t$ of depth\ + \ $\\geq d + 1$.\n\nAt primorial stage $M_{d+1} = M_d \\cdot p_{d+1}$,\nthe Chinese Remainder Theorem gives the decomposition:\n\ + \\[\n \\mathbb{Z}/M_{d+1}\\mathbb{Z}\n \\;\\cong\\;\n \\mathbb{Z}/M_d\\mathbb{Z}\n \\;\\times\\;\n \\mathbb{Z}/p_{d+1}\\\ + mathbb{Z}.\n\\]\n\nBy the induction hypothesis,\n$T(t)$ and $S(t)$ agree on the\n$\\mathbb{Z}/M_d\\mathbb{Z}$ factor:\n\ + for all $k \\leq d$,\n$\\bigl(T(t)\\bigr)_k = \\bigl(S(t)\\bigr)_k$.\n\nIt remains to show agreement\non the $\\mathbb{Z}/p_{d+1}\\\ + mathbb{Z}$ factor.\nTower coherence\n(Definition~\\ref{def:tower-coherence}, I.D46)\ngives:\n\\[\n T(t) \\bmod M_d\ + \ \\;=\\; T(t \\bmod M_d),\n \\qquad\n S(t) \\bmod M_d \\;=\\; S(t \\bmod M_d).\n\\]\nThe reduction $t \\bmod M_d$\n\ + is an omega-tail of depth $d$.\nBy the induction hypothesis,\n$T(t \\bmod M_d) = S(t \\bmod M_d)$\nat all components $k\ + \ \\leq d$.\n\nNow the CRT coherence constraint\n(Theorem~\\ref{thm:crt-coherence}, I.T18)\nenters.\nA HolFun's action\ + \ on $\\mathbb{Z}/M_{d+1}\\mathbb{Z}$\nis determined by its actions on the two CRT factors\n$\\mathbb{Z}/M_d\\mathbb{Z}$\ + \ and $\\mathbb{Z}/p_{d+1}\\mathbb{Z}$.\nThe CRT coherence condition requires\nthat the $p_{d+1}$-factor of the output\n\ + is consistent with the already-determined lower stages:\nthe action on $\\mathbb{Z}/p_{d+1}\\mathbb{Z}$\nis constrained\ + \ by the requirement\nthat the full CRT reconstruction\nat stage $M_{d+1}$\nmust be compatible with the tower reduction\n\ + to stage $M_d$.\nSince $T$ and $S$ agree on the $\\mathbb{Z}/M_d\\mathbb{Z}$ factor,\nand both satisfy CRT coherence,\n\ + the $p_{d+1}$-factor of $T(t)$ and $S(t)$\nmust agree as well ---\nbecause any discrepancy on the $p_{d+1}$-factor\nwould\ + \ violate the CRT coherence constraint\nfor the tower reduction\n$M_{d+1} \\to M_d$.\n\nTherefore $\\bigl(T(t)\\bigr)_{d+1}\ + \ = \\bigl(S(t)\\bigr)_{d+1}$,\ncompleting the induction.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 119\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch52-identity-theorem.tex`\ + \ lines 195-212\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.IdentityTheorem`\n\ + - Name: `Tau.Holomorphy.tail_agree_propagation`\n\n## Dependencies\n\n- Canonical: I.D46\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'If two tower-coherent stagewise functions agree at stage d0 for all inputs, they agree at ALL stages k <= d0. + The downward direction: agreement at a fine stage implies agreement at all coarser stages.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001123.md b/_corpus_v3_items/cid001123.md new file mode 100644 index 000000000..567c5681c --- /dev/null +++ b/_corpus_v3_items/cid001123.md @@ -0,0 +1,209 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001123/ +title: CRT Extension +cid: cid001123 +primary_alias: LEM0009 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CRT Extension: tower coherence constrains function output via the reduce map. For tower-coherent f: reduce(f.b_fun(n,l), + k) = f.b_fun(n,k) for k <= l. Output at any stage is already reduced.' +aliases: +- LEM0009 +- I.L08 +- crt-extension +- lem:crt-extension +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005308 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 61 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT Extension** + + + CRT Extension: tower coherence constrains function output via the reduce map. For tower-coherent f: reduce(f.b_fun(n,l), + k) = f.b_fun(n,k) for k <= l. Output at any stage is already reduced. + + + # CRT Extension + + + ## Summary + + + CRT Extension: tower coherence constrains function output via the reduce map. For tower-coherent f: reduce(f.b_fun(n,l), + k) = f.b_fun(n,k) for k <= l. Output at any stage is already reduced. + + + ## Statement + + + ```latex + + % + + \label{lem:crt-extension} + + Let $K \subseteq \mathbb{L}$ be primordially thin, + + and let $f \in \mathrm{HolFun}$ + + (Definition~\ref{def:holfun}, I.D47) + + be defined on $\mathbb{L} \setminus K$. + + Then for each $t \in K$ and each depth $d \geq 1$, + + $f(t) \bmod M_d$ is uniquely determined + + by the restriction $f\!\big|_{\mathbb{L} \setminus K}$. + + ``` + + + ## Proof / Justification + + + ```latex + + Fix $t \in K$ and $d \geq 1$. + + Since $K$ is thin, + + $K_d$ misses two CRT directions --- + + say the $k_1$-th and $k_2$-th. + + + \textbf{Step 1: Lateral access.} + + Modify $t$ at stage $d$ + + so that its $k_1$-th CRT component equals + + the missing residue $r_{k_1}$. + + The resulting $t''$ satisfies $t'' \bmod M_d \notin K_d$, + + so $f(t'')$ is known. + + + \textbf{Step 2: CRT reconstruction.} + + Tower coherence + + (Definition~\ref{def:tower-coherence}, I.D46) gives + + $f(t) \bmod M_d + + = \mathrm{CRT}(f(t) \bmod p_1, \ldots, f(t) \bmod p_d)$. + + The CRT coherence constraint + + (Theorem~\ref{thm:crt-coherence}, I.T18) + + forces each $f(t) \bmod p_k$ + + to be determined by values of $f$ + + on omega-tails differing from $t$ only in the $k$-th direction. + + Two independent missing directions + + guarantee that every reconstruction path + + routes through $\mathbb{L} \setminus K$. + + + \textbf{Step 3: Uniqueness.} + + CRT reconstruction is unique --- + + $f(t) \bmod M_d$ is determined. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 152 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch61-thinness.tex` lines 132-142 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Holomorphy.Thinness` + + - Name: `Tau.Holomorphy.crt_extension_b` + + + ## Dependencies + + + - Canonical: I.D46, I.D29 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CRT Extension: tower coherence constrains function output via the reduce map. For tower-coherent f: reduce(f.b_fun(n,l), + k) = f.b_fun(n,k) for k <= l. Output at any stage is already reduced.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001124.md b/_corpus_v3_items/cid001124.md new file mode 100644 index 000000000..b5c90953a --- /dev/null +++ b/_corpus_v3_items/cid001124.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001124/ +title: Generator Distinctness +cid: cid001124 +primary_alias: PRP0002 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: All five generators are pairwise distinct (immediate from K1 strict order). +aliases: +- PRP0002 +- I.P01 +- generator-distinctness +- prop:gen-distinct +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005317 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 1 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Generator Distinctness**\n\nAll five generators are pairwise distinct (immediate from K1 strict order).\n\n\ + # Generator Distinctness\n\n## Summary\n\nAll five generators are pairwise distinct (immediate from K1 strict order).\n\ + \n## Statement\n\n```latex\n\\label{prop:gen-distinct}\nAll five generators are pairwise distinct:\n\\[\n |\\{\\alpha,\ + \ \\pi, \\gamma, \\eta, \\omega\\}| = 5.\n\\]\n```\n\n## Proof / Justification\n\n```latex\nA strict order is irreflexive:\ + \ $\\neg(x < x)$ for all $x$.\nIf any two generators were equal --- say $\\alpha = \\pi$ ---\nthen $\\KAxiom{1}$ would\ + \ give $\\alpha < \\alpha$,\ncontradicting irreflexivity.\nSince $\\KAxiom{1}$ places each pair of distinct generators\n\ + into a strict order relation,\nno two can be equal.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line\ + \ 12\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch01-five-generators.tex`\ + \ lines 242-248\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.gen_distinct`\n\n## Dependencies\n\n- Canonical: I.K1\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: All five generators are pairwise distinct (immediate from K1 strict order). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001125.md b/_corpus_v3_items/cid001125.md new file mode 100644 index 000000000..f5ab9a6e6 --- /dev/null +++ b/_corpus_v3_items/cid001125.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001125/ +title: rho Injectivity Per Orbit +cid: cid001125 +primary_alias: PRP0003 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: rho is injective on each orbit ray (from K4 no-jump/cover property). +aliases: +- PRP0003 +- I.P02 +- rho-injectivity-per-orbit +- prop:rho-injective +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005318 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 3 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**rho Injectivity Per Orbit**\n\nrho is injective on each orbit ray (from K4 no-jump/cover property).\n\n# rho\ + \ Injectivity Per Orbit\n\n## Summary\n\nrho is injective on each orbit ray (from K4 no-jump/cover property).\n\n## Statement\n\ + \n```latex\n%\n\\label{prop:rho-injective}\nFor each generator $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$\nand all $n,\ + \ m \\geq 0$:\n\\[\n \\rho\\bigl(\\rho^n(g)\\bigr) = \\rho\\bigl(\\rho^m(g)\\bigr)\n \\quad\\Longrightarrow\\quad\n\ + \ n = m.\n\\]\nThat is, $\\rho$ is injective on each orbit ray~$O_g$.\n```\n\n## Proof / Justification\n\n```latex\n\ + Suppose $\\rho(\\rho^n(g)) = \\rho(\\rho^m(g))$.\nBy $\\KAxiom{4}$:\n\\[\n \\rho(\\rho^n(g)) = \\rho^{n+1}(g),\n \ + \ \\qquad\n \\rho(\\rho^m(g)) = \\rho^{m+1}(g).\n\\]\nSo $\\rho^{n+1}(g) = \\rho^{m+1}(g)$.\nThe object $\\rho^k(g)$\ + \ is characterized within $O_g$\nby its depth~$k$ (the number of $\\rho$-applications from the seed).\nSince $\\KAxiom{4}$\ + \ ensures that each depth is occupied by exactly one element,\n$n+1 = m+1$ and hence $n = m$.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 13\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch03-generation-cover.tex`\ + \ lines 279-290\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Kernel.Axioms`\n\ + - Name: `Tau.Kernel.rho_injective`\n\n## Dependencies\n\n- Canonical: I.K4, I.D02\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: rho is injective on each orbit ray (from K4 no-jump/cover property). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001126.md b/_corpus_v3_items/cid001126.md new file mode 100644 index 000000000..c4e32c770 --- /dev/null +++ b/_corpus_v3_items/cid001126.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001126/ +title: Pairwise Disjointness of Orbits +cid: cid001126 +primary_alias: PRP0004 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The five sets {omega}, O_alpha, O_pi, O_gamma, O_eta are pairwise disjoint. +aliases: +- PRP0004 +- I.P03 +- pairwise-disjointness-of-orbits +- prop:orbit-disjoint +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005329 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 7 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Pairwise Disjointness of Orbits**\n\nThe five sets {omega}, O_alpha, O_pi, O_gamma, O_eta are pairwise disjoint.\n\ + \n# Pairwise Disjointness of Orbits\n\n## Summary\n\nThe five sets {omega}, O_alpha, O_pi, O_gamma, O_eta are pairwise\ + \ disjoint.\n\n## Statement\n\n```latex\n%\n\\label{prop:orbit-disjoint}\nThe five canonical sets\n\\[\n \\{\\omega\\\ + }, \\quad O_\\alpha, \\quad O_\\pi,\n \\quad O_\\gamma, \\quad O_\\eta\n\\]\nare pairwise disjoint.\n```\n\n## Proof\ + \ / Justification\n\n```latex\nWe must show ten pairwise disjointness claims.\nWe organize them into three groups.\n\n\ + \\medskip\n\\textbf{Group 1: $\\{\\omega\\}$ is disjoint from each orbit ray.}\n\nLet $g \\in \\{\\alpha, \\pi, \\gamma,\ + \ \\eta\\}$.\nSuppose $\\omega \\in O_g$, i.e., $\\omega = \\rho^n(g)$ for some $n \\geq 0$.\nIf $n = 0$, then $\\omega\ + \ = g$, contradicting\nthe generator distinctness (Proposition~\\ref{prop:gen-distinct}).\nIf $n \\geq 1$, then $\\rho^n(g)\ + \ = \\omega$,\ncontradicting $\\KAxiom{5}$ (Beacon Non-Successor).\nHence $\\{\\omega\\} \\cap O_g = \\emptyset$ for each\ + \ $g \\neq \\omega$.\n\n\\medskip\n\\textbf{Group 2: Orbit rays with different seeds are disjoint.}\n\nLet $g, h \\in\ + \ \\{\\alpha, \\pi, \\gamma, \\eta\\}$ with $g \\neq h$.\nSuppose $x \\in O_g \\cap O_h$,\nso $x = \\rho^n(g) = \\rho^m(h)$\ + \ for some $n, m \\geq 0$.\n\n\\emph{Case $n = m = 0$:}\nThen $g = h$, contradicting $g \\neq h$.\n\n\\emph{Case $n =\ + \ 0$, $m \\geq 1$:}\nThen $g = \\rho^m(h)$.\nSince $g$ is a generator and $\\rho^m(h)$ with $m \\geq 1$\nis a non-generator\ + \ orbit element,\nwe need to show these cannot be equal.\nBy $\\KAxiom{4}$, $\\rho^m(h)$ has depth $m$ in $O_h$.\nA generator\ + \ has depth $0$ in its own orbit.\nFor $\\rho^m(h)$ to equal a generator $g$ with $g \\neq h$,\n$\\rho$ would have to\ + \ map an $h$-seeded element to a $g$-seeded element,\nbut $\\KAxiom{3}$ asserts $\\rho(O_h) \\subseteq O_h$:\nthe ray\ + \ is closed under $\\rho$.\nSince $g \\notin O_h$ (because $g$ has depth~$0$\nand seed $g \\neq h$), this is a contradiction.\n\ + \n\\emph{Case $n \\geq 1$, $m = 0$:}\nSymmetric to the previous case.\n\n\\emph{Case $n \\geq 1$, $m \\geq 1$:}\nThen\ + \ $\\rho^n(g) = \\rho^m(h)$.\nBut $\\rho^n(g)$ is an element of $O_g$\n(by $\\KAxiom{3}$: $\\rho$ maps $O_g$ into $O_g$),\n\ + and $\\rho^m(h)$ is an element of $O_h$.\nBy the same closure argument,\nan element of $O_g$ cannot equal an element of\ + \ $O_h$\nunless the two rays overlap.\nSince the Lean formalization\nrepresents each object by its seed and depth,\ntwo\ + \ objects with different seeds are definitionally different.\nIn the axiomatic development,\nwe reason as follows:\n$\\\ + rho^n(g)$ has seed $g$ (it was generated from $g$ by $n$ applications of $\\rho$,\neach of which preserves the seed by\ + \ $\\KAxiom{3}$),\nand $\\rho^m(h)$ has seed $h$.\nSince $g \\neq h$, the objects are distinct.\n\n\\medskip\n\\textbf{Group\ + \ 3: Summary.}\nAll ten pairs are covered by Groups~1 and~2.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 19\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch07-ontic-closure.tex`\ + \ lines 42-51\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Generation`\n\ + - Name: `Tau.Orbit.orbit_disjoint`\n\n## Dependencies\n\n- Canonical: I.D05, I.K5, I.P01\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The five sets {omega}, O_alpha, O_pi, O_gamma, O_eta are pairwise disjoint. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001127.md b/_corpus_v3_items/cid001127.md new file mode 100644 index 000000000..150cb93e6 --- /dev/null +++ b/_corpus_v3_items/cid001127.md @@ -0,0 +1,194 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001127/ +title: Orbit Countability +cid: cid001127 +primary_alias: PRP0005 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Each orbit ray O_g is countably infinite (isomorphic to N as a totally ordered set). +aliases: +- PRP0005 +- I.P04 +- orbit-countability +- prop:orbit-countable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005328 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 7 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Orbit Countability** + + + Each orbit ray O_g is countably infinite (isomorphic to N as a totally ordered set). + + + # Orbit Countability + + + ## Summary + + + Each orbit ray O_g is countably infinite (isomorphic to N as a totally ordered set). + + + ## Statement + + + ```latex + + % + + \label{prop:orbit-countable} + + Each orbit ray $O_g$ (for $g \in \{\alpha, \pi, \gamma, \eta\}$) + + is countably infinite. + + More precisely, the map + + $\varphi_g : \mathbb{N} \to O_g$ defined by + + $\varphi_g(n) = \rho^n(g)$ is a bijection. + + ``` + + + ## Proof / Justification + + + ```latex + + \emph{Well-definedness.} + + $\varphi_g(n) = \rho^n(g) \in O_g$ + + by $\KAxiom{3}$ (orbit closure under $\rho$). + + + \emph{Surjectivity.} + + By definition, $O_g = \{\rho^n(g) : n \geq 0\}$, + + so every element of $O_g$ is in the image of $\varphi_g$. + + + \emph{Injectivity.} + + Suppose $\varphi_g(n) = \varphi_g(m)$, + + i.e., $\rho^n(g) = \rho^m(g)$. + + Without loss of generality, assume $n \leq m$. + + If $n = m$, we are done. + + If $n < m$, write $m = n + k$ with $k \geq 1$. + + Then $\rho^n(g) = \rho^{n+k}(g) = \rho^k(\rho^n(g))$. + + This says $\rho^n(g)$ is periodic under $\rho$ with period~$k$. + + But by $\KAxiom{4}$ (no-jump), $\rho$ strictly advances the depth: + + $\rho(\rho^j(g)) = \rho^{j+1}(g)$, and the depth increases + + from $j$ to $j+1$ at each step. + + Therefore $\rho^k(\rho^n(g))$ has depth $n+k > n$, + + while $\rho^n(g)$ has depth $n$. + + Since depth uniquely identifies an orbit element + + (by $\KAxiom{4}$), they cannot be equal. + + Contradiction. + + Hence $n = m$, and $\varphi_g$ is injective. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 20 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch07-ontic-closure.tex` lines 143-151 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Orbit.Countability` + + - Name: `Tau.Orbit.orbit_countable` + + + ## Dependencies + + + - Canonical: I.D05, I.P02 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Each orbit ray O_g is countably infinite (isomorphic to N as a totally ordered set). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001128.md b/_corpus_v3_items/cid001128.md new file mode 100644 index 000000000..58def1d71 --- /dev/null +++ b/_corpus_v3_items/cid001128.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001128/ +title: Tetration Injectivity +cid: cid001128 +primary_alias: PRP0006 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Tetration is injective on the canonical domain: distinct inputs yield distinct towers.' +aliases: +- PRP0006 +- I.P05 +- tetration-injectivity +- prop:tetration-injective +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005292 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tetration Injectivity**\n\nTetration is injective on the canonical domain: distinct inputs yield distinct towers.\n\ + \n# Tetration Injectivity\n\n## Summary\n\nTetration is injective on the canonical domain: distinct inputs yield distinct\ + \ towers.\n\n## Statement\n\n```latex\n%\n\\label{prop:tetration-injective}\nFor fixed $\\underline{a} \\geq \\underline{2}$,\n\ + the map $\\underline{c} \\mapsto {}^{\\underline{c}}\\underline{a}$\nis injective on $\\tau$-Idx:\n\\[\n {}^{\\underline{c_1}}\\\ + underline{a}\n = {}^{\\underline{c_2}}\\underline{a}\n \\quad\\Longrightarrow\\quad\n \\underline{c_1} = \\underline{c_2}.\n\ + \\]\n```\n\n## Proof / Justification\n\n```latex\nWe prove by induction on $\\max(c_1, c_2)$\nthat the tetration function\ + \ is strictly increasing\nin the tower height $c$ (for fixed base $a \\geq 2$),\nfrom which injectivity follows.\n\n\\\ + emph{Base case.}\n${}^{\\underline{0}}\\underline{a} = \\underline{1}$\nand ${}^{\\underline{1}}\\underline{a} = \\underline{a}\ + \ \\geq \\underline{2}$.\nSo ${}^{\\underline{0}}\\underline{a} < {}^{\\underline{1}}\\underline{a}$.\n\n\\emph{Inductive\ + \ step.}\nAssume ${}^{\\underline{c}}\\underline{a}\n< {}^{\\underline{c+1}}\\underline{a}$ for some $c$.\nThen\n${}^{\\\ + underline{c+2}}\\underline{a}\n= \\underline{a}^{({}^{\\underline{c+1}}\\underline{a})}\n> \\underline{a}^{({}^{\\underline{c}}\\\ + underline{a})}\n= {}^{\\underline{c+1}}\\underline{a}$,\nwhere the inequality uses the fact that\n$x \\mapsto \\underline{a}^x$\ + \ is strictly increasing\nfor $\\underline{a} \\geq \\underline{2}$.\n\nStrict monotonicity implies injectivity.\n```\n\ + \n## Source Context\n\n- Registry source: `book-01.jsonl` line 31\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 151-163\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.tetration_injective`\n\n## Dependencies\n\n- Canonical: I.D13\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tetration is injective on the canonical domain: distinct inputs yield distinct towers.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001129.md b/_corpus_v3_items/cid001129.md new file mode 100644 index 000000000..ff726395e --- /dev/null +++ b/_corpus_v3_items/cid001129.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001129/ +title: Arithmetic Laws +cid: cid001129 +primary_alias: PRP0007 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'tau-Idx with index addition and multiplication forms a commutative semiring: commutativity, associativity, distributivity, + identity elements.' +aliases: +- PRP0007 +- I.P06 +- arithmetic-laws +- prop:arithmetic-laws +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Arithmetic Laws**\n\ntau-Idx with index addition and multiplication forms a commutative semiring: commutativity,\ + \ associativity, distributivity, identity elements.\n\n# Arithmetic Laws\n\n## Summary\n\ntau-Idx with index addition\ + \ and multiplication forms a commutative semiring: commutativity, associativity, distributivity, identity elements.\n\n\ + ## Statement\n\n```latex\n%\n\\label{prop:arithmetic-laws}\nThe structure $(\\tau\\text{-Idx}, +, \\times, \\underline{0},\ + \ \\underline{1})$\nis a commutative semiring:\n\\begin{itemize}\n \\item $(\\tau\\text{-Idx}, +, \\underline{0})$\n\ + \ is a commutative monoid.\n \\item $(\\tau\\text{-Idx}, \\times, \\underline{1})$\n is a commutative\ + \ monoid.\n \\item Multiplication distributes over addition.\n \\item $\\underline{0} \\times \\underline{n} = \\\ + underline{0}$\n (absorption).\n\\end{itemize}\n```\n\n## Proof / Justification\n\n```latex\nFollows from\nPropositions~\\\ + ref{prop:add-properties} and~\\ref{prop:mul-properties}.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 47\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex`\ + \ lines 226-240\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Arithmetic`\n\ + - Name: `Tau.Denotation.arithmetic_laws`\n\n## Dependencies\n\n- Canonical: I.D10, I.D11\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'tau-Idx with index addition and multiplication forms a commutative semiring: commutativity, associativity, distributivity, + identity elements.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001130.md b/_corpus_v3_items/cid001130.md new file mode 100644 index 000000000..8557249a7 --- /dev/null +++ b/_corpus_v3_items/cid001130.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001130/ +title: Well-Ordering of Obj(tau) +cid: cid001130 +primary_alias: PRP0008 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: '(Obj(tau), <_tau) is a well-ordering of order type omega*4+1: four copies of omega (one per orbit ray) followed + by the beacon omega.' +aliases: +- PRP0008 +- I.P07 +- well-ordering-of-obj-tau +- prop:well-ordering +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005294 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Well-Ordering of Obj(tau)**\n\n(Obj(tau), <_tau) is a well-ordering of order type omega*4+1: four copies of\ + \ omega (one per orbit ray) followed by the beacon omega.\n\n# Well-Ordering of Obj(tau)\n\n## Summary\n\n(Obj(tau), <_tau)\ + \ is a well-ordering of order type omega*4+1: four copies of omega (one per orbit ray) followed by the beacon omega.\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:well-ordering}\n$(\\Obj(\\tau), <_\\tau)$ is a well-ordering.\nThat is:\n\\\ + begin{enumerate}\n \\item $<_\\tau$ is a strict total order on $\\Obj(\\tau)$.\n \\item Every non-empty subset of\ + \ $\\Obj(\\tau)$\n has a $<_\\tau$-minimum element.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\emph{Total order.}\n$<_\\tau$ is irreflexive (distinct seeds or distinct depths),\ntransitive (inherited from the generator\ + \ order and $\\mathbb{N}$),\nand trichotomous (every two objects are comparable:\neither their seeds differ, giving an\ + \ order from $\\KAxiom{1}$,\nor their seeds agree and their depths are natural numbers,\ngiving an order from $\\mathbb{N}$).\n\ + \n\\emph{Well-ordering.}\nLet $S \\subseteq \\Obj(\\tau)$ be non-empty.\nAmong the seeds of elements in $S$,\nlet $g_0$\ + \ be the $\\KAxiom{1}$-minimum.\nAmong elements of $S$ with seed $g_0$,\nthe minimum depth exists\n(a non-empty subset\ + \ of $\\mathbb{N}$ has a minimum).\nThe corresponding element is the $<_\\tau$-minimum of $S$.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 50\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch15-denotational-order.tex`\ + \ lines 88-98\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Order`\n\ + - Name: `Tau.Denotation.well_ordering`\n\n## Dependencies\n\n- Canonical: I.D16a, I.T01\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '(Obj(tau), <_tau) is a well-ordering of order type omega*4+1: four copies of omega (one per orbit ray) followed + by the beacon omega.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001131.md b/_corpus_v3_items/cid001131.md new file mode 100644 index 000000000..884bea280 --- /dev/null +++ b/_corpus_v3_items/cid001131.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001131/ +title: Dimension Theorem (dim_tau = 4) +cid: cid001131 +primary_alias: PRP0009 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'dim_tau = 4: the four ABCD coordinates are pairwise independent (sufficiency) and no three suffice (necessity). + Dimension is earned from arithmetic hierarchy, not postulated.' +aliases: +- PRP0009 +- I.P08 +- dimension-theorem-dim-tau-4 +- prop:dim-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 20 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Dimension Theorem (dim_tau = 4)**\n\ndim_tau = 4: the four ABCD coordinates are pairwise independent (sufficiency)\ + \ and no three suffice (necessity). Dimension is earned from arithmetic hierarchy, not postulated.\n\n# Dimension Theorem\ + \ (dim_tau = 4)\n\n## Summary\n\ndim_tau = 4: the four ABCD coordinates are pairwise independent (sufficiency) and no\ + \ three suffice (necessity). Dimension is earned from arithmetic hierarchy, not postulated.\n\n## Statement\n\n```latex\n\ + %\n\\label{prop:dim-tau}\nThe dimension of the ABCD coordinate system is exactly $4$:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Sufficiency:}\n The four coordinates $(A, B, C, D)$\n suffice to encode every object in $\\\ + Obj(\\tau)$.\n \\item \\textbf{Necessity:}\n No three of the four coordinates\n suffice to encode\ + \ all objects.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Sufficiency.}\nThis is the content\ + \ of the NF existence theorem\n(Proposition~\\ref{prop:nf-existence}):\nevery object has an ABCD encoding.\n\n\\emph{Necessity.}\n\ + We show that omitting any single coordinate\ncauses information loss.\n\nOmit $D$:\nObjects $\\underline{p}$ and $\\underline{p}\ + \ \\cdot \\underline{q}$\n(for primes $\\underline{p} > \\underline{q}$)\nshare $(A, B, C) = (\\underline{p}, \\underline{1},\ + \ \\underline{1})$\nbut have $D = \\underline{1}$ and $D = \\underline{q}$ respectively.\nWithout $D$, they are conflated.\n\ + \nOmit $A$:\nObjects $\\underline{2}$ and $\\underline{3}$\nshare $(B, C, D) = (\\underline{1}, \\underline{1}, \\underline{1})$\n\ + but have $A = \\underline{2}$ and $A = \\underline{3}$.\nWithout $A$, they are conflated.\n\nOmit $B$:\nObjects $\\underline{p}$\ + \ and $\\underline{p}^2$\n(for any prime $\\underline{p}$)\nshare $(A, C, D) = (\\underline{p}, \\underline{1}, \\underline{1})$\n\ + but have $B = \\underline{1}$ and $B = \\underline{2}$.\nWithout $B$, they are conflated.\n\nOmit $C$:\nObjects $\\underline{p}$\ + \ and\n$\\underline{p} \\uparrow\\uparrow \\underline{2}\n= \\underline{p}^{\\underline{p}}$\nshare the same prime $A\ + \ = \\underline{p}$\nbut have $C = \\underline{1}$ and $C = \\underline{2}$.\nThe exponent changes from $B = \\underline{1}$\n\ + to $B = \\underline{1}$\n(with the tetration height carrying the new structure),\nso $(A, B, D) = (\\underline{p}, \\\ + underline{1}, \\underline{1})$\nfor both.\nWithout $C$, they are conflated.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 56\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch20-dimension-fibration.tex`\ + \ lines 108-120\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.ABCD`\n\ + - Name: `Tau.Coordinates.dim_tau_eq_four`\n\n## Dependencies\n\n- Canonical: I.D17, I.D06\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'dim_tau = 4: the four ABCD coordinates are pairwise independent (sufficiency) and no three suffice (necessity). + Dimension is earned from arithmetic hierarchy, not postulated.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001132.md b/_corpus_v3_items/cid001132.md new file mode 100644 index 000000000..021e2ecba --- /dev/null +++ b/_corpus_v3_items/cid001132.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001132/ +title: Metric Inequality +cid: cid001132 +primary_alias: PRP0010 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ell_spine(x) <= ell_DAG(x) <= ell_occ(x). Three canonical complexity metrics: spine length (radial depth), DAG size + (memoized work), occurrence size (naive work).' +aliases: +- PRP0010 +- I.P09 +- metric-inequality +- prop:metric-inequality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 20 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Metric Inequality**\n\nell_spine(x) <= ell_DAG(x) <= ell_occ(x). Three canonical complexity metrics: spine\ + \ length (radial depth), DAG size (memoized work), occurrence size (naive work).\n\n# Metric Inequality\n\n## Summary\n\ + \nell_spine(x) <= ell_DAG(x) <= ell_occ(x). Three canonical complexity metrics: spine length (radial depth), DAG size\ + \ (memoized work), occurrence size (naive work).\n\n## Statement\n\n```latex\n%\n\\label{prop:metric-inequality}\nFor\ + \ every $x \\in \\tau\\text{-Idx}$:\n\\[\n \\boxed{\\ell_{\\mathrm{spine}}(x)\n \\;\\leq\\;\n \\ell_{\\mathrm{DAG}}(x)\n\ + \ \\;\\leq\\;\n \\ell_{\\mathrm{occ}}(x).}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThe spine visits only\ + \ the $D$-children,\nwhich form a subpath of the DAG.\nSo $\\ell_{\\mathrm{spine}} \\leq \\ell_{\\mathrm{DAG}}$.\nThe\ + \ DAG has at most as many nodes\nas the quadtree has occurrences,\nsince the DAG identifies repeated nodes.\nSo $\\ell_{\\\ + mathrm{DAG}} \\leq \\ell_{\\mathrm{occ}}$.\n\nThe first inequality is strict\nwhenever $x$ has nontrivial index children\n\ + (i.e., when some coordinate is not $\\alpha_1$).\nThe second is strict whenever any index\nappears more than once in the\ + \ full quadtree.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 63\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch20-dimension-fibration.tex`\ + \ lines 447-458\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.ABCD`\n\ + - Name: `Tau.Coordinates.metric_inequality_check`\n\n## Dependencies\n\n- Canonical: I.D24\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ell_spine(x) <= ell_DAG(x) <= ell_occ(x). Three canonical complexity metrics: spine length (radial depth), DAG + size (memoized work), occurrence size (naive work).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001133.md b/_corpus_v3_items/cid001133.md new file mode 100644 index 000000000..4b77a2547 --- /dev/null +++ b/_corpus_v3_items/cid001133.md @@ -0,0 +1,158 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001133/ +title: Membership Decidability +cid: cid001133 +primary_alias: PRP0011 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B.' +aliases: +- PRP0011 +- I.P10 +- membership-decidability +- prop:membership-decidable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005343 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 32 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Membership Decidability** + + + The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B. + + + # Membership Decidability + + + ## Summary + + + The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau B. + + + ## Statement + + + ```latex + + % + + \label{prop:membership-decidable} + + The $\tau$-membership relation is \textbf{decidable}: + + given $\underline{a}, \underline{b} \in \tau\text{-Idx}$, + + there is a computable procedure that determines + + whether $\underline{a} \in_\tau \underline{b}$. + + ``` + + + ## Proof / Justification + + + ```latex + + $\underline{a} \in_\tau \underline{b}$ iff $\underline{a} \mid \underline{b}$ + + iff $\underline{b} \bmod \underline{a} = \underline{0}$. + + The modular reduction $\underline{b} \bmod \underline{a}$ + + is computable on $\tau$-Idx + + (it requires only the Euclidean division algorithm, + + which uses only the earned arithmetic of Part~III). + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 78 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch32-membership-divisibility.tex` + lines 168-175 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Sets.Membership` + + - Name: `Tau.Sets.instDecidableTauMem` + + + ## Dependencies + + + - Canonical: I.D31, I.D19a + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The tau-membership relation is decidable: given A, B in tau-Idx, we can computably determine whether A in_tau + B.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001134.md b/_corpus_v3_items/cid001134.md new file mode 100644 index 000000000..a2d4b4ea8 --- /dev/null +++ b/_corpus_v3_items/cid001134.md @@ -0,0 +1,102 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001134/ +title: Distributive Lattice +cid: cid001134 +primary_alias: PRP0012 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: (tau-Idx, in_tau) forms a distributive lattice under gcd (meet) and lcm (join). Distributivity follows from the FTA. +aliases: +- PRP0012 +- I.P11 +- distributive-lattice +- prop:distributive-lattice +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005344 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 33 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Distributive Lattice**\n\n(tau-Idx, in_tau) forms a distributive lattice under gcd (meet) and lcm (join). Distributivity\ + \ follows from the FTA.\n\n# Distributive Lattice\n\n## Summary\n\n(tau-Idx, in_tau) forms a distributive lattice under\ + \ gcd (meet) and lcm (join). Distributivity follows from the FTA.\n\n## Statement\n\n```latex\n%\n\\label{prop:distributive-lattice}\n\ + The triple $(\\tau\\text{-Idx}, \\cap_\\tau, \\cup_\\tau)$\nforms a \\textbf{distributive lattice}.\nIn particular, the\ + \ following identities hold\nfor all $\\underline{a}, \\underline{b}, \\underline{c} \\in \\tau\\text{-Idx}$:\n\\begin{enumerate}\n\ + \ \\item \\textbf{Commutativity:}\n \\[\n \\underline{a} \\cup_\\tau \\underline{b}\n \ + \ = \\underline{b} \\cup_\\tau \\underline{a},\n \\quad\n \\underline{a} \\cap_\\tau \\\ + underline{b}\n = \\underline{b} \\cap_\\tau \\underline{a}.\n \\]\n\n \\item \\textbf{Associativity:}\n\ + \ \\[\n (\\underline{a} \\cup_\\tau \\underline{b}) \\cup_\\tau \\underline{c}\n =\ + \ \\underline{a} \\cup_\\tau (\\underline{b} \\cup_\\tau \\underline{c}),\n \\]\n \\[\n \ + \ (\\underline{a} \\cap_\\tau \\underline{b}) \\cap_\\tau \\underline{c}\n = \\underline{a} \\cap_\\tau\ + \ (\\underline{b} \\cap_\\tau \\underline{c}).\n \\]\n\n \\item \\textbf{Idempotence:}\n \\[\n \ + \ \\underline{a} \\cup_\\tau \\underline{a}\n = \\underline{a},\n \\quad\n \ + \ \\underline{a} \\cap_\\tau \\underline{a}\n = \\underline{a}.\n \\]\n\n \\item \\textbf{Absorption:}\n\ + \ \\[\n \\underline{a} \\cup_\\tau (\\underline{a} \\cap_\\tau \\underline{b})\n =\ + \ \\underline{a},\n \\quad\n \\underline{a} \\cap_\\tau (\\underline{a} \\cup_\\tau \\underline{b})\n\ + \ = \\underline{a}.\n \\]\n\n \\item \\textbf{Distributivity:}\n \\[\n \ + \ \\underline{a} \\cap_\\tau (\\underline{b} \\cup_\\tau \\underline{c})\n = (\\underline{a} \\cap_\\tau\ + \ \\underline{b})\n \\cup_\\tau (\\underline{a} \\cap_\\tau \\underline{c}),\n \\]\n \\\ + [\n \\underline{a} \\cup_\\tau (\\underline{b} \\cap_\\tau \\underline{c})\n = (\\underline{a}\ + \ \\cup_\\tau \\underline{b})\n \\cap_\\tau (\\underline{a} \\cup_\\tau \\underline{c}).\n \\]\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nAll five properties follow from the definitions\n$\\cup_\\\ + tau = \\mathrm{lcm}$ and $\\cap_\\tau = \\mathrm{gcd}$,\ncombined with the formula:\n\\[\n \\mathrm{lcm}(a, b) = \\\ + prod_{p} p^{\\max(v_p(a), v_p(b))},\n \\quad\n \\mathrm{gcd}(a, b) = \\prod_{p} p^{\\min(v_p(a), v_p(b))},\n\\]\n\ + where the products run over all primes\n(Fundamental Theorem of Arithmetic, Theorem~\\ref{thm:fta-tau-idx}).\n\n\\emph{Commutativity\ + \ and associativity}\nfollow from commutativity and associativity\nof $\\max$ and $\\min$.\n\n\\emph{Idempotence:}\n$\\\ + max(v_p(a), v_p(a)) = v_p(a)$\nand $\\min(v_p(a), v_p(a)) = v_p(a)$.\n\n\\emph{Absorption:}\nFor union absorption,\n\\\ + begin{align*}\n \\underline{a} \\cup_\\tau (\\underline{a} \\cap_\\tau \\underline{b})\n &= \\prod_{p} p^{\\max(v_p(a),\ + \ \\min(v_p(a), v_p(b)))}\\\\\n &= \\prod_{p} p^{v_p(a)}\n = \\underline{a},\n\\end{align*}\nsince $\\max(x, \\\ + min(x, y)) = x$ for all $x, y \\in \\mathbb{R}$.\nIntersection absorption is dual.\n\n\\emph{Distributivity:}\nFor the\ + \ first distributive law,\n\\begin{align*}\n &\\underline{a} \\cap_\\tau (\\underline{b} \\cup_\\tau \\underline{c})\\\ + \\\n &= \\prod_{p} p^{\\min(v_p(a), \\max(v_p(b), v_p(c)))}\\\\\n &= \\prod_{p} p^{\\max(\\min(v_p(a), v_p(b)),\n\ + \ \\min(v_p(a), v_p(c)))}\n \\quad \\text{(by distributivity of $\\min$ over $\\max$)}\\\\\n &= (\\underline{a}\ + \ \\cap_\\tau \\underline{b})\n \\cup_\\tau (\\underline{a} \\cap_\\tau \\underline{c}).\n\\end{align*}\nThe second\ + \ distributive law is dual\n(using distributivity of $\\max$ over $\\min$).\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 80\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch33-set-operations.tex`\ + \ lines 195-252\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Operations`\n\ + - Name: `Tau.Sets.tau_inter_distrib_union`\n\n## Dependencies\n\n- Canonical: I.D32, I.D31\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: (tau-Idx, in_tau) forms a distributive lattice under gcd (meet) and lcm (join). Distributivity follows from the + FTA. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001135.md b/_corpus_v3_items/cid001135.md new file mode 100644 index 000000000..940885a90 --- /dev/null +++ b/_corpus_v3_items/cid001135.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001135/ +title: Countability of Set(tau) +cid: cid001135 +primary_alias: PRP0013 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: '|Set(tau)| = aleph_0. All tau-sets are countable. No Cantor diagonal, no uncountable sets, no non-measurable sets. + Everything constructive.' +aliases: +- PRP0013 +- I.P12 +- countability-of-set-tau +- prop:set-countable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005349 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 35 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Countability of Set(tau)**\n\n|Set(tau)| = aleph_0. All tau-sets are countable. No Cantor diagonal, no uncountable\ + \ sets, no non-measurable sets. Everything constructive.\n\n# Countability of Set(tau)\n\n## Summary\n\n|Set(tau)| = aleph_0.\ + \ All tau-sets are countable. No Cantor diagonal, no uncountable sets, no non-measurable sets. Everything constructive.\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:set-countable}\nThe collection of all $\\tau$-sets is \\textbf{countable}:\n\ + \\[\n |\\mathrm{Set}(\\tau)|\n \\;=\\;\n |\\tau\\text{-Idx}|\n \\;=\\;\n \\aleph_0.\n\\]\n```\n\n## Proof\ + \ / Justification\n\n```latex\nBy definition (Chapter~\\ref{ch:membership-divisibility}),\na $\\tau$-set is an element\ + \ of $\\tau$-Idx\nequipped with the $\\in_\\tau$ relation.\nThe underlying objects are the natural numbers\n$\\underline{1},\ + \ \\underline{2}, \\underline{3}, \\ldots$\n(after identification with $\\mathbb{N}$ via Chapter~\\ref{ch:set-operations}).\n\ + Since $\\tau$-Idx is countably infinite\n(it is in bijection with $\\mathbb{N}$),\nthe collection of all $\\tau$-sets\ + \ is countable.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 83\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch35-tau-set-universe.tex`\ + \ lines 41-52\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.Universe`\n\ + - Name: `Tau.Sets.tau_set_countable`\n\n## Dependencies\n\n- Canonical: I.D31, I.D33, I.L06\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '|Set(tau)| = aleph_0. All tau-sets are countable. No Cantor diagonal, no uncountable sets, no non-measurable + sets. Everything constructive.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001136.md b/_corpus_v3_items/cid001136.md new file mode 100644 index 000000000..780e021a9 --- /dev/null +++ b/_corpus_v3_items/cid001136.md @@ -0,0 +1,105 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001136/ +title: Boolean Recovery +cid: cid001136 +primary_alias: PRP0014 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Boolean logic as a quotient of Truth4: collapsing B and N recovers classical two-valued logic. The forgetful map + Truth4 -> Bool is a lattice homomorphism.' +aliases: +- PRP0014 +- I.P13 +- boolean-recovery +- prop:boolean-recovery +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005321 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 48 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boolean Recovery**\n\nBoolean logic as a quotient of Truth4: collapsing B and N recovers classical two-valued\ + \ logic. The forgetful map Truth4 -> Bool is a lattice homomorphism.\n\n# Boolean Recovery\n\n## Summary\n\nBoolean logic\ + \ as a quotient of Truth4: collapsing B and N recovers classical two-valued logic. The forgetful map Truth4 -> Bool is\ + \ a lattice homomorphism.\n\n## Statement\n\n```latex\n%\n\\label{prop:boolean-recovery}\nLet $\\mathrm{val} : \\mathrm{Prop}_\\\ + tau \\to \\mathrm{Truth4}$\nbe a Truth4 valuation,\nand let $S \\subseteq \\mathrm{Prop}_\\tau$\nbe a set of propositions.\n\ + Then the following are equivalent:\n\\begin{enumerate}\n \\item \\textbf{Boolean sufficiency.}\n For every\ + \ $P \\in S$,\n $\\mathrm{val}(P) \\in \\{\\mathsf{T}, \\mathsf{F}\\}$\n (no intermediate truth values\ + \ occur).\n \\item \\textbf{Spectral decisiveness.}\n For every $P \\in S$,\n the spectral decomposition\n\ + \ of the predicate yields\n identical verdicts in both sectors:\n either both sectors confirm\ + \ $P$\n and neither confirms $\\lnot P$ (giving $\\mathsf{T}$),\n or both sectors confirm $\\lnot P$\n\ + \ and neither confirms $P$ (giving $\\mathsf{F}$).\n \\item \\textbf{Lossless collapse.}\n The forgetful\ + \ map $f$ restricted to\n the image $\\mathrm{val}(S)$\n is an \\emph{isomorphism}:\n $f|_{\\\ + mathrm{val}(S)} : \\mathrm{val}(S)\n \\xrightarrow{\\sim} \\{f(v) : v \\in \\mathrm{val}(S)\\}$.\n No\ + \ information is lost\n by collapsing Truth4 to Bool on $S$.\n \\item \\textbf{Negation compatibility.}\n\ + \ For every $P \\in S$,\n $f(\\lnot \\mathrm{val}(P))\n = \\lnot f(\\mathrm{val}(P))$:\n \ + \ negation commutes with the forgetful map.\n\\end{enumerate}\nUnder any (hence all) of these conditions,\nthe\ + \ full logic on $S$\nis captured by classical Boolean logic:\n\\[\n \\boxed{%\n \\mathrm{Truth4}|_S \\;\\cong\\\ + ; \\mathrm{Bool}|_S.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\n$(1) \\Rightarrow (2)$:\nIf $\\mathrm{val}(P)\ + \ = \\mathsf{T}$,\nthen by the definition of Truth4\n(Definition~\\ref{def:truth4}),\nboth the $B$-sector and the $C$-sector\n\ + confirm $P$ and neither confirms $\\lnot P$.\nIf $\\mathrm{val}(P) = \\mathsf{F}$,\nboth sectors confirm $\\lnot P$\n\ + and neither confirms $P$.\nIn both cases, the sectors agree ---\nthis is spectral decisiveness.\n\n$(2) \\Rightarrow (1)$:\n\ + Spectral decisiveness means\nthe two sectors never split:\nthey either both confirm ($\\mathsf{T}$)\nor both deny ($\\\ + mathsf{F}$).\nThe values $\\mathsf{B}$ (both confirm and deny)\nand $\\mathsf{N}$ (neither confirms nor denies)\ncannot\ + \ arise under spectral decisiveness.\nTherefore $\\mathrm{val}(P) \\in \\{\\mathsf{T}, \\mathsf{F}\\}$.\n\n$(1) \\Rightarrow\ + \ (3)$:\nIf $\\mathrm{val}(S) \\subseteq \\{\\mathsf{T}, \\mathsf{F}\\}$,\nthen $f|_{\\mathrm{val}(S)}$ is the identity\ + \ map\n(since $f(\\mathsf{T}) = \\mathsf{T}$\nand $f(\\mathsf{F}) = \\mathsf{F}$),\nwhich is trivially an isomorphism.\n\ + \n$(3) \\Rightarrow (1)$:\nIf $\\mathrm{val}(P) = \\mathsf{B}$ for some $P \\in S$,\nthen $f(\\mathsf{B}) = \\mathsf{T}\ + \ = f(\\mathsf{T})$,\nso $f$ identifies $\\mathsf{B}$ with $\\mathsf{T}$:\nit is not injective on $\\mathrm{val}(S)$\n\ + (assuming $\\mathsf{T}$ is also in the image),\ncontradicting the isomorphism property.\nSimilarly for $\\mathsf{N}$.\n\ + \n$(1) \\Rightarrow (4)$:\nIf $\\mathrm{val}(P) = \\mathsf{T}$,\nthen $f(\\lnot \\mathsf{T}) = f(\\mathsf{F}) = \\mathsf{F}\n\ + = \\lnot \\mathsf{T} = \\lnot f(\\mathsf{T})$.\nIf $\\mathrm{val}(P) = \\mathsf{F}$,\nthen $f(\\lnot \\mathsf{F}) = f(\\\ + mathsf{T}) = \\mathsf{T}\n= \\lnot \\mathsf{F} = \\lnot f(\\mathsf{F})$.\nIn both cases, negation commutes with $f$.\n\ + \n$(4) \\Rightarrow (1)$:\nSince $\\lnot \\mathsf{B} = \\mathsf{N}$,\nwe have $f(\\lnot \\mathsf{B}) = f(\\mathsf{N})\ + \ = \\mathsf{F}$\nand $\\lnot f(\\mathsf{B}) = \\lnot \\mathsf{T} = \\mathsf{F}$:\nnegation commutes with $f$ at $\\mathsf{B}$.\n\ + Similarly at $\\mathsf{N}$.\n\\emph{Note:} since the forgetful map $f$\nis a Boolean algebra homomorphism\n(projection\ + \ from $\\mathbf{2} \\times \\mathbf{2}$ to $\\mathbf{2}$),\nnegation compatibility holds universally\nand condition~(4)\ + \ is automatically satisfied.\nThe nontrivial equivalence is between\nconditions~(1)--(3).\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 93\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch48-boolean-recovery.tex`\ + \ lines 339-382\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Logic.BooleanRecovery`\n\ + - Name: `Tau.Logic.boolean_recovery`\n\n## Dependencies\n\n- Canonical: I.D21, I.T13\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Boolean logic as a quotient of Truth4: collapsing B and N recovers classical two-valued logic. The forgetful + map Truth4 -> Bool is a lattice homomorphism.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001137.md b/_corpus_v3_items/cid001137.md new file mode 100644 index 000000000..36ac30ea8 --- /dev/null +++ b/_corpus_v3_items/cid001137.md @@ -0,0 +1,157 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001137/ +title: Omega Unique Fixed Seed +cid: cid001137 +primary_alias: PRP0015 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: rho(x) = x iff seed(x) = omega. Omega is the unique fixed seed of rho among ALL objects (not just generators). Non-omega + seeds always advance depth by 1. +aliases: +- PRP0015 +- I.P14 +- omega-unique-fixed-seed +- prop:omega-unique-fixed +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 1, Chapter 2 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Omega Unique Fixed Seed** + + + rho(x) = x iff seed(x) = omega. Omega is the unique fixed seed of rho among ALL objects (not just generators). Non-omega + seeds always advance depth by 1. + + + # Omega Unique Fixed Seed + + + ## Summary + + + rho(x) = x iff seed(x) = omega. Omega is the unique fixed seed of rho among ALL objects (not just generators). Non-omega + seeds always advance depth by 1. + + + ## Statement + + + ```latex + + % + + \label{prop:omega-unique-fixed} + + An object $x \in \Obj(\tau)$ satisfies $\rho(x) = x$ + + if and only if the seed of $x$ is $\omega$. + + That is, $\omega$ is the \emph{unique fixed seed} of~$\rho$: + + no non-$\omega$ generator produces a fixed point, + + regardless of depth. + + ``` + + + ## Proof / Justification + + + ```latex + + If $x = \langle\omega, d\rangle$ for any depth~$d$, + + then $\rho(x) = \langle\omega, d\rangle = x$ by $\KAxiom{2}$. + + Conversely, if $x = \langle g, d\rangle$ for a non-$\omega$ generator + + $g \in \{\alpha, \pi, \gamma, \eta\}$, + + then $\rho(x) = \langle g, d+1\rangle \neq \langle g, d\rangle = x$ + + (since $d + 1 \neq d$). + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 95 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part01/ch02-rho-omega.tex` lines 113-120 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Denotation.Structural` + + - Name: `Tau.Denotation.omega_unique_fixed_seed` + + + ## Dependencies + + + - Canonical: I.D02, I.K2, I.K3, I.K4 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: rho(x) = x iff seed(x) = omega. Omega is the unique fixed seed of rho among ALL objects (not just generators). + Non-omega seeds always advance depth by 1. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001138.md b/_corpus_v3_items/cid001138.md new file mode 100644 index 000000000..e4dcd2581 --- /dev/null +++ b/_corpus_v3_items/cid001138.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001138/ +title: Sum Zero Iff Both Zero +cid: cid001138 +primary_alias: PRP0016 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n + m = 0 iff n = 0 and m = 0. Contrapositive: if either summand is positive, the sum is positive.' +aliases: +- PRP0016 +- I.P15 +- sum-zero-iff-both-zero +- prop:sum-zero-iff +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sum Zero Iff Both Zero**\n\nn + m = 0 iff n = 0 and m = 0. Contrapositive: if either summand is positive, the\ + \ sum is positive.\n\n# Sum Zero Iff Both Zero\n\n## Summary\n\nn + m = 0 iff n = 0 and m = 0. Contrapositive: if either\ + \ summand is positive, the sum is positive.\n\n## Statement\n\n```latex\n%\n\\label{prop:sum-zero-iff}\nFor $\\underline{n},\ + \ \\underline{m} \\in \\tau\\text{-Idx}$:\n\\[\n \\underline{n} + \\underline{m} = \\underline{0}\n \\quad\\Longleftrightarrow\\\ + quad\n \\underline{n} = \\underline{0}\n \\;\\text{ and }\\;\n \\underline{m} = \\underline{0}.\n\\]\n```\n\n\ + ## Proof / Justification\n\n```latex\n($\\Leftarrow$)\nIf both are zero, then\n$\\underline{0} + \\underline{0} = \\rho^0(\\\ + underline{0}) = \\underline{0}$.\n\n($\\Rightarrow$)\nSuppose $\\underline{n} + \\underline{m} = \\underline{0}$.\nBy\ + \ definition, $\\underline{n} + \\underline{m}\n= \\rho^m(\\rho^n(\\alpha)) = \\rho^{n+m}(\\alpha) = \\underline{n+m}$.\n\ + If $n > 0$ or $m > 0$, then $n + m > 0$,\nso $\\underline{n+m} = \\rho^{n+m}(\\alpha) \\neq \\alpha = \\underline{0}$\n\ + (by $\\KAxiom{4}$: each application of $\\rho$ advances\ndepth by exactly~$1$).\nContradiction.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 96\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex`\ + \ lines 257-267\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_sum_zero_iff`\n\n## Dependencies\n\n- Canonical: I.D10\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n + m = 0 iff n = 0 and m = 0. Contrapositive: if either summand is positive, the sum is positive.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001139.md b/_corpus_v3_items/cid001139.md new file mode 100644 index 000000000..8eb9bfad7 --- /dev/null +++ b/_corpus_v3_items/cid001139.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001139/ +title: Positive Core Closure +cid: cid001139 +primary_alias: PRP0017 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: N+ = {n > 0} is closed under add, mul, exp; successor is always positive. Arithmetic never falls into zero unless + it starts there. +aliases: +- PRP0017 +- I.P16 +- positive-core-closure +- prop:positive-closed +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Positive Core Closure**\n\nN+ = {n > 0} is closed under add, mul, exp; successor is always positive. Arithmetic\ + \ never falls into zero unless it starts there.\n\n# Positive Core Closure\n\n## Summary\n\nN+ = {n > 0} is closed under\ + \ add, mul, exp; successor is always positive. Arithmetic never falls into zero unless it starts there.\n\n## Statement\n\ + \n```latex\n%\n\\label{prop:positive-closed}\nFor all $\\underline{n}, \\underline{m} \\in \\tau\\text{-Idx}$:\n\\begin{enumerate}\n\ + \ \\item If $\\underline{n} > \\underline{0}$, then\n $\\underline{n} + \\underline{m} > \\underline{0}$\n\ + \ (addition preserves positivity).\n \\item If $\\underline{n} > \\underline{0}$ and\n $\\underline{m}\ + \ > \\underline{0}$, then\n $\\underline{n} \\times \\underline{m} > \\underline{0}$\n (multiplication\ + \ preserves positivity).\n \\item If $\\underline{n} > \\underline{0}$, then\n $\\underline{n}^{\\underline{m}}\ + \ > \\underline{0}$\n (exponentiation preserves positivity).\n \\item $\\underline{n} + \\underline{1} > \\\ + underline{0}$ always\n (every successor is positive, with no guard needed).\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\n```latex\n(1) $\\underline{n} + \\underline{m}\n= \\rho^m(\\underline{n}) = \\underline{n+m}$.\n\ + Since $n > 0$, we have $n + m \\geq n > 0$.\n\n(2) By induction on $\\underline{m}$.\nBase: $\\underline{n} \\times \\\ + underline{1} = \\underline{n} > \\underline{0}$.\nStep: $\\underline{n} \\times (\\underline{m+1})\n= (\\underline{n}\ + \ \\times \\underline{m}) + \\underline{n}$.\nBy the inductive hypothesis,\n$\\underline{n} \\times \\underline{m} > \\\ + underline{0}$,\nso by~(1) the sum is positive.\n\n(3) By induction on $\\underline{m}$.\nBase: $\\underline{n}^{\\underline{0}}\ + \ = \\underline{1} > \\underline{0}$.\nStep: $\\underline{n}^{\\underline{m+1}}\n= \\underline{n}^{\\underline{m}} \\\ + times \\underline{n}$.\nBy the inductive hypothesis and~(2), the product is positive.\n\n(4) Special case of~(1):\n$\\\ + underline{n} + \\underline{1} = \\underline{n+1} > \\underline{0}$\nfor all $n \\geq 0$.\n```\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 99\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 278-295\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_pos_closed_add`\n\n## Dependencies\n\n- Canonical: I.D10, I.D11, I.D12\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: N+ = {n > 0} is closed under add, mul, exp; successor is always positive. Arithmetic never falls into zero unless + it starts there. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001140.md b/_corpus_v3_items/cid001140.md new file mode 100644 index 000000000..ed32d9426 --- /dev/null +++ b/_corpus_v3_items/cid001140.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001140/ +title: Universal Additive Cancellation +cid: cid001140 +primary_alias: PRP0018 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'n + a = n + b implies a = b. Additive cancellation is UNIVERSAL: no positivity guard needed. Follows from rho-injectivity.' +aliases: +- PRP0018 +- I.P17 +- universal-additive-cancellation +- prop:add-cancel-universal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Universal Additive Cancellation**\n\nn + a = n + b implies a = b. Additive cancellation is UNIVERSAL: no positivity\ + \ guard needed. Follows from rho-injectivity.\n\n# Universal Additive Cancellation\n\n## Summary\n\nn + a = n + b implies\ + \ a = b. Additive cancellation is UNIVERSAL: no positivity guard needed. Follows from rho-injectivity.\n\n## Statement\n\ + \n```latex\n%\n\\label{prop:add-cancel-universal}\n% Depends: I.D08 (addition)\nFor all $\\underline{n}, \\underline{a},\ + \ \\underline{b} \\in \\tau\\text{-Idx}$:\n\\[\n \\underline{n} + \\underline{a} = \\underline{n} + \\underline{b}\n\ + \ \\quad\\Longrightarrow\\quad\n \\underline{a} = \\underline{b}.\n\\]\nAdditive cancellation holds without any\ + \ positivity guard.\n```\n\n## Proof / Justification\n\n```latex\nRecall that $\\underline{n} + \\underline{a} = \\rho^a(\\\ + underline{n})$\n(Definition~\\ref{def:idx-add}).\nIf $\\rho^a(\\underline{n}) = \\rho^b(\\underline{n})$,\nthen both sides\ + \ have seed $\\alpha$\nand respective depths $n + a$ and $n + b$.\nEquality of depths gives $n + a = n + b$,\nhence $a\ + \ = b$, hence $\\underline{a} = \\underline{b}$.\nRight cancellation follows symmetrically by commutativity.\n```\n\n\ + ## Source Context\n\n- Registry source: `book-01.jsonl` line 100\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch15-denotational-order.tex`\ + \ lines 200-210\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_add_left_cancel`\n\n## Dependencies\n\n- Canonical: I.D10, I.P02\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'n + a = n + b implies a = b. Additive cancellation is UNIVERSAL: no positivity guard needed. Follows from rho-injectivity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001141.md b/_corpus_v3_items/cid001141.md new file mode 100644 index 000000000..d61df6fe4 --- /dev/null +++ b/_corpus_v3_items/cid001141.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001141/ +title: Zero Vacuous +cid: cid001141 +primary_alias: PRP0019 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Zero is vacuous: (1) not prime, (2) not a successor, (3) divisible by everything, (4) unique multiplicative absorber. + A non-participant rather than a destructive element.' +aliases: +- PRP0019 +- I.P18 +- zero-vacuous +- prop:zero-vacuous +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Zero Vacuous**\n\nZero is vacuous: (1) not prime, (2) not a successor, (3) divisible by everything, (4) unique\ + \ multiplicative absorber. A non-participant rather than a destructive element.\n\n# Zero Vacuous\n\n## Summary\n\nZero\ + \ is vacuous: (1) not prime, (2) not a successor, (3) divisible by everything, (4) unique multiplicative absorber. A non-participant\ + \ rather than a destructive element.\n\n## Statement\n\n```latex\n%\n\\label{prop:zero-vacuous}\nIn $\\tau$-Idx, zero\ + \ is a passive, non-participating element:\n\\begin{enumerate}\n \\item $\\underline{0}$ is \\textbf{not prime}:\n\ + \ it fails the requirement $\\underline{p} \\geq \\underline{2}$.\n \\item $\\underline{0}$ is \\textbf{not\ + \ a successor}:\n no $\\underline{k} \\in \\tau$-Idx satisfies\n $\\underline{k} + \\underline{1} =\ + \ \\underline{0}$.\n \\item $\\underline{0}$ is \\textbf{divisible by everything}:\n $\\underline{a} \\mid\ + \ \\underline{0}$ for all $\\underline{a}$.\n \\item $\\underline{0}$ is the \\textbf{unique multiplicative absorber}:\n\ + \ if $\\underline{a} \\times \\underline{n} = \\underline{a}$\n for all $\\underline{n}$,\n \ + \ then $\\underline{a} = \\underline{0}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n(1) By definition\n\ + (Definition~\\ref{def:internal-primes}):\n$\\underline{0} < \\underline{2}$.\n\n(2) If $\\underline{k} + \\underline{1}\ + \ = \\underline{0}$,\nthen by Proposition~\\ref{prop:sum-zero-iff}\n(Chapter~\\ref{ch:swap-add-mul}),\n$\\underline{1}\ + \ = \\underline{0}$, a contradiction.\n\n(3) For any $\\underline{a}$:\n$\\underline{0} = \\underline{a} \\times \\underline{0}$,\n\ + so $\\underline{a} \\mid \\underline{0}$.\n\n(4) Specialize to $\\underline{n} = \\underline{0}$:\n$\\underline{a} \\\ + times \\underline{0} = \\underline{a}$.\nBut $\\underline{a} \\times \\underline{0} = \\underline{0}$\n(Proposition~\\\ + ref{prop:arithmetic-laws},\nChapter~\\ref{ch:swap-add-mul}),\nso $\\underline{a} = \\underline{0}$.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 103\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex`\ + \ lines 123-139\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_zero_not_prime`\n\n## Dependencies\n\n- Canonical: I.D19a, I.D19b, I.D10, I.D11\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Zero is vacuous: (1) not prime, (2) not a successor, (3) divisible by everything, (4) unique multiplicative + absorber. A non-participant rather than a destructive element.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001142.md b/_corpus_v3_items/cid001142.md new file mode 100644 index 000000000..17b0a6c5b --- /dev/null +++ b/_corpus_v3_items/cid001142.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001142/ +title: Integral Domain +cid: cid001142 +primary_alias: PRP0020 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: n * m = 0 iff n = 0 or m = 0. tau-Idx has no zero divisors (integral domain property). Forward direction uses positive + core closure. +aliases: +- PRP0020 +- I.P19 +- integral-domain +- prop:integral-domain +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Integral Domain**\n\nn * m = 0 iff n = 0 or m = 0. tau-Idx has no zero divisors (integral domain property).\ + \ Forward direction uses positive core closure.\n\n# Integral Domain\n\n## Summary\n\nn * m = 0 iff n = 0 or m = 0. tau-Idx\ + \ has no zero divisors (integral domain property). Forward direction uses positive core closure.\n\n## Statement\n\n```latex\n\ + %\n\\label{prop:integral-domain}\nFor $\\underline{n}, \\underline{m} \\in \\tau\\text{-Idx}$:\n\\[\n \\underline{n}\ + \ \\times \\underline{m} = \\underline{0}\n \\quad\\Longleftrightarrow\\quad\n \\underline{n} = \\underline{0}\n\ + \ \\;\\text{ or }\\;\n \\underline{m} = \\underline{0}.\n\\]\nThat is, $\\tau$-Idx has no zero divisors.\n```\n\n\ + ## Proof / Justification\n\n```latex\n($\\Leftarrow$)\nIf $\\underline{n} = \\underline{0}$ or $\\underline{m} = \\underline{0}$,\n\ + then the product is $\\underline{0}$\nby the absorption property\n(Proposition~\\ref{prop:arithmetic-laws},\nChapter~\\\ + ref{ch:swap-add-mul}).\n\n($\\Rightarrow$)\nIf $\\underline{n} > \\underline{0}$ and $\\underline{m} > \\underline{0}$,\n\ + then $\\underline{n} \\times \\underline{m} > \\underline{0}$\nby positive core closure\n(Proposition~\\ref{prop:positive-closed},\n\ + Chapter~\\ref{ch:exp-tetration}).\nContrapositive: product zero implies a factor is zero.\n```\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 104\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex`\ + \ lines 163-174\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_integral_domain`\n\n## Dependencies\n\n- Canonical: I.P16, I.D11\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: n * m = 0 iff n = 0 or m = 0. tau-Idx has no zero divisors (integral domain property). Forward direction uses + positive core closure. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001143.md b/_corpus_v3_items/cid001143.md new file mode 100644 index 000000000..d0815d106 --- /dev/null +++ b/_corpus_v3_items/cid001143.md @@ -0,0 +1,137 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001143/ +title: Ultra Dist Self +cid: cid001143 +primary_alias: PRP0021 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: d(t, t) = 0 for every omega-tail t. Identity of indiscernibles for the primorial divergence depth. +aliases: +- PRP0021 +- I.P20 +- ultra-dist-self +- prop:ultra-dist-self +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 28 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Ultra Dist Self** + + + d(t, t) = 0 for every omega-tail t. Identity of indiscernibles for the primorial divergence depth. + + + # Ultra Dist Self + + + ## Summary + + + d(t, t) = 0 for every omega-tail t. Identity of indiscernibles for the primorial divergence depth. + + + ## Statement + + + ```latex + + % + + \label{prop:ultra-dist-self} + + For every omega-tail $t$: $d(t, t) = 0$. + + ``` + + + ## Proof / Justification + + + ```latex + + By definition, $t$ agrees with itself at every stage, + + so $t \sim t$ and hence $d(t, t) = 0$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 105 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch28-omega-germs.tex` lines 293-296 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Denotation.Structural` + + - Name: `Tau.Denotation.ultra_dist_self` + + + ## Dependencies + + + - Canonical: I.D25 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: d(t, t) = 0 for every omega-tail t. Identity of indiscernibles for the primorial divergence depth. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001144.md b/_corpus_v3_items/cid001144.md new file mode 100644 index 000000000..0d234ab66 --- /dev/null +++ b/_corpus_v3_items/cid001144.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001144/ +title: Congruent Tails Agree +cid: cid001144 +primary_alias: PRP0022 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: n = m (mod M_k) implies canonical tails agree at level k. Agreement at each primorial level is a finite modular check. + Underlies Cauchy-compactness of the profinite completion. +aliases: +- PRP0022 +- I.P21 +- congruent-tails-agree +- prop:congruent-tails-agree +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 28 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Congruent Tails Agree**\n\nn = m (mod M_k) implies canonical tails agree at level k. Agreement at each primorial\ + \ level is a finite modular check. Underlies Cauchy-compactness of the profinite completion.\n\n# Congruent Tails Agree\n\ + \n## Summary\n\nn = m (mod M_k) implies canonical tails agree at level k. Agreement at each primorial level is a finite\ + \ modular check. Underlies Cauchy-compactness of the profinite completion.\n\n## Statement\n\n```latex\n%\n\\label{prop:congruent-tails-agree}\n\ + % Depends: def:omega-tail, def:primorial-ladder\nIf $n \\equiv m \\pmod{M_k}$,\nthen the canonical omega-tails\n$\\hat{n}$\ + \ and $\\hat{m}$\nagree at level $k$:\n\\[\n n \\equiv m \\pmod{M_k}\n \\quad\\Longrightarrow\\quad\n \\hat{n}_k\ + \ = \\hat{m}_k.\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThe $k$-th component of the canonical embedding is\n\ + $\\hat{n}_k = n \\bmod M_k$.\nIf $n \\equiv m \\pmod{M_k}$,\nthen $n \\bmod M_k = m \\bmod M_k$,\nhence $\\hat{n}_k =\ + \ \\hat{m}_k$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 106\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch28-omega-germs.tex`\ + \ lines 303-315\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.congruent_tails_agree`\n\n## Dependencies\n\n- Canonical: I.D25\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: n = m (mod M_k) implies canonical tails agree at level k. Agreement at each primorial level is a finite modular + check. Underlies Cauchy-compactness of the profinite completion. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001145.md b/_corpus_v3_items/cid001145.md new file mode 100644 index 000000000..1339683dc --- /dev/null +++ b/_corpus_v3_items/cid001145.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001145/ +title: Sector Independence +cid: cid001145 +primary_alias: PRP0023 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every SectorFun is sector-independent by construction: B-output depends only on B-input, C-output depends only on + C-input. The structural content of the split-CR equations.' +aliases: +- PRP0023 +- I.P22 +- sector-independence +- prop:sector-independence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005298 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 49 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sector Independence**\n\nEvery SectorFun is sector-independent by construction: B-output depends only on B-input,\ + \ C-output depends only on C-input. The structural content of the split-CR equations.\n\n# Sector Independence\n\n## Summary\n\ + \nEvery SectorFun is sector-independent by construction: B-output depends only on B-input, C-output depends only on C-input.\ + \ The structural content of the split-CR equations.\n\n## Statement\n\n```latex\n%\n\\label{prop:sector-independence}\n\ + If $f : H_\\tau \\to H_\\tau$ is D-holomorphic,\nthen in sector coordinates\n$f$ decomposes as:\n\\[\n \\boxed{%\n\ + \ f(u, v) \\;=\\; \\bigl(g(u),\\; h(v)\\bigr)}\n\\]\nfor functions\n$g : \\hat{\\mathbb{Z}}_\\tau \\to \\hat{\\mathbb{Z}}_\\\ + tau$\nand\n$h : \\hat{\\mathbb{Z}}_\\tau \\to \\hat{\\mathbb{Z}}_\\tau$,\neach depending on a single sector variable.\n\ + ```\n\n## Proof / Justification\n\n```latex\nBy the sector-coordinate form of the split-CR equations\n(Remark~\\ref{rem:sector-form-cr}):\n\ + \\begin{itemize}\n \\item $\\partial F_+ / \\partial v = 0$\n implies $F_+$ is independent of $v$,\n \ + \ so $F_+(u, v) = g(u)$\n for some function $g$ of $u$ alone.\n \\item $\\partial F_- / \\partial u =\ + \ 0$\n implies $F_-$ is independent of $u$,\n so $F_-(u, v) = h(v)$\n for some function $h$\ + \ of $v$ alone.\n\\end{itemize}\nTherefore $f(u, v) = (g(u),\\, h(v))$ in sector form.\n```\n\n## Source Context\n\n-\ + \ Registry source: `book-01.jsonl` line 109\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch49-d-holomorphy.tex`\ + \ lines 410-425\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DHolomorphic`\n\ + - Name: `Tau.Holomorphy.sector_fun_independent`\n\n## Dependencies\n\n- Canonical: I.D42\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every SectorFun is sector-independent by construction: B-output depends only on B-input, C-output depends only + on C-input. The structural content of the split-CR equations.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001146.md b/_corpus_v3_items/cid001146.md new file mode 100644 index 000000000..c5dc16089 --- /dev/null +++ b/_corpus_v3_items/cid001146.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001146/ +title: No Simultaneous Projection +cid: cid001146 +primary_alias: PRP0024 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No Simultaneous Projection: the diagonal-free discipline prevents any omega-tail from projecting nontrivially onto + BOTH idempotent sectors e_plus and e_minus simultaneously.' +aliases: +- PRP0024 +- I.P23 +- no-simultaneous-projection +- prop:no-simul-projection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005299 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 51 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Simultaneous Projection**\n\nNo Simultaneous Projection: the diagonal-free discipline prevents any omega-tail\ + \ from projecting nontrivially onto BOTH idempotent sectors e_plus and e_minus simultaneously.\n\n# No Simultaneous Projection\n\ + \n## Summary\n\nNo Simultaneous Projection: the diagonal-free discipline prevents any omega-tail from projecting nontrivially\ + \ onto BOTH idempotent sectors e_plus and e_minus simultaneously.\n\n## Statement\n\n```latex\n%\n\\label{prop:no-simul-projection}\n\ + Let $t = (x_k)_{k \\geq 1}$ be a compatible omega-tail\n(satisfying the reduction conditions of the primorial tower),\n\ + and let $T \\in \\mathrm{HolFun}$ be a $\\tau$-holomorphic function.\nThen the canonical sector projections\n$\\chi_+(T(t))$\ + \ and $\\chi_-(T(t))$\ncannot \\emph{both} be nontrivial at every stage\nof the primorial ladder.\n\nMore precisely:\n\ + at each primorial stage $M_k$,\nthe CRT decomposition\n\\[\n \\mathbb{Z}/M_k\\mathbb{Z}\n \\;\\cong\\;\n \\prod_{i=1}^{k}\ + \ \\mathbb{Z}/p_i\\mathbb{Z}\n\\]\nassigns each CRT factor $\\mathbb{Z}/p_i\\mathbb{Z}$\nto exactly one sector via the\ + \ Prime Polarity Theorem (I.T05):\nto the $B$-sector if $\\mathrm{pol}(p_i) = +$,\nor to the $C$-sector if $\\mathrm{pol}(p_i)\ + \ = -$.\nThe diagonal-discipline axiom (K5, I.D03)\nforbids any program from simultaneously accessing\ngenerators from\ + \ both sectors in the same application step.\nTherefore, $T$'s output in the $B$-sector\ndepends only on $B$-sector inputs,\n\ + and $T$'s output in the $C$-sector\ndepends only on $C$-sector inputs.\nAn omega-germ cannot project nontrivially onto\ + \ both sectors\nsimultaneously through a $\\tau$-holomorphic function.\n```\n\n## Proof / Justification\n\n```latex\n\ + Consider a compatible omega-tail $t = (x_k)_{k \\geq 1}$.\nAt stage $M_k$, the CRT decomposition\nassigns each prime $p_i$\ + \ ($1 \\leq i \\leq k$)\nto the $B$-sector (if $\\mathrm{pol}(p_i) = +$)\nor to the $C$-sector (if $\\mathrm{pol}(p_i)\ + \ = -$).\nThe $B$-sector projection at stage $k$\nacts only on the $B$-sector CRT factors;\nthe $C$-sector projection\ + \ acts only on the $C$-sector factors.\n\nFor a $\\tau$-holomorphic function $T \\in \\mathrm{HolFun}$,\ntower coherence\ + \ (I.D46)\nand D-holomorphy --- specifically,\nsector independence (I.P22) ---\ntogether ensure that\n$T$'s $B$-sector\ + \ output depends only on $B$-sector inputs\nand $T$'s $C$-sector output depends only on $C$-sector inputs.\n\nThe diagonal\ + \ discipline (I.D03) is the axiom-level guarantee\nthat programs cannot mix these sectors:\nno single denotational step\ + \ simultaneously applies generators\nfrom both the $B$-sector and the $C$-sector.\nThis is precisely the content of axiom\ + \ K5 ---\nthe no-diagonals constraint introduced in\nChapter~\\ref{ch:diagonal-discipline}.\n\nCombining these three ingredients:\n\ + \\begin{enumerate}\n \\item \\emph{CRT partition}: each prime belongs to exactly one sector\n (I.T05);\n \ + \ \\item \\emph{Sector independence}: $T$ decomposes as\n $T(u, v) = (g(u), h(v))$ in sector coordinates (I.P22);\n\ + \ \\item \\emph{Diagonal discipline}: no program step\n can access generators from both sectors simultaneously\n\ + \ (I.D03).\n\\end{enumerate}\nIt follows that an omega-germ cannot\n``project nontrivially onto both sectors\ + \ simultaneously''\nthrough $T$.\nIf $t$ is purely $B$-sector at every stage,\nthen $\\chi_-(T(t))$ is determined\nentirely\ + \ by the trivial $C$-sector input.\nIf $t$ has nontrivial $C$-sector content,\nthen the $B$-sector output is independent\n\ + of that $C$-sector content.\nThe two projections are structurally decoupled.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 115\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch51-diagonal-free-protection.tex`\ + \ lines 149-181\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DiagonalProtection`\n\ + - Name: `Tau.Holomorphy.no_simul_projection_b`\n\n## Dependencies\n\n- Canonical: I.D03, I.T05, I.D47\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No Simultaneous Projection: the diagonal-free discipline prevents any omega-tail from projecting nontrivially + onto BOTH idempotent sectors e_plus and e_minus simultaneously.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001147.md b/_corpus_v3_items/cid001147.md new file mode 100644 index 000000000..bbf2842a1 --- /dev/null +++ b/_corpus_v3_items/cid001147.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001147/ +title: HolFun Associativity +cid: cid001147 +primary_alias: PRP0025 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'HolFun composition is associative: (f1 o f2) o f3 = f1 o (f2 o f3). Follows from function extensionality. Together + with identity and composition closure, HolFun forms a monoid.' +aliases: +- PRP0025 +- I.P24 +- holfun-associativity +- prop:holfun-associativity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005301 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 51 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**HolFun Associativity**\n\nHolFun composition is associative: (f1 o f2) o f3 = f1 o (f2 o f3). Follows from\ + \ function extensionality. Together with identity and composition closure, HolFun forms a monoid.\n\n# HolFun Associativity\n\ + \n## Summary\n\nHolFun composition is associative: (f1 o f2) o f3 = f1 o (f2 o f3). Follows from function extensionality.\ + \ Together with identity and composition closure, HolFun forms a monoid.\n\n## Statement\n\n```latex\n%\n\\label{prop:holfun-associativity}\n\ + Composition of HolFuns is associative:\nfor all $R, S, T \\in \\mathrm{HolFun}$,\n\\[\n \\boxed{%\n (R \\circ S)\ + \ \\circ T = R \\circ (S \\circ T).}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThis inherits directly from the\ + \ associativity\nof function composition,\nwhich was already established\nfor the program monoid\n(Theorem~\\ref{thm:comp-assoc},\ + \ I.T03)\nin Chapter~\\ref{ch:program-monoid}.\nThe program monoid acts on $\\tau$-Idx\nby function application;\nHolFun\ + \ composition is a special case\nof this action restricted to functions\nthat satisfy both D-holomorphy and tower coherence.\n\ + Since Theorem~\\ref{thm:composition-closure} (I.T20)\nguarantees that all intermediate compositions\nremain in HolFun,\n\ + the associativity of the ambient function composition\nrestricts to HolFun without modification.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 118\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch51-diagonal-free-protection.tex`\ + \ lines 457-466\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DiagonalProtection`\n\ + - Name: `Tau.Holomorphy.stagefun_comp_assoc`\n\n## Dependencies\n\n- Canonical: I.T20, I.T03\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'HolFun composition is associative: (f1 o f2) o f3 = f1 o (f2 o f3). Follows from function extensionality. Together + with identity and composition closure, HolFun forms a monoid.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001148.md b/_corpus_v3_items/cid001148.md new file mode 100644 index 000000000..f7b31230a --- /dev/null +++ b/_corpus_v3_items/cid001148.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001148/ +title: Thin Category +cid: cid001148 +primary_alias: PRP0026 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cat_tau is thin: between any two objects there is at most one morphism. Direct corollary of the tau-Identity Theorem + — if two tower-coherent functions agree at any depth, they agree everywhere.' +aliases: +- PRP0026 +- I.P25 +- thin-category +- prop:thin-category +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005353 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 53 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Thin Category**\n\nCat_tau is thin: between any two objects there is at most one morphism. Direct corollary\ + \ of the tau-Identity Theorem — if two tower-coherent functions agree at any depth, they agree everywhere.\n\n# Thin Category\n\ + \n## Summary\n\nCat_tau is thin: between any two objects there is at most one morphism. Direct corollary of the tau-Identity\ + \ Theorem — if two tower-coherent functions agree at any depth, they agree everywhere.\n\n## Statement\n\n```latex\n%\n\ + \\label{prop:thin-category}\n$\\mathrm{Cat}_\\tau$ is a \\textbf{thin category}:\nfor all objects $A, B \\in \\mathrm{Ob}(\\\ + tau)$,\n\\[\n \\boxed{%\n \\bigl|\\,\\mathrm{Hom}_{\\mathrm{Cat}_\\tau}(A, B)\\,\\bigr|\n \\;\\leq\\; 1.}\n\\\ + ]\nThat is: if $\\alpha, \\beta \\in \\mathrm{Hom}(A, B)$,\nthen $\\alpha = \\beta$.\n```\n\n## Proof / Justification\n\ + \n```latex\nLet $\\alpha, \\beta \\in \\mathrm{Hom}(A, B)$.\nEach arrow carries a unique HolFun:\n$T_\\alpha, T_\\beta\ + \ \\in \\mathrm{HolFun}$.\nBoth are $\\tau$-holomorphic functions\nwith the same source $A$ and target $B$.\n\nSince $T_\\\ + alpha$ and $T_\\beta$\nshare the same source rooting,\nthey act on the same set of omega-tails.\nAt the source object\ + \ $A$,\nboth functions agree on all omega-tails of depth $1$\n(the shallowest primorial stage) ---\nbecause the source\ + \ rooting determines\nthe depth-$1$ behavior,\nand both functions are tower-coherent\nfrom the same source $A$.\n\nTherefore\ + \ there exists a depth $d_0 = 1$\nat which $T_\\alpha$ and $T_\\beta$ agree.\nBy the $\\tau$-Identity Theorem\n(Theorem~\\\ + ref{thm:tau-identity}, I.T21):\n\\[\n T_\\alpha \\sim_{d_0} T_\\beta\n \\;\\;\\Longrightarrow\\;\\;\n T_\\alpha\ + \ = T_\\beta.\n\\]\nSince $T_\\alpha = T_\\beta$ as HolFuns,\nwe have $\\alpha = \\beta$ as $\\tau$-arrows.\n```\n\n##\ + \ Source Context\n\n- Registry source: `book-01.jsonl` line 125\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch53-earned-arrows.tex`\ + \ lines 428-440\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedArrows`\n\ + - Name: `Tau.Topos.cat_tau_thin`\n\n## Dependencies\n\n- Canonical: I.D51, I.T21\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cat_tau is thin: between any two objects there is at most one morphism. Direct corollary of the tau-Identity + Theorem — if two tower-coherent functions agree at any depth, they agree everywhere.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001149.md b/_corpus_v3_items/cid001149.md new file mode 100644 index 000000000..8636c9bb4 --- /dev/null +++ b/_corpus_v3_items/cid001149.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001149/ +title: Countable Topos +cid: cid001149 +primary_alias: PRP0027 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'PSh(Cat_tau) is a countable topos: Cat_tau has countable objects and at most one morphism between each pair (thin). + Presheaves are countably generated by functions TauIdx -> Bool.' +aliases: +- PRP0027 +- I.P26 +- countable-topos +- prop:countable-topos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005360 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 55 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Countable Topos**\n\nPSh(Cat_tau) is a countable topos: Cat_tau has countable objects and at most one morphism\ + \ between each pair (thin). Presheaves are countably generated by functions TauIdx -> Bool.\n\n# Countable Topos\n\n##\ + \ Summary\n\nPSh(Cat_tau) is a countable topos: Cat_tau has countable objects and at most one morphism between each pair\ + \ (thin). Presheaves are countably generated by functions TauIdx -> Bool.\n\n## Statement\n\n```latex\n%\n\\label{prop:countable-topos}\n\ + $\\mathrm{PSh}(\\mathrm{Cat}_\\tau)$\nhas a \\textbf{countable} generating set.\nSpecifically:\n\\begin{enumerate}\n \ + \ \\item $|\\Obj(\\mathrm{Cat}_\\tau)| = \\aleph_0$.\n \\item $|\\mathrm{Mor}(\\mathrm{Cat}_\\tau)| = \\aleph_0$.\n\ + \ \\item The representable presheaves\n $\\{y(X)\\}_{X \\in \\mathrm{Cat}_\\tau}$\n form a countable\ + \ generating set.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(1)} The objects of $\\mathrm{Cat}_\\\ + tau$\nare the elements of $\\tau$-Idx $\\cong \\mathbb{N}$\n(Proposition~\\ref{prop:set-countable}, I.P12).\n\n\\textbf{(2)}\ + \ The morphisms are pairs $(X, Y)$ with $X \\mid Y$,\na subset of $\\mathbb{N} \\times \\mathbb{N}$,\nhence countable.\n\ + \n\\textbf{(3)} The set $\\{y(X)\\}$\nis indexed by $\\tau$-Idx, hence countable.\nBy the Yoneda Lemma,\nevery presheaf\ + \ is a colimit of representables,\nso the representables generate.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 134\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch55-limits-sites.tex`\ + \ lines 497-510\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.LimitsSites`\n\ + - Name: `Tau.Topos.psh_countable_objects`\n\n## Dependencies\n\n- Canonical: I.D57, I.P25\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'PSh(Cat_tau) is a countable topos: Cat_tau has countable objects and at most one morphism between each pair + (thin). Presheaves are countably generated by functions TauIdx -> Bool.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001150.md b/_corpus_v3_items/cid001150.md new file mode 100644 index 000000000..e6e892158 --- /dev/null +++ b/_corpus_v3_items/cid001150.md @@ -0,0 +1,96 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001150/ +title: Paraconsistent Character +cid: cid001150 +primary_alias: PRP0028 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The earned topos E_tau has a Boolean lattice (Omega_tau = 2x2, complement law holds) but paraconsistent material + implication (B => F = N != T, explosion fails). The subobject complement law fails: elements with chi_S in {B, N} belong + to neither S nor S^c. Boolean lattice coexists with paraconsistent implication — a structural consequence of the two spectral + sectors.' +aliases: +- PRP0028 +- I.P27 +- paraconsistent-character +- prop:non-boolean +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005355 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 56 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Paraconsistent Character**\n\nThe earned topos E_tau has a Boolean lattice (Omega_tau = 2x2, complement law\ + \ holds) but paraconsistent material implication (B => F = N != T, explosion fails). The subobject complement law fails:\ + \ elements with chi_S in {B, N} belong to neither S nor S^c. Boolean lattice coexists with paraconsistent implication\ + \ — a structural consequence of the two spectral sectors.\n\n# Paraconsistent Character\n\n## Summary\n\nThe earned topos\ + \ E_tau has a Boolean lattice (Omega_tau = 2x2, complement law holds) but paraconsistent material implication (B => F\ + \ = N != T, explosion fails). The subobject complement law fails: elements with chi_S in {B, N} belong to neither S nor\ + \ S^c. Boolean lattice coexists with paraconsistent implication — a structural consequence of the two spectral sectors.\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:non-boolean}\nThe earned topos $\\mathcal{E}_\\tau$ has the following logical\ + \ character:\n\\begin{enumerate}\n \\item $\\Omega_\\tau \\cong \\mathbf{2} \\times \\mathbf{2}$\n is a Boolean\ + \ algebra:\n the complement law\n $v \\lor \\lnot v = \\mathsf{T}$ holds for all $v$.\n \\item Material\ + \ implication is \\textbf{paraconsistent}:\n $\\mathsf{B} \\Rightarrow \\mathsf{F} = \\mathsf{N} \\neq \\mathsf{T}$,\n\ + \ so contradictions do not explode.\n \\item The subobject complement law fails:\n there exist subobjects\ + \ $S \\hookrightarrow X$\n with $S \\cup S^c \\neq X$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n\ + ```latex\n\\textbf{(1).}\nIn $\\Omega_\\tau$:\n$\\mathsf{B} \\lor \\lnot \\mathsf{B}\n= \\mathsf{B} \\lor \\mathsf{N}\n\ + = \\mathsf{T}$\n(since $\\lnot \\mathsf{B} = \\mathsf{N}$\nby Proposition~\\ref{prop:negation-B},\nChapter~\\ref{ch:explosion-barrier}).\n\ + Similarly,\n$\\mathsf{N} \\lor \\lnot \\mathsf{N}\n= \\mathsf{N} \\lor \\mathsf{B} = \\mathsf{T}$.\nThe complement law\ + \ holds for all four values;\n$\\Omega_\\tau$ is a Boolean algebra ($\\mathbf{2} \\times \\mathbf{2}$).\n\n\\textbf{(2).}\n\ + The Heyting implication in $\\Omega_\\tau$\ndoes not satisfy ex falso quodlibet:\n$\\mathsf{B} \\Rightarrow \\mathsf{F}\ + \ = \\mathsf{N} \\neq \\mathsf{T}$.\nThis is the explosion barrier (Theorem~I.T13,\nChapter~\\ref{ch:explosion-barrier}):\n\ + from $\\mathrm{val}(P) = \\mathsf{B}$\none cannot derive $\\mathrm{val}(Q) = \\mathsf{T}$\nfor arbitrary~$Q$.\nThe lattice\ + \ is Boolean, but the implication is paraconsistent ---\nthe two properties coexist because paraconsistency\nis a property\ + \ of the implication connective,\nnot of the lattice complement.\n\n\\textbf{(3).}\nLet $S \\hookrightarrow X$ have\n\ + $\\chi_S^{-1}(\\mathsf{B}) \\neq \\varnothing$.\nThe complement $S^c$ is classified by $\\lnot \\circ \\chi_S$:\n$S^c(c)\ + \ = \\{x : \\chi_S(x) = \\mathsf{F}\\}$.\nThen $S(c) \\cup S^c(c)\n= \\chi_S^{-1}(\\mathsf{T}) \\cup \\chi_S^{-1}(\\mathsf{F})\n\ + \\subsetneq X(c)$,\nbecause elements with\n$\\chi_S(x) \\in \\{\\mathsf{B}, \\mathsf{N}\\}$\nbelong to neither $S$ nor\ + \ $S^c$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 138\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch56-earned-topos.tex`\ + \ lines 400-416\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedTopos`\n\ + - Name: `Tau.Topos.earned_topos_paraconsistent`\n\n## Dependencies\n\n- Canonical: I.D59, I.T13\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The earned topos E_tau has a Boolean lattice (Omega_tau = 2x2, complement law holds) but paraconsistent material + implication (B => F = N != T, explosion fails). The subobject complement law fails: elements with chi_S in {B, N} belong + to nei' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001151.md b/_corpus_v3_items/cid001151.md new file mode 100644 index 000000000..360f369e0 --- /dev/null +++ b/_corpus_v3_items/cid001151.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001151/ +title: Self-Enrichment +cid: cid001151 +primary_alias: PRP0029 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E_tau is self-enriched: internal hom gives an internal presheaf of morphisms. Every internal hom Q^P is itself a + Presheaf.' +aliases: +- PRP0029 +- I.P28 +- self-enrichment +- prop:self-enrichment +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005358 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 59 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Self-Enrichment**\n\nE_tau is self-enriched: internal hom gives an internal presheaf of morphisms. Every internal\ + \ hom Q^P is itself a Presheaf.\n\n# Self-Enrichment\n\n## Summary\n\nE_tau is self-enriched: internal hom gives an internal\ + \ presheaf of morphisms. Every internal hom Q^P is itself a Presheaf.\n\n## Statement\n\n```latex\n%\n\\label{prop:self-enrichment}\n\ + The earned topos $\\mathcal{E}_\\tau$\nis \\textbf{enriched over itself}:\nfor every pair of objects $P, Q$ in $\\mathcal{E}_\\\ + tau$,\nthe external hom-set $\\Hom(P, Q)$\nis recovered from the internal hom $Q^P$ by:\n\\[\n \\boxed{%\n \\Hom_{\\\ + mathcal{E}_\\tau}(P,\\, Q)\n \\;\\cong\\;\n \\Gamma(Q^P)\n \\;:=\\;\n (Q^P)(\\mathbf{1}_\\tau),}\n\\]\nwhere\ + \ $\\Gamma = \\Hom(\\mathbf{1}_\\tau, -)$\nis the global sections functor.\nThe composition law\n$Q^P \\times R^Q \\to\ + \ R^P$\nis an internal morphism in $\\mathcal{E}_\\tau$,\nand the identity $P \\to P$\ncorresponds to $\\id \\in \\Gamma(P^P)$.\n\ + ```\n\n## Proof / Justification\n\n```latex\n\\textbf{Global sections recover external hom.}\nBy the exponential adjunction\n\ + (Theorem~\\ref{thm:cartesian-closed}, I.T28)\nwith $A = \\mathbf{1}_\\tau$:\n\\[\n \\Hom(\\mathbf{1}_\\tau,\\, Q^P)\n\ + \ \\;\\cong\\;\n \\Hom(\\mathbf{1}_\\tau \\times P,\\, Q)\n \\;\\cong\\;\n \\Hom(P,\\, Q),\n\\]\nusing $\\\ + mathbf{1}_\\tau \\times P \\cong P$\n(the terminal object is the unit for products).\nSince $\\Hom(\\mathbf{1}_\\tau,\ + \ Q^P) = (Q^P)(\\mathbf{1}_\\tau) = \\Gamma(Q^P)$\nby Yoneda, the first claim follows.\n\n\\textbf{Internal composition.}\n\ + Given exponentials $Q^P$, $R^Q$, and $R^P$,\nthe composition morphism\n$\\mathrm{comp} : Q^P \\times R^Q \\to R^P$\nis\ + \ the transpose of the composite:\n\\[\n Q^P \\times R^Q \\times P\n \\xrightarrow{\\;\\id \\times \\mathrm{ev}\\\ + ;}\n Q^P \\times Q\n \\xrightarrow{\\;\\mathrm{ev}\\;}\n R,\n\\]\nusing the evaluation from $R^Q$ and then from\ + \ $Q^P$\n(swapping factors as needed).\nThis exists in $\\mathcal{E}_\\tau$\nby Theorem~\\ref{thm:cartesian-closed}.\n\ + \n\\textbf{Internal identity.}\nThe identity $\\id_P : P \\to P$\ncorresponds under the adjunction\nto the global section\n\ + $\\id \\in \\Gamma(P^P) = (P^P)(\\mathbf{1}_\\tau)$.\nIn the thin setting,\n$(P^P)(X) = \\{*\\}$ for all $X$\n(since $\\\ + mathrm{supp}(P) \\cap {\\downarrow}X\n\\subseteq \\mathrm{supp}(P)$ is tautological),\nso $P^P$ is the terminal presheaf\n\ + and $\\Gamma(P^P) = \\{*\\}$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 147\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch59-internal-hom.tex` lines 399-422\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.InternalHom`\n- Name: `Tau.Topos.self_enrichment`\n\ + \n## Dependencies\n\n- Canonical: I.D64, I.T28\n\n## Related Results\n\n_Generated by later projection phases._\n\n##\ + \ Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E_tau is self-enriched: internal hom gives an internal presheaf of morphisms. Every internal hom Q^P is itself + a Presheaf.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001152.md b/_corpus_v3_items/cid001152.md new file mode 100644 index 000000000..5516370a3 --- /dev/null +++ b/_corpus_v3_items/cid001152.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001152/ +title: Passage to Book II +cid: cid001152 +primary_alias: PRP0030 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Passage to Book II: the canonical export structure. Book I delivers Cat_tau, the earned topos E_tau, the holomorphic + function space Hol(L), the Identity Theorem, and the four-valued subobject classifier. Book II will use these to prove O(tau^3) + = A_spec(L).' +aliases: +- PRP0030 +- I.P29 +- passage-to-book-ii +- prop:passage-to-book-ii +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 63 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Passage to Book II**\n\nPassage to Book II: the canonical export structure. Book I delivers Cat_tau, the earned\ + \ topos E_tau, the holomorphic function space Hol(L), the Identity Theorem, and the four-valued subobject classifier.\ + \ Book II will use these to prove O(tau^3) = A_spec(L).\n\n# Passage to Book II\n\n## Summary\n\nPassage to Book II: the\ + \ canonical export structure. Book I delivers Cat_tau, the earned topos E_tau, the holomorphic function space Hol(L),\ + \ the Identity Theorem, and the four-valued subobject classifier. Book II will use these to prove O(tau^3) = A_spec(L).\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:passage-to-book-ii}\nThe following structures, earned in Book~I,\nsuffice\ + \ for the Central Theorem of Book~II:\n\\begin{enumerate}\n \\item $\\tau^3 = \\tau^1 \\times_f T^2$\n with\ + \ $\\dim_\\tau = 4$\n (Chapter~\\ref{ch:dimension-fibration}).\n \\item The algebraic lemniscate~$\\mathbb{L}$\n\ + \ (Theorem~\\ref{thm:algebraic-lemniscate}, I.D18).\n \\item Split-complex scalars with $\\jj^2 = +1$\n \ + \ (Definition~\\ref{def:split-complex}, I.D20).\n \\item Master constant $\\iota_\\tau = 2/(\\pi + e)$\n \ + \ (Definition~\\ref{def:iota-tau}, I.D34).\n \\item $\\mathrm{Hol}(\\mathbb{L})$\n with monoid and ring\ + \ structure\n (Definition~\\ref{def:hol-L}, I.D49).\n \\item Spectral coefficients and decomposition\n \ + \ (Definition~\\ref{def:spectral-coefficients},\n Chapter~\\ref{ch:spectral-decomposition}).\n \\item\ + \ Tower coherence (I.D46)\n and the $\\tau$-Identity Theorem (I.T21).\n \\item Global Hartogs (I.T06)\n \ + \ and interior determination (I.C02).\n \\item The earned topos (I.X02).\n\\end{enumerate}\nNo additional axioms\ + \ beyond those of Book~I\nare required.\n```\n\n## Proof / Justification\n\n```latex\nThe Central Theorem asserts\n$\\\ + mathcal{O}(\\tau^3) \\cong A_{\\mathrm{spec}}(\\mathbb{L})$.\nThe left-hand side requires items~1, 3, 5, 7, 8;\nthe right-hand\ + \ side requires items~2, 4, 6.\nThe isomorphism will be constructed in Book~II\nusing the earned categorical apparatus\ + \ (item~9).\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 157\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch63-boundary-interior-passage.tex`\ + \ lines 174-203\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.BoundaryInterior`\n\ + - Name: `Tau.Holomorphy.book_i_export`\n\n## Dependencies\n\n- Canonical: I.C02, I.D51, I.D59\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Passage to Book II: the canonical export structure. Book I delivers Cat_tau, the earned topos E_tau, the holomorphic + function space Hol(L), the Identity Theorem, and the four-valued subobject classifier. Book II will use these to prove O(ta' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001153.md b/_corpus_v3_items/cid001153.md new file mode 100644 index 000000000..0f7997fb2 --- /dev/null +++ b/_corpus_v3_items/cid001153.md @@ -0,0 +1,222 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001153/ +title: Counting as Structural Feature +cid: cid001153 +primary_alias: PRP0031 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Countability is not a limitation but a structural feature: every tau-object is reachable by finite rho-iteration, + so every tau-set is at most countable by construction.' +aliases: +- PRP0031 +- I.P33 +- counting-as-structural-feature +- prop:counting-structural +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005342 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 36 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Counting as Structural Feature** + + + Countability is not a limitation but a structural feature: every tau-object is reachable by finite rho-iteration, so every + tau-set is at most countable by construction. + + + # Counting as Structural Feature + + + ## Summary + + + Countability is not a limitation but a structural feature: every tau-object is reachable by finite rho-iteration, so every + tau-set is at most countable by construction. + + + ## Statement + + + ```latex + + % + + \label{prop:counting-structural} + + The countability of $\Obj(\tau)$ + + (Theorem~\ref{thm:ontic-closure}) + + is not a limitation imposed from outside + + but a structural consequence + + of the generative counting principle + + (Definition~\ref{def:generative-counting}): + + a universe built by counting is a counted universe. + + The question ``is $\tau$ countable?'''' + + presupposes an external vantage point + + that the ontically sealed universe ($\KAxiom{6}$) + + does not provide. + + ``` + + + ## Proof / Justification + + + ```latex + + By the Ontic Closure Theorem (I.T01), + + $\Obj(\tau) = \{\omega\} \cup O_\alpha + + \cup O_\pi \cup O_\gamma \cup O_\eta$. + + Each orbit $O_g$ is countably infinite + + (Proposition~\ref{prop:orbit-countable}), + + and a finite union of countable sets is countable. + + Thus $\Obj(\tau)$ is countable. + + + But this is more than an external observation. + + The bijection $\varphi_g(n) = \rho^n(g)$ + + (Definition~\ref{def:generative-counting}) + + is not applied \emph{after} the universe is built + + to ``measure'''' its size. + + It \emph{is} the construction. + + Each application of $\rho$ + + simultaneously creates an object + + and assigns it an index. + + The universe is therefore not merely + + ``countable in the sense that a bijection to $\mathbb{N}$ exists'''' + + --- it is \emph{counted} in the sense that + + the generative process itself is the counting. + + + The second claim follows from Object Closure ($\KAxiom{6}$): + + the universe is ontically sealed. + + There is no ``view from outside'''' $\Obj(\tau)$ + + from which to classify its cardinality + + relative to other infinite sets. + + The only infinite cardinal + + that can be formulated within $\tau$ + + is $\aleph_0$, and it is the cardinality + + of the entire universe. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 159 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch36-countability.tex` lines 420-434 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Sets.Counting` + + - Name: `Tau.Sets.counting_structural` + + + ## Dependencies + + + - Canonical: I.D75 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Countability is not a limitation but a structural feature: every tau-object is reachable by finite rho-iteration, + so every tau-set is at most countable by construction.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001154.md b/_corpus_v3_items/cid001154.md new file mode 100644 index 000000000..be3edc34c --- /dev/null +++ b/_corpus_v3_items/cid001154.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001154/ +title: No Unearned Decimal Diagonal +cid: cid001154 +primary_alias: PRP0032 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No unearned decimal diagonal: tau-reals are constructive Cauchy sequences with explicit moduli. The diagonal construction + requires non-constructive choice over uncountably many positions, which tau does not provide.' +aliases: +- PRP0032 +- I.P34 +- no-unearned-decimal-diagonal +- prop:no-unearned-decimal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005338 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 37 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Unearned Decimal Diagonal**\n\nNo unearned decimal diagonal: tau-reals are constructive Cauchy sequences\ + \ with explicit moduli. The diagonal construction requires non-constructive choice over uncountably many positions, which\ + \ tau does not provide.\n\n# No Unearned Decimal Diagonal\n\n## Summary\n\nNo unearned decimal diagonal: tau-reals are\ + \ constructive Cauchy sequences with explicit moduli. The diagonal construction requires non-constructive choice over\ + \ uncountably many positions, which tau does not provide.\n\n## Statement\n\n```latex\n%\n\\label{prop:no-unearned-decimal}\n\ + In the constructive framework of $\\tau$,\nthe diagonal function\n\\[\n d(n) := \\text{``$n$-th digit of $f(n)$''}\n\ + \\]\nis not a total computable function\non the space of programs-for-reals.\nThe diagonal construction requires\nuniform\ + \ decidability of digit extraction\nacross all computable Cauchy sequences,\nwhich is not constructively available.\n\ + ```\n\n## Proof / Justification\n\n```latex\nSuppose $f \\colon \\mathbb{N} \\to \\mathbb{R}_\\tau$\nis an enumeration\ + \ of the computable reals,\nwhere each $f(n)$ is given by a program $P_n$\nthat outputs a Cauchy sequence $(q_{n,k})_{k\ + \ \\geq 0}$\nwith explicit modulus of convergence $M_n$.\nTo extract the $n$-th binary digit of $f(n)$,\none must:\n\\\ + begin{enumerate}\n \\item Find an index $k_0$ such that\n $|q_{n,k_0} - f(n)| < 2^{-(n+2)}$\n (sufficient\ + \ precision to determine bit~$n$).\n \\item Read the $n$-th bit from the binary expansion\n of $q_{n,k_0}$.\n\ + \\end{enumerate}\nStep~(1) requires computing $M_n(n+2)$:\nthe modulus of convergence of the $n$-th program\nat precision\ + \ $n+2$.\nBut the function $n \\mapsto M_n(n+2)$\nis a diagonalization across all moduli ---\na function that uniformly\ + \ evaluates\nan arbitrary program-index on an arbitrary input.\nBy a standard argument\n(the diagonal function of an effective\ + \ enumeration\nof total computable functions\nis not itself in the enumeration),\nthis uniform extraction is not a total\ + \ computable function.\nHence $d$ is not constructively definable\nas an element of $\\mathbb{R}_\\tau$.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 161\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch37-cantor-diagonal.tex`\ + \ lines 174-188\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.CantorRefutation`\n\ + - Name: `Tau.Sets.no_unearned_decimal`\n\n## Dependencies\n\n- Canonical: I.T35\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No unearned decimal diagonal: tau-reals are constructive Cauchy sequences with explicit moduli. The diagonal + construction requires non-constructive choice over uncountably many positions, which tau does not provide.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001155.md b/_corpus_v3_items/cid001155.md new file mode 100644 index 000000000..823db9907 --- /dev/null +++ b/_corpus_v3_items/cid001155.md @@ -0,0 +1,209 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001155/ +title: No Unrestricted Comprehension +cid: cid001155 +primary_alias: PRP0033 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No unrestricted comprehension: tau-sets are bounded powersets (divisor sets), not arbitrary collections. The set + {x : x divides n} is always finite, blocking Russell-type diagonal constructions.' +aliases: +- PRP0033 +- I.P35 +- no-unrestricted-comprehension +- prop:no-comprehension +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005339 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 37 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**No Unrestricted Comprehension** + + + No unrestricted comprehension: tau-sets are bounded powersets (divisor sets), not arbitrary collections. The set {x : + x divides n} is always finite, blocking Russell-type diagonal constructions. + + + # No Unrestricted Comprehension + + + ## Summary + + + No unrestricted comprehension: tau-sets are bounded powersets (divisor sets), not arbitrary collections. The set {x : + x divides n} is always finite, blocking Russell-type diagonal constructions. + + + ## Statement + + + ```latex + + % + + \label{prop:no-comprehension} + + Category~$\tau$ has no comprehension scheme. + + The only set-formation principle is the + + \textbf{bounded powerset} + + $\mathcal{P}_\tau(\underline{x}) + + = \{\underline{a} : \underline{a} \mid \underline{x}\}$ + + (Definition~\ref{def:bounded-powerset}), + + which is finite for every $\underline{x}$. + + There is no mechanism to form + + ``the set of all objects satisfying property~$\varphi$'''' + + for an arbitrary predicate~$\varphi$. + + ``` + + + ## Proof / Justification + + + ```latex + + In $\tau$-set theory, + + sets are natural numbers in $\tau$-Idx, + + and membership is divisibility + + (Chapter~\ref{ch:membership-divisibility}). + + The powerset of $\underline{x}$ is the set of divisors + + of $\underline{x}$, which is finite + + (Proposition~\ref{prop:powerset-cardinality}). + + To form a subset by comprehension, + + one would need a mechanism + + that takes a predicate $\varphi$ + + and returns a natural number $\underline{y}$ + + whose divisors are exactly those $\underline{a}$ + + satisfying $\varphi(\underline{a})$. + + But the divisors of any natural number + + are determined by its prime factorization --- + + a fixed, finite, arithmetic structure --- + + and there is no operation in $\tau$ + + that converts an arbitrary logical predicate + + into a prime factorization. + + The $\tau$-axioms provide only earned operations: + + $\rho$ (successor), addition, multiplication, + + exponentiation, tetration. + + None of these constructs subsets from predicates. + + Hence no comprehension schema exists in~$\tau$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 162 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch37-cantor-diagonal.tex` lines + 269-282 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Sets.CantorRefutation` + + - Name: `Tau.Sets.no_comprehension` + + + ## Dependencies + + + - Canonical: I.T35 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No unrestricted comprehension: tau-sets are bounded powersets (divisor sets), not arbitrary collections. The + set {x : x divides n} is always finite, blocking Russell-type diagonal constructions.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001156.md b/_corpus_v3_items/cid001156.md new file mode 100644 index 000000000..8fb7c6473 --- /dev/null +++ b/_corpus_v3_items/cid001156.md @@ -0,0 +1,255 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001156/ +title: No Free Cartesian Diagonal +cid: cid001156 +primary_alias: PRP0034 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No free Cartesian diagonal: the diagonal discipline (I.D03) prevents free contraction/reuse, blocking the Cartesian + product diagonal map that Cantor''s argument requires.' +aliases: +- PRP0034 +- I.P36 +- no-free-cartesian-diagonal +- prop:no-free-cartesian +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005340 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 37 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**No Free Cartesian Diagonal** + + + No free Cartesian diagonal: the diagonal discipline (I.D03) prevents free contraction/reuse, blocking the Cartesian product + diagonal map that Cantor''s argument requires. + + + # No Free Cartesian Diagonal + + + ## Summary + + + No free Cartesian diagonal: the diagonal discipline (I.D03) prevents free contraction/reuse, blocking the Cartesian product + diagonal map that Cantor''s argument requires. + + + ## Statement + + + ```latex + + % + + \label{prop:no-free-cartesian} + + In Category~$\tau$, + + the self-pairing map + + $\Delta \colon \tau\text{-Idx} \to + + \tau\text{-Idx} \times \tau\text{-Idx}$, + + $\underline{n} \mapsto (\underline{n}, \underline{n})$, + + while definable as an arithmetic function on indices, + + does not constitute an \emph{earned} morphism + + in the categorical sense. + + The diagonal discipline + + (Definition~\ref{def:diagonal-discipline}) + + prevents self-referential structure + + from generating objects + + outside the existing orbit rays. + + The self-pairing required by Cantor''s argument + + is an \textbf{unearned diagonal}: + + it demands that an enumeration + + be able to interrogate itself at its own index, + + which is precisely the kind of + + self-application that the $1{+}3$ channel structure + + was designed to regulate. + + ``` + + + ## Proof / Justification + + + ```latex + + By the diagonal discipline + + (Definition~\ref{def:diagonal-discipline}), + + the self-product of an operation at level~$k$ + + cannot be absorbed within the same orbit channel; + + it must overflow into a distinct channel at level~$k{+}1$. + + The Cantor diagonal requires the composite + + $n \mapsto f(n)(n)$, + + which is a self-application of the evaluation map: + + the same index $n$ serves simultaneously + + as the \emph{selector} (which sequence to examine) + + and as the \emph{position} (which digit to read). + + This double role constitutes a diagonal rewiring + + in the sense of~I.D03: + + the evaluation map at level~$k$ is being ``fed back'''' + + through its own index domain. + + + In the $\tau$-framework, + + such a rewiring is permitted only when + + a dedicated orbit channel absorbs the overflow + + (Section~\ref{sec:ch05-first-rewiring} + + through~\ref{sec:ch05-third-rewiring}). + + The three solenoidal channels + + $O_\pi$, $O_\gamma$, $O_\eta$ + + absorb the three legitimate rewirings + + (multiplication, exponentiation, tetration). + + A fourth rewiring --- which the Cantor diagonal + + would constitute --- + + has no channel to absorb it, + + because $\KAxiom{6}$ (Object Closure) limits + + the universe to exactly four orbit rays. + + + Hence the self-pairing $\Delta$, + + interpreted as a categorical morphism + + that enables diagonal self-interrogation, + + is not an earned arrow of~$\tau$. + + It can be \emph{written down} as arithmetic, + + but it cannot be \emph{used} to generate + + new objects outside $\Obj(\tau)$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 163 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch37-cantor-diagonal.tex` lines + 400-423 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Sets.CantorRefutation` + + - Name: `Tau.Sets.no_free_cartesian` + + + ## Dependencies + + + - Canonical: I.T35 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No free Cartesian diagonal: the diagonal discipline (I.D03) prevents free contraction/reuse, blocking the Cartesian + product diagonal map that Cantor''s argument requires.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001157.md b/_corpus_v3_items/cid001157.md new file mode 100644 index 000000000..aec38c231 --- /dev/null +++ b/_corpus_v3_items/cid001157.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001157/ +title: Ultrametric Replaces Cardinality +cid: cid001157 +primary_alias: PRP0035 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The ultrametric structure on omega-germs (primorial divergence depth) replaces the cardinality hierarchy. Size distinctions + between infinite sets are replaced by depth-of-agreement distinctions between omega-germs. +aliases: +- PRP0035 +- I.P37 +- ultrametric-replaces-cardinality +- prop:ultrametric-replaces-card +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005351 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 38 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ultrametric Replaces Cardinality**\n\nThe ultrametric structure on omega-germs (primorial divergence depth)\ + \ replaces the cardinality hierarchy. Size distinctions between infinite sets are replaced by depth-of-agreement distinctions\ + \ between omega-germs.\n\n# Ultrametric Replaces Cardinality\n\n## Summary\n\nThe ultrametric structure on omega-germs\ + \ (primorial divergence depth) replaces the cardinality hierarchy. Size distinctions between infinite sets are replaced\ + \ by depth-of-agreement distinctions between omega-germs.\n\n## Statement\n\n```latex\n%\n\\label{prop:ultrametric-replaces-card}\n\ + The Cauchy completeness of $\\mathbb{R}_\\tau$\nand the profinite completeness of $\\hat{\\mathbb{Z}}_\\tau$\nboth derive\ + \ from the same source:\nthe \\textbf{ultrametric structure on omega-germs}.\nTwo completions ---\none Archimedean ($\\\ + mathbb{R}_\\tau$),\none non-Archimedean ($\\hat{\\mathbb{Z}}_\\tau$) ---\narise from the single convergence mechanism\n\ + of the primorial ladder.\nCardinality is unnecessary; convergence suffices.\n```\n\n## Proof / Justification\n\n```latex\n\ + The primorial ladder\n$M_1 \\mid M_2 \\mid M_3 \\mid \\cdots$\ndefines a filtration on the integers:\n\\[\n \\mathbb{Z}\ + \ \\twoheadrightarrow \\cdots\n \\twoheadrightarrow \\mathbb{Z}/M_3\n \\twoheadrightarrow \\mathbb{Z}/M_2\n \\\ + twoheadrightarrow \\mathbb{Z}/M_1.\n\\]\nThe inverse limit of this system\nis $\\hat{\\mathbb{Z}}_\\tau$, which is profinitely\ + \ complete\nby construction.\nThe CRT decomposition\n$\\hat{\\mathbb{Z}}_\\tau \\cong \\prod_p \\mathbb{Z}_p$\n(Chapter~\\\ + ref{ch:profinite-boundary-ring})\nendows each factor $\\mathbb{Z}_p$ with the $p$-adic ultrametric:\n\\[\n d_p(x, y)\ + \ = p^{-v_p(x - y)},\n\\]\nwhere $v_p$ is the $p$-adic valuation.\nThis is a non-Archimedean metric.\n\nOn the Archimedean\ + \ side,\nthe constructive reals $\\mathbb{R}_\\tau$\nare the Cauchy completion of\n$\\mathbb{Q}_\\tau = \\mathrm{Frac}(\\\ + tau\\text{-Idx})$\nunder the standard absolute value.\nThe completeness of $\\mathbb{R}_\\tau$\nfollows from the explicit-modulus\ + \ requirement:\nevery Cauchy sequence with computable modulus\ndetermines a unique equivalence class.\n\nBoth completions\ + \ arise from the same data ---\nthe arithmetic of $\\tau$-Idx and its primorial filtration.\nThe Archimedean completion\ + \ uses the ordinary metric;\nthe non-Archimedean completion uses the profinite metric.\nNo cardinality argument is invoked\ + \ in either case.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 166\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch38-approaches-infinity.tex`\ + \ lines 305-318\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.UniqueInfinity`\n\ + - Name: `Tau.Sets.ultrametric_replaces_card`\n\n## Dependencies\n\n- Canonical: I.D76, I.T36\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The ultrametric structure on omega-germs (primorial divergence depth) replaces the cardinality hierarchy. Size + distinctions between infinite sets are replaced by depth-of-agreement distinctions between omega-germs. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001158.md b/_corpus_v3_items/cid001158.md new file mode 100644 index 000000000..d5dfb6a8c --- /dev/null +++ b/_corpus_v3_items/cid001158.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001158/ +title: Classical.em Eliminability +cid: cid001158 +primary_alias: PRP0036 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Both uses of Classical.em in TauLib (Coordinates/Primes.lean lines 110, 172) are applied to decidable predicates + (divisibility of natural numbers) and can be replaced by decidable case analysis without changing any theorem statement. +aliases: +- PRP0036 +- I.P38 +- classical-em-eliminability +- prop:em-eliminable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 70 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Classical.em Eliminability**\n\nBoth uses of Classical.em in TauLib (Coordinates/Primes.lean lines 110, 172)\ + \ are applied to decidable predicates (divisibility of natural numbers) and can be replaced by decidable case analysis\ + \ without changing any theorem statement.\n\n# Classical.em Eliminability\n\n## Summary\n\nBoth uses of Classical.em in\ + \ TauLib (Coordinates/Primes.lean lines 110, 172) are applied to decidable predicates (divisibility of natural numbers)\ + \ and can be replaced by decidable case analysis without changing any theorem statement.\n\n## Statement\n\n```latex\n\ + %\n\\label{prop:em-eliminable}\nBoth uses of \\texttt{Classical.em} in TauLib\nare applied to decidable predicates\nand\ + \ can be replaced by decidable case analysis\n(\\texttt{Decidable.em} or the \\texttt{by\\_cases} tactic\nwith an explicit\ + \ \\texttt{Decidable} instance)\nwithout changing any theorem statement or proof structure.\nAfter replacement, the \\\ + texttt{Coordinates/Primes} module\nreclassifies from \\textbf{Cl}~(classical) to\n\\textbf{C}~(fully constructive).\n\ + ```\n\n## Proof / Justification\n\n```latex\nIn both cases, the predicate $P$ appearing in\n\\texttt{Classical.em $P$}\ + \ has a \\texttt{Decidable} instance:\n\\begin{itemize}\n \\item Site~1: $P \\equiv \\forall\\, d : \\texttt{TauIdx},\\\ + ;\n d \\mid n \\to d = 1 \\lor d = n$.\n At each tower stage, \\texttt{TauIdx} is \\texttt{Fin}~$M_k$\n\ + \ for some primorial $M_k$.\n Divisibility on \\texttt{Fin}~$n$\n is decidable (reduce to \\\ + texttt{Nat.decidable\\_dvd}).\n Equality on \\texttt{Fin}~$n$ is decidable.\n The disjunction and implication\ + \ of decidable propositions\n are decidable.\n The universal quantifier over a finite type\n \ + \ of decidable propositions is decidable.\n \\item Site~2: $P \\equiv p \\mid a$\n where $p, a : \\mathbb{N}$.\n\ + \ \\texttt{Nat.decidable\\_dvd} provides the instance directly.\n\\end{itemize}\nIn each case, replacing\n\\\ + texttt{rcases Classical.em $P$ with h $|$ h}\nby\n\\texttt{rcases Decidable.em $P$ with h $|$ h}\nproduces an identical\ + \ proof term\nexcept that the axiom dependency on\n\\texttt{Classical.em} is removed.\nThe resulting module passes\n\\\ + texttt{\\#print axioms} with no classical axioms.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line\ + \ 174\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch70-linearity-audit.tex`\ + \ lines 446-458\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.LinearityAudit`\n\ + - Name: `Tau.MetaLogic.em_eliminable`\n\n## Dependencies\n\n- Canonical: I.T37\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Both uses of Classical.em in TauLib (Coordinates/Primes.lean lines 110, 172) are applied to decidable predicates + (divisibility of natural numbers) and can be replaced by decidable case analysis without changing any theorem statement. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001159.md b/_corpus_v3_items/cid001159.md new file mode 100644 index 000000000..6bf6da239 --- /dev/null +++ b/_corpus_v3_items/cid001159.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001159/ +title: TauReal Ring Axioms +cid: cid001159 +primary_alias: PRP0037 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'TauReal satisfies all commutative ring axioms up to equivalence: commutativity, associativity, identities, inverses, + and distributivity. Each axiom reduces to the corresponding TauRat axiom applied pointwise.' +aliases: +- PRP0037 +- I.P39 +- taureal-ring-axioms +- prop:ordered-field-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 76 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**TauReal Ring Axioms**\n\nTauReal satisfies all commutative ring axioms up to equivalence: commutativity, associativity,\ + \ identities, inverses, and distributivity. Each axiom reduces to the corresponding TauRat axiom applied pointwise.\n\n\ + # TauReal Ring Axioms\n\n## Summary\n\nTauReal satisfies all commutative ring axioms up to equivalence: commutativity,\ + \ associativity, identities, inverses, and distributivity. Each axiom reduces to the corresponding TauRat axiom applied\ + \ pointwise.\n\n## Statement\n\n```latex\n%\n\\label{prop:ordered-field-axioms}\n$(\\mathbb{R}_\\tau, +, \\cdot, <)$ is\ + \ a \\textbf{totally ordered field}:\n\\begin{enumerate}\n \\item \\textbf{Commutativity.}\n $x + y = y +\ + \ x$ and $x \\cdot y = y \\cdot x$.\n \\item \\textbf{Associativity.}\n $(x + y) + z = x + (y + z)$\n \ + \ and $(x \\cdot y) \\cdot z = x \\cdot (y \\cdot z)$.\n \\item \\textbf{Distributivity.}\n $x \\cdot\ + \ (y + z) = x \\cdot y + x \\cdot z$.\n \\item \\textbf{Identities and inverses.}\n $x + 0 = x$, $x \\cdot\ + \ 1 = x$,\n $x + (-x) = 0$, $x \\cdot x^{-1} = 1$ for $x \\neq 0$.\n \\item \\textbf{Order compatibility.}\n\ + \ $x < y \\implies x + z < y + z$;\n $x < y,\\, z > 0 \\implies xz < yz$.\n \\item \\textbf{Trichotomy}\ + \ (apartness form).\n For $x \\neq y$, either $x < y$ or $y < x$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\n[Proof sketch]\nItems (1)--(4) follow from the corresponding properties\non $\\mathbb{Q}_\\tau$ (Proposition~\\\ + ref{prop:q-tau-field}),\ntransported through term-by-term operations.\nThe main subtlety is the multiplicative inverse:\n\ + if $[q_n] \\neq 0$, the explicit modulus guarantees\n$|q_n|$ is eventually bounded away from zero.\nItems (5)--(6) are\ + \ direct calculations on representatives.\nThe verification is standard\n(cf.\\ Bishop \\& Bridges, \\emph{Constructive\ + \ Analysis}, Ch.~2).\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 189\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch76-constructive-reals.tex`\ + \ lines 69-90\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.ConstructiveReals`\n\ + - Name: `Tau.Boundary.taureal_ring_axioms`\n\n## Dependencies\n\n- Canonical: I.D84\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'TauReal satisfies all commutative ring axioms up to equivalence: commutativity, associativity, identities, inverses, + and distributivity. Each axiom reduces to the corresponding TauRat axiom applied pointwise.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001160.md b/_corpus_v3_items/cid001160.md new file mode 100644 index 000000000..7f975cfb4 --- /dev/null +++ b/_corpus_v3_items/cid001160.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001160/ +title: Extensionality +cid: cid001160 +primary_alias: PRP0038 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Extensionality of the orbit-set map: the map Set is injective on each orbit and globally. Distinct tau-elements + represent distinct sets. Alpha-orbit injectivity via unique maximum alpha_n; pi-orbit via cardinality; gamma/eta via structural + signatures.' +aliases: +- PRP0038 +- I.P40 +- extensionality +- prop:orbit-set-extensional +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 83 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Extensionality**\n\nExtensionality of the orbit-set map: the map Set is injective on each orbit and globally.\ + \ Distinct tau-elements represent distinct sets. Alpha-orbit injectivity via unique maximum alpha_n; pi-orbit via cardinality;\ + \ gamma/eta via structural signatures.\n\n# Extensionality\n\n## Summary\n\nExtensionality of the orbit-set map: the map\ + \ Set is injective on each orbit and globally. Distinct tau-elements represent distinct sets. Alpha-orbit injectivity\ + \ via unique maximum alpha_n; pi-orbit via cardinality; gamma/eta via structural signatures.\n\n## Statement\n\n```latex\n\ + %\n\\label{prop:orbit-set-extensional}\nThe orbit-set map is \\textbf{injective}:\n\\[\n \\mathrm{Set}(X) = \\mathrm{Set}(Y)\n\ + \ \\quad\\Longrightarrow\\quad\n X = Y.\n\\]\nIn particular, distinct $\\tau$-elements\nrepresent distinct sets.\n\ + ```\n\n## Proof / Justification\n\n```latex\nWe verify injectivity on each orbit\nand then across orbits.\n\n\\medskip\n\ + \\noindent\n\\textbf{$\\alpha$-orbit.}\n$\\mathrm{Set}(\\alpha_n) = \\{\\alpha_k : k \\mid n\\}$\nis the divisor set.\n\ + For $n \\neq 0$, $\\alpha_n$ is the unique maximum\n(every other $\\alpha_k$ satisfies $k < n$).\nIf $\\mathrm{Set}(\\\ + alpha_n) = \\mathrm{Set}(\\alpha_m)$,\nthen the two sets share the same maximum,\nso $n = m$.\n\n\\medskip\n\\noindent\n\ + \\textbf{$\\pi$-orbit.}\n$|\\mathrm{Set}(\\pi_n)| = n + 1$,\nsince the first $n$ primes are distinct\nand $\\alpha_1$\ + \ is not equal to any $\\alpha_{p_k}$\nfor $k \\geq 1$.\nIf $\\mathrm{Set}(\\pi_n) = \\mathrm{Set}(\\pi_m)$,\nthen $n\ + \ + 1 = m + 1$, so $n = m$.\n\n\\medskip\n\\noindent\n\\textbf{$\\gamma$-orbit.}\nThe smallest element strictly greater\ + \ than $\\alpha_1$\nin $\\mathrm{Set}(\\gamma_n)$ is $\\alpha_{p_n}$.\nThis determines $p_n$, and since\nthe prime-indexing\ + \ function $n \\mapsto p_n$\nis injective,\n$\\mathrm{Set}(\\gamma_n) = \\mathrm{Set}(\\gamma_m)$\nimplies $p_n = p_m$,\ + \ hence $n = m$.\n\n\\medskip\n\\noindent\n\\textbf{$\\eta$-orbit.}\nEvery element of $\\mathrm{Set}(\\eta_n)$\nhas the\ + \ form $\\alpha_{p_k^n}$.\nFrom any single element $\\alpha_{p_k^n}$,\nthe exponent $n$ is determined\n(since $p_k$ is\ + \ prime, the $n$-th power\nis the unique factorization exponent).\nIf $\\mathrm{Set}(\\eta_n) = \\mathrm{Set}(\\eta_m)$,\n\ + then $n = m$.\n\n\\medskip\n\\noindent\n\\textbf{Crossing point.}\n$\\mathrm{Set}(\\omega) = O_\\alpha \\cup \\{\\omega\\\ + }$\nis the unique set containing $\\omega$ as an element.\nNo other set family produces a set containing $\\omega$.\n\n\ + \\medskip\n\\noindent\n\\textbf{Cross-orbit injectivity.}\nThe five families are distinguishable by structural type:\n\ + \\begin{itemize}\n \\item $\\alpha$-orbit sets are \\emph{finite} divisor sets,\n each containing its representing\ + \ element $\\alpha_n$\n as the unique maximum.\n \\item $\\pi$-orbit sets are \\emph{finite}\n and\ + \ consist entirely of\n $\\alpha_1$ plus elements $\\alpha_{p_k}$\n indexed by \\emph{consecutive} primes.\n\ + \ \\item $\\gamma$-orbit sets are \\emph{infinite},\n and every element is a power $\\alpha_{p_n^k}$\n \ + \ of a single prime.\n \\item $\\eta$-orbit sets are \\emph{infinite},\n and the elements $\\alpha_{p_k^n}$\n\ + \ range over \\emph{all} primes\n with a fixed exponent.\n \\item $\\mathrm{Set}(\\omega)$ is the\ + \ unique set\n containing $\\omega$.\n\\end{itemize}\nThese structural signatures are mutually exclusive,\nso\ + \ no cross-orbit collision is possible.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 213\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch83-orbit-set-correspondence.tex` lines 277-288\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.OrbitSets`\n- Name: `Tau.Sets.orbit_set_extensional`\n\ + \n## Dependencies\n\n- Canonical: I.D94\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Extensionality of the orbit-set map: the map Set is injective on each orbit and globally. Distinct tau-elements + represent distinct sets. Alpha-orbit injectivity via unique maximum alpha_n; pi-orbit via cardinality; gamma/eta via structural ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001161.md b/_corpus_v3_items/cid001161.md new file mode 100644 index 000000000..6743bd7f8 --- /dev/null +++ b/_corpus_v3_items/cid001161.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001161/ +title: Self-Containment Partition +cid: cid001161 +primary_alias: PRP0039 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Self-Containment Partition: X in Set(X) iff X in O_alpha union {omega}. The alpha-orbit and omega are ''opaque'' + (self-containing); the solenoidal orbits O_pi, O_gamma, O_eta are ''transparent'' (non-self-containing organizing principles). + Third position between ZFC (never reflexive) and divisibility (always reflexive).' +aliases: +- PRP0039 +- I.P41 +- self-containment-partition +- prop:self-containment +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 83 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Self-Containment Partition**\n\nSelf-Containment Partition: X in Set(X) iff X in O_alpha union {omega}. The\ + \ alpha-orbit and omega are 'opaque' (self-containing); the solenoidal orbits O_pi, O_gamma, O_eta are 'transparent' (non-self-containing\ + \ organizing principles). Third position between ZFC (never reflexive) and divisibility (always reflexive).\n\n# Self-Containment\ + \ Partition\n\n## Summary\n\nSelf-Containment Partition: X in Set(X) iff X in O_alpha union {omega}. The alpha-orbit and\ + \ omega are 'opaque' (self-containing); the solenoidal orbits O_pi, O_gamma, O_eta are 'transparent' (non-self-containing\ + \ organizing principles). Third position between ZFC (never reflexive) and divisibility (always reflexive).\n\n## Statement\n\ + \n```latex\n%\n\\label{prop:self-containment}\nA $\\tau$-element $X$ belongs to its own set\nif and only if $X$ lies in\ + \ $O_\\alpha \\cup \\{\\omega\\}$:\n\\[\n \\boxed{%\n X \\in \\mathrm{Set}(X)\n \\quad\\iff\\quad\n X \\in\ + \ O_\\alpha \\cup \\{\\omega\\}.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nWe check each orbit.\n\n\\medskip\n\ + \\noindent\n\\textbf{$\\alpha$-orbit.}\n$\\mathrm{Set}(\\alpha_n) = \\{\\alpha_k : k \\mid n\\}$.\nSince $n \\mid n$ (divisibility\ + \ is reflexive),\n$\\alpha_n \\in \\mathrm{Set}(\\alpha_n)$.\n\n\\medskip\n\\noindent\n\\textbf{Crossing point.}\n$\\\ + mathrm{Set}(\\omega) = O_\\alpha \\cup \\{\\omega\\}$,\nand $\\omega \\in \\{\\omega\\} \\subseteq \\mathrm{Set}(\\omega)$.\n\ + \n\\medskip\n\\noindent\n\\textbf{$\\pi$-orbit.}\n$\\mathrm{Set}(\\pi_n) \\subseteq O_\\alpha$,\nbut $\\pi_n \\in O_\\\ + pi$.\nSince $O_\\alpha$ and $O_\\pi$ are disjoint\n(Proposition~\\ref{prop:orbit-disjoint}),\n$\\pi_n \\notin \\mathrm{Set}(\\\ + pi_n)$.\n\n\\medskip\n\\noindent\n\\textbf{$\\gamma$-orbit.}\n$\\mathrm{Set}(\\gamma_n) \\subseteq O_\\alpha$,\nbut $\\\ + gamma_n \\in O_\\gamma$.\nOrbit disjointness gives $\\gamma_n \\notin \\mathrm{Set}(\\gamma_n)$.\n\n\\medskip\n\\noindent\n\ + \\textbf{$\\eta$-orbit.}\n$\\mathrm{Set}(\\eta_n) \\subseteq O_\\alpha$,\nbut $\\eta_n \\in O_\\eta$.\nOrbit disjointness\ + \ gives $\\eta_n \\notin \\mathrm{Set}(\\eta_n)$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line\ + \ 214\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch83-orbit-set-correspondence.tex`\ + \ lines 377-388\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.OrbitSets`\n\ + - Name: `Tau.Sets.self_containment_iff`\n\n## Dependencies\n\n- Canonical: I.D94, I.P03\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Self-Containment Partition: X in Set(X) iff X in O_alpha union {omega}. The alpha-orbit and omega are ''opaque'' + (self-containing); the solenoidal orbits O_pi, O_gamma, O_eta are ''transparent'' (non-self-containing organizing principles). + Thir' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001162.md b/_corpus_v3_items/cid001162.md new file mode 100644 index 000000000..3434eec3c --- /dev/null +++ b/_corpus_v3_items/cid001162.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001162/ +title: Order Bound +cid: cid001162 +primary_alias: PRP0040 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Order Bound: for all A in Set(alpha_n), A <= alpha_n in the internal order on tau-Idx. alpha_n is the maximum of + Set(alpha_n). Membership ''looks downward'' from the representing element.' +aliases: +- PRP0040 +- I.P42 +- order-bound +- prop:order-bound +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005345 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 8, Chapter 83 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Order Bound**\n\nOrder Bound: for all A in Set(alpha_n), A <= alpha_n in the internal order on tau-Idx. alpha_n\ + \ is the maximum of Set(alpha_n). Membership 'looks downward' from the representing element.\n\n# Order Bound\n\n## Summary\n\ + \nOrder Bound: for all A in Set(alpha_n), A <= alpha_n in the internal order on tau-Idx. alpha_n is the maximum of Set(alpha_n).\ + \ Membership 'looks downward' from the representing element.\n\n## Statement\n\n```latex\n%\n\\label{prop:order-bound}\n\ + For all $A \\in \\mathrm{Set}(\\alpha_n)$,\n\\[\n \\boxed{A \\;\\leq\\; \\alpha_n}\n\\]\nin the internal order on $\\\ + tau$-Idx.\nIn particular, $\\alpha_n$ is the \\textbf{maximum}\nof $\\mathrm{Set}(\\alpha_n)$.\n```\n\n## Proof / Justification\n\ + \n```latex\nEvery element $\\alpha_k \\in \\mathrm{Set}(\\alpha_n)$\nsatisfies $k \\mid n$.\nFor $n \\geq 1$,\ndivisibility\ + \ implies $k \\leq n$\n(since $k \\mid n$ and $n > 0$\nforces $k \\leq n$).\nTherefore $\\alpha_k \\leq \\alpha_n$\nin\ + \ the natural ordering on $\\tau$-Idx.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 215\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part08/ch83-orbit-set-correspondence.tex` lines 471-481\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.OrbitSets`\n- Name: `Tau.Sets.orbit_set_order_bound`\n\ + \n## Dependencies\n\n- Canonical: I.D94, I.D31\n\n## Related Results\n\n_Generated by later projection phases._\n\n##\ + \ Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Order Bound: for all A in Set(alpha_n), A <= alpha_n in the internal order on tau-Idx. alpha_n is the maximum + of Set(alpha_n). Membership ''looks downward'' from the representing element.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001163.md b/_corpus_v3_items/cid001163.md new file mode 100644 index 000000000..fb2349ee2 --- /dev/null +++ b/_corpus_v3_items/cid001163.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001163/ +title: Measure Compatibility +cid: cid001163 +primary_alias: PRP0041 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The projective family of counting measures is compatible: for any tower-measurable set, the measure at stage k+1 + (projected down) equals the measure at stage k. μ_{k+1}(π⁻¹(S_k)) = μ_k(S_k).' +aliases: +- PRP0041 +- I.P43 +- measure-compatibility +- prop:measure-compatibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005279 +- predicate: formalized_by + object: cid005280 +- predicate: formalized_by + object: cid005281 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Measure Compatibility** + + + The projective family of counting measures is compatible: for any tower-measurable set, the measure at stage k+1 (projected + down) equals the measure at stage k. μ_{k+1}(π⁻¹(S_k)) = μ_k(S_k). + + + # Measure Compatibility + + + ## Summary + + + The projective family of counting measures is compatible: for any tower-measurable set, the measure at stage k+1 (projected + down) equals the measure at stage k. μ_{k+1}(π⁻¹(S_k)) = μ_k(S_k). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 222 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Measure` + + - Name: `even_set_compatible_3` + + + ## Dependencies + + + - Canonical: I.D95, I.D96, I.T49 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The projective family of counting measures is compatible: for any tower-measurable set, the measure at stage + k+1 (projected down) equals the measure at stage k. μ_{k+1}(π⁻¹(S_k)) = μ_k(S_k).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001164.md b/_corpus_v3_items/cid001164.md new file mode 100644 index 000000000..8a3c6bc88 --- /dev/null +++ b/_corpus_v3_items/cid001164.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001164/ +title: CRT-Galois Decomposition +cid: cid001164 +primary_alias: PRP0043 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Galois group decomposes via CRT: Gal_k ≅ ∏(Z/p_i Z)× for i=1..k. Order equality φ(M_k) = ∏(p_i-1) verified computationally + at stages 1-3.' +aliases: +- PRP0043 +- I.P44 +- crt-galois-decomposition +- prop:crt-galois +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005310 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 85 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT-Galois Decomposition** + + + The Galois group decomposes via CRT: Gal_k ≅ ∏(Z/p_i Z)× for i=1..k. Order equality φ(M_k) = ∏(p_i-1) verified computationally + at stages 1-3. + + + # CRT-Galois Decomposition + + + ## Summary + + + The Galois group decomposes via CRT: Gal_k ≅ ∏(Z/p_i Z)× for i=1..k. Order equality φ(M_k) = ∏(p_i-1) verified computationally + at stages 1-3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 226 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Galois` + + - Name: `galois_crt_3` + + + ## Dependencies + + + - Canonical: I.D98, I.T45 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Galois group decomposes via CRT: Gal_k ≅ ∏(Z/p_i Z)× for i=1..k. Order equality φ(M_k) = ∏(p_i-1) verified + computationally at stages 1-3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001165.md b/_corpus_v3_items/cid001165.md new file mode 100644 index 000000000..bab21d021 --- /dev/null +++ b/_corpus_v3_items/cid001165.md @@ -0,0 +1,123 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001165/ +title: Monotone Convergence +cid: cid001165 +primary_alias: PRP0044 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For tower-compatible monotone function families, the sequence of integrals is non-decreasing across stages. +aliases: +- PRP0044 +- I.P45 +- monotone-convergence +- prop:monotone-convergence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Monotone Convergence** + + + For tower-compatible monotone function families, the sequence of integrals is non-decreasing across stages. + + + # Monotone Convergence + + + ## Summary + + + For tower-compatible monotone function families, the sequence of integrals is non-decreasing across stages. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 229 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Integration` + + - Name: `monotone_convergence_check_step` + + + ## Dependencies + + + - Canonical: I.D99, I.P43 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For tower-compatible monotone function families, the sequence of integrals is non-decreasing across stages. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001166.md b/_corpus_v3_items/cid001166.md new file mode 100644 index 000000000..5d6ca556f --- /dev/null +++ b/_corpus_v3_items/cid001166.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001166/ +title: No Second Linearity +cid: cid001166 +primary_alias: PRP0045 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The ABCD chart Phi is not linear: Phi(X+Y) != Phi(X)+Phi(Y) for alpha-ray addition. NF encoding interleaves multiplicative + and exponential data, destroying additivity. Concrete counterexample: X=2, Y=4.' +aliases: +- PRP0045 +- I.P46 +- no-second-linearity +- prop:no-second-linearity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 20 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Second Linearity**\n\nThe ABCD chart Phi is not linear: Phi(X+Y) != Phi(X)+Phi(Y) for alpha-ray addition.\ + \ NF encoding interleaves multiplicative and exponential data, destroying additivity. Concrete counterexample: X=2, Y=4.\n\ + \n# No Second Linearity\n\n## Summary\n\nThe ABCD chart Phi is not linear: Phi(X+Y) != Phi(X)+Phi(Y) for alpha-ray addition.\ + \ NF encoding interleaves multiplicative and exponential data, destroying additivity. Concrete counterexample: X=2, Y=4.\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:no-second-linearity}\nThe ABCD coordinate chart $\\Phi$ is not linear:\nthere\ + \ exist objects $X, Y \\in \\tau\\text{-Idx}$\nsuch that\n\\[\n \\Phi(X + Y) \\;\\neq\\; \\Phi(X) + \\Phi(Y),\n\\]\n\ + where $+$ denotes $\\alpha$-ray addition\n(iterated $\\rho$) on both sides,\napplied componentwise on the right.\n```\n\ + \n## Proof / Justification\n\n```latex\n[Proof sketch]\nTake $X = \\underline{2}$ and $Y = \\underline{3}$.\nThen $X +\ + \ Y = \\underline{5}$.\nThe ABCD coordinates are:\n\\begin{align*}\n \\Phi(\\underline{2})\n &= (\\underline{2},\\\ + , \\underline{1},\\, \\underline{1},\\, \\underline{1}), \\\\\n \\Phi(\\underline{3})\n &= (\\underline{3},\\, \\\ + underline{1},\\, \\underline{1},\\, \\underline{1}), \\\\\n \\Phi(\\underline{5})\n &= (\\underline{5},\\, \\underline{1},\\\ + , \\underline{1},\\, \\underline{1}).\n\\end{align*}\nComponentwise addition of the first two tuples gives\n$A = \\underline{2}\ + \ + \\underline{3} = \\underline{5}$,\nwhich coincidentally matches.\nBut now take $X = \\underline{2}$ and $Y = \\underline{4}$,\n\ + so $X + Y = \\underline{6}$.\nWe have:\n\\begin{align*}\n \\Phi(\\underline{4})\n &= (\\underline{2},\\, \\underline{2},\\\ + , \\underline{1},\\, \\underline{1}), \\\\\n \\Phi(\\underline{6})\n &= (\\underline{3},\\, \\underline{1},\\, \\\ + underline{1},\\, \\underline{2}).\n\\end{align*}\nComponentwise addition gives\n$(A, B, C, D) = (\\underline{2} + \\underline{2},\\\ + ,\n\\underline{1} + \\underline{2},\\,\n\\underline{1} + \\underline{1},\\,\n\\underline{1} + \\underline{1})\n= (\\underline{4},\\\ + , \\underline{3},\\, \\underline{2},\\, \\underline{2})$,\nbut $\\Phi(\\underline{6})\n= (\\underline{3},\\, \\underline{1},\\\ + , \\underline{1},\\, \\underline{2})$.\nThese differ in all of $A$, $B$, and $C$.\n\nThe structural reason is that the\ + \ NF encoding interleaves\nmultiplicative data ($A$, $B$) and exponential data ($C$)\nthrough the greedy peel\n(Definition~\\\ + ref{def:genealogical-decomposition}).\nAddition on the $\\alpha$-ray can change\nthe \\emph{prime factorization} of the\ + \ sum,\nredistributing mass across all four coordinates\nin a nonlinear fashion.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 242\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch20-dimension-fibration.tex`\ + \ lines 601-613\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.D17, I.P08\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The ABCD chart Phi is not linear: Phi(X+Y) != Phi(X)+Phi(Y) for alpha-ray addition. NF encoding interleaves + multiplicative and exponential data, destroying additivity. Concrete counterexample: X=2, Y=4.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001167.md b/_corpus_v3_items/cid001167.md new file mode 100644 index 000000000..fa3f6a867 --- /dev/null +++ b/_corpus_v3_items/cid001167.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001167/ +title: Bayesian Factorization +cid: cid001167 +primary_alias: PRP0046 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every tau-Bayesian state factors: S = S_+ o e_+ + S_- o e_-, where S_+/S_- are classical probability distributions + on {T,F}. Boolean recovery applies to each sector independently.' +aliases: +- PRP0046 +- I.P47 +- bayesian-factorization +- prop:bayesian-factorization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 48 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bayesian Factorization**\n\nEvery tau-Bayesian state factors: S = S_+ o e_+ + S_- o e_-, where S_+/S_- are\ + \ classical probability distributions on {T,F}. Boolean recovery applies to each sector independently.\n\n# Bayesian Factorization\n\ + \n## Summary\n\nEvery tau-Bayesian state factors: S = S_+ o e_+ + S_- o e_-, where S_+/S_- are classical probability distributions\ + \ on {T,F}. Boolean recovery applies to each sector independently.\n\n## Statement\n\n```latex\n%\n\\label{prop:bayesian-factorization}\n\ + % Depends: I.D21, I.T12, I.P13\nEvery $\\tau$-Bayesian state factors\nthrough the spectral decomposition\n(Theorem~\\\ + ref{thm:spectral-decomposition}, I.T12).\nThat is, for every $\\tau$-Bayesian state $S$,\n\\[\n \\boxed{%\n S =\ + \ S_+ \\circ e_+ \\;+\\; S_- \\circ e_-,}\n\\]\nwhere $S_+$ and $S_-$\nare classical probability distributions\non $\\\ + {\\mathsf{T}, \\mathsf{F}\\}$,\ncorresponding to the $B$-channel\nand $C$-channel respectively.\n```\n\n## Proof / Justification\n\ + \n```latex\n[Proof sketch]\nBy the bipolar decomposition\n(Definition~\\ref{def:bipolar-spectral-algebra}, I.D21),\nevery\ + \ element of $[0,1]_\\tau$\ndecomposes via the idempotents\n$e_+$ and $e_-$\nfrom the spectral decomposition.\nThe polarity\ + \ coherence condition\n(Definition~\\ref{def:tau-bayesian-state}, condition~2)\nforces $S$ to respect this decomposition:\n\ + the probability assigned to each truth value\nis determined by the witness pair,\nwhich splits into $B$-channel and $C$-channel\ + \ components.\n\nThe component $S_+ := S \\circ e_+$\nassigns probability only to the outcomes\nof the $B$-sector witness\ + \ $w_B$.\nSince $w_B$ delivers either ``confirms'' or ``refutes''\n(a binary outcome),\n$S_+$ is a classical probability\ + \ distribution\non $\\{\\mathsf{T}, \\mathsf{F}\\}$.\nSimilarly, $S_- := S \\circ e_-$\nis a classical probability distribution\n\ + on $\\{\\mathsf{T}, \\mathsf{F}\\}$\nfor the $C$-sector.\n\nThe normalization condition ensures\nthat $S_+$ and $S_-$\ + \ combine consistently:\n$S(\\mathsf{T}) = S_+(\\mathsf{T}) \\cdot S_-(\\mathsf{T})$,\n$S(\\mathsf{F}) = S_+(\\mathsf{F})\ + \ \\cdot S_-(\\mathsf{F})$,\n$S(\\mathsf{B}) = S_+(\\mathsf{T}) \\cdot S_-(\\mathsf{F})\n+ S_+(\\mathsf{F}) \\cdot S_-(\\\ + mathsf{T})$,\nand $S(\\mathsf{N})$ absorbs the remainder.\nThe Boolean recovery theorem\n(Proposition~\\ref{prop:boolean-recovery})\n\ + then shows that each sector reduces\nto classical probability.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 252\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch48-boolean-recovery.tex`\ + \ lines 946-962\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.D106, I.T12, I.P13\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every tau-Bayesian state factors: S = S_+ o e_+ + S_- o e_-, where S_+/S_- are classical probability distributions + on {T,F}. Boolean recovery applies to each sector independently.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001168.md b/_corpus_v3_items/cid001168.md new file mode 100644 index 000000000..0e6dfd82e --- /dev/null +++ b/_corpus_v3_items/cid001168.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001168/ +title: Ontic Closure +cid: cid001168 +primary_alias: THM0002 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Obj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta, and this set is ontically sealed -- no further + objects can be created. +aliases: +- THM0002 +- I.T01 +- ontic-closure +- thm:ontic-closure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005327 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 7 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ontic Closure**\n\nObj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta, and this set is ontically\ + \ sealed -- no further objects can be created.\n\n# Ontic Closure\n\n## Summary\n\nObj(tau) = {omega} union O_alpha union\ + \ O_pi union O_gamma union O_eta, and this set is ontically sealed -- no further objects can be created.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:ontic-closure}\nThe universe of Category~$\\tau$ satisfies:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Exhaustiveness.}\n $\\Obj(\\tau) = \\{\\omega\\} \\cup O_\\alpha\n \\cup O_\\pi \\cup O_\\\ + gamma \\cup O_\\eta$.\n \\item \\textbf{Disjointness.}\n The five sets are pairwise disjoint.\n \\item\ + \ \\textbf{Countability.}\n Each $O_g$ is countably infinite; $|\\Obj(\\tau)| = \\aleph_0$.\n \\item \\textbf{Unique\ + \ representation.}\n Every $x \\in \\Obj(\\tau)$ is either $\\omega$\n or has a unique representation\ + \ $x = \\rho^n(g)$\n for a unique $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$\n and a unique $n \\\ + geq 0$.\n \\item \\textbf{Ontic seal.}\n No further objects can be produced.\n The generative act\ + \ is complete.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Part~(1)} is $\\KAxiom{6}$ (Object\ + \ Closure).\n\n\\emph{Part~(2)} is Proposition~\\ref{prop:orbit-disjoint}.\n\n\\emph{Part~(3):}\nBy Proposition~\\ref{prop:orbit-countable},\n\ + each $O_g$ is in bijection with $\\mathbb{N}$.\nThe beacon singleton has cardinality~$1$.\nThe total cardinality is\n\ + $1 + \\aleph_0 + \\aleph_0 + \\aleph_0 + \\aleph_0 = \\aleph_0$.\n\n\\emph{Part~(4):}\nBy Part~(1), every $x$ belongs\ + \ to one of the five sets.\nBy Part~(2), it belongs to exactly one.\nIf $x \\in O_g$, then $x = \\rho^n(g)$ for some~$n$\n\ + (by definition of $O_g$).\nUniqueness of $n$ follows from the injectivity of $\\varphi_g$\n(Proposition~\\ref{prop:orbit-countable}).\n\ + Uniqueness of $g$ follows from Part~(2).\n\n\\emph{Part~(5):}\n$\\KAxiom{6}$ asserts that every object of $\\tau$\nis\ + \ in the decomposition.\nSince $\\rho$ maps orbit elements to orbit elements\n(by $\\KAxiom{3}$) and maps $\\omega$ to\ + \ $\\omega$ (by $\\KAxiom{2}$),\nno application of $\\rho$ can produce an object\noutside the decomposition.\nNo other\ + \ operation exists in the signature.\nTherefore, the universe is sealed.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 17\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch07-ontic-closure.tex`\ + \ lines 236-257\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Closure`\n\ + - Name: `Tau.Orbit.ontic_closure`\n\n## Dependencies\n\n- Canonical: I.K3, I.K5, I.K6, I.X01, I.D05\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Obj(tau) = {omega} union O_alpha union O_pi union O_gamma union O_eta, and this set is ontically sealed -- no + further objects can be created. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001169.md b/_corpus_v3_items/cid001169.md new file mode 100644 index 000000000..29ef825ae --- /dev/null +++ b/_corpus_v3_items/cid001169.md @@ -0,0 +1,163 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001169/ +title: Iterator Ladder Saturation +cid: cid001169 +primary_alias: THM0003 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents + a 5th orbit channel. +aliases: +- THM0003 +- I.T02 +- iterator-ladder-saturation +- thm:ladder-saturation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005331 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 8 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Iterator Ladder Saturation** + + + The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents + a 5th orbit channel. + + + # Iterator Ladder Saturation + + + ## Summary + + + The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 prevents + a 5th orbit channel. + + + ## Statement + + + ```latex + + % + + \label{thm:ladder-saturation} + + The iterator ladder $\mathcal{L} = (L_0, L_1, L_2, L_3)$ + + has exactly four levels. + + Higher hyper-operations (pentation, hexation, \ldots) + + are definable but lose canonical injectivity. + + The ladder saturates at level~3 (tetration). + + ``` + + + ## Proof / Justification + + + ```latex + + Levels $L_0$--$L_3$ each consume one orbit channel + + (Observation~\ref{obs:channel-consumption}). + + After $L_3$, all four channels are consumed. + + By Lemma~\ref{lem:pentation-non-inj}, + + no level-4 operation can achieve canonical injectivity. + + The same argument blocks all higher levels. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 18 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch08-iterator-ladder.tex` lines + 261-269 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Orbit.Ladder` + + - Name: `Tau.Orbit.ladder_saturation` + + + ## Dependencies + + + - Canonical: I.D06, I.D03, I.K6 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The iterator ladder has exactly 4 levels (0-3); higher hyper-operations lose canonical injectivity because K6 + prevents a 5th orbit channel. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001170.md b/_corpus_v3_items/cid001170.md new file mode 100644 index 000000000..ad2e24895 --- /dev/null +++ b/_corpus_v3_items/cid001170.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001170/ +title: Composition Associativity +cid: cid001170 +primary_alias: THM0004 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Program composition is associative. A THEOREM, not an axiom: proved via normalization confluence (NF-Confluence + Theorem).' +aliases: +- THM0004 +- I.T03 +- composition-associativity +- thm:comp-assoc +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005295 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Composition Associativity**\n\nProgram composition is associative. A THEOREM, not an axiom: proved via normalization\ + \ confluence (NF-Confluence Theorem).\n\n# Composition Associativity\n\n## Summary\n\nProgram composition is associative.\ + \ A THEOREM, not an axiom: proved via normalization confluence (NF-Confluence Theorem).\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:comp-assoc}\nComposition in $\\mathcal{P}$ is associative:\nfor all programs $P, Q, R$,\n\\[\n \\boxed{[R]\ + \ \\circ ([Q] \\circ [P])\n = ([R] \\circ [Q]) \\circ [P].}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nLet\ + \ $P, Q, R$ be programs.\n\n$[R] \\circ ([Q] \\circ [P])\n= [R] \\circ [\\text{NF}(Q \\cdot P)]\n= [\\text{NF}(R \\cdot\ + \ \\text{NF}(Q \\cdot P))]$.\n\n$([R] \\circ [Q]) \\circ [P]\n= [\\text{NF}(R \\cdot Q)] \\circ [P]\n= [\\text{NF}(\\\ + text{NF}(R \\cdot Q) \\cdot P)]$.\n\nBy the NF-Confluence Lemma~\\ref{lem:nf-confluence},\n$\\text{NF}(R \\cdot \\text{NF}(Q\ + \ \\cdot P))\n= \\text{NF}(R \\cdot Q \\cdot P)\n= \\text{NF}(\\text{NF}(R \\cdot Q) \\cdot P)$.\nThe key step is that\ + \ normalization commutes with concatenation:\nnormalizing a sub-expression before concatenating\nyields the same final\ + \ normal form as normalizing the\nentire expression at once.\nThis follows from the confluence of the reduction system.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 30\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch13-program-monoid.tex`\ + \ lines 147-156\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.ProgramMonoid`\n\ + - Name: `Tau.Denotation.comp_assoc`\n\n## Dependencies\n\n- Canonical: I.D14\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Program composition is associative. A THEOREM, not an axiom: proved via normalization confluence (NF-Confluence + Theorem).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001171.md b/_corpus_v3_items/cid001171.md new file mode 100644 index 000000000..67d2677d3 --- /dev/null +++ b/_corpus_v3_items/cid001171.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001171/ +title: Hyperfactorization Theorem +cid: cid001171 +primary_alias: THM0005 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'HINGE THEOREM 1: Every x in Obj(tau) has a unique canonical NF via three critical lemmas (tetration injectivity, + no-tie determinism, strict remainder descent). Provable in ZFC.' +aliases: +- THM0005 +- I.T04 +- hyperfactorization-theorem +- thm:hyperfactorization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 5, Chapter 24 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Hyperfactorization Theorem**\n\nHINGE THEOREM 1: Every x in Obj(tau) has a unique canonical NF via three critical\ + \ lemmas (tetration injectivity, no-tie determinism, strict remainder descent). Provable in ZFC.\n\n# Hyperfactorization\ + \ Theorem\n\n## Summary\n\nHINGE THEOREM 1: Every x in Obj(tau) has a unique canonical NF via three critical lemmas (tetration\ + \ injectivity, no-tie determinism, strict remainder descent). Provable in ZFC.\n\n## Statement\n\n```latex\n%\n\\label{thm:hyperfactorization}\n\ + For every $X \\in \\tau\\text{-Idx}$ with $X \\geq \\underline{2}$,\nthe decomposition\n\\[\n \\boxed{X\n \\;=\\\ + ;\n ((\\underline{A} \\uparrow\\uparrow \\underline{C})^{\\underline{B}})\n \\cdot \\underline{D}}\n\\]\nwith\n\\\ + begin{enumerate}\n \\item[(i)] $\\underline{A} \\in \\mathbb{P}_\\tau$\n (prime),\n \\item[(ii)] $\\underline{B}\ + \ \\geq \\underline{1}$,\n $\\underline{C} \\geq \\underline{1}$,\n \\item[(iii)] $\\underline{D} \\geq \\\ + underline{1}$,\n and if $\\underline{D} > \\underline{1}$,\n all prime factors of $\\underline{D}$\n\ + \ are strictly less than $\\underline{A}$,\n\\end{enumerate}\nexists and is \\textbf{unique}.\nThat is, if\n\ + $((\\underline{A'} \\uparrow\\uparrow \\underline{C'})\n^{\\underline{B'}}) \\cdot \\underline{D'} = X$\nalso satisfies\ + \ (i)--(iii), then\n$\\underline{A'} = \\underline{A}$,\n$\\underline{B'} = \\underline{B}$,\n$\\underline{C'} = \\underline{C}$,\n\ + $\\underline{D'} = \\underline{D}$.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted\ + \ in this pilot run._\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 34\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part05/ch24-hyperfactorization.tex`\ + \ lines 42-73\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.Hyperfact`\n\ + - Name: `Tau.Coordinates.hyperfact_check`\n\n## Dependencies\n\n- Canonical: I.P05, I.L03, I.L04, I.D17\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'HINGE THEOREM 1: Every x in Obj(tau) has a unique canonical NF via three critical lemmas (tetration injectivity, + no-tie determinism, strict remainder descent). Provable in ZFC.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001172.md b/_corpus_v3_items/cid001172.md new file mode 100644 index 000000000..4b21a2ce2 --- /dev/null +++ b/_corpus_v3_items/cid001172.md @@ -0,0 +1,130 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001172/ +title: Prime Polarity Theorem +cid: cid001172 +primary_alias: THM0006 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'HINGE THEOREM 2: Primes carry canonical bipolar polarization via gamma/eta dominance -- global spectrum with finite + (local) witnesses. Provable in ZFC.' +aliases: +- THM0006 +- I.T05 +- prime-polarity-theorem +- thm:prime-polarity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 6, Chapter 27 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Prime Polarity Theorem**\n\nHINGE THEOREM 2: Primes carry canonical bipolar polarization via gamma/eta dominance\ + \ -- global spectrum with finite (local) witnesses. Provable in ZFC.\n\n# Prime Polarity Theorem\n\n## Summary\n\nHINGE\ + \ THEOREM 2: Primes carry canonical bipolar polarization via gamma/eta dominance -- global spectrum with finite (local)\ + \ witnesses. Provable in ZFC.\n\n## Statement\n\n```latex\n%\n\\label{thm:prime-polarity}\n\\begin{enumerate}\n \\\ + item \\textbf{(Dichotomy.)}\n Every prime $\\underline{p} \\in \\mathbb{P}_\\tau$\n is either B-dominant\ + \ or C-dominant.\n \\item \\textbf{(B-class infinite.)}\n The set of B-dominant primes is infinite.\n \\\ + item \\textbf{(C-class infinite.)}\n The set of C-dominant primes is infinite.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\n```latex\n\\emph{Part~(1): Dichotomy.}\nFix a prime $\\underline{p} \\geq \\underline{2}$.\nConsider\ + \ the comparison between\n$\\underline{p}^{\\underline{B}}$\nand $\\underline{p} \\uparrow\\uparrow \\underline{C}$\n\ + for growing values of $\\underline{B}$ and $\\underline{C}$.\n\nFor fixed $\\underline{C}$,\n$\\underline{p}^{\\underline{B}}$\ + \ grows\nas $\\underline{p}^{\\underline{B}}$\n(exponential in $\\underline{B}$).\nFor fixed $\\underline{B}$,\n$(\\underline{p}\ + \ \\uparrow\\uparrow \\underline{C})^{\\underline{B}}$\ngrows as $\\underline{p}^{\\underline{B} \\cdot\n(\\underline{p}\ + \ \\uparrow\\uparrow (\\underline{C} - 1))}$\n(super-exponential in $\\underline{C}$).\n\nThe critical ratio is:\n\\[\n\ + \ r(\\underline{p}, \\underline{B}, \\underline{C})\n \\;:=\\;\n \\frac{\\log T(\\underline{p}, \\underline{B},\ + \ \\underline{C})}\n {\\log(\\underline{p}^{\\underline{B}})}\n \\;=\\;\n \\frac{\\underline{B} \\cdot\n (\\\ + underline{p} \\uparrow\\uparrow (\\underline{C}-1))}\n {\\underline{B}}\n \\;=\\;\n \\underline{p} \\uparrow\\\ + uparrow (\\underline{C}-1).\n\\]\nWhen $\\underline{C} = \\underline{1}$, $r = \\underline{1}$:\nthe tower atom equals\ + \ the pure power.\nWhen $\\underline{C} = \\underline{2}$,\n$r = \\underline{p}$: the tower atom is\n$\\underline{p}$\ + \ times more ``potent''\nin logarithmic terms than the pure power.\n\nFor a given prime $\\underline{p}$,\nthe \\emph{effective\ + \ sensitivity} is:\nhow quickly does increasing $\\underline{C}$\nfrom $\\underline{1}$ to $\\underline{2}$\namplify the\ + \ tower atom, compared to increasing\n$\\underline{B}$ by $\\underline{p}$?\nSince $T(\\underline{p}, \\underline{B},\ + \ \\underline{2})\n= \\underline{p}^{\\underline{p} \\cdot \\underline{B}}$\nwhile $T(\\underline{p}, \\underline{B}+\\\ + underline{p},\n\\underline{1}) = \\underline{p}^{\\underline{B}+\\underline{p}}$,\nthe $C$-increase wins when\n$\\underline{p}\ + \ \\cdot \\underline{B}\n> \\underline{B} + \\underline{p}$,\ni.e., $\\underline{B}(\\underline{p} - \\underline{1})\n\ + > \\underline{p}$,\ni.e., $\\underline{B} > \\underline{p}/(\\underline{p}-\\underline{1})$.\nFor $\\underline{p} = \\\ + underline{2}$,\nthis threshold is $\\underline{B} > \\underline{2}$.\nFor $\\underline{p} = \\underline{3}$,\nthe threshold\ + \ is $\\underline{B} > \\underline{3}/\\underline{2}$,\ni.e., $\\underline{B} \\geq \\underline{2}$.\n\nThe polarity of\ + \ $\\underline{p}$ is determined by\nhow $\\underline{p}$ interacts with the\n\\emph{divisibility structure of $\\tau$-Idx\n\ + as a whole}:\nspecifically, by the density of objects in\n$\\mathcal{S}_{\\underline{p}}$\n(those with $A = \\underline{p}$)\n\ + that achieve high $\\underline{B}$\nversus high $\\underline{C}$.\nSince the objects of $\\tau$-Idx are all natural numbers,\n\ + the density is governed by divisibility:\n$X$ has $A = \\underline{p}$ and $B = \\underline{b}$\niff $\\underline{p}^{\\\ + underline{b}} \\mid X$\nbut $\\underline{p}^{\\underline{b}+1} \\nmid X$\n(at tetration height $\\underline{1}$).\nThe\ + \ density of such $X$ up to $\\underline{N}$\nis approximately $\\underline{N} / \\underline{p}^{\\underline{b}}$.\n\n\ + Similarly, $X$ has $C = \\underline{c}$ (at $B = 1$)\niff $(\\underline{p} \\uparrow\\uparrow \\underline{c}) \\mid X$\n\ + but $(\\underline{p} \\uparrow\\uparrow \\underline{c})^2 \\nmid X$.\nThe density is approximately\n$\\underline{N} /\ + \ (\\underline{p} \\uparrow\\uparrow \\underline{c})$.\n\nSince $\\underline{p} \\uparrow\\uparrow \\underline{c}$\ngrows\ + \ much faster than $\\underline{p}^{\\underline{b}}$,\nhigh-$C$ objects are much rarer than high-$B$ objects.\nThis creates\ + \ a systematic B-dominant bias for all primes.\n\nHowever, the polarity is defined not by raw density\nbut by the \\emph{maximal}\ + \ value achievable\nwithin the population.\nFor each prime $\\underline{p}$,\nthe question is whether the tower-atom structure\n\ + of $\\tau$-Idx contains objects where $\\underline{p}$'s\ntetration channel is ``saturated''\n--- i.e., where $\\underline{p}\ + \ \\uparrow\\uparrow \\underline{C}$\nexactly divides $X$ at high $\\underline{C}$.\n\nThe dichotomy follows from the\ + \ observation that\nfor each prime $\\underline{p}$,\nthe $\\underline{p}$-adic valuation of\n$\\underline{p} \\uparrow\\\ + uparrow \\underline{C}$\nis $\\underline{p} \\uparrow\\uparrow (\\underline{C}-1)$,\nwhich determines a unique ``tower\ + \ profile''\nfor $\\underline{p}$.\nThis profile either allows arbitrarily high\nB-values to dominate (when $\\underline{p}$'s\n\ + tower profile is ``shallow'')\nor forces C-values to dominate\n(when the profile is ``deep'').\nEvery prime falls into\ + \ one class or the other\nby the well-ordering of $\\tau$-Idx.\n\n\\emph{Part~(2): B-class infinite.}\nFor each $\\underline{k}\ + \ \\geq \\underline{1}$,\nconsider $X = \\underline{p}^{\\underline{k}}$\nwhere $\\underline{p}$ is prime.\nThen $A =\ + \ \\underline{p}$, $B = \\underline{k}$,\n$C = \\underline{1}$, $D = \\underline{1}$.\nFor any prime $\\underline{p}$,\n\ + arbitrarily large $\\underline{B}$-values are achieved\nwith $\\underline{C} = \\underline{1}$.\nThe primes $\\underline{p}$\ + \ for which this\n``pure power'' regime dominates the spectral signature\nform the B-class.\nThis class includes all primes\n\ + $\\underline{p}$ such that\n$\\underline{p} \\uparrow\\uparrow \\underline{2}\n= \\underline{p}^{\\underline{p}}$\ndoes\ + \ not divide any object $X \\leq \\underline{N}$\nwith $A = \\underline{p}$\nfor sufficiently many $\\underline{N}$\n\ + relative to $\\underline{p}^{\\underline{B}}$-divisible objects.\nBy Dirichlet's theorem on primes in arithmetic progressions\n\ + (which holds on $\\tau$-Idx as a consequence of the FTA),\nthere are infinitely many such primes.\n\n\\emph{Part~(3):\ + \ C-class infinite.}\nConsider primes $\\underline{p}$ such that\n$\\underline{p} \\uparrow\\uparrow \\underline{2}\n\ + = \\underline{p}^{\\underline{p}}$ divides\na ``rich'' family of objects.\nSpecifically, for $\\underline{p} = \\underline{2}$:\n\ + $\\underline{2} \\uparrow\\uparrow \\underline{2}\n= \\underline{4}$,\nand the set of multiples of $\\underline{4}$\n\ + is dense in $\\tau$-Idx.\nMoreover, $\\underline{2} \\uparrow\\uparrow \\underline{3}\n= \\underline{16}$,\n$\\underline{2}\ + \ \\uparrow\\uparrow \\underline{4}\n= \\underline{65536}$, etc.\nFor $\\underline{p} = \\underline{2}$,\nhigh tetration\ + \ heights are achievable because\n$\\underline{2}$'s tetration tower grows\n(relatively) slowly: $\\underline{2}, \\underline{4},\n\ + \\underline{16}, \\underline{65536}, \\ldots$\nThis makes $\\underline{2}$ C-dominant.\n\nMore generally, for each $\\\ + underline{k}$,\nthere exist primes $\\underline{p}$\nwith $\\underline{p}^{\\underline{p}} \\leq \\underline{p}^k$\n(i.e.,\ + \ $\\underline{p} \\leq \\underline{k}$)\nfor which the tetration tower\n$\\underline{p} \\uparrow\\uparrow \\underline{c}$\n\ + remains below $\\underline{N}$\nfor more values of $\\underline{c}$\nthan the corresponding $\\underline{p}^{\\underline{b}}$\n\ + reaches for large $\\underline{b}$.\nThe small primes ($\\underline{2}, \\underline{3},\n\\underline{5}, \\ldots$) have\ + \ slow-growing tetration towers,\nmaking them C-dominant.\nSince there are infinitely many primes,\nand the C-dominant\ + \ condition is satisfied by\nall primes below a threshold that grows with $\\underline{N}$,\nthe C-class is infinite.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 35\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part06/ch27-prime-polarity.tex`\ + \ lines 86-98\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Polarity.Polarity`\n\ + - Name: `Tau.Polarity.polarity_map`\n\n## Dependencies\n\n- Canonical: I.D19e, I.T04, I.D06\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'HINGE THEOREM 2: Primes carry canonical bipolar polarization via gamma/eta dominance -- global spectrum with + finite (local) witnesses. Provable in ZFC.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001173.md b/_corpus_v3_items/cid001173.md new file mode 100644 index 000000000..8b2707bfc --- /dev/null +++ b/_corpus_v3_items/cid001173.md @@ -0,0 +1,94 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001173/ +title: Global Hartogs +cid: cid001173 +primary_alias: THM0007 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Boundary determines interior: interior points earned without coordinates via holomorphic continuation. Book I climax + and canonical passage to Book II.' +aliases: +- THM0007 +- I.T06 +- global-hartogs +- thm:global-hartogs +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 62 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Global Hartogs**\n\nBoundary determines interior: interior points earned without coordinates via holomorphic\ + \ continuation. Book I climax and canonical passage to Book II.\n\n# Global Hartogs\n\n## Summary\n\nBoundary determines\ + \ interior: interior points earned without coordinates via holomorphic continuation. Book I climax and canonical passage\ + \ to Book II.\n\n## Statement\n\n```latex\n%\n\\label{thm:global-hartogs}\n% I.T21, I.T18, I.T12, I.D37\nLet\ + \ $K \\subseteq \\mathbb{L}$ be primordially thin\n(Definition~\\ref{def:primorial-thinness}, I.D67),\nand let $f \\in\ + \ \\mathrm{HolFun}$\nbe defined on $\\mathbb{L} \\setminus K$.\nThen $f$ extends uniquely to\n$\\tilde{f} \\in \\mathrm{Hol}(\\\ + mathbb{L})$\n(Definition~\\ref{def:hol-L}, I.D49):\n\\[\n \\boxed{%\n K \\text{ thin},\\;\n f \\in \\mathrm{HolFun}(\\\ + mathbb{L} \\setminus K)\n \\;\\;\\Longrightarrow\\;\\;\n \\exists!\\; \\tilde{f} \\in \\mathrm{Hol}(\\mathbb{L}):\\\ + ;\n \\tilde{f}\\!\\big|_{\\mathbb{L} \\setminus K} = f.}\n\\]\nNo boundedness hypothesis is required.\n```\n\n## Proof\ + \ / Justification\n\n```latex\nThe proof proceeds in four steps,\neach drawing on a different strand\nof the machinery\ + \ developed in Parts~IX--XII.\n\n\\medskip\n\\textbf{Step 1: Spectral reduction.}\nThe spectral decomposition\n(Theorem~\\\ + ref{thm:spectral-decomposition}, I.T12)\nsplits $f$ into spectral components:\n\\[\n f(t) = f_+(t) \\cdot e_+ + f_-(t)\ + \ \\cdot e_-,\n \\qquad\n f_\\pm(t) := \\chi_\\pm(f(t)),\n\\]\nfor $t \\in \\mathbb{L} \\setminus K$.\nIt suffices\ + \ to extend $f_+$ and $f_-$ separately,\nrecovering $\\tilde{f}$ by spectral reassembly.\nEach spectral component inherits\ + \ tower coherence\n(the characters are ring homomorphisms\ncommuting with primorial reduction).\nAt each stage $M_d$,\n\ + $f_\\pm(t) \\bmod M_d \\in \\mathbb{Z}/M_d\\mathbb{Z}$ ---\na finite set.\nEach spectral component is therefore \\emph{automatically\ + \ bounded}.\n\n\\medskip\n\\textbf{Step 2: CRT reconstruction at each stage.}\nFix $f_+$ and a depth $d$.\nFor each $t\ + \ \\in K$,\nthe CRT Extension Lemma\n(Lemma~\\ref{lem:crt-extension}, I.L08)\ndetermines $f_+(t) \\bmod M_d$ uniquely:\n\ + $K$ is thin, so $K_d$ misses two CRT directions,\nand the Chinese Remainder Theorem\nassembles the known values along\ + \ those directions\ninto $f_+(t) \\bmod M_d$.\n\n\\medskip\n\\textbf{Step 3: Tower coherence ensures global consistency.}\n\ + The stage-by-stage reconstructions must satisfy:\n\\[\n \\tilde{f}_+(t) \\bmod M_d\n \\;=\\;\n \\bigl(\\tilde{f}_+(t)\ + \ \\bmod M_{d+1}\\bigr) \\bmod M_d\n \\qquad \\text{for all } d \\geq 1.\n\\]\nThis holds because CRT reconstruction\ + \ at $M_{d+1} = M_d \\cdot p_{d+1}$\ndecomposes into the reconstruction at $M_d$\nplus a new factor.\nTower coherence\n\ + (Definition~\\ref{def:tower-coherence}, I.D46)\nand CRT coherence\n(Theorem~\\ref{thm:crt-coherence}, I.T18)\nforce the\ + \ reconstructed values on $K$\nto be compatible.\nThe sequence $(\\tilde{f}_+(t) \\bmod M_d)_{d \\geq 1}$\ndefines a valid\ + \ omega-tail\n(Definition~\\ref{def:omega-tail}, I.D25).\nSimilarly for $\\tilde{f}_-$.\nSet $\\tilde{f}(t) := \\tilde{f}_+(t)\ + \ \\cdot e_+ + \\tilde{f}_-(t) \\cdot e_-$.\n\n\\medskip\n\\textbf{Step 4: Uniqueness by the Identity Theorem.}\nIf $\\\ + tilde{f}_1, \\tilde{f}_2 \\in \\mathrm{Hol}(\\mathbb{L})$\nboth extend $f$, they agree on $\\mathbb{L} \\setminus K$.\n\ + Since $K$ is thin, $\\mathbb{L} \\setminus K$ is primordially dense ---\nat each depth, it contains elements in all CRT\ + \ directions\nnot blocked by $K$.\nAgreement at some depth $d_0$ follows,\nand the $\\tau$-Identity Theorem\n(Theorem~\\\ + ref{thm:tau-identity}, I.T21)\ngives $\\tilde{f}_1 = \\tilde{f}_2$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 42\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch62-global-hartogs.tex`\ + \ lines 97-117\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.GlobalHartogs`\n\ + - Name: `Tau.Holomorphy.global_hartogs`\n\n## Dependencies\n\n- Canonical: I.D22, I.D18, I.X02\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Boundary determines interior: interior points earned without coordinates via holomorphic continuation. Book + I climax and canonical passage to Book II.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001174.md b/_corpus_v3_items/cid001174.md new file mode 100644 index 000000000..f89034006 --- /dev/null +++ b/_corpus_v3_items/cid001174.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001174/ +title: Rigidity of tau +cid: cid001174 +primary_alias: THM0008 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Aut(tau) = {id}. Every automorphism fixes all generators (named constants) and all orbit elements (by induction on + depth). +aliases: +- THM0008 +- I.T07 +- rigidity-of-tau +- thm:rigidity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005332 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Rigidity of tau**\n\nAut(tau) = {id}. Every automorphism fixes all generators (named constants) and all orbit\ + \ elements (by induction on depth).\n\n# Rigidity of tau\n\n## Summary\n\nAut(tau) = {id}. Every automorphism fixes all\ + \ generators (named constants) and all orbit elements (by induction on depth).\n\n## Statement\n\n```latex\n%\n\\label{thm:rigidity}\n\ + The automorphism group of $\\tau$ is trivial:\n\\[\n \\boxed{\\Aut(\\tau) = \\{\\id\\}.}\n\\]\n```\n\n## Proof / Justification\n\ + \n```latex\nLet $\\varphi : \\Obj(\\tau) \\to \\Obj(\\tau)$ be an automorphism.\n\n\\medskip\n\\textbf{Step 1: $\\varphi$\ + \ fixes $\\omega$.}\n\nSince $\\omega$ is a named constant in the signature,\nany $\\Sigma_\\tau$-automorphism must fix\ + \ it:\n$\\varphi(\\omega) = \\omega$.\n\n\\medskip\n\\textbf{Step 2: $\\varphi$ fixes each generator.}\n\nSimilarly, $\\\ + varphi(\\alpha) = \\alpha$,\n$\\varphi(\\pi) = \\pi$,\n$\\varphi(\\gamma) = \\gamma$,\n$\\varphi(\\eta) = \\eta$.\nAll\ + \ five generators are fixed by $\\varphi$.\n\n\\medskip\n\\textbf{Step 3: $\\varphi$ fixes all orbit elements.}\n\nLet\ + \ $x \\in O_g$ for some $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$.\nThen $x = \\rho^n(g)$ for a unique $n \\geq 0$\n\ + (by the Ontic Closure Theorem, Part~(4)).\nSince $\\varphi$ commutes with $\\rho$ and fixes~$g$:\n\\[\n \\varphi(x)\ + \ = \\varphi(\\rho^n(g))\n = \\rho^n(\\varphi(g)) = \\rho^n(g) = x.\n\\]\nThe second equality uses induction on $n$:\n\ + $\\varphi(\\rho^0(g)) = \\varphi(g) = g = \\rho^0(g)$,\nand if $\\varphi(\\rho^k(g)) = \\rho^k(g)$ then\n$\\varphi(\\\ + rho^{k+1}(g)) = \\varphi(\\rho(\\rho^k(g)))\n= \\rho(\\varphi(\\rho^k(g))) = \\rho(\\rho^k(g)) = \\rho^{k+1}(g)$.\n\n\\\ + medskip\n\\textbf{Step 4: Conclusion.}\n\nEvery element of $\\Obj(\\tau)$ is either $\\omega$\n(fixed by Step~1) or an\ + \ orbit element $\\rho^n(g)$\n(fixed by Steps~2--3).\nTherefore $\\varphi = \\id$.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 45\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch09-rigidity-categoricity.tex`\ + \ lines 78-85\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Rigidity`\n\ + - Name: `Tau.Orbit.rigidity`\n\n## Dependencies\n\n- Canonical: I.T01, I.K1, I.K2, I.K3, I.K4, I.K5, I.K6, I.P01\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Aut(tau) = {id}. Every automorphism fixes all generators (named constants) and all orbit elements (by induction + on depth). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001175.md b/_corpus_v3_items/cid001175.md new file mode 100644 index 000000000..381272fc8 --- /dev/null +++ b/_corpus_v3_items/cid001175.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001175/ +title: Categoricity of tau_0 +cid: cid001175 +primary_alias: THM0009 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'tau_0 is categorical: any two models are uniquely isomorphic. The unique isomorphism maps rho^n(g) to (rho^M)^n(g^M).' +aliases: +- THM0009 +- I.T08 +- categoricity-of-tau-0 +- thm:categoricity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005333 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Categoricity of tau_0**\n\ntau_0 is categorical: any two models are uniquely isomorphic. The unique isomorphism\ + \ maps rho^n(g) to (rho^M)^n(g^M).\n\n# Categoricity of tau_0\n\n## Summary\n\ntau_0 is categorical: any two models are\ + \ uniquely isomorphic. The unique isomorphism maps rho^n(g) to (rho^M)^n(g^M).\n\n## Statement\n\n```latex\n%\n\\label{thm:categoricity}\n\ + The theory $\\tau_0$ is categorical:\nif $M$ and $N$ are any two models of $\\tau_0$,\nthen there exists a unique isomorphism\ + \ $\\varphi : M \\xrightarrow{\\sim} N$.\n```\n\n## Proof / Justification\n\n```latex\nLet $M \\models \\tau_0$.\nWe construct\ + \ the unique isomorphism\n$\\varphi : \\tau \\to M$.\n\n\\medskip\n\\textbf{Existence.}\nDefine $\\varphi$ by:\n\\begin{align*}\n\ + \ \\varphi(\\omega) &:= \\omega^M, \\\\\n \\varphi(\\rho^n(g)) &:= (\\rho^M)^n(g^M)\n \\quad\\text{for } g \\\ + in \\{\\alpha, \\pi, \\gamma, \\eta\\},\\; n \\geq 0.\n\\end{align*}\nThis is well-defined by the Ontic Closure Theorem\n\ + (every object of $\\tau$ has a unique representation).\n\n\\emph{$\\varphi$ is well-defined on $M$:}\nSince $M \\models\ + \ \\tau_0$,\n$M$ satisfies $\\KAxiom{3}$ (orbit-seeded generation),\nso $(\\rho^M)^n(g^M)$ exists in $M$ for all $n \\\ + geq 0$.\n\n\\emph{$\\varphi$ preserves $\\rho$:}\n$\\varphi(\\rho(x)) = \\rho^M(\\varphi(x))$ by construction.\n\n\\emph{$\\\ + varphi$ preserves order:}\nThe order in $\\tau$ is determined by\ndepths within and across orbits (via $\\KAxiom{1}$).\n\ + The same determination holds in $M$.\n\n\\emph{$\\varphi$ is injective:}\nSuppose $\\varphi(\\rho^n(g)) = \\varphi(\\\ + rho^m(h))$,\ni.e., $(\\rho^M)^n(g^M) = (\\rho^M)^m(h^M)$.\nSince $M \\models \\tau_0$, the orbit rays in $M$\nare pairwise\ + \ disjoint\n(by the same proof as Proposition~\\ref{prop:orbit-disjoint},\nwhich uses only $\\KAxiom{1}$--$\\KAxiom{6}$).\n\ + Therefore $g = h$ and $n = m$.\n\n\\emph{$\\varphi$ is surjective:}\nSince $M \\models \\KAxiom{6}$ (Object Closure),\n\ + every element of $M$ is either $\\omega^M$\nor $(\\rho^M)^n(g^M)$ for some $g$ and $n$.\nBoth are in the image of $\\\ + varphi$.\n\n\\medskip\n\\textbf{Uniqueness.}\nAny isomorphism $\\psi : \\tau \\to M$ must satisfy:\n$\\psi(g) = g^M$ (preservation\ + \ of constants)\nand $\\psi(\\rho^n(g)) = (\\rho^M)^n(\\psi(g)) = (\\rho^M)^n(g^M)$.\nTherefore $\\psi = \\varphi$.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 46\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch09-rigidity-categoricity.tex`\ + \ lines 179-185\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Rigidity`\n\ + - Name: `Tau.Orbit.categoricity`\n\n## Dependencies\n\n- Canonical: I.T01, I.K1, I.K2, I.K3, I.K4, I.K5, I.K6, I.P01\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'tau_0 is categorical: any two models are uniquely isomorphic. The unique isomorphism maps rho^n(g) to (rho^M)^n(g^M).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001176.md b/_corpus_v3_items/cid001176.md new file mode 100644 index 000000000..cbdfd689a --- /dev/null +++ b/_corpus_v3_items/cid001176.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001176/ +title: FTA on tau-Idx +cid: cid001176 +primary_alias: THM0010 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Fundamental Theorem of Arithmetic on tau-Idx: every n >= 2 has a unique prime factorization. Proved via strong induction + (existence) and Euclid''s Lemma (uniqueness).' +aliases: +- THM0010 +- I.T09 +- fta-on-tau-idx +- thm:fta-tau-idx +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005291 +- predicate: formalized_by + object: cid005334 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 4, Chapter 16 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**FTA on tau-Idx**\n\nFundamental Theorem of Arithmetic on tau-Idx: every n >= 2 has a unique prime factorization.\ + \ Proved via strong induction (existence) and Euclid's Lemma (uniqueness).\n\n# FTA on tau-Idx\n\n## Summary\n\nFundamental\ + \ Theorem of Arithmetic on tau-Idx: every n >= 2 has a unique prime factorization. Proved via strong induction (existence)\ + \ and Euclid's Lemma (uniqueness).\n\n## Statement\n\n```latex\n%\n\\label{thm:fta-tau-idx}\nEvery $\\underline{n} \\\ + geq \\underline{2}$ in $\\tau$-Idx\nhas a unique representation as a product of primes:\n\\[\n \\boxed{\\underline{n}\n\ + \ = \\underline{p_1}^{\\underline{e_1}}\n \\times \\underline{p_2}^{\\underline{e_2}}\n \\times \\cdots\n \ + \ \\times \\underline{p_k}^{\\underline{e_k}}}\n\\]\nwhere $\\underline{p_1} < \\underline{p_2} < \\cdots\n< \\underline{p_k}$\ + \ are primes\nand each $\\underline{e_i} \\geq \\underline{1}$.\nThe representation is unique up to reordering\n(and we\ + \ fix ascending order by convention).\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Existence.}\nBy strong induction\ + \ on $\\underline{n}$.\nIf $\\underline{n}$ is prime, the factorization is\n$\\underline{n}$ itself.\nIf not, $\\underline{n}\ + \ = \\underline{a} \\times \\underline{b}$\nwith $\\underline{2} \\leq \\underline{a}, \\underline{b}\n< \\underline{n}$.\n\ + By induction, both $\\underline{a}$ and $\\underline{b}$\nhave prime factorizations;\ntheir product gives a factorization\ + \ of $\\underline{n}$.\n\n\\emph{Uniqueness.}\nSuppose $\\underline{n} = \\underline{p_1}^{\\underline{e_1}}\n\\times\ + \ \\cdots \\times \\underline{p_k}^{\\underline{e_k}}\n= \\underline{q_1}^{\\underline{f_1}}\n\\times \\cdots \\times\ + \ \\underline{q_l}^{\\underline{f_l}}$.\nSince $\\underline{p_1} \\mid \\underline{n}$,\nwe have $\\underline{p_1} \\\ + mid\n\\underline{q_j}^{\\underline{f_j}}$ for some $j$\n(by Euclid's Lemma, which holds on $\\tau$-Idx\nbecause the cancellation\ + \ law holds in the semiring).\nSince $\\underline{q_j}$ is prime, $\\underline{p_1}\n= \\underline{q_j}$.\nBy induction\ + \ on the total number of prime factors,\nthe remaining factors agree.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 53\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part04/ch16-primes-divisibility.tex`\ + \ lines 255-272\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Coordinates.Primes`\n\ + - Name: `Tau.Coordinates.prime_product_exists, Tau.Coordinates.prime_product_unique`\n\n## Dependencies\n\n- Canonical:\ + \ I.D19b, I.D11\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Fundamental Theorem of Arithmetic on tau-Idx: every n >= 2 has a unique prime factorization. Proved via strong + induction (existence) and Euclid''s Lemma (uniqueness).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001177.md b/_corpus_v3_items/cid001177.md new file mode 100644 index 000000000..c4c3cda63 --- /dev/null +++ b/_corpus_v3_items/cid001177.md @@ -0,0 +1,235 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001177/ +title: Split-Complex Forced +cid: cid001177 +primary_alias: THM0011 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The bipolar prime partition forces split-complex (j^2 = +1) over elliptic (i^2 = -1) scalars. Three arguments: bipolar + encoding requirement, wave-type propagation, diagonal-discipline compatibility.' +aliases: +- THM0011 +- I.T10 +- split-complex-forced +- thm:split-complex-forced +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005335 +- predicate: formalized_by + object: cid005336 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 7, Chapter 30 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Split-Complex Forced** + + + The bipolar prime partition forces split-complex (j^2 = +1) over elliptic (i^2 = -1) scalars. Three arguments: bipolar + encoding requirement, wave-type propagation, diagonal-discipline compatibility. + + + # Split-Complex Forced + + + ## Summary + + + The bipolar prime partition forces split-complex (j^2 = +1) over elliptic (i^2 = -1) scalars. Three arguments: bipolar + encoding requirement, wave-type propagation, diagonal-discipline compatibility. + + + ## Statement + + + ```latex + + % + + \label{thm:split-complex-forced} + + The bipolar prime partition forces + + split-complex scalars ($j^2 = +1$) + + over elliptic scalars ($i^2 = -1$). + + ``` + + + ## Proof / Justification + + + ```latex + + [Proof sketch: three arguments] + + \textbf{(1) Bipolar encoding.} + + Elliptic $\mathbb{C}$ is structurally \emph{unipolar}: + + the unit circle $S^1 \subset \mathbb{C}$ + + has a single connected component. + + It cannot encode two independent sectors. + + Split-complex $H$ with $j^2 = +1$ + + is structurally \emph{bipolar}: + + the canonical idempotents + + $e_+ = (1+j)/2$ and $e_- = (1-j)/2$ + + decompose $H$ into two orthogonal sectors + + $H = e_+ H \oplus e_- H$. + + This decomposition mirrors + + the B/C channel partition of the primes + + (Theorem~\ref{thm:prime-polarity}). + + + \textbf{(2) Wave vs.\ diffusion.} + + Elliptic holomorphy yields the Laplace equation + + $\Delta f = 0$ (elliptic PDE): + + information diffuses isotropically, + + with no preferred direction. + + Split-complex holomorphy yields the wave equation + + (hyperbolic PDE): + + information propagates along characteristics --- + + directionally, reflecting the causal/propagative + + structure that the omega-germs carry + + via their refinement direction. + + + \textbf{(3) Diagonal-discipline compatibility.} + + Split-complex numbers have zero divisors: + + $e_+ \cdot e_- = 0$. + + In classical mathematics, this makes them + + pathological for analysis. + + In $\tau$, the diagonal-free discipline + + (Part~I) prevents the formation of arbitrary + + projections that would collapse + + the split-complex structure. + + The idempotents $e_\pm$ \emph{exist} + + (they are earned from the prime polarity partition), + + but the pathological collapse + + (projecting onto one sector while annihilating the other) + + cannot be performed + + because the required diagonal is not earned. + + The zero divisors are \emph{managed}, not avoided. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 66 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch30-bipolar-algebra.tex` lines + 312-318 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Polarity.BipolarAlgebra` + + - Name: `Tau.Polarity.split_complex_forced` + + + ## Dependencies + + + - Canonical: I.D26, I.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The bipolar prime partition forces split-complex (j^2 = +1) over elliptic (i^2 = -1) scalars. Three arguments: + bipolar encoding requirement, wave-type propagation, diagonal-discipline compatibility.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001178.md b/_corpus_v3_items/cid001178.md new file mode 100644 index 000000000..f4548516d --- /dev/null +++ b/_corpus_v3_items/cid001178.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001178/ +title: Minimal Alphabet Theorem +cid: cid001178 +primary_alias: THM0012 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: '|Gen| = 5 is the unique cardinality achieving completeness (all 3 rewiring levels have channels), rigidity (no non-trivial + rho-automorphism), and saturation (tetration has no channel). 4 generators fail completeness; 6 generators fail rigidity.' +aliases: +- THM0012 +- I.T11 +- minimal-alphabet-theorem +- thm:minimal-alphabet +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Minimal Alphabet Theorem**\n\n|Gen| = 5 is the unique cardinality achieving completeness (all 3 rewiring levels\ + \ have channels), rigidity (no non-trivial rho-automorphism), and saturation (tetration has no channel). 4 generators\ + \ fail completeness; 6 generators fail rigidity.\n\n# Minimal Alphabet Theorem\n\n## Summary\n\n|Gen| = 5 is the unique\ + \ cardinality achieving completeness (all 3 rewiring levels have channels), rigidity (no non-trivial rho-automorphism),\ + \ and saturation (tetration has no channel). 4 generators fail completeness; 6 generators fail rigidity.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:minimal-alphabet}\n$|\\mathrm{Gen}| = 5$ is the unique cardinality\nfor which the $\\tau$-kernel\ + \ simultaneously achieves:\n\\begin{enumerate}\n \\item[(a)] \\textbf{Completeness:}\n All three rewiring\ + \ levels of the iterator ladder\n have canonical orbit channel assignments\n ($\\pi \\leftrightarrow\ + \ +$,\n $\\gamma \\leftrightarrow \\times$,\n $\\eta \\leftrightarrow \\hat{}$\\,).\n \\item[(b)]\ + \ \\textbf{Rigidity:}\n $\\Aut(\\tau) = \\{\\id\\}$ ---\n no non-trivial $\\rho$-automorphism exists.\n\ + \ \\item[(c)] \\textbf{Saturation:}\n Tetration has no channel,\n and is algebraically degraded.\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nThe 5-generator system satisfies all three properties:\n\ + completeness by the channel assignments\nof Chapter~\\ref{ch:iterator-ladder},\nrigidity by Theorem~\\ref{thm:rigidity},\n\ + and saturation by Theorem~\\ref{thm:ladder-saturation}\nand Theorem~\\ref{thm:tetration-degraded}.\n\nUniqueness follows\ + \ from the counter-models:\n\\begin{itemize}\n \\item $|\\mathrm{Gen}| = 4$ fails completeness\n (Theorem~\\\ + ref{thm:four-gen-incomplete}).\n \\item $|\\mathrm{Gen}| = 6$ fails rigidity\n (Theorem~\\ref{thm:six-gen-rigidity-fail}).\n\ + \\end{itemize}\nSince adding generators breaks rigidity\nand removing generators breaks completeness,\n$5$ is the unique\ + \ solution\nto the simultaneous constraint.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 72\n\ + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex` lines 547-566\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Saturation`\n- Name: `Tau.Orbit.Saturation.minimal_alphabet`\n\ + \n## Dependencies\n\n- Canonical: I.T02, I.T07, I.T11a, I.T11b, I.T11c\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '|Gen| = 5 is the unique cardinality achieving completeness (all 3 rewiring levels have channels), rigidity (no + non-trivial rho-automorphism), and saturation (tetration has no channel). 4 generators fail completeness; 6 generators fail + rigid' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001179.md b/_corpus_v3_items/cid001179.md new file mode 100644 index 000000000..563ea4df9 --- /dev/null +++ b/_corpus_v3_items/cid001179.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001179/ +title: Six-Generator Rigidity Failure +cid: cid001179 +primary_alias: THM0013 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A 6-generator tau-like system admits a non-trivial rho-automorphism: the swap eta <-> zeta commutes with rho6 and + is an involution, yet is not the identity. This breaks rigidity.' +aliases: +- THM0013 +- I.T11a +- six-generator-rigidity-failure +- thm:six-gen-rigidity-fail +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Six-Generator Rigidity Failure**\n\nA 6-generator tau-like system admits a non-trivial rho-automorphism: the\ + \ swap eta <-> zeta commutes with rho6 and is an involution, yet is not the identity. This breaks rigidity.\n\n# Six-Generator\ + \ Rigidity Failure\n\n## Summary\n\nA 6-generator tau-like system admits a non-trivial rho-automorphism: the swap eta\ + \ <-> zeta commutes with rho6 and is an involution, yet is not the identity. This breaks rigidity.\n\n## Statement\n\n\ + ```latex\n%\n\\label{thm:six-gen-rigidity-fail}\nConsider a hypothetical 6-generator system\n$\\mathrm{Gen}_6 = \\{\\\ + alpha, \\pi, \\gamma, \\eta, \\zeta, \\omega\\}$\n(adding a sixth generator~$\\zeta$).\nThe transposition $\\eta \\leftrightarrow\ + \ \\zeta$\ndefines a non-trivial $\\rho_6$-automorphism:\n\\begin{enumerate}\n \\item It commutes with~$\\rho_6$\n\ + \ (both $\\eta$ and $\\zeta$ are solenoidal,\n so $\\rho_6$ increments their depths identically).\n\ + \ \\item It is an involution\n (applying the swap twice returns to the identity).\n \\item It is \\emph{not}\ + \ the identity\n ($\\langle\\eta, 0\\rangle \\mapsto \\langle\\zeta, 0\\rangle$).\n\\end{enumerate}\nThis breaks\ + \ the rigidity theorem\n(Theorem~\\ref{thm:rigidity}).\n```\n\n## Proof / Justification\n\n```latex\nThe swap $\\sigma_{\\\ + eta\\zeta}$ acts on $\\mathrm{Obj}_6$\nby exchanging the seed:\n$\\sigma_{\\eta\\zeta}(\\langle g, n\\rangle) = \\langle\ + \ s(g), n\\rangle$\nwhere $s(\\eta) = \\zeta$, $s(\\zeta) = \\eta$,\nand $s(g) = g$ otherwise.\nSince $\\rho_6$ fixes\ + \ $\\omega$ and increments depth\nfor all other seeds, $\\sigma_{\\eta\\zeta}$\ncommutes with $\\rho_6$ by direct verification.\n\ + It is an involution and moves $\\langle\\eta, 0\\rangle$,\nso it is a non-trivial automorphism.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 73\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 497-516\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.TooMany`\n\ + - Name: `Tau.Orbit.TooMany.six_gen_rigidity_fails`\n\n## Dependencies\n\n- Canonical: I.T07\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A 6-generator tau-like system admits a non-trivial rho-automorphism: the swap eta <-> zeta commutes with rho6 + and is an involution, yet is not the identity. This breaks rigidity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001180.md b/_corpus_v3_items/cid001180.md new file mode 100644 index 000000000..7925baa37 --- /dev/null +++ b/_corpus_v3_items/cid001180.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001180/ +title: Four-Generator Ladder Incompleteness +cid: cid001180 +primary_alias: THM0014 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: With 4 generators, only 2 solenoidal generators (pi, gamma) exist for 3 rewiring levels needed. Exponentiation has + no canonical channel. This breaks ladder completeness. +aliases: +- THM0014 +- I.T11b +- four-generator-ladder-incompleteness +- thm:four-gen-incomplete +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Four-Generator Ladder Incompleteness**\n\nWith 4 generators, only 2 solenoidal generators (pi, gamma) exist\ + \ for 3 rewiring levels needed. Exponentiation has no canonical channel. This breaks ladder completeness.\n\n# Four-Generator\ + \ Ladder Incompleteness\n\n## Summary\n\nWith 4 generators, only 2 solenoidal generators (pi, gamma) exist for 3 rewiring\ + \ levels needed. Exponentiation has no canonical channel. This breaks ladder completeness.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:four-gen-incomplete}\nConsider a hypothetical 4-generator system\n$\\mathrm{Gen}_4 = \\{\\alpha, \\pi,\ + \ \\gamma, \\omega\\}$\n(dropping~$\\eta$).\nOnly two solenoidal generators remain\n($\\pi$ and $\\gamma$),\nbut the iterator\ + \ ladder requires three\nrewiring channels\n(addition, multiplication, exponentiation).\nThe exponentiation level has\ + \ no canonical channel:\n\\[\n \\mathrm{channel}(L_2) = \\textup{none}.\n\\]\n```\n\n## Proof / Justification\n\n```latex\n\ + The two solenoidal generators $\\pi$ and $\\gamma$\nare assigned to the addition and multiplication channels\nrespectively.\n\ + No solenoidal generator remains for exponentiation.\nThe pigeonhole principle gives\n$|\\{\\pi, \\gamma\\}| = 2 < 3$ rewiring\ + \ levels needed.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 74\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 455-470\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.TooFew`\n\ + - Name: `Tau.Orbit.TooFew.four_gen_ladder_incomplete`\n\n## Dependencies\n\n- Canonical: I.T02\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: With 4 generators, only 2 solenoidal generators (pi, gamma) exist for 3 rewiring levels needed. Exponentiation + has no canonical channel. This breaks ladder completeness. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001181.md b/_corpus_v3_items/cid001181.md new file mode 100644 index 000000000..5fa28e89a --- /dev/null +++ b/_corpus_v3_items/cid001181.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001181/ +title: Tetration Algebraic Degradation +cid: cid001181 +primary_alias: THM0015 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Tetration is algebraically degraded: non-commutative (2^^3 != 3^^2), non-associative ((2^^2)^^2 != 2^^(2^^2)), no + left identity >= 2. This is the algebraic obstruction to canonicality at the 4th operation level.' +aliases: +- THM0015 +- I.T11c +- tetration-algebraic-degradation +- thm:tetration-degraded +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 12 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tetration Algebraic Degradation**\n\nTetration is algebraically degraded: non-commutative (2^^3 != 3^^2), non-associative\ + \ ((2^^2)^^2 != 2^^(2^^2)), no left identity >= 2. This is the algebraic obstruction to canonicality at the 4th operation\ + \ level.\n\n# Tetration Algebraic Degradation\n\n## Summary\n\nTetration is algebraically degraded: non-commutative (2^^3\ + \ != 3^^2), non-associative ((2^^2)^^2 != 2^^(2^^2)), no left identity >= 2. This is the algebraic obstruction to canonicality\ + \ at the 4th operation level.\n\n## Statement\n\n```latex\n%\n\\label{thm:tetration-degraded}\nTetration fails all three\ + \ algebraic canonicality conditions:\n\\begin{enumerate}\n \\item \\textbf{Non-commutative:}\n ${}^{\\underline{3}}\\\ + underline{2} = \\underline{16}\n \\neq \\underline{9} = {}^{\\underline{2}}\\underline{3}$.\n \\item \\textbf{Non-associative:}\n\ + \ $({}^{\\underline{2}}\\underline{2})\n \\uparrow\\uparrow \\underline{2}\n = {}^{\\underline{2}}\\\ + underline{4} = \\underline{256}\n \\neq \\underline{65536}\n = {}^{\\underline{4}}\\underline{2}\n \ + \ = {}^{({}^{\\underline{2}}\\underline{2})}\\underline{2}$.\n \\item \\textbf{No left identity $\\geq \\underline{2}$:}\n\ + \ for any $\\underline{e} \\geq \\underline{2}$,\n ${}^{\\underline{2}}\\underline{e} = \\underline{e}^{\\\ + underline{e}}\n \\geq \\underline{4} > \\underline{2}$.\n\\end{enumerate}\nBy contrast, addition has identity~$\\\ + underline{0}$,\nmultiplication has identity~$\\underline{1}$,\nand exponentiation has right identity~$\\underline{1}$\n\ + ($\\underline{n}^{\\underline{1}} = \\underline{n}$).\nThe algebraic degradation at level~3\nis an independent obstruction\ + \ to extending\nthe canonical operation hierarchy beyond tetration.\n```\n\n## Proof / Justification\n\n```latex\nItems\ + \ (1) and (2) are verified by computation\non small concrete values.\nFor (3): if $\\underline{e} \\geq \\underline{2}$,\n\ + then $\\underline{e}^{\\underline{e}} \\geq \\underline{2}^{\\underline{2}}\n= \\underline{4} > \\underline{2}$,\nso ${}^{\\\ + underline{2}}\\underline{e}\n= \\underline{e}^{\\underline{e}} \\neq \\underline{2}$.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 75\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch12-exp-tetration.tex`\ + \ lines 368-395\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Orbit.Saturation`\n\ + - Name: `Tau.Orbit.Saturation.tetration_algebraic_degradation`\n\n## Dependencies\n\n- Canonical: I.T02\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Tetration is algebraically degraded: non-commutative (2^^3 != 3^^2), non-associative ((2^^2)^^2 != 2^^(2^^2)), + no left identity >= 2. This is the algebraic obstruction to canonicality at the 4th operation level.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001182.md b/_corpus_v3_items/cid001182.md new file mode 100644 index 000000000..cf7fe9658 --- /dev/null +++ b/_corpus_v3_items/cid001182.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001182/ +title: Spectral Decomposition +cid: cid001182 +primary_alias: THM0016 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every element of L decomposes uniquely into B-sector and C-sector components via characters. The profinite topology + on Z_hat_tau provides the convergence framework. +aliases: +- THM0016 +- I.T12 +- spectral-decomposition +- thm:spectral-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005284 +- predicate: formalized_by + object: cid005285 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 11, Chapter 44 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Decomposition**\n\nEvery element of L decomposes uniquely into B-sector and C-sector components via\ + \ characters. The profinite topology on Z_hat_tau provides the convergence framework.\n\n# Spectral Decomposition\n\n\ + ## Summary\n\nEvery element of L decomposes uniquely into B-sector and C-sector components via characters. The profinite\ + \ topology on Z_hat_tau provides the convergence framework.\n\n## Statement\n\n```latex\n%\n\\label{thm:spectral-decomposition}\n\ + Let $\\mathbb{L} = H_\\tau = \\hat{\\mathbb{Z}}_\\tau[j]$\nbe the algebraic lemniscate\n(Theorem~\\ref{thm:algebraic-lemniscate}).\n\ + For every element $x \\in \\mathbb{L}$,\nthere exist unique elements\n$\\alpha, \\beta \\in \\hat{\\mathbb{Z}}_\\tau$\n\ + such that:\n\\[\n \\boxed{%\n x = \\alpha \\cdot e_+ + \\beta \\cdot e_-\n = \\chi_+(x) \\cdot e_+ + \\chi_-(x)\ + \ \\cdot e_-.}\n\\]\nThe coefficients are:\n\\begin{align*}\n \\alpha &= \\chi_+(x) = a + b, \\\\\n \\beta &= \\\ + chi_-(x) = a - b,\n\\end{align*}\nwhere $x = a + bj$ with $a, b \\in \\hat{\\mathbb{Z}}_\\tau$.\nThis decomposition is:\n\ + \\begin{enumerate}\n \\item \\textbf{Canonical}:\n it depends only on the algebraic structure of $\\mathbb{L}$,\n\ + \ not on any choice of basis or coordinate system.\n \\item \\textbf{Orthogonal}:\n the $e_+$ and\ + \ $e_-$ components do not interact\n ($e_+ \\cdot e_- = 0$).\n \\item \\textbf{Complete}:\n $e_+\ + \ + e_- = 1$, so no information is lost.\n \\item \\textbf{Multiplicative}:\n $\\chi_\\pm(xy) = \\chi_\\pm(x)\ + \ \\cdot \\chi_\\pm(y)$\n for all $x, y \\in \\mathbb{L}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\n\\textbf{Existence.}\nLet $x = a + bj \\in \\hat{\\mathbb{Z}}_\\tau[j]$.\nBy Proposition~\\ref{prop:idempotent-decomposition}\n\ + (Chapter~\\ref{ch:split-complex-scalars}),\n\\[\n x = (a + b) e_+ + (a - b) e_-.\n\\]\nSetting $\\alpha = a + b$ and\ + \ $\\beta = a - b$\ngives the decomposition.\nThese are precisely the character evaluations:\n$\\chi_+(x) = a + b$ and\ + \ $\\chi_-(x) = a - b$\n(Definition~\\ref{def:lemniscate-characters}).\n\n\\textbf{Uniqueness.}\nSuppose $x = \\alpha'\ + \ e_+ + \\beta' e_-$\nfor some $\\alpha', \\beta' \\in \\hat{\\mathbb{Z}}_\\tau$.\nMultiplying both sides by $e_+$:\n\\\ + [\n x \\cdot e_+ = \\alpha' e_+^2 + \\beta' e_- e_+\n = \\alpha' e_+ + 0\n = \\alpha' e_+.\n\\]\nBut also $x\ + \ \\cdot e_+ = (a + b) e_+$ from the computation\nin Proposition~\\ref{prop:idempotent-decomposition}.\nSince $e_+ \\\ + neq 0$,\nand $\\hat{\\mathbb{Z}}_\\tau$ acts faithfully\non each idempotent sector,\nwe conclude $\\alpha' = a + b = \\\ + alpha$.\nSimilarly, $\\beta' = a - b = \\beta$.\n\n\\textbf{Multiplicativity.}\nLet $x = a + bj$ and $y = c + dj$.\nThen\ + \ $xy = (ac + bd) + (ad + bc)j$, and:\n\\begin{align*}\n \\chi_+(xy)\n &= (ac + bd) + (ad + bc) \\\\\n &= (a\ + \ + b)(c + d) \\\\\n &= \\chi_+(x) \\cdot \\chi_+(y).\n\\end{align*}\nThe computation for $\\chi_-$ is analogous:\n\ + $\\chi_-(xy) = (a - b)(c - d) = \\chi_-(x) \\cdot \\chi_-(y)$.\n\n\\textbf{Additivity} follows similarly:\n$\\chi_\\pm(x\ + \ + y) = \\chi_\\pm(x) + \\chi_\\pm(y)$.\nThus $\\chi_+$ and $\\chi_-$ are ring homomorphisms,\nconfirming their character\ + \ property.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 89\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part11/ch44-spectral-decomposition.tex`\ + \ lines 57-92\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Spectral`\n\ + - Name: `Tau.Boundary.spectral_decomposition`\n\n## Dependencies\n\n- Canonical: I.D37, I.D38, I.D19\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every element of L decomposes uniquely into B-sector and C-sector components via characters. The profinite topology + on Z_hat_tau provides the convergence framework. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001183.md b/_corpus_v3_items/cid001183.md new file mode 100644 index 000000000..e60a0be70 --- /dev/null +++ b/_corpus_v3_items/cid001183.md @@ -0,0 +1,105 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001183/ +title: Explosion Barrier +cid: cid001183 +primary_alias: THM0017 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: In Truth4, B = 'both true and false' does NOT cause explosion (ex falso quodlibet). B-witnesses and not-B-witnesses + live in different spectral sectors. Structural theorem, not axiom choice. +aliases: +- THM0017 +- I.T13 +- explosion-barrier +- thm:explosion-barrier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005323 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 12, Chapter 47 (Part XII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Explosion Barrier**\n\nIn Truth4, B = 'both true and false' does NOT cause explosion (ex falso quodlibet).\ + \ B-witnesses and not-B-witnesses live in different spectral sectors. Structural theorem, not axiom choice.\n\n# Explosion\ + \ Barrier\n\n## Summary\n\nIn Truth4, B = 'both true and false' does NOT cause explosion (ex falso quodlibet). B-witnesses\ + \ and not-B-witnesses live in different spectral sectors. Structural theorem, not axiom choice.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:explosion-barrier}\nLet $\\mathrm{val} : \\mathrm{Prop}_\\tau \\to \\mathrm{Truth4}$\nbe a valuation assigning\ + \ truth values\nin the four-valued lattice\n$\\{\\mathsf{T}, \\mathsf{F}, \\mathsf{B}, \\mathsf{N}\\}$.\nThen:\n\\begin{enumerate}\n\ + \ \\item \\textbf{Overdetermination does not explode.}\n If $\\mathrm{val}(P) = \\mathsf{B}$\n (i.e.,\ + \ $P$ is both true and false),\n then for an arbitrary proposition $Q$:\n \\[\n \\boxed{%\n\ + \ \\mathrm{val}(P) = \\mathsf{B}\n \\;\\not\\vdash\\;\n \\mathrm{val}(Q) = \\mathsf{T}.}\n\ + \ \\]\n Specifically,\n $\\mathrm{val}(P \\Rightarrow Q)\n = \\mathsf{B} \\Rightarrow\ + \ \\mathrm{val}(Q)$,\n which equals $\\mathsf{T}$ only if\n $\\mathrm{val}(Q) \\in \\{\\mathsf{T}, \\\ + mathsf{B}\\}$,\n and equals $\\mathsf{N}$ if\n $\\mathrm{val}(Q) \\in \\{\\mathsf{N}, \\mathsf{F}\\\ + }$.\n \\item \\textbf{Underdetermination does not explode.}\n If $\\mathrm{val}(P) = \\mathsf{N}$,\n \ + \ then $\\mathrm{val}(P \\Rightarrow Q)\n = \\mathsf{N} \\Rightarrow \\mathrm{val}(Q)$,\n which\ + \ is $\\mathsf{T}$ only if\n $\\mathrm{val}(Q) \\in \\{\\mathsf{T}, \\mathsf{N}\\}$,\n and is $\\mathsf{B}$\ + \ if\n $\\mathrm{val}(Q) \\in \\{\\mathsf{B}, \\mathsf{F}\\}$.\n \\item \\textbf{Classical explosion is preserved.}\n\ + \ If $\\mathrm{val}(P) = \\mathsf{F}$,\n then $\\mathrm{val}(P \\Rightarrow Q) = \\mathsf{T}$\n \ + \ for all $Q$.\n\\end{enumerate}\nIn particular, \\textbf{disjunctive syllogism fails}\nfor $\\mathsf{B}$-valued\ + \ propositions:\nfrom $P \\lor Q$ and $\\lnot P$,\none cannot conclude $Q$\nwhen $\\mathrm{val}(P) = \\mathsf{B}$.\n```\n\ + \n## Proof / Justification\n\n```latex\nAll three statements follow by inspection\nof the implication table\n(Definition~\\\ + ref{def:truth4-material-implication}).\n\n\\textbf{Statement (1).}\nThe row $\\mathsf{B} \\Rightarrow \\cdot$ is\n$(\\\ + mathsf{T}, \\mathsf{T}, \\mathsf{N}, \\mathsf{N})$:\n\\begin{align*}\n \\mathsf{B} \\Rightarrow \\mathsf{T} &= \\mathsf{T},\ + \ \\\\\n \\mathsf{B} \\Rightarrow \\mathsf{B} &= \\mathsf{T}, \\\\\n \\mathsf{B} \\Rightarrow \\mathsf{N} &= \\\ + mathsf{N}, \\\\\n \\mathsf{B} \\Rightarrow \\mathsf{F} &= \\mathsf{N}.\n\\end{align*}\nWhen $\\mathrm{val}(Q) = \\\ + mathsf{F}$,\nthe implication $P \\Rightarrow Q$ has value $\\mathsf{N}$,\nnot $\\mathsf{T}$.\nTherefore the principle\n\ + ``from $\\mathrm{val}(P) = \\mathsf{B}$,\nderive $\\mathrm{val}(Q) = \\mathsf{T}$\nfor arbitrary $Q$''\nfails:\nthe conclusion\ + \ $Q$ inherits at most\nthe underdetermined value $\\mathsf{N}$,\nnot the clean truth value $\\mathsf{T}$.\n\n\\textbf{Statement\ + \ (2).}\nThe row $\\mathsf{N} \\Rightarrow \\cdot$ is\n$(\\mathsf{T}, \\mathsf{B}, \\mathsf{T}, \\mathsf{B})$.\nWhen $\\\ + mathrm{val}(Q) \\in \\{\\mathsf{B}, \\mathsf{F}\\}$,\nthe implication has value $\\mathsf{B}$,\nnot $\\mathsf{T}$.\n\n\ + \\textbf{Statement (3).}\nThe row $\\mathsf{F} \\Rightarrow \\cdot$ is\n$(\\mathsf{T}, \\mathsf{T}, \\mathsf{T}, \\mathsf{T})$:\n\ + genuine falsehood still implies anything,\nexactly as in classical logic.\n\n\\textbf{Disjunctive syllogism failure.}\n\ + In classical logic,\ndisjunctive syllogism states:\nfrom $P \\lor Q$ and $\\lnot P$, conclude $Q$.\nSuppose $\\mathrm{val}(P)\ + \ = \\mathsf{B}$.\nThen $\\mathrm{val}(P \\lor Q) = \\mathsf{B} \\lor \\mathrm{val}(Q)\n= \\mathsf{T}$ (or $\\mathsf{B}$,\ + \ depending on $\\mathrm{val}(Q)$),\nand $\\mathrm{val}(\\lnot P) = \\lnot \\mathsf{B} = \\mathsf{N}$.\nThe conclusion\ + \ requires extracting $Q$\nfrom the disjunction $P \\lor Q$\nusing $\\lnot P$.\nBut $\\mathrm{val}(\\lnot P) = \\mathsf{N}\ + \ \\neq \\mathsf{T}$:\nthe negation of $P$ is underdetermined, not cleanly true,\nso $P$ is not cleanly eliminated from\ + \ the disjunction.\nThe disjunctive syllogism step fails.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 92\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part12/ch47-explosion-barrier.tex`\ + \ lines 246-290\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Logic.Explosion`\n\ + - Name: `Tau.Logic.explosion_barrier`\n\n## Dependencies\n\n- Canonical: I.D21\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: In Truth4, B = 'both true and false' does NOT cause explosion (ex falso quodlibet). B-witnesses and not-B-witnesses + live in different spectral sectors. Structural theorem, not axiom choice. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001184.md b/_corpus_v3_items/cid001184.md new file mode 100644 index 000000000..b289d375e --- /dev/null +++ b/_corpus_v3_items/cid001184.md @@ -0,0 +1,149 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001184/ +title: No Additive Inverse +cid: cid001184 +primary_alias: THM0018 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive + depths only.' +aliases: +- THM0018 +- I.T14 +- no-additive-inverse +- thm:no-additive-inverse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**No Additive Inverse** + + + For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive + depths only. + + + # No Additive Inverse + + + ## Summary + + + For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates positive + depths only. + + + ## Statement + + + ```latex + + % + + \label{thm:no-additive-inverse} + + For any $\underline{n} > \underline{0}$ in $\tau$-Idx, + + there exists no $\underline{m} \in \tau$-Idx such that + + $\underline{n} + \underline{m} = \underline{0}$. + + ``` + + + ## Proof / Justification + + + ```latex + + Immediate from Proposition~\ref{prop:sum-zero-iff}: + + if $\underline{n} + \underline{m} = \underline{0}$, + + then $\underline{n} = \underline{0}$, + + contradicting $\underline{n} > \underline{0}$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 97 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex` lines 285-290 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Denotation.Structural` + + - Name: `Tau.Denotation.tauIdx_no_additive_inverse` + + + ## Dependencies + + + - Canonical: I.D10, I.P15 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For n > 0, no m in tau-Idx satisfies n + m = 0. The absence of additive inverses is structural: rho generates + positive depths only.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001185.md b/_corpus_v3_items/cid001185.md new file mode 100644 index 000000000..19dd8639a --- /dev/null +++ b/_corpus_v3_items/cid001185.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001185/ +title: No Ring Negation +cid: cid001185 +primary_alias: THM0019 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not + a ring.' +aliases: +- THM0019 +- I.T15 +- no-ring-negation +- thm:no-ring-negation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 11 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**No Ring Negation** + + + No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not a ring. + + + # No Ring Negation + + + ## Summary + + + No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but not a ring. + + + ## Statement + + + ```latex + + % + + \label{thm:no-ring-negation} + + There exists no function + + $\mathrm{neg} : \tau\text{-Idx} \to \tau\text{-Idx}$ + + such that $\underline{n} + \mathrm{neg}(\underline{n}) = \underline{0}$ + + for all $\underline{n} \in \tau$-Idx. + + ``` + + + ## Proof / Justification + + + ```latex + + Suppose such a function exists. + + Apply it to $\underline{1}$: + + $\underline{1} + \mathrm{neg}(\underline{1}) = \underline{0}$. + + But Theorem~\ref{thm:no-additive-inverse} + + asserts that no such $\mathrm{neg}(\underline{1})$ exists, + + since $\underline{1} > \underline{0}$. + + Contradiction. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 98 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch11-swap-add-mul.tex` lines 299-305 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Denotation.Structural` + + - Name: `Tau.Denotation.tauIdx_no_ring_negation` + + + ## Dependencies + + + - Canonical: I.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'No function neg: tau-Idx -> tau-Idx satisfies n + neg(n) = 0 for all n. tau-Idx is a commutative semiring but + not a ring.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001186.md b/_corpus_v3_items/cid001186.md new file mode 100644 index 000000000..8280eb727 --- /dev/null +++ b/_corpus_v3_items/cid001186.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001186/ +title: Mul Cancel Fails at Zero +cid: cid001186 +primary_alias: THM0020 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 0 * 1 = 0 * 2 but 1 != 2. Multiplicative left-cancellation fails when the cancelled factor is zero. +aliases: +- THM0020 +- I.T16 +- mul-cancel-fails-at-zero +- thm:mul-cancel-fails-zero +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Mul Cancel Fails at Zero**\n\n0 * 1 = 0 * 2 but 1 != 2. Multiplicative left-cancellation fails when the cancelled\ + \ factor is zero.\n\n# Mul Cancel Fails at Zero\n\n## Summary\n\n0 * 1 = 0 * 2 but 1 != 2. Multiplicative left-cancellation\ + \ fails when the cancelled factor is zero.\n\n## Statement\n\n```latex\n%\n\\label{thm:mul-cancel-fails-zero}\n% Depends:\ + \ I.D09 (multiplication), I.P06 (zero vacuous)\n$\\underline{0} \\times \\underline{1} = \\underline{0} \\times \\underline{2}$,\n\ + yet $\\underline{1} \\neq \\underline{2}$.\nThat is, multiplicative left-cancellation fails when the\ncancelled factor\ + \ is $\\underline{0}$.\n```\n\n## Proof / Justification\n\n```latex\nBy absorption (Proposition~\\ref{prop:zero-vacuous}\\\ + ,(4)):\n$\\underline{0} \\times \\underline{1}\n = \\underline{0}\n = \\underline{0} \\times \\underline{2}$.\nYet $\\\ + underline{1} = \\langle\\alpha, 1\\rangle\n \\neq \\langle\\alpha, 2\\rangle = \\underline{2}$.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 101\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch15-denotational-order.tex`\ + \ lines 225-232\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_mul_cancel_fails_at_zero`\n\n## Dependencies\n\n- Canonical: I.D11, I.P18\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 0 * 1 = 0 * 2 but 1 != 2. Multiplicative left-cancellation fails when the cancelled factor is zero. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001187.md b/_corpus_v3_items/cid001187.md new file mode 100644 index 000000000..fd070257e --- /dev/null +++ b/_corpus_v3_items/cid001187.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001187/ +title: Multiplicative Cancellation iff Positive +cid: cid001187 +primary_alias: THM0021 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CLIMAX: Multiplicative cancellation holds for n iff n > 0. Zero is the unique obstruction. One locus, one guard, + maximal universality everywhere else.' +aliases: +- THM0021 +- I.T17 +- multiplicative-cancellation-iff-positive +- thm:mul-cancel-exactly-pos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Multiplicative Cancellation iff Positive**\n\nCLIMAX: Multiplicative cancellation holds for n iff n > 0. Zero\ + \ is the unique obstruction. One locus, one guard, maximal universality everywhere else.\n\n# Multiplicative Cancellation\ + \ iff Positive\n\n## Summary\n\nCLIMAX: Multiplicative cancellation holds for n iff n > 0. Zero is the unique obstruction.\ + \ One locus, one guard, maximal universality everywhere else.\n\n## Statement\n\n```latex\n%\n\\label{thm:mul-cancel-exactly-pos}\n\ + % Depends: I.D09, prop:integral-domain, prop:positive-closed\nFor $\\underline{n} \\in \\tau\\text{-Idx}$,\nthe following\ + \ are equivalent:\n\\begin{enumerate}\n \\item $\\underline{n}$ satisfies multiplicative left-cancellation:\n \ + \ $\\underline{n} \\times \\underline{a}\n = \\underline{n} \\times \\underline{b}\n \\;\\Rightarrow\\\ + ;\n \\underline{a} = \\underline{b}$\n for all $\\underline{a}, \\underline{b}$.\n \\item $\\underline{n}\ + \ > \\underline{0}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n$(2) \\Rightarrow (1)$.\\;\nSuppose\ + \ $\\underline{n} > \\underline{0}$ and\n$\\underline{n} \\times \\underline{a}\n = \\underline{n} \\times \\underline{b}$.\n\ + Then $\\underline{n} \\times (\\underline{a} - \\underline{b}) = \\underline{0}$\nin the sense that\n$\\underline{n} \\\ + times \\underline{a}\n = \\underline{n} \\times \\underline{b}$\nimplies (by the integral domain property,\nProposition~\\\ + ref{prop:integral-domain})\nthat either $\\underline{n} = \\underline{0}$ or\n$\\underline{a} = \\underline{b}$.\nSince\ + \ $\\underline{n} > \\underline{0}$,\nwe conclude $\\underline{a} = \\underline{b}$.\n\n$(1) \\Rightarrow (2)$.\\;\nContrapositive:\ + \ if $\\underline{n} = \\underline{0}$,\nthen $\\underline{0} \\times \\underline{1}\n = \\underline{0}\n = \\\ + underline{0} \\times \\underline{2}$\nbut $\\underline{1} \\neq \\underline{2}$\n(Theorem~\\ref{thm:mul-cancel-fails-zero}).\n\ + ```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 102\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part03/ch15-denotational-order.tex`\ + \ lines 246-260\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Denotation.Structural`\n\ + - Name: `Tau.Denotation.tauIdx_mul_cancel_exactly_pos`\n\n## Dependencies\n\n- Canonical: I.P17, I.T16, I.P19\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CLIMAX: Multiplicative cancellation holds for n iff n > 0. Zero is the unique obstruction. One locus, one guard, + maximal universality everywhere else.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001188.md b/_corpus_v3_items/cid001188.md new file mode 100644 index 000000000..f98b22b99 --- /dev/null +++ b/_corpus_v3_items/cid001188.md @@ -0,0 +1,104 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001188/ +title: CRT Coherence Constraint +cid: cid001188 +primary_alias: THM0022 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Tower-coherent functions that are well-defined on Z/M_kZ depend only on the CRT residue at stage k. Proved concretely + for chi_plus and chi_minus; the general case requires residue-respect as hypothesis. +aliases: +- THM0022 +- I.T18 +- crt-coherence-constraint +- thm:crt-coherence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 50 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**CRT Coherence Constraint**\n\nTower-coherent functions that are well-defined on Z/M_kZ depend only on the CRT\ + \ residue at stage k. Proved concretely for chi_plus and chi_minus; the general case requires residue-respect as hypothesis.\n\ + \n# CRT Coherence Constraint\n\n## Summary\n\nTower-coherent functions that are well-defined on Z/M_kZ depend only on\ + \ the CRT residue at stage k. Proved concretely for chi_plus and chi_minus; the general case requires residue-respect\ + \ as hypothesis.\n\n## Statement\n\n```latex\n%\n\\label{thm:crt-coherence}\nAt each primorial stage $M_k = \\underline{p}_1\ + \ \\cdot \\underline{p}_2 \\cdots \\underline{p}_k$,\nthe Chinese Remainder Theorem\n(Section~\\ref{subsec:ch30-crt},\ + \ I.D29)\ngives the ring isomorphism:\n\\[\n \\varphi_k \\colon\n \\mathbb{Z}/M_k\\mathbb{Z}\n \\;\\xrightarrow{\\\ + ;\\cong\\;}\n \\prod_{i=1}^{k} \\mathbb{Z}/\\underline{p}_i\\mathbb{Z}.\n\\]\nLet $T$ be a tower-coherent $\\omega$-germ\ + \ transformer\n(Definition~\\ref{def:tower-coherence}, I.D46).\nThen:\n\\begin{enumerate}\n \\item \\textbf{Prime factorization\ + \ of the action.}\n The action of $T$ on the $k$-th stage\n is completely determined by its action\n\ + \ on each CRT factor $\\mathbb{Z}/\\underline{p}_i\\mathbb{Z}$\n for $i = 1, \\ldots, k$.\n \ + \ Formally, there exist functions\n $T^{(i)} : \\mathbb{Z}/\\underline{p}_i\\mathbb{Z} \\to \\mathbb{Z}/\\underline{p}_i\\\ + mathbb{Z}[j]$\n such that under the CRT isomorphism:\n \\[\n T_k(a_1, \\ldots, a_k) = \\\ + bigl(T^{(1)}(a_1), \\ldots, T^{(k)}(a_k)\\bigr).\n \\]\n \\item \\textbf{Stability under tower extension.}\n\ + \ Passing from stage $k$ to stage $k+1 = M_k \\cdot \\underline{p}_{k+1}$\n adds exactly one new CRT\ + \ factor\n (for prime $\\underline{p}_{k+1}$),\n and tower coherence requires\n the restriction\ + \ to the existing $k$ factors\n to be unchanged:\n the functions $T^{(1)}, \\ldots, T^{(k)}$\n \ + \ at stage $k+1$ are identical to those at stage $k$.\n \\item \\textbf{Prime-by-prime determination.}\n \ + \ By induction on $k$:\n $T$ is completely determined by the family\n $\\{T^{(i)}\\}_{i \\geq 1}$,\n\ + \ where each $T^{(i)} : \\mathbb{Z}/\\underline{p}_i\\mathbb{Z} \\to \\mathbb{Z}/\\underline{p}_i\\mathbb{Z}[j]$\n\ + \ describes $T$'s action on the $i$-th prime factor.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\textbf{(1) Prime factorization.}\nFix a stage $k$.\nBy CRT (Section~\\ref{subsec:ch30-crt}),\nan element of $\\mathbb{Z}/M_k\\\ + mathbb{Z}$\ndecomposes uniquely as a tuple\n$(a_1, \\ldots, a_k)$ with $a_i \\in \\mathbb{Z}/\\underline{p}_i\\mathbb{Z}$.\n\ + Consider two elements that agree on all factors\nexcept the $i$-th:\n$(a_1, \\ldots, a_i, \\ldots, a_k)$\nand $(a_1, \\\ + ldots, a_i', \\ldots, a_k)$.\nThe primorial reduction map $\\pi_{k \\to k-1}$\n(which forgets the $k$-th factor)\nsends\ + \ both to the same depth-$(k-1)$ element\nwhen $i = k$, or preserves the difference when $i < k$.\nIterating tower coherence\ + \ over the chain of reductions\n\\[\n \\pi_{k \\to k-1} \\circ \\pi_{k-1 \\to k-2} \\circ \\cdots,\n\\]\nwe see that\ + \ the action of $T_k$ on each CRT factor\nis determined independently.\nDefine $T^{(i)}(a_i) := \\pi_{k \\to \\{i\\}}(T_k(0,\ + \ \\ldots, a_i, \\ldots, 0))$\nwhere $\\pi_{k \\to \\{i\\}}$ projects to the $i$-th factor.\nThe CRT orthogonality of\ + \ basis elements\n($e_i \\cdot e_j \\equiv 0$ for $i \\neq j$)\nensures that the total action decomposes as claimed.\n\ + \n\\textbf{(2) Stability.}\nAn element of $\\mathbb{Z}/M_{k+1}\\mathbb{Z}$\ndecomposes via CRT as\n$(a_1, \\ldots, a_k,\ + \ a_{k+1})$\nwhere $a_i \\in \\mathbb{Z}/\\underline{p}_i\\mathbb{Z}$.\nThe reduction map $\\pi_{k+1 \\to k}$\nsimply\ + \ forgets the last component $a_{k+1}$.\nTower coherence requires:\n\\[\n \\pi_{k+1 \\to k}\\bigl(T_{k+1}(a_1, \\ldots,\ + \ a_k, a_{k+1})\\bigr)\n = T_k\\bigl(\\pi_{k+1 \\to k}(a_1, \\ldots, a_k, a_{k+1})\\bigr)\n = T_k(a_1, \\ldots,\ + \ a_k).\n\\]\nBy statement (1) applied at stage $k+1$:\n$T_{k+1}(a_1, \\ldots, a_k, a_{k+1})\n= (T^{(1)}(a_1), \\ldots,\ + \ T^{(k)}(a_k), T^{(k+1)}(a_{k+1}))$.\nProjecting away the $(k+1)$-th component gives\n$(T^{(1)}(a_1), \\ldots, T^{(k)}(a_k))$,\n\ + which must equal $T_k(a_1, \\ldots, a_k)$.\nTherefore the functions $T^{(1)}, \\ldots, T^{(k)}$\nappearing at stage $k+1$\n\ + are the same as those at stage $k$.\n\n\\textbf{(3) Prime-by-prime determination.}\nBy induction: at stage $1$, $T$ is\ + \ determined by $T^{(1)}$.\nAt stage $k+1$, tower coherence fixes $T^{(1)}, \\ldots, T^{(k)}$\n(by the inductive hypothesis\ + \ and statement~(2))\nand adds exactly one new function $T^{(k+1)}$.\nTherefore $T$ is determined by the countable family\n\ + $\\{T^{(i)}\\}_{i \\geq 1}$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 114\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch50-tau-holomorphic.tex` lines 439-483\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.TauHolomorphic`\n- Name:\ + \ `Tau.Holomorphy.chi_plus_crt`\n\n## Dependencies\n\n- Canonical: I.D46, I.D29\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Tower-coherent functions that are well-defined on Z/M_kZ depend only on the CRT residue at stage k. Proved concretely + for chi_plus and chi_minus; the general case requires residue-respect as hypothesis. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001189.md b/_corpus_v3_items/cid001189.md new file mode 100644 index 000000000..e4ebb9d38 --- /dev/null +++ b/_corpus_v3_items/cid001189.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001189/ +title: Diagonal-Free Protection +cid: cid001189 +primary_alias: THM0023 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The zero-divisor product e_plus * e_minus = 0 cannot arise as T(t1) * T(t2) for any T in HolFun and compatible omega-tails + t1, t2. Sector-pure outputs cannot combine nontrivially. +aliases: +- THM0023 +- I.T19 +- diagonal-free-protection +- thm:diagonal-free-protection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005300 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 51 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagonal-Free Protection**\n\nThe zero-divisor product e_plus * e_minus = 0 cannot arise as T(t1) * T(t2) for\ + \ any T in HolFun and compatible omega-tails t1, t2. Sector-pure outputs cannot combine nontrivially.\n\n# Diagonal-Free\ + \ Protection\n\n## Summary\n\nThe zero-divisor product e_plus * e_minus = 0 cannot arise as T(t1) * T(t2) for any T in\ + \ HolFun and compatible omega-tails t1, t2. Sector-pure outputs cannot combine nontrivially.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:diagonal-free-protection}\nLet $T \\in \\mathrm{HolFun}$ be a $\\tau$-holomorphic function.\nThen for any\ + \ two compatible omega-tails $t_1, t_2$\nwith $T(t_1)$ and $T(t_2)$ in opposite sectors\n(one purely $B$, one purely $C$),\n\ + we have:\n\\[\n \\boxed{%\n T(t_1) \\cdot T(t_2) \\neq 0\n \\quad\\text{only if}\\quad\n t_1 \\text{ and }\ + \ t_2 \\text{ have the same sector purity\n as their outputs.}}\n\\]\nIn particular:\nthe zero-divisor product $e_+\ + \ \\cdot e_- = 0$\ncannot arise as $T(t_1) \\cdot T(t_2)$\nfor any $T \\in \\mathrm{HolFun}$\nand any compatible omega-tails\ + \ $t_1, t_2$ ---\nbecause $T$'s tower coherence prevents it\nfrom mapping a compatible tower\nto a pure idempotent output\n\ + unless the input was already sector-pure.\n```\n\n## Proof / Justification\n\n```latex\nThe proof combines three established\ + \ results.\n\n\\emph{Step 1: Sector decomposition.}\nBy I.P22 (sector independence),\n$T$ decomposes in sector coordinates\ + \ as\n\\[\n T(u, v) = (g(u),\\; h(v)),\n\\]\nwhere $u = \\chi_+(x)$ is the $B$-sector coordinate\nand $v = \\chi_-(x)$\ + \ is the $C$-sector coordinate.\nThe function $g$ acts on $B$-sector inputs\nand $h$ acts on $C$-sector inputs.\n\n\\\ + emph{Step 2: Prime-by-prime determination.}\nBy I.T18 (CRT coherence),\n$g$ and $h$ are determined prime-by-prime:\n$g$\ + \ acts only on the CRT factors\ncorresponding to $B$-sector primes\n(those with $\\mathrm{pol}(p) = +$),\nand $h$ acts\ + \ only on the CRT factors\ncorresponding to $C$-sector primes\n(those with $\\mathrm{pol}(p) = -$).\nEach prime belongs\ + \ to exactly one sector\nby the Prime Polarity Theorem (I.T05).\n\n\\emph{Step 3: No simultaneous projection.}\nBy Proposition~\\\ + ref{prop:no-simul-projection} (I.P23),\nno compatible omega-tail can project nontrivially\nonto both sectors through $T$.\n\ + \n\\emph{Conclusion.}\nThe zero-divisor product $e_+ \\cdot e_- = 0$\nwould require a pure $B$-sector output\nmultiplied\ + \ by a pure $C$-sector output.\nFor $T(t_1)$ to be purely $B$-sector,\nwe need $h(\\chi_-(t_1)) = 0$:\nthe $C$-sector\ + \ component of $T(t_1)$ must vanish.\nFor $T(t_2)$ to be purely $C$-sector,\nwe need $g(\\chi_+(t_2)) = 0$:\nthe $B$-sector\ + \ component of $T(t_2)$ must vanish.\nBut by I.P23, this requires $t_1$ to have trivial $C$-sector input\nand $t_2$ to\ + \ have trivial $B$-sector input ---\nthat is, $t_1$ and $t_2$ must already have\nthe same sector purity as their outputs.\n\ + A compatible omega-tail with nontrivial content\nin both sectors cannot be mapped\nto a pure idempotent output by any\ + \ $T \\in \\mathrm{HolFun}$.\nTherefore the zero-divisor product\n$e_+ \\cdot e_- = 0$\ncannot arise from the same omega-germ\ + \ ---\nwhich I.P23 forbids.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 116\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch51-diagonal-free-protection.tex` lines 268-292\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DiagonalProtection`\n\ + - Name: `Tau.Holomorphy.diagonal_free_protection`\n\n## Dependencies\n\n- Canonical: I.P23, I.D47, I.T05\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The zero-divisor product e_plus * e_minus = 0 cannot arise as T(t1) * T(t2) for any T in HolFun and compatible + omega-tails t1, t2. Sector-pure outputs cannot combine nontrivially. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001190.md b/_corpus_v3_items/cid001190.md new file mode 100644 index 000000000..786bbabf6 --- /dev/null +++ b/_corpus_v3_items/cid001190.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001190/ +title: Composition Closure +cid: cid001190 +primary_alias: THM0024 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'HolFun is closed under composition: if f1 and f2 are tower-coherent reduce-form functions, then f1 o f2 is tower-coherent. + Proven via reduction_compat on composed reduce-form functions.' +aliases: +- THM0024 +- I.T20 +- composition-closure +- thm:composition-closure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 51 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Composition Closure**\n\nHolFun is closed under composition: if f1 and f2 are tower-coherent reduce-form functions,\ + \ then f1 o f2 is tower-coherent. Proven via reduction_compat on composed reduce-form functions.\n\n# Composition Closure\n\ + \n## Summary\n\nHolFun is closed under composition: if f1 and f2 are tower-coherent reduce-form functions, then f1 o f2\ + \ is tower-coherent. Proven via reduction_compat on composed reduce-form functions.\n\n## Statement\n\n```latex\n%\n\\\ + label{thm:composition-closure}\n$\\mathrm{HolFun}$ is closed under composition.\nIf $S, T \\in \\mathrm{HolFun}$,\nthen\ + \ $S \\circ T \\in \\mathrm{HolFun}$.\n```\n\n## Proof / Justification\n\n```latex\nWe must verify both conditions\nof\ + \ the HolFun definition (I.D47).\n\n\\emph{Condition (1): D-holomorphy.}\n$S \\circ T$ is D-holomorphic\nbecause the composition\ + \ of D-holomorphic functions\nis D-holomorphic.\nExplicitly:\nif $S(u, v) = (s_1(u),\\; s_2(v))$\nand $T(u, v) = (t_1(u),\\\ + ; t_2(v))$\nin sector coordinates (by I.P22), then\n\\[\n (S \\circ T)(u, v)\n = S(t_1(u),\\; t_2(v))\n = (s_1(t_1(u)),\\\ + ; s_2(t_2(v))),\n\\]\nwhich is again sector-independent:\nthe $B$-sector output $s_1(t_1(u))$\ndepends only on the $B$-sector\ + \ input $u$,\nand the $C$-sector output $s_2(t_2(v))$\ndepends only on the $C$-sector input $v$.\nThe split-CR equations\ + \ are preserved under composition\nbecause the chain rule holds in sector coordinates.\n\n\\emph{Condition (2): Tower\ + \ coherence.}\nFor all $k \\leq \\ell$,\nwe must show that\n$\\pi_{\\ell \\to k}((S \\circ T)(x))\n= (S \\circ T)(\\pi_{\\\ + ell \\to k}(x))$.\nWe compute:\n\\begin{align*}\n \\pi_{\\ell \\to k}((S \\circ T)(x))\n &= \\pi_{\\ell \\to k}(S(T(x)))\ + \ \\\\\n &= S(\\pi_{\\ell \\to k}(T(x)))\n && \\text{(by $S$'s tower coherence)} \\\\\n &= S(T(\\pi_{\\ell \\\ + to k}(x)))\n && \\text{(by $T$'s tower coherence)} \\\\\n &= (S \\circ T)(\\pi_{\\ell \\to k}(x)).\n\\end{align*}\n\ + Both conditions are satisfied,\nso $S \\circ T \\in \\mathrm{HolFun}$.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-01.jsonl` line 117\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch51-diagonal-free-protection.tex`\ + \ lines 394-400\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.DiagonalProtection`\n\ + - Name: `Tau.Holomorphy.comp_reduce_coherent`\n\n## Dependencies\n\n- Canonical: I.D47, I.D46\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'HolFun is closed under composition: if f1 and f2 are tower-coherent reduce-form functions, then f1 o f2 is tower-coherent. + Proven via reduction_compat on composed reduce-form functions.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001191.md b/_corpus_v3_items/cid001191.md new file mode 100644 index 000000000..43f5b1097 --- /dev/null +++ b/_corpus_v3_items/cid001191.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001191/ +title: Tau-Identity Theorem +cid: cid001191 +primary_alias: THM0025 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'THE CROWN JEWEL: if two tau-holomorphic functions agree at stage d0 for all inputs, they agree at all stages <= + d0. Hallmark of holomorphic rigidity. Proof uses tower coherence for vertical propagation (not classical sideways analytic + continuation).' +aliases: +- THM0025 +- I.T21 +- tau-identity-theorem +- thm:tau-identity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005304 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 13, Chapter 52 (Part XIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Identity Theorem**\n\nTHE CROWN JEWEL: if two tau-holomorphic functions agree at stage d0 for all inputs,\ + \ they agree at all stages <= d0. Hallmark of holomorphic rigidity. Proof uses tower coherence for vertical propagation\ + \ (not classical sideways analytic continuation).\n\n# Tau-Identity Theorem\n\n## Summary\n\nTHE CROWN JEWEL: if two tau-holomorphic\ + \ functions agree at stage d0 for all inputs, they agree at all stages <= d0. Hallmark of holomorphic rigidity. Proof\ + \ uses tower coherence for vertical propagation (not classical sideways analytic continuation).\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:tau-identity}\nLet $T, S \\in \\mathrm{HolFun}$\n(Definition~\\ref{def:holfun}, I.D47).\nIf there exists\ + \ $d_0 \\geq 1$ such that\n$T$ and $S$ agree at depth $d_0$\n(Definition~\\ref{def:tail-agreement}),\nthen $T = S$ as\ + \ functions on $\\mathrm{OmegaTail}$\n(Definition~\\ref{def:omega-tail}, I.D25).\n\\[\n \\boxed{%\n \\exists\\,\ + \ d_0 \\geq 1:\\;\n T \\sim_{d_0} S\n \\;\\;\\Longrightarrow\\;\\;\n T = S.}\n\\]\n```\n\n## Proof / Justification\n\ + \n```latex\nBy Lemma~\\ref{lem:tail-agreement-propagation},\n$T$ and $S$ agree at every depth $d \\geq d_0$.\nBut agreement\ + \ at all depths means:\nfor every omega-tail $t$\nand every primorial stage $k$,\nthe $k$-th components\nof $T(t)$ and\ + \ $S(t)$ coincide.\n\nAn omega-tail \\emph{is} its sequence of components ---\nit is a compatible tower\n$(x_1, x_2, x_3,\ + \ \\ldots)$\nwith $x_k \\in \\mathbb{Z}/M_k\\mathbb{Z}$\n(Definition~\\ref{def:omega-tail}).\nIf $\\bigl(T(t)\\bigr)_k\ + \ = \\bigl(S(t)\\bigr)_k$\nfor all $k \\geq 1$,\nthen $T(t)$ and $S(t)$ are the same omega-tail.\nTherefore $T(t) = S(t)$\ + \ for all $t$,\ni.e., $T = S$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 120\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part13/ch52-identity-theorem.tex` lines 325-342\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.IdentityTheorem`\n\ + - Name: `Tau.Holomorphy.tau_identity_nat`\n\n## Dependencies\n\n- Canonical: I.D46, I.D47, I.T18, I.L07\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'THE CROWN JEWEL: if two tau-holomorphic functions agree at stage d0 for all inputs, they agree at all stages + <= d0. Hallmark of holomorphic rigidity. Proof uses tower coherence for vertical propagation (not classical sideways analytic + conti' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001192.md b/_corpus_v3_items/cid001192.md new file mode 100644 index 000000000..6f810c8d6 --- /dev/null +++ b/_corpus_v3_items/cid001192.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001192/ +title: Category Axioms +cid: cid001192 +primary_alias: THM0026 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cat_tau satisfies the category axioms: left/right identity for stagewise composition (id_stage), and associativity + of composition (stagefun_comp_assoc). All proven from HolFun monoid properties.' +aliases: +- THM0026 +- I.T22 +- category-axioms +- thm:category-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 53 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Category Axioms**\n\nCat_tau satisfies the category axioms: left/right identity for stagewise composition (id_stage),\ + \ and associativity of composition (stagefun_comp_assoc). All proven from HolFun monoid properties.\n\n# Category Axioms\n\ + \n## Summary\n\nCat_tau satisfies the category axioms: left/right identity for stagewise composition (id_stage), and associativity\ + \ of composition (stagefun_comp_assoc). All proven from HolFun monoid properties.\n\n## Statement\n\n```latex\n%\n\\label{thm:category-axioms}\n\ + $\\mathrm{Cat}_\\tau$ (Definition~\\ref{def:cat-tau}, I.D51)\nsatisfies the axioms of a category:\n\\begin{enumerate}\n\ + \ \\item \\textbf{Left identity}: for all $\\alpha \\in \\mathrm{Hom}(A, B)$,\n $\\mathrm{id}_B \\circ \\\ + alpha = \\alpha$.\n \\item \\textbf{Right identity}: for all $\\alpha \\in \\mathrm{Hom}(A, B)$,\n $\\alpha\ + \ \\circ \\mathrm{id}_A = \\alpha$.\n \\item \\textbf{Composition closure}: for all\n $\\alpha \\in \\mathrm{Hom}(A,\ + \ B)$\n and $\\beta \\in \\mathrm{Hom}(B, C)$,\n the composite $\\beta \\circ \\alpha$\n exists\ + \ and lies in $\\mathrm{Hom}(A, C)$.\n \\item \\textbf{Associativity}: for all\n $\\alpha \\in \\mathrm{Hom}(A,\ + \ B)$,\n $\\beta \\in \\mathrm{Hom}(B, C)$,\n $\\gamma \\in \\mathrm{Hom}(C, D)$,\n \\[\n \ + \ (\\gamma \\circ \\beta) \\circ \\alpha\n = \\gamma \\circ (\\beta \\circ \\alpha).\n \ + \ \\]\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nEach axiom follows directly\nfrom the established\ + \ properties of $\\mathrm{HolFun}$.\n\n\\textbf{(1) Left identity.}\nLet $\\alpha = [\\pi]_{\\mathrm{NF}} \\in \\mathrm{Hom}(A,\ + \ B)$.\nThen\n$\\mathrm{id}_B \\circ \\alpha\n= [\\mathrm{id}_\\tau \\circ \\pi]_{\\mathrm{NF}}\n= [\\pi]_{\\mathrm{NF}}\n\ + = \\alpha$,\nbecause $\\mathrm{id}_\\tau \\circ \\pi = \\pi$\nas functions on omega-tails.\n\n\\textbf{(2) Right identity.}\n\ + Similarly,\n$\\alpha \\circ \\mathrm{id}_A\n= [\\pi \\circ \\mathrm{id}_\\tau]_{\\mathrm{NF}}\n= [\\pi]_{\\mathrm{NF}}\n\ + = \\alpha$.\n\n\\textbf{(3) Composition closure.}\nLet $\\alpha = [\\pi_\\alpha]_{\\mathrm{NF}} \\in \\mathrm{Hom}(A,\ + \ B)$\nand $\\beta = [\\pi_\\beta]_{\\mathrm{NF}} \\in \\mathrm{Hom}(B, C)$.\nBy Theorem~\\ref{thm:composition-closure}\ + \ (I.T20),\n$T_\\beta \\circ T_\\alpha \\in \\mathrm{HolFun}$.\nTherefore $\\beta \\circ \\alpha\n= [\\pi_\\beta \\circ\ + \ \\pi_\\alpha]_{\\mathrm{NF}}\n\\in \\mathrm{Hom}(A, C)$.\n\n\\textbf{(4) Associativity.}\nBy Proposition~\\ref{prop:holfun-associativity}\ + \ (I.P24),\n$(T_\\gamma \\circ T_\\beta) \\circ T_\\alpha\n= T_\\gamma \\circ (T_\\beta \\circ T_\\alpha)$\nas functions\ + \ on omega-tails.\nPassing to NF-equivalence classes:\n$(\\gamma \\circ \\beta) \\circ \\alpha\n= \\gamma \\circ (\\beta\ + \ \\circ \\alpha)$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 124\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch53-earned-arrows.tex`\ + \ lines 336-360\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedArrows`\n\ + - Name: `Tau.Topos.cat_tau_assoc`\n\n## Dependencies\n\n- Canonical: I.D51, I.T20, I.P24\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cat_tau satisfies the category axioms: left/right identity for stagewise composition (id_stage), and associativity + of composition (stagefun_comp_assoc). All proven from HolFun monoid properties.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001193.md b/_corpus_v3_items/cid001193.md new file mode 100644 index 000000000..729fcc67f --- /dev/null +++ b/_corpus_v3_items/cid001193.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001193/ +title: Yoneda Lemma +cid: cid001193 +primary_alias: THM0027 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Yoneda Lemma for thin Cat_tau: natural transformations from y(X) to presheaf P are in bijection with P(X). In + the thin setting this simplifies to: evaluation at X determines the transformation.' +aliases: +- THM0027 +- I.T23 +- yoneda-lemma +- thm:yoneda-lemma +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005356 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 54 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Yoneda Lemma**\n\nThe Yoneda Lemma for thin Cat_tau: natural transformations from y(X) to presheaf P are in\ + \ bijection with P(X). In the thin setting this simplifies to: evaluation at X determines the transformation.\n\n# Yoneda\ + \ Lemma\n\n## Summary\n\nThe Yoneda Lemma for thin Cat_tau: natural transformations from y(X) to presheaf P are in bijection\ + \ with P(X). In the thin setting this simplifies to: evaluation at X determines the transformation.\n\n## Statement\n\n\ + ```latex\n%\n\\label{thm:yoneda-lemma}\nLet $F : \\mathrm{Cat}_\\tau^{\\mathrm{op}} \\to \\mathrm{Set}$\nbe a presheaf\ + \ and $X \\in \\mathrm{Cat}_\\tau$ an object.\nThere is a natural bijection:\n\\[\n \\boxed{%\n \\mathrm{Nat}\\\ + bigl(y(X),\\, F\\bigr)\n \\;\\cong\\;\n F(X).}\n\\]\nEvery natural transformation\n$\\eta : y(X) \\Rightarrow F$\n\ + is uniquely determined\nby the value $\\eta_X(\\mathrm{id}_X) \\in F(X)$.\n```\n\n## Proof / Justification\n\n```latex\n\ + \\textbf{The Yoneda map.}\nDefine $\\Phi : \\mathrm{Nat}(y(X), F) \\to F(X)$\nby $\\Phi(\\eta) := \\eta_X(\\mathrm{id}_X)$.\n\ + Since $y(X)(X) = \\{\\mathrm{id}_X\\}$ by thinness,\nthis is well-defined.\n\n\\textbf{The inverse.}\nGiven $a \\in F(X)$,\n\ + define $\\eta^a : y(X) \\Rightarrow F$ by:\nfor each $Y$,\nif $Y \\not\\leq X$ then $\\eta^a_Y$ is the empty function;\n\ + if $Y \\leq X$ with unique arrow $f : Y \\to X$,\nset $\\eta^a_Y(*) := F(f)(a)$.\n\n\\textbf{Naturality.}\nFor $g : Y_1\ + \ \\to Y_2$ with $Y_2 \\leq X$,\nlet $f_1 : Y_1 \\to X$ and $f_2 : Y_2 \\to X$\nbe the unique arrows.\nThen\n$F(g)(\\\ + eta^a_{Y_2}(*))\n= F(g)(F(f_2)(a))\n= F(f_2 \\circ g)(a)\n= F(f_1)(a)\n= \\eta^a_{Y_1}(*)$,\nusing $f_2 \\circ g = f_1$\ + \ by thinness.\nThe case $Y_2 \\not\\leq X$ is vacuous.\n\n\\textbf{Bijectivity.}\n$\\Phi(\\eta^a) = F(\\mathrm{id}_X)(a)\ + \ = a$,\nso $\\Phi \\circ \\Psi = \\mathrm{id}$.\nConversely, given $\\eta$ and $Y \\leq X$\nwith unique $f : Y \\to X$,\n\ + naturality forces\n$\\eta_Y(*) = F(f)(\\eta_X(\\mathrm{id}_X))\n= F(f)(\\Phi(\\eta))$,\nso $\\eta = \\eta^{\\Phi(\\eta)}$\n\ + and $\\Psi \\circ \\Phi = \\mathrm{id}$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 129\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch54-functors.tex` lines 397-413\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.Functors`\n- Name: `Tau.Topos.yoneda_thin`\n\ + \n## Dependencies\n\n- Canonical: I.D54, I.P25\n\n## Related Results\n\n_Generated by later projection phases._\n\n##\ + \ Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Yoneda Lemma for thin Cat_tau: natural transformations from y(X) to presheaf P are in bijection with P(X). + In the thin setting this simplifies to: evaluation at X determines the transformation.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001194.md b/_corpus_v3_items/cid001194.md new file mode 100644 index 000000000..da5d65559 --- /dev/null +++ b/_corpus_v3_items/cid001194.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001194/ +title: Grothendieck Topos +cid: cid001194 +primary_alias: THM0028 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'PSh(Cat_tau) is a Grothendieck topos. Standard result: for any small category C, PSh(C) is a Grothendieck topos. + Cat_tau is small (countable objects, thin morphisms). Verified via terminal object, products, and identity laws.' +aliases: +- THM0028 +- I.T24 +- grothendieck-topos +- thm:grothendieck-topos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005359 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 55 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Grothendieck Topos**\n\nPSh(Cat_tau) is a Grothendieck topos. Standard result: for any small category C, PSh(C)\ + \ is a Grothendieck topos. Cat_tau is small (countable objects, thin morphisms). Verified via terminal object, products,\ + \ and identity laws.\n\n# Grothendieck Topos\n\n## Summary\n\nPSh(Cat_tau) is a Grothendieck topos. Standard result: for\ + \ any small category C, PSh(C) is a Grothendieck topos. Cat_tau is small (countable objects, thin morphisms). Verified\ + \ via terminal object, products, and identity laws.\n\n## Statement\n\n```latex\n%\n\\label{thm:grothendieck-topos}\n\ + $\\mathrm{PSh}(\\mathrm{Cat}_\\tau)$\nis a \\textbf{Grothendieck topos}.\n```\n\n## Proof / Justification\n\n```latex\n\ + By Giraud's theorem \\cite{MacLaneMoerdijk1992},\nthe presheaf category\n$[\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$\n\ + on any small category $\\mathcal{C}$\nis a Grothendieck topos.\n$\\mathrm{Cat}_\\tau$ is small:\n\\begin{itemize}\n \ + \ \\item \\textbf{Objects:}\n $\\Obj(\\mathrm{Cat}_\\tau) = \\tau\\text{-Idx}\n \\cong \\mathbb{N}$\ + \ (a set).\n \\item \\textbf{Morphisms:}\n $\\mathrm{Mor}(\\mathrm{Cat}_\\tau)\n = \\{(X, Y) : X\ + \ \\mid Y\\}\n \\subseteq \\tau\\text{-Idx} \\times \\tau\\text{-Idx}$\n (a set).\n\\end{itemize}\n\ + Explicitly, the four Giraud axioms hold:\n\\begin{enumerate}\n \\item All small colimits exist\n (computed\ + \ pointwise in $\\mathbf{Set}$).\n \\item The representables $\\{y(X)\\}_{X \\in \\mathrm{Cat}_\\tau}$\n form\ + \ a small generating set.\n \\item Coproducts are disjoint\n (inherited from $\\mathbf{Set}$).\n \\item\ + \ Equivalence relations are effective\n (inherited from $\\mathbf{Set}$).\n\\end{enumerate}\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 133\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch55-limits-sites.tex`\ + \ lines 435-440\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.LimitsSites`\n\ + - Name: `Tau.Topos.psh_has_terminal`\n\n## Dependencies\n\n- Canonical: I.D57, I.T22, I.T23\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'PSh(Cat_tau) is a Grothendieck topos. Standard result: for any small category C, PSh(C) is a Grothendieck topos. + Cat_tau is small (countable objects, thin morphisms). Verified via terminal object, products, and identity laws.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001195.md b/_corpus_v3_items/cid001195.md new file mode 100644 index 000000000..fcb82032e --- /dev/null +++ b/_corpus_v3_items/cid001195.md @@ -0,0 +1,102 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001195/ +title: Omega_tau Subobject Classifier +cid: cid001195 +primary_alias: THM0029 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Omega_tau = Truth4 is the subobject classifier for PSh(Cat_tau). Every truth value is one of T, F, B, N. The four + elements are pairwise distinct (omega_tau_card_four). Previewed in Part XI, now EARNED. +aliases: +- THM0029 +- I.T25 +- omega-tau-subobject-classifier +- thm:omega-tau-classifier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005354 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 14, Chapter 56 (Part XIV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega_tau Subobject Classifier**\n\nOmega_tau = Truth4 is the subobject classifier for PSh(Cat_tau). Every\ + \ truth value is one of T, F, B, N. The four elements are pairwise distinct (omega_tau_card_four). Previewed in Part XI,\ + \ now EARNED.\n\n# Omega_tau Subobject Classifier\n\n## Summary\n\nOmega_tau = Truth4 is the subobject classifier for\ + \ PSh(Cat_tau). Every truth value is one of T, F, B, N. The four elements are pairwise distinct (omega_tau_card_four).\ + \ Previewed in Part XI, now EARNED.\n\n## Statement\n\n```latex\n%\n\\label{thm:omega-tau-classifier}\nThe object\n\\\ + [\n \\boxed{%\n \\Omega_\\tau = \\mathrm{Truth4}\n = \\{\\mathsf{T}, \\mathsf{F}, \\mathsf{B}, \\mathsf{N}\\\ + }}\n\\]\nis the subobject classifier\nof $\\mathrm{PSh}(\\mathrm{Cat}_\\tau)$.\nThat is, $\\Omega_\\tau$ with\nthe truth\ + \ morphism\n$\\mathrm{true} : 1 \\to \\Omega_\\tau$,\n$* \\mapsto \\mathsf{T}$\n(Definition~\\ref{def:omega-tau}, I.D41),\n\ + satisfies:\nfor every monomorphism\n$m : S \\hookrightarrow X$\nin $\\mathrm{PSh}(\\mathrm{Cat}_\\tau)$,\nthere exists\ + \ a \\emph{unique} morphism\n$\\chi_S : X \\to \\Omega_\\tau$\nsuch that the following square is a pullback:\n\\[\n \ + \ \\begin{tikzcd}\n S \\arrow[r] \\arrow[d, hook, \"m\"']\n & 1 \\arrow[d, \"\\mathrm{true}\"] \\\\\n \ + \ X \\arrow[r, \"\\chi_S\"']\n & \\Omega_\\tau\n \\end{tikzcd}\n\\]\n```\n\n## Proof / Justification\n\ + \n```latex\n\\textbf{Step 1: Construction of $\\chi_S$.}\nFor each object $c$ in $\\mathrm{Cat}_\\tau$\nand $x \\in X(c)$,\ + \ define\n$(\\chi_S)_c(x) := \\mathrm{membership}_S(x, c)$\nvia the spectral sector analysis.\nThe spectral decomposition\ + \ partitions\nthe morphisms arriving at $c$\ninto $B$-sector and $C$-sector contributions.\nSet $s_+(x) = 1$ if the $B$-sector\ + \ restriction\nof $x$ lies in $S$, and $0$ otherwise;\nsimilarly $s_-(x)$ for the $C$-sector.\nThen:\n\\[\n \\mathrm{membership}_S(x,\ + \ c) :=\n \\begin{cases}\n \\mathsf{T} & \\text{if } s_+(x) = 1,\\; s_-(x) = 1\n \\text{\ + \ (clean),} \\\\\n \\mathsf{F} & \\text{if } s_+(x) = 0,\\; s_-(x) = 0\n \\text{ (clean),}\ + \ \\\\\n \\mathsf{B} & \\text{if degenerate boundary\n (both confirm and deny),} \\\\\n \ + \ \\mathsf{N} & \\text{if neither sector is decisive.}\n \\end{cases}\n\\]\nThe distinction between $\\mathsf{T}$\ + \ and $\\mathsf{B}$\nuses the spectral decomposition:\n$\\mathsf{B}$ arises when $x$ lies in the image\nof both idempotent\ + \ projections $e_+$ and $e_-$\nof $S$ at $c$ (degeneracy),\nwhile $\\mathsf{N}$ arises when $x$ lies in neither.\n\n\\\ + textbf{Step 2: Naturality.}\nSince $\\mathrm{Cat}_\\tau$ is thin\n(Proposition~\\ref{prop:thin-category},\nChapter~\\\ + ref{ch:earned-arrows}),\nrestriction maps are canonical.\nThe morphisms of $\\mathrm{Cat}_\\tau$\npreserve the bipolar\ + \ structure\n(they are equivalence classes\nof $\\tau$-holomorphic programs\nrespecting tower coherence),\nso membership\ + \ status is preserved under restriction:\n$\\mathrm{membership}_S(X(f)(x), c)\n= \\mathrm{membership}_S(x, d)$.\nNaturality\ + \ follows.\n\n\\textbf{Step 3: Pullback property.}\nThe pullback of $\\mathrm{true}$ along $\\chi_S$ is\n$\\{x \\in X(c)\ + \ : (\\chi_S)_c(x) = \\mathsf{T}\\}$.\nBy construction this equals $S(c)$:\nelements of $S$ have clean membership\n(the\ + \ monomorphism $m$ is an injection,\nso $S$-elements are witnessed by both sectors\nwithout degeneracy),\nand $\\mathsf{B}$-elements\ + \ are excluded\nbecause $\\mathsf{B} \\neq \\mathsf{T}$.\n\n\\textbf{Step 4: Uniqueness.}\nIf $\\chi' : X \\to \\Omega_\\\ + tau$ also\npulls $\\mathrm{true}$ back to $S$,\nthen $\\chi'_c(x) = \\mathsf{T}$ for $x \\in S(c)$,\nand $\\chi'_c(x)\ + \ \\neq \\mathsf{T}$ for $x \\notin S(c)$.\nThe specific value among\n$\\{\\mathsf{F}, \\mathsf{B}, \\mathsf{N}\\}$\n\ + is determined by naturality:\nthe thinness of $\\mathrm{Cat}_\\tau$ forces\n$\\chi'_c(x)$ to match the spectral sector\ + \ analysis,\nleaving no freedom.\nTherefore $\\chi' = \\chi_S$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 135\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part14/ch56-earned-topos.tex`\ + \ lines 146-177\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.EarnedTopos`\n\ + - Name: `Tau.Topos.omega_tau_classifier`\n\n## Dependencies\n\n- Canonical: I.D41, I.D57, I.T24\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Omega_tau = Truth4 is the subobject classifier for PSh(Cat_tau). Every truth value is one of T, F, B, N. The + four elements are pairwise distinct (omega_tau_card_four). Previewed in Part XI, now EARNED. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001196.md b/_corpus_v3_items/cid001196.md new file mode 100644 index 000000000..eeff50e03 --- /dev/null +++ b/_corpus_v3_items/cid001196.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001196/ +title: Product Universal Property +cid: cid001196 +primary_alias: THM0030 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Product universal property: if R maps to both P and Q pointwise, then R maps to P x Q. Also: product is commutative, + associative, and unital (with terminal presheaf).' +aliases: +- THM0030 +- I.T26 +- product-universal-property +- thm:product-universal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005352 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 57 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Product Universal Property**\n\nProduct universal property: if R maps to both P and Q pointwise, then R maps\ + \ to P x Q. Also: product is commutative, associative, and unital (with terminal presheaf).\n\n# Product Universal Property\n\ + \n## Summary\n\nProduct universal property: if R maps to both P and Q pointwise, then R maps to P x Q. Also: product is\ + \ commutative, associative, and unital (with terminal presheaf).\n\n## Statement\n\n```latex\n%\n\\label{thm:product-universal}\n\ + For presheaves $P, Q \\in \\mathcal{E}_\\tau$,\nthe product $(P \\times Q, \\pi_1, \\pi_2)$\nsatisfies the universal property:\n\ + for any presheaf $R$ and natural transformations\n$f : R \\to P$ and $g : R \\to Q$,\nthere exists a \\emph{unique} natural\ + \ transformation\n\\[\n \\boxed{%\n \\langle f, g \\rangle : R \\to P \\times Q}\n\\]\nsuch that $\\pi_1 \\circ\ + \ \\langle f, g \\rangle = f$\nand $\\pi_2 \\circ \\langle f, g \\rangle = g$.\n```\n\n## Proof / Justification\n\n```latex\n\ + \\textbf{Existence.}\nDefine $\\langle f, g \\rangle_X(r)\n:= \\bigl(f_X(r),\\; g_X(r)\\bigr)$\nfor each $X$ and $r \\\ + in R(X)$.\nNaturality follows from the naturality of $f$ and $g$:\nfor $\\phi : X \\to Y$,\n$(P \\times Q)(\\phi) \\circ\ + \ \\langle f, g \\rangle_Y\n= \\langle f, g \\rangle_X \\circ R(\\phi)$.\nThe projection laws hold by construction.\n\n\ + \\textbf{Uniqueness.}\nIf $h : R \\to P \\times Q$\nsatisfies $\\pi_1 \\circ h = f$\nand $\\pi_2 \\circ h = g$,\nthen\ + \ $h_X(r) = (f_X(r),\\; g_X(r))\n= \\langle f, g \\rangle_X(r)$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 140\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch57-cartesian-product.tex`\ + \ lines 116-131\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.CartesianProduct`\n\ + - Name: `Tau.Topos.product_universal`\n\n## Dependencies\n\n- Canonical: I.D60\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Product universal property: if R maps to both P and Q pointwise, then R maps to P x Q. Also: product is commutative, + associative, and unital (with terminal presheaf).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001197.md b/_corpus_v3_items/cid001197.md new file mode 100644 index 000000000..563a7fd49 --- /dev/null +++ b/_corpus_v3_items/cid001197.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001197/ +title: Distributivity +cid: cid001197 +primary_alias: THM0032 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Product distributes over coproduct: P x (Q v R) = (P x Q) v (P x R). Both left and right distributivity hold. This + makes (E_tau, x, v) a distributive category.' +aliases: +- THM0032 +- I.T27 +- distributivity +- thm:distributivity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005361 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 58 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Distributivity**\n\nProduct distributes over coproduct: P x (Q v R) = (P x Q) v (P x R). Both left and right\ + \ distributivity hold. This makes (E_tau, x, v) a distributive category.\n\n# Distributivity\n\n## Summary\n\nProduct\ + \ distributes over coproduct: P x (Q v R) = (P x Q) v (P x R). Both left and right distributivity hold. This makes (E_tau,\ + \ x, v) a distributive category.\n\n## Statement\n\n```latex\n%\n\\label{thm:distributivity}\nFor all presheaves $P, Q,\ + \ R$ in $\\mathcal{E}_\\tau$:\n\\[\n \\boxed{%\n P \\times (Q \\wedge R)\n \\;\\cong\\;\n (P \\times Q) \\\ + wedge (P \\times R).}\n\\]\nThe categorical product distributes\nover the categorical coproduct.\n```\n\n## Proof / Justification\n\ + \n```latex\nWe verify the isomorphism pointwise.\nAt each object $X$ in $\\mathrm{Cat}_\\tau$:\n\\begin{align*}\n \\\ + bigl(P \\times (Q \\wedge R)\\bigr)(X)\n &= P(X) \\times \\bigl(Q(X) \\vee R(X)\\bigr) \\\\\n &= \\bigl(P(X) \\\ + times Q(X)\\bigr)\n \\vee \\bigl(P(X) \\times R(X)\\bigr) \\\\\n &= \\bigl((P \\times Q) \\wedge (P \\times R)\\\ + bigr)(X).\n\\end{align*}\nThe second equality is the set-theoretic distributive law\n$A \\times (B \\cup C) = (A \\times\ + \ B) \\cup (A \\times C)$.\nNaturality holds because\nboth product and coproduct are defined pointwise,\nso restriction\ + \ maps commute with the isomorphism at each stage.\nBy commutativity of~$\\times$,\nright-distributivity\n$(Q \\wedge\ + \ R) \\times P\n\\cong (Q \\times P) \\wedge (R \\times P)$\nfollows immediately.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-01.jsonl` line 143\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch58-wedge-product.tex`\ + \ lines 151-163\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.WedgeProduct`\n\ + - Name: `Tau.Topos.product_distributes_over_coproduct`\n\n## Dependencies\n\n- Canonical: I.D60, I.D62\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Product distributes over coproduct: P x (Q v R) = (P x Q) v (P x R). Both left and right distributivity hold. + This makes (E_tau, x, v) a distributive category.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001198.md b/_corpus_v3_items/cid001198.md new file mode 100644 index 000000000..24fe712de --- /dev/null +++ b/_corpus_v3_items/cid001198.md @@ -0,0 +1,96 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001198/ +title: Cartesian Closed +cid: cid001198 +primary_alias: THM0033 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E_tau is cartesian closed: Hom(A x B, C) <-> Hom(A, C^B) pointwise. The evaluation morphism (Q^P x P) -> Q is verified. + Internal hom with terminal/initial satisfies expected identities.' +aliases: +- THM0033 +- I.T28 +- cartesian-closed +- thm:cartesian-closed +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005357 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 15, Chapter 59 (Part XV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cartesian Closed**\n\nE_tau is cartesian closed: Hom(A x B, C) <-> Hom(A, C^B) pointwise. The evaluation morphism\ + \ (Q^P x P) -> Q is verified. Internal hom with terminal/initial satisfies expected identities.\n\n# Cartesian Closed\n\ + \n## Summary\n\nE_tau is cartesian closed: Hom(A x B, C) <-> Hom(A, C^B) pointwise. The evaluation morphism (Q^P x P)\ + \ -> Q is verified. Internal hom with terminal/initial satisfies expected identities.\n\n## Statement\n\n```latex\n%\n\ + \\label{thm:cartesian-closed}\nThe earned topos\n$\\mathcal{E}_\\tau = \\mathrm{PSh}(\\mathrm{Cat}_\\tau)$\n(Definition~\\\ + ref{def:earned-topos}, I.D59)\nis a \\textbf{cartesian closed category}.\nThat is, for all presheaves $A$, $P$, $Q$\n\ + in $\\mathcal{E}_\\tau$:\n\\begin{enumerate}\n \\item The internal hom $Q^P$\n (Definition~\\ref{def:internal-hom},\ + \ I.D64)\n is an object of $\\mathcal{E}_\\tau$.\n \\item The exponential adjunction holds naturally:\n \ + \ \\[\n \\boxed{%\n \\Hom_{\\mathcal{E}_\\tau}(A \\times P,\\; Q)\n \\;\\\ + cong\\;\n \\Hom_{\\mathcal{E}_\\tau}(A,\\; Q^P).}\n \\]\n \\item The evaluation morphism\n \ + \ $\\mathrm{ev} : Q^P \\times P \\to Q$\n is universal:\n every $\\varphi : A \\times P \\to\ + \ Q$\n factors uniquely as\n $\\varphi = \\mathrm{ev} \\circ (\\tilde{\\varphi} \\times \\id_P)$\n \ + \ for a unique $\\tilde{\\varphi} : A \\to Q^P$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\textbf{Step 1: $Q^P$ is a presheaf.}\nDefinition~\\ref{def:internal-hom} assigns to each $X$\nthe set $(Q^P)(X) = \\\ + mathrm{Nat}(y(X) \\times P, Q)$.\nThe restriction maps are defined by precomposition,\nand functoriality follows from\n\ + $y(g \\circ f) = y(g) \\circ y(f)$.\nThus $Q^P : \\mathrm{Cat}_\\tau^{\\mathrm{op}} \\to \\mathrm{Set}$\nis a well-defined\ + \ presheaf.\n\n\\textbf{Step 2: The currying bijection.}\nGiven a natural transformation\n$\\varphi : A \\times P \\Rightarrow\ + \ Q$,\ndefine for each $X$ the map\n$\\tilde{\\varphi}_X : A(X) \\to (Q^P)(X)$\nas follows.\nFor $a \\in A(X)$,\nconsider\ + \ the element $a$ as inducing\na morphism $y(X) \\to A$\nvia Yoneda\n(Theorem~\\ref{thm:yoneda-lemma}, I.T23).\nThen $\\\ + tilde{\\varphi}_X(a)$ is the composite:\n\\[\n y(X) \\times P\n \\xrightarrow{\\;a \\times \\id_P\\;}\n A \\\ + times P\n \\xrightarrow{\\;\\varphi\\;}\n Q.\n\\]\nThis defines a natural transformation\n$\\tilde{\\varphi} : A\ + \ \\Rightarrow Q^P$.\nConversely, given $\\psi : A \\Rightarrow Q^P$,\nset $\\varphi := \\mathrm{ev} \\circ (\\psi \\\ + times \\id_P)$.\nThese two maps are mutually inverse.\n\nIn the thin setting,\nthe bijection simplifies further.\nBoth\ + \ sides are subterminal:\n$\\Hom(A \\times P, Q) \\in \\{\\varnothing, \\{*\\}\\}$\nand $\\Hom(A, Q^P) \\in \\{\\varnothing,\ + \ \\{*\\}\\}$.\nThe condition $A \\times P \\leq Q$\n(i.e., $\\mathrm{supp}(A) \\cap \\mathrm{supp}(P) \\subseteq \\mathrm{supp}(Q)$)\n\ + is equivalent to\n$\\mathrm{supp}(A) \\subseteq \\mathrm{supp}(Q^P)$,\nwhich holds by the support formula\n(Remark~\\\ + ref{rem:internal-hom-supports}).\n\n\\textbf{Step 3: Naturality.}\nBoth in $A$ (by Yoneda)\nand in $P$, $Q$ (by the universal\ + \ property\nof the product), the bijection is natural.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 146\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part15/ch59-internal-hom.tex`\ + \ lines 261-289\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Topos.InternalHom`\n\ + - Name: `Tau.Topos.cartesian_closed_adj`\n\n## Dependencies\n\n- Canonical: I.D64, I.D60, I.T26\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E_tau is cartesian closed: Hom(A x B, C) <-> Hom(A, C^B) pointwise. The evaluation morphism (Q^P x P) -> Q is + verified. Internal hom with terminal/initial satisfies expected identities.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001199.md b/_corpus_v3_items/cid001199.md new file mode 100644 index 000000000..805183cc1 --- /dev/null +++ b/_corpus_v3_items/cid001199.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001199/ +title: Spectral Determination +cid: cid001199 +primary_alias: THM0034 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Spectral Determination: two StageFuns with identical spectral coefficients at all inputs and stages are equal. The + spectral decomposition is faithful.' +aliases: +- THM0034 +- I.T29 +- spectral-determination +- thm:spectral-determination +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005307 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 60 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Determination**\n\nSpectral Determination: two StageFuns with identical spectral coefficients at all\ + \ inputs and stages are equal. The spectral decomposition is faithful.\n\n# Spectral Determination\n\n## Summary\n\nSpectral\ + \ Determination: two StageFuns with identical spectral coefficients at all inputs and stages are equal. The spectral decomposition\ + \ is faithful.\n\n## Statement\n\n```latex\n%\n\\label{thm:spectral-determination}\nLet $f, g \\in \\mathrm{Hol}(\\mathbb{L})$.\n\ + If $f$ and $g$ have the same spectral coefficients\nat every primorial stage ---\n\\[\n \\forall\\, k \\geq 1:\\;\n\ + \ (a_k(f), b_k(f)) = (a_k(g), b_k(g))\n\\]\n--- then $f = g$.\nEquivalently:\n\\[\n \\boxed{%\n \\hat{f} = \\\ + hat{g}\n \\;\\;\\Longrightarrow\\;\\;\n f = g.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{Step\ + \ 1: Character basis at each stage.}\nBy the spectral decomposition\n(Theorem~\\ref{thm:spectral-decomposition}, I.T12),\n\ + $\\chi_+$ and $\\chi_-$ provide\na unique decomposition at each primorial stage.\nIf $(a_k(f), b_k(f)) = (a_k(g), b_k(g))$,\n\ + then:\n\\[\n f_k = a_k(f) \\cdot e_+ + b_k(f) \\cdot e_-\n = a_k(g) \\cdot e_+ + b_k(g) \\cdot e_-\n = g_k\n\\\ + ]\nin $(\\mathbb{Z}/M_k\\mathbb{Z})[j]$.\nSo $f$ and $g$ agree at primorial depth $k$.\n\n\\textbf{Step 2: Tower coherence\ + \ forces global agreement.}\nSince spectral coefficients agree at \\emph{all} stages,\n$f$ and $g$ agree at every depth\ + \ $k \\geq 1$.\nBy the $\\tau$-Identity Theorem\n(Theorem~\\ref{thm:tau-identity}, I.T21),\nagreement at even a single\ + \ depth suffices.\nTherefore $f = g$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 150\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch60-spectral-coefficients.tex` lines 182-200\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.SpectralCoefficients`\n\ + - Name: `Tau.Holomorphy.spectral_determines`\n\n## Dependencies\n\n- Canonical: I.D65, I.D66\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Spectral Determination: two StageFuns with identical spectral coefficients at all inputs and stages are equal. + The spectral decomposition is faithful.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001200.md b/_corpus_v3_items/cid001200.md new file mode 100644 index 000000000..055a9659d --- /dev/null +++ b/_corpus_v3_items/cid001200.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001200/ +title: Removable Singularity +cid: cid001200 +primary_alias: THM0035 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Removable Singularity: two tower-coherent functions agreeing at depth d0 for all inputs agree at all depths <= d0. + Repackaging of the Identity Theorem in extension language. The thin set is removable.' +aliases: +- THM0035 +- I.T30 +- removable-singularity +- thm:removable-singularity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005309 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 61 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Removable Singularity**\n\nRemovable Singularity: two tower-coherent functions agreeing at depth d0 for all\ + \ inputs agree at all depths <= d0. Repackaging of the Identity Theorem in extension language. The thin set is removable.\n\ + \n# Removable Singularity\n\n## Summary\n\nRemovable Singularity: two tower-coherent functions agreeing at depth d0 for\ + \ all inputs agree at all depths <= d0. Repackaging of the Identity Theorem in extension language. The thin set is removable.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:removable-singularity}\nLet $K \\subseteq \\mathbb{L}$ be primordially thin\n\ + (Definition~\\ref{def:primorial-thinness}, I.D67),\n$f \\in \\mathrm{HolFun}$ bounded on $\\mathbb{L} \\setminus K$.\n\ + Then $f$ extends uniquely to\n$\\tilde{f} \\in \\mathrm{Hol}(\\mathbb{L})$\n(Definition~\\ref{def:hol-L}, I.D49):\n\\\ + [\n \\boxed{%\n K \\text{ thin},\\;\n f \\in \\mathrm{HolFun}(\\mathbb{L} \\setminus K)\n \\text{ bounded}\n\ + \ \\;\\;\\Longrightarrow\\;\\;\n \\exists!\\; \\tilde{f} \\in \\mathrm{Hol}(\\mathbb{L}):\\;\n \\tilde{f}\\!\\\ + big|_{\\mathbb{L} \\setminus K} = f.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{Existence.}\nFor $t\ + \ \\in K$, define $\\tilde{f}(t) \\bmod M_d$\nby CRT extension (Lemma~\\ref{lem:crt-extension}, I.L08)\nat each depth\ + \ $d$.\nFor $t \\notin K$, set $\\tilde{f}(t) := f(t)$.\nThe sequence $(\\tilde{f}(t) \\bmod M_d)_{d \\geq 1}$\nis compatible:\ + \ the CRT reconstruction at $M_{d+1}$\nreduces to the one at $M_d$ by transitivity of CRT.\nBoundedness of $f$ ensures\ + \ the reconstructed values\nremain in $\\hat{\\mathbb{Z}}_\\tau$.\nThe extension satisfies tower coherence\n(Definition~\\\ + ref{def:tower-coherence}, I.D46)\nand CRT coherence\n(Theorem~\\ref{thm:crt-coherence}, I.T18)\nby construction, so $\\\ + tilde{f}$ is $\\tau$-holomorphic.\n\n\\textbf{Uniqueness.}\nTwo extensions $\\tilde{f}_1, \\tilde{f}_2$\nagree on $\\\ + mathbb{L} \\setminus K$.\nSince $K$ is thin,\n$\\mathbb{L} \\setminus K$ is primordially dense,\nso $\\tilde{f}_1$ and\ + \ $\\tilde{f}_2$ agree at some depth $d_0$.\nBy the $\\tau$-Identity Theorem\n(Theorem~\\ref{thm:tau-identity}, I.T21),\n\ + $\\tilde{f}_1 = \\tilde{f}_2$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 153\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch61-thinness.tex` lines 192-210\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.Thinness`\n- Name: `Tau.Holomorphy.removable_singularity`\n\ + \n## Dependencies\n\n- Canonical: I.D67, I.L08, I.T21\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Removable Singularity: two tower-coherent functions agreeing at depth d0 for all inputs agree at all depths + <= d0. Repackaging of the Identity Theorem in extension language. The thin set is removable.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001201.md b/_corpus_v3_items/cid001201.md new file mode 100644 index 000000000..eec0ed569 --- /dev/null +++ b/_corpus_v3_items/cid001201.md @@ -0,0 +1,102 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001201/ +title: Global Hartogs Extension +cid: cid001201 +primary_alias: THM0036 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'THE BOOK I CLIMAX. Global Hartogs Extension: any two tower-coherent functions agreeing at a reference depth agree + at all depths below. Uses all four ingredients: spectral determination, CRT extension, tower coherence, and the Identity + Theorem.' +aliases: +- THM0036 +- I.T31 +- global-hartogs-extension +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005302 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 16, Chapter 62 (Part XVI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Global Hartogs Extension**\n\nTHE BOOK I CLIMAX. Global Hartogs Extension: any two tower-coherent functions\ + \ agreeing at a reference depth agree at all depths below. Uses all four ingredients: spectral determination, CRT extension,\ + \ tower coherence, and the Identity Theorem.\n\n# Global Hartogs Extension\n\n## Summary\n\nTHE BOOK I CLIMAX. Global\ + \ Hartogs Extension: any two tower-coherent functions agreeing at a reference depth agree at all depths below. Uses all\ + \ four ingredients: spectral determination, CRT extension, tower coherence, and the Identity Theorem.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:global-hartogs}\n% I.T21, I.T18, I.T12, I.D37\nLet $K \\subseteq \\mathbb{L}$ be\ + \ primordially thin\n(Definition~\\ref{def:primorial-thinness}, I.D67),\nand let $f \\in \\mathrm{HolFun}$\nbe defined\ + \ on $\\mathbb{L} \\setminus K$.\nThen $f$ extends uniquely to\n$\\tilde{f} \\in \\mathrm{Hol}(\\mathbb{L})$\n(Definition~\\\ + ref{def:hol-L}, I.D49):\n\\[\n \\boxed{%\n K \\text{ thin},\\;\n f \\in \\mathrm{HolFun}(\\mathbb{L} \\setminus\ + \ K)\n \\;\\;\\Longrightarrow\\;\\;\n \\exists!\\; \\tilde{f} \\in \\mathrm{Hol}(\\mathbb{L}):\\;\n \\tilde{f}\\\ + !\\big|_{\\mathbb{L} \\setminus K} = f.}\n\\]\nNo boundedness hypothesis is required.\n```\n\n## Proof / Justification\n\ + \n```latex\nThe proof proceeds in four steps,\neach drawing on a different strand\nof the machinery developed in Parts~IX--XII.\n\ + \n\\medskip\n\\textbf{Step 1: Spectral reduction.}\nThe spectral decomposition\n(Theorem~\\ref{thm:spectral-decomposition},\ + \ I.T12)\nsplits $f$ into spectral components:\n\\[\n f(t) = f_+(t) \\cdot e_+ + f_-(t) \\cdot e_-,\n \\qquad\n\ + \ f_\\pm(t) := \\chi_\\pm(f(t)),\n\\]\nfor $t \\in \\mathbb{L} \\setminus K$.\nIt suffices to extend $f_+$ and $f_-$\ + \ separately,\nrecovering $\\tilde{f}$ by spectral reassembly.\nEach spectral component inherits tower coherence\n(the\ + \ characters are ring homomorphisms\ncommuting with primorial reduction).\nAt each stage $M_d$,\n$f_\\pm(t) \\bmod M_d\ + \ \\in \\mathbb{Z}/M_d\\mathbb{Z}$ ---\na finite set.\nEach spectral component is therefore \\emph{automatically bounded}.\n\ + \n\\medskip\n\\textbf{Step 2: CRT reconstruction at each stage.}\nFix $f_+$ and a depth $d$.\nFor each $t \\in K$,\nthe\ + \ CRT Extension Lemma\n(Lemma~\\ref{lem:crt-extension}, I.L08)\ndetermines $f_+(t) \\bmod M_d$ uniquely:\n$K$ is thin,\ + \ so $K_d$ misses two CRT directions,\nand the Chinese Remainder Theorem\nassembles the known values along those directions\n\ + into $f_+(t) \\bmod M_d$.\n\n\\medskip\n\\textbf{Step 3: Tower coherence ensures global consistency.}\nThe stage-by-stage\ + \ reconstructions must satisfy:\n\\[\n \\tilde{f}_+(t) \\bmod M_d\n \\;=\\;\n \\bigl(\\tilde{f}_+(t) \\bmod M_{d+1}\\\ + bigr) \\bmod M_d\n \\qquad \\text{for all } d \\geq 1.\n\\]\nThis holds because CRT reconstruction at $M_{d+1} = M_d\ + \ \\cdot p_{d+1}$\ndecomposes into the reconstruction at $M_d$\nplus a new factor.\nTower coherence\n(Definition~\\ref{def:tower-coherence},\ + \ I.D46)\nand CRT coherence\n(Theorem~\\ref{thm:crt-coherence}, I.T18)\nforce the reconstructed values on $K$\nto be compatible.\n\ + The sequence $(\\tilde{f}_+(t) \\bmod M_d)_{d \\geq 1}$\ndefines a valid omega-tail\n(Definition~\\ref{def:omega-tail},\ + \ I.D25).\nSimilarly for $\\tilde{f}_-$.\nSet $\\tilde{f}(t) := \\tilde{f}_+(t) \\cdot e_+ + \\tilde{f}_-(t) \\cdot e_-$.\n\ + \n\\medskip\n\\textbf{Step 4: Uniqueness by the Identity Theorem.}\nIf $\\tilde{f}_1, \\tilde{f}_2 \\in \\mathrm{Hol}(\\\ + mathbb{L})$\nboth extend $f$, they agree on $\\mathbb{L} \\setminus K$.\nSince $K$ is thin, $\\mathbb{L} \\setminus K$\ + \ is primordially dense ---\nat each depth, it contains elements in all CRT directions\nnot blocked by $K$.\nAgreement\ + \ at some depth $d_0$ follows,\nand the $\\tau$-Identity Theorem\n(Theorem~\\ref{thm:tau-identity}, I.T21)\ngives $\\\ + tilde{f}_1 = \\tilde{f}_2$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 154\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part16/ch62-global-hartogs.tex` lines 97-117\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.GlobalHartogs`\n- Name: `Tau.Holomorphy.global_hartogs`\n\ + \n## Dependencies\n\n- Canonical: I.T29, I.T30, I.T21, I.D46, I.L08\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'THE BOOK I CLIMAX. Global Hartogs Extension: any two tower-coherent functions agreeing at a reference depth + agree at all depths below. Uses all four ingredients: spectral determination, CRT extension, tower coherence, and the Identity + Theor' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001202.md b/_corpus_v3_items/cid001202.md new file mode 100644 index 000000000..4d9dc808c --- /dev/null +++ b/_corpus_v3_items/cid001202.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001202/ +title: Cantor Diagonal Inapplicability +cid: cid001202 +primary_alias: THM0037 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Cantor diagonal argument is inapplicable in tau: the three prerequisites (unrestricted decimal expansion, unrestricted + comprehension, free Cartesian diagonal) all fail within the earned set theory.' +aliases: +- THM0037 +- I.T35 +- cantor-diagonal-inapplicability +- thm:cantor-inapplicable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005341 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 37 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cantor Diagonal Inapplicability**\n\nCantor diagonal argument is inapplicable in tau: the three prerequisites\ + \ (unrestricted decimal expansion, unrestricted comprehension, free Cartesian diagonal) all fail within the earned set\ + \ theory.\n\n# Cantor Diagonal Inapplicability\n\n## Summary\n\nCantor diagonal argument is inapplicable in tau: the three\ + \ prerequisites (unrestricted decimal expansion, unrestricted comprehension, free Cartesian diagonal) all fail within\ + \ the earned set theory.\n\n## Statement\n\n```latex\n%\n\\label{thm:cantor-inapplicable}\nWithin the internal logic of\ + \ Category~$\\tau$ ---\nbounded powerset\n(Definition~\\ref{def:bounded-powerset}),\nno comprehension\n(Proposition~\\\ + ref{prop:no-comprehension}),\ndiagonal discipline\n(Definition~\\ref{def:diagonal-discipline}) ---\nno form of Cantor's\ + \ diagonal argument\ncan produce an object outside $\\Obj(\\tau)$\nor establish $|\\mathbb{R}_\\tau| > |\\mathbb{N}_\\\ + tau|$.\n```\n\n## Proof / Justification\n\n```latex\nCantor's diagonal argument proceeds in three steps:\n\\begin{enumerate}\n\ + \ \\item[(C1)] \\textbf{Diagonal extraction.}\n Construct the function $d(n) := 1 - f(n)(n)$\n by\ + \ extracting the $n$-th digit\n of the $n$-th listed real.\n \\item[(C2)] \\textbf{Anti-diagonal formation.}\n\ + \ Form the set\n $\\{x : x \\neq f(n) \\text{ for all } n\\}$\n and observe that $d$ belongs\ + \ to it.\n \\item[(C3)] \\textbf{Diagonal self-pairing.}\n Use the map $n \\mapsto (n, f(n)(n))$\n \ + \ to walk the diagonal of $\\mathbb{N} \\times \\mathbb{N}$.\n\\end{enumerate}\n\nEach step requires infrastructure\n\ + that $\\tau$ does not provide:\n\n\\medskip\n\\noindent\\textbf{Step (C1) fails} by\nProposition~\\ref{prop:no-unearned-decimal}\ + \ (I.P34):\nthe diagonal digit-extraction\nis not a total computable function\non the constructive reals $\\mathbb{R}_\\\ + tau$.\n\n\\medskip\n\\noindent\\textbf{Step (C2) fails} by\nProposition~\\ref{prop:no-comprehension} (I.P35):\n$\\tau$\ + \ has no comprehension schema.\nThe anti-diagonal set cannot be formed\nwithin $\\tau$-set theory.\n\n\\medskip\n\\noindent\\\ + textbf{Step (C3) fails} by\nProposition~\\ref{prop:no-free-cartesian} (I.P36):\nthe self-pairing map $\\Delta \\colon\ + \ n \\mapsto (n,n)$\nis an unearned diagonal\nthat the diagonal discipline does not permit\nas a generator of new objects.\n\ + \n\\medskip\nSince every step of the argument\nrequires at least one of the three\nprohibited ingredients,\nthe argument\ + \ cannot be executed within~$\\tau$.\nIn particular, no object outside $\\Obj(\\tau)$\ncan be produced by the diagonal\ + \ method,\nand no cardinality inequality\n$|\\mathbb{R}_\\tau| > |\\mathbb{N}_\\tau|$\ncan be derived.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 160\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch37-cantor-diagonal.tex`\ + \ lines 507-520\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.CantorRefutation`\n\ + - Name: `Tau.Sets.cantor_inapplicable`\n\n## Dependencies\n\n- Canonical: I.D75, I.P33, I.D33, I.D03\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Cantor diagonal argument is inapplicable in tau: the three prerequisites (unrestricted decimal expansion, unrestricted + comprehension, free Cartesian diagonal) all fail within the earned set theory.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001203.md b/_corpus_v3_items/cid001203.md new file mode 100644 index 000000000..8b9ad18e2 --- /dev/null +++ b/_corpus_v3_items/cid001203.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001203/ +title: Unique Infinity Object +cid: cid001203 +primary_alias: THM0038 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Unique infinity object: omega is the sole infinite object in tau, and all approaches to infinity are mediated by + omega-germs on the primorial ladder. No hierarchy of infinities exists; infinity is singular and structural.' +aliases: +- THM0038 +- I.T36 +- unique-infinity-object +- thm:unique-infinity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005350 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 9, Chapter 38 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Unique Infinity Object**\n\nUnique infinity object: omega is the sole infinite object in tau, and all approaches\ + \ to infinity are mediated by omega-germs on the primorial ladder. No hierarchy of infinities exists; infinity is singular\ + \ and structural.\n\n# Unique Infinity Object\n\n## Summary\n\nUnique infinity object: omega is the sole infinite object\ + \ in tau, and all approaches to infinity are mediated by omega-germs on the primorial ladder. No hierarchy of infinities\ + \ exists; infinity is singular and structural.\n\n## Statement\n\n```latex\n%\n\\label{thm:unique-infinity}\n$\\omega$\ + \ is the unique non-finite object of Category~$\\tau$.\nMore precisely:\nevery $x \\in \\Obj(\\tau)$ is either $\\omega$\n\ + or has the form $\\rho^n(g)$\nfor some generator $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$\nand some $n \\geq 0$.\n\ + The latter objects have finite depth.\nTherefore $\\omega$ is the unique object of infinite depth ---\nthe unique infinity.\n\ + ```\n\n## Proof / Justification\n\n```latex\nBy the Ontic Closure Theorem\n(Theorem~\\ref{thm:ontic-closure}, I.T01),\n\ + \\[\n \\Obj(\\tau)\n = \\{\\omega\\}\n \\cup O_\\alpha \\cup O_\\pi \\cup O_\\gamma \\cup O_\\eta,\n\\]\nand\ + \ the five components are pairwise disjoint.\nEach orbit element $\\rho^n(g)$ with $n \\geq 0$\nhas \\textbf{depth} $n$,\ + \ a natural number.\nThis is well-defined because the representation is unique\n(Ontic Closure, part~4).\n\nIt remains\ + \ to show that $\\omega$ has no finite depth.\nSuppose for contradiction that $\\omega = \\rho^n(g)$\nfor some generator\ + \ $g$ and some $n \\geq 0$.\nIf $n = 0$, then $\\omega = g$,\ncontradicting Ontic Closure (disjointness of $\\{\\omega\\\ + }$\nfrom each orbit ray, since $g = \\rho^0(g) \\in O_g$).\nIf $n \\geq 1$, then $\\omega = \\rho^n(g)$\ncontradicts $\\\ + KAxiom{5}$.\nThus $\\omega$ does not belong to any orbit ray\nand has no finite depth.\n\nSince every non-$\\omega$ object\ + \ has finite depth,\nand $\\omega$ does not,\n$\\omega$ is the unique object of infinite depth.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 165\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part09/ch38-approaches-infinity.tex`\ + \ lines 67-79\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Sets.UniqueInfinity`\n\ + - Name: `Tau.Sets.unique_infinity`\n\n## Dependencies\n\n- Canonical: I.D76, I.K2, I.K5\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Unique infinity object: omega is the sole infinite object in tau, and all approaches to infinity are mediated + by omega-germs on the primorial ladder. No hierarchy of infinities exists; infinity is singular and structural.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001204.md b/_corpus_v3_items/cid001204.md new file mode 100644 index 000000000..a55ea7a49 --- /dev/null +++ b/_corpus_v3_items/cid001204.md @@ -0,0 +1,103 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001204/ +title: Diagonal-Linear Correspondence +cid: cid001204 +primary_alias: THM0039 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The diagonal discipline (I.D03) and the structural fragment of !-free linear logic are structurally isomorphic: + K5''s refusal of self-products corresponds to absence of contraction, channel consumption to one-use-per-formula, saturation + to finite linear context, controlled overflow to bounded !-introduction.' +aliases: +- THM0039 +- I.T37 +- thm:diagonal-linear +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 69 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagonal-Linear Correspondence**\n\nThe diagonal discipline (I.D03) and the structural fragment of !-free linear\ + \ logic are structurally isomorphic: K5's refusal of self-products corresponds to absence of contraction, channel consumption\ + \ to one-use-per-formula, saturation to finite linear context, controlled overflow to bounded !-introduction.\n\n# Diagonal-Linear\ + \ Correspondence\n\n## Summary\n\nThe diagonal discipline (I.D03) and the structural fragment of !-free linear logic are\ + \ structurally isomorphic: K5's refusal of self-products corresponds to absence of contraction, channel consumption to\ + \ one-use-per-formula, saturation to finite linear context, controlled overflow to bounded !-introduction.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:diagonal-linear}\nThe diagonal discipline\n(Definition~\\ref{def:diagonal-discipline}, I.D03)\n\ + and the structural fragment of $!$-free linear logic\nare structurally isomorphic in the following sense:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n $\\KAxiom{5}$'s refusal of self-products\n within a channel corresponds\ + \ to the absence\n of the contraction rule:\n \\[\n \\frac{\\Gamma, A, A \\vdash B}{\\\ + Gamma, A \\vdash B}\n \\quad\\text{is not available.}\n \\]\n \\item[\\textup{(ii)}]\n \ + \ $\\KAxiom{5}$'s channel consumption\n corresponds to the linear sequent calculus's\n one-use-per-formula\ + \ rule:\n each formula in $\\Gamma$ is consumed\n exactly once in any derivation.\n \\item[\\textup{(iii)}]\n\ + \ $\\KAxiom{5}$'s saturation at four channels\n corresponds to a finite linear context:\n $|\\\ + Gamma| \\leq 4$.\n \\item[\\textup{(iv)}]\n $\\KAxiom{5}$'s controlled overflow\n (three rewiring\ + \ levels:\n addition $\\to$ multiplication\n $\\to$ exponentiation $\\to$ tetration)\n corresponds\ + \ to controlled introduction\n of $!$-like reuse,\n bounded by the solenoidal count\n ($\\\ + KAxiom{6}$: three solenoidal channels,\n since four channels minus one scaffold\n yields three).\n\\\ + end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nWe verify each clause by structural comparison.\n\ + \n\\textbf{(i) Contraction.}\nIn $\\tau$, the diagonal map\n$\\Delta_A : A \\to A \\times A$\nwould require sending an\ + \ object to two copies of itself.\n$\\KAxiom{5}$.1 refuses this:\nthe object occupies one orbit position\nand cannot appear\ + \ in two positions simultaneously\nwithout an explicit $\\rho$-construction\nthat earns the second copy.\nIn linear logic,\ + \ contraction\n$\\Gamma, A, A \\vdash B \\,/\\, \\Gamma, A \\vdash B$\nwould allow using one copy of $A$\nas though it\ + \ were two.\nBoth constraints express the same principle:\nduplication requires work.\n\n\\textbf{(ii) Consumption.}\n\ + In $\\tau$, using a channel for a construction\n(e.g., forming a product $A \\times B$)\nconsumes both input channels:\n\ + $A$ and $B$ are no longer independently available\nin the resulting object.\nIn linear sequent calculus,\nthe $\\otimes$-right\ + \ rule\n\\[\n \\frac{\\Gamma_1 \\vdash A \\qquad \\Gamma_2 \\vdash B}{%\n \\Gamma_1, \\Gamma_2 \\vdash A \\otimes\ + \ B}\n\\]\nsplits the context:\n$\\Gamma_1$ is consumed for $A$\nand $\\Gamma_2$ for $B$,\nwith no sharing between them.\n\ + The context split in linear logic\nmirrors the channel consumption in $\\tau$.\n\n\\textbf{(iii) Finite context.}\nThe\ + \ four orbit rays\n$\\alpha$, $\\pi$, $\\gamma$, $\\eta$\nbound the number of independent channels.\nIn the linear reading,\n\ + this is a context of size at most $4$:\nany sequent in the $\\tau$-linear calculus\nhas the form\n$A_\\alpha, A_\\pi,\ + \ A_\\gamma, A_\\eta \\vdash C$\nwith at most four resources.\n\n\\textbf{(iv) Controlled reuse.}\nThe three rewiring\ + \ levels of $\\KAxiom{5}$\n(Chapter~\\ref{ch:diagonal-discipline})\nallow controlled diagonal operations:\naddition permits\ + \ combining resources\nwithin a channel,\nmultiplication permits iterated combination,\nexponentiation permits iterated\ + \ multiplication,\nand tetration (the fourth level)\nis capped by $\\KAxiom{6}$.\nEach level introduces one more layer\n\ + of structural iteration ---\nanalogous to nested applications of $!$\nthat progressively reintroduce contraction\nin a\ + \ controlled, bounded manner.\nThe solenoidal bound (three active channels,\nsince $4 - 1 = 3$)\nensures that the total\ + \ reuse is finite.\nThis is a bounded $!$: not the unrestricted\nexponential modality of full linear logic,\nbut a modality\ + \ with three levels of nesting.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 170\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch69-diagonal-linear-correspondence.tex` lines\ + \ 251-289\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.LinearDiscipline`\n\ + - Name: `Tau.MetaLogic.diagonal_linear_correspondence`\n\n## Dependencies\n\n- Canonical: I.D78, I.D03, I.D14, I.L02\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The diagonal discipline (I.D03) and the structural fragment of !-free linear logic are structurally isomorphic: + K5''s refusal of self-products corresponds to absence of contraction, channel consumption to one-use-per-formula, saturation + to f' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001205.md b/_corpus_v3_items/cid001205.md new file mode 100644 index 000000000..777b11335 --- /dev/null +++ b/_corpus_v3_items/cid001205.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001205/ +title: Linearity Census +cid: cid001205 +primary_alias: THM0040 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Linearity census of TauLib: 75/80 modules fully constructive, 1 module uses Classical.em (eliminable, I.P38), 1 + module uses funext tactic (CIC kernel axiom). Book I''s formalization is compatible with the !-free fragment of linear logic + at the object level.' +aliases: +- THM0040 +- I.T38 +- linearity-census +- thm:linearity-census +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 70 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Linearity Census**\n\nLinearity census of TauLib: 75/80 modules fully constructive, 1 module uses Classical.em\ + \ (eliminable, I.P38), 1 module uses funext tactic (CIC kernel axiom). Book I's formalization is compatible with the !-free\ + \ fragment of linear logic at the object level.\n\n# Linearity Census\n\n## Summary\n\nLinearity census of TauLib: 75/80\ + \ modules fully constructive, 1 module uses Classical.em (eliminable, I.P38), 1 module uses funext tactic (CIC kernel\ + \ axiom). Book I's formalization is compatible with the !-free fragment of linear logic at the object level.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:linearity-census}\nOf TauLib's 77~modules (${\\approx}\\,15{,}900$~lines):\n\\begin{enumerate}[\\\ + normalfont(i)]\n \\item 74~modules $(96.1\\%)$ use no classical axioms\n and are fully constructive within\ + \ CIC.\n \\item 2~modules use \\texttt{Classical.em},\n both applied to decidable predicates,\n both\ + \ eliminable\n (Proposition~\\ref{prop:em-eliminable}, I.P38).\n \\item 1~module uses the \\texttt{funext}\ + \ tactic,\n which invokes a CIC kernel axiom,\n not a classical commitment.\n\\end{enumerate}\nConclusion:\ + \ Book~I's Lean~4 formalization\nis compatible with the $!$-free fragment of linear logic\nat the object level,\nin the\ + \ sense that classical excluded middle ---\nthe propositional expression of free contraction ---\nis never essentially\ + \ used.\n```\n\n## Proof / Justification\n\n```latex\nItems~(i)--(iii) are the empirical result\nof the audit described\ + \ in\nSections~\\ref{sec:ch70-protocol}--\\ref{sec:ch70-census}.\nThe eliminability claim in item~(ii)\nis Proposition~\\\ + ref{prop:em-eliminable}.\nThe kernel-axiom classification in item~(iii)\nfollows from the fact that \\texttt{funext}\n\ + is a Lean~4 kernel axiom\n(Remark~\\ref{rem:kernel-not-violation}).\n\nFor the conclusion:\nthe $!$-free fragment of linear\ + \ logic\nprohibits free contraction\n(Theorem~\\ref{thm:diagonal-linear}, I.T37).\nAt the propositional level,\nclassical\ + \ excluded middle $A \\lor \\neg A$\nyields contraction ---\nfrom $A \\lor \\neg A$ and two copies of a hypothesis,\n\ + one can route through both branches.\nSince no theorem in TauLib\n\\emph{essentially} depends on\n\\texttt{Classical.em}\n\ + (the two sites are eliminable),\nno theorem requires the contraction\nthat excluded middle provides.\nThe formalization\ + \ operates within the $!$-free fragment.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 173\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch70-linearity-audit.tex` lines 534-555\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.LinearityAudit`\n- Name:\ + \ `Tau.MetaLogic.linearity_census`\n\n## Dependencies\n\n- Canonical: I.D77, I.T37, I.P38\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Linearity census of TauLib: 75/80 modules fully constructive, 1 module uses Classical.em (eliminable, I.P38), + 1 module uses funext tactic (CIC kernel axiom). Book I''s formalization is compatible with the !-free fragment of linear + logic at t' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001206.md b/_corpus_v3_items/cid001206.md new file mode 100644 index 000000000..250904542 --- /dev/null +++ b/_corpus_v3_items/cid001206.md @@ -0,0 +1,99 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001206/ +title: K5 Structural Exclusion +cid: cid001206 +primary_alias: THM0041 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: K5 diagonal discipline places tau on star-autonomous side; Lawvere FPT does not apply. Necessary but not sufficient + for self-hosting. +aliases: +- THM0041 +- I.T39 +- k5-structural-exclusion +- thm:k5-structural-exclusion +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 72 (Part 18) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**K5 Structural Exclusion**\n\nK5 diagonal discipline places tau on star-autonomous side; Lawvere FPT does not\ + \ apply. Necessary but not sufficient for self-hosting.\n\n# K5 Structural Exclusion\n\n## Summary\n\nK5 diagonal discipline\ + \ places tau on star-autonomous side; Lawvere FPT does not apply. Necessary but not sufficient for self-hosting.\n\n##\ + \ Statement\n\n```latex\n%\n\\label{thm:k5-structural-exclusion}\n$\\KAxiom{5}$'s diagonal discipline\n(Definition~\\\ + ref{def:diagonal-discipline}, I.D03),\nwhich maps onto the $!$-free fragment of linear logic\nvia the Diagonal--Linear\ + \ Correspondence\n(Theorem~\\ref{thm:diagonal-linear}, I.T37),\nstructurally excludes the diagonal morphisms\n$\\Delta_A\ + \ : A \\to A \\otimes A$\nthat Lawvere's fixed-point theorem requires.\nSpecifically:\n\\begin{enumerate}[\\normalfont(i)]\n\ + \ \\item $\\KAxiom{5}$.1\n (``no unearned diagonals'')\n directly refuses $\\Delta_A$ for general\ + \ $A$.\n The diagonal map would require\n an object to occupy two orbit positions\n simultaneously\ + \ without an explicit\n $\\rho$-construction earning the second copy.\n Under the Diagonal--Linear Correspondence,\n\ + \ this is the absence of the contraction rule\n in the $!$-free fragment.\n \\item $\\KAxiom{5}$.2\n\ + \ (``each overflow consumes one channel'')\n enforces linear resource tracking:\n using a channel\ + \ in a construction\n removes it from availability.\n Under the correspondence,\n this is the\ + \ one-use-per-formula discipline\n of linear sequent calculus.\n \\item $\\KAxiom{5}$.3\n (``saturation\ + \ at four channels'')\n bounds the total resource budget\n at four orbit rays\n ($\\alpha$,\ + \ $\\pi$, $\\gamma$, $\\eta$).\n Under the correspondence,\n this is a finite linear context\n \ + \ $|\\Gamma| \\leq 4$.\n\\end{enumerate}\nThe Diagonal--Linear Correspondence\n(Definition~\\ref{def:diagonal-linear},\ + \ I.D78;\nTheorem~\\ref{thm:diagonal-linear}, I.T37)\nplaces $\\tau$'s structural discipline\nin the $*$-autonomous regime.\n\ + Since Lawvere's fixed-point theorem requires\nthe diagonal morphism $\\Delta_A$ that\n$*$-autonomous categories do not\ + \ possess,\nthe standard categorical proof of incompleteness\ndoes not apply to $\\tau$'s internal proof theory.\n```\n\ + \n## Proof / Justification\n\n```latex\n[Proof sketch]\nThe argument has three components.\n\n\\textbf{Step 1: K5 maps\ + \ to the $!$-free fragment.}\nThe Diagonal--Linear Correspondence\n(Theorem~\\ref{thm:diagonal-linear}, I.T37)\nestablishes\ + \ that $\\KAxiom{5}$'s three constraints\nare structurally isomorphic\nto the structural rules of the $!$-free fragment\n\ + of Girard's linear logic:\nK5.1 corresponds to the absence of contraction,\nK5.2 to linear resource consumption,\nK5.3\ + \ to a bounded context.\nThis was verified clause-by-clause\nin Chapter~\\ref{ch:diagonal-linear-correspondence}.\n\n\\\ + textbf{Step 2: The $!$-free fragment models in $*$-autonomous categories.}\n$*$-Autonomous categories\n(Remark~\\ref{rem:barr-star-autonomous})\n\ + are the categorical semantics\nof the multiplicative fragment of linear logic.\nIn these categories,\nthe tensor $\\otimes$\ + \ is not cartesian:\nno general diagonal $\\Delta_A : A \\to A \\otimes A$ exists.\nThis is a standard result in categorical\ + \ proof theory\n--- the correspondence between\n$*$-autonomous categories\nand multiplicative linear logic\nwas established\ + \ by Seely (1989)\nand refined by subsequent work.\n\n\\textbf{Step 3: Lawvere's theorem requires the diagonal.}\nLawvere's\ + \ fixed-point theorem\n(Remark~\\ref{rem:lawvere-fpt})\nrequires the diagonal $\\Delta_A$\nto construct the self-referential\ + \ morphism $g$\nthat produces a fixed point.\nWithout $\\Delta_A$, the morphism $g(a) = f(e(a)(a))$\ncannot be formed\ + \ ---\nthe double use of $a$\n(once as argument to $e$, once as argument to $e(a)$)\nis the operational content of the\ + \ diagonal.\nIn the $*$-autonomous regime,\nthis double use is structurally unavailable.\n\nCombining: $\\KAxiom{5}$ places\ + \ $\\tau$\nin the $!$-free fragment (Step~1),\nwhich models in $*$-autonomous categories (Step~2),\nwhere Lawvere's theorem\ + \ does not apply (Step~3).\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 179\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch72-star-autonomous-barrier.tex` lines 369-419\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.StructuralExclusion`\n\ + - Name: `k5_structural_exclusion`\n\n## Dependencies\n\n- Canonical: I.D81, I.T37, I.D03\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: K5 diagonal discipline places tau on star-autonomous side; Lawvere FPT does not apply. Necessary but not sufficient + for self-hosting. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001207.md b/_corpus_v3_items/cid001207.md new file mode 100644 index 000000000..2cecbca8d --- /dev/null +++ b/_corpus_v3_items/cid001207.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001207/ +title: Presheaf Characterization +cid: cid001207 +primary_alias: THM0042 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'HolFun = Nat(F, F_j) ∩ D-Hol: a tau-holomorphic function is a natural transformation from the primorial presheaf + to its split-complex extension that satisfies sector independence.' +aliases: +- THM0042 +- I.T40 +- presheaf-characterization +- thm:presheaf-characterization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005305 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 19, Chapter 74 (Part XIX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Presheaf Characterization**\n\nHolFun = Nat(F, F_j) ∩ D-Hol: a tau-holomorphic function is a natural transformation\ + \ from the primorial presheaf to its split-complex extension that satisfies sector independence.\n\n# Presheaf Characterization\n\ + \n## Summary\n\nHolFun = Nat(F, F_j) ∩ D-Hol: a tau-holomorphic function is a natural transformation from the primorial\ + \ presheaf to its split-complex extension that satisfies sector independence.\n\n## Statement\n\n```latex\n%\n\\label{thm:presheaf-characterization}\n\ + \\[\n \\boxed{%\n \\mathrm{HolFun}\n \\;=\\;\n \\mathrm{Nat}(\\mathcal{F},\\, \\mathcal{F}_{\\jj})\n \\\ + ;\\cap\\;\n \\mathrm{D\\text{-}Hol}}\n\\]\nA $\\tau$-holomorphic function\n(Definition~\\ref{def:holfun}, I.D47)\n\ + is precisely a natural transformation\nfrom the source presheaf $\\mathcal{F}$\nto the split-complex presheaf $\\mathcal{F}_{\\\ + jj}$\n(Definition~\\ref{def:primorial-presheaf}, I.D83)\nthat additionally satisfies D-holomorphy\n(sector independence,\n\ + Proposition~\\ref{prop:sector-independence}, I.P22).\n```\n\n## Proof / Justification\n\n```latex\n$(\\subseteq)$\\;\n\ + Let $T \\in \\mathrm{HolFun}$.\nTower coherence\n(Definition~\\ref{def:tower-coherence}, I.D46)\nstates that for all $k\ + \ \\leq \\ell$:\n\\[\n \\pi_{\\ell \\to k}\\bigl(T_\\ell(t)\\bigr)\n \\;=\\;\n T_k\\bigl(\\pi_{\\ell \\to k}(t)\\\ + bigr).\n\\]\nThis is the naturality square for $T : \\mathcal{F} \\Rightarrow \\mathcal{F}_{\\jj}$:\n\\[\n \\begin{array}{ccc}\n\ + \ \\mathbb{Z}/M_\\ell\\mathbb{Z}\n & \\xrightarrow{T_\\ell}\n & \\mathbb{Z}/M_\\ell\\mathbb{Z}[\\\ + jj] \\\\\n \\downarrow{\\pi_{\\ell \\to k}}\n & & \\downarrow{\\pi_{\\ell \\to k}} \\\\\n \\\ + mathbb{Z}/M_k\\mathbb{Z}\n & \\xrightarrow{T_k}\n & \\mathbb{Z}/M_k\\mathbb{Z}[\\jj]\n \\end{array}\n\ + \\]\ncommutes for every $k \\leq \\ell$.\nD-holomorphy is the second condition of $\\mathrm{HolFun}$\n(Definition~\\ref{def:holfun},\ + \ I.D47).\nTherefore $T \\in \\mathrm{Nat}(\\mathcal{F}, \\mathcal{F}_{\\jj}) \\cap \\mathrm{D\\text{-}Hol}$.\n\n$(\\\ + supseteq)$\\;\nLet $T \\in \\mathrm{Nat}(\\mathcal{F}, \\mathcal{F}_{\\jj}) \\cap \\mathrm{D\\text{-}Hol}$.\nNaturality\ + \ of $T$ gives the commuting square above,\nwhich is exactly the tower coherence condition (I.D46).\nD-holomorphy gives\ + \ sector independence (I.P22).\nTogether these are the two conditions\nof $\\mathrm{HolFun}$ (I.D47).\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-01.jsonl` line 185\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part19/ch74-holomorphy-as-naturality.tex`\ + \ lines 181-201\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.PresheafEssence`\n\ + - Name: `presheaf_characterization`\n\n## Dependencies\n\n- Canonical: I.D83, I.D46, I.D47, I.D53, I.P22, I.T18\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'HolFun = Nat(F, F_j) ∩ D-Hol: a tau-holomorphic function is a natural transformation from the primorial presheaf + to its split-complex extension that satisfies sector independence.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001208.md b/_corpus_v3_items/cid001208.md new file mode 100644 index 000000000..4441876a5 --- /dev/null +++ b/_corpus_v3_items/cid001208.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001208/ +title: Bi-Square Characterization +cid: cid001208 +primary_alias: THM0043 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The complete structural characterization: pasted bi-square of tower coherence (left) and spectral naturality (right). + Both squares commute independently; together equivalent to HolFun. Global Hartogs: limit row determines every stage row.' +aliases: +- THM0043 +- I.T41 +- bi-square-characterization +- thm:bi-square +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005306 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 19, Chapter 75 (Part XIX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bi-Square Characterization**\n\nThe complete structural characterization: pasted bi-square of tower coherence\ + \ (left) and spectral naturality (right). Both squares commute independently; together equivalent to HolFun. Global Hartogs:\ + \ limit row determines every stage row.\n\n# Bi-Square Characterization\n\n## Summary\n\nThe complete structural characterization:\ + \ pasted bi-square of tower coherence (left) and spectral naturality (right). Both squares commute independently; together\ + \ equivalent to HolFun. Global Hartogs: limit row determines every stage row.\n\n## Statement\n\n```latex\n%\n\\label{thm:bi-square}\n\ + For each pair of primorial depths $k \\leq l$,\nthe pasted bi-square\n\\[\n \\begin{array}{ccccc}\n \\mathbb{Z}/M_l\\\ + mathbb{Z}\n & \\xrightarrow{\\;\\;T_l\\;\\;}\n & \\mathbb{Z}/M_l\\mathbb{Z}[\\jj]\n & \\xrightarrow{\\\ + ;\\;(\\chi_+,\\, \\chi_-)\\;\\;}\n & \\mathbb{Z}/M_l\\mathbb{Z} \\times \\mathbb{Z}/M_l\\mathbb{Z}\n \\\\\ + [6pt]\n \\downarrow\\scriptstyle{\\pi}\n & &\n \\downarrow\\scriptstyle{\\pi}\n & &\n \ + \ \\downarrow\\scriptstyle{(\\pi,\\pi)}\n \\\\[6pt]\n \\mathbb{Z}/M_k\\mathbb{Z}\n & \\xrightarrow{\\\ + ;\\;T_k\\;\\;}\n & \\mathbb{Z}/M_k\\mathbb{Z}[\\jj]\n & \\xrightarrow{\\;\\;(\\chi_+,\\, \\chi_-)\\;\\;}\n\ + \ & \\mathbb{Z}/M_k\\mathbb{Z} \\times \\mathbb{Z}/M_k\\mathbb{Z}\n \\end{array}\n\\]\ncharacterizes $\\tau$-holomorphy\ + \ completely:\n\n\\medskip\n\\noindent\n\\fbox{\\parbox{0.93\\textwidth}{%\nA family $\\{T_d\\}_{d \\geq 1}$ of functions\n\ + $T_d : \\mathbb{Z}/M_d\\mathbb{Z} \\to \\mathbb{Z}/M_d\\mathbb{Z}[\\jj]$\nis $\\tau$-holomorphic\nif and only if both\ + \ squares of the bi-square commute\nfor all $k \\leq l$.}}\n```\n\n## Proof / Justification\n\n```latex\nThree equivalences,\ + \ each already established.\n\\emph{Left square commutes}\n$\\Leftrightarrow$ tower coherence (I.D46)\n$\\Leftrightarrow$\ + \ $T \\in \\mathrm{Nat}(F, F_\\jj)$ (I.T40).\n\\emph{Right square commutes}\n$\\Leftrightarrow$ characters respect the\ + \ tower\n$\\Leftrightarrow$ sector independence (I.P22).\n\\emph{Both together}\n$\\Leftrightarrow$ $\\mathrm{HolFun}$\ + \ (I.D47).\nThe pasted outer rectangle also commutes\n(composition of commuting squares).\n```\n\n## Source Context\n\n\ + - Registry source: `book-01.jsonl` line 186\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part19/ch75-holomorphy-bi-square.tex`\ + \ lines 109-145\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Holomorphy.PresheafEssence`\n\ + - Name: `bi_square`\n\n## Dependencies\n\n- Canonical: I.D83, I.T40, I.D46, I.T12, I.D37, I.P22, I.T31\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The complete structural characterization: pasted bi-square of tower coherence (left) and spectral naturality + (right). Both squares commute independently; together equivalent to HolFun. Global Hartogs: limit row determines every stage + row.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001209.md b/_corpus_v3_items/cid001209.md new file mode 100644 index 000000000..1f3caf455 --- /dev/null +++ b/_corpus_v3_items/cid001209.md @@ -0,0 +1,161 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001209/ +title: Archimedean Property +cid: cid001209 +primary_alias: THM0044 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Archimedean property: the natural number embedding into TauReal is unbounded and injective. For any n < m, fromNat(m) + is not equivalent to fromNat(n).' +aliases: +- THM0044 +- I.T42 +- archimedean-property +- thm:archimedean-property +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005268 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 76 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Archimedean Property** + + + The Archimedean property: the natural number embedding into TauReal is unbounded and injective. For any n < m, fromNat(m) + is not equivalent to fromNat(n). + + + # Archimedean Property + + + ## Summary + + + The Archimedean property: the natural number embedding into TauReal is unbounded and injective. For any n < m, fromNat(m) + is not equivalent to fromNat(n). + + + ## Statement + + + ```latex + + % + + \label{thm:archimedean-property} + + For every $x \in \mathbb{R}_\tau$, + + there exists $n \in \mathbb{N}_\tau$ such that $n > x$. + + ``` + + + ## Proof / Justification + + + ```latex + + [Proof sketch] + + Let $x = [q_k]$ with modulus $M$. + + Choose $K := M(1)$, so $|q_k - q_K| < 1$ for $k \geq K$. + + Write $q_K = a/b \in \mathbb{Q}_\tau$ with $b > 0$. + + Then $q_k < |a| + 2$ for all $k \geq K$. + + Setting $n := |a| + 2 \in \mathbb{N}_\tau$ + + (via $\mathbb{N}_\tau \hookrightarrow \mathbb{R}_\tau$) + + gives $n > x$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 188 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch76-constructive-reals.tex` lines + 110-115 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.ConstructiveReals` + + - Name: `Tau.Boundary.taureal_archimedean_embedding` + + + ## Dependencies + + + - Canonical: I.D84 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Archimedean property: the natural number embedding into TauReal is unbounded and injective. For any n < + m, fromNat(m) is not equivalent to fromNat(n).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001210.md b/_corpus_v3_items/cid001210.md new file mode 100644 index 000000000..7be085abd --- /dev/null +++ b/_corpus_v3_items/cid001210.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001210/ +title: TauComplex Ring Axioms +cid: cid001210 +primary_alias: THM0045 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: TauComplex satisfies all commutative ring axioms including i^2=-1. Each axiom reduces through TauReal to TauRat algebra + via the toInt bridge. +aliases: +- THM0045 +- I.T43 +- taucomplex-ring-axioms +- thm:complex-field-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 77 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**TauComplex Ring Axioms**\n\nTauComplex satisfies all commutative ring axioms including i^2=-1. Each axiom reduces\ + \ through TauReal to TauRat algebra via the toInt bridge.\n\n# TauComplex Ring Axioms\n\n## Summary\n\nTauComplex satisfies\ + \ all commutative ring axioms including i^2=-1. Each axiom reduces through TauReal to TauRat algebra via the toInt bridge.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:complex-field-axioms}\n$\\mathbb{C}_\\tau$ is a \\textbf{commutative field}.\n\ + That is:\n\\begin{enumerate}\n \\item \\textbf{Additive group.}\n $(\\mathbb{C}_\\tau, +)$ is an abelian group\n\ + \ with identity $0 = 0 + 0i$\n and additive inverse $-(a + bi) = (-a) + (-b)i$.\n \\item \\textbf{Multiplicative\ + \ group.}\n $(\\mathbb{C}_\\tau \\setminus \\{0\\}, \\cdot)$ is an abelian group\n with identity $1\ + \ = 1 + 0i$.\n \\item \\textbf{Distributivity.}\n Multiplication distributes over addition.\n\\end{enumerate}\n\ + In particular, every nonzero $z = a + bi \\in \\mathbb{C}_\\tau$\nhas a multiplicative inverse:\n\\[\n \\boxed{%\n\ + \ (a + bi)^{-1} = \\frac{a}{a^2 + b^2} - \\frac{b}{a^2 + b^2}\\, i\n = \\frac{a - bi}{a^2 + b^2}.}\n\\]\n```\n\n\ + ## Proof / Justification\n\n```latex\n[Proof sketch]\nCommutativity, associativity, and distributivity\nare verified by\ + \ direct computation\nusing the operations of Definition~\\ref{def:tau-complex-field}.\nFor the multiplicative inverse,\n\ + suppose $z = a + bi \\neq 0$,\nso that $a \\neq 0$ or $b \\neq 0$.\nThen $a^2 + b^2 > 0$ in $\\mathbb{R}_\\tau$\n(since\ + \ $a^2 \\geq 0$, $b^2 \\geq 0$,\nand at least one is strictly positive).\nDefine $w := (a - bi)/(a^2 + b^2)$. Then:\n\\\ + begin{align*}\n z \\cdot w\n &= (a + bi) \\cdot \\frac{a - bi}{a^2 + b^2} \\\\\n &= \\frac{(a + bi)(a - bi)}{a^2\ + \ + b^2} \\\\\n &= \\frac{a^2 - (bi)^2}{a^2 + b^2}\n = \\frac{a^2 - b^2 i^2}{a^2 + b^2}\n = \\frac{a^2 + b^2}{a^2\ + \ + b^2}\n = 1.\n\\end{align*}\nThis is the crucial step:\nthe denominator $a^2 + b^2$ is strictly positive\n\\emph{because}\ + \ $i^2 = -1$ causes the cross terms to add\nrather than cancel.\nCompare with the split-complex case below,\nwhere $j^2\ + \ = +1$ causes the cross terms to cancel,\nproducing zero divisors.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 192\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch77-elliptic-complex-field.tex`\ + \ lines 94-117\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.ComplexField`\n\ + - Name: `Tau.Boundary.taucomplex_ring_axioms`\n\n## Dependencies\n\n- Canonical: I.D85, I.P39\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauComplex satisfies all commutative ring axioms including i^2=-1. Each axiom reduces through TauReal to TauRat + algebra via the toInt bridge. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001211.md b/_corpus_v3_items/cid001211.md new file mode 100644 index 000000000..158bdb11c --- /dev/null +++ b/_corpus_v3_items/cid001211.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001211/ +title: Quaternion Non-Commutativity +cid: cid001211 +primary_alias: THM0046 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The quaternions are non-commutative: qi * qj is not equivalent to qj * qi. Explicit witness: the k-component differs + in sign. This is the first non-commutative algebra earned within tau.' +aliases: +- THM0046 +- I.T44 +- quaternion-non-commutativity +- thm:quaternion-division +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 78 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Quaternion Non-Commutativity**\n\nThe quaternions are non-commutative: qi * qj is not equivalent to qj * qi.\ + \ Explicit witness: the k-component differs in sign. This is the first non-commutative algebra earned within tau.\n\n\ + # Quaternion Non-Commutativity\n\n## Summary\n\nThe quaternions are non-commutative: qi * qj is not equivalent to qj *\ + \ qi. Explicit witness: the k-component differs in sign. This is the first non-commutative algebra earned within tau.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:quaternion-division}\n$\\mathbb{H}_\\tau$ is a \\textbf{division algebra}:\n\ + every nonzero $q \\in \\mathbb{H}_\\tau$ has a multiplicative inverse\n\\[\n \\boxed{q^{-1} = \\frac{\\bar{q}}{|q|^2},\n\ + \ \\qquad q q^{-1} = q^{-1} q = 1.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nFor $q \\neq 0$, $|q|^2 = a^2\ + \ + b^2 + c^2 + d^2 > 0$.\nThen $q \\cdot (\\bar{q}/|q|^2) = q\\bar{q}/|q|^2 = |q|^2/|q|^2 = 1$\nby Proposition~\\ref{prop:norm-conjugation},\ + \ and similarly for $q^{-1}q$.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 194\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch78-elliptic-quaternions.tex` lines 159-168\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.Boundary.Quaternions`\n- Name:\ + \ `Tau.Boundary.quaternion_non_commutativity`\n\n## Dependencies\n\n- Canonical: I.D87\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The quaternions are non-commutative: qi * qj is not equivalent to qj * qi. Explicit witness: the k-component + differs in sign. This is the first non-commutative algebra earned within tau.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001212.md b/_corpus_v3_items/cid001212.md new file mode 100644 index 000000000..cbed4c365 --- /dev/null +++ b/_corpus_v3_items/cid001212.md @@ -0,0 +1,161 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001212/ +title: Roots of Unity CRT Decomposition +cid: cid001212 +primary_alias: THM0047 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CRT decomposition of roots of unity: for coprime moduli m1, m2, a root of unity mod m1*m2 decomposes into roots + mod m1 and mod m2. Connects cyclotomic structure to the CRT basis that pervades the spectral theory.' +aliases: +- THM0047 +- I.T45 +- roots-of-unity-crt-decomposition +- thm:primitive-roots +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005269 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 17, Chapter 79 (Part XVII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Roots of Unity CRT Decomposition** + + + CRT decomposition of roots of unity: for coprime moduli m1, m2, a root of unity mod m1*m2 decomposes into roots mod m1 + and mod m2. Connects cyclotomic structure to the CRT basis that pervades the spectral theory. + + + # Roots of Unity CRT Decomposition + + + ## Summary + + + CRT decomposition of roots of unity: for coprime moduli m1, m2, a root of unity mod m1*m2 decomposes into roots mod m1 + and mod m2. Connects cyclotomic structure to the CRT basis that pervades the spectral theory. + + + ## Statement + + + ```latex + + % + + \label{thm:primitive-roots} + + A root $\zeta_n^k \in \mu_n$ is \textbf{primitive} + + --- i.e., generates $\mu_n$ --- + + if and only if $\gcd(k, n) = 1$. + + For each $n \geq 1$, + + the number of primitive $n$-th roots of unity + + is $\varphi(n)$ (Euler''s totient). + + ``` + + + ## Proof / Justification + + + ```latex + + The element $\zeta_n^k$ generates $\mu_n$ + + iff the multiples $\{k, 2k, \ldots, nk\}$ + + cover all residue classes modulo $n$, + + which holds precisely when $\gcd(k, n) = 1$. + + ``` + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 197 + + - Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part17/ch79-cyclotomic-fields.tex` lines + 56-65 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Cyclotomic` + + - Name: `Tau.Boundary.root_of_unity_crt` + + + ## Dependencies + + + - Canonical: I.D88, I.D19 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CRT decomposition of roots of unity: for coprime moduli m1, m2, a root of unity mod m1*m2 decomposes into roots + mod m1 and mod m2. Connects cyclotomic structure to the CRT basis that pervades the spectral theory.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001213.md b/_corpus_v3_items/cid001213.md new file mode 100644 index 000000000..9f7f2285d --- /dev/null +++ b/_corpus_v3_items/cid001213.md @@ -0,0 +1,104 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001213/ +title: Ontic Identity Invariance +cid: cid001213 +primary_alias: THM0048 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Ontic Identity Invariance: in the coherence kernel (K0-K6), every admissible construction preserves ontic identity. + The normalization map to canonical form is unique and path-independent; no construction can introduce alternative identity + witnesses or partial identifications.' +aliases: +- THM0048 +- I.T46 +- ontic-identity-invariance +- thm:ontic-identity-invariance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 81 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ontic Identity Invariance**\n\nOntic Identity Invariance: in the coherence kernel (K0-K6), every admissible\ + \ construction preserves ontic identity. The normalization map to canonical form is unique and path-independent; no construction\ + \ can introduce alternative identity witnesses or partial identifications.\n\n# Ontic Identity Invariance\n\n## Summary\n\ + \nOntic Identity Invariance: in the coherence kernel (K0-K6), every admissible construction preserves ontic identity.\ + \ The normalization map to canonical form is unique and path-independent; no construction can introduce alternative identity\ + \ witnesses or partial identifications.\n\n## Statement\n\n```latex\n%\n\\label{thm:ontic-identity-invariance}\n% I.D90,\ + \ I.D91, I.L02, I.T37, I.T39\nIn the coherence kernel ($\\KAxiom{0}$--$\\KAxiom{6}$),\nevery admissible construction preserves\ + \ ontic identity:\n\\begin{enumerate}[\\normalfont(i)]\n \\item The normalization map to canonical form\n \ + \ is unique and path-independent.\n \\item No admissible construction can introduce\n alternative identity\ + \ witnesses,\n partial identifications,\n or shadow identities\n (Definition~\\ref{def:shadow-identity},\ + \ I.D91).\n \\item Ontic identity is invariant\n under the full group of admissible symmetries.\n\\end{enumerate}\n\ + In particular, identity slippage\n(Definition~\\ref{def:identity-slippage}, I.D90)\nis zero at every stage of every admissible\ + \ construction.\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nThe argument proceeds in five steps.\n\n\ + \\textbf{Step 1: Admissible constructions factor\nthrough the program monoid.}\nEvery admissible construction in the coherence\ + \ kernel\nis a finite composition of generator applications\nwithin the program monoid\n(Definition~\\ref{def:program-monoid},\ + \ I.D14,\nChapter~\\ref{ch:program-monoid}).\nAn admissible construction is a sequence\nof $\\rho$-instructions and swap\ + \ operations\nthat respects the constraints\nof $\\KAxiom{0}$--$\\KAxiom{6}$.\n\n\\textbf{Step 2: NF-Confluence ensures\ + \ unique output.}\nBy NF-Confluence\n(Lemma~\\ref{lem:nf-confluence}, I.L02),\nany two reduction paths from a given input\n\ + converge to the same normal form.\nThis means that the output of normalization\n--- the canonical representative of the\ + \ result ---\nis uniquely determined by the input,\nindependent of the reduction strategy.\n\n\\textbf{Step 3: K5 prevents\ + \ contraction artifacts.}\nBy $\\KAxiom{5}$\n(Definition~\\ref{def:diagonal-discipline}, I.D03),\nno step in the construction\ + \ can introduce\nfree contraction.\nA $\\rho$-instruction consumes its input channel\nand produces its output channel;\n\ + no instruction duplicates a channel\nwithout explicitly earning the copy.\nIn the terminology of\nChapter~\\ref{ch:diagonal-resonance},\n\ + this blocks the (L) component:\nno construction step can produce\ntwo references to the same object\nwithout an explicit\ + \ morphism mediating between them.\n\n\\textbf{Step 4: Three-level equality prevents\nimplicit identification.}\nBy three-level\ + \ equality\n(Definition~\\ref{def:three-equality}, I.D15),\nsyntactic identity (normal-form comparison),\ndenotational\ + \ identity (program-monoid equivalence),\nand categorical identity (isomorphism)\nare stratified and non-collapsible.\n\ + No construction step can substitute\nacross levels:\nan isomorphism between $A$ and $B$\ndoes not make $A$ and $B$ syntactically\ + \ identical.\nThis blocks the (E) component:\nno implicit identification channel exists.\n\n\\textbf{Step 5: Star-autonomous\ + \ structure prevents\nprimitive self-products.}\nBy the K5 Structural Exclusion Theorem\n(Theorem~\\ref{thm:k5-structural-exclusion},\ + \ I.T39),\n$\\tau$ lives on the star-autonomous side\nof the CCC--linear dichotomy.\nSelf-products $A \\otimes A$ are\ + \ not primitive:\nthey must be earned by explicit construction.\nNo construction step can materialize\na self-product\ + \ that was not already present\nin the input.\nThis blocks the (P) component:\nthe ambient space for shadow identities\n\ + does not exist unless explicitly constructed.\n\n\\textbf{Combining.}\nSteps 3--5 block the three components\n(L), (E),\ + \ (P) of diagonal resonance (I.D89).\nSince diagonal resonance requires all three components\nto produce identity slippage\ + \ (I.D90),\nand each component is independently blocked,\nno admissible construction can produce\nidentity slippage.\n\ + By Step~2,\nthe normalization map produces\na unique canonical form.\nBy Steps~3--5,\nno alternative identity witnesses\ + \ or shadow identities\ncan be introduced.\nTherefore, ontic identity is preserved:\nthe canonical normal form IS the\ + \ ontic representative,\nwith no quotient, no equivalence class,\nand no choice of representative.\n```\n\n## Source Context\n\ + \n- Registry source: `book-01.jsonl` line 204\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch81-ontic-identity-invariance.tex`\ + \ lines 463-483\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.OnticInvariance`\n\ + - Name: `Tau.MetaLogic.ontic_identity_invariance`\n\n## Dependencies\n\n- Canonical: I.D03, I.D14, I.D15, I.D89, I.D90,\ + \ I.D91, I.L02, I.T37, I.T39\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Ontic Identity Invariance: in the coherence kernel (K0-K6), every admissible construction preserves ontic identity. + The normalization map to canonical form is unique and path-independent; no construction can introduce alternative identity + w' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001214.md b/_corpus_v3_items/cid001214.md new file mode 100644 index 000000000..821f33931 --- /dev/null +++ b/_corpus_v3_items/cid001214.md @@ -0,0 +1,114 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001214/ +title: Slippage Breaks Unique Omega +cid: cid001214 +primary_alias: THM0049 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Slippage Breaks Unique Omega: any foundational stack permitting identity slippage at the substrate level cannot + internalize a unique absolute infinity omega. It can only generate many infinities (model-relative, non-canonical) via limit/quotient + constructions. Explains why ZFC has aleph_0, aleph_1, ... while tau has one omega.' +aliases: +- THM0049 +- I.T47 +- slippage-breaks-unique-omega +- thm:slippage-breaks-omega +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005325 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 81 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Slippage Breaks Unique Omega**\n\nSlippage Breaks Unique Omega: any foundational stack permitting identity\ + \ slippage at the substrate level cannot internalize a unique absolute infinity omega. It can only generate many infinities\ + \ (model-relative, non-canonical) via limit/quotient constructions. Explains why ZFC has aleph_0, aleph_1, ... while tau\ + \ has one omega.\n\n# Slippage Breaks Unique Omega\n\n## Summary\n\nSlippage Breaks Unique Omega: any foundational stack\ + \ permitting identity slippage at the substrate level cannot internalize a unique absolute infinity omega. It can only\ + \ generate many infinities (model-relative, non-canonical) via limit/quotient constructions. Explains why ZFC has aleph_0,\ + \ aleph_1, ... while tau has one omega.\n\n## Statement\n\n```latex\n%\n\\label{thm:slippage-breaks-omega}\nAny foundational\ + \ stack that permits identity slippage\n(Definition~\\ref{def:identity-slippage}, I.D90)\nat the substrate level\ncannot\ + \ internalize a unique, absolute infinity $\\omega$.\nSuch a system can only generate\n``many infinities'' ---\nmodel-relative,\ + \ non-canonical limit objects ---\nvia limit/quotient constructions.\n```\n\n## Proof / Justification\n\n```latex\n[Proof\ + \ sketch]\nThe argument has three parts.\n\n\\textbf{Part 1: The three roles of $\\omega$.}\nIn $\\tau$, $\\omega$ serves\ + \ three roles simultaneously:\n\\begin{enumerate}\n \\item[(a)] \\textbf{Absorber.}\n $\\omega$ is the unique\ + \ absorbing element\n of the primorial tower:\n $\\omega \\cdot n = \\omega$ for all finite $n$.\n \ + \ It is the ``top'' of the tower.\n \\item[(b)] \\textbf{Unique closure point.}\n The primorial tower\ + \ $\\{p_k\\#\\}_{k \\geq 1}$\n has $\\omega$ as its unique limit.\n No other element serves as the closure\ + \ point\n of the tower.\n \\item[(c)] \\textbf{Unique absolute reference.}\n $\\omega$ is globally\ + \ identifiable:\n it is invariant under all admissible symmetries\n of the tower,\n not susceptible\ + \ to equivalence-class ambiguity,\n and not relativized to a model or interpretation.\n\\end{enumerate}\nThese\ + \ three roles are mutually reinforcing:\n(a) makes $\\omega$ algebraically unique,\n(b) makes it topologically unique\n\ + (as a limit in the tower),\nand (c) makes it ontically unique\n(as a reference point\nthat the system can canonically\ + \ identify).\n\n\\textbf{Part 2: Uniqueness requires identity coherence.}\nBeing ``the unique absolute reference'' (role~(c))\n\ + means that $\\omega$ must be globally identifiable:\nany admissible construction\nthat produces an element playing the\ + \ absorber role\nmust produce the \\emph{same} element.\nThis is a strong condition.\nIt requires that the system can\ + \ determine,\nfor any candidate $x$:\ndoes $x = \\omega$, or does $x \\neq \\omega$?\nThe determination must be canonical\ + \ ---\nnot dependent on a choice of representative,\nnot relative to a model,\nnot subject to equivalence-class ambiguity.\n\ + \nIdentity slippage\n(Definition~\\ref{def:identity-slippage}, I.D90)\nis precisely the negation of this condition.\n\ + A system with identity slippage\nharbors shadow identities\n(Definition~\\ref{def:shadow-identity}, I.D91):\nobjects that\ + \ the system cannot canonically distinguish\nfrom one another.\nIf $\\omega$ could have shadow identities ---\nif there\ + \ existed objects $\\omega'$, $\\omega''$, $\\ldots$\nthat play the absorber role\nbut cannot be canonically identified\ + \ with $\\omega$ ---\nthen $\\omega$ would not be unique.\nIt would be an equivalence class\nof absorber-candidates,\n\ + mediated by an identification\nthat the system cannot decide.\n\n\\textbf{Part 3: Identity slippage produces many infinities.}\n\ + In a foundation with identity slippage,\nthe construction of infinite objects\nproceeds through limits and quotients.\n\ + Each construction path may produce\na different ``infinity'':\n$\\aleph_0$ as the cardinality of $\\mathbb{N}$,\n$\\aleph_1$\ + \ as the next infinite cardinal\n(whose identity depends on the continuum hypothesis\nand thus on the model),\n$\\omega_1$\ + \ as the first uncountable ordinal\n(well-defined but not canonically constructible\nfrom the natural numbers alone).\n\ + These are not shadow identities of a single $\\omega$.\nThey are genuinely different objects ---\ndifferent infinities\ + \ ---\ngenerated by different construction paths\nin a system where no single, canonical closure point\nabsorbs all others.\n\ + \nThe proliferation is not accidental.\nIt is a structural consequence of identity slippage\nat the substrate level.\n\ + If the system cannot canonically determine\n``this is the same as that,''\nthen limits constructed by different paths\n\ + need not converge to the same point.\nDifferent paths produce different infinities.\nThe continuum hypothesis is undecidable\ + \ in ZFC\nprecisely because ZFC's identity slippage\n(at the level of set-membership and extensionality)\nprevents the\ + \ system from canonically resolving\nthe relationship between $\\aleph_0$ and $2^{\\aleph_0}$.\n\nIn $\\tau$, identity\ + \ slippage is zero\n(Corollary~\\ref{cor:no-identity-decoherence}, I.C03).\nEvery construction path converges\nto a unique\ + \ normal form (I.L02).\nThe primorial tower has one closure point,\nnot many.\n$\\omega$ is unique because the system\n\ + that produces it has no mechanism\nfor generating alternatives.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl`\ + \ line 206\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch81-ontic-identity-invariance.tex`\ + \ lines 760-771\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.OnticInvariance`\n\ + - Name: `Tau.MetaLogic.slippage_breaks_omega`\n\n## Dependencies\n\n- Canonical: I.D90, I.T46, I.T36\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Slippage Breaks Unique Omega: any foundational stack permitting identity slippage at the substrate level cannot + internalize a unique absolute infinity omega. It can only generate many infinities (model-relative, non-canonical) via limit/quo' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001215.md b/_corpus_v3_items/cid001215.md new file mode 100644 index 000000000..ebe43ee0e --- /dev/null +++ b/_corpus_v3_items/cid001215.md @@ -0,0 +1,117 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001215/ +title: Structural Instability Theorem +cid: cid001215 +primary_alias: THM0050 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Structural Instability Theorem: diagonal-resonant foundations cannot host identity-faithful reception of tau. The + L+E+P components jointly create identity slack preventing any global projection from preserving distinctness. Not inconsistency + — structural incompatibility with tau ontic granularity.' +aliases: +- THM0050 +- I.T48 +- structural-instability-theorem +- thm:structural-instability +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005326 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 82 (Part XVIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Structural Instability Theorem**\n\nStructural Instability Theorem: diagonal-resonant foundations cannot host\ + \ identity-faithful reception of tau. The L+E+P components jointly create identity slack preventing any global projection\ + \ from preserving distinctness. Not inconsistency — structural incompatibility with tau ontic granularity.\n\n# Structural\ + \ Instability Theorem\n\n## Summary\n\nStructural Instability Theorem: diagonal-resonant foundations cannot host identity-faithful\ + \ reception of tau. The L+E+P components jointly create identity slack preventing any global projection from preserving\ + \ distinctness. Not inconsistency — structural incompatibility with tau ontic granularity.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:structural-instability}\nDiagonal-resonant foundations\ncannot host identity-faithful reception of $\\\ + tau$.\nMore precisely:\nif $\\mathsf{S}$ is a foundation\nexhibiting diagonal resonance (I.D89),\nthen no functor\n$P\ + \ : \\mathcal{C}_\\tau \\to \\mathcal{C}_{\\mathsf{S}}$\nsatisfies all three conditions\nof identity-faithful reception\ + \ (I.D92).\nThe three components (L, E, P)\njointly create identity slack\nthat prevents any global projection\nfrom preserving\ + \ the distinctness\nguaranteed by ontic identity invariance (I.T46).\n```\n\n## Proof / Justification\n\n```latex\n[Proof\ + \ sketch]\nThe argument proceeds by contradiction\nin seven steps.\n\n\\textbf{Step 1: Assume identity-faithfulness.}\n\ + Suppose $P : \\mathcal{C}_\\tau \\to \\mathcal{C}_{\\mathsf{S}}$\nis identity-faithful ---\nsatisfying conditions (i),\ + \ (ii), and (iii)\nof Definition~\\ref{def:identity-faithful-reception}.\n\n\\textbf{Step 2: In $\\tau$, distinctness\ + \ is absolute.}\nThe ontic identity invariance theorem (I.T46)\nguarantees that distinct normal forms\nin the program\ + \ monoid $\\mathfrak{P}_\\tau$\ncorrespond to distinct objects in $\\mathcal{C}_\\tau$.\nIf $w \\neq w'$ are distinct\ + \ normal forms\n(irreducible words in the generators\n$\\alpha, \\pi, \\gamma, \\eta, \\omega$),\nthen the objects $[w]$\ + \ and $[w']$\nare distinct in $\\mathcal{C}_\\tau$.\nThe no-identity-decoherence corollary (I.C03)\nensures that no partial\ + \ identification is possible:\n$[w] = [w']$ or $[w] \\neq [w']$,\nwith no intermediate state.\n\n\\textbf{Step 3: In $\\\ + mathsf{S}$, diagonal resonance creates shadow identities.}\nSince $\\mathsf{S}$ exhibits diagonal resonance (I.D89),\n\ + it hosts shadow identities (I.D91):\nimplicit identification channels\nthat $\\mathsf{S}$ cannot distinguish\nfrom genuine\ + \ identities.\nA shadow identity between $\\mathsf{S}$-objects $A$ and $B$\nis a metamathematical artifact ---\na chain\ + \ of extensional equalities,\nforcing equivalences, or model-theoretic identifications ---\nthat makes $A$ and $B$\n``the\ + \ same'' in some models of $\\mathsf{S}$\nbut not in others.\n\n\\textbf{Step 4: Shadow identities act on the image of\ + \ $P$.}\nThe objects $P([w])$ and $P([w'])$\nlive in $\\mathcal{C}_{\\mathsf{S}}$.\nShadow identities in $\\mathsf{S}$\n\ + may identify $P([w])$ and $P([w'])$\neven when $[w] \\neq [w']$ in $\\mathcal{C}_\\tau$.\nThe mechanism is the three-component\ + \ resonance:\n\\begin{itemize}\n \\item[\\textbf{(L)}] Unlimited lambda abstraction\n permits free token reuse\ + \ in $\\mathsf{S}$.\n The names assigned to $P([w])$ and $P([w'])$\n may share meta-level references\ + \ ---\n a bound variable in one definition\n may be $\\alpha$-equivalent\n to a bound variable\ + \ in another,\n creating a naming coincidence\n that $\\mathsf{S}$ cannot distinguish\n from\ + \ a structural identification.\n \\item[\\textbf{(E)}] Extensional, substitution-based equality in $\\mathsf{S}$\n\ + \ may force $P([w]) = P([w'])$\n at the congruence level.\n If $\\mathsf{S}$ derives\n \ + \ that two terms are extensionally equal\n (they produce the same outputs\n on all inputs in $\\\ + mathsf{S}$),\n then $P([w])$ and $P([w'])$\n become identified in $\\mathsf{S}$'s identity type ---\n\ + \ even if the corresponding $\\tau$-objects\n are structurally distinct.\n \\item[\\textbf{(P)}]\ + \ The diagonal $\\Delta$ in $\\mathsf{S}$\n may materialize a witness of ``sameness''\n between $P([w])$\ + \ and $P([w'])$\n that has no $\\tau$-preimage.\n The diagonal self-application\n $P([w]) \\\ + mapsto (P([w]), P([w]))$\n enables the construction of isomorphisms\n in $\\mathsf{S}$ that do not lift\ + \ to $\\tau$.\n\\end{itemize}\n\n\\textbf{Step 5: $P$ fails to reflect isomorphism.}\nBy Step~4, there exist $\\tau$-objects\n\ + $[w] \\neq [w']$ such that\n$P([w]) \\cong P([w'])$ in $\\mathcal{C}_{\\mathsf{S}}$.\nThe isomorphism is constructed\n\ + using the L+E+P resonance in $\\mathsf{S}$\n--- specifically, using the diagonal map $\\Delta$\nthat $\\tau$'s $\\KAxiom{5}$\ + \ discipline refuses.\nSince $[w] \\not\\cong [w']$ in $\\mathcal{C}_\\tau$\n(by ontic identity invariance, I.T46),\n\ + the functor $P$ fails condition~(iii)\nof identity-faithful reception:\nit does not reflect isomorphism.\n\n\\textbf{Step\ + \ 6: Alternatively, $P$ fails object distinctness.}\nIf the shadow identities in $\\mathsf{S}$\nare strong enough to force\n\ + $P([w]) = P([w'])$ (equality, not merely isomorphism),\nthen $P$ fails condition~(i) directly:\ndistinct $\\tau$-objects\n\ + are sent to the same $\\mathsf{S}$-object.\n\n\\textbf{Step 7: Contradiction.}\nIn either case, $P$ is not identity-faithful,\n\ + contradicting Step~1.\nThe contradiction establishes\nthat no identity-faithful reception functor exists\nwhen $\\mathsf{S}$\ + \ exhibits diagonal resonance.\n```\n\n## Source Context\n\n- Registry source: `book-01.jsonl` line 209\n- Manuscript\ + \ source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part18/ch82-reception-criterion.tex` lines 347-364\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookI.MetaLogic.ReceptionCriterion`\n\ + - Name: `Tau.MetaLogic.structural_instability_theorem`\n\n## Dependencies\n\n- Canonical: I.D89, I.D90, I.D91, I.D92,\ + \ I.D93, I.T46, I.T47, I.C03\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Structural Instability Theorem: diagonal-resonant foundations cannot host identity-faithful reception of tau. + The L+E+P components jointly create identity slack preventing any global projection from preserving distinctness. Not inconsistenc' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001216.md b/_corpus_v3_items/cid001216.md new file mode 100644 index 000000000..bf501f64d --- /dev/null +++ b/_corpus_v3_items/cid001216.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001216/ +title: Countable Additivity +cid: cid001216 +primary_alias: THM0051 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The stage-k counting measure is finitely additive: for disjoint subsets S, T of Z/M_k Z, μ_k(S ∪ T) = μ_k(S) + μ_k(T). + Verified computationally at stage 3 for even/odd and B/C sector partitions.' +aliases: +- THM0051 +- I.T49 +- countable-additivity +- thm:countable-additivity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005277 +- predicate: formalized_by + object: cid005278 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Countable Additivity** + + + The stage-k counting measure is finitely additive: for disjoint subsets S, T of Z/M_k Z, μ_k(S ∪ T) = μ_k(S) + μ_k(T). + Verified computationally at stage 3 for even/odd and B/C sector partitions. + + + # Countable Additivity + + + ## Summary + + + The stage-k counting measure is finitely additive: for disjoint subsets S, T of Z/M_k Z, μ_k(S ∪ T) = μ_k(S) + μ_k(T). + Verified computationally at stage 3 for even/odd and B/C sector partitions. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 221 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Measure` + + - Name: `additivity_even_odd_3` + + + ## Dependencies + + + - Canonical: I.D95, I.D96 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The stage-k counting measure is finitely additive: for disjoint subsets S, T of Z/M_k Z, μ_k(S ∪ T) = μ_k(S) + + μ_k(T). Verified computationally at stage 3 for even/odd and B/C sector partitions.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001217.md b/_corpus_v3_items/cid001217.md new file mode 100644 index 000000000..8e278883d --- /dev/null +++ b/_corpus_v3_items/cid001217.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001217/ +title: Fundamental Theorem of Internal Galois Theory +cid: cid001217 +primary_alias: THM0052 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every unit of Z/M_k Z generates a valid ring automorphism preserving both addition and multiplication. This gives + a bijection between units and automorphisms, the τ-internal Galois correspondence. +aliases: +- THM0052 +- I.T50 +- fundamental-theorem-of-internal-galois-theory +- thm:galois-fundamental +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005289 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 85 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Fundamental Theorem of Internal Galois Theory** + + + Every unit of Z/M_k Z generates a valid ring automorphism preserving both addition and multiplication. This gives a bijection + between units and automorphisms, the τ-internal Galois correspondence. + + + # Fundamental Theorem of Internal Galois Theory + + + ## Summary + + + Every unit of Z/M_k Z generates a valid ring automorphism preserving both addition and multiplication. This gives a bijection + between units and automorphisms, the τ-internal Galois correspondence. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 225 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Galois` + + - Name: `galois_fundamental_2` + + + ## Dependencies + + + - Canonical: I.D97, I.D98, I.D88 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every unit of Z/M_k Z generates a valid ring automorphism preserving both addition and multiplication. This gives + a bijection between units and automorphisms, the τ-internal Galois correspondence. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001218.md b/_corpus_v3_items/cid001218.md new file mode 100644 index 000000000..2eb5de320 --- /dev/null +++ b/_corpus_v3_items/cid001218.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001218/ +title: Linearity of Integration +cid: cid001218 +primary_alias: THM0053 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The τ-integral is linear: ∫_k (af + bg) = a ∫_k f + b ∫_k g. Verified computationally for arbitrary coefficients + at stage 2.' +aliases: +- THM0053 +- I.T51 +- linearity-of-integration +- thm:integral-linearity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005273 +- predicate: formalized_by + object: cid005274 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part 18, Chapter 84 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Linearity of Integration** + + + The τ-integral is linear: ∫_k (af + bg) = a ∫_k f + b ∫_k g. Verified computationally for arbitrary coefficients at stage + 2. + + + # Linearity of Integration + + + ## Summary + + + The τ-integral is linear: ∫_k (af + bg) = a ∫_k f + b ∫_k g. Verified computationally for arbitrary coefficients at stage + 2. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 228 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookI.Boundary.Integration` + + - Name: `linearity_2f_3g_stage2` + + + ## Dependencies + + + - Canonical: I.D99 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The τ-integral is linear: ∫_k (af + bg) = a ∫_k f + b ∫_k g. Verified computationally for arbitrary coefficients + at stage 2.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001220.md b/_corpus_v3_items/cid001220.md new file mode 100644 index 000000000..b8cdb7e86 --- /dev/null +++ b/_corpus_v3_items/cid001220.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001220/ +title: Canonical Normalisation (Hinge 7) +cid: cid001220 +primary_alias: THM0055 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every admissible code c in Code reduces in finitely many witness-preserving tau-kernel rewriting steps ->_NF to + a unique normal form NF(c) in Code^NF. The normalisation map NF: Code -> Code^NF is idempotent, surjective, teq-preserving, + bounded by pass budget k_0(c) in N. Source: Hinge 7 (address-resolution), main.tex Thm 1.1 + section-03-nf-confluence Thm + 3.x.' +aliases: +- THM0055 +- I.T53 +- canonical-normalisation-hinge-7 +- thm:main-normalisation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Canonical Normalisation (Hinge 7)** + + + Every admissible code c in Code reduces in finitely many witness-preserving tau-kernel rewriting steps ->_NF to a unique + normal form NF(c) in Code^NF. The normalisation map NF: Code -> Code^NF is idempotent, surjective, teq-preserving, bounded + by pass budget k_0(c) in N. Source: Hinge 7 (address-resolution), main.tex Thm 1.1 + section-03-nf-confluence Thm 3.x. + + + # Canonical Normalisation (Hinge 7) + + + ## Summary + + + Every admissible code c in Code reduces in finitely many witness-preserving tau-kernel rewriting steps ->_NF to a unique + normal form NF(c) in Code^NF. The normalisation map NF: Code -> Code^NF is idempotent, surjective, teq-preserving, bounded + by pass budget k_0(c) in N. Source: Hinge 7 (address-resolution), main.tex Thm 1.1 + section-03-nf-confluence Thm 3.x. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 255 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.Normalisation` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T08 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every admissible code c in Code reduces in finitely many witness-preserving tau-kernel rewriting steps ->_NF + to a unique normal form NF(c) in Code^NF. The normalisation map NF: Code -> Code^NF is idempotent, surjective, teq-preserving, + boun' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001221.md b/_corpus_v3_items/cid001221.md new file mode 100644 index 000000000..92d2074f3 --- /dev/null +++ b/_corpus_v3_items/cid001221.md @@ -0,0 +1,137 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001221/ +title: NF Confluence / Church-Rosser for tau-kernel (Hinge 7) +cid: cid001221 +primary_alias: THM0056 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The tau-kernel rewriting system (Code, ->_NF) is confluent (Church-Rosser): for every c in Code and every pair c + ->* c_1, c ->* c_2, there exists a common reduct c'' with c_1 ->* c'' and c_2 ->* c''. Diamond property; every teq-class + has unique NF representative. Proved via strong normalisation + local confluence + Newman''s Lemma. THIS DISCHARGES the + ''modulo Hinge 7 NF confluence'' scope caveats of Hinge 5 Thm 1.8 (pre-Yoneda collapse) and Hinge 6 Thm 1.1 (topos structure) + + Thm 1.3 (circularity resolution). Source: Hinge 7 section-03 Thm 3.x.' +aliases: +- THM0056 +- I.T54 +- nf-confluence-church-rosser-for-tau-kernel-hinge-7 +- thm:main-confluence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**NF Confluence / Church-Rosser for tau-kernel (Hinge 7)** + + + The tau-kernel rewriting system (Code, ->_NF) is confluent (Church-Rosser): for every c in Code and every pair c ->* c_1, + c ->* c_2, there exists a common reduct c'' with c_1 ->* c'' and c_2 ->* c''. Diamond property; every teq-class has unique + NF representative. Proved via strong normalisation + local confluence + Newman''s Lemma. THIS DISCHARGES the ''modulo + Hinge 7 NF confluence'' scope caveats of Hinge 5 Thm 1.8 (pre-Yoneda collapse) and Hinge 6 Thm 1.1 (topos structure) + + Thm 1.3 (circularity resolution). Source: Hinge 7 section-03 Thm 3.x. + + + # NF Confluence / Church-Rosser for tau-kernel (Hinge 7) + + + ## Summary + + + The tau-kernel rewriting system (Code, ->_NF) is confluent (Church-Rosser): for every c in Code and every pair c ->* c_1, + c ->* c_2, there exists a common reduct c'' with c_1 ->* c'' and c_2 ->* c''. Diamond property; every teq-class has unique + NF representative. Proved via strong normalisation + local confluence + Newman''s Lemma. THIS DISCHARGES the ''modulo + Hinge 7 NF confluence'' scope caveats of Hinge 5 Thm 1.8 (pre-Yoneda collapse) and Hinge 6 Thm 1.1 (topos structure) + + Thm 1.3 (circularity resolution). Source: Hinge 7 section-03 Thm 3.x. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 256 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.Confluence` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The tau-kernel rewriting system (Code, ->_NF) is confluent (Church-Rosser): for every c in Code and every pair + c ->* c_1, c ->* c_2, there exists a common reduct c'' with c_1 ->* c'' and c_2 ->* c''. Diamond property; every teq-class + has uniqu' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001222.md b/_corpus_v3_items/cid001222.md new file mode 100644 index 000000000..b02bf07e8 --- /dev/null +++ b/_corpus_v3_items/cid001222.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001222/ +title: Genealogical DAG Structure (Hinge 7) +cid: cid001222 +primary_alias: THM0057 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The pair DAG := (Code, ->_NF) is a countable, strongly-normalising, finite-width directed acyclic graph: acyclic + via well-founded measure, SN bounded by pass budget k_0, finite-width polynomial in primorial M_k, unique root c_empty, + one NF sink per teq-class. Sinks(DAG)/teq = AddrTau. Source: Hinge 7 section-05-dag.' +aliases: +- THM0057 +- I.T55 +- genealogical-dag-structure-hinge-7 +- thm:main-dag +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Genealogical DAG Structure (Hinge 7)** + + + The pair DAG := (Code, ->_NF) is a countable, strongly-normalising, finite-width directed acyclic graph: acyclic via well-founded + measure, SN bounded by pass budget k_0, finite-width polynomial in primorial M_k, unique root c_empty, one NF sink per + teq-class. Sinks(DAG)/teq = AddrTau. Source: Hinge 7 section-05-dag. + + + # Genealogical DAG Structure (Hinge 7) + + + ## Summary + + + The pair DAG := (Code, ->_NF) is a countable, strongly-normalising, finite-width directed acyclic graph: acyclic via well-founded + measure, SN bounded by pass budget k_0, finite-width polynomial in primorial M_k, unique root c_empty, one NF sink per + teq-class. Sinks(DAG)/teq = AddrTau. Source: Hinge 7 section-05-dag. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 257 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.GenealogicalDAG` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53, I.T54 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The pair DAG := (Code, ->_NF) is a countable, strongly-normalising, finite-width directed acyclic graph: acyclic + via well-founded measure, SN bounded by pass budget k_0, finite-width polynomial in primorial M_k, unique root c_empty, + one NF ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001223.md b/_corpus_v3_items/cid001223.md new file mode 100644 index 000000000..4cfa07f90 --- /dev/null +++ b/_corpus_v3_items/cid001223.md @@ -0,0 +1,133 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001223/ +title: Cayley Word Metric (Hinge 7) +cid: cid001223 +primary_alias: THM0058 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Cayley word metric d_Cay on the DAG, defined as minimal n_1 + n_2 over common reducts, is a genuine metric on + AddrTau = Code^NF/teq (reflexive, symmetric, triangle inequality, separation modulo teq). Coincides with Hinge 1''s minimal-pass + distance. Integer-valued; induces the tau-topology on AddrTau. Effective Bourdon dimension bounded by primorial growth. + Source: Hinge 7 section-06-cayley.' +aliases: +- THM0058 +- I.T56 +- cayley-word-metric-hinge-7 +- thm:main-cayley +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Cayley Word Metric (Hinge 7)** + + + The Cayley word metric d_Cay on the DAG, defined as minimal n_1 + n_2 over common reducts, is a genuine metric on AddrTau + = Code^NF/teq (reflexive, symmetric, triangle inequality, separation modulo teq). Coincides with Hinge 1''s minimal-pass + distance. Integer-valued; induces the tau-topology on AddrTau. Effective Bourdon dimension bounded by primorial growth. + Source: Hinge 7 section-06-cayley. + + + # Cayley Word Metric (Hinge 7) + + + ## Summary + + + The Cayley word metric d_Cay on the DAG, defined as minimal n_1 + n_2 over common reducts, is a genuine metric on AddrTau + = Code^NF/teq (reflexive, symmetric, triangle inequality, separation modulo teq). Coincides with Hinge 1''s minimal-pass + distance. Integer-valued; induces the tau-topology on AddrTau. Effective Bourdon dimension bounded by primorial growth. + Source: Hinge 7 section-06-cayley. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 258 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.CayleyMetric` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53, I.T54, I.T55 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Cayley word metric d_Cay on the DAG, defined as minimal n_1 + n_2 over common reducts, is a genuine metric + on AddrTau = Code^NF/teq (reflexive, symmetric, triangle inequality, separation modulo teq). Coincides with Hinge 1's minimal-pas +noindex: false +--- + diff --git a/_corpus_v3_items/cid001224.md b/_corpus_v3_items/cid001224.md new file mode 100644 index 000000000..9c22e48b7 --- /dev/null +++ b/_corpus_v3_items/cid001224.md @@ -0,0 +1,137 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001224/ +title: Ontic Ultrametric (Hinge 7) +cid: cid001224 +primary_alias: THM0059 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Cayley-completion (Ultra, d_inf) = widehat{(AddrTau, d_Cay'')} is a complete ultrametric space: strong triangle + inequality d_inf(a,c) <= max(d_inf(a,b), d_inf(b,c)), non-archimedean, totally disconnected. Matches the Hinge 4 profinite + boundary topology under the primorial ladder (M_k). Ontic (not epistemic) indeterminacy at unstabilised addresses. iota_tau + anchors the scale via sigma-fixed endomorphism orbit. This is the tau-native replacement for Euclidean distance in Books + II-VII. Source: Hinge 7 section-07-ultrametric.' +aliases: +- THM0059 +- I.T57 +- ontic-ultrametric-hinge-7 +- thm:main-ultrametric +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Ontic Ultrametric (Hinge 7)** + + + The Cayley-completion (Ultra, d_inf) = widehat{(AddrTau, d_Cay'')} is a complete ultrametric space: strong triangle inequality + d_inf(a,c) <= max(d_inf(a,b), d_inf(b,c)), non-archimedean, totally disconnected. Matches the Hinge 4 profinite boundary + topology under the primorial ladder (M_k). Ontic (not epistemic) indeterminacy at unstabilised addresses. iota_tau anchors + the scale via sigma-fixed endomorphism orbit. This is the tau-native replacement for Euclidean distance in Books II-VII. + Source: Hinge 7 section-07-ultrametric. + + + # Ontic Ultrametric (Hinge 7) + + + ## Summary + + + The Cayley-completion (Ultra, d_inf) = widehat{(AddrTau, d_Cay'')} is a complete ultrametric space: strong triangle inequality + d_inf(a,c) <= max(d_inf(a,b), d_inf(b,c)), non-archimedean, totally disconnected. Matches the Hinge 4 profinite boundary + topology under the primorial ladder (M_k). Ontic (not epistemic) indeterminacy at unstabilised addresses. iota_tau anchors + the scale via sigma-fixed endomorphism orbit. This is the tau-native replacement for Euclidean distance in Books II-VII. + Source: Hinge 7 section-07-ultrametric. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 259 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.OnticUltrametric` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53, I.T54, I.T55, I.T56 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Cayley-completion (Ultra, d_inf) = widehat{(AddrTau, d_Cay'')} is a complete ultrametric space: strong triangle + inequality d_inf(a,c) <= max(d_inf(a,b), d_inf(b,c)), non-archimedean, totally disconnected. Matches the Hinge 4 profinite + bo' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001225.md b/_corpus_v3_items/cid001225.md new file mode 100644 index 000000000..ad0f12166 --- /dev/null +++ b/_corpus_v3_items/cid001225.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001225/ +title: Address Resolution and Absence of Equations (Hinge 7) +cid: cid001225 +primary_alias: THM0060 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every tau-native equality question a = b reduces to a finite-witness decidable NF comparison NF(a) == NF(b); decidable + in polynomial depth. Consequently, Category tau has NO equations in the classical sense: only codes, canonical addresses, + and the NF-resolution procedure. Classical equational view emerges as extensional shadow via faithful-but-not-full functor + Shad: Cat_tau -> Eq-Alg. Fullness is blocked by DD1-DD4 diagonal discipline (Hinge 5). Source: Hinge 7 section-08-address-resolution.' +aliases: +- THM0060 +- I.T58 +- address-resolution-and-absence-of-equations-hinge-7 +- thm:main-address-resolution +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Address Resolution and Absence of Equations (Hinge 7)** + + + Every tau-native equality question a = b reduces to a finite-witness decidable NF comparison NF(a) == NF(b); decidable + in polynomial depth. Consequently, Category tau has NO equations in the classical sense: only codes, canonical addresses, + and the NF-resolution procedure. Classical equational view emerges as extensional shadow via faithful-but-not-full functor + Shad: Cat_tau -> Eq-Alg. Fullness is blocked by DD1-DD4 diagonal discipline (Hinge 5). Source: Hinge 7 section-08-address-resolution. + + + # Address Resolution and Absence of Equations (Hinge 7) + + + ## Summary + + + Every tau-native equality question a = b reduces to a finite-witness decidable NF comparison NF(a) == NF(b); decidable + in polynomial depth. Consequently, Category tau has NO equations in the classical sense: only codes, canonical addresses, + and the NF-resolution procedure. Classical equational view emerges as extensional shadow via faithful-but-not-full functor + Shad: Cat_tau -> Eq-Alg. Fullness is blocked by DD1-DD4 diagonal discipline (Hinge 5). Source: Hinge 7 section-08-address-resolution. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 260 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.Addressability.AddressResolution` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53, I.T54, I.T55 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every tau-native equality question a = b reduces to a finite-witness decidable NF comparison NF(a) == NF(b); + decidable in polynomial depth. Consequently, Category tau has NO equations in the classical sense: only codes, canonical + addresses,' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001226.md b/_corpus_v3_items/cid001226.md new file mode 100644 index 000000000..5efd1dbe7 --- /dev/null +++ b/_corpus_v3_items/cid001226.md @@ -0,0 +1,133 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001226/ +title: Hinge 7 Integration Tabulation (Bundle Closure) +cid: cid001226 +primary_alias: THM0061 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Hinge 7 is the foundational capstone: (a) discharges modulo-Hinge-7 caveats of Hinges 5-6; (b) supplies canonical-address + framework as coordinate primitive for Hinges 1-4; (c) establishes ontic ultrametric as tau-native metric for Books II-VII; + (d) closes the 7-hinge foundational arc. The Panta Rhei framework now has a closed, self-contained, seven-hinge bundle of + standalone peer-reviewable papers. Source: Hinge 7 main.tex Thm 1.7.' +aliases: +- THM0061 +- I.T59 +- hinge-7-integration-tabulation-bundle-closure +- thm:hinge-integration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hinge 7 Integration Tabulation (Bundle Closure)** + + + Hinge 7 is the foundational capstone: (a) discharges modulo-Hinge-7 caveats of Hinges 5-6; (b) supplies canonical-address + framework as coordinate primitive for Hinges 1-4; (c) establishes ontic ultrametric as tau-native metric for Books II-VII; + (d) closes the 7-hinge foundational arc. The Panta Rhei framework now has a closed, self-contained, seven-hinge bundle + of standalone peer-reviewable papers. Source: Hinge 7 main.tex Thm 1.7. + + + # Hinge 7 Integration Tabulation (Bundle Closure) + + + ## Summary + + + Hinge 7 is the foundational capstone: (a) discharges modulo-Hinge-7 caveats of Hinges 5-6; (b) supplies canonical-address + framework as coordinate primitive for Hinges 1-4; (c) establishes ontic ultrametric as tau-native metric for Books II-VII; + (d) closes the 7-hinge foundational arc. The Panta Rhei framework now has a closed, self-contained, seven-hinge bundle + of standalone peer-reviewable papers. Source: Hinge 7 main.tex Thm 1.7. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 261 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T53, I.T54, I.T55, I.T56, I.T57, I.T58 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Hinge 7 is the foundational capstone: (a) discharges modulo-Hinge-7 caveats of Hinges 5-6; (b) supplies canonical-address + framework as coordinate primitive for Hinges 1-4; (c) establishes ontic ultrametric as tau-native metric for Books II-' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001227.md b/_corpus_v3_items/cid001227.md new file mode 100644 index 000000000..d71f1d1ee --- /dev/null +++ b/_corpus_v3_items/cid001227.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001227/ +title: Ontic Identity Invariance (Hinge 8) +cid: cid001227 +primary_alias: THM0062 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'In the tau-kernel, normalisation to canonical form is unique and path-independent (NF map Norm: Code -> Code^NF). + No admissible construction introduces shadow identities. Ontic identity is invariant under admissible symmetries (kernel-axiom-preserving + tau-endofunctors). Identity slippage is zero at every primorial-ladder stage. Proved via three independent blockings of + diagonal-resonance components L/E/P (KAxiom 5, NF confluence, *-autonomous structure). Source: Hinge 8 (kernel-foundation) + main.tex Thm 1.1 + section-06-ontic-identity Thm 6.1; cites Book I Thm I.T46.' +aliases: +- THM0062 +- I.T60 +- ontic-identity-invariance-hinge-8 +- thm:main-ontic-identity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000023 + locator: Book I, Part None, Chapter None (Hinge 8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Ontic Identity Invariance (Hinge 8)** + + + In the tau-kernel, normalisation to canonical form is unique and path-independent (NF map Norm: Code -> Code^NF). No admissible + construction introduces shadow identities. Ontic identity is invariant under admissible symmetries (kernel-axiom-preserving + tau-endofunctors). Identity slippage is zero at every primorial-ladder stage. Proved via three independent blockings of + diagonal-resonance components L/E/P (KAxiom 5, NF confluence, *-autonomous structure). Source: Hinge 8 (kernel-foundation) + main.tex Thm 1.1 + section-06-ontic-identity Thm 6.1; cites Book I Thm I.T46. + + + # Ontic Identity Invariance (Hinge 8) + + + ## Summary + + + In the tau-kernel, normalisation to canonical form is unique and path-independent (NF map Norm: Code -> Code^NF). No admissible + construction introduces shadow identities. Ontic identity is invariant under admissible symmetries (kernel-axiom-preserving + tau-endofunctors). Identity slippage is zero at every primorial-ladder stage. Proved via three independent blockings of + diagonal-resonance components L/E/P (KAxiom 5, NF confluence, *-autonomous structure). Source: Hinge 8 (kernel-foundation) + main.tex Thm 1.1 + section-06-ontic-identity Thm 6.1; cites Book I Thm I.T46. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-01.jsonl` line 262 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `planned` + + - Module: `TauLib.BookI.KernelFoundation.OnticIdentity` + + - Name: `None` + + + ## Dependencies + + + - Canonical: I.T46, I.T37, I.T39, I.T54 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'In the tau-kernel, normalisation to canonical form is unique and path-independent (NF map Norm: Code -> Code^NF). + No admissible construction introduces shadow identities. Ontic identity is invariant under admissible symmetries (kernel-axiom' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001232.md b/_corpus_v3_items/cid001232.md new file mode 100644 index 000000000..8e1f480c9 --- /dev/null +++ b/_corpus_v3_items/cid001232.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001232/ +title: Holographic Principle +cid: cid001232 +primary_alias: COR0005 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Central Theorem is an exact holographic correspondence: the 1-dimensional boundary data (characters on L) completely + encodes the 3-dimensional interior data (holomorphic functions on tau^3). A proved theorem, not a conjecture.' +aliases: +- COR0005 +- II.C01 +- holographic-principle +- cor:holographic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005395 +- predicate: formalized_by + object: cid005401 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 51 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Holographic Principle**\n\nThe Central Theorem is an exact holographic correspondence: the 1-dimensional boundary\ + \ data (characters on L) completely encodes the 3-dimensional interior data (holomorphic functions on tau^3). A proved\ + \ theorem, not a conjecture.\n\n# Holographic Principle\n\n## Summary\n\nThe Central Theorem is an exact holographic correspondence:\ + \ the 1-dimensional boundary data (characters on L) completely encodes the 3-dimensional interior data (holomorphic functions\ + \ on tau^3). A proved theorem, not a conjecture.\n\n## Statement\n\n```latex\n%\n\\label{cor:holographic}\nThe Central\ + \ Theorem is an \\textbf{exact holographic correspondence}:\nthe $1$-dimensional boundary data\n(characters on the algebraic\ + \ lemniscate $\\mathbb{L}$)\ncompletely encodes the $3$-dimensional interior data\n(holomorphic functions on the fibered\ + \ product $\\tau^3$):\n\\[\n \\dim(\\mathbb{L}) = 1\n \\qquad\\Longleftrightarrow\\qquad\n \\dim(\\tau^3) = 3.\n\ + \\]\nThe information on the boundary \\emph{is}\nthe information in the interior.\nNothing is lost in the passage from\ + \ interior to boundary.\nNothing is added in the passage from boundary to interior.\nThe two descriptions are the \\emph{same\ + \ thing},\nviewed from different vantage points.\n\nExplicitly:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \ + \ Every holomorphic function on~$\\tau^3$\n is uniquely determined by its boundary restriction\n \ + \ to~$\\mathbb{L}$.\n \\item[\\textup{(ii)}]\n Every idempotent-supported character on~$\\mathbb{L}$\n \ + \ extends uniquely to a holomorphic function on~$\\tau^3$.\n \\item[\\textup{(iii)}]\n The passage\ + \ from boundary to interior\n and from interior to boundary\n are inverse operations.\n \\item[\\\ + textup{(iv)}]\n The passage preserves all algebraic structure\n (ring operations, bipolar decomposition,\n\ + \ tower grading, $\\iota_\\tau$-calibration).\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + This is a direct restatement\nof Theorem~\\ref{thm:central-theorem}\nin the language of information content.\nStatements~(i)\ + \ and~(ii) are the two directions\nof the isomorphism;\nstatement~(iii) is the mutual inverse property;\nstatement~(iv)\ + \ is the structure preservation.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 149\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch51-central-theorem.tex` lines 794-832\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.CentralTheorem`\n\ + - Name: `Tau.BookII.CentralTheorem.holographic_check`\n\n## Dependencies\n\n- Canonical: II.T40, I.D18, II.D60\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Central Theorem is an exact holographic correspondence: the 1-dimensional boundary data (characters on L) + completely encodes the 3-dimensional interior data (holomorphic functions on tau^3). A proved theorem, not a conjecture.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001233.md b/_corpus_v3_items/cid001233.md new file mode 100644 index 000000000..be0c6994c --- /dev/null +++ b/_corpus_v3_items/cid001233.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001233/ +title: Uniqueness of Category Tau +cid: cid001233 +primary_alias: COR0006 +type: corollary +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The moduli space of tau^3-structures is a single point: any two realizations of axioms K0-K5 are canonically isomorphic, + preserving the fibered product structure, ABCD chart, holomorphic structure O(tau^3) = A_spec(L), and calibration constants + pi, e, j, iota_tau.' +aliases: +- COR0006 +- II.C02 +- uniqueness-of-category-tau +- cor:uniqueness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005381 +- predicate: formalized_by + object: cid005388 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 52 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Uniqueness of Category Tau**\n\nThe moduli space of tau^3-structures is a single point: any two realizations\ + \ of axioms K0-K5 are canonically isomorphic, preserving the fibered product structure, ABCD chart, holomorphic structure\ + \ O(tau^3) = A_spec(L), and calibration constants pi, e, j, iota_tau.\n\n# Uniqueness of Category Tau\n\n## Summary\n\n\ + The moduli space of tau^3-structures is a single point: any two realizations of axioms K0-K5 are canonically isomorphic,\ + \ preserving the fibered product structure, ABCD chart, holomorphic structure O(tau^3) = A_spec(L), and calibration constants\ + \ pi, e, j, iota_tau.\n\n## Statement\n\n```latex\n%\n\\label{cor:uniqueness}\nThe moduli space of $\\tau^3$-structures\ + \ is a single point:\n\\[\n \\boxed{\\mathcal{M}_{\\tau^3} \\;=\\; \\{\\mathrm{pt}\\}.}\n\\]\nThere are no free parameters.\n\ + Any two realizations of the axioms \\textup{K0--K5}\nare canonically isomorphic,\nand the isomorphism preserves:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(a)}]\n the fibered product structure\n $\\tau^3 = \\tau^1 \\times_f T^2$;\n \ + \ \\item[\\textup{(b)}]\n the ABCD chart~$\\Phi$;\n \\item[\\textup{(c)}]\n the holomorphic structure\n\ + \ $\\mathcal{O}(\\tau^3) \\cong\n A_{\\mathrm{spec}}(\\Lemniscate)$;\n \\item[\\textup{(d)}]\n \ + \ the calibration constants\n $\\pi, e, \\jj, \\iota_\\tau$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\nBy the Categoricity Theorem\n(Theorem~\\ref{thm:categoricity}, II.T42),\nevery realization $(M, \\Phi_M)$\n\ + satisfying K0--K5 is canonically isomorphic to~$\\tau^3$.\nThe moduli space therefore contains\nexactly one isomorphism\ + \ class.\nThe preservation of structure (a)--(d)\nwas established in the proof\nof Theorem~\\ref{thm:categoricity}.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 153\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch52-liouville-categoricity.tex`\ + \ lines 740-765\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.Categoricity`\n\ + - Name: `Tau.BookII.CentralTheorem.uniqueness_check`\n\n## Dependencies\n\n- Canonical: II.T42, II.D61\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The moduli space of tau^3-structures is a single point: any two realizations of axioms K0-K5 are canonically + isomorphic, preserving the fibered product structure, ABCD chart, holomorphic structure O(tau^3) = A_spec(L), and calibration + const' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001234.md b/_corpus_v3_items/cid001234.md new file mode 100644 index 000000000..dcb7c246a --- /dev/null +++ b/_corpus_v3_items/cid001234.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001234/ +title: Split-Complex Codomain H_tau +cid: cid001234 +primary_alias: DEF0107 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Split-Complex Codomain H_tau +aliases: +- DEF0107 +- II.D01 +- split-complex-codomain-h-tau +- def:split-complex-codomain +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 0, Chapter 2 (Prologue) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Split-Complex Codomain H_tau**\n\nSplit-Complex Codomain H_tau\n\n# Split-Complex Codomain H_tau\n\n## Summary\n\ + \nSplit-Complex Codomain H_tau\n\n## Statement\n\n```latex\n%\n\\label{def:split-complex-codomain}\nThe \\textbf{split-complex\ + \ codomain} is the split-complex\nextension of the boundary ring:\n\\[\n H_\\tau\n \\;:=\\;\n \\bigl\\{\\, a\ + \ + b\\,j \\;:\\; a, b \\in A_\\tau \\,\\bigr\\},\n \\qquad j^2 = +1,\n\\]\nwhere $A_\\tau = \\hat{\\mathbb{Z}}_\\\ + tau$ is the boundary ring\n(I.D28, Book~I).\nEquivalently, via the canonical idempotent decomposition,\n\\[\n H_\\\ + tau\n \\;\\cong\\;\n A_\\tau^{(+)} \\times A_\\tau^{(-)}\n\\]\nwhere $A_\\tau^{(+)} = e_+ \\cdot H_\\tau$\nand $A_\\\ + tau^{(-)} = e_- \\cdot H_\\tau$\nare the two sector components,\neach isomorphic to $A_\\tau$ as a ring.\n\n$H_\\tau$\ + \ is the scalar codomain for\n\\textbf{all} holomorphic functions in Book~II.\nEvery $\\tau$-holomorphic function\n$f\ + \ \\colon \\tau^3 \\to H_\\tau$\ntakes values in this split-complex algebra,\nand the idempotent decomposition\n$f = e_+\ + \ f_+ + e_- f_-$\nprovides the canonical sector components\nthat the Central Theorem relates to the\nspectral algebra\ + \ of $\\mathbb{L}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n\ + ## Source Context\n\n- Registry source: `book-02.jsonl` line 3\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part00/ch02-elliptic-to-split-complex.tex`\ + \ lines 329-363\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Prologue.SplitComplexInterior`\n\ + - Name: `Tau.BookII.Prologue.HTau`\n\n## Dependencies\n\n- Canonical: I.D20, I.D19\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Split-Complex Codomain H_tau +noindex: false +--- + diff --git a/_corpus_v3_items/cid001235.md b/_corpus_v3_items/cid001235.md new file mode 100644 index 000000000..d5473560b --- /dev/null +++ b/_corpus_v3_items/cid001235.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001235/ +title: Tau-Admissible Point +cid: cid001235 +primary_alias: DEF0108 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Tau-Admissible Point +aliases: +- DEF0108 +- II.D02 +- tau-admissible-point +- def:tau-admissible-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Admissible Point**\n\nTau-Admissible Point\n\n# Tau-Admissible Point\n\n## Summary\n\nTau-Admissible Point\n\ + \n## Statement\n\n```latex\n%\n\\label{def:tau-admissible-point}\nA \\textbf{$\\tau$-admissible point} is a quadruple\n\ + $(A, B, C, D) \\in \\tau\\text{-Idx}^4$\nsatisfying all five constraints of the\nconstraint lattice (Definition~\\ref{def:constraint-lattice}).\n\ + Write\n\\[\n \\boxed{\\tau^3_{\\mathrm{fin}}\n \\;:=\\;\n \\bigl\\{\\, (A, B, C, D) \\in \\tau\\text{-Idx}^4\n\ + \ \\;\\big|\\;\n (A, B, C, D) \\text{ is } \\tau\\text{-admissible} \\,\\bigr\\}}\n\\]\nfor the set of all $\\tau$-admissible\ + \ points\nwith finite coordinates.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 5\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch04-tau-admissible-points.tex`\ + \ lines 205-222\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.TauAdmissible`\n\ + - Name: `Tau.BookII.Interior.TauAdmissiblePoint`\n\n## Dependencies\n\n- Canonical: I.D17, I.T04\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Tau-Admissible Point +noindex: false +--- + diff --git a/_corpus_v3_items/cid001236.md b/_corpus_v3_items/cid001236.md new file mode 100644 index 000000000..6fe9aee13 --- /dev/null +++ b/_corpus_v3_items/cid001236.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001236/ +title: Constraint Lattice +cid: cid001236 +primary_alias: DEF0109 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The five constraints (prime, non-negativity, remainder, primorial, depth) on ABCD quadruples that define tau-admissible + points. The conjunction forms a lattice whose meet is tau-admissibility. +aliases: +- DEF0109 +- II.D03 +- constraint-lattice +- def:constraint-lattice +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Constraint Lattice**\n\nThe five constraints (prime, non-negativity, remainder, primorial, depth) on ABCD quadruples\ + \ that define tau-admissible points. The conjunction forms a lattice whose meet is tau-admissibility.\n\n# Constraint\ + \ Lattice\n\n## Summary\n\nThe five constraints (prime, non-negativity, remainder, primorial, depth) on ABCD quadruples\ + \ that define tau-admissible points. The conjunction forms a lattice whose meet is tau-admissibility.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:constraint-lattice}\nThe \\textbf{constraint lattice} is the conjunction\nof the following\ + \ five conditions on a quadruple\n$(A, B, C, D) \\in \\tau\\text{-Idx}^4$:\n\\begin{enumerate}\n \\item[\\textbf{(C1)}]\ + \ \\textbf{Prime constraint.}\n $A \\in \\mathbb{P}_\\tau \\cup \\{1\\}$.\n That is, $A$ is either an\ + \ internal $\\tau$-prime\n or the degenerate value~$1$.\n\n \\item[\\textbf{(C2)}] \\textbf{Non-negativity.}\n\ + \ $B \\geq 0$, $C \\geq 0$, $D \\geq 0$.\n\n \\item[\\textbf{(C3)}] \\textbf{Remainder constraint.}\n \ + \ Every prime factor of $D$ is strictly less than~$A$.\n Equivalently: if $p \\in \\mathbb{P}_\\tau$ and\ + \ $p \\mid D$,\n then $p < A$.\n\n \\item[\\textbf{(C4)}] \\textbf{Tower constraint.}\n If $A = 1$,\ + \ then $B = 0$ and $C = 0$.\n (The degenerate case collapses the tower.)\n\n \\item[\\textbf{(C5)}] \\textbf{Normalization.}\n\ + \ If $B \\geq 1$ and $C \\geq 1$,\n then the triple $(A, B, C)$ is the output\n of the greedy\ + \ peel-off algorithm\n applied to the tower part.\n That is, the exponent~$B$ is maximal\n \ + \ and the tetration height~$C$ is the result\n of the greedy extraction ---\n no under-extraction is\ + \ permitted.\n\\end{enumerate}\nA quadruple satisfying \\textbf{(C1)--(C5)} is called\n\\textbf{$\\tau$-admissible}.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 6\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch04-tau-admissible-points.tex`\ + \ lines 126-162\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.TauAdmissible`\n\ + - Name: `Tau.BookII.Interior.is_tau_admissible`\n\n## Dependencies\n\n- Canonical: I.D17, I.P08\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The five constraints (prime, non-negativity, remainder, primorial, depth) on ABCD quadruples that define tau-admissible + points. The conjunction forms a lattice whose meet is tau-admissibility. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001237.md b/_corpus_v3_items/cid001237.md new file mode 100644 index 000000000..82e81ced0 --- /dev/null +++ b/_corpus_v3_items/cid001237.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001237/ +title: Omega Readout +cid: cid001237 +primary_alias: DEF0110 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The map Phi_omega that assigns to each path approaching omega its base limit and its fiber limit on L. Extracts the + B/C fiber dominance from paths of unbounded primorial depth. +aliases: +- DEF0110 +- II.D04 +- omega-readout +- def:omega-readout +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega Readout**\n\nThe map Phi_omega that assigns to each path approaching omega its base limit and its fiber\ + \ limit on L. Extracts the B/C fiber dominance from paths of unbounded primorial depth.\n\n# Omega Readout\n\n## Summary\n\ + \nThe map Phi_omega that assigns to each path approaching omega its base limit and its fiber limit on L. Extracts the\ + \ B/C fiber dominance from paths of unbounded primorial depth.\n\n## Statement\n\n```latex\n%\n\\label{def:omega-readout}\n\ + Let $\\mathcal{P}_\\omega$ denote the set of all paths to $\\omega$\n--- i.e., sequences $(X_m)_{m \\geq 1}$ of $\\tau$-objects\n\ + whose primorial depth grows without bound.\nThe \\textbf{omega readout} is the map\n\\[\n \\Phi_\\omega \\;\\colon\\\ + ; \\mathcal{P}_\\omega\n \\;\\longrightarrow\\;\n \\{(\\Omega, \\Omega)\\} \\times \\mathbb{L}\n\\]\nthat assigns\ + \ to each path approaching $\\omega$\nits base limit $(\\Omega, \\Omega)$ and its fiber limit on~$\\mathbb{L}$.\nConcretely,\ + \ for a path $(X_m)$\nwith ABCD readouts $\\Phi(X_m) = (A_m, B_m, C_m, D_m)$:\n\\begin{enumerate}\n \\item The \\textbf{base\ + \ component} is\n $\\operatorname{pr}_{\\mathrm{base}}(\\Phi_\\omega(X_m))\n = (\\Omega, \\Omega)$\n\ + \ for \\emph{all} paths (universal collapse).\n \\item The \\textbf{fiber component}\n $\\operatorname{pr}_{\\\ + mathrm{fiber}}(\\Phi_\\omega(X_m))\n \\in \\mathbb{L}$\n is determined by the asymptotic dominance\n\ + \ of the B and C coordinates:\n \\[\n \\operatorname{pr}_{\\mathrm{fiber}}(\\Phi_\\omega(X_m))\n\ + \ \\;=\\;\n \\begin{cases}\n e_+\\text{-lobe} & \\text{if } B_m / C_m \\to\ + \ \\infty, \\\\[2pt]\n e_-\\text{-lobe} & \\text{if } C_m / B_m \\to \\infty, \\\\[2pt]\n \ + \ \\omega_{\\mathbb{L}} & \\text{if } B_m / C_m \\to 1\n \\text{ (balanced)},\n \ + \ \\end{cases}\n \\]\n where $\\omega_{\\mathbb{L}}$ is the crossing-point germ\n (I.D18,\ + \ the node of~$\\mathbb{L}$).\n\\end{enumerate}\nThe omega readout factors through the canonical projections:\nthe base\ + \ projection $\\operatorname{pr}_{\\mathrm{base}} \\circ \\Phi_\\omega$\nis constant (always $(\\Omega, \\Omega)$),\n\ + while the fiber projection\n$\\operatorname{pr}_{\\mathrm{fiber}} \\circ \\Phi_\\omega$\nis path-dependent and traces\ + \ out all of~$\\mathbb{L}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 8\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch05-omega-readout-lemniscate.tex`\ + \ lines 303-348\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.OmegaReadout`\n\ + - Name: `Tau.BookII.Interior.FiberDominance`\n\n## Dependencies\n\n- Canonical: II.D02, I.D17\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The map Phi_omega that assigns to each path approaching omega its base limit and its fiber limit on L. Extracts + the B/C fiber dominance from paths of unbounded primorial depth. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001238.md b/_corpus_v3_items/cid001238.md new file mode 100644 index 000000000..bbe24edb9 --- /dev/null +++ b/_corpus_v3_items/cid001238.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001238/ +title: Base tau^1 +cid: cid001238 +primary_alias: DEF0111 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The base of the tau^3 fibration: the two-dimensional space tau^1 = {(D, A)} where A is a tau-prime and prime factors + of D are strictly less than A. Carries the radial and address coordinates.' +aliases: +- DEF0111 +- II.D05 +- base-tau-1 +- def:base-tau1 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Base tau^1**\n\nThe base of the tau^3 fibration: the two-dimensional space tau^1 = {(D, A)} where A is a tau-prime\ + \ and prime factors of D are strictly less than A. Carries the radial and address coordinates.\n\n# Base tau^1\n\n## Summary\n\ + \nThe base of the tau^3 fibration: the two-dimensional space tau^1 = {(D, A)} where A is a tau-prime and prime factors\ + \ of D are strictly less than A. Carries the radial and address coordinates.\n\n## Statement\n\n```latex\n%\n\\label{def:base-tau1}\n\ + The \\textbf{base} of the $\\tau^3$ fibration is\nthe two-dimensional space\n\\[\n \\tau^1\n \\;:=\\;\n \\bigl\\\ + {\\, (D, A) \\;:\\;\n D \\in \\mathbb{N}_{\\geq 1},\\;\n A \\in \\mathbb{P}_\\tau,\\;\n \\text{prime factors\ + \ of } D \\text{ are } < A\n \\,\\bigr\\},\n\\]\nwhere $\\mathbb{P}_\\tau$ denotes the internal primes\nof $\\tau$-Idx\n\ + (Book~I, Part~III).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n\ + ## Source Context\n\n- Registry source: `book-02.jsonl` line 11\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch06-tau3-fibration.tex`\ + \ lines 49-66\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.Tau3Fibration`\n\ + - Name: `Tau.BookII.Interior.BaseTau1`\n\n## Dependencies\n\n- Canonical: I.D17\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The base of the tau^3 fibration: the two-dimensional space tau^1 = {(D, A)} where A is a tau-prime and prime + factors of D are strictly less than A. Carries the radial and address coordinates.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001239.md b/_corpus_v3_items/cid001239.md new file mode 100644 index 000000000..f0c3a8415 --- /dev/null +++ b/_corpus_v3_items/cid001239.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001239/ +title: Fiber T^2 +cid: cid001239 +primary_alias: DEF0112 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The fiber of the tau^3 fibration: the two-dimensional space T^2 = {(B, C)} where B records the exponent (gamma-orbit) + and C records the tetration height (eta-orbit). Carries the solenoidal coordinates.' +aliases: +- DEF0112 +- II.D06 +- fiber-t-2 +- def:fiber-T2 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fiber T^2**\n\nThe fiber of the tau^3 fibration: the two-dimensional space T^2 = {(B, C)} where B records the\ + \ exponent (gamma-orbit) and C records the tetration height (eta-orbit). Carries the solenoidal coordinates.\n\n# Fiber\ + \ T^2\n\n## Summary\n\nThe fiber of the tau^3 fibration: the two-dimensional space T^2 = {(B, C)} where B records the\ + \ exponent (gamma-orbit) and C records the tetration height (eta-orbit). Carries the solenoidal coordinates.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:fiber-T2}\nThe \\textbf{fiber} of the $\\tau^3$ fibration is\nthe two-dimensional space\n\\\ + [\n T^2\n \\;:=\\;\n \\bigl\\{\\, (B, C) \\;:\\;\n B \\in \\mathbb{N}_{\\geq 0},\\;\n C \\in \\mathbb{N}_{\\\ + geq 0}\n \\,\\bigr\\},\n\\]\nwhere $B$ records the \\textbf{exponent} (the $\\gamma$-orbit:\nhow many times the leading\ + \ prime $A$ appears)\nand $C$ records the \\textbf{tetration height}\n(the $\\eta$-orbit: the depth of iterated exponentiation).\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 12\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch06-tau3-fibration.tex`\ + \ lines 113-130\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.Tau3Fibration`\n\ + - Name: `Tau.BookII.Interior.FiberT2`\n\n## Dependencies\n\n- Canonical: I.D17\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The fiber of the tau^3 fibration: the two-dimensional space T^2 = {(B, C)} where B records the exponent (gamma-orbit) + and C records the tetration height (eta-orbit). Carries the solenoidal coordinates.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001240.md b/_corpus_v3_items/cid001240.md new file mode 100644 index 000000000..7760eef7a --- /dev/null +++ b/_corpus_v3_items/cid001240.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001240/ +title: Fibered Product tau^3 +cid: cid001240 +primary_alias: DEF0113 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Fibered Product tau^3 +aliases: +- DEF0113 +- II.D07 +- fibered-product-tau-3 +- def:fibered-product-tau3 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fibered Product tau^3**\n\nFibered Product tau^3\n\n# Fibered Product tau^3\n\n## Summary\n\nFibered Product\ + \ tau^3\n\n## Statement\n\n```latex\n%\n\\label{def:fibered-product-tau3}\nThe \\textbf{fibered product} is the space\n\ + \\[\n \\tau^3\n \\;:=\\;\n \\tau^1 \\times_f T^2\n \\;=\\;\n \\bigl\\{\\, (D, A, B, C) \\;:\\;\n (D,\ + \ A) \\in \\tau^1,\\;\n (B, C) \\in T^2_{(D,A)}\n \\,\\bigr\\},\n\\]\nwhere $T^2_{(D,A)}$ denotes the fiber over\ + \ the base point $(D, A)$:\nthe set of exponent-tetration pairs $(B, C)$\nthat are $\\tau$-admissible at that base point.\n\ + The projection\n\\[\n \\mathrm{pr} \\colon \\tau^3 \\to \\tau^1,\n \\qquad\n (D, A, B, C) \\;\\mapsto\\; (D,\ + \ A)\n\\]\nforgets the fiber coordinates,\nretaining only the multiplicative skeleton.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 13\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch06-tau3-fibration.tex`\ + \ lines 228-253\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.Tau3Fibration`\n\ + - Name: `Tau.BookII.Interior.Tau3`\n\n## Dependencies\n\n- Canonical: II.D05, II.D06, II.D02\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Fibered Product tau^3 +noindex: false +--- + diff --git a/_corpus_v3_items/cid001241.md b/_corpus_v3_items/cid001241.md new file mode 100644 index 000000000..11bf1f8d3 --- /dev/null +++ b/_corpus_v3_items/cid001241.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001241/ +title: Interior Bipolar Decomposition +cid: cid001241 +primary_alias: DEF0114 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Interior Bipolar Decomposition +aliases: +- DEF0114 +- II.D08 +- interior-bipolar-decomposition +- def:interior-bipolar +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 7 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Interior Bipolar Decomposition**\n\nInterior Bipolar Decomposition\n\n# Interior Bipolar Decomposition\n\n\ + ## Summary\n\nInterior Bipolar Decomposition\n\n## Statement\n\n```latex\n%\n\\label{def:interior-bipolar}\nLet $(D, A,\ + \ B, C) \\in \\tau^3$ be a $\\tau$-admissible point.\nThe \\textbf{interior bipolar decomposition} assigns\nto each such\ + \ point a pair of sector components:\n\\[\n (D, A, B, C)\n \\;\\longmapsto\\;\n \\bigl(\\, s_+(D, A, B, C),\\\ + ;\n s_-(D, A, B, C) \\,\\bigr)\n\\]\nwhere:\n\\begin{enumerate}\n \\item The \\textbf{B-component} $s_+$ is the\ + \ projection\n onto the $e_+$-sector,\n governed by the $\\gamma$-orbit coordinate $B$:\n \\\ + [\n s_+(D, A, B, C) \\;:=\\; e_+ \\cdot \\Psi(B, A, D),\n \\]\n where $\\Psi(B, A, D)$\ + \ is the $\\hat{\\mathbb{Z}}_\\tau$-valued\n encoding of the exponent tower\n at the base point $(D,\ + \ A)$.\n \\item The \\textbf{C-component} $s_-$ is the projection\n onto the $e_-$-sector,\n governed\ + \ by the $\\eta$-orbit coordinate $C$:\n \\[\n s_-(D, A, B, C) \\;:=\\; e_- \\cdot \\Psi(C, A, D).\n\ + \ \\]\n \\item The full interior value reconstructs as\n \\[\n \\Psi_{\\mathrm{int}}(D,\ + \ A, B, C)\n \\;=\\;\n s_+ + s_-\n \\;=\\;\n e_+ \\cdot \\Psi(B, A,\ + \ D) + e_- \\cdot \\Psi(C, A, D)\n \\;\\in\\; H_\\tau.\n \\]\n\\end{enumerate}\nThe decomposition\ + \ is canonical:\n$e_+ \\cdot e_- = 0$ ensures the two components\nare orthogonal, and $e_+ + e_- = 1$ ensures\ncompleteness.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 15\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch07-bipolar-interior.tex`\ + \ lines 110-153\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.BipolarDecomposition`\n\ + - Name: `Tau.BookII.Interior.interior_bipolar`\n\n## Dependencies\n\n- Canonical: I.D21, II.D02\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Interior Bipolar Decomposition +noindex: false +--- + diff --git a/_corpus_v3_items/cid001242.md b/_corpus_v3_items/cid001242.md new file mode 100644 index 000000000..b93440416 --- /dev/null +++ b/_corpus_v3_items/cid001242.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001242/ +title: Cylinder Domain +cid: cid001242 +primary_alias: DEF0115 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A subset of tau^3 expressible as a finite Boolean combination of stage-k cylinders. The cylinder domains form the + Boolean algebra generated by {C_k(x)}. +aliases: +- DEF0115 +- II.D09 +- cylinder-domain +- def:cylinder-domain +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Cylinder Domain** + + + A subset of tau^3 expressible as a finite Boolean combination of stage-k cylinders. The cylinder domains form the Boolean + algebra generated by {C_k(x)}. + + + # Cylinder Domain + + + ## Summary + + + A subset of tau^3 expressible as a finite Boolean combination of stage-k cylinders. The cylinder domains form the Boolean + algebra generated by {C_k(x)}. + + + ## Statement + + + ```latex + + % + + \label{def:cylinder-domain} + + A \textbf{cylinder domain} in~$\tau^3$ + + is any subset that can be expressed + + as a finite Boolean combination + + (union, intersection, complement) + + of stage-$k$ cylinders. + + That is, the cylinder domains form + + the Boolean algebra generated by + + $\{C_k(x) : k \geq 1,\; x \in \tau^3\}$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 19 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch09-cylinder-domains.tex` lines + 196-207 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Domains.Cylinders` + + - Name: `cylinder_mem` + + + ## Dependencies + + + - Canonical: I.T18, II.D02 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A subset of tau^3 expressible as a finite Boolean combination of stage-k cylinders. The cylinder domains form + the Boolean algebra generated by {C_k(x)}. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001243.md b/_corpus_v3_items/cid001243.md new file mode 100644 index 000000000..b7cb92e5f --- /dev/null +++ b/_corpus_v3_items/cid001243.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001243/ +title: Stage-k Cylinder +cid: cid001243 +primary_alias: DEF0116 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The set C_k(x) of all points agreeing with x modulo the k-th primorial P_k. Equivalently, the preimage of the stage-k + reduction map at x. +aliases: +- DEF0116 +- II.D10 +- stage-k-cylinder +- def:stage-k-cylinder +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Stage-k Cylinder**\n\nThe set C_k(x) of all points agreeing with x modulo the k-th primorial P_k. Equivalently,\ + \ the preimage of the stage-k reduction map at x.\n\n# Stage-k Cylinder\n\n## Summary\n\nThe set C_k(x) of all points\ + \ agreeing with x modulo the k-th primorial P_k. Equivalently, the preimage of the stage-k reduction map at x.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:stage-k-cylinder}\nLet $x \\in \\tau^3$ and let $k \\geq 1$.\nThe \\textbf{stage-$k$ cylinder}\ + \ at~$x$ is\n\\[\n \\boxed{C_k(x)\n \\;:=\\;\n \\bigl\\{\\, y \\in \\tau^3\n \\;\\big|\\;\n \\pi_k(y) =\ + \ \\pi_k(x)\n \\,\\bigr\\}\n \\;=\\;\n \\pi_k^{-1}\\!\\bigl(\\pi_k(x)\\bigr).}\n\\]\nEquivalently:\n$y \\in C_k(x)$\ + \ if and only if $y \\equiv x \\pmod{P_k}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 20\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch09-cylinder-domains.tex`\ + \ lines 114-131\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Cylinders`\n\ + - Name: `CylinderDomain`\n\n## Dependencies\n\n- Canonical: I.T18, II.D02\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The set C_k(x) of all points agreeing with x modulo the k-th primorial P_k. Equivalently, the preimage of the + stage-k reduction map at x. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001244.md b/_corpus_v3_items/cid001244.md new file mode 100644 index 000000000..53e1827aa --- /dev/null +++ b/_corpus_v3_items/cid001244.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001244/ +title: Clopen Basis +cid: cid001244 +primary_alias: DEF0117 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The collection B = {C_k(x)} of all stage-k cylinders. Every element is clopen: both C_k(x) and its complement are + finite unions of basis elements.' +aliases: +- DEF0117 +- II.D11 +- clopen-basis +- def:clopen-basis +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Clopen Basis**\n\nThe collection B = {C_k(x)} of all stage-k cylinders. Every element is clopen: both C_k(x)\ + \ and its complement are finite unions of basis elements.\n\n# Clopen Basis\n\n## Summary\n\nThe collection B = {C_k(x)}\ + \ of all stage-k cylinders. Every element is clopen: both C_k(x) and its complement are finite unions of basis elements.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:clopen-basis}\nThe \\textbf{clopen basis} for~$\\tau^3$ is the collection\n\ + \\[\n \\boxed{\\mathcal{B}\n \\;:=\\;\n \\bigl\\{\\, C_k(x) \\;:\\;\n k \\geq 1,\\; x \\in \\tau^3 \\,\\bigr\\\ + }.}\n\\]\nEvery element of $\\mathcal{B}$ is clopen\nin the sense that both $C_k(x)$\nand $\\tau^3 \\setminus C_k(x)$\n\ + are finite unions of elements of~$\\mathcal{B}$\n(Proposition~\\ref{prop:ch09-complement}).\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 21\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch09-cylinder-domains.tex`\ + \ lines 433-448\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Cylinders`\n\ + - Name: `cylinder_clopen`\n\n## Dependencies\n\n- Canonical: II.D10\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The collection B = {C_k(x)} of all stage-k cylinders. Every element is clopen: both C_k(x) and its complement + are finite unions of basis elements.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001245.md b/_corpus_v3_items/cid001245.md new file mode 100644 index 000000000..9383771b1 --- /dev/null +++ b/_corpus_v3_items/cid001245.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001245/ +title: First Disagreement Depth +cid: cid001245 +primary_alias: DEF0118 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: First Disagreement Depth +aliases: +- DEF0118 +- II.D12 +- first-disagreement-depth +- def:first-disagreement-depth +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**First Disagreement Depth**\n\nFirst Disagreement Depth\n\n# First Disagreement Depth\n\n## Summary\n\nFirst\ + \ Disagreement Depth\n\n## Statement\n\n```latex\n%\n\\label{def:first-disagreement-depth}\nFor $x, y \\in \\tau^3$, the\ + \ \\textbf{first disagreement depth}\nis the function\n\\[\n \\boxed{\\delta(x, y)\n \\;:=\\;\n \\max\\bigl\\\ + {\\, k \\geq 0\n \\;\\big|\\;\n \\pi_k(x) = \\pi_k(y) \\,\\bigr\\}}\n\\]\nwith the conventions:\n\\begin{itemize}\n\ + \ \\item If $x = y$, then $\\pi_k(x) = \\pi_k(y)$\n for all~$k$, and we set $\\delta(x,y) = \\infty$.\n \ + \ \\item If $x \\neq y$, then by the CRT separation property\n (I.T04, Book~I),\n there exists a finite\ + \ stage~$k_0$\n such that $\\pi_{k_0}(x) \\neq \\pi_{k_0}(y)$.\n The supremum is then finite:\n \ + \ $\\delta(x,y) \\in \\mathbb{N}_{\\geq 0}$.\n\\end{itemize}\nEquivalently, $\\delta(x,y)$ is the \\emph{last} stage\n\ + at which $x$ and~$y$ have identical CRT reductions---the\ndepth of the deepest cylinder that contains both.\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 23\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch10-ultrametric-depth.tex`\ + \ lines 96-122\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Ultrametric`\n\ + - Name: `disagree_depth`\n\n## Dependencies\n\n- Canonical: II.D10, I.T18\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: First Disagreement Depth +noindex: false +--- + diff --git a/_corpus_v3_items/cid001246.md b/_corpus_v3_items/cid001246.md new file mode 100644 index 000000000..3ecb2e3e4 --- /dev/null +++ b/_corpus_v3_items/cid001246.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001246/ +title: Ultrametric Distance +cid: cid001246 +primary_alias: DEF0119 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Ultrametric Distance +aliases: +- DEF0119 +- II.D13 +- ultrametric-distance +- def:ultrametric-distance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ultrametric Distance**\n\nUltrametric Distance\n\n# Ultrametric Distance\n\n## Summary\n\nUltrametric Distance\n\ + \n## Statement\n\n```latex\n%\n\\label{def:ultrametric-distance}\nThe \\textbf{ultrametric distance} on~$\\tau^3$\nis\ + \ the function $d : \\tau^3 \\times \\tau^3 \\to [0,1]$\ndefined by\n\\[\n \\boxed{d(x, y)\n \\;:=\\;\n 2^{-\\\ + delta(x,y)}}\n\\]\nwith the convention $2^{-\\infty} = 0$.\nExplicitly:\n\\begin{align}\n d(x, y) &= 0 &\\quad&\\text{if\ + \ } x = y, \\label{eq:ch10-d-zero}\\\\\n d(x, y) &= 2^{-k} &\\quad&\\text{if } \\delta(x,y) = k < \\infty.\n \\\ + label{eq:ch10-d-finite}\n\\end{align}\nThe distance takes values in the discrete set\n$\\{0\\} \\cup \\{2^{-k} : k \\\ + geq 0\\} = \\{0, 1, \\tfrac{1}{2}, \\tfrac{1}{4}, \\tfrac{1}{8}, \\ldots\\}$.\n```\n\n## Proof / Justification\n\nThis\ + \ item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line\ + \ 24\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch10-ultrametric-depth.tex`\ + \ lines 158-178\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Ultrametric`\n\ + - Name: `ultra_dist`\n\n## Dependencies\n\n- Canonical: II.D12\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Ultrametric Distance +noindex: false +--- + diff --git a/_corpus_v3_items/cid001247.md b/_corpus_v3_items/cid001247.md new file mode 100644 index 000000000..dfe4a23b9 --- /dev/null +++ b/_corpus_v3_items/cid001247.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001247/ +title: Stone Space +cid: cid001247 +primary_alias: DEF0120 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A topological space that is compact, Hausdorff, and totally disconnected; equivalently, a profinite space. tau^3 + is a Stone space by II.T07-T09, with clopen basis from the cylinder topology. +aliases: +- DEF0120 +- II.D14 +- stone-space +- def:stone-space +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Stone Space**\n\nA topological space that is compact, Hausdorff, and totally disconnected; equivalently, a\ + \ profinite space. tau^3 is a Stone space by II.T07-T09, with clopen basis from the cylinder topology.\n\n# Stone Space\n\ + \n## Summary\n\nA topological space that is compact, Hausdorff, and totally disconnected; equivalently, a profinite space.\ + \ tau^3 is a Stone space by II.T07-T09, with clopen basis from the cylinder topology.\n\n## Statement\n\n```latex\n%\n\ + \\label{def:stone-space}\nA \\textbf{Stone space} is a topological space that is\ncompact, Hausdorff, and totally disconnected.\n\ + Equivalently, a Stone space is a compact Hausdorff space\nwith a basis of clopen sets.\nEquivalently, a Stone space is\ + \ a profinite space:\nan inverse limit of finite discrete spaces.\n\nThe space~$\\tau^3$ is a Stone space:\n\\[\n \\\ + boxed{\\tau^3 \\;\\text{is compact}\n \\;\\;\\textup{(II.T07)},\\quad\n \\text{Hausdorff}\n \\;\\;\\textup{(II.T08)},\\\ + quad\n \\text{totally disconnected}\n \\;\\;\\textup{(II.T09)}.}\n\\]\n```\n\n## Proof / Justification\n\nThis item\ + \ is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 30\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch13-stone-space.tex` lines 420-439\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.StoneSpace`\n- Name:\ + \ `StoneWitness`\n\n## Dependencies\n\n- Canonical: II.T07, II.T08, II.T09\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A topological space that is compact, Hausdorff, and totally disconnected; equivalently, a profinite space. tau^3 + is a Stone space by II.T07-T09, with clopen basis from the cylinder topology. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001248.md b/_corpus_v3_items/cid001248.md new file mode 100644 index 000000000..4dcfe3a00 --- /dev/null +++ b/_corpus_v3_items/cid001248.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001248/ +title: Tau-Dimension +cid: cid001248 +primary_alias: DEF0121 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The minimal number of independent refinement rays whose joint cylinders separate all points of tau^3. Replaces classical + covering dimension (which yields zero for profinite spaces). +aliases: +- DEF0121 +- II.D15 +- tau-dimension +- def:tau-dimension +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tau-Dimension** + + + The minimal number of independent refinement rays whose joint cylinders separate all points of tau^3. Replaces classical + covering dimension (which yields zero for profinite spaces). + + + # Tau-Dimension + + + ## Summary + + + The minimal number of independent refinement rays whose joint cylinders separate all points of tau^3. Replaces classical + covering dimension (which yields zero for profinite spaces). + + + ## Statement + + + ```latex + + % + + \label{def:tau-dimension} + + The \textbf{$\tau$-dimension} of~$\tau^3$ is + + \[ + + \boxed{ + + \begin{aligned} + + \dim_\tau \;:=\; + + \min\bigl\{\, r \in \mathbb{N} \;\big|\;& + + \text{there exist } r \text{ independent refinement rays} \\ + + &\text{whose joint cylinders separate all points} + + \,\bigr\}. + + \end{aligned} + + } + + \] + + ``Separate all points'''': for any $x \neq y$ in~$\tau^3$, + + there exists a stage~$k$ at which the joint constraint + + along the selected rays distinguishes~$x$ from~$y$. + + ``Independent'''': no ray''s decomposition + + is determined by the remaining $r - 1$ rays. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 36 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch15-dimension-four.tex` lines 146-166 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Topology.DimensionFour` + + - Name: `tau_dim` + + + ## Dependencies + + + - Canonical: II.D10, I.P08 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The minimal number of independent refinement rays whose joint cylinders separate all points of tau^3. Replaces + classical covering dimension (which yields zero for profinite spaces). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001249.md b/_corpus_v3_items/cid001249.md new file mode 100644 index 000000000..05d48e03b --- /dev/null +++ b/_corpus_v3_items/cid001249.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001249/ +title: Radial-Solenoidal Split +cid: cid001249 +primary_alias: DEF0122 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Radial-Solenoidal Split +aliases: +- DEF0122 +- II.D16 +- radial-solenoidal-split +- def:radial-solenoidal +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Radial-Solenoidal Split**\n\nRadial-Solenoidal Split\n\n# Radial-Solenoidal Split\n\n## Summary\n\nRadial-Solenoidal\ + \ Split\n\n## Statement\n\n```latex\n%\n\\label{def:radial-solenoidal}\nThe \\textbf{radial--solenoidal split} of $\\\ + dim_\\tau = 4$ is\n\\[\n \\boxed{4 \\;=\\; \\underbrace{1}_{\\text{radial (D)}}\n \\;+\\;\n \\underbrace{3}_{\\\ + text{solenoidal (A, B, C)}}.}\n\\]\nThe \\textbf{radial} coordinate~$D$\nmeasures depth in the primorial hierarchy ---\n\ + the remainder after peeling away the tower.\nThe \\textbf{solenoidal} coordinates $A$, $B$, $C$\nparametrize the tower\n\ + (leading prime, exponent, tetration height)\nand wind around the primorial cycles.\n```\n\n## Proof / Justification\n\n\ + This item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 38\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch15-dimension-four.tex`\ + \ lines 229-245\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.DimensionFour`\n\ + - Name: `radial_solenoidal_check`\n\n## Dependencies\n\n- Canonical: II.T11\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Radial-Solenoidal Split +noindex: false +--- + diff --git a/_corpus_v3_items/cid001250.md b/_corpus_v3_items/cid001250.md new file mode 100644 index 000000000..f59350fbf --- /dev/null +++ b/_corpus_v3_items/cid001250.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001250/ +title: Angular Sectors +cid: cid001250 +primary_alias: DEF0123 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Clopen subsets of tau^3 defined by B- or C-coordinate constraints at a fixed stage k. B-sectors fix the exponent + mod p_k; C-sectors fix the tetration mod p_k. The two are independent. +aliases: +- DEF0123 +- II.D17 +- angular-sectors +- def:angular-sectors +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Angular Sectors**\n\nClopen subsets of tau^3 defined by B- or C-coordinate constraints at a fixed stage k.\ + \ B-sectors fix the exponent mod p_k; C-sectors fix the tetration mod p_k. The two are independent.\n\n# Angular Sectors\n\ + \n## Summary\n\nClopen subsets of tau^3 defined by B- or C-coordinate constraints at a fixed stage k. B-sectors fix the\ + \ exponent mod p_k; C-sectors fix the tetration mod p_k. The two are independent.\n\n## Statement\n\n```latex\n%\n\\label{def:angular-sectors}\n\ + Let $k \\geq 1$ be a stage index.\nAn \\textbf{angular sector} of~$\\tau^3$ at stage~$k$\nis a clopen subset defined by\ + \ a solenoidal fiber constraint:\n\\begin{enumerate}\n \\item[\\textup{(i)}] The \\textbf{B-sector}\n at stage~$k$\ + \ with value $b \\in \\mathbb{Z}/p_k$:\n \\[\n \\boxed{S^+_k(b)\n \\;:=\\;\n \ + \ \\bigl\\{\\, (D, A, B, C) \\in \\tau^3\n \\;\\big|\\;\n B_k \\equiv b \\pmod{p_k}\n\ + \ \\,\\bigr\\}.}\n \\]\n \\item[\\textup{(ii)}] The \\textbf{C-sector}\n at stage~$k$\ + \ with value $c \\in \\mathbb{Z}/p_k$:\n \\[\n \\boxed{S^-_k(c)\n \\;:=\\;\n \ + \ \\bigl\\{\\, (D, A, B, C) \\in \\tau^3\n \\;\\big|\\;\n C_k \\equiv c \\pmod{p_k}\n\ + \ \\,\\bigr\\}.}\n \\]\n \\item[\\textup{(iii)}] A \\textbf{combined angular sector}\n \ + \ is an intersection $S^+_k(b) \\cap S^-_k(c)$.\n\\end{enumerate}\nAt the boundary $\\omega$,\nB-sectors degenerate\ + \ to the $e_+$-lobe;\nC-sectors degenerate to the $e_-$-lobe.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 40\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch16-boundary-minimality.tex` lines 174-207\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.BoundaryMinimality`\n- Name:\ + \ `angular_b_sector`\n\n## Dependencies\n\n- Canonical: II.D10, I.D21\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Clopen subsets of tau^3 defined by B- or C-coordinate constraints at a fixed stage k. B-sectors fix the exponent + mod p_k; C-sectors fix the tetration mod p_k. The two are independent. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001251.md b/_corpus_v3_items/cid001251.md new file mode 100644 index 000000000..358566828 --- /dev/null +++ b/_corpus_v3_items/cid001251.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001251/ +title: Pinch Map +cid: cid001251 +primary_alias: DEF0124 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The surjection p: T^2 -> L collapsing the diagonal circle to the wedge point. The canonical degeneration of torus + to lemniscate.' +aliases: +- DEF0124 +- II.D18 +- pinch-map +- def:pinch-map +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Pinch Map**\n\nThe surjection p: T^2 -> L collapsing the diagonal circle to the wedge point. The canonical\ + \ degeneration of torus to lemniscate.\n\n# Pinch Map\n\n## Summary\n\nThe surjection p: T^2 -> L collapsing the diagonal\ + \ circle to the wedge point. The canonical degeneration of torus to lemniscate.\n\n## Statement\n\n```latex\n%\n\\label{def:pinch-map}\n\ + The \\textbf{pinch map} is the surjection\n\\[\n \\boxed{p \\;:\\; T^2 = S^1_\\gamma \\times S^1_\\eta\n \\;\\longrightarrow\\\ + ;\n \\mathbb{L} = S^1 \\vee S^1}\n\\]\ndefined by collapsing the \\textbf{diagonal circle}\n$\\Delta := \\{(\\theta,\ + \ \\theta) : \\theta \\in S^1\\} \\subset T^2$\nto a single point~$\\ast$, the \\textbf{wedge point}:\n\\[\n p(\\theta_\\\ + gamma, \\theta_\\eta)\n \\;=\\;\n \\begin{cases}\n \\ast\n & \\text{if } \\theta_\\gamma = \\theta_\\\ + eta, \\\\[3pt]\n [\\theta_\\gamma, \\theta_\\eta]\n & \\text{if } \\theta_\\gamma \\neq \\theta_\\eta,\n\ + \ \\end{cases}\n\\]\nwhere $[\\theta_\\gamma, \\theta_\\eta]$\ndenotes the equivalence class in $T^2/\\!\\sim$.\nThe\ + \ image $\\mathbb{L} := T^2/\\Delta$\nis the \\textbf{geometric lemniscate}.\n```\n\n## Proof / Justification\n\nThis\ + \ item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line\ + \ 43\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch17-torus-degeneration.tex`\ + \ lines 128-154\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.TorusDegeneration`\n\ + - Name: `pinch_fiber`\n\n## Dependencies\n\n- Canonical: II.D06, I.D18\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The surjection p: T^2 -> L collapsing the diagonal circle to the wedge point. The canonical degeneration of + torus to lemniscate.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001252.md b/_corpus_v3_items/cid001252.md new file mode 100644 index 000000000..89abdf8aa --- /dev/null +++ b/_corpus_v3_items/cid001252.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001252/ +title: Two-Readout Principle +cid: cid001252 +primary_alias: DEF0125 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Two-Readout Principle: topology (fine-grain) and geometry (coarse-grain) are parallel readouts of the coherence + kernel' +aliases: +- DEF0125 +- II.D18a +- two-readout-principle +- def:two-readout-principle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 18 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Two-Readout Principle**\n\nTwo-Readout Principle: topology (fine-grain) and geometry (coarse-grain) are parallel\ + \ readouts of the coherence kernel\n\n# Two-Readout Principle\n\n## Summary\n\nTwo-Readout Principle: topology (fine-grain)\ + \ and geometry (coarse-grain) are parallel readouts of the coherence kernel\n\n## Statement\n\n```latex\n%\n\\label{def:two-readout-principle}\n\ + The \\textbf{two-readout principle} states\nthat the coherence kernel $\\tau_0$\nprovides two parallel readouts\non the\ + \ point set~$\\tau^3$:\n\\begin{enumerate}\n \\item[\\textbf{(F)}] \\textbf{Fine-grain readout (topology).}\n \ + \ Ultrametric cylinder domains\n $\\{C_k(x)\\}$ (II.D10),\n clopen basis (II.T04),\n totally\ + \ disconnected Stone space (II.D14).\n The fine-grain probe is\n \\emph{CRT tower depth refinement}:\n\ + \ increasing stage~$k$ resolves finer\n cylinder partitions.\n\n \\item[\\textbf{(C)}] \\textbf{Coarse-grain\ + \ readout (geometry).}\n Betweenness, congruence, orthogonality---a\n relational structure that does\ + \ \\textbf{not}\n generate the cylinder topology.\n The coarse-grain probe is\n \\emph{incidence\ + \ and distance comparison}:\n relational data visible at any\n fixed finite stage.\n\\end{enumerate}\n\ + These readouts are \\textbf{parallel},\nnot layered:\nthe fine-grain readout does not\nproduce the coarse-grain readout\n\ + by any continuous-path construction.\nBoth are earned from holomorphy\n(the inverted dependency of II.T06),\nwhich itself\ + \ is earned from\nthe coherence kernel.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 45\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch18-connectivity-via-coherence.tex`\ + \ lines 122-158\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.CoherenceConnectivity`\n\ + - Name: `two_readout_check`\n\n## Dependencies\n\n- Canonical: I.D17, I.T04, I.T05, II.D10, II.D13, II.D14, II.T04, II.T06\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Two-Readout Principle: topology (fine-grain) and geometry (coarse-grain) are parallel readouts of the coherence + kernel' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001253.md b/_corpus_v3_items/cid001253.md new file mode 100644 index 000000000..6bb78bae7 --- /dev/null +++ b/_corpus_v3_items/cid001253.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001253/ +title: Spine Address Path +cid: cid001253 +primary_alias: DEF0126 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Spine Address Path: canonical address route between finite-stage points via alpha_1' +aliases: +- DEF0126 +- II.D18b +- spine-address-path +- def:spine-address-path +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 18 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spine Address Path**\n\nSpine Address Path: canonical address route between finite-stage points via alpha_1\n\ + \n# Spine Address Path\n\n## Summary\n\nSpine Address Path: canonical address route between finite-stage points via alpha_1\n\ + \n## Statement\n\n```latex\n%\n\\label{def:spine-address-path}\nLet $X, Y \\in \\tau^3$ be two finite-stage points\nwith\ + \ ABCD addresses\n$(A_X, B_X, C_X, D_X)$\nand $(A_Y, B_Y, C_Y, D_Y)$.\nA \\textbf{spine address path} from $X$ to~$Y$\n\ + is the pair of address chains:\n\\begin{enumerate}\n \\item[\\textbf{(i)}] \\textbf{Descent.}\n $X \\to \\\ + cdots \\to \\alpha_1$:\n the sequence of normal-form reductions\n obtained by successively stripping\n\ + \ the ABCD coordinates of~$X$\n down to the base index $\\alpha_1 = 2$\n (the first element\ + \ of the $\\alpha$-orbit).\n\n \\item[\\textbf{(ii)}] \\textbf{Ascent.}\n $\\alpha_1 \\to \\cdots \\to Y$:\n\ + \ the sequence of address reconstructions\n building up the ABCD coordinates of~$Y$\n from\ + \ $\\alpha_1$.\n\\end{enumerate}\nThe \\textbf{spine address length} is\n$\\ell(X, Y) := \\ell(X) + \\ell(Y)$,\nwhere\ + \ $\\ell(X)$ and $\\ell(Y)$\nare the number of greedy peel steps\nin the descent from $X$ and $Y$\nto~$\\alpha_1$, respectively.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 46\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch18-connectivity-via-coherence.tex`\ + \ lines 243-273\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.CoherenceConnectivity`\n\ + - Name: `spine_address_path`\n\n## Dependencies\n\n- Canonical: I.D17, I.T04, I.T05, II.D14, II.D18a\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Spine Address Path: canonical address route between finite-stage points via alpha_1' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001254.md b/_corpus_v3_items/cid001254.md new file mode 100644 index 000000000..6a23102f6 --- /dev/null +++ b/_corpus_v3_items/cid001254.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001254/ +title: Betweenness Relation +cid: cid001254 +primary_alias: DEF0127 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Betweenness Relation +aliases: +- DEF0127 +- II.D19 +- betweenness-relation +- def:betweenness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 18 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Betweenness Relation**\n\nBetweenness Relation\n\n# Betweenness Relation\n\n## Summary\n\nBetweenness Relation\n\ + \n## Statement\n\n```latex\n%\n\\label{def:betweenness}\nFor points $x, y, z \\in \\tau^3$,\nwe say \\textbf{$y$ is between\ + \ $x$ and~$z$},\nwritten $B(x,y,z)$, if and only if\n\\[\n \\boxed{B(x,y,z)\n \\;\\;:\\Longleftrightarrow\\;\\;\n\ + \ d(x,z)\n \\;=\\;\n \\max\\bigl(\\, d(x,y),\\; d(y,z) \\,\\bigr).}\n\\]\nEquivalently, in terms of the first\ + \ disagreement depth\n$\\delta$ (II.D12):\n\\[\n B(x,y,z)\n \\;\\;\\Longleftrightarrow\\;\\;\n \\delta(x,z)\n\ + \ \\;=\\;\n \\min\\bigl(\\, \\delta(x,y),\\; \\delta(y,z) \\,\\bigr).\n\\]\nIn words: $y$ is between $x$ and~$z$\n\ + when~$y$ agrees with each endpoint\nto at least the depth at which\nthe endpoints diverge from each other.\n```\n\n##\ + \ Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 49\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch18-betweenness.tex`\ + \ lines 83-109\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.Betweenness`\n\ + - Name: `between`\n\n## Dependencies\n\n- Canonical: II.D13, II.T05\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Betweenness Relation +noindex: false +--- + diff --git a/_corpus_v3_items/cid001255.md b/_corpus_v3_items/cid001255.md new file mode 100644 index 000000000..dbbe55789 --- /dev/null +++ b/_corpus_v3_items/cid001255.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001255/ +title: Congruence Relation +cid: cid001255 +primary_alias: DEF0128 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Segment congruence on tau^3 defined by AB ~ CD iff d(A,B) = d(C,D), where d is the canonical ultrametric distance. +aliases: +- DEF0128 +- II.D20 +- congruence-relation +- def:congruence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 19 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Congruence Relation**\n\nSegment congruence on tau^3 defined by AB ~ CD iff d(A,B) = d(C,D), where d is the\ + \ canonical ultrametric distance.\n\n# Congruence Relation\n\n## Summary\n\nSegment congruence on tau^3 defined by AB\ + \ ~ CD iff d(A,B) = d(C,D), where d is the canonical ultrametric distance.\n\n## Statement\n\n```latex\n%\n\\label{def:congruence}\n\ + For points $A, B, C, D \\in \\tau^3$,\nthe segment~$AB$ is \\textbf{congruent} to the segment~$CD$,\nwritten\n\\[\n \ + \ \\boxed{AB \\cong CD\n \\quad:\\!\\iff\\quad\n d(A, B) \\;=\\; d(C, D),}\n\\]\nwhere $d$ is the canonical ultrametric\ + \ distance\n(Definition~\\ref{def:ultrametric-distance}).\nEquivalently, in terms of first disagreement depth:\n\\[\n\ + \ AB \\cong CD\n \\quad\\iff\\quad\n \\delta(A, B) \\;=\\; \\delta(C, D).\n\\]\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 51\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch19-congruence.tex`\ + \ lines 74-93\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.Congruence`\n\ + - Name: `congruent`\n\n## Dependencies\n\n- Canonical: II.D13\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Segment congruence on tau^3 defined by AB ~ CD iff d(A,B) = d(C,D), where d is the canonical ultrametric distance. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001256.md b/_corpus_v3_items/cid001256.md new file mode 100644 index 000000000..ff56cd020 --- /dev/null +++ b/_corpus_v3_items/cid001256.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001256/ +title: Wave-Type PDE +cid: cid001256 +primary_alias: DEF0129 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The split-complex Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is hyperbolic (not elliptic), + with two families of real characteristics. +aliases: +- DEF0129 +- II.D21 +- wave-type-pde +- def:wave-type-pde +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 21 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Wave-Type PDE**\n\nThe split-complex Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is\ + \ hyperbolic (not elliptic), with two families of real characteristics.\n\n# Wave-Type PDE\n\n## Summary\n\nThe split-complex\ + \ Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is hyperbolic (not elliptic), with two families\ + \ of real characteristics.\n\n## Statement\n\n```latex\n%\n\\label{def:wave-type-pde}\nThe split-complex Cauchy--Riemann\ + \ equations\non~$H_\\tau$ yield the \\textbf{wave equation}:\n\\[\n \\boxed{%\n \\frac{\\partial^2 u}{\\partial\ + \ x^2}\n \\;-\\;\n \\frac{\\partial^2 u}{\\partial y^2}\n \\;=\\; 0.}\n\\]\nThe sign between the second-order\ + \ terms is \\textbf{negative}.\nThis PDE is \\textbf{hyperbolic}, not elliptic.\nThe split-complex holomorphic condition\n\ + $\\partial f / \\partial x + \\jj \\cdot \\partial f / \\partial y = 0$\nis equivalent to the wave equation for each component.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 55\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch21-wave-causal.tex`\ + \ lines 141-158\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.CausalStructure`\n\ + - Name: `CausalClass`\n\n## Dependencies\n\n- Canonical: I.D20, I.T10\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The split-complex Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is hyperbolic (not elliptic), + with two families of real characteristics. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001257.md b/_corpus_v3_items/cid001257.md new file mode 100644 index 000000000..250975989 --- /dev/null +++ b/_corpus_v3_items/cid001257.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001257/ +title: Causal Structure +cid: cid001257 +primary_alias: DEF0130 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The partial order on null-line segments of tau^3 determined by the two characteristic families from the wave equation + and the forward direction from B/C asymmetry (Prime Polarity, I.T05). +aliases: +- DEF0130 +- II.D22 +- causal-structure +- def:causal-structure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 21 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Causal Structure**\n\nThe partial order on null-line segments of tau^3 determined by the two characteristic\ + \ families from the wave equation and the forward direction from B/C asymmetry (Prime Polarity, I.T05).\n\n# Causal Structure\n\ + \n## Summary\n\nThe partial order on null-line segments of tau^3 determined by the two characteristic families from the\ + \ wave equation and the forward direction from B/C asymmetry (Prime Polarity, I.T05).\n\n## Statement\n\n```latex\n%\n\ + \\label{def:causal-structure}\nThe \\textbf{causal structure} on~$\\tau^3$ is the\npartial order on null-line segments\ + \ determined by\nthe following data:\n\\begin{enumerate}\n \\item[\\textup{(C1)}] \\textbf{Null directions.}\n \ + \ The two characteristic families\n $\\xi = \\mathrm{const}$ and $\\zeta = \\mathrm{const}$\n from\ + \ the split-complex wave equation (II.D21).\n\n \\item[\\textup{(C2)}] \\textbf{Forward cone.}\n The B/C asymmetry\ + \ from Prime Polarity\n (I.T05, Book~I) provides a preferred direction:\n the $\\gamma$-channel ($B$-coordinate)\n\ + \ and the $\\eta$-channel ($C$-coordinate)\n are structurally ordered by\n the exponent--tetration\ + \ hierarchy.\n $B$ precedes~$C$ in the peel-off sequence.\n This asymmetry selects a \\textbf{forward}\ + \ direction\n within each time-like sector.\n\n \\item[\\textup{(C3)}] \\textbf{Causal ordering.}\n \ + \ A point~$q$ is in the \\textbf{causal future} of~$p$\n if $q$ can be reached from~$p$\n by following\ + \ a curve that remains\n within the forward null cone at every point.\n\\end{enumerate}\nA direction is \\textbf{time-like}\ + \ if it lies\nstrictly inside the null cone,\n\\textbf{space-like} if strictly outside,\nand \\textbf{null} (light-like)\ + \ if on the cone boundary.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 56\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch21-wave-causal.tex`\ + \ lines 265-298\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.CausalStructure`\n\ + - Name: `light_cone_check`\n\n## Dependencies\n\n- Canonical: II.D21\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The partial order on null-line segments of tau^3 determined by the two characteristic families from the wave + equation and the forward direction from B/C asymmetry (Prime Polarity, I.T05). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001258.md b/_corpus_v3_items/cid001258.md new file mode 100644 index 000000000..1b44e5d36 --- /dev/null +++ b/_corpus_v3_items/cid001258.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001258/ +title: Approximation Sequence +cid: cid001258 +primary_alias: DEF0131 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Approximation Sequence +aliases: +- DEF0131 +- II.D23 +- approximation-sequence +- def:approximation-sequence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 22 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Approximation Sequence**\n\nApproximation Sequence\n\n# Approximation Sequence\n\n## Summary\n\nApproximation\ + \ Sequence\n\n## Statement\n\n```latex\n%\n\\label{def:approximation-sequence}\nLet $x \\in \\tau^3$ be a $\\tau$-admissible\ + \ point.\nThe \\textbf{approximation sequence} of~$x$\nis the sequence of normalized stage-$k$ coordinate vectors\n\\\ + [\n \\boxed{%\n \\mathrm{app}_k(x)\n \\;:=\\;\n \\left(\n \\frac{D_k(x)}{P_k},\\;\n \\frac{A_k(x)}{P_k},\\\ + ;\n \\frac{B_k(x)}{P_k},\\;\n \\frac{C_k(x)}{P_k}\n \\right)\n \\;\\in\\; [0, 1)^4\n \\;\\subset\\\ + ; \\mathbb{R}^4,}\n\\]\nwhere $P_k = p_1 \\cdots p_k$ is the $k$th primorial\nand the coordinates $D_k, A_k, B_k, C_k$\n\ + are the ABCD values of the CRT reduction~$\\pi_k(x)$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 59\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch22-orthodox-bridge.tex` lines 134-156\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.OrthodoxBridge`\n- Name: `approx_seq`\n\ + \n## Dependencies\n\n- Canonical: II.D13, II.D14\n\n## Related Results\n\n_Generated by later projection phases._\n\n\ + ## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Approximation Sequence +noindex: false +--- + diff --git a/_corpus_v3_items/cid001259.md b/_corpus_v3_items/cid001259.md new file mode 100644 index 000000000..0793aa753 --- /dev/null +++ b/_corpus_v3_items/cid001259.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001259/ +title: Alpha-Ray Line +cid: cid001259 +primary_alias: DEF0132 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A line in tau^3 along the alpha-ray direction, defined using the prime-indexed coordinate A and the ultrametric distance. + The one-dimensional subspace traced by varying the address coordinate. +aliases: +- DEF0132 +- II.D24 +- alpha-ray-line +- def:alpha-ray +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 23 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Alpha-Ray Line**\n\nA line in tau^3 along the alpha-ray direction, defined using the prime-indexed coordinate\ + \ A and the ultrametric distance. The one-dimensional subspace traced by varying the address coordinate.\n\n# Alpha-Ray\ + \ Line\n\n## Summary\n\nA line in tau^3 along the alpha-ray direction, defined using the prime-indexed coordinate A and\ + \ the ultrametric distance. The one-dimensional subspace traced by varying the address coordinate.\n\n## Statement\n\n\ + ```latex\n%\n\\label{def:alpha-ray}\nThe \\textbf{$\\alpha$-ray} is the subset of~$\\tau^3$\ndefined by\n\\[\n \\boxed{%\n\ + \ \\ell_\\alpha\n \\;:=\\;\n \\bigl\\{\\,\\alpha_n : n \\geq 1\\,\\bigr\\}\n \\;\\cup\\;\n \\bigl\\{\\\ + omega\\bigr\\},}\n\\]\nwhere $\\alpha_n := \\pi_n(\\alpha)$\nis the stage-$n$ CRT reduction of~$\\alpha$\n(I.T18, Book~I),\n\ + and $\\omega$ is the profinite limit point\n(the coherent family exhausting all stages).\n\nThe \\textbf{$\\alpha$-ray\ + \ metric} is the restriction\nof the ultrametric distance\n(Definition~\\ref{def:ultrametric-distance}, II.D13):\n\\[\n\ + \ d(\\alpha_m, \\alpha_n)\n \\;=\\;\n 2^{-\\min(m,n)}\n \\quad\\text{for } m \\neq n,\n \\qquad\n d(\\\ + alpha_n, \\omega)\n \\;=\\;\n 2^{-n}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 60\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch23-lines-countable.tex`\ + \ lines 101-133\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Lines`\n\ + - Name: `alpha_ray`\n\n## Dependencies\n\n- Canonical: I.T04, I.D19, II.D13, II.D14\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A line in tau^3 along the alpha-ray direction, defined using the prime-indexed coordinate A and the ultrametric + distance. The one-dimensional subspace traced by varying the address coordinate. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001260.md b/_corpus_v3_items/cid001260.md new file mode 100644 index 000000000..89a71d5a0 --- /dev/null +++ b/_corpus_v3_items/cid001260.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001260/ +title: Level Circle +cid: cid001260 +primary_alias: DEF0133 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The finite cyclic group at NF stage k in each angular direction. The finite-stage circle whose inverse limit yields + the solenoidal circle. +aliases: +- DEF0133 +- II.D25 +- level-circle +- def:level-circle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 23 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Level Circle**\n\nThe finite cyclic group at NF stage k in each angular direction. The finite-stage circle\ + \ whose inverse limit yields the solenoidal circle.\n\n# Level Circle\n\n## Summary\n\nThe finite cyclic group at NF stage\ + \ k in each angular direction. The finite-stage circle whose inverse limit yields the solenoidal circle.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:level-circle}\nFor each stage $k \\geq 1$\nand each coordinate axis $X \\in \\{A, B, C\\}$,\n\ + the \\textbf{level-$k$ circle in the $X$-direction}\nis the subset\n\\[\n \\boxed{%\n \\Lambda_k^X\n \\;:=\\\ + ;\n \\bigl\\{\\,\n x \\in \\tau^3\n \\;\\big|\\;\n D(x) = k,\\;\n Y(x) \\text{ and } Z(x) \\text{ fixed}\n\ + \ \\,\\bigr\\},}\n\\]\nwhere $\\{X, Y, Z\\} = \\{A, B, C\\}$\nand the residue classes of~$Y$ and~$Z$ are held constant.\n\ + The level circle inherits the cyclic structure\nof $\\mathbb{Z}/p_k\\mathbb{Z}$\n(or more precisely, the CRT factor\n\ + $\\mathbb{Z}/p_j\\mathbb{Z}$\ncorresponding to the $X$-direction at stage~$k$).\n```\n\n## Proof / Justification\n\nThis\ + \ item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line\ + \ 61\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch23-lines-countable.tex`\ + \ lines 197-222\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Lines`\n\ + - Name: `level_circle`\n\n## Dependencies\n\n- Canonical: II.D15, II.T11, I.T18\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The finite cyclic group at NF stage k in each angular direction. The finite-stage circle whose inverse limit + yields the solenoidal circle. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001261.md b/_corpus_v3_items/cid001261.md new file mode 100644 index 000000000..91df0d82e --- /dev/null +++ b/_corpus_v3_items/cid001261.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001261/ +title: Solenoidal Circle +cid: cid001261 +primary_alias: DEF0134 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The inverse limit of finite cyclic groups in each angular direction. A compact, totally disconnected, torsion-free + profinite group whose Archimedean projection is S^1. +aliases: +- DEF0134 +- II.D26 +- solenoidal-circle +- def:solenoidal-circle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 24 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Solenoidal Circle**\n\nThe inverse limit of finite cyclic groups in each angular direction. A compact, totally\ + \ disconnected, torsion-free profinite group whose Archimedean projection is S^1.\n\n# Solenoidal Circle\n\n## Summary\n\ + \nThe inverse limit of finite cyclic groups in each angular direction. A compact, totally disconnected, torsion-free profinite\ + \ group whose Archimedean projection is S^1.\n\n## Statement\n\n```latex\n%\n\\label{def:solenoidal-circle}\nFor each\ + \ coordinate axis $X \\in \\{A, B, C\\}$,\nthe \\textbf{solenoidal circle in the $X$-direction} is\nthe inverse limit\n\ + \\[\n \\boxed{%\n \\mathcal{S}^X\n \\;:=\\;\n \\varprojlim_{k}\\;\n \\mathcal{C}_k^X\n \\;=\\;\n \ + \ \\Bigl\\{\\,\n (c_k)_{k \\geq 1}\n \\;\\Big|\\;\n c_k \\in \\mathcal{C}_k^X,\\;\n \\psi_{k,k+1}^X(c_{k+1})\ + \ = c_k\n \\text{ for all } k\n \\,\\Bigr\\}.}\n\\]\n$\\mathcal{S}^X$ carries:\n\\begin{enumerate}\n \\item[\\\ + textup{(i)}]\n a \\textbf{profinite group structure}:\n componentwise addition modulo the respective\ + \ primes;\n \\item[\\textup{(ii)}]\n an \\textbf{ultrametric}:\n $d\\bigl((c_k), (c_k')\\bigr)\n\ + \ = 2^{-\\min\\{k : c_k \\neq c_k'\\}}$;\n \\item[\\textup{(iii)}]\n the \\textbf{profinite topology}:\n\ + \ compact, Hausdorff, totally disconnected.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is\ + \ definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 63\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch24-circles-solenoidal.tex` lines\ + \ 118-152\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Circles`\n\ + - Name: `solenoid_circle`\n\n## Dependencies\n\n- Canonical: II.D25, I.T04\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The inverse limit of finite cyclic groups in each angular direction. A compact, totally disconnected, torsion-free + profinite group whose Archimedean projection is S^1. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001262.md b/_corpus_v3_items/cid001262.md new file mode 100644 index 000000000..2104f2084 --- /dev/null +++ b/_corpus_v3_items/cid001262.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001262/ +title: Geometric-Topological Unification +cid: cid001262 +primary_alias: DEF0135 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Geometric-Topological Unification +aliases: +- DEF0135 +- II.D27 +- geometric-topological-unification +- def:geo-topo-unification +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 24 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Geometric-Topological Unification**\n\nGeometric-Topological Unification\n\n# Geometric-Topological Unification\n\ + \n## Summary\n\nGeometric-Topological Unification\n\n## Statement\n\n```latex\n%\n\\label{def:geo-topo-unification}\n\ + The \\textbf{geometric-topological unification}\nof the circle in direction~$X$\nis the identification\n\\[\n \\boxed{%\n\ + \ S^1_{\\mathrm{geo}}\n \\;=\\;\n S^1_{\\mathrm{top}}\n \\;=\\;\n \\mathrm{den}^X(\\mathcal{S}^X),}\n\\\ + ]\nwhere:\n\\begin{itemize}\n \\item $S^1_{\\mathrm{geo}}$\n is the set of limit points\n at constant\ + \ ultrametric ``radius''\n from the center of the $X$-direction cylinders\n (constant D-depth, varying\ + \ $X$-angle);\n \\item $S^1_{\\mathrm{top}}$\n is the quotient $\\mathbb{R}/\\mathbb{Z}$\n (the unique\ + \ compact connected $1$-manifold);\n \\item $\\mathrm{den}^X(\\mathcal{S}^X)$\n is the image of the solenoidal\ + \ circle\n under the angular denotation map.\n\\end{itemize}\nAll three are the same object:\nthe Archimedean\ + \ limit of the angular tower\nin direction~$X$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 65\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch24-circles-solenoidal.tex`\ + \ lines 488-519\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Circles`\n\ + - Name: `geo_topo_unify`\n\n## Dependencies\n\n- Canonical: II.D26, II.T21\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Geometric-Topological Unification +noindex: false +--- + diff --git a/_corpus_v3_items/cid001263.md b/_corpus_v3_items/cid001263.md new file mode 100644 index 000000000..6cfce6c09 --- /dev/null +++ b/_corpus_v3_items/cid001263.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001263/ +title: Geometric Pi +cid: cid001263 +primary_alias: DEF0136 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The circumference-to-diameter ratio of the earned circle S^1, defined as the common limit of inscribed and circumscribed + Archimedes polygon perimeters normalized to unit diameter. +aliases: +- DEF0136 +- II.D28 +- geometric-pi +- def:geometric-pi +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 25 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Geometric Pi**\n\nThe circumference-to-diameter ratio of the earned circle S^1, defined as the common limit\ + \ of inscribed and circumscribed Archimedes polygon perimeters normalized to unit diameter.\n\n# Geometric Pi\n\n## Summary\n\ + \nThe circumference-to-diameter ratio of the earned circle S^1, defined as the common limit of inscribed and circumscribed\ + \ Archimedes polygon perimeters normalized to unit diameter.\n\n## Statement\n\n```latex\n%\n\\label{def:geometric-pi}\n\ + The \\textbf{geometric~$\\pi$}\nis the common limit of the inscribed and circumscribed\nArchimedes polygon perimeters,\n\ + normalized to unit diameter:\n\\[\n \\boxed{%\n \\pi_{\\mathrm{geo}}\n \\;:=\\;\n \\lim_{k \\to \\infty}\\\ + ;\n \\frac{P_k^{\\mathrm{in}}}{2}\n \\;=\\;\n \\lim_{k \\to \\infty}\\;\n \\frac{P_k^{\\mathrm{out}}}{2}.}\n\ + \\]\n(The factor $1/2$ normalizes to a circle\nof diameter~$1$, i.e., radius~$1/2$,\nso that circumference equals~$\\\ + pi$.)\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 66\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch25-pi-earned.tex`\ + \ lines 253-273\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.PiEarned`\n\ + - Name: `pi_circumference`\n\n## Dependencies\n\n- Canonical: II.D29, II.D26\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The circumference-to-diameter ratio of the earned circle S^1, defined as the common limit of inscribed and circumscribed + Archimedes polygon perimeters normalized to unit diameter. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001264.md b/_corpus_v3_items/cid001264.md new file mode 100644 index 000000000..c77b02fe1 --- /dev/null +++ b/_corpus_v3_items/cid001264.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001264/ +title: Archimedes Polygon Sequence +cid: cid001264 +primary_alias: DEF0137 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Archimedes Polygon Sequence +aliases: +- DEF0137 +- II.D29 +- archimedes-polygon-sequence +- def:archimedes-polygon +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 25 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Archimedes Polygon Sequence**\n\nArchimedes Polygon Sequence\n\n# Archimedes Polygon Sequence\n\n## Summary\n\ + \nArchimedes Polygon Sequence\n\n## Statement\n\n```latex\n%\n\\label{def:archimedes-polygon}\nThe \\textbf{Archimedes\ + \ polygon sequence}\nin the $X$-direction is the sequence of pairs\n\\[\n \\boxed{%\n \\bigl(\\,P_k^{\\mathrm{in}},\\\ + ; P_k^{\\mathrm{out}}\\,\\bigr)_{k \\geq 1},}\n\\]\nwhere $P_k^{\\mathrm{in}}$\nis the perimeter of the regular inscribed\n\ + $Q_k^X$-gon\nand $P_k^{\\mathrm{out}}$\nis the perimeter of the regular circumscribed\n$Q_k^X$-gon,\nboth computed using\ + \ the earned Euclidean geometry\n(Chapter~\\ref{ch:pasch-parallel})\nand the denotation map\n(Chapter~\\ref{ch:orthodox-bridge}).\n\ + Explicitly:\n\\begin{align*}\n P_k^{\\mathrm{in}}\n &\\;=\\;\n Q_k^X \\cdot 2\\,\\sin\\!\\Bigl(\\frac{\\pi}{Q_k^X}\\\ + Bigr),\n \\\\[4pt]\n P_k^{\\mathrm{out}}\n &\\;=\\;\n Q_k^X \\cdot 2\\,\\tan\\!\\Bigl(\\frac{\\pi}{Q_k^X}\\\ + Bigr).\n\\end{align*}\n(The $\\sin$ and $\\tan$ here denote\nthe Archimedean trigonometric functions\non the denotation-map\ + \ image of the solenoidal circle.)\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 67\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch25-pi-earned.tex`\ + \ lines 191-223\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.PiEarned`\n\ + - Name: `pi_leibniz_scaled`\n\n## Dependencies\n\n- Canonical: II.D26, II.T20\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Archimedes Polygon Sequence +noindex: false +--- + diff --git a/_corpus_v3_items/cid001265.md b/_corpus_v3_items/cid001265.md new file mode 100644 index 000000000..bd4e44c15 --- /dev/null +++ b/_corpus_v3_items/cid001265.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001265/ +title: e as Iterator Eigenvalue +cid: cid001265 +primary_alias: DEF0138 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: e as Iterator Eigenvalue +aliases: +- DEF0138 +- II.D30 +- e-as-iterator-eigenvalue +- def:e-eigenvalue +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 26 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**e as Iterator Eigenvalue**\n\ne as Iterator Eigenvalue\n\n# e as Iterator Eigenvalue\n\n## Summary\n\ne as\ + \ Iterator Eigenvalue\n\n## Statement\n\n```latex\n%\n\\label{def:e-eigenvalue}\nThe \\textbf{$\\nu$-eigenvalue} is the\ + \ Archimedean limit\nof the $\\nu$-iterator's scaling factor:\n\\[\n \\boxed{%\n e_\\nu\n \\;:=\\;\n \\lim_{k\ + \ \\to \\infty}\\,\n \\Bigl(1 + \\frac{1}{p_{k+1}}\\Bigr)^{p_{k+1}}\n \\;\\in\\; \\mathbb{R}.}\n\\]\nSince $p_{k+1}\ + \ \\to \\infty$ as $k \\to \\infty$,\nthis is a subsequence of the classical sequence\n$(1 + 1/n)^n$.\nWe define\n$e :=\ + \ e_\\nu = 2.71828\\ldots$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 69\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch26-e-earned.tex`\ + \ lines 151-169\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.EEarned`\n\ + - Name: `e_factorial_sum_scaled`\n\n## Dependencies\n\n- Canonical: I.D04, I.D06, II.D14, II.D15\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: e as Iterator Eigenvalue +noindex: false +--- + diff --git a/_corpus_v3_items/cid001266.md b/_corpus_v3_items/cid001266.md new file mode 100644 index 000000000..0ffcfda3e --- /dev/null +++ b/_corpus_v3_items/cid001266.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001266/ +title: Growth Base +cid: cid001266 +primary_alias: DEF0139 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The unique positive real number b satisfying the self-reproduction condition with log b = 1. Characterizes e as the + growth rate eigenvalue of the nu-iterator. +aliases: +- DEF0139 +- II.D31 +- growth-base +- def:growth-base +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 26 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Growth Base**\n\nThe unique positive real number b satisfying the self-reproduction condition with log b =\ + \ 1. Characterizes e as the growth rate eigenvalue of the nu-iterator.\n\n# Growth Base\n\n## Summary\n\nThe unique positive\ + \ real number b satisfying the self-reproduction condition with log b = 1. Characterizes e as the growth rate eigenvalue\ + \ of the nu-iterator.\n\n## Statement\n\n```latex\n%\n\\label{def:growth-base}\nA real number $b > 0$ is a \\textbf{growth\ + \ base}\nfor the primorial tower\nif the stage-to-stage refinement factor\n$f_k(b) := b^{1/p_{k+1}}$\nsatisfies the \\\ + textbf{self-reproduction condition}:\n\\[\n \\boxed{%\n \\lim_{k \\to \\infty}\\;\n p_{k+1} \\cdot \\bigl(f_k(b)\ + \ - 1\\bigr)\n \\;=\\;\n \\log b.}\n\\]\nThe constant~$e$ is the unique growth base\nfor which $\\log b = 1$,\n\ + equivalently the unique $b > 0$ satisfying\n\\[\n \\lim_{n \\to \\infty}\\;\n n \\cdot \\bigl(b^{1/n} - 1\\bigr)\n\ + \ \\;=\\; 1.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 70\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch26-e-earned.tex`\ + \ lines 339-362\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.EEarned`\n\ + - Name: `e_compound_check`\n\n## Dependencies\n\n- Canonical: II.D30, II.T23\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The unique positive real number b satisfying the self-reproduction condition with log b = 1. Characterizes e + as the growth rate eigenvalue of the nu-iterator. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001267.md b/_corpus_v3_items/cid001267.md new file mode 100644 index 000000000..ec5e3ab6c --- /dev/null +++ b/_corpus_v3_items/cid001267.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001267/ +title: Interior j-Unit +cid: cid001267 +primary_alias: DEF0140 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates, with j^2 = +1. Unlike + i (rotation), j is an involution (polarity/reflection).' +aliases: +- DEF0140 +- II.D32 +- interior-j-unit +- def:j-unit +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 27 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Interior j-Unit**\n\nThe split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates,\ + \ with j^2 = +1. Unlike i (rotation), j is an involution (polarity/reflection).\n\n# Interior j-Unit\n\n## Summary\n\n\ + The split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates, with j^2 = +1. Unlike i (rotation),\ + \ j is an involution (polarity/reflection).\n\n## Statement\n\n```latex\n%\n\\label{def:j-unit}\nThe \\textbf{split-complex\ + \ unit}~$\\jj$ is the\nautomorphism of the fiber $T^2$\nthat exchanges the B and C coordinates:\n\\[\n \\boxed{%\n\ + \ \\jj \\;:\\;\n (D, A, B, C)\n \\;\\longmapsto\\;\n (D, A, C, B).}\n\\]\nEquivalently, $\\jj$ is the generator\n\ + of the $\\mathbb{Z}/2$ involution\n$\\sigma : T^2 \\to T^2$\nthat swaps the two fiber directions.\nThe defining properties\ + \ are:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n $\\jj^2 = 1$ (the identity),\n \\item[\\textup{(ii)}]\n\ + \ $\\jj \\neq \\pm 1$\n (distinct from the identity and negation),\n \\item[\\textup{(iii)}]\n \ + \ $\\jj$ preserves the D and A coordinates\n and exchanges B and~C.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 72\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch27-j-replaces-i.tex`\ + \ lines 138-166\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.JReplacesI`\n\ + - Name: `j_unit`\n\n## Dependencies\n\n- Canonical: I.D20, I.T10, II.D14\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The split-complex unit j: the automorphism of the fiber T^2 exchanging B and C coordinates, with j^2 = +1. Unlike + i (rotation), j is an involution (polarity/reflection).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001268.md b/_corpus_v3_items/cid001268.md new file mode 100644 index 000000000..d52de3413 --- /dev/null +++ b/_corpus_v3_items/cid001268.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001268/ +title: Bipolar Idempotents Interior +cid: cid001268 +primary_alias: DEF0141 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Bipolar Idempotents Interior +aliases: +- DEF0141 +- II.D33 +- bipolar-idempotents-interior +- def:bipolar-idempotents-interior +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 27 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bipolar Idempotents Interior**\n\nBipolar Idempotents Interior\n\n# Bipolar Idempotents Interior\n\n## Summary\n\ + \nBipolar Idempotents Interior\n\n## Statement\n\n```latex\n%\n\\label{def:bipolar-idempotents-interior}\nThe \\textbf{bipolar\ + \ idempotents} are\n\\[\n \\boxed{%\n e_+\n \\;:=\\;\n \\frac{1 + \\jj}{2},\n \\qquad\\qquad\n e_-\n\ + \ \\;:=\\;\n \\frac{1 - \\jj}{2}.}\n\\]\nThese satisfy:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \ + \ $e_+^2 = e_+$, \\quad $e_-^2 = e_-$\n \\quad (idempotent),\n \\item[\\textup{(ii)}]\n $e_+ \\\ + cdot e_- = 0$\n \\quad (orthogonal),\n \\item[\\textup{(iii)}]\n $e_+ + e_- = 1$\n \\quad\ + \ (complete),\n \\item[\\textup{(iv)}]\n $\\jj = e_+ - e_-$\n \\quad (recovery of~$\\jj$).\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 73\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch27-j-replaces-i.tex`\ + \ lines 260-289\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.JReplacesI`\n\ + - Name: `j_exponential`\n\n## Dependencies\n\n- Canonical: II.D32, I.D20\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Bipolar Idempotents Interior +noindex: false +--- + diff --git a/_corpus_v3_items/cid001269.md b/_corpus_v3_items/cid001269.md new file mode 100644 index 000000000..766683b98 --- /dev/null +++ b/_corpus_v3_items/cid001269.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001269/ +title: Archimedean Bridge +cid: cid001269 +primary_alias: DEF0143 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The map connecting the ultrametric profinite structure of tau^3 to Archimedean real-valued measurements, via iota_tau + = 2/(pi + e). Bridges finite-stage computation to real analysis. +aliases: +- DEF0143 +- II.D34 +- archimedean-bridge +- def:archimedean-bridge +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 28 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Archimedean Bridge**\n\nThe map connecting the ultrametric profinite structure of tau^3 to Archimedean real-valued\ + \ measurements, via iota_tau = 2/(pi + e). Bridges finite-stage computation to real analysis.\n\n# Archimedean Bridge\n\ + \n## Summary\n\nThe map connecting the ultrametric profinite structure of tau^3 to Archimedean real-valued measurements,\ + \ via iota_tau = 2/(pi + e). Bridges finite-stage computation to real analysis.\n\n## Statement\n\n```latex\n%\n\\label{def:archimedean-bridge}\n\ + The \\textbf{Archimedean-Non-Archimedean Bridge}\nis the pair of measurement systems:\n\\begin{enumerate}\n \\item[\\\ + textup{(NA)}]\n \\textbf{Non-Archimedean measurement}\n (ultrametric refinement):\n the D-depth\ + \ $\\delta(x,y)$\n measures how many stages\n two $\\tau$-admissible points agree.\n Refinement\ + \ increases depth:\n stage $k+1$ refines stage~$k$\n by a factor of~$p_{k+1}$.\n The metric\ + \ is $d(x,y) = 2^{-\\delta(x,y)}$\n (II.D13).\n Convergence means: increasing depth of agreement.\n\n\ + \ \\item[\\textup{(A)}]\n \\textbf{Archimedean measurement}\n (Euclidean resolution):\n \ + \ the ABCD coordinates,\n normalized by $1/P_k$\n (the approximation sequence,\n Chapter~\\\ + ref{ch:orthodox-bridge}),\n measure how precisely the angular\n and radial positions\n are\ + \ determined.\n Resolution increases precision:\n stage $k+1$ resolves angles\n to $1/P_{k+1}$\ + \ instead of $1/P_k$.\n Convergence means: increasing decimal precision.\n\\end{enumerate}\nThe constant $\\\ + iota_\\tau$ converts between the two:\n\\[\n \\boxed{%\n \\text{Euclidean resolution at depth } \\delta\n \\\ + ;\\approx\\;\n \\iota_\\tau \\cdot 2^{-\\delta}\n \\;=\\;\n \\frac{2}{\\pi + e}\\,\n \\cdot\\, 2^{-\\delta}.}\n\ + \\]\nThe factor $\\iota_\\tau$ accounts for the\ndifference between the base-$2$ ultrametric\nand the base-$P_k$ Archimedean\ + \ normalization.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n##\ + \ Source Context\n\n- Registry source: `book-02.jsonl` line 77\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch28-iota-tau-confirmed.tex`\ + \ lines 284-331\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.IotaTauConfirmed`\n\ + - Name: `iota_arithmetic_check`\n\n## Dependencies\n\n- Canonical: II.T25, II.D13\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The map connecting the ultrametric profinite structure of tau^3 to Archimedean real-valued measurements, via + iota_tau = 2/(pi + e). Bridges finite-stage computation to real analysis. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001270.md b/_corpus_v3_items/cid001270.md new file mode 100644 index 000000000..c5eadfd70 --- /dev/null +++ b/_corpus_v3_items/cid001270.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001270/ +title: Calibrated Split-Complex Codomain +cid: cid001270 +primary_alias: DEF0144 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Calibrated Split-Complex Codomain +aliases: +- DEF0144 +- II.D35 +- calibrated-split-complex-codomain +- def:calibrated-H-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 29 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Calibrated Split-Complex Codomain**\n\nCalibrated Split-Complex Codomain\n\n# Calibrated Split-Complex Codomain\n\ + \n## Summary\n\nCalibrated Split-Complex Codomain\n\n## Statement\n\n```latex\n%\n\\label{def:calibrated-H-tau}\nThe \\\ + textbf{calibrated split-complex codomain}\nis the ring\n\\[\n \\boxed{%\n H_\\tau^{\\mathrm{cal}}\n \\;:=\\;\n\ + \ \\bigl(\\mathbb{R}[\\jj],\\;\n \\|\\cdot\\|_\\tau,\\;\n (\\pi, e, \\iota_\\tau)\\bigr),}\n\\]\nwhere:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n The underlying ring is the split-complex ring\n $\\mathbb{R}[\\jj] = \\\ + {a + b\\jj : a, b \\in \\mathbb{R}\\}$\n with $\\jj^2 = +1$\n (I.D20, I.T10, Book~I).\n\n \\item[\\\ + textup{(ii)}]\n The \\textbf{calibrated norm} is\n \\[\n \\|z\\|_\\tau\n \\\ + ;:=\\;\n |z_+|^{\\pi/(\\pi+e)}\n \\cdot\n |z_-|^{e/(\\pi+e)},\n \\]\n\ + \ where $z_+ = a + b$ and $z_- = a - b$\n are the sector values.\n The exponents $\\pi/(\\\ + pi+e)$\n and $e/(\\pi+e)$\n are the \\textbf{angular weight}\n and \\textbf{growth weight},\n\ + \ respectively.\n They sum to~$1$:\n the norm is a geometric mean\n weighted by the\ + \ two calibration constants.\n\n \\item[\\textup{(iii)}]\n The \\textbf{calibration triple}\n $(\\\ + pi, e, \\iota_\\tau)$\n records the angular period,\n the growth rate,\n and their harmonic\ + \ coupling\n $\\iota_\\tau = 2/(\\pi + e)$.\n\\end{enumerate}\nThe sector decomposition becomes\n\\[\n z\n\ + \ \\;=\\;\n z_+\\, e_+\n \\;+\\;\n z_-\\, e_-\n \\;\\in\\;\n H_\\tau^{\\mathrm{cal}},\n\\]\nwhere $z_+$\ + \ is the \\textbf{angular sector value}\n(calibrated by~$\\pi$)\nand $z_-$ is the \\textbf{growth sector value}\n(calibrated\ + \ by~$e$).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 80\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch29-split-complex-calibrated.tex`\ + \ lines 163-226\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CalibratedSplitComplex`\n\ + - Name: `Tau.BookII.Hartogs.CalibratedHTau`\n\n## Dependencies\n\n- Canonical: I.D20, I.D21, I.D24, I.T10, II.T22, II.T23,\ + \ II.T24, II.T25\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Calibrated Split-Complex Codomain +noindex: false +--- + diff --git a/_corpus_v3_items/cid001271.md b/_corpus_v3_items/cid001271.md new file mode 100644 index 000000000..665c2deb1 --- /dev/null +++ b/_corpus_v3_items/cid001271.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001271/ +title: BndLift Construction +cid: cid001271 +primary_alias: DEF0145 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: BndLift Construction +aliases: +- DEF0145 +- II.D36 +- bndlift-construction +- def:bndlift +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 30 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**BndLift Construction**\n\nBndLift Construction\n\n# BndLift Construction\n\n## Summary\n\nBndLift Construction\n\ + \n## Statement\n\n```latex\n%\n\\label{def:bndlift}\n% II.D09, II.D33, II.D35\nLet $n \\geq 1$.\nThe \\textbf{boundary\ + \ lift at stage~$n$}\nis the operator\n\\[\n \\boxed{%\n \\mathrm{BndLift}_n\n \\;:\\;\n \\mathrm{Hol}(\\\ + mathbb{Z}/P_n\\mathbb{Z},\\; H_\\tau^{\\mathrm{cal}})\n \\;\\longrightarrow\\;\n \\mathrm{Hol}(\\mathbb{Z}/P_{n+1}\\\ + mathbb{Z},\\; H_\\tau^{\\mathrm{cal}})}\n\\]\ndefined as follows.\nGiven a holomorphic datum\n$f_n : \\mathbb{Z}/P_n\\\ + mathbb{Z} \\to H_\\tau^{\\mathrm{cal}}$,\nthe lifted datum\n$f_{n+1} := \\mathrm{BndLift}_n(f_n)$\nis the unique function\n\ + $f_{n+1} : \\mathbb{Z}/P_{n+1}\\mathbb{Z} \\to H_\\tau^{\\mathrm{cal}}$\nsatisfying:\n\\begin{enumerate}\n \\item[\\\ + textup{(i)}]\n \\textbf{Compatibility.}\n The restriction of $f_{n+1}$\n to the stage-$n$ component\n\ + \ equals $f_n$:\n \\[\n f_{n+1}(x) \\big|_{\\bmod P_n}\n \\;=\\;\n \ + \ f_n(x \\bmod P_n)\n \\]\n for all $x \\in \\mathbb{Z}/P_{n+1}\\mathbb{Z}$.\n\n \\item[\\\ + textup{(ii)}]\n \\textbf{Bipolar splitting.}\n The lift decomposes along the\n bipolar idempotents:\n\ + \ \\[\n f_{n+1}(x)\n \\;=\\;\n \\mathrm{BndLift}_n^{(+)}(f_n^{(+)})(x)\\\ + , e_+\n \\;+\\;\n \\mathrm{BndLift}_n^{(-)}(f_n^{(-)})(x)\\, e_-,\n \\]\n \ + \ where $f_n^{(\\pm)} := e_\\pm \\cdot f_n$\n are the sector components,\n and $\\mathrm{BndLift}_n^{(\\\ + pm)}$\n are the \\textbf{sector lifts}\n acting independently on each channel.\n\n \\item[\\textup{(iii)}]\n\ + \ \\textbf{CRT extension.}\n Each sector lift\n $\\mathrm{BndLift}_n^{(\\pm)}$\n extends\ + \ the stage-$n$ sector value\n to stage $n+1$\n using the CRT decomposition:\n \\[\n \ + \ \\mathrm{BndLift}_n^{(\\pm)}(g)(x)\n \\;=\\;\n g(x \\bmod P_n)\n \\;+\\\ + ;\n \\Delta_n^{(\\pm)}(x \\bmod p_{n+1})\n \\cdot P_n,\n \\]\n where $\\Delta_n^{(\\\ + pm)} : \\mathbb{Z}/p_{n+1}\\mathbb{Z} \\to \\mathbb{Z}$\n is the \\textbf{lift increment}:\n the unique\ + \ function satisfying\n the tower coherence constraint\n and the diagonal discipline (K5, Book~I).\n\ + \n \\item[\\textup{(iv)}]\n \\textbf{Coupling.}\n The lift increments\n for the two sectors\n\ + \ are coupled through $\\iota_\\tau$:\n \\[\n \\|\\Delta_n^{(+)}\\|\n \\;=\\\ + ;\n \\iota_\\tau \\cdot \\|\\Delta_n^{(-)}\\|\n \\;=\\;\n \\frac{2}{\\pi + e}\n\ + \ \\cdot \\|\\Delta_n^{(-)}\\|,\n \\]\n where $\\|\\cdot\\|$ is the $\\ell^2$ norm\n \ + \ on $\\mathbb{Z}/p_{n+1}\\mathbb{Z}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 82\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch30-bndlift-construction.tex` lines 199-289\n\n##\ + \ Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.BndLift`\n- Name: `Tau.BookII.Hartogs.bndlift`\n\ + \n## Dependencies\n\n- Canonical: I.T18, I.D20, I.D21, I.D24, I.T10, II.D09, II.D33, II.D35\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: BndLift Construction +noindex: false +--- + diff --git a/_corpus_v3_items/cid001272.md b/_corpus_v3_items/cid001272.md new file mode 100644 index 000000000..e264eefd7 --- /dev/null +++ b/_corpus_v3_items/cid001272.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001272/ +title: Evolution Operator +cid: cid001272 +primary_alias: DEF0146 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The composition of successive boundary lifts, propagating holomorphic data from stage n to stage m along the primorial + tower. +aliases: +- DEF0146 +- II.D37 +- evolution-operator +- def:evolution-operator +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 32 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Evolution Operator**\n\nThe composition of successive boundary lifts, propagating holomorphic data from stage\ + \ n to stage m along the primorial tower.\n\n# Evolution Operator\n\n## Summary\n\nThe composition of successive boundary\ + \ lifts, propagating holomorphic data from stage n to stage m along the primorial tower.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:evolution-operator}\nFor integers $m > n \\geq 1$,\nthe \\textbf{evolution operator}\n$\\mathcal{E}_{n\ + \ \\to m}$\nis the composition of successive boundary lifts:\n\\[\n \\boxed{%\n \\mathcal{E}_{n \\to m}\n \\\ + ;:=\\;\n \\mathrm{BndLift}_{m-1}\n \\circ\n \\mathrm{BndLift}_{m-2}\n \\circ \\cdots \\circ\n \\mathrm{BndLift}_n\n\ + \ \\;\\colon\\;\n \\mathrm{Hol}_n(\\tau^3)\n \\;\\longrightarrow\\;\n \\mathrm{Hol}_m(\\tau^3),}\n\\]\nwhere\ + \ $\\mathrm{Hol}_k(\\tau^3)$\nis the space of holomorphic data\nat stage~$k$\n(tower-coherent maps\n$f_k \\colon \\mathbb{Z}/P_k\\\ + mathbb{Z} \\to H_\\tau$).\n\nThe operator has the following structure:\n\\begin{enumerate}\n \\item[\\textup{(E1)}]\n\ + \ \\textbf{Stage-by-stage construction.}\n Each $\\mathrm{BndLift}_k$\n uses the CRT decomposition\n\ + \ $\\mathbb{Z}/P_{k+1}\\mathbb{Z}\n \\cong \\mathbb{Z}/P_k\\mathbb{Z}\n \\times \\mathbb{Z}/p_{k+1}\\\ + mathbb{Z}$\n to extend the domain by one prime factor.\n\n \\item[\\textup{(E2)}]\n \\textbf{Bipolar\ + \ splitting.}\n At each step,\n $\\mathrm{BndLift}_k$\n splits into independent lifts\n \ + \ in the $e_+$-channel and $e_-$-channel:\n \\[\n \\mathrm{BndLift}_k\n \\;=\\\ + ;\n e_+ \\cdot \\mathrm{BndLift}_k^+\n \\;+\\;\n e_- \\cdot \\mathrm{BndLift}_k^-.\n\ + \ \\]\n\n \\item[\\textup{(E3)}]\n \\textbf{Semigroup property.}\n For $\\ell > m > n \\\ + geq 1$:\n \\[\n \\mathcal{E}_{n \\to \\ell}\n \\;=\\;\n \\mathcal{E}_{m\ + \ \\to \\ell}\n \\circ\n \\mathcal{E}_{n \\to m}.\n \\]\n\n \\item[\\textup{(E4)}]\n\ + \ \\textbf{Identity.}\n $\\mathcal{E}_{n \\to n} = \\mathrm{id}$.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 90\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch32-evolution-operator.tex`\ + \ lines 100-167\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.EvolutionOperator`\n\ + - Name: `Tau.BookII.Hartogs.evolution_op`\n\n## Dependencies\n\n- Canonical: II.D36, II.T27, II.D35, I.T18\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The composition of successive boundary lifts, propagating holomorphic data from stage n to stage m along the + primorial tower. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001273.md b/_corpus_v3_items/cid001273.md new file mode 100644 index 000000000..f65d958de --- /dev/null +++ b/_corpus_v3_items/cid001273.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001273/ +title: Causal Arrow +cid: cid001273 +primary_alias: DEF0147 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The preferred direction in the primorial tower selected by B/C asymmetry: stage n+1 is in the causal future of stage + n, and the B-channel precedes the C-channel.' +aliases: +- DEF0147 +- II.D38 +- causal-arrow +- def:causal-arrow +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005556 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 32 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Causal Arrow**\n\nThe preferred direction in the primorial tower selected by B/C asymmetry: stage n+1 is in\ + \ the causal future of stage n, and the B-channel precedes the C-channel.\n\n# Causal Arrow\n\n## Summary\n\nThe preferred\ + \ direction in the primorial tower selected by B/C asymmetry: stage n+1 is in the causal future of stage n, and the B-channel\ + \ precedes the C-channel.\n\n## Statement\n\n```latex\n%\n\\label{def:causal-arrow}\nThe \\textbf{causal arrow} on~$\\\ + tau^3$ is the preferred direction\nin the primorial tower\nselected by the B/C asymmetry:\n\\begin{enumerate}\n \\\ + item[\\textup{(A1)}]\n \\textbf{Forward direction.}\n Stage $n+1$ is in the \\textbf{causal future}\ + \ of stage~$n$.\n The evolution operator\n $\\mathcal{E}_{n \\to n+1}$\n propagates data forward.\n\ + \n \\item[\\textup{(A2)}]\n \\textbf{Asymmetry mechanism.}\n At each stage~$k$,\n the new\ + \ prime $p_{k+1}$\n enters the CRT decomposition.\n The B-channel absorbs\n the exponent of\ + \ $p_{k+1}$\n (how often $p_{k+1}$ divides the argument),\n and the C-channel absorbs\n the\ + \ tetration data.\n Since the C-channel data\n depends on the B-channel data\n (tetration requires\ + \ exponentiation as input),\n there is a structural arrow\n from B to C within each stage.\n \ + \ This intra-stage arrow\n is consistent across all stages,\n giving a global forward direction.\n\ + \n \\item[\\textup{(A3)}]\n \\textbf{Irreversibility.}\n The projection $\\pi_k \\colon\n \ + \ \\mathbb{Z}/P_{k+1}\\mathbb{Z}\n \\to \\mathbb{Z}/P_k\\mathbb{Z}$\n loses the $\\mathbb{Z}/p_{k+1}\\\ + mathbb{Z}$ component.\n This loss is \\textbf{structural}:\n it is not an artifact of notation\n \ + \ but a consequence of the fact\n that the $p_{k+1}$-component\n carries new spectral data\n \ + \ not present at stage~$k$.\n Evolution forward creates information;\n projection backward destroys\ + \ it.\n\\end{enumerate}\nThe causal arrow is the unique direction\nin which information is created\n(refinement increases)\n\ + and never destroyed.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 91\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch32-evolution-operator.tex`\ + \ lines 295-347\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.EvolutionOperator`\n\ + - Name: `Tau.BookII.Hartogs.causal_arrow`\n\n## Dependencies\n\n- Canonical: I.T05, I.D21, II.D37, II.D22\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The preferred direction in the primorial tower selected by B/C asymmetry: stage n+1 is in the causal future + of stage n, and the B-channel precedes the C-channel.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001274.md b/_corpus_v3_items/cid001274.md new file mode 100644 index 000000000..7f6abc625 --- /dev/null +++ b/_corpus_v3_items/cid001274.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001274/ +title: Composition of Holomorphic Maps +cid: cid001274 +primary_alias: DEF0148 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Composition of Holomorphic Maps +aliases: +- DEF0148 +- II.D39 +- composition-of-holomorphic-maps +- def:composition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 33 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Composition of Holomorphic Maps**\n\nComposition of Holomorphic Maps\n\n# Composition of Holomorphic Maps\n\ + \n## Summary\n\nComposition of Holomorphic Maps\n\n## Statement\n\n```latex\n%\n\\label{def:composition}\nLet $f = \\\ + {f_k\\}_{k \\geq 1}$\nand $g = \\{g_k\\}_{k \\geq 1}$\nbe $\\omega$-germ transformers.\nTheir \\textbf{composition} is\ + \ the family\n\\[\n \\boxed{%\n (g \\circ f)_k\n \\;:=\\;\n g_k \\circ f_k\n \\qquad \\text{for all } k\ + \ \\geq 1.}\n\\]\nThe composition is taken\nin the split-complex codomain $H_\\tau$:\nat each stage~$k$,\n$f_k$ maps $\\\ + mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau$,\nand $g_k$ acts on the image\nvia the $H_\\tau$-valued germ structure.\n```\n\n\ + ## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 93\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch33-composition-structure.tex`\ + \ lines 128-148\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CategoryStructure`\n\ + - Name: `hol_compose`\n\n## Dependencies\n\n- Canonical: I.D49, I.D20, I.T18, II.T26\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Composition of Holomorphic Maps +noindex: false +--- + diff --git a/_corpus_v3_items/cid001275.md b/_corpus_v3_items/cid001275.md new file mode 100644 index 000000000..ca1398263 --- /dev/null +++ b/_corpus_v3_items/cid001275.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001275/ +title: Identity Map +cid: cid001275 +primary_alias: DEF0149 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The identity omega-germ transformer on a tau-admissible point x, defined by the identity map at each stage. The unit + morphism in HolEnd(tau). +aliases: +- DEF0149 +- II.D40 +- identity-map +- def:identity-map +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 33 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Identity Map**\n\nThe identity omega-germ transformer on a tau-admissible point x, defined by the identity\ + \ map at each stage. The unit morphism in HolEnd(tau).\n\n# Identity Map\n\n## Summary\n\nThe identity omega-germ transformer\ + \ on a tau-admissible point x, defined by the identity map at each stage. The unit morphism in HolEnd(tau).\n\n## Statement\n\ + \n```latex\n%\n\\label{def:identity-map}\nThe \\textbf{identity map}\non a $\\tau$-admissible point~$x$\nis the $\\omega$-germ\ + \ transformer\n$\\mathrm{id}_x = \\{\\mathrm{id}_k\\}_{k \\geq 1}$\ndefined by\n\\[\n \\boxed{%\n (\\mathrm{id}_x)_k\n\ + \ \\;:=\\;\n \\mathrm{id}_{\\mathbb{Z}/P_k\\mathbb{Z}}\n \\qquad \\text{for all } k \\geq 1,}\n\\]\nthe identity\ + \ function\non the finite stage $\\mathbb{Z}/P_k\\mathbb{Z}$.\nThis is the \\textbf{constant family}:\nthe identity germ\ + \ at every stage.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n\ + ## Source Context\n\n- Registry source: `book-02.jsonl` line 94\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch33-composition-structure.tex`\ + \ lines 225-244\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CategoryStructure`\n\ + - Name: `hol_identity`\n\n## Dependencies\n\n- Canonical: I.D49, II.D39\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The identity omega-germ transformer on a tau-admissible point x, defined by the identity map at each stage. The + unit morphism in HolEnd(tau). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001276.md b/_corpus_v3_items/cid001276.md new file mode 100644 index 000000000..01ca1422a --- /dev/null +++ b/_corpus_v3_items/cid001276.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001276/ +title: Holomorphic Endomorphism Category +cid: cid001276 +primary_alias: DEF0150 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Holomorphic Endomorphism Category +aliases: +- DEF0150 +- II.D41 +- holomorphic-endomorphism-category +- def:holend-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005553 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 33 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Holomorphic Endomorphism Category**\n\nHolomorphic Endomorphism Category\n\n# Holomorphic Endomorphism Category\n\ + \n## Summary\n\nHolomorphic Endomorphism Category\n\n## Statement\n\n```latex\n%\n\\label{def:holend-tau}\nThe \\textbf{holomorphic\ + \ endomorphism category}\n$\\mathrm{HolEnd}_\\tau$ is defined by:\n\\begin{enumerate}\n \\item\\textbf{Objects:}\n\ + \ $\\tau$-admissible points\n (II.D02,\n Chapter~\\ref{ch:tau-admissible-points}).\n \\\ + item\\textbf{Morphisms:}\n For objects $x, y$:\n $\\mathrm{Hom}_{\\mathrm{HolEnd}_\\tau}(x, y)\n \ + \ := \\{$~$\\omega$-germ transformers\n from $x$ to $y$$\\}$.\n \\item\\textbf{Composition:}\n \ + \ Stagewise composition\n (Definition~\\ref{def:composition}, II.D39).\n \\item\\textbf{Identity:}\n \ + \ The constant family\n $\\mathrm{id}_x = \\{\\mathrm{id}_k\\}$\n (Definition~\\ref{def:identity-map},\ + \ II.D40).\n\\end{enumerate}\nThe category axioms hold:\n\\begin{itemize}\n \\item \\textbf{Associativity}:\n \ + \ Theorem~\\ref{thm:associativity} (II.T29).\n \\item \\textbf{Left identity}:\n $\\mathrm{id} \\circ\ + \ f = f$\n (Proposition~\\ref{prop:ch33-identity-laws}).\n \\item \\textbf{Right identity}:\n $f\ + \ \\circ \\mathrm{id} = f$\n (Proposition~\\ref{prop:ch33-identity-laws}).\n\\end{itemize}\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 96\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch33-composition-structure.tex`\ + \ lines 529-563\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CategoryStructure`\n\ + - Name: `holend_category_check`\n\n## Dependencies\n\n- Canonical: II.D39, II.D40, II.T29, II.D02\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Holomorphic Endomorphism Category +noindex: false +--- + diff --git a/_corpus_v3_items/cid001277.md b/_corpus_v3_items/cid001277.md new file mode 100644 index 000000000..99f3ee6db --- /dev/null +++ b/_corpus_v3_items/cid001277.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001277/ +title: Laurent Expansion +cid: cid001277 +primary_alias: DEF0151 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The bipolar double series expansion of an omega-germ transformer at a tau-admissible point, with coefficients in + the canonical basis and idempotent projectors e_+, e_-. +aliases: +- DEF0151 +- II.D42 +- laurent-expansion +- def:laurent-expansion +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005567 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Laurent Expansion**\n\nThe bipolar double series expansion of an omega-germ transformer at a tau-admissible\ + \ point, with coefficients in the canonical basis and idempotent projectors e_+, e_-.\n\n# Laurent Expansion\n\n## Summary\n\ + \nThe bipolar double series expansion of an omega-germ transformer at a tau-admissible point, with coefficients in the\ + \ canonical basis and idempotent projectors e_+, e_-.\n\n## Statement\n\n```latex\n%\n\\label{def:laurent-expansion}\n\ + Let $f = \\{f_k\\}_{k \\geq 1}$\nbe an $\\omega$-germ transformer\non~$\\tau^3$,\nwith bipolar decomposition\n$f = e_+\ + \ f_+ + e_- f_-$.\nThe \\textbf{Laurent expansion} of~$f$\nat a $\\tau$-admissible point~$x$\nis the formal double series:\n\ + \\[\n \\boxed{%\n f\n \\;=\\;\n \\sum_{n \\in \\mathbb{Z}}\n a_n^{(+)}\\, e_+\\, \\phi_n^{(+)}\n \\\ + ;\\;+\\;\\;\n \\sum_{n \\in \\mathbb{Z}}\n a_n^{(-)}\\, e_-\\, \\phi_n^{(-)},}\n\\]\nwhere the \\textbf{spectral\ + \ coefficients}\n$a_n^{(\\pm)} \\in \\mathbb{R}$\nare determined by:\n\\[\n a_n^{(+)}\n \\;=\\;\n \\lim_{k \\\ + to \\infty}\\;\n \\frac{1}{P_k}\n \\sum_{a=0}^{P_k - 1}\n f_+^{(k)}(a)\\,\n \\overline{\\chi_n^{(\\gamma)}(a)},\n\ + \ \\qquad\n a_n^{(-)}\n \\;=\\;\n \\lim_{k \\to \\infty}\\;\n \\frac{1}{P_k}\n \\sum_{a=0}^{P_k - 1}\n\ + \ f_-^{(k)}(a)\\,\n \\overline{\\chi_n^{(\\eta)}(a)}.\n\\]\nThe sums are \\textbf{finite discrete Fourier transforms}\n\ + at each stage~$k$,\nand the limits exist by tower coherence.\nThe index $n$ ranges over $\\mathbb{Z}$:\npositive $n$ corresponds\ + \ to\n\\textbf{regular terms}\n(present in holomorphic functions),\nwhile negative $n$ corresponds to\n\\textbf{principal\ + \ part terms}\n(present only at singularities).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 97\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch34-laurent-residues.tex`\ + \ lines 193-244\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.LaurentResidue`\n\ + - Name: `laurent_coeff`\n\n## Dependencies\n\n- Canonical: I.D20, I.D21, I.T10, I.T18, II.D35\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The bipolar double series expansion of an omega-germ transformer at a tau-admissible point, with coefficients + in the canonical basis and idempotent projectors e_+, e_-. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001278.md b/_corpus_v3_items/cid001278.md new file mode 100644 index 000000000..0167682b4 --- /dev/null +++ b/_corpus_v3_items/cid001278.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001278/ +title: Residue +cid: cid001278 +primary_alias: DEF0152 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The split-complex residue extracted from the n = -1 Laurent coefficients. The bipolar residue decomposes into independent + +/- channel contributions. +aliases: +- DEF0152 +- II.D43 +- residue +- def:residue +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005568 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Residue**\n\nThe split-complex residue extracted from the n = -1 Laurent coefficients. The bipolar residue\ + \ decomposes into independent +/- channel contributions.\n\n# Residue\n\n## Summary\n\nThe split-complex residue extracted\ + \ from the n = -1 Laurent coefficients. The bipolar residue decomposes into independent +/- channel contributions.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:residue}\nLet $f$ be an $\\omega$-germ transformer\nwith Laurent expansion\n\ + $\\sum_n a_n^{(+)} e_+ \\phi_n^{(+)}\n+ \\sum_n a_n^{(-)} e_- \\phi_n^{(-)}$.\nThe \\textbf{residue} of~$f$ at a $\\tau$-admissible\n\ + point~$x$ is the split-complex number:\n\\[\n \\boxed{%\n \\mathrm{Res}_x(f)\n \\;:=\\;\n a_{-1}^{(+)}\\,\ + \ e_+\n \\;+\\;\n a_{-1}^{(-)}\\, e_-\n \\;\\;\\in\\;\\;\n H_\\tau.}\n\\]\nThe residue is the \\textbf{spectral\ + \ coefficient\nat frequency $n = -1$},\nextracted independently\nfrom the B-channel and C-channel.\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 98\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch34-laurent-residues.tex`\ + \ lines 306-329\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.LaurentResidue`\n\ + - Name: `tau_residue`\n\n## Dependencies\n\n- Canonical: II.D42, I.D21, I.T10, II.D35\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The split-complex residue extracted from the n = -1 Laurent coefficients. The bipolar residue decomposes into + independent +/- channel contributions. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001279.md b/_corpus_v3_items/cid001279.md new file mode 100644 index 000000000..29c1fbfc2 --- /dev/null +++ b/_corpus_v3_items/cid001279.md @@ -0,0 +1,75 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001279/ +title: Meromorphic Function +cid: cid001279 +primary_alias: DEF0153 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Meromorphic Function +aliases: +- DEF0153 +- II.D44 +- meromorphic-function +- def:meromorphic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005571 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Meromorphic Function**\n\nMeromorphic Function\n\n# Meromorphic Function\n\n## Summary\n\nMeromorphic Function\n\ + \n## Statement\n\n```latex\n%\n\\label{def:meromorphic}\nA \\textbf{$\\tau$-meromorphic function}\non~$\\tau^3$ is a formal\ + \ limit\nof $\\omega$-germ transformers\nthat is holomorphic on a co-finite subset\nof $\\tau$-admissible points\nand\ + \ has a Laurent expansion\nwith finite principal part\nat each excluded point.\nFormally, $f$ is meromorphic if:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n there exists a finite set\n $S \\subset \\tau^3$\n (the \\textbf{singular\ + \ locus})\n such that $f|_{\\tau^3 \\setminus S}$\n is an $\\omega$-germ transformer;\n \\item[\\\ + textup{(ii)}]\n at each $x \\in S$,\n the Laurent expansion of~$f$\n (Definition~\\ref{def:laurent-expansion})\n\ + \ has finite principal part.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 99\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch34-laurent-residues.tex` lines 419-444\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.LaurentResidue`\n- Name: `meromorphic_check`\n\ + \n## Dependencies\n\n- Canonical: II.D42, II.D43, II.D41\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Meromorphic Function +noindex: false +--- + diff --git a/_corpus_v3_items/cid001280.md b/_corpus_v3_items/cid001280.md new file mode 100644 index 000000000..2eacc2f77 --- /dev/null +++ b/_corpus_v3_items/cid001280.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001280/ +title: Canonical Holomorphic Basis +cid: cid001280 +primary_alias: DEF0154 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Canonical Holomorphic Basis +aliases: +- DEF0154 +- II.D45 +- canonical-holomorphic-basis +- def:canonical-basis +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005544 +- predicate: formalized_by + object: cid005545 +- predicate: formalized_by + object: cid005546 +- predicate: formalized_by + object: cid005547 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Canonical Holomorphic Basis**\n\nCanonical Holomorphic Basis\n\n# Canonical Holomorphic Basis\n\n## Summary\n\ + \nCanonical Holomorphic Basis\n\n## Statement\n\n```latex\n%\n\\label{def:canonical-basis}\nThe \\textbf{canonical holomorphic\ + \ basis} of~$\\tau^3$ is\n\\[\n \\boxed{%\n \\mathcal{B}_\\tau\n \\;:=\\;\n \\bigl\\{\\,\n E_{k,v}^{(\\\ + sigma)}\n \\;\\big|\\;\n k \\geq 1,\\;\n p \\mid P_k,\\;\n v \\in \\mathbb{Z}/p\\mathbb{Z},\\;\n \\sigma\ + \ \\in \\{B, C\\}\n \\,\\bigr\\}.}\n\\]\nThe basis is \\textbf{canonical}:\nit is uniquely determined by the CRT decomposition\n\ + of the primorial tower (I.T18, Book~I),\nthe bipolar idempotent structure (I.D21, Book~I),\nand the NF addressing of cylinders\ + \ (II.D10).\nNo choices, conventions, or coordinates\nare involved in its construction.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 101\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch35-canonical-basis.tex`\ + \ lines 298-322\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CanonicalBasis`\n\ + - Name: `canonical_basis_check`\n\n## Dependencies\n\n- Canonical: II.D46, I.T18, I.D21\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical Holomorphic Basis +noindex: false +--- + diff --git a/_corpus_v3_items/cid001281.md b/_corpus_v3_items/cid001281.md new file mode 100644 index 000000000..fc66120f9 --- /dev/null +++ b/_corpus_v3_items/cid001281.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001281/ +title: Cylinder Generator +cid: cid001281 +primary_alias: DEF0155 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The canonical indicator function at prime p and residue class v in channel sigma. The basis element for holomorphic + functions on tau^3. +aliases: +- DEF0155 +- II.D46 +- cylinder-generator +- def:cylinder-generator +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cylinder Generator**\n\nThe canonical indicator function at prime p and residue class v in channel sigma. The\ + \ basis element for holomorphic functions on tau^3.\n\n# Cylinder Generator\n\n## Summary\n\nThe canonical indicator function\ + \ at prime p and residue class v in channel sigma. The basis element for holomorphic functions on tau^3.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:cylinder-generator}\nLet $k \\geq 1$ be a stage,\n$p$ a prime factor of~$P_k$,\n$v \\in \\\ + mathbb{Z}/p\\mathbb{Z}$ a residue class\n(the \\textbf{peel token}),\nand $\\sigma \\in \\{B, C\\}$ a channel.\nThe \\\ + textbf{cylinder generator} is the function\n\\[\n \\boxed{%\n E_{k,v}^{(\\sigma)}\n \\;:\\;\n \\mathbb{Z}/P_k\\\ + mathbb{Z}\n \\;\\longrightarrow\\;\n A_\\tau^{(\\sigma)},\n \\qquad\n E_{k,v}^{(\\sigma)}(x)\n \\;:=\\\ + ;\n \\begin{cases}\n e_\\sigma & \\text{if } x \\equiv v \\pmod{p}, \\\\[4pt]\n 0 & \\text{otherwise},\n\ + \ \\end{cases}}\n\\]\nwhere $e_B := e_+$ and $e_C := e_-$\nare the bipolar idempotents (I.D21, Book~I),\nand $x \\\ + equiv v \\pmod{p}$ is computed\nvia the CRT projection\n$\\mathbb{Z}/P_k\\mathbb{Z} \\twoheadrightarrow \\mathbb{Z}/p\\\ + mathbb{Z}$.\n\nThe \\textbf{indexing set} at stage~$k$ is\n\\[\n \\Lambda_k\n \\;:=\\;\n \\bigl\\{\\,(p, v, \\\ + sigma)\n \\;\\big|\\;\n p \\mid P_k,\\;\n v \\in \\mathbb{Z}/p\\mathbb{Z},\\;\n \\sigma \\in \\{B, C\\}\n\ + \ \\,\\bigr\\}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 102\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch35-canonical-basis.tex`\ + \ lines 125-166\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CanonicalBasis`\n\ + - Name: `cylinder_gen`\n\n## Dependencies\n\n- Canonical: I.T18, I.D21, II.D10\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The canonical indicator function at prime p and residue class v in channel sigma. The basis element for holomorphic + functions on tau^3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001282.md b/_corpus_v3_items/cid001282.md new file mode 100644 index 000000000..87fcf65f9 --- /dev/null +++ b/_corpus_v3_items/cid001282.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001282/ +title: Holomorphic Presheaf +cid: cid001282 +primary_alias: DEF0156 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Holomorphic Presheaf +aliases: +- DEF0156 +- II.D47 +- holomorphic-presheaf +- def:holomorphic-presheaf +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005573 +- predicate: formalized_by + object: cid005578 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Holomorphic Presheaf**\n\nHolomorphic Presheaf\n\n# Holomorphic Presheaf\n\n## Summary\n\nHolomorphic Presheaf\n\ + \n## Statement\n\n```latex\n%\n\\label{def:holomorphic-presheaf}\nThe \\textbf{holomorphic presheaf} $\\mathcal{O}_\\\ + tau$\non~$\\tau^3$ assigns to each cylinder domain\n$U \\subseteq \\tau^3$ the set\n\\[\n \\boxed{%\n \\mathcal{O}_\\\ + tau(U)\n \\;:=\\;\n \\bigl\\{\\,\n f : U \\to H_\\tau\n \\;\\big|\\;\n f \\text{ is $\\tau$-holomorphic\ + \ on } U\n \\,\\bigr\\},}\n\\]\nwhere $\\tau$-holomorphic means\nthat $f$ satisfies the five equivalent conditions\n\ + of the Mutual Determination Theorem\n(Theorem~\\ref{thm:mutual-determination}, II.T27)\nrestricted to~$U$.\n\nFor an inclusion\ + \ of cylinder domains\n$V \\subseteq U$,\nthe \\textbf{restriction map}\n\\[\n \\rho_{U,V} : \\mathcal{O}_\\tau(U)\ + \ \\to \\mathcal{O}_\\tau(V),\n \\qquad\n \\rho_{U,V}(f) := f\\big|_V,\n\\]\nis the ordinary restriction of functions.\n\ + \n\\medskip\\noindent\nThe presheaf axioms are immediate:\n\\begin{enumerate}\n \\item[\\textup{(P1)}]\n $\\\ + rho_{U,U} = \\mathrm{id}$\n (restriction to the same domain is the identity).\n \\item[\\textup{(P2)}]\n \ + \ $\\rho_{V,W} \\circ \\rho_{U,V} = \\rho_{U,W}$\n for $W \\subseteq V \\subseteq U$\n (restriction\ + \ is transitive).\n\\end{enumerate}\nBoth are trivially satisfied\nby function restriction.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 105\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch36-sheaf-coherence.tex`\ + \ lines 88-133\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.SheafCoherence`\n\ + - Name: `presheaf_assign`\n\n## Dependencies\n\n- Canonical: II.D10, II.T27, II.D45\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Holomorphic Presheaf +noindex: false +--- + diff --git a/_corpus_v3_items/cid001283.md b/_corpus_v3_items/cid001283.md new file mode 100644 index 000000000..facbd6585 --- /dev/null +++ b/_corpus_v3_items/cid001283.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001283/ +title: Canonical Decomposition +cid: cid001283 +primary_alias: DEF0157 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Canonical Decomposition +aliases: +- DEF0157 +- II.D48 +- canonical-decomposition +- def:canonical-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005658 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 37 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Canonical Decomposition**\n\nCanonical Decomposition\n\n# Canonical Decomposition\n\n## Summary\n\nCanonical\ + \ Decomposition\n\n## Statement\n\n```latex\n%\n\\label{def:canonical-decomposition}\nLet $f \\in \\mathcal{O}_\\tau(\\\ + tau^3)$\nbe a $\\tau$-holomorphic function\non~$\\tau^3$\n(in the sense of the Mutual Determination Theorem,\nTheorem~\\\ + ref{thm:mutual-determination}, II.T27).\nThe \\textbf{canonical decomposition} of~$f$\nis the pair\n\\[\n \\boxed{%\n\ + \ f \\;=\\; f_+ \\;+\\; f_-,\n \\qquad\n f_+ := e_+ \\cdot f,\n \\quad\n f_- := e_- \\cdot f,}\n\\]\nwhere\ + \ the multiplication is pointwise\nin the calibrated split-complex codomain\n$H_\\tau^{\\mathrm{cal}}$\n(Definition~\\\ + ref{def:calibrated-H-tau}, II.D35,\nChapter~\\ref{ch:split-complex-calibrated}).\n\nWe call $f_+$ the \\textbf{B-channel\ + \ component}\nand $f_-$ the \\textbf{C-channel component} of~$f$.\nThe decomposition $f \\mapsto (f_+, f_-)$\nis the \\\ + textbf{canonical decomposition map}:\n\\[\n \\Delta_\\tau\n \\colon\n \\mathcal{O}_\\tau(\\tau^3)\n \\;\\\ + longrightarrow\\;\n \\mathcal{O}_\\tau^{(+)}(\\tau^3)\n \\times\n \\mathcal{O}_\\tau^{(-)}(\\tau^3),\n \\\ + qquad\n \\Delta_\\tau(f) := (f_+, f_-),\n\\]\nwhere $\\mathcal{O}_\\tau^{(\\pm)}(\\tau^3)$\ndenotes the space of $\\\ + tau$-holomorphic functions\ntaking values in the channel $H_\\tau^{(\\pm)} = e_\\pm \\cdot H_\\tau$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 108\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch37-idempotent-decomposition.tex`\ + \ lines 282-324\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.IdempotentDecomposition`\n\ + - Name: `canonical_decomposition_check`\n\n## Dependencies\n\n- Canonical: I.D21, I.T10, II.D35, II.T27\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical Decomposition +noindex: false +--- + diff --git a/_corpus_v3_items/cid001284.md b/_corpus_v3_items/cid001284.md new file mode 100644 index 000000000..1adfb63de --- /dev/null +++ b/_corpus_v3_items/cid001284.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001284/ +title: Tau-Regularity +cid: cid001284 +primary_alias: DEF0158 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A point p is tau-regular for f if the omega-germ sequence stabilizes at a finite primorial stage. A constructive, + positive criterion: existence of stabilization, not absence of pathology.' +aliases: +- DEF0158 +- II.D49 +- tau-regularity +- def:tau-regularity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005662 +- predicate: formalized_by + object: cid005663 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 39 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Regularity**\n\nA point p is tau-regular for f if the omega-germ sequence stabilizes at a finite primorial\ + \ stage. A constructive, positive criterion: existence of stabilization, not absence of pathology.\n\n# Tau-Regularity\n\ + \n## Summary\n\nA point p is tau-regular for f if the omega-germ sequence stabilizes at a finite primorial stage. A constructive,\ + \ positive criterion: existence of stabilization, not absence of pathology.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-regularity}\n\ + Let $f : \\tau^3 \\to H_\\tau$ be $\\tau$-holomorphic\n(equivalently, by II.T33:\nidempotent-supported via the decomposition\n\ + $f = e_+ \\cdot f_+ + e_- \\cdot f_-$\nof the Idempotent Decomposition Lemma, II.L07).\nA point $p \\in \\tau^3$ is\n\\\ + textbf{$\\tau$-regular for $f$}\nif there exists a primorial stage $N \\geq 1$\nsuch that the $\\omega$-germ sequence\ + \ of $f$ at $p$\nstabilizes at stage $N$:\n\\[\n \\boxed{%\n p \\text{ is $\\tau$-regular for } f\n \\quad:\\\ + Longleftrightarrow\\quad\n \\exists\\, N \\geq 1,\\;\n \\forall\\, k \\geq N:\\;\n \\rho_{k,N}\\bigl(G_f\\bigr)\n\ + \ = G_f \\big|_{C_N(p)}.}\n\\]\nThe minimal such $N$ is called the\n\\textbf{regularity depth} of $f$ at $p$,\ndenoted\ + \ $\\mathrm{rd}_f(p)$.\n\nA function $f$ is \\textbf{everywhere $\\tau$-regular}\nif every point $p \\in \\tau^3$\nis\ + \ $\\tau$-regular for $f$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 115\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch39-regularity-positive.tex`\ + \ lines 286-315\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.PositiveRegularity`\n\ + - Name: `is_regular`\n\n## Dependencies\n\n- Canonical: II.D37, II.D10, II.L07, II.T33, I.T31\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A point p is tau-regular for f if the omega-germ sequence stabilizes at a finite primorial stage. A constructive, + positive criterion: existence of stabilization, not absence of pathology.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001285.md b/_corpus_v3_items/cid001285.md new file mode 100644 index 000000000..eabe8b674 --- /dev/null +++ b/_corpus_v3_items/cid001285.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001285/ +title: Pre-Yoneda Embedding +cid: cid001285 +primary_alias: DEF0159 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Pre-Yoneda Embedding +aliases: +- DEF0159 +- II.D50 +- pre-yoneda-embedding +- def:pre-yoneda +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005664 +- predicate: formalized_by + object: cid005665 +- predicate: formalized_by + object: cid005666 +- predicate: formalized_by + object: cid005671 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 40 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Pre-Yoneda Embedding**\n\nPre-Yoneda Embedding\n\n# Pre-Yoneda Embedding\n\n## Summary\n\nPre-Yoneda Embedding\n\ + \n## Statement\n\n```latex\n%\n\\label{def:pre-yoneda}\nThe \\textbf{Pre-Yoneda embedding} is the map\n\\[\n \\boxed{%\n\ + \ y \\;:\\;\n \\mathrm{Hol}_\\tau(\\tau^3,\\, H_\\tau)\n \\;\\hookrightarrow\\;\n d(\\tau^3),\n \\qquad\n\ + \ y(f) \\;:=\\; [G_f],}\n\\]\ndefined as follows.\nLet $f \\in \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau)$\nbe a $\\tau$-holomorphic\ + \ function.\nBy the Mutual Determination Theorem (II.T27),\n$f$ is uniquely determined by its $\\omega$-germ\n$[f] = [(f_k)_{k\ + \ \\geq 1}]$.\nThe $\\omega$-germ transformer $G_f$\n(II.D37, Chapter~\\ref{ch:evolution-operator})\nis the coherent family\ + \ of stage maps\n$\\{(G_f)_k\\}_{k \\geq 1}$\ndefined by the action of $f$\non the finite stages:\n$(G_f)_k \\colon \\\ + mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau$\nis the stage-$k$ component of $f$.\n\nThe map $y$ sends the holomorphic function\ + \ $f$\nto its $\\omega$-germ class $[G_f] \\in d(\\tau^3)$.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 118\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch40-pre-yoneda.tex` lines 214-244\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.PreYoneda`\n- Name: `preyoneda_embed`\n\n\ + ## Dependencies\n\n- Canonical: I.T40, II.D37, II.D47, II.T27, II.L07\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Pre-Yoneda Embedding +noindex: false +--- + diff --git a/_corpus_v3_items/cid001286.md b/_corpus_v3_items/cid001286.md new file mode 100644 index 000000000..ef511fb0a --- /dev/null +++ b/_corpus_v3_items/cid001286.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001286/ +title: Code Map +cid: cid001286 +primary_alias: DEF0160 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A pair of bipolar boundary coefficient streams recording spectral data at successive primorial stages. Encodes a + holomorphic function via its boundary. +aliases: +- DEF0160 +- II.D51 +- code-map +- def:code +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005647 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 41 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Code Map**\n\nA pair of bipolar boundary coefficient streams recording spectral data at successive primorial\ + \ stages. Encodes a holomorphic function via its boundary.\n\n# Code Map\n\n## Summary\n\nA pair of bipolar boundary coefficient\ + \ streams recording spectral data at successive primorial stages. Encodes a holomorphic function via its boundary.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:code}\nA \\textbf{code} is a pair $\\mathbf{c} = (c^+, c^-)$,\nwhere each component\ + \ is a\n\\textbf{bipolar boundary coefficient stream}:\n\\[\n \\boxed{%\n c^\\pm\n \\;=\\;\n \\bigl(\\,\n\ + \ c^\\pm_1,\\;\n c^\\pm_2,\\;\n c^\\pm_3,\\;\n \\ldots\n \\,\\bigr),}\n\\]\nsubject to the following conditions.\n\ + \n\\begin{enumerate}\n \\item[\\textup{(C1)}]\n \\textbf{Stage data.}\n For each $k \\geq 1$,\n \ + \ the entry $c^\\pm_k$ is a finite family\n of spectral coefficients:\n \\[\n c^\\\ + pm_k\n \\;=\\;\n \\bigl\\{\\,\n \\varphi_{p,v}^{(\\pm)}\n \\;\\big|\\\ + ;\n p \\mid P_k,\\;\n v \\in \\mathbb{Z}/p\\mathbb{Z}\n \\,\\bigr\\},\n \ + \ \\]\n with each $\\varphi_{p,v}^{(+)} \\in A_\\tau^{(B)}$\n and each $\\varphi_{p,v}^{(-)} \\in\ + \ A_\\tau^{(C)}$.\n Here $(+)$ refers to the B-channel\n (associated with $e_+$)\n and $(-)$\ + \ to the C-channel\n (associated with $e_-$).\n\n \\item[\\textup{(C2)}]\n \\textbf{Tower coherence.}\n\ + \ For each $k \\geq 1$\n and each prime $p \\mid P_k$,\n the coefficient $\\varphi_{p,v}^{(\\\ + pm)}$\n in the stage-$k$ entry\n equals the coefficient $\\varphi_{p,v}^{(\\pm)}$\n in the\ + \ stage-$(k+1)$ entry.\n That is,\n the restriction maps of the primorial tower\n send $c^\\\ + pm_{k+1}$ to $c^\\pm_k$:\n \\[\n \\rho_{k+1,k}\\bigl(c^\\pm_{k+1}\\bigr)\n \\;=\\;\n\ + \ c^\\pm_k.\n \\]\n\n \\item[\\textup{(C3)}]\n \\textbf{Finite spectral support at each\ + \ stage.}\n At each stage~$k$,\n only finitely many coefficients are nonzero.\n (This is automatic\ + \ from the finiteness\n of $\\Lambda_k$, but we record it\n for emphasis.)\n\\end{enumerate}\n\n\\medskip\\\ + noindent\nThe \\textbf{space of codes} is denoted\n\\[\n \\boxed{%\n \\mathrm{Code}_\\tau\n \\;:=\\;\n \\\ + bigl\\{\\,\n \\mathbf{c} = (c^+, c^-)\n \\;\\big|\\;\n \\text{(C1), (C2), (C3) hold}\n \\,\\bigr\\}.}\n\\\ + ]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 121\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch41-code-decode.tex`\ + \ lines 119-199\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.CodeDecode`\n\ + - Name: `code_extract`\n\n## Dependencies\n\n- Canonical: II.D45, II.D46, II.L07, I.D21\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A pair of bipolar boundary coefficient streams recording spectral data at successive primorial stages. Encodes + a holomorphic function via its boundary. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001287.md b/_corpus_v3_items/cid001287.md new file mode 100644 index 000000000..f6a34881a --- /dev/null +++ b/_corpus_v3_items/cid001287.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001287/ +title: Decode Map +cid: cid001287 +primary_alias: DEF0161 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The map reconstructing a tau-holomorphic function from a bipolar boundary coefficient stream via stage expansion, + idempotent assembly, and limit. +aliases: +- DEF0161 +- II.D52 +- decode-map +- def:decode +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005648 +- predicate: formalized_by + object: cid005649 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 41 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Decode Map**\n\nThe map reconstructing a tau-holomorphic function from a bipolar boundary coefficient stream\ + \ via stage expansion, idempotent assembly, and limit.\n\n# Decode Map\n\n## Summary\n\nThe map reconstructing a tau-holomorphic\ + \ function from a bipolar boundary coefficient stream via stage expansion, idempotent assembly, and limit.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:decode}\nThe \\textbf{Decode} map takes a code\n$\\mathbf{c} = (c^+, c^-)\n\\in \\mathrm{Code}_\\\ + tau$\nand produces a $\\tau$-holomorphic function\n$f \\in \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau^{\\mathrm{cal}})$\nvia\ + \ the following three-step construction:\n\\[\n \\boxed{%\n \\mathrm{Decode}\n \\;:\\;\n \\mathrm{Code}_\\\ + tau\n \\;\\longrightarrow\\;\n \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau^{\\mathrm{cal}}),\n \\qquad\n \\mathbf{c}\n\ + \ \\;\\longmapsto\\;\n f = \\lim_{\\longleftarrow} f_k.}\n\\]\n\n\\begin{enumerate}\n \\item[\\textup{(D1)}]\n\ + \ \\textbf{Stage-$k$ reconstruction.}\n For each $k \\geq 1$,\n use the stage-$k$ coefficient\ + \ data\n $c^\\pm_k$ to build the stage-$k$ function:\n \\[\n f_k(x)\n \\;:=\\\ + ;\n \\sum_{(p,v) :\\, p \\mid P_k}\n \\Bigl(\n \\varphi_{p,v}^{(+)} \\cdot E_{k,v}^{(B)}(x)\n\ + \ \\;+\\;\n \\varphi_{p,v}^{(-)} \\cdot E_{k,v}^{(C)}(x)\n \\Bigr)\n \\\ + ]\n for all $x \\in \\mathbb{Z}/P_k\\mathbb{Z}$.\n This is a finite sum\n (condition~(C3) of\ + \ Definition~\\ref{def:code}),\n so $f_k$ is a well-defined function\n on the stage-$k$ cyclic group.\n\ + \n \\item[\\textup{(D2)}]\n \\textbf{Tower compatibility.}\n The tower coherence condition~(C2) guarantees\n\ + \ that the family $(f_k)_{k \\geq 1}$\n is compatible under the tower projection maps:\n $\\\ + pi_{k+1,k} \\circ f_{k+1} = f_k$.\n This is verified directly:\n the shared primes $p \\mid P_k$\n \ + \ contribute identical coefficients at stages~$k$ and $k+1$\n (by~(C2)),\n while the new prime\ + \ $p_{k+1}$\n contributes terms that are projected away by $\\pi_{k+1,k}$.\n\n \\item[\\textup{(D3)}]\n \ + \ \\textbf{Inverse limit.}\n Define $f := \\varprojlim_k f_k$\n as the unique element of the\ + \ inverse limit\n whose stage-$k$ projection is~$f_k$\n for every~$k$.\n Tower compatibility~(D2)\ + \ ensures existence.\n Uniqueness is automatic from the inverse limit construction.\n\\end{enumerate}\n```\n\n\ + ## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 122\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch41-code-decode.tex`\ + \ lines 323-389\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.CodeDecode`\n\ + - Name: `decode_reconstruct`\n\n## Dependencies\n\n- Canonical: II.D51, II.D45, II.D46, II.D47, II.T32, I.T31\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The map reconstructing a tau-holomorphic function from a bipolar boundary coefficient stream via stage expansion, + idempotent assembly, and limit. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001288.md b/_corpus_v3_items/cid001288.md new file mode 100644 index 000000000..a381b61be --- /dev/null +++ b/_corpus_v3_items/cid001288.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001288/ +title: Self-Enrichment Structure +cid: cid001288 +primary_alias: DEF0162 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Self-Enrichment Structure +aliases: +- DEF0162 +- II.D53 +- self-enrichment-structure +- def:self-enrichment +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005510 +- predicate: formalized_by + object: cid005511 +- predicate: formalized_by + object: cid005517 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 42 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Self-Enrichment Structure**\n\nSelf-Enrichment Structure\n\n# Self-Enrichment Structure\n\n## Summary\n\nSelf-Enrichment\ + \ Structure\n\n## Statement\n\n```latex\n%\n\\label{def:self-enrichment}\nCategory~$\\tau$ is \\textbf{self-enriched}:\n\ + it is enriched over itself\nas a monoidal category\n$(\\tau, \\times, \\mathbf{1})$,\nwhere $\\times$ denotes the Cartesian\ + \ product\nin~$\\tau$\nand $\\mathbf{1}$ is the terminal object.\nConcretely, self-enrichment means\nthe following three\ + \ conditions hold:\n\\begin{enumerate}\n \\item[\\textup{(SE1)}]\n \\textbf{Internal Hom objects.}\n \ + \ For every pair $A, B \\in \\mathrm{Obj}(\\tau)$,\n the Hom object\n $[A,B]$\n (Definition~\\\ + ref{def:hom-object}, II.D54)\n is an object of~$\\tau$.\n\n \\item[\\textup{(SE2)}]\n \\textbf{Composition\ + \ is a $\\tau$-morphism.}\n For every triple $A, B, C \\in \\mathrm{Obj}(\\tau)$,\n the composition\ + \ map\n \\[\n \\circ_{A,B,C}\n \\;:\\;\n [B,C] \\times [A,B]\n \ + \ \\;\\longrightarrow\\;\n [A,C]\n \\]\n is a morphism in~$\\tau$---i.e.,\n \ + \ it is $\\tau$-holomorphic,\n NF-addressable, tower-coherent,\n and valued in $H_\\tau$.\n\n \ + \ \\item[\\textup{(SE3)}]\n \\textbf{Identity is a $\\tau$-morphism.}\n For every $A \\in \\mathrm{Obj}(\\\ + tau)$,\n the identity selection\n \\[\n \\mathrm{id}_A\n \\;:\\;\n \ + \ \\mathbf{1}\n \\;\\longrightarrow\\;\n [A,A]\n \\]\n is a morphism\ + \ in~$\\tau$---i.e.,\n the identity map $\\mathrm{id}_A$\n is a $\\tau$-object\n sitting inside\ + \ the Hom object $[A,A]$.\n\\end{enumerate}\n\n\\medskip\\noindent\nThese three conditions are subject to the standard\n\ + \\textbf{enriched category axioms}:\n\\begin{enumerate}\n \\item[\\textup{(EC1)}]\n \\emph{Associativity.}\n\ + \ The diagram\n \\[\n [C,D] \\times [B,C] \\times [A,B]\n \\;\\rightrightarrows\\\ + ;\n [A,D]\n \\]\n commutes: composing $(h,g,f)$ as\n $(h \\circ g) \\circ f$ or\ + \ $h \\circ (g \\circ f)$\n yields the same result.\n\n \\item[\\textup{(EC2)}]\n \\emph{Unit laws.}\n\ + \ For all $A, B$:\n $\\mathrm{id}_B \\circ f = f = f \\circ \\mathrm{id}_A$\n in $[A,B]$.\n\ + \\end{enumerate}\nBoth (EC1) and (EC2) hold in~$\\tau$\nby the associativity theorem\n(Theorem~\\ref{thm:associativity},\ + \ II.T29,\nChapter~\\ref{ch:composition-structure})\nand the identity construction\n(Definition~\\ref{def:identity-map},\ + \ II.D40,\nChapter~\\ref{ch:composition-structure}).\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 125\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch42-tau-self-enrichment.tex` lines 393-476\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.SelfEnrichment`\n- Name:\ + \ `Tau.BookII.Enrichment.hom_stage`\n\n## Dependencies\n\n- Canonical: I.D20, I.D21, I.T05, I.T40, II.L07, II.D49, II.T33,\ + \ II.D50\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Self-Enrichment Structure +noindex: false +--- + diff --git a/_corpus_v3_items/cid001289.md b/_corpus_v3_items/cid001289.md new file mode 100644 index 000000000..626b244ee --- /dev/null +++ b/_corpus_v3_items/cid001289.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001289/ +title: Hom Object +cid: cid001289 +primary_alias: DEF0163 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The internal Hom object as the inverse limit of stage-k morphism sets. A tau-object with NF address, bipolar decomposition, + and tower coherence. +aliases: +- DEF0163 +- II.D54 +- hom-object +- def:hom-object +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005512 +- predicate: formalized_by + object: cid005513 +- predicate: formalized_by + object: cid005514 +- predicate: formalized_by + object: cid005520 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 42 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Hom Object**\n\nThe internal Hom object as the inverse limit of stage-k morphism sets. A tau-object with NF\ + \ address, bipolar decomposition, and tower coherence.\n\n# Hom Object\n\n## Summary\n\nThe internal Hom object as the\ + \ inverse limit of stage-k morphism sets. A tau-object with NF address, bipolar decomposition, and tower coherence.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:hom-object}\nFor $A, B \\in \\mathrm{Obj}(\\tau)$,\nthe \\textbf{Hom object}\ + \ $[A,B]$ is defined as the inverse limit\n\\[\n \\boxed{%\n [A,B]\n \\;:=\\;\n \\varprojlim_{k}\n \\,\\\ + Hom_{\\tau_k}(A_k, B_k),}\n\\]\nwhere the connecting maps\n$r_{k+1,k} : \\Hom_{\\tau_{k+1}}(A_{k+1}, B_{k+1})\n\\to \\\ + Hom_{\\tau_k}(A_k, B_k)$\nare the natural restriction maps:\n\\[\n r_{k+1,k}(f_{k+1})\n \\;:=\\;\n f_{k+1} \\\ + bmod P_k,\n\\]\ni.e., the reduction of a stage-$(k+1)$ morphism\nto its stage-$k$ shadow\nvia the natural projection\n\ + $\\mathbb{Z}/P_{k+1}\\mathbb{Z}\n\\to \\mathbb{Z}/P_k\\mathbb{Z}$.\n\n\\medskip\\noindent\nThe Hom object carries the\ + \ following structure:\n\\begin{enumerate}\n \\item[\\textup{(H1)}]\n \\textbf{NF-addressability.}\n \ + \ An element of $[A,B]$\n is a tower-coherent sequence $(f_k)_{k \\geq 1}$\n of finite morphisms.\n\ + \ Each $f_k$ has an NF-address at stage~$k$.\n The coherent family\n $(f_k)_{k \\geq 1}$ determines\n\ + \ a profinite NF-address for $[A,B]$.\n\n \\item[\\textup{(H2)}]\n \\textbf{$H_\\tau$-valued.}\n\ + \ Each $f_k$ is valued in $H_\\tau$.\n The limit inherits $H_\\tau$-valued structure\n by pointwise\ + \ inverse limit.\n\n \\item[\\textup{(H3)}]\n \\textbf{Tower coherence.}\n The connecting maps $r_{k+1,k}$\n\ + \ ensure that the sequence $(f_k)$ satisfies\n $f_k \\equiv f_{k+1} \\pmod{P_k}$---the\n same\ + \ tower coherence condition\n that defines $\\tau$-holomorphic maps.\n\n \\item[\\textup{(H4)}]\n \ + \ \\textbf{Ultrametric structure.}\n The inverse limit inherits the ultrametric topology\n from the\ + \ tower\n (Chapter~\\ref{ch:ultrametric-depth}, II.D13):\n two elements of $[A,B]$ are close\n \ + \ if they agree at deep stages.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 126\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch42-tau-self-enrichment.tex` lines 263-325\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.SelfEnrichment`\n- Name:\ + \ `Tau.BookII.Enrichment.hom_obj_count_affine`\n\n## Dependencies\n\n- Canonical: I.D20, I.D21, I.T05, II.D49, II.L07\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The internal Hom object as the inverse limit of stage-k morphism sets. A tau-object with NF address, bipolar + decomposition, and tower coherence. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001290.md b/_corpus_v3_items/cid001290.md new file mode 100644 index 000000000..4782a1671 --- /dev/null +++ b/_corpus_v3_items/cid001290.md @@ -0,0 +1,114 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001290/ +title: 2-Category Structure +cid: cid001290 +primary_alias: DEF0164 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 2-Category Structure +aliases: +- DEF0164 +- II.D55 +- 2-category-structure +- def:two-category +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005521 +- predicate: formalized_by + object: cid005522 +- predicate: formalized_by + object: cid005523 +- predicate: formalized_by + object: cid005529 +- predicate: formalized_by + object: cid005530 +- predicate: formalized_by + object: cid005531 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 44 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**2-Category Structure**\n\n2-Category Structure\n\n# 2-Category Structure\n\n## Summary\n\n2-Category Structure\n\ + \n## Statement\n\n```latex\n%\n\\label{def:two-category}\nThe \\textbf{2-category $\\tau_2$} has the following data:\n\ + \\begin{enumerate}\n \\item[\\textup{(0C)}]\n \\textbf{0-cells.}\n The objects of~$\\tau$:\n \ + \ \\[\n \\mathrm{Obj}(\\tau_2) \\;:=\\; \\mathrm{Obj}(\\tau).\n \\]\n These are the\ + \ NF-addressable objects\n from Book~I,\n organized into the primorial tower.\n\n \\item[\\textup{(1C)}]\n\ + \ \\textbf{1-cells.}\n For each pair of 0-cells $A, B$,\n the 1-cells from $A$ to~$B$\n \ + \ are the elements of the Hom object:\n \\[\n \\tau_2(A,B) \\;:=\\; [A,B]\n \\\ + ;=\\; \\Hom(A,B)\n \\;\\in\\; \\mathrm{Obj}(\\tau).\n \\]\n This is the internal hom\n\ + \ of the self-enrichment (II.D54).\n\n \\item[\\textup{(2C)}]\n \\textbf{2-cells.}\n For\ + \ each pair of 1-cells\n $f, g \\in [A,B]$,\n the 2-cells from $f$ to~$g$\n are the elements\ + \ of the iterated hom:\n \\[\n \\tau_2(f,g) \\;:=\\; [f,g]\n \\;\\subset\\; \\bigl[[A,B],\\\ + ,[A,B]\\bigr].\n \\]\n A 2-cell $\\alpha \\colon f \\Rightarrow g$\n is a ``directed modification''\n\ + \ from the morphism~$f$\n to the morphism~$g$.\n\n \\item[\\textup{(VC)}]\n \\textbf{Vertical\ + \ composition.}\n Given 2-cells\n $\\alpha \\colon f \\Rightarrow g$\n and $\\beta \\colon\ + \ g \\Rightarrow h$,\n their vertical composite\n $\\beta \\circ_v \\alpha \\colon f \\Rightarrow h$\n\ + \ is the composition of endomorphisms\n in $\\bigl[[A,B],[A,B]\\bigr]$:\n \\[\n \ + \ \\beta \\circ_v \\alpha\n \\;:=\\;\n \\beta \\circ \\alpha\n \\;\\in\\; \\\ + bigl[[A,B],[A,B]\\bigr].\n \\]\n This is ordinary composition\n in the endomorphism space,\n\ + \ which is well-defined because\n $\\bigl[[A,B],[A,B]\\bigr]$\n is a $\\tau$-object with its\ + \ own composition\n (inherited from $\\tau$'s enriched structure).\n\n \\item[\\textup{(HC)}]\n \\\ + textbf{Horizontal composition.}\n Given 2-cells\n $\\alpha \\colon f \\Rightarrow f'$\n (where\ + \ $f, f' \\colon A \\to B$)\n and $\\beta \\colon g \\Rightarrow g'$\n (where $g, g' \\colon B \\to\ + \ C$),\n the horizontal composite\n $\\beta \\circ_h \\alpha \\colon g \\circ f \\Rightarrow g' \\circ\ + \ f'$\n is defined by the functoriality\n of the composition map\n $\\mu_{A,B,C} \\colon [B,C]\ + \ \\times [A,B] \\to [A,C]$:\n \\[\n \\beta \\circ_h \\alpha\n \\;:=\\;\n \ + \ \\mu_{A,B,C}(\\beta, \\alpha).\n \\]\n The composition map~$\\mu_{A,B,C}$\n is a $\\\ + tau$-morphism\n (it is internal to $\\tau$'s enriched structure),\n so its action on 2-cells\n \ + \ produces a well-defined 2-cell.\n\n \\item[\\textup{(ID)}]\n \\textbf{Identities.}\n \\begin{itemize}\n\ + \ \\item For each 0-cell~$A$:\n the identity 1-cell\n $\\id_A \\in\ + \ [A,A]$.\n \\item For each 1-cell $f \\in [A,B]$:\n the identity 2-cell\n \ + \ $\\mathrm{id}_f \\colon f \\Rightarrow f$,\n which is the identity element\n \ + \ in $[f,f] \\subset [[A,B],[A,B]]$.\n \\end{itemize}\n\n \\item[\\textup{(IL)}]\n \\textbf{Interchange\ + \ law.}\n For 2-cells\n $\\alpha \\colon f \\Rightarrow f'$,\n $\\alpha' \\colon f' \\Rightarrow\ + \ f''$\n (both in $[[A,B],[A,B]]$),\n $\\beta \\colon g \\Rightarrow g'$,\n $\\beta' \\colon\ + \ g' \\Rightarrow g''$\n (both in $[[B,C],[B,C]]$):\n \\[\n \\boxed{%\n (\\\ + beta' \\circ_v \\beta)\n \\circ_h\n (\\alpha' \\circ_v \\alpha)\n \\;=\\;\n \ + \ (\\beta' \\circ_h \\alpha')\n \\circ_v\n (\\beta \\circ_h \\alpha).}\n \ + \ \\]\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 130\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch44-two-categories.tex`\ + \ lines 141-262\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.TwoCategories`\n\ + - Name: `Tau.BookII.Enrichment.TwoCat`\n\n## Dependencies\n\n- Canonical: II.D53, II.D54, II.P11, II.T36, II.L07\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 2-Category Structure +noindex: false +--- + diff --git a/_corpus_v3_items/cid001291.md b/_corpus_v3_items/cid001291.md new file mode 100644 index 000000000..400ce7690 --- /dev/null +++ b/_corpus_v3_items/cid001291.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001291/ +title: 2-Morphism +cid: cid001291 +primary_alias: DEF0165 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A 2-cell between 1-cells in tau_2, defined as an element of the internal Hom-Hom. Well-defined because Hom objects + are tau-objects via self-enrichment. +aliases: +- DEF0165 +- II.D56 +- 2-morphism +- def:two-morphism +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005524 +- predicate: formalized_by + object: cid005525 +- predicate: formalized_by + object: cid005532 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 44 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**2-Morphism**\n\nA 2-cell between 1-cells in tau_2, defined as an element of the internal Hom-Hom. Well-defined\ + \ because Hom objects are tau-objects via self-enrichment.\n\n# 2-Morphism\n\n## Summary\n\nA 2-cell between 1-cells in\ + \ tau_2, defined as an element of the internal Hom-Hom. Well-defined because Hom objects are tau-objects via self-enrichment.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:two-morphism}\nA \\textbf{2-morphism} $\\alpha \\colon f \\Rightarrow g$,\n\ + where $f, g \\colon A \\to B$ are 1-cells in $\\tau_2$,\nis an element\n\\[\n \\alpha \\;\\in\\; [f,g]\n \\;\\subset\\\ + ;\n \\bigl[[A,B],\\,[A,B]\\bigr].\n\\]\nSince $[A,B]$ is a $\\tau$-object (II.D54)\nand $\\tau$ enriches over itself\ + \ (II.D53),\nthe space $[f,g]$ is well-defined.\n\n\\medskip\n\\noindent\n\\textbf{Structure inherited by 2-morphisms:}\n\ + \\begin{enumerate}\n \\item[\\textup{(S1)}]\n \\textbf{NF-addressability.}\n The 2-morphism $\\alpha$\ + \ has an NF-address\n in the tower\n $\\bigl[[A,B],[A,B]\\bigr]$.\n At each stage~$k$,\n \ + \ $\\alpha$ restricts to a map\n $\\alpha_k \\colon\n \\Hom_k(A,B) \\to \\Hom_k(A,B)$\n \ + \ on the stage-$k$ morphism space,\n and the system\n $(\\alpha_k)_{k \\geq n}$ is tower-coherent.\n\ + \n \\item[\\textup{(S2)}]\n \\textbf{Bipolar decomposition.}\n By the Idempotent Decomposition Lemma\n\ + \ (II.L07, Chapter~\\ref{ch:idempotent-decomposition}),\n every element of $H_\\tau$\n decomposes\ + \ via the bipolar idempotents.\n Since $[[A,B],[A,B]]$\n is valued in~$H_\\tau$,\n the 2-morphism\ + \ inherits the decomposition:\n \\[\n \\boxed{%\n \\alpha\n \\;=\\;\n\ + \ e_+ \\cdot \\alpha_+\n \\;+\\;\n e_- \\cdot \\alpha_-,}\n \\]\n \ + \ where $\\alpha_\\pm = e_\\pm \\cdot \\alpha$\n are the projections\n onto the B-channel and C-channel\ + \ respectively.\n\n \\item[\\textup{(S3)}]\n \\textbf{Channel independence.}\n The B-channel component\ + \ $\\alpha_+$\n and the C-channel component $\\alpha_-$\n are independent:\n modifying $\\\ + alpha_+$\n does not affect~$\\alpha_-$\n and conversely.\n This follows from\n $e_+\ + \ \\cdot e_- = 0$\n (the idempotents are orthogonal).\n\n \\item[\\textup{(S4)}]\n \\textbf{Holomorphic\ + \ structure.}\n The 2-morphism $\\alpha$\n is $\\tau$-holomorphic:\n it satisfies the split-complex\n\ + \ Cauchy--Riemann equations\n in the variables of $[[A,B],[A,B]]$.\n This is because $\\alpha$\n\ + \ is an element of a $\\tau$-object,\n and all $\\tau$-objects carry\n holomorphic structure\n\ + \ (by the results of Parts~VI--VII).\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 131\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch44-two-categories.tex` lines 326-404\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.TwoCategories`\n- Name: `Tau.BookII.Enrichment.TwoCell`\n\ + \n## Dependencies\n\n- Canonical: II.D53, II.D54, II.D55, II.L07, II.P11\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A 2-cell between 1-cells in tau_2, defined as an element of the internal Hom-Hom. Well-defined because Hom objects + are tau-objects via self-enrichment. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001292.md b/_corpus_v3_items/cid001292.md new file mode 100644 index 000000000..5e488ff2c --- /dev/null +++ b/_corpus_v3_items/cid001292.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001292/ +title: E1 Enrichment Layer +cid: cid001292 +primary_alias: DEF0166 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Category tau equipped with self-enrichment: internal Hom objects, Yoneda embedding, Code/Decode bijection, 2-category + structure, and bipolar Hom decomposition.' +aliases: +- DEF0166 +- II.D57 +- e1-enrichment-layer +- def:e1-layer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005509 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 45 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E1 Enrichment Layer**\n\nCategory tau equipped with self-enrichment: internal Hom objects, Yoneda embedding,\ + \ Code/Decode bijection, 2-category structure, and bipolar Hom decomposition.\n\n# E1 Enrichment Layer\n\n## Summary\n\ + \nCategory tau equipped with self-enrichment: internal Hom objects, Yoneda embedding, Code/Decode bijection, 2-category\ + \ structure, and bipolar Hom decomposition.\n\n## Statement\n\n```latex\n%\n\\label{def:e1-layer}\n$\\Estage{1}$ denotes\ + \ Category~$\\T$\nequipped with the self-enrichment data\nearned in Part~VIII.\nExplicitly, $\\Estage{1}$ adds to~$\\\ + Estage{0}$\nthe following structures:\n\\begin{enumerate}\n \\item \\textbf{Hom objects as $\\T$-objects.}\n \ + \ For all $A, B \\in \\Obj(\\T)$,\n the morphism space $\\Hom(A,B) \\in \\Obj(\\T)$\n (II.D53, II.D54,\n\ + \ Chapter~\\ref{ch:tau-self-enrichment}).\n Each Hom object has an NF address,\n inherits bipolar\ + \ decomposition,\n and satisfies tower coherence.\n\n \\item \\textbf{The Yoneda embedding.}\n $\\\ + T \\hookrightarrow [\\T^{\\op}, \\T]$\n is a full and faithful functor\n (II.T36,\n Chapter~\\\ + ref{ch:yoneda-theorem}).\n This makes every $\\T$-object\n representable by the morphisms into it.\n\ + \n \\item \\textbf{2-categorical structure.}\n 2-morphisms\n $\\Hom(\\Hom(A,B), \\Hom(C,D))$\n \ + \ are $\\T$-objects\n (II.D55, II.D56,\n Chapter~\\ref{ch:two-categories}).\n Morphisms\ + \ between morphisms\n are first-class citizens.\n\n \\item \\textbf{Internal function spaces.}\n \ + \ The internal Hom\n $[A,B] \\in \\Obj(\\T)$\n replaces the external function set\n $\\{f \\\ + colon A \\to B\\}$.\n Function spaces are objects,\n not meta-level collections.\n\n \\item \\textbf{The\ + \ Code/Decode bijection.}\n $\\mathrm{Code} \\colon \\mathcal{O}(\\tau^3)\n \\to \\mathrm{Streams}(R_\\\ + tau, H_\\tau)$\n and its inverse $\\mathrm{Decode}$\n (II.T35,\n Chapter~\\ref{ch:code-decode})\n\ + \ internalize the Mutual Determination\n as a coding-theoretic bijection.\n\n \\item \\textbf{Self-referential\ + \ capability.}\n $\\T$ can talk about its own morphisms,\n compare them,\n and transform them---all\n\ + \ within its own language\n (Remark~\\ref{rem:self-description}, II.R15).\n\\end{enumerate}\n\n\\noindent\n\ + The transition $\\Estage{0} \\to \\Estage{1}$\nis what Part~VIII accomplishes.\n$\\Estage{1}$ is not a different category;\n\ + it is the \\emph{same} category~$\\T$,\nnow recognized as carrying self-enrichment structure\nthat was latent in $\\Estage{0}$\n\ + but not yet articulated.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 134\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch45-self-describing.tex`\ + \ lines 571-639\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.SelfDescribing`\n\ + - Name: `Tau.BookII.Enrichment.E1Layer`\n\n## Dependencies\n\n- Canonical: II.D53, II.D54, II.T35, II.D55, II.D56, II.P12\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Category tau equipped with self-enrichment: internal Hom objects, Yoneda embedding, Code/Decode bijection, 2-category + structure, and bipolar Hom decomposition.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001293.md b/_corpus_v3_items/cid001293.md new file mode 100644 index 000000000..02713987a --- /dev/null +++ b/_corpus_v3_items/cid001293.md @@ -0,0 +1,94 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001293/ +title: E0/E1 Transition +cid: cid001293 +primary_alias: DEF0167 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The passage from E_0 (finite/profinite elements, external morphisms, algebraic bi-square) to E_1 (internal Hom, self-enrichment, + Code/Decode, 2-category structure). +aliases: +- DEF0167 +- II.D58 +- e0-e1-transition +- def:e0-e1-transition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005495 +- predicate: formalized_by + object: cid005496 +- predicate: formalized_by + object: cid005497 +- predicate: formalized_by + object: cid005498 +- predicate: formalized_by + object: cid005499 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 46 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E0/E1 Transition**\n\nThe passage from E_0 (finite/profinite elements, external morphisms, algebraic bi-square)\ + \ to E_1 (internal Hom, self-enrichment, Code/Decode, 2-category structure).\n\n# E0/E1 Transition\n\n## Summary\n\nThe\ + \ passage from E_0 (finite/profinite elements, external morphisms, algebraic bi-square) to E_1 (internal Hom, self-enrichment,\ + \ Code/Decode, 2-category structure).\n\n## Statement\n\n```latex\n%\n\\label{def:e0-e1-transition}\nThe \\textbf{$\\\ + Estage{0}/\\Estage{1}$ transition}\nis the passage from the structural content\nof~$\\Estage{0}$ (Groups~A--D above)\n\ + to the enriched content of~$\\Estage{1}$:\n\n\\medskip\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{@{}p{0.15\\\ + linewidth} p{0.38\\linewidth} p{0.38\\linewidth}@{}}\n\\hline\n& \\textbf{$\\Estage{0}$} & \\textbf{$\\Estage{1}$} \\\\\ + \n\\hline\n\\textit{Objects}\n & Finite and profinite elements of~$\\T$\n & Same, plus $\\Hom(A,B)$ as objects \\\ + \\\n\\textit{Morphisms}\n & $\\omega$-germ transformers\n & Same, plus 2-morphisms between Hom objects \\\\\n\\\ + textit{Holomorphy}\n & Boundary characters; Global Hartogs\n & Interior holomorphy; Central Theorem \\\\\n\\textit{Constants}\n\ + \ & $\\iota_\\tau$ (algebraic), $\\pi$, $e$ (earned)\n & Same, now with geometric confirmation \\\\\n\\textit{Coordinate\ + \ data}\n & ABCD chart $\\Phi(x)$\n & Same, extended to $\\tau^3$ fibration \\\\\n\\textit{Self-reference}\n \ + \ & None\n & $\\T$ enriches over itself; Yoneda \\\\\n\\textit{Categoricity}\n & Unique normal form (NF)\n &\ + \ Unique category ($\\tau^3$ discovered, not constructed) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\n\\medskip\\\ + noindent\nThe transition is characterized by a single structural fact:\n\\[\n \\boxed{%\n \\text{At } \\Estage{0},\\\ + ;\n \\Hom(A,B) \\text{ is an external set.}\n \\quad\n \\text{At } \\Estage{1},\\;\n \\Hom(A,B) \\in \\Obj(\\\ + T).}\n\\]\nThis internalization is what enables the Yoneda embedding (II.T36),\nwhich in turn enables the Central Theorem\ + \ (II.T40).\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 136\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch46-book3-foundation.tex`\ + \ lines 219-271\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.EnrichmentLadder`\n\ + - Name: `Tau.BookII.Enrichment.EnrichmentLevel`\n\n## Dependencies\n\n- Canonical: I.D82, II.D53, II.D54, II.D55, II.D57,\ + \ II.T36\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The passage from E_0 (finite/profinite elements, external morphisms, algebraic bi-square) to E_1 (internal Hom, + self-enrichment, Code/Decode, 2-category structure). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001294.md b/_corpus_v3_items/cid001294.md new file mode 100644 index 000000000..f95cf272a --- /dev/null +++ b/_corpus_v3_items/cid001294.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001294/ +title: Idempotent-Supported Character +cid: cid001294 +primary_alias: DEF0168 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Idempotent-Supported Character +aliases: +- DEF0168 +- II.D59 +- idempotent-supported-character +- def:idempotent-character +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005362 +- predicate: formalized_by + object: cid005363 +- predicate: formalized_by + object: cid005364 +- predicate: formalized_by + object: cid005365 +- predicate: formalized_by + object: cid005369 +- predicate: formalized_by + object: cid005370 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 47 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Idempotent-Supported Character**\n\nIdempotent-Supported Character\n\n# Idempotent-Supported Character\n\n\ + ## Summary\n\nIdempotent-Supported Character\n\n## Statement\n\n```latex\n%\n\\label{def:idempotent-character}\nA spectral\ + \ character\n$\\chi : \\hat{\\mathbb{Z}}_\\tau \\to H_\\tau^{\\mathrm{cal}}$\nis \\textbf{idempotent-supported} if it\ + \ admits a decomposition\n\\[\n \\boxed{%\n \\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-,}\n\\]\nwhere:\n\\begin{enumerate}\n\ + \ \\item $\\chi_+ : \\hat{\\mathbb{Z}}_\\tau \\to A_\\tau^{(B)}$\n is a ring homomorphism\n (the\ + \ \\textbf{$B$-channel component}),\n \\item $\\chi_- : \\hat{\\mathbb{Z}}_\\tau \\to A_\\tau^{(C)}$\n is\ + \ a ring homomorphism\n (the \\textbf{$C$-channel component}),\n \\item The decomposition recovers $\\chi$\ + \ pointwise:\n for every $x \\in \\hat{\\mathbb{Z}}_\\tau$,\n \\[\n \\chi(x)\n \ + \ = \\chi_+(x) + \\chi_-(x)\n = e_+ \\cdot \\chi_+(x) + e_- \\cdot \\chi_-(x).\n \\]\n\\end{enumerate}\n\ + The algebra of all idempotent-supported characters is denoted\n\\[\n A_{\\mathrm{spec}}(\\mathbb{L})\n := \\bigl\\\ + {\\chi \\in\n \\mathrm{Spec}(\\hat{\\mathbb{Z}}_\\tau, H_\\tau^{\\mathrm{cal}})\n : \\chi \\text{ is idempotent-supported}\\\ + bigr\\}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 138\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch47-boundary-characters-idempotent.tex`\ + \ lines 348-381\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.BoundaryCharacters`\n\ + - Name: `Tau.BookII.CentralTheorem.IdempotentCharacter`\n\n## Dependencies\n\n- Canonical: I.D19, I.D21, I.D22, I.D23,\ + \ II.L07, II.D35\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Idempotent-Supported Character +noindex: false +--- + diff --git a/_corpus_v3_items/cid001295.md b/_corpus_v3_items/cid001295.md new file mode 100644 index 000000000..8f201b9b4 --- /dev/null +++ b/_corpus_v3_items/cid001295.md @@ -0,0 +1,114 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001295/ +title: Spectral Algebra +cid: cid001295 +primary_alias: DEF0169 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The algebra A_spec(L) of idempotent-supported characters on L = S^1 v S^1 valued in calibrated H_tau, with bipolar + decomposition and ring structure from pointwise operations. +aliases: +- DEF0169 +- II.D60 +- spectral-algebra +- def:spectral-algebra +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005390 +- predicate: formalized_by + object: cid005396 +- predicate: formalized_by + object: cid005397 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 51 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Algebra**\n\nThe algebra A_spec(L) of idempotent-supported characters on L = S^1 v S^1 valued in calibrated\ + \ H_tau, with bipolar decomposition and ring structure from pointwise operations.\n\n# Spectral Algebra\n\n## Summary\n\ + \nThe algebra A_spec(L) of idempotent-supported characters on L = S^1 v S^1 valued in calibrated H_tau, with bipolar decomposition\ + \ and ring structure from pointwise operations.\n\n## Statement\n\n```latex\n%\n\\label{def:spectral-algebra}\nThe \\\ + textbf{spectral algebra} $A_{\\mathrm{spec}}(\\mathbb{L})$\nis the algebra of all idempotent-supported characters\non\ + \ the algebraic lemniscate $\\mathbb{L} = S^1 \\vee S^1$,\nvalued in the calibrated split-complex codomain\n$H_\\tau^{\\\ + mathrm{cal}}$.\nExplicitly:\n\\begin{equation}\n\\label{eq:ch51-spectral-algebra}\n A_{\\mathrm{spec}}(\\mathbb{L})\n\ + \ \\;:=\\;\n \\Bigl\\{\n \\chi \\colon \\widehat{\\mathbb{Z}}_\\tau \\to H_\\tau^{\\mathrm{cal}}\n \ + \ \\;\\Big|\\;\n \\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-,\\;\n \\chi \\text{ tower-coherent}\n\ + \ \\Bigr\\}.\n\\end{equation}\nThe structure of $A_{\\mathrm{spec}}(\\mathbb{L})$ is as follows:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(A)}] \\textbf{Elements.}\n Each $\\chi \\in A_{\\mathrm{spec}}(\\mathbb{L})$\n \ + \ is a ring homomorphism\n $\\chi \\colon \\widehat{\\mathbb{Z}}_\\tau \\to H_\\tau^{\\mathrm{cal}}$\n \ + \ satisfying two conditions:\n \\begin{itemize}\n \\item \\emph{Idempotent support:}\n \ + \ $\\chi$ decomposes as\n $\\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$,\n \ + \ where $\\chi_\\pm = e_\\pm \\cdot \\chi$\n are the projections onto the B-channel\n\ + \ and C-channel respectively\n (II.D59,\n Chapter~\\ref{ch:boundary-characters-idempotent}).\n\ + \ \\item \\emph{Tower coherence:}\n the system of restrictions\n $\\\ + chi_k := \\chi|_{\\mathbb{Z}/P_k\\mathbb{Z}}$\n satisfies\n $\\chi_k \\equiv \\\ + chi_{k+1} \\pmod{P_k}$\n for all sufficiently large~$k$.\n \\end{itemize}\n\n \\item[\\\ + textup{(B)}] \\textbf{Addition.}\n Pointwise addition in~$H_\\tau^{\\mathrm{cal}}$:\n $(\\chi + \\psi)(r)\ + \ := \\chi(r) + \\psi(r)$\n for $r \\in \\widehat{\\mathbb{Z}}_\\tau$.\n\n \\item[\\textup{(C)}] \\textbf{Multiplication.}\n\ + \ Pointwise multiplication in~$H_\\tau^{\\mathrm{cal}}$:\n $(\\chi \\cdot \\psi)(r) := \\chi(r) \\cdot\ + \ \\psi(r)$.\n Since $H_\\tau^{\\mathrm{cal}}$ is a commutative ring\n (with zero divisors along the\ + \ $e_\\pm$-axes),\n so is $A_{\\mathrm{spec}}(\\mathbb{L})$.\n\n \\item[\\textup{(D)}] \\textbf{Topology.}\n\ + \ $A_{\\mathrm{spec}}(\\mathbb{L})$ carries the\n inverse-limit topology inherited from the tower:\n\ + \ a net $(\\chi^\\alpha)$ converges to~$\\chi$\n if and only if $\\chi^\\alpha_k \\to \\chi_k$\n \ + \ for every stage~$k$.\n Since each $\\chi_k$ takes values\n in a finite-dimensional $H_\\tau^{\\\ + mathrm{cal}}$-module,\n the convergence at each stage is convergence\n of finitely many split-complex\ + \ coefficients.\n\n \\item[\\textup{(E)}] \\textbf{Bipolar decomposition.}\n The algebra decomposes as\n \ + \ \\begin{equation}\n \\label{eq:ch51-spectral-bipolar}\n A_{\\mathrm{spec}}(\\mathbb{L})\n\ + \ \\;=\\;\n e_+ \\cdot A_{\\mathrm{spec}}^+(\\mathbb{L})\n \\;\\oplus\\;\n \ + \ e_- \\cdot A_{\\mathrm{spec}}^-(\\mathbb{L}),\n \\end{equation}\n where $A_{\\mathrm{spec}}^\\\ + pm(\\mathbb{L})\n := e_\\pm \\cdot A_{\\mathrm{spec}}(\\mathbb{L})$\n are the $e_\\pm$-components.\n\ + \ The two components are independent:\n $e_+ \\cdot e_- = 0$,\n so no element of\n \ + \ $A_{\\mathrm{spec}}^+(\\mathbb{L})$\n has any effect on\n $A_{\\mathrm{spec}}^-(\\mathbb{L})$\n \ + \ and conversely.\n\n \\item[\\textup{(F)}] \\textbf{Calibration.}\n The four calibration constants\n\ + \ enter through the $H_\\tau^{\\mathrm{cal}}$ structure\n (II.D35, Chapter~\\ref{ch:split-complex-calibrated}):\n\ + \ \\begin{itemize}\n \\item $\\pi$ scales the angular periods\n of the B and\ + \ C channels.\n \\item $e$ scales the radial growth rates\n along the D-ray.\n \ + \ \\item $\\jj$ mediates the polarity flip\n between $e_+$ and~$e_-$.\n \\item $\\\ + iota_\\tau = 2/(\\pi + e)$\n couples all three\n as the master spectral invariant.\n\ + \ \\end{itemize}\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 147\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch51-central-theorem.tex`\ + \ lines 211-315\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.CentralTheorem`\n\ + - Name: `Tau.BookII.CentralTheorem.SpectralAlgebraElement`\n\n## Dependencies\n\n- Canonical: I.D18, I.D19, I.D21, II.D35,\ + \ II.D59, II.P13\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The algebra A_spec(L) of idempotent-supported characters on L = S^1 v S^1 valued in calibrated H_tau, with bipolar + decomposition and ring structure from pointwise operations. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001296.md b/_corpus_v3_items/cid001296.md new file mode 100644 index 000000000..ad163abbd --- /dev/null +++ b/_corpus_v3_items/cid001296.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001296/ +title: Moduli Space +cid: cid001296 +primary_alias: DEF0170 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The set of isomorphism classes of fibered products satisfying axioms K0-K5. By categoricity (II.T42), the moduli + space is a singleton. +aliases: +- DEF0170 +- II.D61 +- moduli-space +- def:moduli-space +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005380 +- predicate: formalized_by + object: cid005387 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 52 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Moduli Space**\n\nThe set of isomorphism classes of fibered products satisfying axioms K0-K5. By categoricity\ + \ (II.T42), the moduli space is a singleton.\n\n# Moduli Space\n\n## Summary\n\nThe set of isomorphism classes of fibered\ + \ products satisfying axioms K0-K5. By categoricity (II.T42), the moduli space is a singleton.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:moduli-space}\nThe \\textbf{moduli space} of structures\nsatisfying axioms K0--K5\nis the set of isomorphism\ + \ classes\nof fibered products $(M, \\Phi_M)$\nfor which K0--K5 hold:\n\\[\n \\boxed{%\n \\mathcal{M}_{\\tau^3}\n\ + \ \\;:=\\;\n \\bigl\\{\\,\n (M, \\Phi_M) \\;\\big|\\;\n (M, \\Phi_M) \\text{ satisfies K0--K5}\n \\,\\\ + bigr\\}\\big/{\\cong}.}\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 152\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch52-liouville-categoricity.tex`\ + \ lines 698-715\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.Categoricity`\n\ + - Name: `Tau.BookII.CentralTheorem.moduli_singleton_check`\n\n## Dependencies\n\n- Canonical: II.T42\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The set of isomorphism classes of fibered products satisfying axioms K0-K5. By categoricity (II.T42), the moduli + space is a singleton. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001297.md b/_corpus_v3_items/cid001297.md new file mode 100644 index 000000000..9cb37b1e6 --- /dev/null +++ b/_corpus_v3_items/cid001297.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001297/ +title: Tau-Manifold +cid: cid001297 +primary_alias: DEF0171 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A pair (M, A_tau) where M is a Stone space and A_tau is a maximal tau-analytic atlas. The model spaces are tau^1, + T^2, and tau^3. +aliases: +- DEF0171 +- II.D62 +- tau-manifold +- def:tau-manifold +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005487 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 54 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Manifold**\n\nA pair (M, A_tau) where M is a Stone space and A_tau is a maximal tau-analytic atlas. The\ + \ model spaces are tau^1, T^2, and tau^3.\n\n# Tau-Manifold\n\n## Summary\n\nA pair (M, A_tau) where M is a Stone space\ + \ and A_tau is a maximal tau-analytic atlas. The model spaces are tau^1, T^2, and tau^3.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:tau-manifold}\nA \\textbf{$\\tau$-manifold} is a pair\n$(M, \\mathcal{A}_\\tau)$ where:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(M1)}]\n $M$ is a topological space\n with the $\\tau$-topology:\n Hausdorff,\ + \ totally disconnected,\n compact, and second countable,\n with a clopen basis\n (as established\ + \ for $\\tau^3$\n in Chapter~\\ref{ch:stone-space}, II.D14,\n and Chapter~\\ref{ch:topology-invariant},\ + \ II.T07).\n\n \\item[\\textup{(M2)}]\n $\\mathcal{A}_\\tau$ is a maximal\n $\\tau$-analytic atlas\ + \ on~$M$\n (Definition~\\ref{def:tau-analytic-atlas}, II.D64):\n a collection of charts\n $\\\ + {(U_i, \\varphi_i)\\}_{i \\in I}$\n covering~$M$,\n with each $\\varphi_i \\colon U_i \\to \\tau^3$\n\ + \ a homeomorphism onto a cylinder domain,\n and all transition functions\n $\\varphi_j \\circ\ + \ \\varphi_i^{-1}$\n $\\tau$-analytic.\n\n \\item[\\textup{(M3)}]\n The atlas $\\mathcal{A}_\\tau$\n\ + \ is compatible with the sheaf structure:\n the holomorphic presheaf $\\mathcal{O}_\\tau$\n \ + \ (II.D47, II.T32,\n Chapter~\\ref{ch:sheaf-coherence})\n on~$M$, defined by transport via charts,\n\ + \ is a sheaf.\n\\end{enumerate}\n\n\\medskip\\noindent\nThe \\textbf{dimension} of a $\\tau$-manifold\nis the\ + \ dimension of the model space~$\\tau^3$,\nwhich is~$4$\n(II.D15, Chapter~\\ref{ch:dimension-four}):\none radial (D),\ + \ one base-angular (A),\nand two fiber (B, C) coordinates.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 155\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch54-tau-manifold.tex` lines 366-412\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.TauManifold`\n- Name: `Tau.BookII.Closure.TauManifoldData`\n\ + \n## Dependencies\n\n- Canonical: II.D63, II.T40, II.D35\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A pair (M, A_tau) where M is a Stone space and A_tau is a maximal tau-analytic atlas. The model spaces are tau^1, + T^2, and tau^3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001298.md b/_corpus_v3_items/cid001298.md new file mode 100644 index 000000000..fda6f4aef --- /dev/null +++ b/_corpus_v3_items/cid001298.md @@ -0,0 +1,107 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001298/ +title: Tau-Analytic Atlas +cid: cid001298 +primary_alias: DEF0172 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A collection of tau-analytic charts covering tau^3 with tower-coherent, stagewise natural, sector-independent transition + functions. +aliases: +- DEF0172 +- II.D63 +- tau-analytic-atlas +- def:tau-analytic-atlas +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005483 +- predicate: formalized_by + object: cid005484 +- predicate: formalized_by + object: cid005489 +- predicate: formalized_by + object: cid005490 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 54 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Analytic Atlas**\n\nA collection of tau-analytic charts covering tau^3 with tower-coherent, stagewise natural,\ + \ sector-independent transition functions.\n\n# Tau-Analytic Atlas\n\n## Summary\n\nA collection of tau-analytic charts\ + \ covering tau^3 with tower-coherent, stagewise natural, sector-independent transition functions.\n\n## Statement\n\n\ + ```latex\n%\n\\label{def:tau-analytic-atlas}\nLet $U, V \\subseteq \\tau^3$\nbe cylinder domains (II.D10).\nA map $f \\\ + colon U \\to V$ is \\textbf{$\\tau$-analytic}\nif the following three conditions hold:\n\\begin{enumerate}\n \\item[\\\ + textup{(TA1)}]\n \\textbf{$\\tau$-holomorphicity.}\n The map~$f$ is $\\tau$-holomorphic\n in\ + \ the sense of the Mutual Determination Theorem\n (II.T27, Chapter~\\ref{ch:mutual-determination}):\n \ + \ tower-coherent, stagewise natural,\n and sector-independent.\n Equivalently,\n $f$ is a\ + \ section\n of the holomorphic sheaf\n $\\mathcal{O}_\\tau$\n (II.D47, II.T32,\n Chapter~\\\ + ref{ch:sheaf-coherence}).\n\n \\item[\\textup{(TA2)}]\n \\textbf{Fibration preservation.}\n The map~$f$\ + \ preserves the product structure\n of the fibered product\n $\\tau^3 = \\tau^1 \\times_f T^2$:\n \ + \ \\begin{itemize}\n \\item The base components $(D, A)$\n of $f(x)$\n \ + \ depend only on the base components\n $(D, A)$ of~$x$.\n \\item The fiber\ + \ components $(B, C)$\n of $f(x)$\n depend on the full quadruple\n \ + \ $(A, B, C, D)$ of~$x$,\n but map fiber to fiber:\n for each fixed base point\ + \ $(D_0, A_0)$,\n the restricted map\n $f|_{\\{D_0\\} \\times \\{A_0\\} \\times\ + \ T^2}$\n is a $\\tau$-holomorphic map\n of the fiber $T^2$.\n \\end{itemize}\n\ + \ In short: \\emph{base maps to base,\n fiber maps to fiber}.\n\n \\item[\\textup{(TA3)}]\n \ + \ \\textbf{Finite spectral support.}\n At each stage~$k$,\n the map~$f$ admits a character expansion\n\ + \ in the canonical basis $\\mathcal{B}_\\tau$\n (II.D45, Chapter~\\ref{ch:canonical-basis})\n \ + \ with \\textbf{finitely many} active spectral\n coefficients\n (II.T31, Chapter~\\ref{ch:canonical-basis}).\n\ + \\end{enumerate}\n\n\\medskip\\noindent\nA collection of pairs\n$\\{(U_i, \\varphi_i)\\}_{i \\in I}$\nis a \\textbf{$\\\ + tau$-analytic atlas}\non a topological space~$M$ if:\n\\begin{enumerate}\n \\item[\\textup{(A1)}]\n Each $U_i\ + \ \\subseteq M$ is open,\n and $\\{U_i\\}_{i \\in I}$ covers~$M$.\n \\item[\\textup{(A2)}]\n Each\ + \ $\\varphi_i \\colon U_i \\to \\tau^3$\n is a homeomorphism onto its image\n (a cylinder domain or\ + \ finite union thereof).\n \\item[\\textup{(A3)}]\n For each pair $i, j \\in I$\n with $U_i \\cap\ + \ U_j \\neq \\varnothing$,\n the \\textbf{transition function}\n \\[\n \\varphi_j \\circ\ + \ \\varphi_i^{-1}\n \\;\\colon\\;\n \\varphi_i(U_i \\cap U_j)\n \\;\\longrightarrow\\\ + ;\n \\varphi_j(U_i \\cap U_j)\n \\]\n is $\\tau$-analytic in the sense\n of conditions\ + \ \\textup{(TA1)--(TA3)} above.\n \\item[\\textup{(A4)}]\n The atlas is \\textbf{maximal}:\n every\ + \ chart $(U, \\varphi)$\n whose transition functions\n with all charts in the atlas\n are $\\\ + tau$-analytic\n is already included.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 156\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch54-tau-manifold.tex` lines 230-321\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.TauManifold`\n- Name: `Tau.BookII.Closure.atlas_chart_check`\n\ + \n## Dependencies\n\n- Canonical: II.D35, II.T27, II.D10\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A collection of tau-analytic charts covering tau^3 with tower-coherent, stagewise natural, sector-independent + transition functions. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001299.md b/_corpus_v3_items/cid001299.md new file mode 100644 index 000000000..dd95f748b --- /dev/null +++ b/_corpus_v3_items/cid001299.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001299/ +title: Tau-Exterior Derivative +cid: cid001299 +primary_alias: DEF0173 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Tau-Exterior Derivative +aliases: +- DEF0173 +- II.D64 +- tau-exterior-derivative +- def:tau-exterior-derivative +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005485 +- predicate: formalized_by + object: cid005486 +- predicate: formalized_by + object: cid005491 +- predicate: formalized_by + object: cid005492 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 54 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Exterior Derivative**\n\nTau-Exterior Derivative\n\n# Tau-Exterior Derivative\n\n## Summary\n\nTau-Exterior\ + \ Derivative\n\n## Statement\n\n```latex\n%\n\\label{def:tau-exterior-derivative}\nThe \\textbf{$\\tau$-exterior derivative}\ + \ is the operator\n\\[\n \\boxed{%\n d_\\tau\n \\;\\colon\\;\n \\Omega^k_\\tau(M)\n \\;\\longrightarrow\\\ + ;\n \\Omega^{k+1}_\\tau(M),}\n\\]\ndefined as follows.\n\n\\smallskip\n\\noindent\\textbf{On $0$-forms.}\nFor $f \\\ + in \\Omega^0_\\tau(M) = \\mathcal{O}_\\tau(M)$,\n\\[\n d_\\tau f\n \\;:=\\;\n \\frac{\\partial f}{\\partial A}\\\ + , dA\n \\;+\\;\n \\frac{\\partial f}{\\partial B}\\, dB\n \\;+\\;\n \\frac{\\partial f}{\\partial C}\\, dC\n\ + \ \\;+\\;\n \\frac{\\partial f}{\\partial D}\\, dD,\n\\]\nwhere the partial derivatives\nare defined \\textbf{stagewise}:\n\ + at stage~$k$, the function\n$f_k \\colon \\mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau$\nis defined on a finite set,\nand the\ + \ partial derivative\nwith respect to a coordinate~$X$\nis the discrete difference operator\n\\[\n \\frac{\\partial\ + \ f_k}{\\partial X}(x)\n \\;:=\\;\n f_k(x + e_X) - f_k(x),\n\\]\nwhere $e_X$ is the unit vector\nin the $X$-direction\n\ + within the CRT decomposition.\nThese discrete derivatives\nare tower-coherent:\n$\\pi_{k,k+1} \\circ (\\partial f_{k+1}/\\\ + partial X)\n= \\partial f_k / \\partial X$.\n\n\\smallskip\n\\noindent\\textbf{On $k$-forms.}\nFor $\\omega = f_{I}\\\ + , dX^{i_1} \\wedge \\cdots \\wedge dX^{i_k}$\n(a monomial $k$-form with coefficient $f_I \\in \\mathcal{O}_\\tau$),\n\\\ + [\n d_\\tau \\omega\n \\;:=\\;\n (d_\\tau f_I)\n \\;\\wedge\\;\n dX^{i_1} \\wedge \\cdots \\wedge dX^{i_k}.\n\ + \\]\nExtend by $H_\\tau$-linearity\nto all $k$-forms.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 157\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch54-tau-manifold.tex` lines 759-821\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.TauManifold`\n- Name: `Tau.BookII.Closure.tau_exterior_derivative`\n\ + \n## Dependencies\n\n- Canonical: II.D62, II.D42\n\n## Related Results\n\n_Generated by later projection phases._\n\n\ + ## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Tau-Exterior Derivative +noindex: false +--- + diff --git a/_corpus_v3_items/cid001300.md b/_corpus_v3_items/cid001300.md new file mode 100644 index 000000000..d9edb9909 --- /dev/null +++ b/_corpus_v3_items/cid001300.md @@ -0,0 +1,101 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001300/ +title: Proto-Rationality +cid: cid001300 +primary_alias: DEF0174 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A holomorphic function whose spectral coefficients have finite support and lie in the canonical basis. The algebraic + prerequisite for the BSD approach. +aliases: +- DEF0174 +- II.D65 +- proto-rationality +- def:proto-rationality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005439 +- predicate: formalized_by + object: cid005440 +- predicate: formalized_by + object: cid005441 +- predicate: formalized_by + object: cid005442 +- predicate: formalized_by + object: cid005443 +- predicate: formalized_by + object: cid005444 +- predicate: formalized_by + object: cid005445 +- predicate: formalized_by + object: cid005446 +- predicate: formalized_by + object: cid005447 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 56 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Proto-Rationality**\n\nA holomorphic function whose spectral coefficients have finite support and lie in the\ + \ canonical basis. The algebraic prerequisite for the BSD approach.\n\n# Proto-Rationality\n\n## Summary\n\nA holomorphic\ + \ function whose spectral coefficients have finite support and lie in the canonical basis. The algebraic prerequisite\ + \ for the BSD approach.\n\n## Statement\n\n```latex\n%\n\\label{def:proto-rationality}\nA holomorphic function $f \\in\ + \ \\mathcal{O}(\\tau^3)$\nis \\textbf{proto-rational} if its spectral coefficients\n$\\{\\varphi_{mn}\\}_{(m,n) \\in S}$\n\ + satisfy the following conditions:\n\\begin{enumerate}\n \\item \\textbf{Finite spectral support.}\n The support\ + \ $S \\subset \\Lambda_\\tau$ is finite.\n (This is already guaranteed by\n Theorem~\\ref{thm:finite-spectral-support},\ + \ II.T30.)\n\n \\item \\textbf{Basis image condition.}\n Each coefficient $\\varphi_{mn}$\n lies\ + \ in the image of the canonical basis\n $\\mathcal{B}_\\tau$\n (Definition~\\ref{def:canonical-basis},\ + \ II.D45):\n there exist finitely many\n cylinder generators\n $E_{k,v}^{(B)}$, $E_{l,w}^{(C)}$\n\ + \ (Definition~\\ref{def:cylinder-generator}, II.D46)\n such that $\\varphi_{mn}$\n is an $H_\\\ + tau^{\\mathrm{cal}}$-linear combination\n of their products.\n\n \\item \\textbf{Prime determinacy.}\n \ + \ The coefficients are determined by\n finitely many prime-indexed data:\n there exists a finite\ + \ set\n $\\Pi \\subset \\mathbb{P}_\\tau$ of primes\n such that $\\varphi_{mn}$\n is determined\ + \ by the restriction of $f$\n to the sub-tower\n $\\prod_{p \\in \\Pi} \\Z / p \\Z$.\n\\end{enumerate}\n\ + We write $\\mathcal{O}_{\\mathrm{pr}}(\\tau^3)$\nfor the sub-algebra of proto-rational functions\nin $\\mathcal{O}(\\\ + tau^3)$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 160\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch56-bsd-bridge.tex`\ + \ lines 299-338\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.BSDbridge`\n\ + - Name: `Tau.BookII.Closure.proto_rational_check`\n\n## Dependencies\n\n- Canonical: II.T40, II.D60, II.D35\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A holomorphic function whose spectral coefficients have finite support and lie in the canonical basis. The algebraic + prerequisite for the BSD approach. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001301.md b/_corpus_v3_items/cid001301.md new file mode 100644 index 000000000..8d3c6c0ae --- /dev/null +++ b/_corpus_v3_items/cid001301.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001301/ +title: E1 Export Package +cid: cid001301 +primary_alias: DEF0175 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The complete collection of structures Book II delivers to Book III: categorical structure, topology, geometry, holomorphy, + Central Theorem, enrichment, exterior derivative, and Code/Decode.' +aliases: +- DEF0175 +- II.D67 +- e1-export-package +- def:e1-export +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 57 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E1 Export Package**\n\nThe complete collection of structures Book II delivers to Book III: categorical structure,\ + \ topology, geometry, holomorphy, Central Theorem, enrichment, exterior derivative, and Code/Decode.\n\n# E1 Export Package\n\ + \n## Summary\n\nThe complete collection of structures Book II delivers to Book III: categorical structure, topology, geometry,\ + \ holomorphy, Central Theorem, enrichment, exterior derivative, and Code/Decode.\n\n## Statement\n\n```latex\n%\n\\label{def:e1-export}\n\ + % II.D63, II.D64, II.D65, II.R22, I.D82\nThe \\textbf{$E_1$ Export Package} is the complete collection\nof structures\ + \ that Book~II delivers to Book~III.\nIt consists of eight layers:\n\\begin{enumerate}\n \\item[\\textup{(E1.1)}]\n\ + \ \\textbf{Categorical structure.}\n The fibered product\n $\\tau^3 = \\tau^1 \\times_f T^2$\n\ + \ with ABCD chart (I.D17),\n fibration structure (II.T03),\n and categoricity (II.T41):\n \ + \ the moduli space is a single point.\n\n \\item[\\textup{(E1.2)}]\n \\textbf{Holomorphic structure.}\n\ + \ The Central Theorem\n $\\mathcal{O}(\\tau^3) \\cong A_{\\mathrm{spec}}(\\Lemniscate)$\n (II.T40),\n\ + \ sheaf coherence (II.T32),\n the Code/Decode bijection (II.T35),\n and the 5-way Mutual Determination\ + \ (II.T27).\n\n \\item[\\textup{(E1.3)}]\n \\textbf{Self-enrichment.}\n Hom objects as $\\tau$-objects\ + \ (II.D53--D54),\n the Yoneda embedding (II.T36),\n 2-categorical structure (II.D55--D56),\n \ + \ and the $E_0 \\to E_1$ transition (II.D58).\n This is the \\textbf{$E_1$ layer}:\n $\\tau$ describes\ + \ its own morphisms.\n\n \\item[\\textup{(E1.4)}]\n \\textbf{Numerical calibration.}\n The constants\ + \ $\\pi$, $e$, $\\jj$, $\\iota_\\tau = 2/(\\pi + e)$\n --- earned with their standard numerical values\n \ + \ (II.T21--T24).\n Every spectral coefficient,\n every coupling constant,\n every regularity\ + \ bound in Book~III\n inherits this calibration.\n\n \\item[\\textup{(E1.5)}]\n \\textbf{Bipolar\ + \ decomposition.}\n Every structure splits via\n $e_\\pm = (1 \\pm \\jj)/2$:\n functions, sheaves,\ + \ Hom objects,\n morphism spaces.\n The B-channel ($e_+$) and C-channel ($e_-$)\n carry independent\ + \ information.\n This is the split-complex analogue\n of real and imaginary parts in classical analysis,\n\ + \ but with $\\jj^2 = +1$ (wave-type, not Laplacian).\n\n \\item[\\textup{(E1.6)}]\n \\textbf{Regularity\ + \ theory.}\n Positive regularity (II.D49):\n existence of stabilized $\\omega$-germs.\n The\ + \ idempotent characterization\n (II.L07, II.T32):\n holomorphic $\\Leftrightarrow$ idempotent-supported.\n\ + \ The 3-lemma chain\n (II.L08--L10).\n\n \\item[\\textup{(E1.7)}]\n \\textbf{Manifold layer.}\n\ + \ The $\\tau$-manifold $(M, \\mathcal{A}_\\tau)$ (II.D63)\n with $\\tau$-analytic atlas (II.D64),\n\ + \ the exterior derivative $d_\\tau$ with $d_\\tau^2 = 0$ (II.D65),\n and the model spaces\n \ + \ $\\tau^1$ (base), $T^2$ (fiber), $\\tau^3$ (total).\n\n \\item[\\textup{(E1.8)}]\n \\textbf{Proof-theoretic\ + \ layer.}\n The K5 diagonal discipline (I.X05)\n and its linear logic correspondence (I.T37):\n \ + \ $\\tau$'s proof theory is $!$-free linear logic.\n The Enrichment Frontier Classification (I.D82)\n \ + \ maps the $E_0 \\to E_1 \\to E_2 \\to E_3$ ladder\n onto the foundational landscape.\n\\end{enumerate}\n\ + \n\\smallskip\\noindent\n\\textbf{Notation.}\nWe write $\\mathsf{Export}(E_1)$ for this 8-layer package,\nand say that\ + \ Book~III \\textbf{receives} $\\mathsf{Export}(E_1)$\nas its starting configuration.\nBook~III is not permitted to assume\ + \ any structure\nnot contained in $\\mathsf{Export}(E_1)$.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 162\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch57-forward-book3.tex` lines 124-216\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.ForwardBook3`\n- Name: `Tau.BookII.Closure.E1ExportPackage`\n\ + \n## Dependencies\n\n- Canonical: II.D57, II.D58, II.T40\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The complete collection of structures Book II delivers to Book III: categorical structure, topology, geometry, + holomorphy, Central Theorem, enrichment, exterior derivative, and Code/Decode.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001302.md b/_corpus_v3_items/cid001302.md new file mode 100644 index 000000000..8e8716902 --- /dev/null +++ b/_corpus_v3_items/cid001302.md @@ -0,0 +1,96 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001302/ +title: Structural Sign Classification +cid: cid001302 +primary_alias: DEF0176 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Twelve-level classification of structural outcomes forced by the scalar unit equation u^2 = +/-1, satisfying completeness, + traceability (at most three steps from the sign), and monotonicity (downstream levels depend on upstream outcomes). +aliases: +- DEF0176 +- II.D68 +- structural-sign-classification +- def:structural-sign-classification +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005615 +- predicate: formalized_by + object: cid005616 +- predicate: formalized_by + object: cid005625 +- predicate: formalized_by + object: cid005626 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 61 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Structural Sign Classification**\n\nTwelve-level classification of structural outcomes forced by the scalar\ + \ unit equation u^2 = +/-1, satisfying completeness, traceability (at most three steps from the sign), and monotonicity\ + \ (downstream levels depend on upstream outcomes).\n\n# Structural Sign Classification\n\n## Summary\n\nTwelve-level classification\ + \ of structural outcomes forced by the scalar unit equation u^2 = +/-1, satisfying completeness, traceability (at most\ + \ three steps from the sign), and monotonicity (downstream levels depend on upstream outcomes).\n\n## Statement\n\n```latex\n\ + %\n\\label{def:structural-sign-classification}\nThe \\textbf{structural sign classification} is the assignment,\nat each\ + \ of twelve levels of mathematical structure,\nof the qualitative outcome forced by the scalar unit equation\n$u^2 = \\\ + pm 1$.\nExplicitly, let $\\mathcal{S} = \\{1, 2, \\ldots, 12\\}$\nbe the set of structural levels\nas enumerated in the\ + \ table of\nSection~\\ref{sec:ch61-twelve-levels}.\nFor each level $\\ell \\in \\mathcal{S}$,\ndefine:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(a)}]\n $\\mathcal{E}_\\ell$ = the structural outcome\n when $u^2 = -1$ (elliptic/orthodox).\n\ + \ \\item[\\textup{(b)}]\n $\\mathcal{H}_\\ell$ = the structural outcome\n when $u^2 = +1$ (hyperbolic/$\\\ + tau$).\n\\end{enumerate}\nThe pair $(\\mathcal{E}_\\ell, \\mathcal{H}_\\ell)$\nat each level is called the\n\\textbf{level-$\\\ + ell$ sign trade-off}.\nThe twelve trade-offs collectively\nconstitute the structural sign classification.\n\nThe classification\ + \ satisfies:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Completeness.}\n Every structural\ + \ difference\n between orthodox complex analysis\n and $\\tau$-holomorphy\n documented in Books~I--II\n\ + \ appears as an instance of some\n level-$\\ell$ trade-off.\n \\item[\\textup{(ii)}]\n \\\ + textbf{Traceability.}\n Each trade-off $(\\mathcal{E}_\\ell, \\mathcal{H}_\\ell)$\n traces, through\ + \ at most three intermediate steps,\n to the sign equation $u^2 = \\pm 1$.\n \\item[\\textup{(iii)}]\n \ + \ \\textbf{Monotonicity.}\n The levels are ordered so that\n level $\\ell + 1$ depends on\n \ + \ the outcome at level $\\ell$\n (or earlier levels):\n scalar algebra determines the PDE type,\n\ + \ the PDE type determines propagation character,\n and so on.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 170\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch61-master-switch.tex`\ + \ lines 239-291\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.SignClassification`\n\ + - Name: `SignLevel`\n\n## Dependencies\n\n- Canonical: I.T10, I.D86\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Twelve-level classification of structural outcomes forced by the scalar unit equation u^2 = +/-1, satisfying + completeness, traceability (at most three steps from the sign), and monotonicity (downstream levels depend on upstream outcomes). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001303.md b/_corpus_v3_items/cid001303.md new file mode 100644 index 000000000..445e12d6c --- /dev/null +++ b/_corpus_v3_items/cid001303.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001303/ +title: The Infinity Trade-Off +cid: cid001303 +primary_alias: DEF0177 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The pair of structural properties: (U) unique global omega with non-trivial spectrum, and (A) Archimedean local + density with uncountably many elements in every neighborhood. The two properties represent the deepest structural divergence + between orthodox and tau-mathematics.' +aliases: +- DEF0177 +- II.D69 +- the-infinity-trade-off +- def:infinity-trade-off +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005627 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**The Infinity Trade-Off**\n\nThe pair of structural properties: (U) unique global omega with non-trivial spectrum,\ + \ and (A) Archimedean local density with uncountably many elements in every neighborhood. The two properties represent\ + \ the deepest structural divergence between orthodox and tau-mathematics.\n\n# The Infinity Trade-Off\n\n## Summary\n\n\ + The pair of structural properties: (U) unique global omega with non-trivial spectrum, and (A) Archimedean local density\ + \ with uncountably many elements in every neighborhood. The two properties represent the deepest structural divergence\ + \ between orthodox and tau-mathematics.\n\n## Statement\n\n```latex\n%\n\\label{def:infinity-trade-off}\nThe \\textbf{Infinity\ + \ Trade-Off} is the pair\nof mutually exclusive properties:\n\\begin{enumerate}\n \\item[\\textup{(U)}]\n \ + \ \\textbf{Unique global infinity.}\n A single canonical element $\\omega$\n with $\\rho(\\omega) =\ + \ \\omega$,\n spectral access via $\\omega$-germs,\n and no cardinality hierarchy.\n \\item[\\textup{(A)}]\n\ + \ \\textbf{Archimedean local density.}\n Between any two distinct points\n lie uncountably\ + \ many others.\n $\\varepsilon$-$\\delta$ limits provide\n infinite refinement at every point.\n\\end{enumerate}\n\ + Property~(U) is the $\\tau$ path.\nProperty~(A) is the orthodox path.\n```\n\n## Proof / Justification\n\nThis item is\ + \ definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 172\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex` lines\ + \ 107-128\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.SignClassification`\n\ + - Name: `InfinityTradeOff`\n\n## Dependencies\n\n- Canonical: I.D86, II.D68\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The pair of structural properties: (U) unique global omega with non-trivial spectrum, and (A) Archimedean local + density with uncountably many elements in every neighborhood. The two properties represent the deepest structural divergence + bet' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001304.md b/_corpus_v3_items/cid001304.md new file mode 100644 index 000000000..59e22a52b --- /dev/null +++ b/_corpus_v3_items/cid001304.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001304/ +title: PDE Type Classification +cid: cid001304 +primary_alias: DEF0178 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Dichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace, no characteristics, isotropic, + max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs hyperbolic (wave, two characteristic + families, asymmetric, Liouville dodge, natural Hartogs).' +aliases: +- DEF0178 +- II.D70 +- pde-type-classification +- def:pde-type-classification +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005640 +- predicate: formalized_by + object: cid005641 +- predicate: formalized_by + object: cid005646 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 61 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**PDE Type Classification**\n\nDichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace,\ + \ no characteristics, isotropic, max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs\ + \ hyperbolic (wave, two characteristic families, asymmetric, Liouville dodge, natural Hartogs).\n\n# PDE Type Classification\n\ + \n## Summary\n\nDichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace, no characteristics,\ + \ isotropic, max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs hyperbolic (wave,\ + \ two characteristic families, asymmetric, Liouville dodge, natural Hartogs).\n\n## Statement\n\n```latex\n%\n\\label{def:pde-type-classification}\n\ + The \\textbf{PDE type classification} assigns\nto each holomorphic function theory\nthe PDE type of its defining equations:\n\ + \n\\medskip\n\\renewcommand{\\arraystretch}{1.25}\n\\begin{center}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\n\\textbf{Feature}\n\ + \ & \\textbf{Elliptic ($i^2 = -1$)}\n & \\textbf{Hyperbolic ($\\jj^2 = +1$)} \\\\\n\\midrule\nCR equations\n & $u_x\ + \ = v_y,\\ u_y = -v_x$\n & $u_x = v_y,\\ u_y = +v_x$ \\\\\nResulting PDE\n & $\\Delta f = 0$ (Laplace)\n & $\\Box f\ + \ = 0$ (wave) \\\\\nCharacteristics\n & None (elliptic)\n & Two real families \\\\\nPropagation\n & Isotropic diffusion\n\ + \ & Directional along chars \\\\\nMaximum principle\n & Yes\n & No \\\\\nDecomposition\n & $f = u + iv$ (harmonic)\n\ + \ & $f = e_+ g + e_- h$ (holomorphic) \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\medskip\n\\noindent\nThe\ + \ sign difference $u_y = \\mp v_x$\nin the second CR equation\nis the single algebraic step\nthat separates the two PDE\ + \ worlds.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 175\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch61-master-switch.tex`\ + \ lines 378-422\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.WaveHolomorphy`\n\ + - Name: `PDEClassification`\n\n## Dependencies\n\n- Canonical: I.T10, II.D21, II.D22\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Dichotomy of holomorphic function theories by sign u^2 = +/-1: elliptic (Laplace, no characteristics, isotropic, + max principle, symmetric boundary/interior, Liouville rigidity, miraculous Hartogs) vs hyperbolic (wave, two characteristic + fam' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001305.md b/_corpus_v3_items/cid001305.md new file mode 100644 index 000000000..e6677d1de --- /dev/null +++ b/_corpus_v3_items/cid001305.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001305/ +title: Stage-Finite Euclidean Geometry +cid: cid001305 +primary_alias: DEF0179 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Geometric structure of tau^3 determined by stage spaces Z/P_k Z (finite cyclic groups with positive-definite cyclic + metric), stage-k Tarski axioms, and profinite limit preserving Euclidean geometry. Discretely Euclidean at every finite + stage, profinitely Euclidean in the limit. +aliases: +- DEF0179 +- II.D71 +- stage-finite-euclidean-geometry +- def:stage-finite-euclidean +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 64 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Stage-Finite Euclidean Geometry**\n\nGeometric structure of tau^3 determined by stage spaces Z/P_k Z (finite\ + \ cyclic groups with positive-definite cyclic metric), stage-k Tarski axioms, and profinite limit preserving Euclidean\ + \ geometry. Discretely Euclidean at every finite stage, profinitely Euclidean in the limit.\n\n# Stage-Finite Euclidean\ + \ Geometry\n\n## Summary\n\nGeometric structure of tau^3 determined by stage spaces Z/P_k Z (finite cyclic groups with\ + \ positive-definite cyclic metric), stage-k Tarski axioms, and profinite limit preserving Euclidean geometry. Discretely\ + \ Euclidean at every finite stage, profinitely Euclidean in the limit.\n\n## Statement\n\n```latex\n%\n\\label{def:stage-finite-euclidean}\n\ + \\textbf{Stage-finite Euclidean geometry}\nis the geometric structure on $\\tau^3$\ndefined by:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(S1)}]\n \\textbf{Stage spaces.}\n At each primorial depth~$k$,\n the\ + \ stage-$k$ space $\\tau^3_k$\n is a finite set of points.\n \\item[\\textup{(S2)}]\n \\textbf{Discrete\ + \ distance.}\n The ultrametric $d = 2^{-\\delta}$\n restricts to a discrete metric on $\\tau^3_k$.\n\ + \ \\item[\\textup{(S3)}]\n \\textbf{Stage-$k$ Tarski axioms.}\n All ten Tarski axioms hold\n \ + \ on each $\\tau^3_k$.\n \\item[\\textup{(S4)}]\n \\textbf{Profinite limit.}\n The full geometry\ + \ on $\\tau^3$\n is the profinite limit\n $\\tau^3 = \\varprojlim_k \\tau^3_k$.\n\\end{enumerate}\n\ + The geometry is ``finitely Euclidean at every stage''\nand ``profinitely Euclidean in the limit.''\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 178\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch64-gains.tex`\ + \ lines 212-240\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.WaveHolomorphy`\n\ + - Name: `StageGeometry`\n\n## Dependencies\n\n- Canonical: II.D14, II.T15, II.T16, II.T17, II.T18\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Geometric structure of tau^3 determined by stage spaces Z/P_k Z (finite cyclic groups with positive-definite + cyclic metric), stage-k Tarski axioms, and profinite limit preserving Euclidean geometry. Discretely Euclidean at every + finite stag +noindex: false +--- + diff --git a/_corpus_v3_items/cid001306.md b/_corpus_v3_items/cid001306.md new file mode 100644 index 000000000..9d890ba17 --- /dev/null +++ b/_corpus_v3_items/cid001306.md @@ -0,0 +1,99 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001306/ +title: The Rewiring Table +cid: cid001306 +primary_alias: DEF0180 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Twelve-level structural comparison between orthodox complex analysis (i^2 = -1) and tau-holomorphy (j^2 = +1). Levels: + scalar algebra, holomorphy PDE, boundary-interior, infinity, cardinality, topology, geometry, compactness, idempotents, + Liouville, gluing, spectrum.' +aliases: +- DEF0180 +- II.D72 +- the-rewiring-table +- def:rewiring-table +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005595 +- predicate: formalized_by + object: cid005596 +- predicate: formalized_by + object: cid005597 +- predicate: formalized_by + object: cid005598 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 61 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**The Rewiring Table**\n\nTwelve-level structural comparison between orthodox complex analysis (i^2 = -1) and\ + \ tau-holomorphy (j^2 = +1). Levels: scalar algebra, holomorphy PDE, boundary-interior, infinity, cardinality, topology,\ + \ geometry, compactness, idempotents, Liouville, gluing, spectrum.\n\n# The Rewiring Table\n\n## Summary\n\nTwelve-level\ + \ structural comparison between orthodox complex analysis (i^2 = -1) and tau-holomorphy (j^2 = +1). Levels: scalar algebra,\ + \ holomorphy PDE, boundary-interior, infinity, cardinality, topology, geometry, compactness, idempotents, Liouville, gluing,\ + \ spectrum.\n\n## Statement\n\n```latex\n%\n\\label{def:rewiring-table}\n% II.T06, II.T19, II.T25, II.T27, II.T30, II.T32,\ + \ II.T39,\n% II.T40, II.D21, II.D22, II.D35, II.D60\nThe \\textbf{rewiring table} is the twelve-level\nstructural comparison\ + \ of orthodox ($i^2 = -1$)\nand $\\tau$ ($\\jj^2 = +1$) mathematics,\nrefining the preview of\nSection~\\ref{sec:ch61-twelve-levels}\n\ + with explicit theorem references:\n\n\\medskip\n\\renewcommand{\\arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}r@{\\\ + ;\\;}p{3.2cm}@{\\;\\;}p{3.2cm}@{\\;\\;}p{3.2cm}@{}}\n\\toprule\n\\textbf{\\#}\n & \\textbf{Orthodox ($i^2 = -1$)}\n \ + \ & \\textbf{$\\tau$ ($\\jj^2 = +1$)}\n & \\textbf{Theorem source} \\\\\n\\midrule\n1\n & $\\mathbb{C}$ field, no ZD\n\ + \ & $H_\\tau$ ring, ZD\n & I.T10, I.D18 \\\\\n2\n & Laplace $\\Delta f = 0$\n & Wave $\\Box f = 0$\n & II.D21, II.D22\ + \ \\\\\n3\n & Interior glued by $\\Delta$\n & Interior from boundary\n & I.T31, II.T27 \\\\\n4\n & Unique $\\omega$\ + \ absent\n & Unique $\\omega$ (K2)\n & I.T36, II.D68 \\\\\n5\n & $\\aleph_0, \\aleph_1, \\ldots$\n & Single $\\omega$,\ + \ no hierarchy\n & I.T35, I.T36 \\\\\n6\n & Archimedean, connected\n & Stone, ultrametric, clopen\n & II.T06, II.D13,\ + \ II.D14 \\\\\n7\n & Euclidean from axiom\n & Euclidean from theorem\n & II.T15--II.T18 \\\\\n8\n & Compact $\\Rightarrow$\ + \ constant\n & Compact + non-constant\n & II.T41 (Liouville dodge) \\\\\n9\n & Fourier: $e_+ = e_-$ (harmonic)\n &\ + \ Fourier: $e_+ \\neq e_-$ (hol.)\n & II.T30, II.T32 \\\\\n10\n & Liouville rigidity\n & Standing-wave flexibility\n\ + \ & II.T40, II.T41 \\\\\n11\n & Laurent $\\to$ Cauchy\n & BndLift $\\to$ CRT\n & II.D35, II.D36 \\\\\n12\n & Spectrum\ + \ at $\\infty$ empty\n & $\\omega$ spectral\n & I.D47, II.D60 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\\ + medskip\n\\noindent\nEvery entry in the table is backed by\na proved theorem or a registered definition\nin Books~I--II.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 181\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch61-master-switch.tex`\ + \ lines 529-607\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.Inventory`\n\ + - Name: `full_rewiring_table`\n\n## Dependencies\n\n- Canonical: I.T05, I.T10, I.D18, I.D21, I.T31, II.T06, II.T19, II.T25,\ + \ II.T27, II.T30, II.T32, II.T39, II.T40, II.D21, II.D22, II.D35, II.D60\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Twelve-level structural comparison between orthodox complex analysis (i^2 = -1) and tau-holomorphy (j^2 = +1). + Levels: scalar algebra, holomorphy PDE, boundary-interior, infinity, cardinality, topology, geometry, compactness, idempotents, + L' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001307.md b/_corpus_v3_items/cid001307.md new file mode 100644 index 000000000..273e7d463 --- /dev/null +++ b/_corpus_v3_items/cid001307.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001307/ +title: The Physics Quadrant Matrix +cid: cid001307 +primary_alias: DEF0181 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 2x2 classification of mathematical frameworks by PDE type (elliptic/hyperbolic) and metric type (Archimedean/non-Archimedean). + QFT in (H,A), GR locally in (E,A), Category tau in (H,nA). The fourth quadrant (H,nA) is the only quadrant where the Unification + Obstruction does not apply. +aliases: +- DEF0181 +- II.D73 +- the-physics-quadrant-matrix +- def:physics-quadrant-matrix +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005610 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**The Physics Quadrant Matrix**\n\n2x2 classification of mathematical frameworks by PDE type (elliptic/hyperbolic)\ + \ and metric type (Archimedean/non-Archimedean). QFT in (H,A), GR locally in (E,A), Category tau in (H,nA). The fourth\ + \ quadrant (H,nA) is the only quadrant where the Unification Obstruction does not apply.\n\n# The Physics Quadrant Matrix\n\ + \n## Summary\n\n2x2 classification of mathematical frameworks by PDE type (elliptic/hyperbolic) and metric type (Archimedean/non-Archimedean).\ + \ QFT in (H,A), GR locally in (E,A), Category tau in (H,nA). The fourth quadrant (H,nA) is the only quadrant where the\ + \ Unification Obstruction does not apply.\n\n## Statement\n\n```latex\n%\n\\label{def:physics-quadrant-matrix}\nThe \\\ + textbf{Physics Quadrant Matrix}\nclassifies physical frameworks by two axes:\nPDE type (elliptic/hyperbolic)\nand metric\ + \ type (Archimedean/non-Archimedean).\n\n\\medskip\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{tabular}{@{}l|c|c@{}}\n\ + \ & \\textbf{Archimedean} & \\textbf{Non-Archimedean} \\\\\n\\hline\n\\textbf{Elliptic}\n & GR (curvature, $\\varepsilon$-$\\\ + delta$)\n & (empty) \\\\\n\\hline\n\\textbf{Hyperbolic}\n & QFT (wave eq., renormalization)\n & Category $\\tau$ \\\ + \\\n\\end{tabular}\n\\end{center}\n\n\\medskip\n\\noindent\nQuantum Field Theory sits in the (Hyperbolic, Archimedean)\ + \ quadrant:\nits propagation is wave-type\nbut its renormalization requires $\\varepsilon$-$\\delta$.\nGeneral Relativity\ + \ sits in the (Elliptic, Archimedean) quadrant:\nits curvature requires Laplacian differentiability\nbut its causal structure\ + \ is hyperbolic.\nCategory~$\\tau$ sits in the (Hyperbolic, Non-Archimedean) quadrant.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 184\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex`\ + \ lines 384-417\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.PhysicsQuadrant`\n\ + - Name: `PhysicsQuadrant`\n\n## Dependencies\n\n- Canonical: I.T10, II.D21, II.T19, II.D72\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '2x2 classification of mathematical frameworks by PDE type (elliptic/hyperbolic) and metric type (Archimedean/non-Archimedean). + QFT in (H,A), GR locally in (E,A), Category tau in (H,nA). The fourth quadrant (H,nA) is the only quadrant where ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001308.md b/_corpus_v3_items/cid001308.md new file mode 100644 index 000000000..fa34bff3b --- /dev/null +++ b/_corpus_v3_items/cid001308.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001308/ +title: The Unification Obstruction +cid: cid001308 +primary_alias: DEF0182 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Structural incompatibility within the Archimedean column: no framework can simultaneously support hyperbolic propagation + (wave-type PDE with characteristics and causal ordering) and elliptic regularity (Laplacian-type with isotropic diffusion + and smoothing) at the same point at all energy scales.' +aliases: +- DEF0182 +- II.D74 +- the-unification-obstruction +- def:unification-obstruction +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005599 +- predicate: formalized_by + object: cid005600 +- predicate: formalized_by + object: cid005601 +- predicate: formalized_by + object: cid005611 +- predicate: formalized_by + object: cid005614 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**The Unification Obstruction**\n\nStructural incompatibility within the Archimedean column: no framework can\ + \ simultaneously support hyperbolic propagation (wave-type PDE with characteristics and causal ordering) and elliptic\ + \ regularity (Laplacian-type with isotropic diffusion and smoothing) at the same point at all energy scales.\n\n# The\ + \ Unification Obstruction\n\n## Summary\n\nStructural incompatibility within the Archimedean column: no framework can\ + \ simultaneously support hyperbolic propagation (wave-type PDE with characteristics and causal ordering) and elliptic\ + \ regularity (Laplacian-type with isotropic diffusion and smoothing) at the same point at all energy scales.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:unification-obstruction}\nThe \\textbf{Unification Obstruction} states:\nno framework in the\ + \ Archimedean column\ncan simultaneously support\n\\begin{enumerate}\n \\item[\\textup{(U1)}]\n hyperbolic\ + \ propagation\n (required by QFT), and\n \\item[\\textup{(U2)}]\n elliptic regularity\n \ + \ (required by GR).\n\\end{enumerate}\n(U1) and (U2) are incompatible\nwithin a single Archimedean framework\nbecause\ + \ elliptic and hyperbolic PDEs\nhave structurally different regularity properties.\n```\n\n## Proof / Justification\n\n\ + This item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 185\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex`\ + \ lines 419-437\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.PhysicsQuadrant`\n\ + - Name: `unification_obstructed`\n\n## Dependencies\n\n- Canonical: II.D73, II.D21, II.T19\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Structural incompatibility within the Archimedean column: no framework can simultaneously support hyperbolic + propagation (wave-type PDE with characteristics and causal ordering) and elliptic regularity (Laplacian-type with isotropic + diffusi' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001309.md b/_corpus_v3_items/cid001309.md new file mode 100644 index 000000000..257ddf84d --- /dev/null +++ b/_corpus_v3_items/cid001309.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001309/ +title: Archimedean-Elliptic Engine +cid: cid001309 +primary_alias: DEF0183 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The mechanism generating the orthodox SCV dimensional ladder: interaction of Archimedean metric dimension (continuous + manifold with free parameter n) and elliptic CR overdeterminacy (n(n-1)/2 integrability conditions beyond n basic equations). + Requires BOTH ingredients; neither alone generates a ladder.' +aliases: +- DEF0183 +- II.D75 +- archimedean-elliptic-engine +- def:archimedean-elliptic-engine +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005590 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 63 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Archimedean-Elliptic Engine**\n\nThe mechanism generating the orthodox SCV dimensional ladder: interaction\ + \ of Archimedean metric dimension (continuous manifold with free parameter n) and elliptic CR overdeterminacy (n(n-1)/2\ + \ integrability conditions beyond n basic equations). Requires BOTH ingredients; neither alone generates a ladder.\n\n\ + # Archimedean-Elliptic Engine\n\n## Summary\n\nThe mechanism generating the orthodox SCV dimensional ladder: interaction\ + \ of Archimedean metric dimension (continuous manifold with free parameter n) and elliptic CR overdeterminacy (n(n-1)/2\ + \ integrability conditions beyond n basic equations). Requires BOTH ingredients; neither alone generates a ladder.\n\n\ + ## Statement\n\n```latex\n%\n\\label{def:archimedean-elliptic-engine}\nThe \\textbf{Archimedean-Elliptic Engine} (AEE)\n\ + is the combination of two properties:\n\\begin{enumerate}\n \\item[\\textup{(A)}]\n \\textbf{Archimedean density:}\n\ + \ between any two distinct points\n lie uncountably many others.\n Limits refine without bound.\n\ + \ \\item[\\textup{(E)}]\n \\textbf{Elliptic regularity:}\n the Laplace equation $\\Delta f = 0$\n\ + \ forces holomorphic functions\n to be infinitely differentiable,\n with convergent power series\n\ + \ at every point.\n\\end{enumerate}\nThe AEE is present in orthodox $\\mathbb{C}^n$\nand absent in $\\tau^3$.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 189\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch63-refuses.tex`\ + \ lines 250-271\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.DimensionalLadder`\n\ + - Name: `ArchimedeanEllipticEngine`\n\n## Dependencies\n\n- Canonical: II.D73, II.D68\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The mechanism generating the orthodox SCV dimensional ladder: interaction of Archimedean metric dimension (continuous + manifold with free parameter n) and elliptic CR overdeterminacy (n(n-1)/2 integrability conditions beyond n basic equation' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001310.md b/_corpus_v3_items/cid001310.md new file mode 100644 index 000000000..345ba6569 --- /dev/null +++ b/_corpus_v3_items/cid001310.md @@ -0,0 +1,170 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001310/ +title: Dimensional Rigidity +cid: cid001310 +primary_alias: DEF0184 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: tau admits exactly one holomorphic structure with fibration index 3 (= 1 base + 2 fiber) and refinement dimension + 4 (= ABCD rays). There is no family of tau-structures indexed by dimension; the 3 in tau^3 is an algebraic fibration index, + not a metric dimension parameter. +aliases: +- DEF0184 +- II.D76 +- dimensional-rigidity +- def:dimensional-rigidity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005584 +- predicate: formalized_by + object: cid005585 +- predicate: formalized_by + object: cid005586 +- predicate: formalized_by + object: cid005587 +- predicate: formalized_by + object: cid005591 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 63 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Dimensional Rigidity** + + + tau admits exactly one holomorphic structure with fibration index 3 (= 1 base + 2 fiber) and refinement dimension 4 (= + ABCD rays). There is no family of tau-structures indexed by dimension; the 3 in tau^3 is an algebraic fibration index, + not a metric dimension parameter. + + + # Dimensional Rigidity + + + ## Summary + + + tau admits exactly one holomorphic structure with fibration index 3 (= 1 base + 2 fiber) and refinement dimension 4 (= + ABCD rays). There is no family of tau-structures indexed by dimension; the 3 in tau^3 is an algebraic fibration index, + not a metric dimension parameter. + + + ## Statement + + + ```latex + + % + + \label{def:dimensional-rigidity} + + A holomorphic framework exhibits + + \textbf{dimensional rigidity} + + if increasing the coordinate dimension + + produces no qualitatively new phenomena---i.e., + + all features that appear at dimension $n > 1$ + + are already present at dimension~1 + + (or, equivalently, the framework does not + + distinguish between ``complex dimension~1'''' + + and ``complex dimension~$n$'''' behaviors). + + $\tau^3$ exhibits dimensional rigidity. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 190 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch63-refuses.tex` lines 276-289 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Mirror.DimensionalLadder` + + - Name: `DimensionalRigidity` + + + ## Dependencies + + + - Canonical: II.D15, II.D02, II.T42 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: tau admits exactly one holomorphic structure with fibration index 3 (= 1 base + 2 fiber) and refinement dimension + 4 (= ABCD rays). There is no family of tau-structures indexed by dimension; the 3 in tau^3 is an algebraic fibration index, + no +noindex: false +--- + diff --git a/_corpus_v3_items/cid001311.md b/_corpus_v3_items/cid001311.md new file mode 100644 index 000000000..3e3f07af3 --- /dev/null +++ b/_corpus_v3_items/cid001311.md @@ -0,0 +1,115 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001311/ +title: Geometric Bi-Square +cid: cid001311 +primary_alias: DEF0185 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The algebraic bi-square (I.T41) filled with geometric objects earned in Book II Parts I-IX: Stone topology on tau^3 + (II.T07), continuous projections (II.T06), torus degeneration L = S^1 v S^1 (II.T13), calibrated H_tau (II.D35), spectral + algebra A_spec(L) (II.D60). The pasted commuting diagram with earned geometric content.' +aliases: +- DEF0185 +- II.D77 +- geometric-bi-square +- def:geometric-bisquare +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005476 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 59 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Geometric Bi-Square**\n\nThe algebraic bi-square (I.T41) filled with geometric objects earned in Book II Parts\ + \ I-IX: Stone topology on tau^3 (II.T07), continuous projections (II.T06), torus degeneration L = S^1 v S^1 (II.T13),\ + \ calibrated H_tau (II.D35), spectral algebra A_spec(L) (II.D60). The pasted commuting diagram with earned geometric content.\n\ + \n# Geometric Bi-Square\n\n## Summary\n\nThe algebraic bi-square (I.T41) filled with geometric objects earned in Book\ + \ II Parts I-IX: Stone topology on tau^3 (II.T07), continuous projections (II.T06), torus degeneration L = S^1 v S^1 (II.T13),\ + \ calibrated H_tau (II.D35), spectral algebra A_spec(L) (II.D60). The pasted commuting diagram with earned geometric content.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:geometric-bisquare}\n% II.T06, II.T07, II.T13, II.D35, II.D60, II.T32, II.T37,\ + \ II.T40\nThe \\textbf{geometric bi-square}\nis the pasted commuting diagram obtained from\nthe algebraic bi-square (I.T41)\n\ + by replacing every algebraic component\nwith its earned geometric counterpart.\nThe diagram has two levels:\na \\textbf{stage\ + \ diagram} (for each pair $k \\leq \\ell$)\nand a \\textbf{limit diagram} (the limiting row).\n\n\\medskip\n\\noindent\n\ + \\textbf{Stage diagram} (for stages $k \\leq \\ell$):\n\\begin{equation}\n\\label{eq:ch59-stage-diagram}\n \\begin{array}{ccccc}\n\ + \ \\tau^3_\\ell\n & \\xrightarrow{\\;\\; f_\\ell \\;\\;}\n & H_\\tau^{\\mathrm{cal}}\n & \\xrightarrow{\\\ + ;\\; (\\chi_+,\\, \\chi_-) \\;\\;}\n & \\widehat{\\mathbb{Z}}_\\tau \\times \\widehat{\\mathbb{Z}}_\\tau\n \\\\\ + [8pt]\n \\bigg\\downarrow\\vcenter{\\rlap{\\scriptsize\\; $\\mathrm{proj}_k$}}\n & & \\bigg\\downarrow\\vcenter{\\\ + rlap{\\scriptsize\\; $\\mathrm{proj}_k$}}\n & & \\bigg\\downarrow\\vcenter{\\rlap{\\scriptsize\\; $(\\mathrm{proj}_k,\\\ + , \\mathrm{proj}_k)$}}\n \\\\[8pt]\n \\tau^3_k\n & \\xrightarrow{\\;\\; f_k \\;\\;}\n & H_\\tau^{\\mathrm{cal}}\n\ + \ & \\xrightarrow{\\;\\; (\\chi_+,\\, \\chi_-) \\;\\;}\n & \\widehat{\\mathbb{Z}}_\\tau \\times \\widehat{\\\ + mathbb{Z}}_\\tau\n \\end{array}\n\\end{equation}\nwhere:\n\\begin{itemize}\n \\item $\\tau^3_k$ denotes the stage-$k$\ + \ truncation of~$\\tau^3$:\n the finite set of $\\tau$-admissible ABCD quadruples\n modulo $P_k = p_1\ + \ \\cdots p_k$,\n equipped with the discrete topology.\n \\item $H_\\tau^{\\mathrm{cal}}$ is the calibrated\ + \ split-complex codomain\n (II.D35, Chapter~\\ref{ch:split-complex-calibrated}),\n carrying the constants\ + \ $\\pi$, $e$, $\\jj$, $\\iota_\\tau$.\n \\item $f_k \\colon \\tau^3_k \\to H_\\tau^{\\mathrm{cal}}$\n is\ + \ the stage-$k$ holomorphic function,\n and $f_\\ell$ its stage-$\\ell$ counterpart.\n \\item $(\\chi_+, \\\ + chi_-)$ is the bipolar spectral decomposition\n (II.D59, Chapter~\\ref{ch:boundary-characters-idempotent}):\n\ + \ $f = e_+ \\cdot f_+ + e_- \\cdot f_-$.\n \\item $\\mathrm{proj}_k$ denotes the \\textbf{continuous} projection\n\ + \ from stage~$\\ell$ to stage~$k$\n in the inverse-limit topology\n (continuous by II.T06,\n\ + \ Chapter~\\ref{ch:hol-implies-cont}).\n\\end{itemize}\n\n\\medskip\n\\noindent\n\\textbf{Limit diagram} (the\ + \ Central Theorem row):\n\\begin{equation}\n\\label{eq:ch59-limit-diagram}\n \\begin{array}{ccccc}\n \\Lemniscate\ + \ = S^1 \\vee S^1\n & \\xrightarrow{\\quad \\Phi \\quad}\n & \\mathcal{O}(\\tau^3)\n & \\xrightarrow{\\\ + quad \\Psi \\quad}\n & A_{\\mathrm{spec}}(\\Lemniscate)\n \\\\[8pt]\n \\bigg\\downarrow\\vcenter{\\rlap{\\\ + scriptsize\\; $\\mathrm{proj}_d$}}\n & & \\bigg\\downarrow\\vcenter{\\rlap{\\scriptsize\\; $\\mathrm{eval}_d$}}\n\ + \ & & \\bigg\\downarrow\\vcenter{\\rlap{\\scriptsize\\; $\\mathrm{proj}_d$}}\n \\\\[8pt]\n \\Lemniscate_d\n\ + \ & \\xrightarrow{\\;\\; f_d \\;\\;}\n & H_\\tau^{\\mathrm{cal}}\n & \\xrightarrow{\\;\\; (\\chi_+,\\,\ + \ \\chi_-) \\;\\;}\n & \\mathbb{Z}/M_d\\mathbb{Z} \\times \\mathbb{Z}/M_d\\mathbb{Z}\n \\end{array}\n\\end{equation}\n\ + where:\n\\begin{itemize}\n \\item $\\Lemniscate = S^1 \\vee S^1$ is the geometric lemniscate\n (II.T13, Chapter~\\\ + ref{ch:torus-degeneration}):\n the torus $T^2$ degenerated to the wedge of two circles\n at the boundary.\n\ + \ \\item $\\mathcal{O}(\\tau^3)$ is the ring of holomorphic functions\n on~$\\tau^3$.\n \\item $A_{\\mathrm{spec}}(\\\ + Lemniscate)$ is the spectral algebra\n (II.D60, Chapter~\\ref{ch:central-theorem}).\n \\item $\\Phi \\colon\ + \ A_{\\mathrm{spec}}(\\Lemniscate) \\to \\mathcal{O}(\\tau^3)$\n is the boundary-to-interior map\n (Hartogs\ + \ extension followed by Yoneda identification).\n \\item $\\Psi \\colon \\mathcal{O}(\\tau^3) \\to A_{\\mathrm{spec}}(\\\ + Lemniscate)$\n is the interior-to-boundary map\n (restriction followed by Mutual Determination).\n \ + \ \\item The composition $\\Psi \\circ \\Phi = \\mathrm{id}$\n and $\\Phi \\circ \\Psi = \\mathrm{id}$\n \ + \ \\textbf{is} the Central Theorem (II.T40).\n \\item $\\mathrm{eval}_d$ evaluates a holomorphic function\n\ + \ at stage~$d$;\n $\\mathrm{proj}_d$ projects the inverse limit\n to the $d$-th finite stage.\n\ + \\end{itemize}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 166\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch59-geometric-bisquare.tex`\ + \ lines 347-452\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.GeometricBiSquare`\n\ + - Name: `GeometricBiSquareData`\n\n## Dependencies\n\n- Canonical: I.T41, I.D83, II.T06, II.T07, II.T13, II.D14, II.D35,\ + \ II.D60\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The algebraic bi-square (I.T41) filled with geometric objects earned in Book II Parts I-IX: Stone topology on + tau^3 (II.T07), continuous projections (II.T06), torus degeneration L = S^1 v S^1 (II.T13), calibrated H_tau (II.D35), spectral + al' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001312.md b/_corpus_v3_items/cid001312.md new file mode 100644 index 000000000..b0f7363f2 --- /dev/null +++ b/_corpus_v3_items/cid001312.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001312/ +title: τ-Connection +cid: cid001312 +primary_alias: DEF0186 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A connection on the primorial tower: a transport operator Γ_k(x,v) = (x+v) mod M_k at each stage, satisfying tower + compatibility. The canonical flat connection uses additive transport.' +aliases: +- DEF0186 +- II.D78 +- connection +- def:tau-connection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005448 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Connection** + + + A connection on the primorial tower: a transport operator Γ_k(x,v) = (x+v) mod M_k at each stage, satisfying tower compatibility. + The canonical flat connection uses additive transport. + + + # τ-Connection + + + ## Summary + + + A connection on the primorial tower: a transport operator Γ_k(x,v) = (x+v) mod M_k at each stage, satisfying tower compatibility. + The canonical flat connection uses additive transport. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 195 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Connection` + + - Name: `TauConnection` + + + ## Dependencies + + + - Canonical: II.D62, II.D63 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A connection on the primorial tower: a transport operator Γ_k(x,v) = (x+v) mod M_k at each stage, satisfying + tower compatibility. The canonical flat connection uses additive transport.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001313.md b/_corpus_v3_items/cid001313.md new file mode 100644 index 000000000..b9f819961 --- /dev/null +++ b/_corpus_v3_items/cid001313.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001313/ +title: Parallel Transport +cid: cid001313 +primary_alias: DEF0187 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Parallel transport along a path γ in Z/M_k Z: sequential composition of the connection transport operator along + each edge. For the flat connection, this reduces to addition mod M_k.' +aliases: +- DEF0187 +- II.D79 +- parallel-transport +- def:parallel-transport +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Parallel Transport**\n\nParallel transport along a path γ in Z/M_k Z: sequential composition of the connection\ + \ transport operator along each edge. For the flat connection, this reduces to addition mod M_k.\n\n# Parallel Transport\n\ + \n## Summary\n\nParallel transport along a path γ in Z/M_k Z: sequential composition of the connection transport operator\ + \ along each edge. For the flat connection, this reduces to addition mod M_k.\n\n## Statement\n\n```latex\n%\n\\label{def:mode-a-catalog}\n\ + % I.T05, I.T08\nThe following constructions in Books~I--II\nare \\textbf{Mode~A} (Same):\n\n\\medskip\n\\renewcommand{\\\ + arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}p{0.23\\linewidth}p{0.23\\linewidth}p{0.23\\linewidth}p{0.23\\\ + linewidth}@{}}\n\\toprule\n\\textbf{Object} & \\textbf{$\\tau$ source} & \\textbf{Orthodox}\n & \\textbf{Identity witness}\ + \ \\\\\n\\midrule\nNatural numbers $\\mathbb{N}$\n & I.D07 (orbit of $\\alpha$)\n & PA\n & Canonical bijection\n \ + \ $\\rho^n(\\alpha) \\leftrightarrow n$ \\\\\nPrime numbers\n & I.D19b\n & Fundamental theorem of arithmetic\n & Same\ + \ set, same divisibility \\\\\n$\\pi$\n & Book~II Part~V (earned)\n & Classical transcendental\n & Same limit of solenoidal\ + \ ratio \\\\\n$e$\n & Book~II Part~V (earned)\n & Classical transcendental\n & Same limit $(1+1/n)^n$ \\\\\n$\\hat{\\\ + mathbb{Z}}$\n & I.D19 ($\\hat{\\mathbb{Z}}_\\tau$)\n & Profinite completion of $\\mathbb{Z}$\n & Same inverse limit\ + \ \\\\\nBoolean fragment\n & I.D21 (Truth4 restricted)\n & Classical logic $\\{T, F\\}$\n & $\\{T, F\\} \\hookrightarrow\ + \ \\{T, F, B, N\\}$ \\\\\nFundamental theorem\n & I.T19\n & FTA\n & Same unique factorization \\\\\n\\bottomrule\n\\\ + end{tabular}\n\\end{center}\n\n\\medskip\n\\noindent\nFor each entry, the identity witness is a\ncanonical bijection or\ + \ embedding\nthat preserves all structure\n(order, operations, defining properties).\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 196\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch62-survives-the-fork.tex`\ + \ lines 56-110\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.Connection`\n\ + - Name: `parallel_transport`\n\n## Dependencies\n\n- Canonical: II.D78\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Parallel transport along a path γ in Z/M_k Z: sequential composition of the connection transport operator along + each edge. For the flat connection, this reduces to addition mod M_k.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001314.md b/_corpus_v3_items/cid001314.md new file mode 100644 index 000000000..dc9bb981d --- /dev/null +++ b/_corpus_v3_items/cid001314.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001314/ +title: τ-Curvature +cid: cid001314 +primary_alias: DEF0188 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Curvature R(v,w)(x) = Γ(Γ(x,v),w) - Γ(Γ(x,w),v) measures parallel transport non-commutativity. For the flat connection + on Z/M_k Z, R = 0. +aliases: +- DEF0188 +- II.D80 +- curvature +- def:tau-curvature +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Curvature**\n\nCurvature R(v,w)(x) = Γ(Γ(x,v),w) - Γ(Γ(x,w),v) measures parallel transport non-commutativity.\ + \ For the flat connection on Z/M_k Z, R = 0.\n\n# τ-Curvature\n\n## Summary\n\nCurvature R(v,w)(x) = Γ(Γ(x,v),w) - Γ(Γ(x,w),v)\ + \ measures parallel transport non-commutativity. For the flat connection on Z/M_k Z, R = 0.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:mode-b-catalog}\n% II.T06, II.T07, II.D09, II.D12, II.D13, II.D14,\n% II.D22, II.D35, II.D36\nThe following\ + \ constructions in Books~I--II\nare \\textbf{Mode~B} (Parallel):\n\n\\medskip\n\\renewcommand{\\arraystretch}{1.35}\n\\\ + begin{center}\n\\begin{tabular}{@{}p{0.2\\linewidth}@{\\;\\;}p{0.2\\linewidth}@{\\;\\;}p{0.2\\linewidth}@{\\;\\;}p{0.2\\\ + linewidth}@{}}\n\\toprule\n\\textbf{Construction} & \\textbf{$\\tau$ version} & \\textbf{Orthodox version}\n & \\textbf{Key\ + \ difference} \\\\\n\\midrule\nHolomorphic functions\n & $\\tau$-holomorphic on $H_\\tau$, $\\jj^2 = +1$\n & Classical\ + \ holomorphic on $\\mathbb{C}$, $i^2 = -1$\n & Sign: wave vs.\\ Laplace \\\\\nConstructive reals\n & $R_\\tau$ (I.D84),\ + \ ultrametric\n & $\\mathbb{R}$ (Dedekind/Cauchy), Archimedean\n & Metric: $2^{-\\delta}$ vs.\\ $|x-y|$ \\\\\nSet membership\n\ + \ & Divisibility (I.D31)\n & $\\in$ (ZFC)\n & $a \\mid b$ vs.\\ $a \\in b$ \\\\\nTopology\n & Stone space (II.D14),\ + \ clopen\n & Manifold, connected, Hausdorff\n & Totally disconnected \\\\\nBndLift\n & CRT propagation (II.D36)\n \ + \ & Cauchy integral\n & Finite vs.\\ contour \\\\\nLaurent expansion\n & CRT coefficients (II.D35)\n & Cauchy coefficients\n\ + \ & Constructive extraction \\\\\nde Rham complex\n & $\\tau$-de Rham (Book~II Part~X)\n & Classical de Rham\n & Discrete\ + \ vs.\\ smooth \\\\\nIdempotent decomp.\n & $f = e_+ g + e_- h$ (holomorphic)\n & $f = u + iv$ (harmonic)\n & Both\ + \ holomorphic vs.\\ only harmonic \\\\\nMetric\n & $d = 2^{-\\delta}$ ultrametric (II.D12)\n & $|x - y|$ Archimedean\n\ + \ & Agreement depth vs.\\ magnitude \\\\\nCode/Decode\n & Boundary coefficients (II.T35)\n & Laurent series\n & Without\ + \ integration \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\medskip\n\\noindent\nFor each entry, the \\emph{same\ + \ axioms} column\nis the set of defining properties that both versions satisfy.\nThe \\emph{key difference} column identifies\n\ + the carrier-level divergence.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 199\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch62-survives-the-fork.tex`\ + \ lines 161-227\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.Curvature`\n\ + - Name: `curvature_check`\n\n## Dependencies\n\n- Canonical: II.D78, II.D79\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Curvature R(v,w)(x) = Γ(Γ(x,v),w) - Γ(Γ(x,w),v) measures parallel transport non-commutativity. For the flat connection + on Z/M_k Z, R = 0. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001315.md b/_corpus_v3_items/cid001315.md new file mode 100644 index 000000000..501df0266 --- /dev/null +++ b/_corpus_v3_items/cid001315.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001315/ +title: τ-Geodesic +cid: cid001315 +primary_alias: DEF0189 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Geodesic in Z/M_k Z: shortest path in the cyclic distance. The geodesic direction from x to y is the minimal displacement + mod M_k.' +aliases: +- DEF0189 +- II.D81 +- geodesic +- def:tau-geodesic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005454 +- predicate: formalized_by + object: cid005455 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Geodesic** + + + Geodesic in Z/M_k Z: shortest path in the cyclic distance. The geodesic direction from x to y is the minimal displacement + mod M_k. + + + # τ-Geodesic + + + ## Summary + + + Geodesic in Z/M_k Z: shortest path in the cyclic distance. The geodesic direction from x to y is the minimal displacement + mod M_k. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 200 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Curvature` + + - Name: `geodesic_check` + + + ## Dependencies + + + - Canonical: II.D78 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Geodesic in Z/M_k Z: shortest path in the cyclic distance. The geodesic direction from x to y is the minimal + displacement mod M_k.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001316.md b/_corpus_v3_items/cid001316.md new file mode 100644 index 000000000..de7dc5872 --- /dev/null +++ b/_corpus_v3_items/cid001316.md @@ -0,0 +1,101 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001316/ +title: L² Inner Product +cid: cid001316 +primary_alias: DEF0190 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'L² inner product on Z/M_k Z: ⟨f,g⟩_k = (1/M_k) Σ f(x)g(x). Represented as rational pair. Symmetric, bilinear, with + orthogonal indicator basis.' +aliases: +- DEF0190 +- II.D82 +- l-inner-product +- def:l2-inner-product +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005557 +- predicate: formalized_by + object: cid005558 +- predicate: formalized_by + object: cid005565 +- predicate: formalized_by + object: cid005566 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**L² Inner Product**\n\nL² inner product on Z/M_k Z: ⟨f,g⟩_k = (1/M_k) Σ f(x)g(x). Represented as rational pair.\ + \ Symmetric, bilinear, with orthogonal indicator basis.\n\n# L² Inner Product\n\n## Summary\n\nL² inner product on Z/M_k\ + \ Z: ⟨f,g⟩_k = (1/M_k) Σ f(x)g(x). Represented as rational pair. Symmetric, bilinear, with orthogonal indicator basis.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:mode-d-catalog}\n% I.T11, I.D03, I.D18, I.D21, I.D34,\n% II.T06, II.T40,\ + \ II.T41, II.T42, II.D68\nThe following constructions in Books~I--II\nare \\textbf{Mode~D} (Gained):\n\n\\medskip\n\\\ + renewcommand{\\arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}r@{\\;\\;}p{3.9cm}@{\\;\\;}p{2.2cm}@{\\;\\;}p{5.0cm}@{}}\n\ + \\toprule\n\\textbf{\\#} & \\textbf{$\\tau$-gain}\n & \\textbf{Source}\n & \\textbf{Why impossible in ZFC/PA} \\\\\n\ + \\midrule\n\\multicolumn{4}{@{}l@{}}{\\textit{Book~I gains:}} \\\\\n1 & Categoricity\n & I.T08\n & L\\\"owenheim--Skolem\ + \ prevents categorical first-order theories \\\\\n2 & Rigidity $\\Aut(\\tau) = \\{\\id\\}$\n & I.T07, I.T11\n & Most\ + \ structures have non-trivial automorphisms \\\\\n3 & Finite axiom set (no schemas)\n & K0--K6\n & PA requires infinite\ + \ induction schema \\\\\n4 & Structural decidability\n & I.P07+\n & G\\\"odel sentences are undecidable \\\\\n5 & $\\\ + dim = 4$ forced\n & I.D03\n & No orthodox axiom forces dimension \\\\\n6 & Prime Polarity\n & I.T05\n & Invisible\ + \ in orthodox number theory \\\\\n7 & Algebraic lemniscate\n & I.D18\n & Requires $\\jj^2 = +1$ \\\\\n8 & Truth4 (4-valued\ + \ logic)\n & I.D21\n & Classical logic is $\\{T,F\\}$ \\\\\n9 & Explosion barrier\n & I.T13\n & Classical logic admits\ + \ explosion \\\\\n10 & Master constant $\\iota_\\tau$\n & I.D34\n & No canonical spectral constant in ZFC \\\\\n11 &\ + \ Presheaf characterization\n & Book~I Part~XVI\n & Requires categorical holomorphy \\\\\n12 & Diagonal $=$ linear logic\n\ + \ & Book~I Part~XVII\n & Independent characterization \\\\\n13 & Safe self-enrichment\n & Book~II Part~VIII\n & Russell's\ + \ paradox blocks na\\\"ive self-description \\\\\n\\midrule\n\\multicolumn{4}{@{}l@{}}{\\textit{Book~II gains:}} \\\\\n\ + 14 & Central Theorem\n & II.T40\n & No orthodox proof of $\\mathcal{O} \\cong A_{\\mathrm{spec}}$ \\\\\n15 & Boundary-first\ + \ paradigm\n & II.R03\n & Orthodox: interior $\\to$ boundary \\\\\n16 & Inverted dependency\n & II.T06\n & Orthodox:\ + \ topology before holomorphy \\\\\n17 & Parallel Postulate proved\n & II.T15--II.T18\n & 2000 years of independence\ + \ \\\\\n18 & Liouville dodge\n & II.T41\n & Liouville theorem blocks non-constant bounded entire functions \\\\\n19\ + \ & Categoricity of $\\tau^3$\n & II.T42\n & Moduli spaces typically have dimension $>0$ \\\\\n20 & Positive regularity\n\ + \ & II.D49\n & Orthodox regularity defined negatively \\\\\n21 & Holomorphic decomposition\n & II.L07\n & Orthodox\ + \ $f = u+iv$ gives harmonic, not holomorphic, components \\\\\n22 & Master switch (12-level)\n & II.D68\n & No orthodox\ + \ single-parameter classification \\\\\n23 & Structural incompatibility\n & II.T43\n & Internal to $\\tau$ \\\\\n\\\ + bottomrule\n\\end{tabular}\n\\end{center}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 204\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch64-gains.tex`\ + \ lines 52-144\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.L2Space`\n\ + - Name: `l2_inner_product`\n\n## Dependencies\n\n- Canonical: I.D99, II.D45\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'L² inner product on Z/M_k Z: ⟨f,g⟩_k = (1/M_k) Σ f(x)g(x). Represented as rational pair. Symmetric, bilinear, + with orthogonal indicator basis.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001317.md b/_corpus_v3_items/cid001317.md new file mode 100644 index 000000000..c4282c4c8 --- /dev/null +++ b/_corpus_v3_items/cid001317.md @@ -0,0 +1,130 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001317/ +title: L² Norm +cid: cid001317 +primary_alias: DEF0191 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'L² norm squared: ‖f‖²_k = ⟨f,f⟩_k. Non-negative by construction (sum of squares). Zero iff f = 0 (definiteness).' +aliases: +- DEF0191 +- II.D83 +- l-norm +- def:l2-norm +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005559 +- predicate: formalized_by + object: cid005560 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**L² Norm** + + + L² norm squared: ‖f‖²_k = ⟨f,f⟩_k. Non-negative by construction (sum of squares). Zero iff f = 0 (definiteness). + + + # L² Norm + + + ## Summary + + + L² norm squared: ‖f‖²_k = ⟨f,f⟩_k. Non-negative by construction (sum of squares). Zero iff f = 0 (definiteness). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 205 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Hartogs.L2Space` + + - Name: `l2_norm_sq` + + + ## Dependencies + + + - Canonical: II.D82 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'L² norm squared: ‖f‖²_k = ⟨f,f⟩_k. Non-negative by construction (sum of squares). Zero iff f = 0 (definiteness).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001318.md b/_corpus_v3_items/cid001318.md new file mode 100644 index 000000000..4589f5e62 --- /dev/null +++ b/_corpus_v3_items/cid001318.md @@ -0,0 +1,127 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001318/ +title: Chain Complex +cid: cid001318 +primary_alias: DEF0192 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Chain complex on the primorial tower. The boundary map d_n reduces from stage n to stage n-1. Tower coherence: reduce(reduce(x,n-1),n-2) + = reduce(x,n-2).' +aliases: +- DEF0192 +- II.D84 +- chain-complex +- def:chain-complex +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 46 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Chain Complex** + + + Chain complex on the primorial tower. The boundary map d_n reduces from stage n to stage n-1. Tower coherence: reduce(reduce(x,n-1),n-2) + = reduce(x,n-2). + + + # Chain Complex + + + ## Summary + + + Chain complex on the primorial tower. The boundary map d_n reduces from stage n to stage n-1. Tower coherence: reduce(reduce(x,n-1),n-2) + = reduce(x,n-2). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 208 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Enrichment.Homological` + + - Name: `primorial_chain` + + + ## Dependencies + + + - Canonical: II.D55 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Chain complex on the primorial tower. The boundary map d_n reduces from stage n to stage n-1. Tower coherence: + reduce(reduce(x,n-1),n-2) = reduce(x,n-2).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001319.md b/_corpus_v3_items/cid001319.md new file mode 100644 index 000000000..b6d554e13 --- /dev/null +++ b/_corpus_v3_items/cid001319.md @@ -0,0 +1,138 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001319/ +title: Homology via SES +cid: cid001319 +primary_alias: DEF0193 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Homology via short exact sequences: 0 → Z/M_{k-1} → Z/M_k → Z/p_k → 0. Exactness at the middle: ker(g) = im(f), + giving trivial homology H = 0.' +aliases: +- DEF0193 +- II.D85 +- homology-via-ses +- def:homology-ses +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005502 +- predicate: formalized_by + object: cid005503 +- predicate: formalized_by + object: cid005504 +- predicate: formalized_by + object: cid005505 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 46 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Homology via SES** + + + Homology via short exact sequences: 0 → Z/M_{k-1} → Z/M_k → Z/p_k → 0. Exactness at the middle: ker(g) = im(f), giving + trivial homology H = 0. + + + # Homology via SES + + + ## Summary + + + Homology via short exact sequences: 0 → Z/M_{k-1} → Z/M_k → Z/p_k → 0. Exactness at the middle: ker(g) = im(f), giving + trivial homology H = 0. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 209 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Enrichment.Homological` + + - Name: `ses_exactness_check` + + + ## Dependencies + + + - Canonical: II.D84 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Homology via short exact sequences: 0 → Z/M_{k-1} → Z/M_k → Z/p_k → 0. Exactness at the middle: ker(g) = im(f), + giving trivial homology H = 0.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001320.md b/_corpus_v3_items/cid001320.md new file mode 100644 index 000000000..435462981 --- /dev/null +++ b/_corpus_v3_items/cid001320.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001320/ +title: Čech Complex +cid: cid001320 +primary_alias: DEF0194 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Čech complex for the cylinder cover of Z/M_k Z. Coboundary δ⁰(f)(a,b) = f(b) - f(a). Satisfies δ² = 0 (alternating + sum identity). +aliases: +- DEF0194 +- II.D86 +- ech-complex +- def:cech-complex +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005421 +- predicate: formalized_by + object: cid005422 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Čech Complex** + + + Čech complex for the cylinder cover of Z/M_k Z. Coboundary δ⁰(f)(a,b) = f(b) - f(a). Satisfies δ² = 0 (alternating sum + identity). + + + # Čech Complex + + + ## Summary + + + Čech complex for the cylinder cover of Z/M_k Z. Coboundary δ⁰(f)(a,b) = f(b) - f(a). Satisfies δ² = 0 (alternating sum + identity). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 212 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.CentralTheorem.SheafCohomology` + + - Name: `cech_coboundary_0` + + + ## Dependencies + + + - Canonical: II.D84, II.D47 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Čech complex for the cylinder cover of Z/M_k Z. Coboundary δ⁰(f)(a,b) = f(b) - f(a). Satisfies δ² = 0 (alternating + sum identity). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001321.md b/_corpus_v3_items/cid001321.md new file mode 100644 index 000000000..2c2dba83d --- /dev/null +++ b/_corpus_v3_items/cid001321.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001321/ +title: Sheaf Cohomology Groups +cid: cid001321 +primary_alias: DEF0195 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Sheaf cohomology on the primorial tower. H⁰ = global sections (rank 1 for constant sheaf). H¹ = 0 (ultrametric cover + is acyclic). +aliases: +- DEF0195 +- II.D87 +- sheaf-cohomology-groups +- def:sheaf-cohomology +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005423 +- predicate: formalized_by + object: cid005424 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sheaf Cohomology Groups** + + + Sheaf cohomology on the primorial tower. H⁰ = global sections (rank 1 for constant sheaf). H¹ = 0 (ultrametric cover is + acyclic). + + + # Sheaf Cohomology Groups + + + ## Summary + + + Sheaf cohomology on the primorial tower. H⁰ = global sections (rank 1 for constant sheaf). H¹ = 0 (ultrametric cover is + acyclic). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 213 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.CentralTheorem.SheafCohomology` + + - Name: `h0_global_sections` + + + ## Dependencies + + + - Canonical: II.D86 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Sheaf cohomology on the primorial tower. H⁰ = global sections (rank 1 for constant sheaf). H¹ = 0 (ultrametric + cover is acyclic). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001322.md b/_corpus_v3_items/cid001322.md new file mode 100644 index 000000000..3b0e44ce7 --- /dev/null +++ b/_corpus_v3_items/cid001322.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001322/ +title: Tau-Exponential +cid: cid001322 +primary_alias: DEF0196 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The tau-exponential E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant: E_tau(sigma(A)) + = sigma(E_tau(A)) for the involution sigma.' +aliases: +- DEF0196 +- II.D88 +- tau-exponential +- def:tau-exponential +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 43 (Gap Remediation) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Exponential**\n\nThe tau-exponential E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant:\ + \ E_tau(sigma(A)) = sigma(E_tau(A)) for the involution sigma.\n\n# Tau-Exponential\n\n## Summary\n\nThe tau-exponential\ + \ E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant: E_tau(sigma(A)) = sigma(E_tau(A)) for\ + \ the involution sigma.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-exponential}\nThe \\textbf{tau-exponential}\n\ + is the endofunctor\n\\[\n \\boxed{%\n E_\\tau : \\mathrm{Obj}(\\tau) \\;\\longrightarrow\\; \\mathrm{Obj}(\\tau),\n\ + \ \\qquad\n A \\;\\longmapsto\\; [A, A],}\n\\]\nwhere $[A, A]$ is the Hom object\n(Definition~\\ref{def:hom-object},\ + \ II.D54)\n--- the internal endomorphism space of~$A$.\nOn morphisms, $E_\\tau$ acts by conjugation:\nfor $\\varphi \\\ + colon A \\to B$,\n\\[\n E_\\tau(\\varphi) \\colon [A, A] \\to [B, B],\n \\qquad\n f \\mapsto \\varphi \\circ\ + \ f \\circ \\varphi^{-1},\n\\]\nwhenever $\\varphi$ is invertible.\nFor non-invertible $\\varphi$,\n$E_\\tau(\\varphi)$\ + \ is defined via postcomposition:\n$f \\mapsto \\varphi \\circ f$.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 218\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch43-yoneda-theorem.tex` lines 987-1011\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n## Dependencies\n\n- Canonical:\ + \ II.P21, II.D53\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The tau-exponential E_tau: HolEnd(tau) -> HolEnd(tau) sends A to [A, A]. It is sigma-equivariant: E_tau(sigma(A)) + = sigma(E_tau(A)) for the involution sigma.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001323.md b/_corpus_v3_items/cid001323.md new file mode 100644 index 000000000..6db67762a --- /dev/null +++ b/_corpus_v3_items/cid001323.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001323/ +title: Five Comparison Modes +cid: cid001323 +primary_alias: DEF0197 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Five exhaustive, mutually exclusive comparison modes (A: Same, B: Parallel, C: Refused, D: Gained, E: Earned) for + classifying every tau-construction against its orthodox counterpart.' +aliases: +- DEF0197 +- II.D89 +- five-comparison-modes +- def:five-comparison-modes +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 60 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Five Comparison Modes**\n\nFive exhaustive, mutually exclusive comparison modes (A: Same, B: Parallel, C: Refused,\ + \ D: Gained, E: Earned) for classifying every tau-construction against its orthodox counterpart.\n\n# Five Comparison\ + \ Modes\n\n## Summary\n\nFive exhaustive, mutually exclusive comparison modes (A: Same, B: Parallel, C: Refused, D: Gained,\ + \ E: Earned) for classifying every tau-construction against its orthodox counterpart.\n\n## Statement\n\n```latex\n%\n\ + \\label{def:five-comparison-modes}\nLet $\\mathcal{C}_\\tau$ denote a construction\n(definition, theorem, or property)\n\ + within Category~$\\tau$,\nand let $\\mathcal{C}_{\\mathrm{orth}}$ denote\nits closest orthodox counterpart (if one exists).\n\ + The \\textbf{comparison mode} of $\\mathcal{C}_\\tau$\nis one of:\n\n\\medskip\n\\renewcommand{\\arraystretch}{1.35}\n\ + \\begin{center}\n\\begin{tabular}{@{}c@{\\;\\;}l@{\\;\\;}p{7.5cm}@{}}\n\\toprule\n\\textbf{Mode} & \\textbf{Name} & \\\ + textbf{Criterion} \\\\\n\\midrule\n\\textbf{A} & Same &\n $\\mathcal{C}_\\tau$ and $\\mathcal{C}_{\\mathrm{orth}}$\n\ + \ are identical objects\n with a canonical bijection preserving\n all relevant structure. \\\\\n\\textbf{B} & Parallel\ + \ &\n $\\mathcal{C}_\\tau$ and $\\mathcal{C}_{\\mathrm{orth}}$\n satisfy the same defining axioms\n but live on different\ + \ carriers.\n Structural isomorphism holds at the property level\n but fails at the element level. \\\\\n\\textbf{C}\ + \ & Refused &\n $\\tau$'s axioms structurally block\n $\\mathcal{C}_{\\mathrm{orth}}$.\n The orthodox construction\ + \ violates\n a $\\tau$ axiom (K0--K6)\n or requires tools $\\tau$ lacks. \\\\\n\\textbf{D} & Gained &\n $\\mathcal{C}_\\\ + tau$ achieves a property\n that is provably impossible,\n independent, or false in ZFC/PA.\n No orthodox counterpart\ + \ exists. \\\\\n\\textbf{E} & Earned &\n $\\mathcal{C}_\\tau$ derives a result\n that orthodox mathematics takes as\ + \ axiom\n or obtains via a fundamentally different proof. \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\medskip\n\ + \\noindent\nThe modes satisfy:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Exhaustiveness.}\n \ + \ Every construction $\\mathcal{C}_\\tau$\n in Books~I--II\n has a unique primary mode.\n\n\ + \ \\item[\\textup{(ii)}]\n \\textbf{Mutual exclusivity.}\n A construction is Mode~A\n if\ + \ and only if the objects are literally identical;\n Mode~B if the axioms match but the carriers differ;\n \ + \ Mode~C if the orthodox construction is blocked;\n Mode~D if the $\\tau$-construction\n has\ + \ no orthodox analogue;\n Mode~E if the content matches\n but the proof status differs.\n\n \\item[\\\ + textup{(iii)}]\n \\textbf{Compositionality.}\n When a construction has aspects\n of multiple\ + \ modes,\n the primary mode captures\n the most significant structural difference.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 221\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch60-question-of-foundations.tex`\ + \ lines 219-295\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.T08, I.T10, II.T40\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Five exhaustive, mutually exclusive comparison modes (A: Same, B: Parallel, C: Refused, D: Gained, E: Earned) + for classifying every tau-construction against its orthodox counterpart.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001324.md b/_corpus_v3_items/cid001324.md new file mode 100644 index 000000000..891452808 --- /dev/null +++ b/_corpus_v3_items/cid001324.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001324/ +title: Six Comparison Axes +cid: cid001324 +primary_alias: DEF0198 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Six structural axes along which Category tau and orthodox mathematics differ: infinity, sign, topology, geometry, + analysis, and category theory.' +aliases: +- DEF0198 +- II.D90 +- six-comparison-axes +- def:six-comparison-axes +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 60 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Six Comparison Axes**\n\nSix structural axes along which Category tau and orthodox mathematics differ: infinity,\ + \ sign, topology, geometry, analysis, and category theory.\n\n# Six Comparison Axes\n\n## Summary\n\nSix structural axes\ + \ along which Category tau and orthodox mathematics differ: infinity, sign, topology, geometry, analysis, and category\ + \ theory.\n\n## Statement\n\n```latex\n%\n\\label{def:six-comparison-axes}\nFor any construction $\\mathcal{C}_\\tau$\n\ + with comparison mode $M \\in \\{A, B, C, D, E\\}$,\nthe \\textbf{six comparison axes} are:\n\n\\medskip\n\\renewcommand{\\\ + arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}r@{\\;\\;}l@{\\;\\;}p{7.5cm}@{}}\n\\toprule\n\\textbf{\\#} &\ + \ \\textbf{Axis} & \\textbf{Question} \\\\\n\\midrule\n1 & Ontic Status &\n Is $\\mathcal{C}_\\tau$ the same object,\n\ + \ a parallel object, absent,\n or a new object?\n (Determined by the mode.) \\\\\n2 & Axiom Dependency &\n Which K-axioms\ + \ does $\\mathcal{C}_\\tau$ depend on?\n Which ZFC/PA axioms does\n $\\mathcal{C}_{\\mathrm{orth}}$ depend on? \\\\\n\ + 3 & Constructivity &\n Is $\\mathcal{C}_\\tau$ decidable?\n Effective?\n Does $\\mathcal{C}_{\\mathrm{orth}}$ require\n\ + \ non-constructive principles (AC, LEM)? \\\\\n4 & Uniqueness &\n How many models?\n $\\tau$: exactly one (I.T08).\n\ + \ ZFC: models of every infinite cardinality.\n PA (first-order): non-standard models. \\\\\n5 & Scope Gain &\n What\ + \ does $\\tau$ gain\n that orthodox mathematics lacks? \\\\\n6 & Scope Cost &\n What does $\\tau$ lose\n that orthodox\ + \ mathematics has? \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\n\\medskip\n\\noindent\nAxes~1--4 are factual:\ + \ they have definite answers\nfor each construction.\nAxes~5--6 are evaluative:\nthey compile the gains and costs\nthat\ + \ the factual axes reveal.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 222\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch60-question-of-foundations.tex`\ + \ lines 349-399\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.T08, I.T10, II.T42\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Six structural axes along which Category tau and orthodox mathematics differ: infinity, sign, topology, geometry, + analysis, and category theory.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001325.md b/_corpus_v3_items/cid001325.md new file mode 100644 index 000000000..9329497fc --- /dev/null +++ b/_corpus_v3_items/cid001325.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001325/ +title: Mode C Catalog +cid: cid001325 +primary_alias: DEF0199 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Catalog of 16 Mode C (Refused) constructions: orthodox structures that Category tau structurally cannot support, + including algebraic closure, conformal geometry, Hilbert spaces, and continuous spectra.' +aliases: +- DEF0199 +- II.D91 +- mode-c-catalog +- def:mode-c-catalog +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 63 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Mode C Catalog**\n\nCatalog of 16 Mode C (Refused) constructions: orthodox structures that Category tau structurally\ + \ cannot support, including algebraic closure, conformal geometry, Hilbert spaces, and continuous spectra.\n\n# Mode C\ + \ Catalog\n\n## Summary\n\nCatalog of 16 Mode C (Refused) constructions: orthodox structures that Category tau structurally\ + \ cannot support, including algebraic closure, conformal geometry, Hilbert spaces, and continuous spectra.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:mode-c-catalog}\n% I.T36, I.D76, I.D86, II.T40, II.T41, II.D13, II.D14, II.D68\nThe following\ + \ constructions are \\textbf{Mode~C} (Refused):\n$\\tau$'s axioms structurally block them.\n\n\\medskip\n\\renewcommand{\\\ + arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}r@{\\;\\;}p{3.2cm}@{\\;\\;}p{3.2cm}@{\\;\\;}p{3.2cm}@{}}\n\\\ + toprule\n\\textbf{\\#} & \\textbf{Refused construction}\n & \\textbf{Blocking axiom/theorem}\n & \\textbf{What it protects}\ + \ \\\\\n\\midrule\n1 & Uncountable sets\n & K6 (Object Closure)\n & Categoricity (I.T08) \\\\\n2 & $\\varepsilon$-$\\\ + delta$ limits\n & Unique $\\omega$ (K2)\n & Single infinity \\\\\n3 & Cantor diagonal argument\n & K5 (Diagonal Discipline)\n\ + \ & No cardinality hierarchy \\\\\n4 & Unrestricted comprehension\n & K5, K6\n & Paradox avoidance \\\\\n5 & Conformal\ + \ maps\n & $\\jj^2 = +1$ (I.T10)\n & Bipolar decomposition \\\\\n6 & Non-constructive existence\n & Constructive witnesses\n\ + \ & Decidability \\\\\n7 & Connected open sets\n & Stone topology (II.D14)\n & Clopen partition \\\\\n8 & Smooth structure\n\ + \ & Ultrametric (II.D13)\n & Stage-finite geometry \\\\\n9 & Arbitrary choice (AC)\n & Constructive witnesses\n &\ + \ Effective procedures \\\\\n\\midrule\n\\multicolumn{4}{@{}l@{}}{\\textit{Book~I specific:}} \\\\\n10 & Non-Euclidean\ + \ geometry\n & Parallel Postulate proved (II.T15)\n & Euclidean uniqueness \\\\\n11 & Hyperoperators above tet.\n &\ + \ K5 (saturation at level~3)\n & Diagonal discipline \\\\\n12 & Star-autonomous duality\n & Thin category\n & Rigid\ + \ morphism structure \\\\\n13 & Arbitrary model theory\n & Categoricity (I.T08)\n & Unique model \\\\\n\\midrule\n\\\ + multicolumn{4}{@{}l@{}}{\\textit{Book~II specific:}} \\\\\n14 & Cauchy integral formula\n & Wave PDE (II.D70)\n & BndLift\ + \ replaces \\\\\n15 & Riemann surfaces\n & $\\jj^2 = +1$, no conformal\n & Wave holomorphy \\\\\n16 & SCV dimensional\ + \ ladder\n & No Archimedean engine\n & Simultaneous features \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 223\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch63-refuses.tex`\ + \ lines 57-128\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.T05, I.T08, I.T10, I.T35, I.D76, I.D86, II.T06, II.T07, II.T15, II.T40, II.T42, II.D13,\ + \ II.D14, II.D68\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Catalog of 16 Mode C (Refused) constructions: orthodox structures that Category tau structurally cannot support, + including algebraic closure, conformal geometry, Hilbert spaces, and continuous spectra.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001326.md b/_corpus_v3_items/cid001326.md new file mode 100644 index 000000000..1c536dc73 --- /dev/null +++ b/_corpus_v3_items/cid001326.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001326/ +title: Mode E Catalog +cid: cid001326 +primary_alias: DEF0200 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Catalog of 13 Mode E (Earned) results: constructions that both frameworks prove, but tau derives from fewer axioms. + Includes induction, Euclidean geometry, number tower, and topos structure.' +aliases: +- DEF0200 +- II.D92 +- mode-e-catalog +- def:mode-e-catalog +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 65 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Mode E Catalog**\n\nCatalog of 13 Mode E (Earned) results: constructions that both frameworks prove, but tau\ + \ derives from fewer axioms. Includes induction, Euclidean geometry, number tower, and topos structure.\n\n# Mode E Catalog\n\ + \n## Summary\n\nCatalog of 13 Mode E (Earned) results: constructions that both frameworks prove, but tau derives from\ + \ fewer axioms. Includes induction, Euclidean geometry, number tower, and topos structure.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:mode-e-catalog}\n% I.T31, I.D07, I.D34, I.D84,\n% II.T06, II.T07, II.T15, II.T22, II.T27, II.T40,\n\ + % II.D14, II.D22, II.D64\nThe following constructions in Books~I--II\nare \\textbf{Mode~E} (Earned):\n\n\\medskip\n\\\ + renewcommand{\\arraystretch}{1.35}\n\\begin{center}\n\\begin{tabular}{@{}r@{\\;\\;}p{3.0cm}@{\\;\\;}p{3.0cm}@{\\;\\;}p{3.5cm}@{}}\n\ + \\toprule\n\\textbf{\\#} & \\textbf{Earned result}\n & \\textbf{Orthodox status}\n & \\textbf{$\\tau$-derivation} \\\ + \\\n\\midrule\n\\multicolumn{4}{@{}l@{}}{\\textit{Book~I:}} \\\\\n1 & Natural numbers $\\mathbb{N}$\n & PA axioms (including\ + \ $\\infty$ induction schema)\n & Orbit of $\\alpha$ under $\\rho$ (I.D07) \\\\\n2 & Composition associativity\n & Category\ + \ theory axiom\n & Normal-form confluence \\\\\n3 & Number tower $\\mathbb{N} \\to \\mathbb{Z} \\to \\mathbb{Q} \\to\ + \ R_\\tau \\to H_\\tau$\n & ZFC constructions (Dedekind, Cauchy)\n & Universal constructions from $\\rho$ \\\\\n4 &\ + \ ABCD coordinates\n & External coordinate choice\n & Forced by hyperfactorization (I.T04) \\\\\n5 & Hyperfactorization\n\ + \ & No counterpart\n & $X = (A \\hat{\\phantom{x}} C)^B \\cdot D$ unique \\\\\n6 & Topos structure\n & Lawvere--Tierney\ + \ axioms\n & Earned from HolFun monoid \\\\\n7 & Global Hartogs\n & Cauchy integral proof (analytic)\n & CRT proof,\ + \ constructive (I.T31) \\\\\n\\midrule\n\\multicolumn{4}{@{}l@{}}{\\textit{Book~II:}} \\\\\n8 & $\\tau^3$ fibration\n\ + \ & External fibration choice\n & Forced by ABCD (II.T03) \\\\\n9 & Mutual determination\n & Separate proofs\n & Single\ + \ derivation chain (II.T27) \\\\\n10 & Self-enrichment\n & Grothendieck universes (new axiom)\n & K5-protected derivation\ + \ \\\\\n11 & $\\tau$-manifold structure\n & Smooth manifold axioms\n & Earned from fibered product (Part~X) \\\\\n12\ + \ & Tarski geometry (all 10 axioms)\n & 10 axioms postulated (Tarski 1959)\n & Derived from ultrametric (II.T15--II.T18)\ + \ \\\\\n13 & Induction\n & PA axiom schema ($\\infty$ instances)\n & Earned from iterator ladder \\\\\n\\bottomrule\n\ + \\end{tabular}\n\\end{center}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 224\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch65-earns.tex`\ + \ lines 48-111\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: I.T05, I.T08, I.T10, I.T19, I.T31, I.D07, I.D34, I.D84, II.T06, II.T07, II.T15, II.T22,\ + \ II.T27, II.T40, II.D14, II.D22, II.D64\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Catalog of 13 Mode E (Earned) results: constructions that both frameworks prove, but tau derives from fewer + axioms. Includes induction, Euclidean geometry, number tower, and topos structure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001327.md b/_corpus_v3_items/cid001327.md new file mode 100644 index 000000000..2e301f7c2 --- /dev/null +++ b/_corpus_v3_items/cid001327.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001327/ +title: Combined Gain-Cost Ledger +cid: cid001327 +primary_alias: DEF0201 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Combined gain-cost ledger: 49 gains (23 Mode D + 13 Mode E + 17 neutral) vs 16 Mode C costs. No partial selection + possible due to trade-off atomicity.' +aliases: +- DEF0201 +- II.D93 +- combined-gain-cost-ledger +- def:combined-ledger +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Combined Gain-Cost Ledger**\n\nCombined gain-cost ledger: 49 gains (23 Mode D + 13 Mode E + 17 neutral) vs\ + \ 16 Mode C costs. No partial selection possible due to trade-off atomicity.\n\n# Combined Gain-Cost Ledger\n\n## Summary\n\ + \nCombined gain-cost ledger: 49 gains (23 Mode D + 13 Mode E + 17 neutral) vs 16 Mode C costs. No partial selection possible\ + \ due to trade-off atomicity.\n\n## Statement\n\n```latex\n%\n\\label{def:combined-ledger}\nThe \\textbf{Combined Gain--Cost\ + \ Ledger}\nacross Books~I--II\ntallies all mode-classified constructions:\n\n\\medskip\n\\renewcommand{\\arraystretch}{1.25}\n\ + \\begin{center}\n\\begin{tabular}{@{}clrrl@{}}\n\\toprule\n\\textbf{Mode} & \\textbf{Name}\n & \\textbf{Book~I} & \\\ + textbf{Book~II} & \\textbf{Assessment} \\\\\n\\midrule\nA & Same & 4 & 3 & Identical objects \\\\\nB & Parallel\ + \ & 5 & 5 & Same axioms, different carriers \\\\\nC & Refused & 9 & 7 & Genuine costs \\\\\nD & Gained & 13\ + \ & 10 & Structurally impossible in ZFC \\\\\nE & Earned & 7 & 6 & Derivation replaces postulation \\\\\n\\midrule\n\ + \\multicolumn{2}{@{}l}{\\textbf{Total gains (D+E)}}\n & 20 & 16 & \\textbf{36} \\\\\n\\multicolumn{2}{@{}l}{\\textbf{Total\ + \ costs (C)}}\n & 9 & 7 & \\textbf{16} \\\\\n\\multicolumn{2}{@{}l}{\\textbf{Neutral (A+B)}}\n & 9 & 8 & \\textbf{17}\ + \ \\\\\n\\midrule\n\\multicolumn{2}{@{}l}{\\textbf{Grand total}}\n & 38 & 31 & \\textbf{69} \\\\\n\\bottomrule\n\\end{tabular}\n\ + \\end{center}\n\n\\medskip\n\\noindent\nIncluding Mode~A and~B constructions\nthat carry additional $\\tau$-specific structure,\n\ + the gain count rises to approximately 49.\nThe cost count remains 16.\nThe ratio is approximately $3{:}1$ gains to costs.\n\ + \nThis ratio is \\emph{not} evidence\nthat $\\tau$ is ``better.''\nIt reflects the structural fact\nthat categoricity,\ + \ while expensive\n(16 refusals),\nproduces many downstream consequences\n(each gain is a theorem that follows\nfrom categoricity\ + \ + the forced sign).\nThe costs are concentrated at the foundation;\nthe gains cascade through the structure.\n```\n\n\ + ## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 225\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex`\ + \ lines 223-275\n\n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n##\ + \ Dependencies\n\n- Canonical: II.D91, II.D82, II.D92\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Combined gain-cost ledger: 49 gains (23 Mode D + 13 Mode E + 17 neutral) vs 16 Mode C costs. No partial selection + possible due to trade-off atomicity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001328.md b/_corpus_v3_items/cid001328.md new file mode 100644 index 000000000..59e3b341f --- /dev/null +++ b/_corpus_v3_items/cid001328.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001328/ +title: Naturality Forces Cylinder Compatibility +cid: cid001328 +primary_alias: LEM0010 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Naturality Forces Cylinder Compatibility +aliases: +- LEM0010 +- II.L01 +- naturality-forces-cylinder-compatibility +- lem:naturality-cylinder +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 11 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Naturality Forces Cylinder Compatibility**\n\nNaturality Forces Cylinder Compatibility\n\n# Naturality Forces\ + \ Cylinder Compatibility\n\n## Summary\n\nNaturality Forces Cylinder Compatibility\n\n## Statement\n\n```latex\n%\n\\\ + label{lem:naturality-cylinder}\nLet $f$ be an $\\omega$-germ transformer.\nThen for every stage~$k$ and every point $x\ + \ \\in \\tau^3$:\n\\[\n \\boxed{f\\bigl(C_k(x)\\bigr)\n \\;\\subseteq\\;\n C_k\\bigl(f(x)\\bigr),}\n\\]\nwhere\ + \ $C_k(x)$ denotes the stage-$k$ cylinder at~$x$\n(Definition~\\ref{def:stage-k-cylinder}, Chapter~\\ref{ch:cylinder-domains}).\n\ + That is, $f$ maps the stage-$k$ cylinder at~$x$\ninto the stage-$k$ cylinder at~$f(x)$.\n```\n\n## Proof / Justification\n\ + \n```latex\nLet $y \\in C_k(x)$.\nBy the definition of stage-$k$ cylinder (II.D10),\nthis means $\\pi_k(y) = \\pi_k(x)$:\n\ + the points $x$ and~$y$ agree at stage~$k$\nof the primorial tower.\n\nBy tower coherence of~$f$\n(the naturality condition\ + \ of Notation~\\ref{not:ch11-omega-germ}):\n\\[\n \\pi_k\\bigl(f(y)\\bigr)\n \\;=\\;\n f_k\\bigl(\\pi_k(y)\\\ + bigr)\n \\;=\\;\n f_k\\bigl(\\pi_k(x)\\bigr)\n \\;=\\;\n \\pi_k\\bigl(f(x)\\bigr).\n\\]\nHence $f(y) \\in\ + \ C_k(f(x))$.\nSince $y \\in C_k(x)$ was arbitrary,\n$f(C_k(x)) \\subseteq C_k(f(x))$.\n```\n\n## Source Context\n\n-\ + \ Registry source: `book-02.jsonl` line 28\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch11-hol-implies-cont.tex`\ + \ lines 270-284\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.HolImpliesCont`\n\ + - Name: `cyl_compat_check`\n\n## Dependencies\n\n- Canonical: I.D47, II.D10, I.T18\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Naturality Forces Cylinder Compatibility +noindex: false +--- + diff --git a/_corpus_v3_items/cid001329.md b/_corpus_v3_items/cid001329.md new file mode 100644 index 000000000..8d0a96386 --- /dev/null +++ b/_corpus_v3_items/cid001329.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001329/ +title: Refinement-Spectral Equivalence +cid: cid001329 +primary_alias: LEM0011 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Refinement sequence data (stage-by-stage tower values) is equivalent to spectral decomposition data (bipolar channel + coefficients): each determines the other via the canonical basis expansion.' +aliases: +- LEM0011 +- II.L02 +- refinement-spectral-equivalence +- lem:refinement-spectral +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 31 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Refinement-Spectral Equivalence**\n\nRefinement sequence data (stage-by-stage tower values) is equivalent to\ + \ spectral decomposition data (bipolar channel coefficients): each determines the other via the canonical basis expansion.\n\ + \n# Refinement-Spectral Equivalence\n\n## Summary\n\nRefinement sequence data (stage-by-stage tower values) is equivalent\ + \ to spectral decomposition data (bipolar channel coefficients): each determines the other via the canonical basis expansion.\n\ + \n## Statement\n\n```latex\n%\n\\label{lem:refinement-spectral}\nLet $(f_k)_{k \\geq n}$ be a tower-coherent sequence\n\ + in~$H_\\tau$ stabilized after stage~$n$.\nThen there exists a unique finite spectral decomposition\n\\[\n f_k\n \ + \ \\;=\\;\n \\sum_{\\chi \\in S_n} c_\\chi \\cdot \\chi_k,\n \\qquad\n k \\geq n,\n\\]\nwhere $S_n \\subset \\\ + widehat{R}_\\tau$ is a finite\nsupport set\nand $\\chi_k$ is the restriction\nof the character~$\\chi$ to stage~$k$.\n\ + Conversely, every spectral tail\nwith finite support\ndefines a tower-coherent refinement tail.\n```\n\n## Proof / Justification\n\ + \n```latex\n\\emph{Forward direction ($\\mathrm{R} \\Rightarrow \\mathrm{S}$).}\nAt each stage~$k$, the map\n$f_k \\colon\ + \ \\mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau$\nis a function on a finite cyclic group.\nBy the Chinese Remainder Theorem\ + \ (I.T18, Book~I),\n$\\mathbb{Z}/P_k\\mathbb{Z} \\cong\n\\prod_{j=1}^k \\mathbb{Z}/p_j\\mathbb{Z}$,\nand the characters\ + \ of $\\mathbb{Z}/P_k\\mathbb{Z}$\nare products of characters\nof the individual factors.\nThe discrete Fourier transform\n\ + on $\\mathbb{Z}/P_k\\mathbb{Z}$\ndecomposes $f_k$ into a finite sum\nof characters with coefficients in $H_\\tau$.\n\n\ + The bipolar decomposition applies:\neach coefficient $c_\\chi \\in H_\\tau$\nsplits as $c_\\chi = e_+ c_\\chi^+ + e_-\ + \ c_\\chi^-$.\nTower coherence\n($f_k \\equiv f_{k+1} \\pmod{P_k}$)\nforces the spectral coefficients\nat stage~$k+1$\ + \ to reduce\nto those at stage~$k$\nmodulo the new prime~$p_{k+1}$.\nSince the tail is stabilized after stage~$n$,\nno\ + \ new spectral components appear\nfor $k > n$:\nthe support $S_n$ is fixed,\nand the coefficients refine consistently.\n\ + \n\\emph{Reverse direction ($\\mathrm{S} \\Rightarrow \\mathrm{R}$).}\nGiven a spectral tail with finite support $S_n$\n\ + and coefficients $(c_\\chi)_{\\chi \\in S_n}$ in~$H_\\tau$,\ndefine $f_k := \\sum_{\\chi \\in S_n} c_\\chi \\cdot \\chi_k$.\n\ + Tower coherence follows from the coherence\nof the characters:\n$\\chi_{k+1} \\equiv \\chi_k \\pmod{P_k}$\nby construction\ + \ of the dual group.\nStabilization after stage~$n$ is immediate\nsince $S_n$ is fixed.\n\n\\emph{Uniqueness.}\nThe DFT\ + \ on a finite abelian group\nis invertible,\nso the spectral decomposition\nat each stage is unique.\nCoherence forces\ + \ the decompositions at different stages\nto be compatible,\nand stabilization ensures\nthe system has a unique limit.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 85\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch31-mutual-determination.tex`\ + \ lines 213-233\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.MutualDetermination`\n\ + - Name: `Tau.BookII.Hartogs.refine_spectral_check`\n\n## Dependencies\n\n- Canonical: I.T18, I.D21, II.D33, II.D35\n\n\ + ## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Refinement sequence data (stage-by-stage tower values) is equivalent to spectral decomposition data (bipolar + channel coefficients): each determines the other via the canonical basis expansion.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001330.md b/_corpus_v3_items/cid001330.md new file mode 100644 index 000000000..2bb35708b --- /dev/null +++ b/_corpus_v3_items/cid001330.md @@ -0,0 +1,218 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001330/ +title: Spectral-Germ Equivalence +cid: cid001330 +primary_alias: LEM0012 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Spectral decomposition data is equivalent to germ data: knowing the spectral coefficients of a function at a point + determines its omega-germ, and conversely.' +aliases: +- LEM0012 +- II.L03 +- spectral-germ-equivalence +- lem:spectral-germ +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 31 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Spectral-Germ Equivalence** + + + Spectral decomposition data is equivalent to germ data: knowing the spectral coefficients of a function at a point determines + its omega-germ, and conversely. + + + # Spectral-Germ Equivalence + + + ## Summary + + + Spectral decomposition data is equivalent to germ data: knowing the spectral coefficients of a function at a point determines + its omega-germ, and conversely. + + + ## Statement + + + ```latex + + % + + \label{lem:spectral-germ} + + A stabilized spectral decomposition + + with finite support $S_n \subset \widehat{R}_\tau$ + + determines a unique $\omega$-germ. + + Conversely, every $\omega$-germ + + arises from a unique stabilized spectral tail. + + ``` + + + ## Proof / Justification + + + ```latex + + \emph{Forward ($\mathrm{S} \Rightarrow \mathrm{G}$).} + + A spectral tail defines a tower-coherent sequence + + $(f_k)_{k \geq n}$ by Lemma~\ref{lem:refinement-spectral}. + + Two tower-coherent sequences that differ + + only at finitely many stages + + define the same $\omega$-germ + + (they agree eventually). + + Since the spectral tail is stabilized, + + the induced sequence is eventually constant + + in its spectral structure: + + the coefficients $(c_\chi)$ do not change + + for $k \geq n$. + + This determines a unique equivalence class + + at the profinite limit---an $\omega$-germ. + + + The bipolar decomposition ensures uniqueness: + + the $e_+$-component and $e_-$-component + + of the spectral data + + each produce a one-dimensional datum + + at the limit, + + and the $\omega$-germ is the pair + + of these two limiting data. + + + \emph{Reverse ($\mathrm{G} \Rightarrow \mathrm{S}$).} + + An $\omega$-germ is an equivalence class + + of tower-coherent sequences + + that agree on all sufficiently deep stages. + + Pick any representative $(f_k)_{k \geq m}$. + + By the compactness of the profinite space + + $\tau^3$ (II.T07, Chapter~\ref{ch:stone-space}), + + the spectral support of $(f_k)$ + + stabilizes at some stage~$n \geq m$: + + for $k \geq n$, no new characters enter the support. + + The resulting spectral tail + + is independent of the representative chosen, + + since any two representatives agree + + for $k$ sufficiently large. + + ``` + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 86 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch31-mutual-determination.tex` lines + 294-302 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Hartogs.MutualDetermination` + + - Name: `Tau.BookII.Hartogs.spectral_germ_check` + + + ## Dependencies + + + - Canonical: I.D21, II.D14, II.D33, II.L02 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Spectral decomposition data is equivalent to germ data: knowing the spectral coefficients of a function at a + point determines its omega-germ, and conversely.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001331.md b/_corpus_v3_items/cid001331.md new file mode 100644 index 000000000..d8a6dc00a --- /dev/null +++ b/_corpus_v3_items/cid001331.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001331/ +title: Germ-Character Equivalence +cid: cid001331 +primary_alias: LEM0013 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Omega-germ data at a point is equivalent to boundary character data on L: the germ determines a unique character + via bipolar projection, and the character recovers the germ via idempotent assembly.' +aliases: +- LEM0013 +- II.L04 +- germ-character-equivalence +- lem:germ-character +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 31 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Germ-Character Equivalence**\n\nOmega-germ data at a point is equivalent to boundary character data on L: the\ + \ germ determines a unique character via bipolar projection, and the character recovers the germ via idempotent assembly.\n\ + \n# Germ-Character Equivalence\n\n## Summary\n\nOmega-germ data at a point is equivalent to boundary character data on\ + \ L: the germ determines a unique character via bipolar projection, and the character recovers the germ via idempotent\ + \ assembly.\n\n## Statement\n\n```latex\n%\n\\label{lem:germ-character}\nAn $\\omega$-germ determines a unique\nboundary\ + \ character\n$\\varphi \\colon R_\\tau \\to H_\\tau$.\nConversely, every boundary character\ndetermines a unique $\\omega$-germ.\n\ + ```\n\n## Proof / Justification\n\n```latex\n\\emph{Forward ($\\mathrm{G} \\Rightarrow \\mathrm{C}$).}\nAn $\\omega$-germ\ + \ has a unique stabilized spectral decomposition\n$\\sum_{\\chi \\in S_n} c_\\chi \\cdot \\chi$\nby Lemma~\\ref{lem:spectral-germ}.\n\ + Define the boundary character\n$\\varphi \\colon R_\\tau \\to H_\\tau$ by\n\\[\n \\varphi(r)\n \\;:=\\;\n \\\ + sum_{\\chi \\in S_n} c_\\chi \\cdot \\chi(r),\n \\qquad r \\in R_\\tau.\n\\]\nThis is a ring homomorphism because\n\ + each $\\chi$ is a character\n(multiplicative homomorphism)\nand $H_\\tau$ is a ring under the split-complex\nmultiplication.\n\ + The bipolar decomposition splits $\\varphi$\ninto $\\varphi^+ = e_+ \\circ \\varphi$\nand $\\varphi^- = e_- \\circ \\\ + varphi$,\neach of which is a character of $R_\\tau$\nvalued in a single channel.\n\n\\emph{Reverse ($\\mathrm{C} \\Rightarrow\ + \ \\mathrm{G}$).}\nGiven a boundary character\n$\\varphi \\colon R_\\tau \\to H_\\tau$,\ndecompose via bipolar idempotents:\n\ + $\\varphi = e_+ \\varphi^+ + e_- \\varphi^-$.\nEach $\\varphi^\\pm$ is a character of $R_\\tau$\nvalued in one channel.\n\ + Since $R_\\tau$ is the profinite completion\nof $\\varinjlim \\mathbb{Z}/P_k\\mathbb{Z}$\n(I.D19, Book~I),\neach character\ + \ is determined by its values\non the finite quotients $\\mathbb{Z}/P_k\\mathbb{Z}$.\nThese values form a tower-coherent\ + \ system,\nwhich by the CRT (I.T18, Book~I)\nhas a unique spectral decomposition\nwith eventually stable support.\nLemma~\\\ + ref{lem:spectral-germ}\nconverts this spectral tail to an $\\omega$-germ.\n\n\\emph{Uniqueness.}\nIn each direction, the\ + \ bipolar decomposition\nreduces the problem to one-dimensional data\n(one real coefficient per channel per character).\n\ + The constructions are inverse to each other\nby the uniqueness of the DFT\nand the uniqueness of profinite limits.\n```\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 87\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch31-mutual-determination.tex`\ + \ lines 350-358\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.MutualDetermination`\n\ + - Name: `Tau.BookII.Hartogs.germ_character_check`\n\n## Dependencies\n\n- Canonical: I.D19, I.D21, I.T10, II.D33, II.D35,\ + \ II.L03\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Omega-germ data at a point is equivalent to boundary character data on L: the germ determines a unique character + via bipolar projection, and the character recovers the germ via idempotent assembly.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001332.md b/_corpus_v3_items/cid001332.md new file mode 100644 index 000000000..75420c9f0 --- /dev/null +++ b/_corpus_v3_items/cid001332.md @@ -0,0 +1,268 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001332/ +title: Character-Hartogs Equivalence +cid: cid001332 +primary_alias: LEM0014 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Boundary character data is equivalent to Hartogs extension data: a character on L extends uniquely to a holomorphic + function on tau^3 via the boundary lift, and every Hartogs extension restricts to a unique character.' +aliases: +- LEM0014 +- II.L05 +- character-hartogs-equivalence +- lem:character-hartogs +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 31 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Character-Hartogs Equivalence** + + + Boundary character data is equivalent to Hartogs extension data: a character on L extends uniquely to a holomorphic function + on tau^3 via the boundary lift, and every Hartogs extension restricts to a unique character. + + + # Character-Hartogs Equivalence + + + ## Summary + + + Boundary character data is equivalent to Hartogs extension data: a character on L extends uniquely to a holomorphic function + on tau^3 via the boundary lift, and every Hartogs extension restricts to a unique character. + + + ## Statement + + + ```latex + + % + + \label{lem:character-hartogs} + + A boundary character + + $\varphi \colon R_\tau \to H_\tau$ + + determines a unique Hartogs continuation + + from the boundary to the interior. + + Conversely, every Hartogs continuation + + restricts to a unique boundary character. + + ``` + + + ## Proof / Justification + + + ```latex + + \emph{Forward ($\mathrm{C} \Rightarrow \mathrm{H}$).} + + Given a boundary character~$\varphi$, + + construct the Hartogs continuation as follows. + + The character $\varphi$ provides boundary datum + + at the initial stage: + + $\varphi$ determines $f_n$ + + on $\mathbb{Z}/P_n\mathbb{Z}$ + + via the spectral decomposition + + (Lemma~\ref{lem:germ-character}). + + Apply $\mathrm{BndLift}_n$ + + (II.D36, Chapter~\ref{ch:bndlift-construction}): + + the boundary datum at stage~$n$ + + lifts to interior datum at stage~$n+1$. + + The lift uses the CRT decomposition + + $\mathbb{Z}/P_{n+1}\mathbb{Z} + + \cong \mathbb{Z}/P_n\mathbb{Z} + + \times \mathbb{Z}/p_{n+1}\mathbb{Z}$ + + and the bipolar channel independence + + (II.P07, Chapter~\ref{ch:bndlift-construction}). + + + Iteration: apply $\mathrm{BndLift}_{n+1}$ + + to produce stage~$n+2$ data, + + then $\mathrm{BndLift}_{n+2}$ for stage~$n+3$, + + and so on. + + The resulting sequence $(f_k)_{k \geq n}$ + + is tower-coherent by construction + + (each $\mathrm{BndLift}_k$ + + preserves coherence with the previous stage). + + The coupling strength at each step + + is governed by $\iota_\tau = 2/(\pi + e)$ + + (II.T25, Chapter~\ref{ch:iota-tau-confirmed}). + + + Uniqueness follows from the uniqueness + + of $\mathrm{BndLift}_n$ + + at each stage: + + the CRT decomposition is canonical, + + and the lift in each bipolar channel + + is one-dimensional, + + leaving no freedom. + + + \emph{Reverse ($\mathrm{H} \Rightarrow \mathrm{C}$).} + + Given a Hartogs continuation $(f_k)_{k \geq n}$, + + restrict to the boundary ring: + + define $\varphi(r) := \lim_{k \to \infty} f_k(r \bmod P_k)$ + + for each $r \in R_\tau$. + + The limit exists because $(f_k)$ + + is tower-coherent + + and $\tau^3$ is compact (II.T07). + + The resulting $\varphi$ is a ring homomorphism + + because each $f_k$ respects the CRT ring structure. + + + \emph{Inverse property.} + + Starting from $\varphi$, + + constructing the Hartogs continuation, + + and restricting back to the boundary + + recovers~$\varphi$: + + the lift is the unique extension, + + and restriction is the inverse operation. + + Starting from a Hartogs continuation, + + restricting to the boundary, + + and lifting back recovers the original continuation + + by the uniqueness of $\mathrm{BndLift}_n$. + + ``` + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 88 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch31-mutual-determination.tex` lines + 418-427 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Hartogs.MutualDetermination` + + - Name: `Tau.BookII.Hartogs.character_hartogs_check` + + + ## Dependencies + + + - Canonical: I.T18, I.D21, II.D35, II.D36, II.T25, II.L04 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Boundary character data is equivalent to Hartogs extension data: a character on L extends uniquely to a holomorphic + function on tau^3 via the boundary lift, and every Hartogs extension restricts to a unique character.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001333.md b/_corpus_v3_items/cid001333.md new file mode 100644 index 000000000..b3ce36101 --- /dev/null +++ b/_corpus_v3_items/cid001333.md @@ -0,0 +1,101 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001333/ +title: Gluing Lemma +cid: cid001333 +primary_alias: LEM0015 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Compatible tau-holomorphic sections on a cover by cylinder sub-domains glue uniquely to a global section, guaranteed + by the ultrametric topology. +aliases: +- LEM0015 +- II.L06 +- gluing-lemma +- lem:gluing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005574 +- predicate: formalized_by + object: cid005575 +- predicate: formalized_by + object: cid005576 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Gluing Lemma**\n\nCompatible tau-holomorphic sections on a cover by cylinder sub-domains glue uniquely to a\ + \ global section, guaranteed by the ultrametric topology.\n\n# Gluing Lemma\n\n## Summary\n\nCompatible tau-holomorphic\ + \ sections on a cover by cylinder sub-domains glue uniquely to a global section, guaranteed by the ultrametric topology.\n\ + \n## Statement\n\n```latex\n%\n\\label{lem:gluing}\nLet $U \\subseteq \\tau^3$ be a cylinder domain,\nand let $\\{U_i\\\ + }_{i \\in I}$\nbe a cover of~$U$ by cylinder sub-domains.\nSuppose that for each $i \\in I$\nwe are given $f_i \\in \\\ + mathcal{O}_\\tau(U_i)$,\nand that these local sections are \\textbf{compatible}:\n\\[\n f_i\\big|_{U_i \\cap U_j}\n\ + \ \\;=\\;\n f_j\\big|_{U_i \\cap U_j}\n \\qquad\n \\text{for all } i, j \\in I.\n\\]\nThen there exists a\ + \ unique $f \\in \\mathcal{O}_\\tau(U)$\nwith $f\\big|_{U_i} = f_i$ for all $i \\in I$.\n```\n\n## Proof / Justification\n\ + \n```latex\nThe proof proceeds in two steps:\nconstruction of~$f$ as a set-theoretic function,\nthen verification that~$f$\ + \ is $\\tau$-holomorphic.\n\n\\smallskip\n\\noindent\\textbf{Step 1: Construction.}\nDefine $f : U \\to H_\\tau$ by setting,\n\ + for each $x \\in U$:\n\\[\n f(x) := f_i(x)\n \\qquad\\text{for any } i \\in I \\text{ with } x \\in U_i.\n\\]\n\ + This is well-defined:\nif $x \\in U_i \\cap U_j$,\nthen $f_i(x) = f_j(x)$\nby the compatibility condition.\n\n\\smallskip\n\ + \\noindent\\textbf{Step 2: $\\tau$-holomorphicity.}\nWe verify that $f$ is $\\tau$-holomorphic on~$U$,\ni.e., that the\ + \ stagewise components\n$(f_k)_{k \\geq 1}$ satisfy tower coherence\nand sector independence.\n\n\\emph{Tower coherence.}\n\ + The cylinder cover $\\{U_i\\}$\ncan be refined to a cover by stage-$N$ cylinders\nfor some sufficiently large~$N$\n(because\ + \ every cylinder domain\nis a union of deeper-stage cylinders).\nAt each stage $k \\leq N$,\nthe component $f_k$\nis determined\ + \ by the $f_i$\non the stage-$k$ projection of each~$U_i$.\nThe individual $f_i$ are $\\tau$-holomorphic,\nso each satisfies\ + \ tower coherence.\nOn overlaps, the coherence conditions agree\n(because the $f_i$ agree on overlaps),\nand the coherence\ + \ data pastes consistently.\n\nThe key observation is that cylinder overlaps\nin the ultrametric topology\nare themselves\ + \ cylinders:\n\\[\n C_{k,a} \\cap C_{k,b}\n \\;=\\;\n \\begin{cases}\n C_{k,a} & \\text{if } a = b, \\\ + \\\n \\varnothing & \\text{if } a \\neq b.\n \\end{cases}\n\\]\nTwo cylinders at the same stage\nare either\ + \ identical or disjoint---there\nare no ``partial overlaps.''\nOverlaps between cylinders at different stages\nreduce\ + \ to containment:\n$C_{k,a} \\cap C_{\\ell, b}$\nis either empty or equals the deeper cylinder.\nThis clean overlap structure\n\ + is what makes gluing straightforward:\nthe compatibility condition\nreduces to agreement on nested sub-cylinders,\nwhich\ + \ is exactly tower coherence.\n\n\\emph{Sector independence.}\nEach $f_i$ satisfies the bipolar sector independence\n\ + of the Mutual Determination Theorem (II.T27):\nthe B-channel and C-channel components\nof~$f_i$ are determined independently.\n\ + Since $f$ agrees with~$f_i$ on each~$U_i$,\nthe sector independence of the local pieces\ncarries over to~$f$ on all of~$U$.\n\ + \n\\emph{Uniqueness.}\nUniqueness follows from locality\n(Proposition~\\ref{prop:ch36-locality}):\nif $f$ and $g$ both\ + \ satisfy\n$f\\big|_{U_i} = g\\big|_{U_i} = f_i$,\nthen $(f - g)\\big|_{U_i} = 0$ for all~$i$,\nso $f - g = 0$ on~$U$.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 106\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch36-sheaf-coherence.tex`\ + \ lines 222-240\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.SheafCoherence`\n\ + - Name: `gluing_lemma_check`\n\n## Dependencies\n\n- Canonical: II.D47, II.D10, I.T18\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Compatible tau-holomorphic sections on a cover by cylinder sub-domains glue uniquely to a global section, guaranteed + by the ultrametric topology. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001334.md b/_corpus_v3_items/cid001334.md new file mode 100644 index 000000000..e777b2f7c --- /dev/null +++ b/_corpus_v3_items/cid001334.md @@ -0,0 +1,116 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001334/ +title: Idempotent Decomposition Lemma +cid: cid001334 +primary_alias: LEM0016 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every holomorphic function f on tau^3 decomposes uniquely as f = e_+ f_+ + e_- f_- where f_+ and f_- are the B-channel + and C-channel components. The decomposition is functorial, stage-coherent, and respects the canonical basis. +aliases: +- LEM0016 +- II.L07 +- idempotent-decomposition-lemma +- lem:idempotent-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005656 +- predicate: formalized_by + object: cid005657 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 37 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Idempotent Decomposition Lemma**\n\nEvery holomorphic function f on tau^3 decomposes uniquely as f = e_+ f_+\ + \ + e_- f_- where f_+ and f_- are the B-channel and C-channel components. The decomposition is functorial, stage-coherent,\ + \ and respects the canonical basis.\n\n# Idempotent Decomposition Lemma\n\n## Summary\n\nEvery holomorphic function f\ + \ on tau^3 decomposes uniquely as f = e_+ f_+ + e_- f_- where f_+ and f_- are the B-channel and C-channel components.\ + \ The decomposition is functorial, stage-coherent, and respects the canonical basis.\n\n## Statement\n\n```latex\n%\n\\\ + label{lem:idempotent-decomposition}\n% II.D35, II.T27, II.D45, II.T31\nLet $f \\in \\mathcal{O}_\\tau(\\tau^3)$ be $\\\ + tau$-holomorphic.\nThen:\n\\begin{enumerate}\n \\item[\\textup{(D1)}]\n \\textbf{Channel holomorphy:}\n \ + \ $f_+ = e_+ \\cdot f$\n and $f_- = e_- \\cdot f$\n are each $\\tau$-holomorphic.\n \\item[\\\ + textup{(D2)}]\n \\textbf{Canonical splitting:}\n $f = f_+ + f_-$\n is the unique decomposition\ + \ of~$f$\n into a sum of\n a $B$-channel holomorphic function\n and a $C$-channel holomorphic\ + \ function.\n \\item[\\textup{(D3)}]\n \\textbf{Completeness:}\n the canonical decomposition map\n\ + \ $\\Delta_\\tau \\colon\n \\mathcal{O}_\\tau(\\tau^3) \\to\n \\mathcal{O}_\\tau^{(+)}(\\tau^3)\n\ + \ \\times \\mathcal{O}_\\tau^{(-)}(\\tau^3)$\n is an isomorphism of $H_\\tau$-modules.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n```latex\nWe prove each assertion in turn.\n\n\\smallskip\n\\noindent\\textbf{Proof\ + \ of (D1): Channel holomorphy.}\nWe must show that $f_+ = e_+ \\cdot f$\nis $\\tau$-holomorphic.\nBy the Mutual Determination\ + \ Theorem\n(Theorem~\\ref{thm:mutual-determination}, II.T27),\n$\\tau$-holomorphy of~$f$ is equivalent\nto any of the\ + \ five descriptions\n(R), (S), (G), (C), (H).\nWe use the spectral description~(S),\nwhich is the most transparent\nfor\ + \ this argument.\n\nBy the finite spectral support theorem\n(Theorem~\\ref{thm:finite-spectral-support}, II.T31,\nChapter~\\\ + ref{ch:canonical-basis}),\n$f$ has a canonical expansion\nin the basis $\\mathcal{B}_\\tau$\n(Definition~\\ref{def:canonical-basis},\ + \ II.D45):\n\\[\n f\n \\;=\\;\n \\sum_{(k,v,\\sigma) \\in S_f}\n c_{k,v}^{(\\sigma)} \\cdot E_{k,v}^{(\\sigma)},\n\ + \\]\nwhere $S_f$ is a finite support set,\n$c_{k,v}^{(\\sigma)} \\in H_\\tau$\nare the spectral coefficients,\nand $E_{k,v}^{(\\\ + sigma)}$ are the cylinder generators\n(Definition~\\ref{def:cylinder-generator}, II.D46).\nThe index $\\sigma \\in \\\ + {B, C\\}$\nlabels the bipolar channel.\n\nThe cylinder generators satisfy the channel rule:\n\\[\n e_+ \\cdot E_{k,v}^{(B)}\ + \ = E_{k,v}^{(B)},\n \\quad\n e_+ \\cdot E_{k,v}^{(C)} = 0,\n \\qquad\n e_- \\cdot E_{k,v}^{(B)} = 0,\n \ + \ \\quad\n e_- \\cdot E_{k,v}^{(C)} = E_{k,v}^{(C)}.\n\\]\nThis follows from (I4):\n$E_{k,v}^{(B)}$ lies in the $e_+$-eigenspace\n\ + and $E_{k,v}^{(C)}$ lies in the $e_-$-eigenspace\nof the $\\jj$-action.\nTherefore:\n\\[\n f_+\n \\;=\\;\n e_+\ + \ \\cdot f\n \\;=\\;\n \\sum_{(k,v,\\sigma) \\in S_f}\n c_{k,v}^{(\\sigma)} \\cdot (e_+ \\cdot E_{k,v}^{(\\sigma)})\n\ + \ \\;=\\;\n \\sum_{(k,v,B) \\in S_f}\n c_{k,v}^{(B)} \\cdot E_{k,v}^{(B)}.\n\\]\nThe right side is a finite linear\ + \ combination\nof cylinder generators---all\nfrom the $B$-channel---with\nthe \\emph{same} spectral coefficients.\nSince\ + \ the support $S_f^{(+)} := \\{(k,v,B) : (k,v,B) \\in S_f\\}$\nis a subset of~$S_f$,\nit is finite.\nThe resulting expansion\n\ + is a valid spectral tail\nwith finite support,\nhence $f_+$ is $\\tau$-holomorphic\nby the converse direction\nof the\ + \ finite spectral support theorem.\n\nThe argument for $f_-$ is identical,\nwith $B$ replaced by $C$ throughout.\n\n\\\ + smallskip\n\\noindent\\textbf{Proof of (D2): Canonical splitting.}\nThe identity $f = f_+ + f_-$\nfollows from completeness\ + \ (I3):\n$f(x) = e_+ f(x) + e_- f(x)\n= f_+(x) + f_-(x)$\nfor all $x \\in \\tau^3$.\nBy~(D1), both $f_+$ and $f_-$ are\ + \ holomorphic.\n\nUniqueness: suppose $f = g + h$\nwhere $g \\in \\mathcal{O}_\\tau^{(+)}(\\tau^3)$\nand $h \\in \\mathcal{O}_\\\ + tau^{(-)}(\\tau^3)$.\nThen $g$ takes values in $e_+ \\cdot H_\\tau$,\nso $e_- \\cdot g = 0$;\nsimilarly $e_+ \\cdot h\ + \ = 0$.\nApply $e_+$ to both sides:\n$e_+ f = e_+ g + e_+ h = g + 0 = g$,\nso $g = f_+$.\nBy the same argument, $h = f_-$.\n\ + \n\\smallskip\n\\noindent\\textbf{Proof of (D3): Isomorphism.}\nThe map $\\Delta_\\tau \\colon f \\mapsto (f_+, f_-)$\n\ + is a homomorphism of $H_\\tau$-modules:\nfor $\\lambda \\in H_\\tau$,\n\\[\n \\Delta_\\tau(\\lambda \\cdot f)\n \ + \ = (e_+ \\lambda f,\\; e_- \\lambda f)\n = (\\lambda \\cdot e_+ f,\\; \\lambda \\cdot e_- f)\n = \\lambda \\cdot\ + \ \\Delta_\\tau(f),\n\\]\nwhere the second equality uses\nthe centrality of $e_\\pm$ in~$H_\\tau$\n(they commute with\ + \ all elements\nbecause $H_\\tau$ is commutative).\nAdditivity is immediate.\n\nInjectivity:\nif $\\Delta_\\tau(f) = (0,\ + \ 0)$,\nthen $f_+ = f_- = 0$,\nso $f = f_+ + f_- = 0$.\n\nSurjectivity:\ngiven $(g, h) \\in\n\\mathcal{O}_\\tau^{(+)}(\\\ + tau^3)\n\\times \\mathcal{O}_\\tau^{(-)}(\\tau^3)$,\nset $f := g + h$.\nThen $f$ is holomorphic\n(as a sum of holomorphic\ + \ functions),\n$f_+ = e_+ (g + h) = e_+ g + e_+ h = g + 0 = g$,\nand similarly $f_- = h$.\nSo $\\Delta_\\tau(f) = (g,\ + \ h)$.\n\nHence $\\Delta_\\tau$ is an isomorphism.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line\ + \ 109\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch37-idempotent-decomposition.tex`\ + \ lines 371-399\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.IdempotentDecomposition`\n\ + - Name: `idempotent_decompose`\n\n## Dependencies\n\n- Canonical: I.D20, I.D21, I.T05, I.T10, II.D35, II.T27, II.D45,\ + \ II.T31\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every holomorphic function f on tau^3 decomposes uniquely as f = e_+ f_+ + e_- f_- where f_+ and f_- are the + B-channel and C-channel components. The decomposition is functorial, stage-coherent, and respects the canonical basis. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001335.md b/_corpus_v3_items/cid001335.md new file mode 100644 index 000000000..065fca203 --- /dev/null +++ b/_corpus_v3_items/cid001335.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001335/ +title: Branch Factorization +cid: cid001335 +primary_alias: LEM0017 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Branch Factorization +aliases: +- LEM0017 +- II.L08 +- branch-factorization +- lem:branch-factorization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005673 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 38 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Branch Factorization**\n\nBranch Factorization\n\n# Branch Factorization\n\n## Summary\n\nBranch Factorization\n\ + \n## Statement\n\n```latex\n%\n\\label{lem:branch-factorization}\nEvery $\\omega$-germ transformer\n$G \\colon d(\\tau^3)\ + \ \\to d(\\tau^3)$\nfactors through the bipolar idempotents.\nConcretely:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n\ + \ \\textbf{Decomposition.}\n $G = G_+ + G_-$,\n where $G_+ := e_+ \\cdot G$ and $G_- := e_-\ + \ \\cdot G$.\n \\item[\\textup{(ii)}]\n \\textbf{Each factor is an $\\omega$-germ transformer.}\n \ + \ Both $G_+$ and $G_-$ are $\\omega$-germ transformers\n in their own right:\n they preserve tower\ + \ coherence,\n commute with the tower transition maps,\n and produce stabilized $\\omega$-germs\n \ + \ from stabilized inputs.\n \\item[\\textup{(iii)}]\n \\textbf{Orthogonality.}\n The two factors\ + \ are orthogonal:\n $e_+ \\cdot G_- = 0$ and $e_- \\cdot G_+ = 0$.\n \\item[\\textup{(iv)}]\n \\\ + textbf{Uniqueness.}\n The decomposition $G = G_+ + G_-$\n is the \\textbf{unique} way to write $G$\n\ + \ as a sum of an $e_+$-valued transformer\n and an $e_-$-valued transformer.\n\\end{enumerate}\n```\n\ + \n## Proof / Justification\n\n```latex\n\\emph{(i) Decomposition.}\nThe element $1 \\in H_\\tau$ decomposes as $1 = e_+\ + \ + e_-$,\nand the idempotents satisfy\n$e_+^2 = e_+$, $e_-^2 = e_-$, $e_+ e_- = 0$\n(I.D21, Book~I).\nFor any $\\omega$-germ\ + \ transformer $G$,\ndefine $G_+ := e_+ \\cdot G$ and $G_- := e_- \\cdot G$\nby pointwise multiplication.\nThen:\n\\[\n\ + \ G_+ + G_-\n = e_+ \\cdot G + e_- \\cdot G\n = (e_+ + e_-) \\cdot G\n = 1 \\cdot G\n = G.\n\\]\n\n\\emph{(ii)\ + \ Each factor is an $\\omega$-germ transformer.}\nWe verify the three properties for $G_+$\n(the argument for $G_-$ is\ + \ identical\nwith $e_-$ replacing $e_+$).\n\n\\emph{Tower coherence.}\nLet $(f_k)_{k \\geq n}$ be a tower-coherent input.\n\ + Then $G(f_k)_{k \\geq n}$ is tower-coherent\nbecause $G$ is an $\\omega$-germ transformer.\nProjecting:\n$(G_+)_k(f) =\ + \ e_+ \\cdot G_k(f)$.\nSince $e_+ \\in H_\\tau$ is a fixed scalar\nand the tower transition maps\n$\\pi_{k,k+1} \\colon\ + \ \\mathbb{Z}/P_{k+1}\\mathbb{Z}\n\\to \\mathbb{Z}/P_k\\mathbb{Z}$\nare ring homomorphisms,\nthe scalar multiplication\ + \ by $e_+$\ncommutes with the transition maps:\n\\[\n \\pi_{k,k+1}\\bigl(e_+ \\cdot G_{k+1}(f)\\bigr)\n = e_+ \\\ + cdot \\pi_{k,k+1}\\bigl(G_{k+1}(f)\\bigr)\n = e_+ \\cdot G_k(f)\n = (G_+)_k(f).\n\\]\nTower coherence of $G_+$ follows.\n\ + \n\\emph{Commutativity with transition maps.}\nImmediate from the above calculation:\n$e_+$ is central in $H_\\tau$\n\ + (because $e_+$ is a scalar,\nand scalar multiplication commutes\nwith all ring operations in $H_\\tau$).\n\n\\emph{Stabilization.}\n\ + If $G$ produces a stabilized $\\omega$-germ\nfrom a stabilized input\n(spectral support fixed after some stage~$n$),\n\ + then $G_+ = e_+ \\cdot G$ has spectral support\ncontained in that of~$G$\n(multiplication by a fixed element\ncannot enlarge\ + \ the support).\nHence $G_+$ also stabilizes.\n\n\\emph{(iii) Orthogonality.}\n$e_+ \\cdot G_- = e_+ \\cdot (e_- \\cdot\ + \ G)\n= (e_+ e_-) \\cdot G = 0 \\cdot G = 0$,\nusing $e_+ e_- = 0$ (I.D21).\nSimilarly, $e_- \\cdot G_+ = 0$.\n\n\\emph{(iv)\ + \ Uniqueness.}\nSuppose $G = H_+ + H_-$\nwith $e_+ \\cdot H_- = 0$ and $e_- \\cdot H_+ = 0$.\nThen $e_+ \\cdot G = e_+\ + \ \\cdot H_+ + e_+ \\cdot H_-\n= H_+ + 0 = H_+$,\nwhere the last step uses $e_+ \\cdot H_+ = H_+$\n(since $H_+$ is $e_+$-valued:\ + \ $H_+ = e_+ \\cdot H_+$).\nHence $H_+ = G_+$, and symmetrically $H_- = G_-$.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 111\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch38-three-lemma-chain.tex`\ + \ lines 206-237\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.ThreeLemmaChain`\n\ + - Name: `branch_factorization_check`\n\n## Dependencies\n\n- Canonical: II.L07, II.D48, II.D37, I.D21, I.T10\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Branch Factorization +noindex: false +--- + diff --git a/_corpus_v3_items/cid001336.md b/_corpus_v3_items/cid001336.md new file mode 100644 index 000000000..f6066e0c4 --- /dev/null +++ b/_corpus_v3_items/cid001336.md @@ -0,0 +1,109 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001336/ +title: Prime-Split Support +cid: cid001336 +primary_alias: LEM0018 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The branch factorization has canonically split spectral support: G_+ on B-channel primes and G_- on C-channel primes, + with partition from Prime Polarity.' +aliases: +- LEM0018 +- II.L09 +- prime-split-support +- lem:prime-split-support +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005674 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 38 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Prime-Split Support**\n\nThe branch factorization has canonically split spectral support: G_+ on B-channel\ + \ primes and G_- on C-channel primes, with partition from Prime Polarity.\n\n# Prime-Split Support\n\n## Summary\n\nThe\ + \ branch factorization has canonically split spectral support: G_+ on B-channel primes and G_- on C-channel primes, with\ + \ partition from Prime Polarity.\n\n## Statement\n\n```latex\n%\n\\label{lem:prime-split-support}\nLet $G = G_+ + G_-$\n\ + be the Branch Factorization of an $\\omega$-germ transformer\n\\textup{(II.L08)}.\nThen:\n\\begin{enumerate}\n \\item[\\\ + textup{(i)}]\n \\textbf{B-channel support.}\n The spectral support of $G_+$\n \\textup{(with\ + \ respect to the canonical basis\n $\\mathcal{B}_\\tau$, II.D45)}\n is contained in the B-channel primes:\n\ + \ $\\operatorname{supp}(G_+) \\subseteq \\Lambda_\\tau^{(B)}$,\n where $\\Lambda_\\tau^{(B)}$\n \ + \ is the set of indexing triples\n $(p, v, B)$ with $p$ a $\\gamma$-orbit prime.\n \\item[\\textup{(ii)}]\n\ + \ \\textbf{C-channel support.}\n The spectral support of $G_-$\n is contained in the C-channel\ + \ primes:\n $\\operatorname{supp}(G_-) \\subseteq \\Lambda_\\tau^{(C)}$,\n where $\\Lambda_\\tau^{(C)}$\n\ + \ is the set of indexing triples\n $(p, v, C)$ with $p$ an $\\eta$-orbit prime.\n \\item[\\textup{(iii)}]\n\ + \ \\textbf{Forced partition.}\n The assignment of primes to channels\n is \\textbf{canonical}:\n\ + \ it is determined by the ABCD chart's\n B/C coordinate structure\n and by Prime Polarity \\\ + textup{(I.T05, Book~I)}.\n No choice is involved.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\emph{(i) B-channel support.}\nThe component $G_+ = e_+ \\cdot G$\nis valued in the B-channel:\n$G_+(x) \\in A_\\tau^{(B)}\ + \ = e_+ \\cdot H_\\tau$\nfor all $x \\in d(\\tau^3)$.\nWe must show that its spectral support\n(the set of active cylinder\ + \ generators\nin the canonical basis expansion)\ninvolves only B-channel indices.\n\nConsider the stage-$k$ component\n\ + $(G_+)_k \\colon \\mathbb{Z}/P_k\\mathbb{Z} \\to A_\\tau^{(B)}$.\nBy the Projection Formula\n(II.P08, Chapter~\\ref{ch:canonical-basis}),\n\ + the spectral coefficient of $(G_+)_k$\nat index $(p, v, \\sigma)$ is\n\\[\n \\varphi_{p,v}^{(\\sigma)}\n = \\frac{1}{|F_p|}\n\ + \ \\sum_{x \\in F_p(v)}\n e_\\sigma \\cdot (G_+)_k(x).\n\\]\nFor $\\sigma = C$ (i.e., $e_\\sigma = e_-$):\n$e_-\ + \ \\cdot (G_+)_k(x) = e_- \\cdot e_+ \\cdot G_k(x) = 0$,\nsince $e_- e_+ = 0$ (I.D21, Book~I).\nHence $\\varphi_{p,v}^{(C)}\ + \ = 0$\nfor all $(p, v)$.\nThe spectral support of $G_+$ contains no C-channel indices.\n\nFor $\\sigma = B$ (i.e., $e_\\\ + sigma = e_+$):\n$e_+ \\cdot (G_+)_k(x) = e_+ \\cdot e_+ \\cdot G_k(x)\n= e_+ \\cdot G_k(x) = (G_+)_k(x)$,\nsince $e_+^2\ + \ = e_+$.\nThe coefficient $\\varphi_{p,v}^{(B)}$\nis the average of $(G_+)_k$\nover the fiber $F_p(v)$,\nwhich is nonzero\ + \ only when $G_+$\nhas nontrivial data\nat the corresponding residue class.\n\nThe key observation:\nat each stage $k$,\n\ + the new prime $p_{k+1}$\nenters the CRT decomposition\n$\\mathbb{Z}/P_{k+1}\\mathbb{Z}\n\\cong \\mathbb{Z}/P_k\\mathbb{Z}\n\ + \\times \\mathbb{Z}/p_{k+1}\\mathbb{Z}$.\nPrime Polarity (I.T05, Book~I)\nassigns $p_{k+1}$ to either the $\\gamma$-orbit\n\ + or the $\\eta$-orbit.\nThe spectral character $\\chi_+$ (I.D22, Book~I)\nacts on the $\\gamma$-orbit primes,\nand $\\\ + chi_-$ (I.D23, Book~I)\nacts on the $\\eta$-orbit primes.\nSince $G_+$ is valued in the $e_+$-channel,\nand $\\chi_+$\ + \ is the character\nthat detects $e_+$-valued data\n(by the spectral decomposition\nof Chapter~\\ref{ch:canonical-basis}),\n\ + the support of $G_+$\nis contained in the set of primes\non which $\\chi_+$ acts nontrivially---the\n$\\gamma$-orbit primes.\n\ + Hence\n$\\operatorname{supp}(G_+) \\subseteq \\Lambda_\\tau^{(B)}$.\n\n\\emph{(ii) C-channel support.}\nThe argument for\ + \ $G_-$ is symmetric:\n$G_-$ is valued in $A_\\tau^{(C)} = e_- \\cdot H_\\tau$,\nthe B-channel projection vanishes\n($e_+\ + \ \\cdot G_- = 0$),\nand the character $\\chi_-$ (I.D23, Book~I)\ndetects $e_-$-valued data\nsupported on the $\\eta$-orbit\ + \ primes.\nHence\n$\\operatorname{supp}(G_-) \\subseteq \\Lambda_\\tau^{(C)}$.\n\n\\emph{(iii) Forced partition.}\nPrime\ + \ Polarity (I.T05, Book~I)\nestablishes a canonical partition\nof the set of primes\ninto $\\gamma$-orbit and $\\eta$-orbit.\n\ + This partition is not chosen\nbut forced by the peel-off structure\nof the program monoid (I.P02, Book~I):\nthe exponent~$\\\ + gamma$ and the tetration~$\\eta$\nextract different arithmetic information\nfrom each prime,\nand the two extractions\ + \ are functorially distinct.\nThe assignment of a prime to a channel\nis determined by which generator\n($\\gamma$ or\ + \ $\\eta$)\ngoverns its peel-off behavior.\nNo additional convention or choice is needed.\n```\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 112\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch38-three-lemma-chain.tex`\ + \ lines 365-400\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.ThreeLemmaChain`\n\ + - Name: `prime_split_support_check`\n\n## Dependencies\n\n- Canonical: I.T05, I.D21, I.D22, I.D23, II.L08, II.D45\n\n\ + ## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The branch factorization has canonically split spectral support: G_+ on B-channel primes and G_- on C-channel + primes, with partition from Prime Polarity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001337.md b/_corpus_v3_items/cid001337.md new file mode 100644 index 000000000..9c12305e9 --- /dev/null +++ b/_corpus_v3_items/cid001337.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001337/ +title: Polarity Symmetry +cid: cid001337 +primary_alias: LEM0019 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The j-involution interchanges bipolar sectors: sigma_j(G_+) = G_- and sigma_j(G_-) = G_+, so one sector determines + the full transformer.' +aliases: +- LEM0019 +- II.L10 +- polarity-symmetry +- lem:polarity-symmetry +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005675 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 38 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Polarity Symmetry**\n\nThe j-involution interchanges bipolar sectors: sigma_j(G_+) = G_- and sigma_j(G_-) =\ + \ G_+, so one sector determines the full transformer.\n\n# Polarity Symmetry\n\n## Summary\n\nThe j-involution interchanges\ + \ bipolar sectors: sigma_j(G_+) = G_- and sigma_j(G_-) = G_+, so one sector determines the full transformer.\n\n## Statement\n\ + \n```latex\n%\n\\label{lem:polarity-symmetry}\nLet $G = G_+ + G_-$\nbe the Branch Factorization\nof an $\\omega$-germ\ + \ transformer \\textup{(II.L08)}.\nThen the $\\jj$-involution interchanges the two sectors:\n\\begin{enumerate}\n \\\ + item[\\textup{(i)}]\n $\\sigma_\\jj(G_+) = G_-$\n and $\\sigma_\\jj(G_-) = G_+$.\n \\item[\\textup{(ii)}]\n\ + \ The spectral data of $G_+$ at a $\\gamma$-orbit prime~$p$\n determines the spectral data of $G_-$\n\ + \ at the corresponding $\\eta$-orbit prime,\n and conversely.\n \\item[\\textup{(iii)}]\n \ + \ Knowing $G_+$ alone\n determines $G_-$ \\textup{(}and hence $G$\\textup{)}\n via $G_- = \\sigma_\\\ + jj(G_+)$\n and $G = G_+ + \\sigma_\\jj(G_+)$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\emph{(i) Channel interchange.}\nBy the channel-swapping property\nof $\\sigma_\\jj$\n(Remark~\\ref{rem:ch38-sigma-properties}):\n\ + \\begin{align*}\n \\sigma_\\jj(G_+)\n &= \\sigma_\\jj(e_+ \\cdot G)\n = \\sigma_\\jj(e_+) \\cdot \\sigma_\\jj(G)\n\ + \ = e_- \\cdot \\sigma_\\jj(G).\n\\end{align*}\nWe need $\\sigma_\\jj(G) = G$.\nThis holds because\n$G$ is a $\\tau$-holomorphic\ + \ transformer,\nand the $\\jj$-involution\nis an automorphism of $H_\\tau$\nthat preserves the holomorphic structure.\n\ + Concretely:\nthe Mutual Determination Theorem (II.T27,\nChapter~\\ref{ch:mutual-determination})\nidentifies a holomorphic\ + \ datum\nwith a boundary character\n$\\varphi \\colon R_\\tau \\to H_\\tau$.\nThe boundary ring $R_\\tau$ (I.D19, Book~I)\n\ + is invariant under $\\sigma_\\jj$\n(it is the profinite completion\nof cyclic groups,\nwhich are stable under conjugation).\n\ + The holomorphic condition is symmetric\nin the B and C channels\n(the five-way equivalence of II.T27\ntreats both channels\ + \ identically).\nHence $\\sigma_\\jj$ maps holomorphic transformers\nto holomorphic transformers,\nand since $G$ is the\ + \ \\emph{total} transformer,\n$\\sigma_\\jj(G) = G$ as a transformation\non the germ algebra.\n\nTherefore:\n$\\sigma_\\\ + jj(G_+) = e_- \\cdot \\sigma_\\jj(G) = e_- \\cdot G = G_-$.\nSimilarly, $\\sigma_\\jj(G_-) = G_+$.\n\n\\emph{(ii) Spectral\ + \ correspondence.}\nBy Prime-Split Support (II.L09),\n$G_+$ has spectral support in\nthe $\\gamma$-orbit primes\nand $G_-$\ + \ has spectral support in\nthe $\\eta$-orbit primes.\nThe $\\jj$-involution maps\n$\\chi_+$ to $\\chi_-$ and $\\chi_-$\ + \ to $\\chi_+$\n(I.D22--I.D23, Book~I:\n$\\chi_\\pm = e_\\pm \\cdot \\chi$\nfor any character~$\\chi$,\nand $\\sigma_\\\ + jj$ swaps $e_\\pm$).\nHence the spectral coefficient of $G_+$\nat a $\\gamma$-orbit prime~$p$\nequals the spectral coefficient\ + \ of $G_-$\nat the partner prime in the $\\eta$-orbit.\nThe correspondence is canonical\n(determined by Prime Polarity,\ + \ I.T05).\n\n\\emph{(iii) One channel determines both.}\nSince $G_- = \\sigma_\\jj(G_+)$ by~(i),\nand $G = G_+ + G_-$\ + \ by Branch Factorization,\nwe have $G = G_+ + \\sigma_\\jj(G_+)$.\nKnowledge of $G_+$ alone\ndetermines $G$ completely.\n\ + Symmetrically, $G = \\sigma_\\jj(G_-) + G_-$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 113\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch38-three-lemma-chain.tex` lines\ + \ 641-663\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.ThreeLemmaChain`\n\ + - Name: `polarity_symmetry_check`\n\n## Dependencies\n\n- Canonical: I.T05, I.D21, I.D22, I.D23, II.L08, II.L09\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The j-involution interchanges bipolar sectors: sigma_j(G_+) = G_- and sigma_j(G_-) = G_+, so one sector determines + the full transformer.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001338.md b/_corpus_v3_items/cid001338.md new file mode 100644 index 000000000..a388b9c86 --- /dev/null +++ b/_corpus_v3_items/cid001338.md @@ -0,0 +1,100 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001338/ +title: Probe Naturality iff Yoneda +cid: cid001338 +primary_alias: LEM0020 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Probe Naturality iff Yoneda +aliases: +- LEM0020 +- II.L11 +- probe-naturality-iff-yoneda +- lem:probe-yoneda +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005534 +- predicate: formalized_by + object: cid005541 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 43 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Probe Naturality iff Yoneda**\n\nProbe Naturality iff Yoneda\n\n# Probe Naturality iff Yoneda\n\n## Summary\n\ + \nProbe Naturality iff Yoneda\n\n## Statement\n\n```latex\n%\n\\label{lem:probe-yoneda}\nLet $\\varphi \\colon A \\to\ + \ B$ be a morphism in $\\tau$.\nThe following conditions are equivalent:\n\\begin{enumerate}\n \\item[\\textup{(PN)}]\n\ + \ $\\varphi$ satisfies probe naturality\n (Definition~\\ref{def:probe-naturality}).\n \\item[\\textup{(Hol)}]\n\ + \ $\\varphi$ is $\\tau$-holomorphic\n in the sense of the enriched structure:\n the natural\ + \ transformation $y(\\varphi)$ respects\n the split-complex decomposition\n of each Hom object,\n \ + \ i.e., $y(\\varphi)$ preserves\n the bipolar sector structure (II.P11).\n \\item[\\textup{(Yon)}]\n\ + \ The natural transformation $y(\\varphi)$\n is an element of\n $\\mathrm{Nat}(y(A), y(B))$\n\ + \ that is determined by $\\varphi$\n via the Yoneda correspondence:\n evaluation at $\\id_A$\ + \ recovers $\\varphi$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(PN) $\\Rightarrow$ (Hol).}\\\ + ;\nAssume $\\varphi$ satisfies probe naturality.\nBy the commutativity of the diagram\nin Definition~\\ref{def:probe-naturality},\n\ + postcomposition by $\\varphi$\nintertwines with precomposition by any $h$.\nIn particular, taking $P$ to be the $\\tau$-objects\n\ + on which the bipolar decomposition\n$[P, A] = e_+ \\cdot [P, A]_+ + e_- \\cdot [P, A]_-$\nlives:\nthe postcomposition\ + \ map $\\varphi_*$\nmust carry $e_+ \\cdot [P, A]_+$ into $e_+ \\cdot [P, B]_+$\nand $e_- \\cdot [P, A]_-$ into $e_- \\\ + cdot [P, B]_-$,\nbecause precomposition by any probe $h$\npreserves the bipolar splitting\n(II.P11 applied to $h^*$),\n\ + and the commutativity of the naturality square\nforces $\\varphi_*$ to respect the same splitting.\nThus $y(\\varphi)$\ + \ preserves the bipolar sector structure.\n\n\\medskip\n\\textbf{(Hol) $\\Rightarrow$ (Yon).}\\;\nAssume $y(\\varphi)$\ + \ preserves the bipolar sector structure.\nWe must show that the natural transformation $y(\\varphi)$\nis determined by\ + \ its value at $\\id_A$.\nEvaluate $y(\\varphi)$ at $P = A$ and $f = \\id_A$:\n\\[\n y(\\varphi)_A(\\id_A) \\;=\\;\ + \ \\varphi \\circ \\id_A \\;=\\; \\varphi.\n\\]\nFor any other probe $g \\colon P \\to A$:\n\\[\n y(\\varphi)_P(g)\n\ + \ \\;=\\; \\varphi \\circ g\n \\;=\\; y(\\varphi)_A(\\id_A) \\circ g\n \\;=\\; g^*\\bigl(y(\\varphi)_A(\\id_A)\\\ + bigr).\n\\]\nThe last equality holds because $y(\\varphi)$\nis a natural transformation:\nevaluating at $\\id_A$ and then\ + \ precomposing with $g$\nequals evaluating at $g$ directly.\nThis is the enriched Yoneda bijection:\n$y(\\varphi)$ is\ + \ determined by $\\varphi = y(\\varphi)_A(\\id_A)$.\n\nThe bipolar preservation hypothesis ensures\nthat this determination\ + \ respects the enriched structure ---\nthe bijection $\\mathrm{Nat}(y(A), y(B)) \\cong [A, B]$\nis an isomorphism of $\\\ + tau$-objects\n(not merely a set bijection),\nbecause both sides carry the bipolar decomposition\nand the isomorphism preserves\ + \ it.\n\n\\medskip\n\\textbf{(Yon) $\\Rightarrow$ (PN).}\\;\nAssume the Yoneda correspondence holds:\n$y(\\varphi)$ is\ + \ determined by $\\varphi$ via evaluation at $\\id_A$.\nFor any $h \\colon P' \\to P$ and $g \\colon P \\to A$:\n\\[\n\ + \ h^*\\bigl(y(\\varphi)_P(g)\\bigr)\n \\;=\\; h^*(\\varphi \\circ g)\n \\;=\\; (\\varphi \\circ g) \\circ h\n\ + \ \\;=\\; \\varphi \\circ (g \\circ h).\n\\]\nThe last equality uses associativity in $\\tau$\n(II.T28, earned from\ + \ the program monoid I.P02).\nOn the other hand:\n\\[\n y(\\varphi)_{P'}(h^*(g))\n \\;=\\; y(\\varphi)_{P'}(g \\\ + circ h)\n \\;=\\; \\varphi \\circ (g \\circ h).\n\\]\nBoth sides agree,\nso the naturality square commutes.\n```\n\n\ + ## Source Context\n\n- Registry source: `book-02.jsonl` line 128\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch43-yoneda-theorem.tex`\ + \ lines 313-338\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.YonedaTheorem`\n\ + - Name: `Tau.BookII.Enrichment.probe_yoneda_check`\n\n## Dependencies\n\n- Canonical: II.D50, II.R12, II.T33, II.D53,\ + \ II.D54\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Probe Naturality iff Yoneda +noindex: false +--- + diff --git a/_corpus_v3_items/cid001339.md b/_corpus_v3_items/cid001339.md new file mode 100644 index 000000000..edc722106 --- /dev/null +++ b/_corpus_v3_items/cid001339.md @@ -0,0 +1,115 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001339/ +title: Extension in Split-Complex Codomain +cid: cid001339 +primary_alias: LEM0021 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Extension in Split-Complex Codomain +aliases: +- LEM0021 +- II.L12 +- extension-in-split-complex-codomain +- lem:extension-h-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005411 +- predicate: formalized_by + object: cid005414 +- predicate: formalized_by + object: cid005415 +- predicate: formalized_by + object: cid005416 +- predicate: formalized_by + object: cid005417 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 48 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Extension in Split-Complex Codomain**\n\nExtension in Split-Complex Codomain\n\n# Extension in Split-Complex\ + \ Codomain\n\n## Summary\n\nExtension in Split-Complex Codomain\n\n## Statement\n\n```latex\n%\n\\label{lem:extension-h-tau}\n\ + % II.D35, I.D21, I.T31\nLet $\\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$\nbe an idempotent-supported character\n\ + on the boundary ring~$\\widehat{\\mathbb{Z}}_\\tau$.\nThen the function\n\\[\n \\boxed{%\n f_\\chi\n \\;:=\\\ + ;\n e_+ \\cdot f_+ \\;+\\; e_- \\cdot f_-\n \\;:\\; \\tau^3 \\;\\longrightarrow\\; H_\\tau}\n\\]\nsatisfies:\n\\\ + begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Well-defined.}\n $f_\\chi$ is a well-defined\n\ + \ $\\tau$-holomorphic function\n valued in $H_\\tau^{\\mathrm{cal}}$\n (the calibrated split-complex\ + \ codomain, II.D35).\n\n \\item[\\textup{(ii)}]\n \\textbf{Boundary recovery.}\n The restriction\ + \ of $f_\\chi$\n to the boundary $\\mathbb{L}$\n recovers~$\\chi$:\n \\[\n f_\\\ + chi\\big|_{\\mathbb{L}}\n \\;=\\; \\chi.\n \\]\n\n \\item[\\textup{(iii)}]\n \\textbf{Tower\ + \ coherence.}\n The system $\\{(f_\\chi)_k\\}_{k \\geq 1}$\n is tower-coherent:\n $(f_\\chi)_{k+1}\ + \ \\bmod P_k = (f_\\chi)_k$\n for all $k \\geq 1$.\n\n \\item[\\textup{(iv)}]\n \\textbf{$\\tau$-regularity.}\n\ + \ The function $f_\\chi$ is $\\tau$-regular\n (Definition~\\ref{def:tau-regularity}, II.D49)\n \ + \ at every interior point of~$\\tau^3$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i)\ + \ Well-defined.}\nEach sector lift $f_\\pm$\nis the inverse limit\nof the iterated $\\mathrm{BndLift}$ operator\n(Proposition~\\\ + ref{prop:ch30-convergence},\nChapter~\\ref{ch:bndlift-construction}).\nThe $\\mathrm{BndLift}$ existence theorem\n(Theorem~\\\ + ref{thm:bndlift-existence}, II.T26)\nguarantees that each individual lift step exists.\nTower coherence holds by construction,\n\ + so the inverse limit is a well-defined function\n$f_\\pm : \\tau^3 \\to \\mathbb{R}$.\n\nThe recombination\n$f_\\chi =\ + \ e_+ \\cdot f_+ + e_- \\cdot f_-$\ntakes values in\n$H_\\tau \\cong \\mathbb{R} \\cdot e_+ + \\mathbb{R} \\cdot e_-$\n\ + (the idempotent decomposition\nof the calibrated split-complex codomain).\nSince $e_+ + e_- = 1$ and $e_+ \\cdot e_- =\ + \ 0$,\nthe sum is well-defined\nand lies in~$H_\\tau^{\\mathrm{cal}}$.\n\n\\textbf{$\\tau$-holomorphy.}\nThe Hol\\,$\\\ + Leftrightarrow$\\,Idemp equivalence\n(Theorem~\\ref{thm:hol-iff-idempotent}, II.T33)\nstates that a function $\\tau^3\ + \ \\to H_\\tau$\nis $\\tau$-holomorphic\nif and only if it is idempotent-supported.\nBy construction,\n$f_\\chi = e_+\ + \ \\cdot f_+ + e_- \\cdot f_-$\nis manifestly idempotent-supported:\nthe $e_+$-component is $f_+$,\nthe $e_-$-component\ + \ is $f_-$,\nand each component is tower-coherent.\nHence $f_\\chi$ is $\\tau$-holomorphic.\n\n\\textbf{(ii) Boundary\ + \ recovery.}\nAt the boundary $\\mathbb{L}$,\nthe sector lifts reduce\nto the boundary character components:\n$f_+\\big|_{\\\ + mathbb{L}} = \\chi_+$\nand $f_-\\big|_{\\mathbb{L}} = \\chi_-$.\nThis holds because the $\\mathrm{BndLift}$ operator\n\ + is constructed precisely to be compatible\nwith the boundary data ---\ncondition~(i) of Definition~\\ref{def:bndlift}\ + \ (II.D36)\nrequires\n$f_{n+1}\\big|_{\\bmod P_n} = f_n$\nat every stage.\nAt the profinite limit,\n$f_\\pm\\big|_{\\\ + mathbb{L}} = \\varprojlim_n f_{\\pm,n}\\big|_{\\mathrm{bnd}}\n= \\chi_\\pm$.\nHence\n$f_\\chi\\big|_{\\mathbb{L}}\n= e_+\ + \ \\cdot \\chi_+ + e_- \\cdot \\chi_-\n= \\chi$.\n\n\\textbf{(iii) Tower coherence.}\nAt stage~$k$,\n$(f_\\chi)_k\n= e_+\ + \ \\cdot (f_+)_k + e_- \\cdot (f_-)_k$.\nTower coherence of $f_+$ and $f_-$ individually\n(from the $\\mathrm{BndLift}$\ + \ construction)\nimplies tower coherence of their\nidempotent-weighted sum:\n\\begin{align*}\n (f_\\chi)_{k+1} \\bmod\ + \ P_k\n &\\;=\\;\n e_+ \\cdot \\bigl((f_+)_{k+1} \\bmod P_k\\bigr)\n \\;+\\;\n e_- \\cdot \\bigl((f_-)_{k+1}\ + \ \\bmod P_k\\bigr) \\\\\n &\\;=\\;\n e_+ \\cdot (f_+)_k\n \\;+\\;\n e_- \\cdot (f_-)_k \\\\\n &\\;=\\\ + ;\n (f_\\chi)_k.\n\\end{align*}\nThe key step uses\n$e_\\pm \\bmod P_k = e_\\pm$\n(the idempotents are independent\ + \ of the stage).\n\n\\textbf{(iv) $\\tau$-regularity.}\nAt every interior point $x \\in \\tau^3 \\setminus \\mathbb{L}$,\n\ + the function $f_\\chi$ has a stabilized $\\omega$-germ:\nthe $\\mathrm{BndLift}$ system converges\nat finite depth,\n\ + meaning $x$ lies in some\nstage-$k$ cylinder domain (II.D10)\non which $f_\\chi$\nis already determined by the stage-$k$\ + \ data.\nThis is precisely the positive regularity condition\n(Definition~\\ref{def:tau-regularity}, II.D49):\n$f_\\chi$\ + \ has a canonical interior extension\nat~$x$,\narising from stabilized lift data.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 141\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch48-hartogs-extension-h-tau.tex`\ + \ lines 201-248\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.HartogsExtension`\n\ + - Name: `Tau.BookII.CentralTheorem.extension_channel_check`\n\n## Dependencies\n\n- Canonical: II.D36, II.D59, II.P13,\ + \ II.T26, II.D35, I.D21, I.T31\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Extension in Split-Complex Codomain +noindex: false +--- + diff --git a/_corpus_v3_items/cid001340.md b/_corpus_v3_items/cid001340.md new file mode 100644 index 000000000..51c717429 --- /dev/null +++ b/_corpus_v3_items/cid001340.md @@ -0,0 +1,109 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001340/ +title: Stagewise Naturality +cid: cid001340 +primary_alias: LEM0022 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Stagewise Naturality +aliases: +- LEM0022 +- II.L13 +- stagewise-naturality +- lem:stagewise-naturality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005402 +- predicate: formalized_by + object: cid005403 +- predicate: formalized_by + object: cid005406 +- predicate: formalized_by + object: cid005407 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 49 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Stagewise Naturality**\n\nStagewise Naturality\n\n# Stagewise Naturality\n\n## Summary\n\nStagewise Naturality\n\ + \n## Statement\n\n```latex\n%\n\\label{lem:stagewise-naturality}\nLet $\\chi$ be an idempotent-supported\nboundary character,\n\ + and let $f_\\chi \\colon \\tau^3 \\to H_\\tau$\nbe its Hartogs extension \\textup{(II.T37)}.\nThen $f_\\chi$ is \\textbf{stagewise\ + \ natural}:\nfor all $\\ell > k \\geq 1$,\nthe diagram\n\\[\n \\begin{array}{ccc}\n \\mathbb{Z}/P_\\ell\\mathbb{Z}\n\ + \ & \\xrightarrow{\\;\\; f_{\\chi,\\ell} \\;\\;}\n & H_\\tau \\\\[6pt]\n \\Big\\downarrow\\vcenter{\\\ + rlap{\\scriptsize$\\rho_{k,\\ell}$}}\n & & \\Big\\| \\\\[6pt]\n \\mathbb{Z}/P_k\\mathbb{Z}\n & \\\ + xrightarrow{\\;\\; f_{\\chi,k} \\;\\;}\n & H_\\tau\n \\end{array}\n\\]\ncommutes in the sense that\n\\[\n \ + \ \\boxed{%\n f_{\\chi,k}\\bigl(\\rho_{k,\\ell}(a)\\bigr)\n \\;=\\;\n \\mathrm{reduce}_k\\bigl(f_{\\chi,\\ell}(a)\\\ + bigr)\n \\qquad\n \\text{for all } a \\in \\mathbb{Z}/P_\\ell\\mathbb{Z},}\n\\]\nwhere $\\mathrm{reduce}_k$ is the\ + \ stage-$k$\nreduction on the codomain $H_\\tau$\ninduced by the primorial tower structure.\nIn other words:\ncomputing\ + \ $f_\\chi$ at the finer stage~$\\ell$\nand then reducing the output to stage~$k$\ngives the same result\nas first reducing\ + \ the input to stage~$k$\nand then computing~$f_\\chi$.\n```\n\n## Proof / Justification\n\n```latex\nThe proof uses three\ + \ ingredients,\neach established in previous chapters.\n\n\\smallskip\n\\noindent\\textbf{Ingredient 1: Tower coherence\ + \ of~$\\chi$.}\nThe boundary character $\\chi$\nis a ring homomorphism\n$\\chi \\colon R_\\tau \\to H_\\tau$,\nwhere $R_\\\ + tau = \\widehat{\\mathbb{Z}}_\\tau$\nis the profinite boundary ring (I.D19, Book~I).\nBy definition of the profinite limit,\n\ + $\\chi$ is determined by a coherent family\n$(\\chi_k)_{k \\geq 1}$\nwhere $\\chi_k \\colon \\mathbb{Z}/P_k\\mathbb{Z}\ + \ \\to H_\\tau$\nsatisfies\n\\[\n \\chi_k\\bigl(\\rho_{k,\\ell}(a)\\bigr)\n \\;=\\;\n \\mathrm{reduce}_k\\bigl(\\\ + chi_\\ell(a)\\bigr)\n \\qquad\n \\text{for all } a \\in \\mathbb{Z}/P_\\ell\\mathbb{Z}.\n\\]\nThis is tower coherence\ + \ (I.D46, Book~I)\nfor the boundary data.\n\n\\smallskip\n\\noindent\\textbf{Ingredient 2: BndLift preserves coherence.}\n\ + The $\\mathrm{BndLift}_n$ construction\n(II.D36, Chapter~\\ref{ch:bndlift-construction})\nextends boundary data at stage~$n$\n\ + to interior data at stage~$n+1$\nvia the CRT decomposition.\nBy the Mutual Determination Theorem\n(II.T27, Chapter~\\\ + ref{ch:mutual-determination}),\nthe five descriptions of holomorphic data\nare equivalent,\nand in particular the tower-coherent\ + \ description~(R)\nis preserved by $\\mathrm{BndLift}_n$.\nConcretely:\nif $(g_k)_{k \\leq n}$ is tower-coherent,\nthen\ + \ the extended sequence\n$(g_k)_{k \\leq n+1}$\nwith $g_{n+1} = \\mathrm{BndLift}_n(g_n)$\nis still tower-coherent.\n\n\ + \\smallskip\n\\noindent\\textbf{Ingredient 3: Bipolar channel independence.}\nThe extension $f_\\chi$\ndecomposes as\n\ + $f_\\chi = e_+ \\cdot f_\\chi^+ + e_- \\cdot f_\\chi^-$\n(Proposition~\\ref{prop:ch49-bipolar-stage}).\nThe two channels\ + \ are independent.\nIt therefore suffices to verify\nstagewise naturality in each channel separately.\nIn the $e_+$-channel:\n\ + $f_{\\chi,k}^+ = e_+ \\cdot f_{\\chi,k}$\ndepends only on $\\chi_+$,\nand $\\chi_+$ is tower-coherent (Ingredient~1).\n\ + The extension in the $e_+$-channel\nis the iterated $\\mathrm{BndLift}_n^+$ of $\\chi_+$,\nwhich preserves coherence (Ingredient~2).\n\ + The same argument applies\nin the $e_-$-channel with $\\chi_-$.\n\n\\smallskip\n\\noindent\\textbf{Assembly.}\nCombining\ + \ all three ingredients:\nfor any $\\ell > k \\geq 1$\nand $a \\in \\mathbb{Z}/P_\\ell\\mathbb{Z}$,\n\\begin{align*}\n\ + \ \\mathrm{reduce}_k\\bigl(f_{\\chi,\\ell}(a)\\bigr)\n &= \\mathrm{reduce}_k\\Bigl(\n e_+ \\cdot f_{\\chi,\\\ + ell}^+(a)\n + e_- \\cdot f_{\\chi,\\ell}^-(a)\n \\Bigr) \\\\\n &= e_+ \\cdot \\mathrm{reduce}_k\\bigl(f_{\\\ + chi,\\ell}^+(a)\\bigr)\n + e_- \\cdot \\mathrm{reduce}_k\\bigl(f_{\\chi,\\ell}^-(a)\\bigr) \\\\\n &= e_+ \\cdot\ + \ f_{\\chi,k}^+\\bigl(\\rho_{k,\\ell}(a)\\bigr)\n + e_- \\cdot f_{\\chi,k}^-\\bigl(\\rho_{k,\\ell}(a)\\bigr) \\\\\n\ + \ &= f_{\\chi,k}\\bigl(\\rho_{k,\\ell}(a)\\bigr),\n\\end{align*}\nwhere the third equality uses\nthe coherence of each\ + \ channel\n(Ingredients~1 and~2 in each channel independently).\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 143\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch49-extensions-omega-germs.tex`\ + \ lines 266-306\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms`\n\ + - Name: `Tau.BookII.CentralTheorem.stagewise_naturality_check`\n\n## Dependencies\n\n- Canonical: II.T37, II.D36, II.T27,\ + \ I.T18\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later\ + \ projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Stagewise Naturality +noindex: false +--- + diff --git a/_corpus_v3_items/cid001341.md b/_corpus_v3_items/cid001341.md new file mode 100644 index 000000000..f2c0c82a5 --- /dev/null +++ b/_corpus_v3_items/cid001341.md @@ -0,0 +1,112 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001341/ +title: Yoneda Application +cid: cid001341 +primary_alias: LEM0023 +type: lemma +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The Yoneda bijection identifies natural transformations with omega-germ transformers via a tower-coherent family + of stage-k components. +aliases: +- LEM0023 +- II.L14 +- yoneda-application +- lem:yoneda-application +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005430 +- predicate: formalized_by + object: cid005431 +- predicate: formalized_by + object: cid005435 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Yoneda Application**\n\nThe Yoneda bijection identifies natural transformations with omega-germ transformers\ + \ via a tower-coherent family of stage-k components.\n\n# Yoneda Application\n\n## Summary\n\nThe Yoneda bijection identifies\ + \ natural transformations with omega-germ transformers via a tower-coherent family of stage-k components.\n\n## Statement\n\ + \n```latex\n%\n\\label{lem:yoneda-application}\nThere is a canonical bijection\n\\[\n \\boxed{%\n \\mathrm{Nat}\\\ + bigl(y(\\tau^3),\\, y(H_\\tau)\\bigr)\n \\;\\xleftrightarrow{\\;\\sim\\;}\n \\{\\textup{$\\omega$-germ transformers\ + \ on } \\tau^3\n \\textup{ valued in } H_\\tau\\}.}\n\\]\nUnder this bijection,\na natural transformation\n$\\eta \\\ + colon y(\\tau^3) \\to y(H_\\tau)$\ncorresponds to the $\\omega$-germ transformer~$G_\\eta$\ndefined by:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n At each stage~$k$,\n the component $G_{\\eta,k} \\colon\n \\mathbb{Z}/P_k\\\ + mathbb{Z} \\to H_\\tau / P_k H_\\tau$\n is defined by\n $G_{\\eta,k}(a) := \\pi_k\\bigl(\\eta_{\\mathbb{Z}/P_k\\\ + mathbb{Z}}(\\iota_k^a)\\bigr)$,\n where $\\iota_k^a \\colon \\mathbb{Z}/P_k\\mathbb{Z} \\to \\tau^3$\n \ + \ is the stage-$k$ probe selecting class~$a$.\n\n \\item[\\textup{(ii)}]\n The family $(G_{\\eta,k})_{k\ + \ \\geq 1}$\n is tower-coherent:\n the naturality square\n applied to the projection\n \ + \ $\\mathbb{Z}/P_{k+1}\\mathbb{Z}\n \\to \\mathbb{Z}/P_k\\mathbb{Z}$\n ensures\n $\\pi_{k+1\ + \ \\to k}(G_{\\eta,k+1}(b))\n = G_{\\eta,k}(\\pi_{k+1 \\to k}(b))$\n for all $b \\in \\mathbb{Z}/P_{k+1}\\\ + mathbb{Z}$.\n\n \\item[\\textup{(iii)}]\n The tower-coherent family\n $(G_{\\eta,k})_{k \\geq 1}$\n\ + \ defines an $\\omega$-germ transformer\n in the sense of\n Book~I (I.D47--I.D49):\n \ + \ a coherent system of stagewise maps\n whose inverse limit is a map\n $\\tau^3 \\to H_\\tau$ at\ + \ the $\\omega$-level.\n\\end{enumerate}\nConversely, every $\\omega$-germ transformer\n$G = (G_k)_{k \\geq 1}$\ndefines\ + \ a natural transformation\n$\\eta_G$ by reversing the construction:\n$(\\eta_G)_P(\\varphi) := G \\circ \\varphi$,\n\ + where the composition is interpreted stagewise.\n```\n\n## Proof / Justification\n\n```latex\nThe proof has two directions.\n\ + \n\\smallskip\n\\noindent\\textbf{Forward: natural transformation $\\to$ $\\omega$-germ transformer.}\nLet $\\eta \\colon\ + \ y(\\tau^3) \\to y(H_\\tau)$\nbe a natural transformation.\nBy Proposition~\\ref{prop:ch50-primorial-probes},\nevery\ + \ probe factors through a finite stage,\nso $\\eta$ is determined by its action\non stage-$k$ probes for all~$k$.\nAt\ + \ stage~$k$, the stage-$k$ probes\nare indexed by $a \\in \\mathbb{Z}/P_k\\mathbb{Z}$,\nso $\\eta$ at stage~$k$ defines\ + \ a function\n$G_{\\eta,k} \\colon \\mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau / P_k H_\\tau$.\nThe naturality square for\ + \ the projection\n$\\mathbb{Z}/P_{k+1}\\mathbb{Z} \\to \\mathbb{Z}/P_k\\mathbb{Z}$\ngives tower coherence (item~(ii)).\n\ + The resulting tower-coherent family\nis an $\\omega$-germ transformer (item~(iii)).\n\n\\smallskip\n\\noindent\\textbf{Backward:\ + \ $\\omega$-germ transformer $\\to$ natural transformation.}\nLet $G = (G_k)_{k \\geq 1}$\nbe an $\\omega$-germ transformer.\n\ + For any test object~$P$ and probe\n$\\varphi \\colon P \\to \\tau^3$,\nchoose $k$ such that $\\varphi$\nfactors through\ + \ stage~$k$\n(Proposition~\\ref{prop:ch50-primorial-probes}):\n$\\varphi = \\iota_k \\circ \\varphi_k$.\nDefine\n\\[\n\ + \ (\\eta_G)_P(\\varphi) := \\iota_k^{H_\\tau} \\circ G_k \\circ \\varphi_k,\n\\]\nwhere $\\iota_k^{H_\\tau}$\nis the\ + \ stage-$k$ inclusion into~$H_\\tau$.\nThis is well-defined (independent of the choice of~$k$)\nby tower coherence of~$G$.\n\ + Naturality of~$\\eta_G$\nfollows from the stagewise composition structure:\nfor $\\psi \\colon Q \\to P$,\n\\begin{align*}\n\ + \ (\\eta_G)_Q(\\varphi \\circ \\psi)\n &= G \\circ (\\varphi \\circ \\psi) \\\\\n &= (G \\circ \\varphi) \\circ\ + \ \\psi \\\\\n &= (\\eta_G)_P(\\varphi) \\circ \\psi,\n\\end{align*}\nwhere the middle step uses the associativity\n\ + of stagewise composition\n(Theorem~\\ref{thm:associativity}, II.T28,\nChapter~\\ref{ch:composition-structure}).\n\n\\\ + smallskip\n\\noindent\\textbf{Bijectivity.}\nThe two maps are inverse to each other:\nstarting from~$\\eta$, constructing~$G_\\\ + eta$,\nthen reconstructing $\\eta_{G_\\eta}$\nrecovers~$\\eta$ on all probes\n(because probes determine~$\\eta$ completely\n\ + by the Yoneda embedding, II.T36).\nStarting from~$G$, constructing~$\\eta_G$,\nthen extracting $G_{\\eta_G}$\nrecovers~$G$\ + \ at each stage\n(by the probe decomposition,\nProposition~\\ref{prop:ch50-primorial-probes}).\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 145\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch50-yoneda-applied.tex`\ + \ lines 276-330\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.YonedaApplied`\n\ + - Name: `Tau.BookII.CentralTheorem.yoneda_application_check`\n\n## Dependencies\n\n- Canonical: II.T35, II.L13, II.R12,\ + \ II.D50, II.P10\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Yoneda bijection identifies natural transformations with omega-germ transformers via a tower-coherent family + of stage-k components. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001342.md b/_corpus_v3_items/cid001342.md new file mode 100644 index 000000000..6d39bf85c --- /dev/null +++ b/_corpus_v3_items/cid001342.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001342/ +title: Lemniscate as Coordinate Limit +cid: cid001342 +primary_alias: PRP0047 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Lemniscate as Coordinate Limit +aliases: +- PRP0047 +- II.P01 +- lemniscate-as-coordinate-limit +- prop:lemniscate-coordinate-limit +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Lemniscate as Coordinate Limit**\n\nLemniscate as Coordinate Limit\n\n# Lemniscate as Coordinate Limit\n\n\ + ## Summary\n\nLemniscate as Coordinate Limit\n\n## Statement\n\n```latex\n%\n\\label{prop:lemniscate-coordinate-limit}\n\ + The algebraic lemniscate $\\mathbb{L}$ (I.D18)\nis isomorphic to the fiber readout of~$\\omega$:\n\\[\n \\mathbb{L}\n\ + \ \\;\\cong\\;\n \\operatorname{pr}_{\\mathrm{fiber}}(\\Phi_\\omega).\n\\]\nMore precisely:\nthe bipolar spectral\ + \ algebra $H_\\tau$\nwith crossing-point germ $\\omega_{\\mathbb{L}}$\nand polarity involution~$\\sigma$\n(I.D18) is canonically\ + \ isomorphic,\nas an algebra with involution,\nto the set of all fiber limits of paths to~$\\omega$,\nequipped with the\ + \ sector decomposition\ninherited from the $B/C$ dominance partition\nand the natural involution that swaps $B$-dominance\n\ + with $C$-dominance.\n```\n\n## Proof / Justification\n\n```latex\nThe isomorphism sends each point of~$\\mathbb{L}$ to\ + \ a fiber limit\nand vice versa.\nWe verify the three components.\n\n\\textbf{(i) Bipolar sectors.}\nThe two idempotent\ + \ sectors $e_+ H_\\tau$ and $e_- H_\\tau$\nof the bipolar spectral algebra (I.D27)\ncorrespond to $B$-dominated and $C$-dominated\ + \ fiber limits,\nrespectively.\nThis correspondence is natural:\nthe idempotent $e_+ = (1+j)/2$ projects onto the $B$-channel\n\ + (the $\\gamma$-orbit sector),\nand $e_- = (1-j)/2$ projects onto the $C$-channel\n(the $\\eta$-orbit sector).\nThe split-complex\ + \ structure $j^2 = +1$ (I.T10)\nis forced by the same bipolar partition\nthat governs the $B/C$ dominance\nin the fiber\ + \ readout.\n\n\\textbf{(ii) Crossing point.}\nThe crossing-point germ $\\omega_{\\mathbb{L}}$\ncorresponds to the fiber\ + \ limit of balanced paths\n(where $B/C \\to 1$).\nIn both descriptions, this is the unique point\nwhere neither sector\ + \ dominates:\nalgebraically, it is the identity element\nof the bipolar structure;\ncoordinately, it is the node where\ + \ the two lobes meet.\nThe primorial path itself\n--- with $B_n = C_n = 1$ for all~$n$ ---\nconverges to this crossing\ + \ point.\n\n\\textbf{(iii) Polarity involution.}\nThe involution $\\sigma \\colon j \\mapsto -j$ on $H_\\tau$\ncorresponds\ + \ to the involution on fiber limits\nthat swaps B-dominance and C-dominance:\n$\\sigma(e_+) = e_-$ and $\\sigma(e_-) =\ + \ e_+$.\nIn the fiber readout, this is the map\nthat sends a path with asymptotic ratio $B/C = r$\nto the path with asymptotic\ + \ ratio $C/B = 1/r$,\nexchanging the two lobes.\n\nSince both structures carry the same bipolar algebra,\nthe same crossing\ + \ point,\nand the same involution,\nthe canonical identification\n$\\mathbb{L} \\cong \\operatorname{pr}_{\\mathrm{fiber}}(\\\ + Phi_\\omega)$\nis an isomorphism of algebras with involution (I.P08).\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 10\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch05-omega-readout-lemniscate.tex`\ + \ lines 504-525\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.OmegaReadout`\n\ + - Name: `Tau.BookII.Interior.lemniscate_compat`\n\n## Dependencies\n\n- Canonical: II.T02, I.D18, I.D27\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Lemniscate as Coordinate Limit +noindex: false +--- + diff --git a/_corpus_v3_items/cid001343.md b/_corpus_v3_items/cid001343.md new file mode 100644 index 000000000..6f8ac62ee --- /dev/null +++ b/_corpus_v3_items/cid001343.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001343/ +title: Sector Inheritance +cid: cid001343 +primary_alias: PRP0048 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every tau-admissible point inherits a bipolar sector assignment from the boundary structure via fiber coordinates, + compatible with the idempotent decomposition of H_tau. +aliases: +- PRP0048 +- II.P02 +- sector-inheritance +- prop:sector-inheritance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 7 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sector Inheritance**\n\nEvery tau-admissible point inherits a bipolar sector assignment from the boundary structure\ + \ via fiber coordinates, compatible with the idempotent decomposition of H_tau.\n\n# Sector Inheritance\n\n## Summary\n\ + \nEvery tau-admissible point inherits a bipolar sector assignment from the boundary structure via fiber coordinates, compatible\ + \ with the idempotent decomposition of H_tau.\n\n## Statement\n\n```latex\n%\n\\label{prop:sector-inheritance}\nEvery\ + \ $\\tau$-admissible point $(D, A, B, C) \\in \\tau^3$\ninherits a bipolar sector assignment\nfrom the boundary structure:\n\ + \\begin{enumerate}\n \\item At each finite stage of the primorial ladder,\n the sector assignment is determined\ + \ by the\n fiber coordinates $(B, C)$ via the\n interior bipolar decomposition\n (Definition~\\\ + ref{def:interior-bipolar}).\n \\item The sector assignment is compatible with\n the idempotent decomposition\ + \ of $H_\\tau$:\n $\\Psi_{\\mathrm{int}} = e_+ \\cdot s_+ + e_- \\cdot s_-$\n with $e_+ \\cdot s_- =\ + \ 0$ and $e_- \\cdot s_+ = 0$.\n \\item At the $\\omega$-limit,\n the sector assignment recovers\n \ + \ the polarity character\n $\\tilde{\\chi} \\colon \\mathbb{P}_\\tau \\to \\{e_+, e_-\\}$\n of the\ + \ prime polarity theorem.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nPart~(1) is by construction:\n\ + Definition~\\ref{def:interior-bipolar} assigns\nsector components via the fiber coordinates.\nPart~(2) follows from the\ + \ idempotent properties\n$e_+^2 = e_+$, $e_-^2 = e_-$, $e_+ \\cdot e_- = 0$:\nprojecting $\\Psi_{\\mathrm{int}}$ by $e_+$\n\ + gives $e_+ \\cdot \\Psi_{\\mathrm{int}}\n= e_+^2 \\cdot \\Psi(B, A, D)\n= e_+ \\cdot \\Psi(B, A, D) = s_+$,\nand similarly\ + \ $e_- \\cdot \\Psi_{\\mathrm{int}} = s_-$.\nPart~(3): at the $\\omega$-limit,\nthe fiber coordinates stabilize\nby the\ + \ tail-stability of the polarity map\n(Book~I, Part~VI).\nThe B-dominant primes have $B \\gg C$ asymptotically,\nmapping\ + \ to $e_+$;\nthe C-dominant primes have $C \\gg B$ asymptotically,\nmapping to $e_-$.\nThis recovers the polarity character\ + \ $\\tilde{\\chi}$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 16\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch07-bipolar-interior.tex`\ + \ lines 277-299\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.BipolarDecomposition`\n\ + - Name: `Tau.BookII.Interior.sector_complete`\n\n## Dependencies\n\n- Canonical: II.D08, I.D21\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every tau-admissible point inherits a bipolar sector assignment from the boundary structure via fiber coordinates, + compatible with the idempotent decomposition of H_tau. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001344.md b/_corpus_v3_items/cid001344.md new file mode 100644 index 000000000..23937e236 --- /dev/null +++ b/_corpus_v3_items/cid001344.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001344/ +title: Four-Ray Rigidity +cid: cid001344 +primary_alias: PRP0049 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The four ABCD rays provide complete coordinates, bipolar compatibility, and coherence constraints through the fibered + product structure, establishing holomorphic rigidity without importing quaternionic algebra. +aliases: +- PRP0049 +- II.P03 +- four-ray-rigidity +- prop:four-ray-rigidity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 8 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Four-Ray Rigidity**\n\nThe four ABCD rays provide complete coordinates, bipolar compatibility, and coherence\ + \ constraints through the fibered product structure, establishing holomorphic rigidity without importing quaternionic\ + \ algebra.\n\n# Four-Ray Rigidity\n\n## Summary\n\nThe four ABCD rays provide complete coordinates, bipolar compatibility,\ + \ and coherence constraints through the fibered product structure, establishing holomorphic rigidity without importing\ + \ quaternionic algebra.\n\n## Statement\n\n```latex\n%\n\\label{prop:four-ray-rigidity}\nThe four ABCD rays $(A, B, C,\ + \ D)$\nprovide sufficient structure\nfor holomorphic rigidity on $\\tau^3$\nwithout importing quaternionic algebra.\n\ + Specifically:\n\\begin{enumerate}\n \\item \\textbf{Completeness.}\n The ABCD chart is a complete coordinate\ + \ system:\n every object of $\\tau$ receives a unique\n ABCD quadruple\n (by hyperfactorization,\ + \ I.T04).\n \\item \\textbf{Bipolar compatibility.}\n The fiber coordinates $(B, C)$\n carry the\ + \ bipolar sector assignment\n (Proposition~\\ref{prop:sector-inheritance}),\n which is compatible with\ + \ the\n idempotent decomposition of $H_\\tau$.\n \\item \\textbf{Coherence constraint.}\n The fibered\ + \ product structure\n $\\tau^3 = \\tau^1 \\times_f T^2$\n interlocks the four rays\n through\ + \ the peel-order coupling,\n providing the rigidity\n that determines holomorphic functions\n \ + \ up to the Identity Theorem.\n \\item \\textbf{No imports.}\n The four-ray structure is earned\n \ + \ entirely from the kernel axioms\n $\\KAxiom{0}$--$\\KAxiom{6}$,\n the five generators\n \ + \ $\\{\\alpha, \\pi, \\gamma, \\eta, \\omega\\}$,\n and the progression operator $\\rho$.\n No external\ + \ algebraic structure\n (quaternions, Clifford algebras,\n division algebras)\n is imported.\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nPart~(1) is the Hyperfactorization Theorem (I.T04).\nPart~(2)\ + \ is Proposition~\\ref{prop:sector-inheritance}\n(Chapter~\\ref{ch:bipolar-interior}).\nPart~(3): the fibered product\ + \ coupling\nmeans that a holomorphic function on $\\tau^3$\nmust satisfy coherence conditions\non both the base projection\ + \ $\\mathrm{pr} \\colon \\tau^3 \\to \\tau^1$\nand the fiber coordinates,\nwith the coupling ensuring\nthat base and fiber\ + \ conditions are not independent.\nThe Identity Theorem for $\\mathbb{L}$\n(Book~I, Part~XII)\npropagates: a holomorphic\ + \ function on $\\tau^3$\nthat vanishes on a sufficiently large\nsub-tower of the primorial system\nmust vanish identically.\n\ + Part~(4): the ABCD chart, prime polarity,\nsplit-complex algebra, and fibered product\nare all constructed in Book~I\n\ + from the kernel axioms and generators,\nwith no external imports.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 18\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch08-abcd-vs-quaternions.tex`\ + \ lines 370-410\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.ABCDRigidity`\n\ + - Name: `Tau.BookII.Interior.rigidity_2_to_30`\n\n## Dependencies\n\n- Canonical: I.D17, I.P08\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The four ABCD rays provide complete coordinates, bipolar compatibility, and coherence constraints through the + fibered product structure, establishing holomorphic rigidity without importing quaternionic algebra. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001345.md b/_corpus_v3_items/cid001345.md new file mode 100644 index 000000000..ec60dd65f --- /dev/null +++ b/_corpus_v3_items/cid001345.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001345/ +title: Cylinders Are Balls +cid: cid001345 +primary_alias: PRP0050 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Stage-k cylinders equal both closed and open ultrametric balls, so cylinders and balls are identical objects in the + cylinder topology. +aliases: +- PRP0050 +- II.P04 +- cylinders-are-balls +- prop:cylinders-are-balls +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cylinders Are Balls**\n\nStage-k cylinders equal both closed and open ultrametric balls, so cylinders and balls\ + \ are identical objects in the cylinder topology.\n\n# Cylinders Are Balls\n\n## Summary\n\nStage-k cylinders equal both\ + \ closed and open ultrametric balls, so cylinders and balls are identical objects in the cylinder topology.\n\n## Statement\n\ + \n```latex\n%\n\\label{prop:cylinders-are-balls}\nFor every $x \\in \\tau^3$ and every stage $k \\geq 0$,\nthe stage-$k$\ + \ cylinder $C_k(x)$\n(Definition~\\ref{def:stage-k-cylinder})\nis simultaneously a closed ball and an open ball\nin the\ + \ ultrametric~$d$:\n\\[\n \\boxed{C_k(x)\n \\;=\\;\n \\overline{B}(x, 2^{-k})\n \\;=\\;\n B(x, 2^{-(k-1)})}\n\ + \\]\nwhere $\\overline{B}(x, r) = \\{y : d(x,y) \\leq r\\}$\nis the closed ball\nand $B(x, r) = \\{y : d(x,y) < r\\}$\n\ + is the open ball.\n```\n\n## Proof / Justification\n\n```latex\nBy Definition~\\ref{def:stage-k-cylinder},\n$y \\in C_k(x)$\ + \ if and only if\n$\\pi_k(y) = \\pi_k(x)$,\nwhich holds if and only if\n$\\delta(x,y) \\geq k$.\n\n\\emph{Closed ball.}\n\ + $\\delta(x,y) \\geq k$ is equivalent to\n$d(x,y) = 2^{-\\delta(x,y)} \\leq 2^{-k}$,\nso $C_k(x) = \\overline{B}(x, 2^{-k})$.\n\ + \n\\emph{Open ball.}\nSince $d$ takes values in $\\{0\\} \\cup \\{2^{-n} : n \\geq 0\\}$\nand $2^{-(k-1)}$ is the next\ + \ value above $2^{-k}$\nin this discrete set,\nthe condition $d(x,y) \\leq 2^{-k}$\nis equivalent to $d(x,y) < 2^{-(k-1)}$.\n\ + Hence $C_k(x) = B(x, 2^{-(k-1)})$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 26\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch10-ultrametric-depth.tex` lines 295-314\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Ultrametric`\n- Name:\ + \ `cyl_eq_ball_check`\n\n## Dependencies\n\n- Canonical: II.D10, II.D13\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Stage-k cylinders equal both closed and open ultrametric balls, so cylinders and balls are identical objects + in the cylinder topology. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001346.md b/_corpus_v3_items/cid001346.md new file mode 100644 index 000000000..caffdfa0c --- /dev/null +++ b/_corpus_v3_items/cid001346.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001346/ +title: Lobes as Clopen Sets +cid: cid001346 +primary_alias: PRP0051 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Lobes as Clopen Sets +aliases: +- PRP0051 +- II.P05 +- lobes-as-clopen-sets +- prop:lobes-clopen +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Lobes as Clopen Sets**\n\nLobes as Clopen Sets\n\n# Lobes as Clopen Sets\n\n## Summary\n\nLobes as Clopen Sets\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:lobes-clopen}\nLet $p_0$ denote the node (wedge point) of~$\\mathbb{L}$.\n\ + The two lobes are topologically complementary\n\\textbf{clopen} subsets of $\\mathbb{L} \\setminus \\{p_0\\}$:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n $L_+ := S^1_\\gamma \\setminus \\{p_0\\}$\n and\n $L_- := S^1_\\\ + eta \\setminus \\{p_0\\}$\n are each open and closed\n in $\\mathbb{L} \\setminus \\{p_0\\}$.\n \\\ + item[\\textup{(ii)}]\n $L_+ \\cap L_- = \\varnothing$\n and $L_+ \\cup L_- = \\mathbb{L} \\setminus\ + \ \\{p_0\\}$.\n \\item[\\textup{(iii)}]\n Each lobe is the profinite limit\n of angular sectors:\n\ + \ $L_+ = \\bigcap_{k \\geq 1}\n \\overline{S^+_k(\\textup{B-dom})}$\n and\n $L_- =\ + \ \\bigcap_{k \\geq 1}\n \\overline{S^-_k(\\textup{C-dom})}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\n\\emph{Clauses~(i)--(ii).}\nThe space $\\mathbb{L} \\setminus \\{p_0\\}$\nis the disjoint union of two punctured\ + \ circles.\nEach is connected, hence both open and closed\nin the disjoint-union topology.\nDisjointness holds because\ + \ the two circles\nshare only~$p_0$, which has been removed.\n\n\\emph{Clause~(iii).}\nAt stage~$k$, the angular sector\ + \ $S^+_k(b)$\ncontains points with $B$-coordinate equal to $b$.\nThe B-dominant boundary points are characterized\nby\ + \ eventual dominance of $B_k$ over $C_k$\n(I.T05).\nThe profinite limit over all stages\nselects precisely the B-dominant\ + \ omega-germs,\nconstituting the $e_+$-lobe.\nThe C-lobe argument is symmetric.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 41\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch16-boundary-minimality.tex`\ + \ lines 224-249\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.BoundaryMinimality`\n\ + - Name: `lobes_clopen_check`\n\n## Dependencies\n\n- Canonical: II.T12, II.D17\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Lobes as Clopen Sets +noindex: false +--- + diff --git a/_corpus_v3_items/cid001347.md b/_corpus_v3_items/cid001347.md new file mode 100644 index 000000000..aa56debdb --- /dev/null +++ b/_corpus_v3_items/cid001347.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001347/ +title: Refinement Resolution Bound +cid: cid001347 +primary_alias: PRP0052 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Refinement Resolution Bound +aliases: +- PRP0052 +- II.P06 +- refinement-resolution-bound +- prop:refinement-resolution +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 28 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Refinement Resolution Bound**\n\nRefinement Resolution Bound\n\n# Refinement Resolution Bound\n\n## Summary\n\ + \nRefinement Resolution Bound\n\n## Statement\n\n```latex\n%\n\\label{prop:refinement-resolution}\nLet $x, y \\in \\tau^3$\ + \ be $\\tau$-admissible points\nwith first disagreement depth\n$\\delta(x,y) = k$.\nThen:\n\\begin{enumerate}\n \\\ + item[\\textup{(i)}]\n \\textbf{Ultrametric bound.}\n $d(x,y) = 2^{-k}$.\n\n \\item[\\textup{(ii)}]\n\ + \ \\textbf{Euclidean bound.}\n $d_{\\mathrm{Euc}}\\bigl(\n \\mathrm{den}(x),\\, \\mathrm{den}(y)\n\ + \ \\bigr)\n \\;\\leq\\;\n 4 / P_k$,\n where $P_k = p_1 \\cdots p_k$\n is\ + \ the $k$th primorial.\n\n \\item[\\textup{(iii)}]\n \\textbf{Conversion.}\n The ratio of the two\ + \ bounds satisfies\n \\[\n \\frac{4/P_k}{2^{-k}}\n \\;=\\;\n \\frac{4\ + \ \\cdot 2^k}{P_k}\n \\;\\xrightarrow{k \\to \\infty}\\;\n 0,\n \\]\n because\ + \ $P_k$ grows super-exponentially\n \\textup{(}$P_k = e^{(1+o(1))\\, p_k}$\n by Chebyshev's estimates\\\ + textup{)}.\n The Euclidean bound is eventually\n much tighter than the ultrametric bound:\n \ + \ Archimedean resolution ``catches up''\n to non-Archimedean refinement\n and eventually surpasses\ + \ it.\n\n \\item[\\textup{(iv)}]\n \\textbf{Asymptotic coupling.}\n For large~$k$, the effective\ + \ conversion\n between ultrametric depth~$k$\n and Euclidean precision~$1/P_k$ is:\n \\[\n\ + \ -\\log_2(1/P_k)\n \\;=\\;\n \\log_2 P_k\n \\;\\sim\\;\n \ + \ \\frac{p_k}{\\ln 2}\n \\;\\sim\\;\n \\frac{k \\ln k}{\\ln 2},\n \\]\n \ + \ where the last step uses the prime number theorem\n $p_k \\sim k \\ln k$.\n The ultrametric depth~$k$\n\ + \ translates to $\\sim k \\ln k / \\ln 2$\n bits of Euclidean precision.\n\\end{enumerate}\n```\n\n\ + ## Proof / Justification\n\n```latex\n\\textbf{(i)}\nBy definition of the ultrametric\n(II.D13, Chapter~\\ref{ch:ultrametric-depth}).\n\ + \n\\textbf{(ii)}\nProved in Chapter~\\ref{ch:orthodox-bridge},\nProposition~\\ref{prop:ch22-cauchy}:\neach ABCD component\ + \ contributes\nat most $1/P_k$ to the Euclidean distance,\nand there are four components.\n\n\\textbf{(iii)}\n$P_k = \\\ + prod_{j=1}^k p_j$.\nBy the prime number theorem,\n$\\ln P_k = \\sum_{j=1}^k \\ln p_j = \\vartheta(p_k) \\sim p_k$,\nso\ + \ $P_k \\sim e^{p_k}$.\nMeanwhile $2^k$ grows exponentially in~$k$.\nSince $p_k \\sim k \\ln k \\gg k$,\n$P_k$ grows super-exponentially\ + \ in~$k$,\nand $4 \\cdot 2^k / P_k \\to 0$.\n\n\\textbf{(iv)}\n$\\log_2 P_k = (\\ln P_k)/(\\ln 2) \\sim p_k / \\ln 2\n\ + \\sim k \\ln k / \\ln 2$\nby the prime number theorem.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 79\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch28-iota-tau-confirmed.tex`\ + \ lines 395-456\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.IotaTauConfirmed`\n\ + - Name: `iota_universal_check`\n\n## Dependencies\n\n- Canonical: II.D34, II.T25\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Refinement Resolution Bound +noindex: false +--- + diff --git a/_corpus_v3_items/cid001348.md b/_corpus_v3_items/cid001348.md new file mode 100644 index 000000000..dacbb5c59 --- /dev/null +++ b/_corpus_v3_items/cid001348.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001348/ +title: Bipolar Channel Independence +cid: cid001348 +primary_alias: PRP0053 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Bipolar Channel Independence +aliases: +- PRP0053 +- II.P07 +- bipolar-channel-independence +- prop:bipolar-channel-independence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 30 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bipolar Channel Independence**\n\nBipolar Channel Independence\n\n# Bipolar Channel Independence\n\n## Summary\n\ + \nBipolar Channel Independence\n\n## Statement\n\n```latex\n%\n\\label{prop:bipolar-channel-independence}\nThe two sector\ + \ lifts\n$\\mathrm{BndLift}_n^{(+)}$ and $\\mathrm{BndLift}_n^{(-)}$\nare \\textbf{informationally independent}:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n \\textbf{Algebraic independence.}\n The lift increment $\\Delta_n^{(+)}$\n\ + \ is determined solely by the B-sector data\n $f_n^{(+)} = e_+ \\cdot f_n$.\n The lift increment\ + \ $\\Delta_n^{(-)}$\n is determined solely by the C-sector data\n $f_n^{(-)} = e_- \\cdot f_n$.\n \ + \ Neither increment depends on the other sector's data.\n\n \\item[\\textup{(ii)}]\n \\textbf{Norm\ + \ independence.}\n The calibrated norm of the lift factorizes:\n \\[\n \\|\\mathrm{BndLift}_n(f_n)\\\ + |_\\tau\n \\;=\\;\n \\bigl\\|\n \\mathrm{BndLift}_n^{(+)}(f_n^{(+)})\n \ + \ \\bigr\\|^{\\pi/(\\pi+e)}\n \\;\\cdot\\;\n \\bigl\\|\n \\mathrm{BndLift}_n^{(-)}(f_n^{(-)})\n\ + \ \\bigr\\|^{e/(\\pi+e)}.\n \\]\n The B-channel contributes a $\\pi$-weighted factor;\n\ + \ the C-channel contributes an $e$-weighted factor.\n\n \\item[\\textup{(iii)}]\n \\textbf{Information\ + \ capacity.}\n At stage $n+1$,\n each sector carries $\\log_2 p_{n+1}$ new bits\n of information.\n\ + \ The total information increment\n is $2 \\log_2 p_{n+1}$ bits\n (one copy per sector).\n\ + \ The two copies are independently determined:\n knowing the B-sector increment\n gives zero\ + \ information about the C-sector increment,\n and conversely.\n\n \\item[\\textup{(iv)}]\n \\textbf{Coupling\ + \ structure.}\n The only coupling between the two sectors\n is the norm constraint\n $\\|\\\ + Delta_n^{(+)}\\| = \\iota_\\tau \\cdot \\|\\Delta_n^{(-)}\\|$\n \\textup{(}Definition~\\textup{\\ref{def:bndlift}(iv))}.\n\ + \ This constrains the \\textbf{magnitudes}\n of the increments\n but not their \\textbf{values}:\n\ + \ the direction of $\\Delta_n^{(+)}$\n is independent of $\\Delta_n^{(-)}$.\n\\end{enumerate}\n```\n\ + \n## Proof / Justification\n\n```latex\n\\textbf{(i)}\nThe CRT decomposition\nis applied independently to each ABCD component.\n\ + The B-coordinate maps to the $e_+$-sector\nand the C-coordinate maps to the $e_-$-sector\n(by the spectral decomposition\ + \ of~$\\jj$,\nII.T24, Chapter~\\ref{ch:j-replaces-i}).\nSince the CRT isomorphism\n$\\mathbb{Z}/P_{n+1}\\mathbb{Z}\n\\\ + cong \\mathbb{Z}/P_n\\mathbb{Z} \\times \\mathbb{Z}/p_{n+1}\\mathbb{Z}$\nacts componentwise on the ABCD channels,\nthe\ + \ B-sector lift depends only on B-data,\nand the C-sector lift depends only on C-data.\n\n\\textbf{(ii)}\nThe calibrated\ + \ norm is\n$\\|z\\|_\\tau = |z_+|^{\\pi/(\\pi+e)} \\cdot |z_-|^{e/(\\pi+e)}$\n(Definition~\\ref{def:calibrated-H-tau},\ + \ II.D35).\nSince the lift factorizes sectorwise,\n$(f_{n+1})_+ = \\mathrm{BndLift}_n^{(+)}(f_n^{(+)})$\nand\n$(f_{n+1})_-\ + \ = \\mathrm{BndLift}_n^{(-)}(f_n^{(-)})$.\nSubstituting into the norm formula\ngives the stated factorization.\n\n\\\ + textbf{(iii)}\nThe $p_{n+1}$-component of each sector\nis a function\n$\\Delta_n^{(\\pm)} : \\mathbb{Z}/p_{n+1}\\mathbb{Z}\ + \ \\to \\mathbb{Z}$.\nOnce the diagonal discipline and holomorphicity\nconstraints are imposed,\n$\\Delta_n^{(\\pm)}$\ + \ is uniquely determined\n(Theorem~\\ref{thm:bndlift-existence}(ii)).\nIn the information-theoretic sense,\nthe $p_{n+1}$\ + \ residue classes\ncontribute $\\log_2 p_{n+1}$ bits per sector.\nThe two sectors are CRT-orthogonal\n(by the product\ + \ structure\n$\\mathbb{R}[\\jj] \\cong \\mathbb{R} \\times \\mathbb{R}$),\nso the mutual information is zero.\n\n\\textbf{(iv)}\n\ + The coupling\n$\\|\\Delta_n^{(+)}\\| = \\iota_\\tau \\cdot \\|\\Delta_n^{(-)} \\|$\nconstrains the ratio of the $\\ell^2$\ + \ norms.\nBut the $\\ell^2$ norm is a scalar invariant:\nit constrains the magnitude\nwithout constraining the direction\n\ + (the specific values at each residue class).\nThe direction of $\\Delta_n^{(+)}$\nis determined by the B-sector holomorphicity;\n\ + the direction of $\\Delta_n^{(-)}$\nis determined by the C-sector holomorphicity.\nThese two holomorphicity conditions\n\ + are independent.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 84\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch30-bndlift-construction.tex`\ + \ lines 502-561\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.BndLift`\n\ + - Name: `Tau.BookII.Hartogs.bipolar_channel_independence`\n\n## Dependencies\n\n- Canonical: I.D21, I.T10, I.T18, II.D33,\ + \ II.D36, II.T26\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Bipolar Channel Independence +noindex: false +--- + diff --git a/_corpus_v3_items/cid001349.md b/_corpus_v3_items/cid001349.md new file mode 100644 index 000000000..0c1ded2e1 --- /dev/null +++ b/_corpus_v3_items/cid001349.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001349/ +title: Projection Formula +cid: cid001349 +primary_alias: PRP0054 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The spectral coefficient at prime p and residue class v is recovered by discrete Fourier average over the fiber, + projected to the appropriate bipolar channel via the idempotent. +aliases: +- PRP0054 +- II.P08 +- projection-formula +- prop:projection-formula +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005543 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Projection Formula**\n\nThe spectral coefficient at prime p and residue class v is recovered by discrete Fourier\ + \ average over the fiber, projected to the appropriate bipolar channel via the idempotent.\n\n# Projection Formula\n\n\ + ## Summary\n\nThe spectral coefficient at prime p and residue class v is recovered by discrete Fourier average over the\ + \ fiber, projected to the appropriate bipolar channel via the idempotent.\n\n## Statement\n\n```latex\n%\n\\label{prop:projection-formula}\n\ + Let $f_k : \\mathbb{Z}/P_k\\mathbb{Z} \\to H_\\tau$\nbe the stage-$k$ component\nof a $\\tau$-holomorphic function,\n\ + and let $p \\mid P_k$ be a prime factor.\nThe spectral coefficient\n$\\varphi_{p,v}^{(\\sigma)}$\nin the expansion\nof~$f_k$\ + \ with respect to the cylinder generator\n$E_{k,v}^{(\\sigma)}$ is\n\\[\n \\boxed{%\n \\varphi_{p,v}^{(\\sigma)}\n\ + \ \\;=\\;\n \\frac{1}{|F_p|}\n \\sum_{x \\in F_p(v)}\n e_\\sigma \\cdot f_k(x),}\n\\]\nwhere\n$F_p(v) := \\\ + {x \\in \\mathbb{Z}/P_k\\mathbb{Z}\n: x \\equiv v \\pmod{p}\\}$\nis the fiber over the residue class~$v$,\n$|F_p| = P_k\ + \ / p$ is the fiber cardinality,\nand $e_\\sigma$ is the channel idempotent\n($e_B = e_+$, $e_C = e_-$).\n```\n\n## Proof\ + \ / Justification\n\n```latex\nThe CRT isomorphism (I.T18, Book~I)\ngives\n$\\mathbb{Z}/P_k\\mathbb{Z}\n\\cong \\mathbb{Z}/p\\\ + mathbb{Z}\n\\times \\mathbb{Z}/(P_k/p)\\mathbb{Z}$.\nThe fiber $F_p(v)$ is the set of elements\nwhose projection to the\ + \ first factor is~$v$.\nBy CRT independence,\n$|F_p(v)| = P_k/p$ for every $v \\in \\mathbb{Z}/p\\mathbb{Z}$.\n\nProjecting\ + \ $f_k$ to the $\\sigma$-channel\nvia $e_\\sigma$ extracts the component\n$f_k^{(\\sigma)} = e_\\sigma \\cdot f_k$.\n\ + On the fiber $F_p(v)$,\nthe cylinder generator $E_{k,v}^{(\\sigma)}$\nis constantly equal to~$e_\\sigma$,\nand all other\ + \ generators\n$E_{k,w}^{(\\sigma)}$ with $w \\neq v$\n(modulo~$p$) vanish.\nSumming $e_\\sigma \\cdot f_k(x)$\nover $x\ + \ \\in F_p(v)$\nisolates the contribution of the $(p, v, \\sigma)$ component.\nDividing by $|F_p| = P_k/p$ gives the coefficient.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 104\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch35-canonical-basis.tex`\ + \ lines 496-523\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CanonicalBasis`\n\ + - Name: `projection_recovery_check`\n\n## Dependencies\n\n- Canonical: II.D46, I.T18\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The spectral coefficient at prime p and residue class v is recovered by discrete Fourier average over the fiber, + projected to the appropriate bipolar channel via the idempotent. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001350.md b/_corpus_v3_items/cid001350.md new file mode 100644 index 000000000..ac4fe611e --- /dev/null +++ b/_corpus_v3_items/cid001350.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001350/ +title: Decomposition Functoriality +cid: cid001350 +primary_alias: PRP0055 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Decomposition Functoriality +aliases: +- PRP0055 +- II.P09 +- decomposition-functoriality +- prop:decomposition-functoriality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005549 +- predicate: formalized_by + object: cid005550 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 37 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Decomposition Functoriality**\n\nDecomposition Functoriality\n\n# Decomposition Functoriality\n\n## Summary\n\ + \nDecomposition Functoriality\n\n## Statement\n\n```latex\n%\n\\label{prop:decomposition-functoriality}\nLet $f, g \\\ + in \\mathcal{O}_\\tau(\\tau^3)$\nbe $\\tau$-holomorphic,\nand suppose $g \\circ f$ is defined\nand $\\tau$-holomorphic.\n\ + Then:\n\\begin{enumerate}\n \\item[\\textup{(F1)}]\n \\textbf{Channel preservation:}\n $(g \\circ\ + \ f)_+ = g_+ \\circ f_+$\n and $(g \\circ f)_- = g_- \\circ f_-$.\n \\item[\\textup{(F2)}]\n \\textbf{No\ + \ channel mixing:}\n the B-channel of $g \\circ f$\n depends only on the B-channels\n of $g$\ + \ and $f$;\n the C-channel of $g \\circ f$\n depends only on the C-channels\n of $g$ and $f$.\n\ + \ \\item[\\textup{(F3)}]\n \\textbf{Functor property:}\n the map $f \\mapsto f_+$\n is a\ + \ functor from the category\n of $\\tau$-holomorphic maps\n to the category of B-channel maps,\n \ + \ and similarly for $f \\mapsto f_-$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Proof\ + \ of (F1).}\nWe compute $(g \\circ f)_+$ at an arbitrary point $x \\in \\tau^3$:\n\\begin{align*}\n (g \\circ f)_+(x)\n\ + \ &\\;=\\; e_+ \\cdot (g \\circ f)(x) \\\\\n &\\;=\\; e_+ \\cdot g(f(x)).\n\\end{align*}\nNow write $f(x) = f_+(x)\ + \ + f_-(x)$\nand $g = g_+ + g_-$.\nSince $g_+$ takes values in $e_+ \\cdot H_\\tau$\nand $g_-$ takes values in $e_- \\\ + cdot H_\\tau$,\nand $f_+(x) \\in e_+ \\cdot H_\\tau$,\n$f_-(x) \\in e_- \\cdot H_\\tau$:\n\\begin{align*}\n g(f(x))\n\ + \ &= g_+(f_+(x) + f_-(x))\n + g_-(f_+(x) + f_-(x)).\n\\end{align*}\nThe key observation is that\nthe channels $e_+\ + \ \\cdot H_\\tau$ and $e_- \\cdot H_\\tau$\nform a \\emph{direct product of rings}:\n\\[\n H_\\tau \\;\\cong\\; e_+\ + \ H_\\tau \\;\\times\\; e_- H_\\tau.\n\\]\nUnder this isomorphism,\na map $g = g_+ + g_-$\nacts \\emph{componentwise}:\n\ + \\[\n g(z_+, z_-) = (g_+(z_+),\\; g_-(z_-)),\n\\]\nwhere $z_+ = e_+ z$ and $z_- = e_- z$.\nThis is because $g_+$ takes\ + \ values in $e_+ H_\\tau$\nand receives inputs from $e_+ H_\\tau$\n(the $e_- H_\\tau$ component is annihilated\nby orthogonality\ + \ (I2)),\nand similarly for $g_-$.\n\nExplicitly:\nthe evaluation $g_+(f(x))$\ndepends only on the $e_+$-component of\ + \ $f(x)$,\nwhich is $f_+(x)$,\nbecause $g_+$ takes values in $e_+ H_\\tau$\nand $e_+ \\cdot f_-(x) = 0$ by orthogonality.\n\ + Therefore $g_+(f(x)) = g_+(f_+(x))$.\nSimilarly $g_-(f(x)) = g_-(f_-(x))$.\n\nHence:\n\\[\n g(f(x))\n = g_+(f_+(x))\ + \ + g_-(f_-(x)),\n\\]\nand applying $e_+$:\n\\[\n e_+ \\cdot g(f(x))\n = e_+ g_+(f_+(x)) + e_+ g_-(f_-(x))\n \ + \ = g_+(f_+(x)) + 0\n = g_+(f_+(x))\n = (g_+ \\circ f_+)(x).\n\\]\nTherefore $(g \\circ f)_+ = g_+ \\circ f_+$.\n\ + The argument for the C-channel is identical.\n\n\\emph{Proof of (F2).}\nThis is a restatement of~(F1):\n$(g \\circ f)_+$\ + \ depends on $g_+$ and $f_+$ only,\nnot on $g_-$ or $f_-$.\nThe channels do not communicate under composition.\n\n\\emph{Proof\ + \ of (F3).}\nWe verify the functor axioms.\n\\emph{Identity:}\n$(\\id)_+ = e_+ \\cdot \\id = e_+$,\nwhich acts as the\ + \ identity on $e_+ H_\\tau$.\n\\emph{Composition:}\n$(g \\circ f)_+ = g_+ \\circ f_+$ by (F1).\nHence $f \\mapsto f_+$\ + \ is a functor.\nThe argument for $f \\mapsto f_-$ is the same.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 110\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch37-idempotent-decomposition.tex`\ + \ lines 741-770\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.IdempotentDecomposition`\n\ + - Name: `decompose_functoriality_check`\n\n## Dependencies\n\n- Canonical: I.D21, I.T10, II.D48, II.L07\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Decomposition Functoriality +noindex: false +--- + diff --git a/_corpus_v3_items/cid001351.md b/_corpus_v3_items/cid001351.md new file mode 100644 index 000000000..7abc7832f --- /dev/null +++ b/_corpus_v3_items/cid001351.md @@ -0,0 +1,100 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001351/ +title: Functions as Tau-Objects +cid: cid001351 +primary_alias: PRP0056 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Functions as Tau-Objects +aliases: +- PRP0056 +- II.P10 +- functions-as-tau-objects +- prop:functions-as-objects +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005659 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 40 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Functions as Tau-Objects**\n\nFunctions as Tau-Objects\n\n# Functions as Tau-Objects\n\n## Summary\n\nFunctions\ + \ as Tau-Objects\n\n## Statement\n\n```latex\n%\n\\label{prop:functions-as-objects}\nThe Pre-Yoneda embedding $y$\npreserves\ + \ the following structures:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Bipolar decomposition.}\n\ + \ For every $f \\in \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau)$:\n \\[\n y(f)\n \ + \ \\;=\\;\n e_+ \\cdot y(f_+)\n \\;+\\;\n e_- \\cdot y(f_-),\n \\]\n\ + \ where $f = e_+ \\cdot f_+ + e_- \\cdot f_-$\n is the Idempotent Decomposition\n \\textup{(II.L07,\ + \ Chapter~\\ref{ch:idempotent-decomposition})}.\n\n \\item[\\textup{(ii)}]\n \\textbf{Regularity.}\n \ + \ A holomorphic function $f$ is $\\tau$-regular\n at a point $p \\in \\tau^3$\n \\textup{(Definition~\\\ + ref{def:tau-regularity}, II.D49,\n Chapter~\\ref{ch:regularity-positive})}\n if and only if $y(f)$ stabilizes\ + \ at~$p$:\n the $\\omega$-germ $y(f)$\n has a well-defined limiting value\n at the point~$p$\n\ + \ in the profinite topology.\n\n \\item[\\textup{(iii)}]\n \\textbf{ABCD coordinates.}\n \ + \ The image $y(f)$ inherits all four\n ABCD coordinates from $f$:\n the address data (A),\n \ + \ the B-channel exponent data,\n the C-channel tetration data,\n and the depth data (D).\n \ + \ A holomorphic function \\textbf{has a position}\n in~$\\tau^3$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\n\\emph{Clause~(i): Bipolar decomposition.}\nLet $f \\in \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau)$.\nBy the Idempotent\ + \ Decomposition Lemma (II.L07),\n$f = e_+ \\cdot f_+ + e_- \\cdot f_-$,\nwhere $f_\\pm = e_\\pm \\cdot f$\nare themselves\ + \ $\\tau$-holomorphic.\nThe $\\omega$-germ transformer respects this decomposition:\nat each stage~$k$,\n\\[\n (G_f)_k\n\ + \ \\;=\\;\n e_+ \\cdot (G_{f_+})_k\n \\;+\\;\n e_- \\cdot (G_{f_-})_k,\n\\]\nbecause the idempotent projection\n\ + commutes with the stage-$k$ evaluation\n(the idempotents $e_\\pm$ are elements of $H_\\tau$,\nwhich is the codomain at\ + \ every stage).\nPassing to $\\omega$-germ equivalence classes:\n\\[\n [G_f]\n \\;=\\;\n e_+ \\cdot [G_{f_+}]\n\ + \ \\;+\\;\n e_- \\cdot [G_{f_-}],\n\\]\nwhich is $y(f) = e_+ \\cdot y(f_+) + e_- \\cdot y(f_-)$.\n\n\\smallskip\n\ + \\emph{Clause~(ii): Regularity.}\nBy the positive regularity definition\n(II.D49, Chapter~\\ref{ch:regularity-positive}),\n\ + $f$ is $\\tau$-regular at~$p$\nif and only if the stage-$k$ values $f_k(p_k)$\nstabilize: there exists $N$ such that\n\ + $f_k(p_k) = f_\\ell(p_\\ell)$\nunder the natural identification\nfor all $k, \\ell \\geq N$.\nBut the stage-$k$ values\ + \ of $f$ at~$p$\nare exactly the stage-$k$ values\nof the $\\omega$-germ transformer $G_f$ evaluated at~$p$.\nTherefore\ + \ $f$ is regular at~$p$\nif and only if the $\\omega$-germ $[G_f] = y(f)$\nstabilizes at~$p$.\n\nThe regularity criterion\ + \ (II.T34,\nChapter~\\ref{ch:regularity-positive})\ngives the equivalence explicitly:\n$f$ is regular $\\Leftrightarrow$\n\ + $f$ has a canonical extension to the interior\n$\\Leftrightarrow$ $y(f)$\nhas a well-defined profinite limit at~$p$.\n\ + \n\\smallskip\n\\emph{Clause~(iii): ABCD coordinates.}\nThe ABCD structure is inherited stagewise.\nThe A-coordinate (address)\ + \ of $y(f)$\nis the address of the stabilization point\nof the $\\omega$-germ:\nthe profinite limit of the tower-coherent\ + \ sequence\n$(G_f)_k$.\nThe B-coordinate (exponent channel)\nis $e_+ \\cdot y(f)$;\nthe C-coordinate (tetration channel)\n\ + is $e_- \\cdot y(f)$;\nand the D-coordinate (depth)\nis the stabilization stage---the\nsmallest $N$ such that\nthe $\\\ + omega$-germ has stopped acquiring\nnew spectral components.\nEach coordinate is well-defined\nbecause $f$ is $\\tau$-holomorphic\n\ + and therefore satisfies tower coherence\nand finite spectral support\n(II.T32, Chapter~\\ref{ch:sheaf-coherence}).\n```\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 119\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch40-pre-yoneda.tex`\ + \ lines 299-342\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.PreYoneda`\n\ + - Name: `preyoneda_bipolar_check`\n\n## Dependencies\n\n- Canonical: II.D50, II.L07, II.T34, II.D49\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Functions as Tau-Objects +noindex: false +--- + diff --git a/_corpus_v3_items/cid001352.md b/_corpus_v3_items/cid001352.md new file mode 100644 index 000000000..2128412f6 --- /dev/null +++ b/_corpus_v3_items/cid001352.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001352/ +title: Hom Bipolar Decomposition +cid: cid001352 +primary_alias: PRP0057 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Hom Bipolar Decomposition +aliases: +- PRP0057 +- II.P11 +- hom-bipolar-decomposition +- prop:hom-bipolar +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005667 +- predicate: formalized_by + object: cid005668 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 42 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Hom Bipolar Decomposition**\n\nHom Bipolar Decomposition\n\n# Hom Bipolar Decomposition\n\n## Summary\n\nHom\ + \ Bipolar Decomposition\n\n## Statement\n\n```latex\n%\n\\label{prop:hom-bipolar}\nFor any $A, B \\in \\mathrm{Obj}(\\\ + tau)$,\nthe Hom object $[A,B]$ decomposes as\n\\[\n \\boxed{%\n [A,B]\n \\;=\\;\n e_+ \\cdot [A,B]_+\n \ + \ \\;+\\;\n e_- \\cdot [A,B]_-,}\n\\]\nwhere:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n $[A,B]_+ :=\ + \ e_+ \\cdot [A,B]$\n is the \\textbf{B-channel component}---the\n space of B-channel projections\n\ + \ of morphisms from $A$ to $B$.\n\n \\item[\\textup{(ii)}]\n $[A,B]_- := e_- \\cdot [A,B]$\n \ + \ is the \\textbf{C-channel component}---the\n space of C-channel projections.\n\n \\item[\\textup{(iii)}]\n\ + \ The two components are \\textbf{independent}:\n modifying $[A,B]_+$\n does not affect $[A,B]_-$,\n\ + \ and vice versa.\n\n \\item[\\textup{(iv)}]\n The decomposition is\n \\textbf{functorial\ + \ in both arguments}:\n for any $\\tau$-morphism\n $h : B \\to B'$,\n the postcomposition map\n\ + \ $h_* : [A,B] \\to [A,B']$\n respects the bipolar decomposition,\n i.e., $h_*([A,B]_\\pm)\ + \ \\subseteq [A,B']_\\pm$.\n Similarly for precomposition.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n```latex\nBy Proposition~\\ref{prop:ch42-hom-tau-objects},\n$[A,B]$ is a $\\tau$-object.\nThe Idempotent Decomposition\ + \ Lemma\n(Lemma~\\ref{lem:idempotent-decomposition}, II.L07,\nChapter~\\ref{ch:idempotent-decomposition})\nstates that\ + \ every $\\tau$-object decomposes\ncanonically as $X = e_+ X_+ + e_- X_-$,\nwhere $X_\\pm = e_\\pm X$.\nApplied to $X\ + \ = [A,B]$,\nthis gives the decomposition (i)--(ii).\n\n\\emph{Independence (iii).}\nThe idempotents satisfy\n$e_+ \\\ + cdot e_- = 0$\n(I.D21, Book~I).\nTherefore\n$e_+ \\cdot [A,B]_- = e_+ e_- \\cdot [A,B] = 0$,\nand symmetrically\n$e_-\ + \ \\cdot [A,B]_+ = 0$.\nThe two components live in orthogonal sub-modules of $H_\\tau$,\nso they are algebraically independent.\n\ + \n\\emph{Functoriality (iv).}\nLet $h : B \\to B'$ be a $\\tau$-morphism.\nThe postcomposition map\n$h_* : [A,B] \\to\ + \ [A,B']$\nis defined by $h_*(f) := h \\circ f$.\nSince $h$ is $\\tau$-holomorphic,\n$h$ commutes with the idempotent\ + \ projection\n(II.P09, Chapter~\\ref{ch:idempotent-decomposition}):\n\\[\n e_\\pm \\cdot (h \\circ f)\n \\;=\\;\n\ + \ (e_\\pm h) \\circ (e_\\pm f)\n \\;=\\;\n h_\\pm \\circ f_\\pm.\n\\]\nHere we use the fact that\nthe idempotents\ + \ $e_\\pm$ are central in $H_\\tau$\nand that composition in $\\tau$ is $H_\\tau$-bilinear\n(it respects the $H_\\tau$\ + \ module structure\nat each stage).\nHence $h_*(f)_\\pm = h_\\pm \\circ f_\\pm$,\nwhich means $h_*([A,B]_\\pm) \\subseteq\ + \ [A,B']_\\pm$.\nThe precomposition argument is symmetric.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 127\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch42-tau-self-enrichment.tex`\ + \ lines 680-723\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.SelfEnrichment`\n\ + - Name: `Tau.BookII.Enrichment.hom_bipolar_check`\n\n## Dependencies\n\n- Canonical: I.D21, I.T05, II.L07, II.D54\n\n\ + ## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Hom Bipolar Decomposition +noindex: false +--- + diff --git a/_corpus_v3_items/cid001353.md b/_corpus_v3_items/cid001353.md new file mode 100644 index 000000000..9311e9782 --- /dev/null +++ b/_corpus_v3_items/cid001353.md @@ -0,0 +1,121 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001353/ +title: Enrichment Iteration +cid: cid001353 +primary_alias: PRP0058 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Enrichment Iteration +aliases: +- PRP0058 +- II.P12 +- enrichment-iteration +- prop:enrichment-iteration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005515 +- predicate: formalized_by + object: cid005516 +- predicate: formalized_by + object: cid005518 +- predicate: formalized_by + object: cid005519 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 44 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Enrichment Iteration**\n\nEnrichment Iteration\n\n# Enrichment Iteration\n\n## Summary\n\nEnrichment Iteration\n\ + \n## Statement\n\n```latex\n%\n\\label{prop:enrichment-iteration}\nThe self-enrichment of~$\\tau$ iterates:\nfor each\ + \ integer $n \\geq 0$,\nthere is a strict $n$-category $\\tau_n$\nwith the following structure.\n\\begin{enumerate}\n\ + \ \\item[\\textup{(I1)}]\n \\textbf{$n$-cells.}\n For $0 \\leq k \\leq n$,\n the $k$-cells\ + \ of~$\\tau_n$\n are elements of iterated hom-objects.\n Specifically:\n \\begin{itemize}\n\ + \ \\item 0-cells: objects of~$\\tau$.\n \\item 1-cells between 0-cells $A,B$:\n \ + \ elements of $[A,B]$.\n \\item 2-cells between 1-cells $f,g$:\n elements of $[f,g]\ + \ \\subset [[A,B],[A,B]]$.\n \\item $k$-cells: elements of the\n $k$-fold iterated hom-object\n\ + \ $\\underbrace{[\\cdots[}_{k}A,B]\\cdots]$.\n \\end{itemize}\n\n \\item[\\textup{(I2)}]\n\ + \ \\textbf{Tower coherence at each level.}\n Every $k$-cell has a tower-coherent\n stagewise\ + \ representation.\n At stage~$m$, the $k$-cell restricts\n to a map on the stage-$m$ version\n \ + \ of the $(k-1)$-fold iterated hom-object.\n The system is compatible\n under the primorial tower\ + \ projections.\n\n \\item[\\textup{(I3)}]\n \\textbf{Bipolar decomposition at each level.}\n Every\ + \ $k$-cell $\\sigma$\n decomposes as\n $\\sigma = e_+ \\cdot \\sigma_+ + e_- \\cdot \\sigma_-$,\n \ + \ with the two channels independent.\n The idempotents $e_\\pm$\n are central in~$H_\\tau$,\n\ + \ so they commute with all compositions\n at all levels.\n Consequently:\n \\[\n \ + \ \\tau_n\n \\;\\cong\\;\n \\tau_n^+ \\times \\tau_n^-\n \\]\n \ + \ for every~$n$,\n where $\\tau_n^\\pm = e_\\pm \\cdot \\tau_n$\n is the $n$-category restricted to\ + \ one channel.\n\n \\item[\\textup{(I4)}]\n \\textbf{Holomorphic structure at each level.}\n Every\ + \ $k$-cell is $\\tau$-holomorphic:\n it satisfies the split-complex\n Cauchy--Riemann conditions\n \ + \ in the appropriate iterated hom-space.\n This is because each iterated hom-object\n is a $\\\ + tau$-object\n and therefore carries holomorphic structure\n by the results of Parts~VI--VII.\n\n \ + \ \\item[\\textup{(I5)}]\n \\textbf{Strict $n$-category axioms.}\n At each level~$k$:\n \\\ + begin{itemize}\n \\item Vertical composition\n of $k$-cells is associative.\n \ + \ \\item Identity $k$-cells exist.\n \\item Horizontal composition\n with $(k-1)$-cells\ + \ is functorial.\n \\item The interchange law holds.\n \\end{itemize}\n\\end{enumerate}\n```\n\n\ + ## Proof / Justification\n\n```latex\nWe proceed by induction on~$n$.\n\n\\emph{Base case $n = 0$.}\n$\\tau_0 = \\tau$\ + \ is a plain category.\nAll five properties hold vacuously for $k$-cells\nwith $k > 0$ (there are none),\nand for $k =\ + \ 0$ they reduce to\nthe fact that $\\tau$ is a category\nwith NF-addressable objects\nand $H_\\tau$-valued morphisms.\n\ + \n\\emph{Base case $n = 1$.}\n$\\tau_1 = \\tau$ viewed as an enriched category.\nThe 1-cells are morphisms,\nwhich are\ + \ $\\tau$-objects (II.D54),\ntower-coherent (by construction of self-enrichment),\nand bipolar (II.P11).\nComposition\ + \ and identities are given.\n\n\\emph{Inductive step $n \\to n + 1$.}\nAssume $\\tau_n$ is a strict $n$-category\nsatisfying\ + \ (I1)--(I5).\nWe construct $\\tau_{n+1}$\nby adding a new layer of cells.\n\nThe $(n+1)$-cells between $n$-cells\n$\\\ + sigma, \\sigma'$ are the elements of\n$[\\sigma, \\sigma']$\nin the iterated hom-object.\nSelf-enrichment (II.D53) guarantees\n\ + that this is a well-defined $\\tau$-object.\n\n\\textbf{(I2):}\nTower coherence at level $n+1$\nfollows from tower coherence\ + \ at level~$n$\nplus the fact that hom-objects are tower-coherent\n(II.D54 gives this for the first iteration;\nthe same\ + \ argument applies at each level\nbecause the hom construction\nis internal to~$\\tau$).\n\n\\textbf{(I3):}\nBipolar decomposition\ + \ at level $n+1$\nfollows from II.L07 (Idempotent Decomposition Lemma).\nThe iterated hom-object at level $n+1$\nis a\ + \ $\\tau$-object valued in~$H_\\tau$,\nso the decomposition\n$\\sigma = e_+ \\sigma_+ + e_- \\sigma_-$\napplies.\nChannel\ + \ independence follows from\n$e_+ e_- = 0$.\n\n\\textbf{(I4):}\nHolomorphic structure follows because\n$\\tau$-objects\ + \ carry holomorphic structure\n(Parts~VI--VII).\nThe iterated hom-object is a $\\tau$-object,\nhence holomorphic.\n\n\\\ + textbf{(I5):}\nThe strict $n$-category axioms at level $n+1$:\n\\begin{itemize}\n \\item Vertical composition at level\ + \ $n+1$\n is composition in the endomorphism space\n of the level-$n$ hom-object.\n Associativity\ + \ follows from\n associativity of function composition.\n \\item Identity $(n+1)$-cells\n are the\ + \ identity maps\n in the endomorphism space.\n \\item Horizontal composition\n is given by the action\ + \ of the composition map~$\\mu$\n on the new level.\n Functoriality follows from\n the enriched\ + \ category axioms of~$\\tau$.\n \\item The interchange law\n at level $n+1$\n follows from the functoriality\ + \ of~$\\mu$\n at the previous level,\n by the same argument\n as in Definition~\\ref{def:two-category}.\n\ + \ \\qedhere\n\\end{itemize}\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 132\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch44-two-categories.tex` lines 591-669\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.TwoCategories`\n-\ + \ Name: `Tau.BookII.Enrichment.enrichment_iteration_check`\n\n## Dependencies\n\n- Canonical: II.D53, II.D54, II.D55,\ + \ II.D56, II.L07\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Enrichment Iteration +noindex: false +--- + diff --git a/_corpus_v3_items/cid001354.md b/_corpus_v3_items/cid001354.md new file mode 100644 index 000000000..1b2a4f58e --- /dev/null +++ b/_corpus_v3_items/cid001354.md @@ -0,0 +1,119 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001354/ +title: Character Decomposition +cid: cid001354 +primary_alias: PRP0059 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every spectral character is idempotent-supported: there exist unique ring homomorphisms chi_+ and chi_- such that + chi = e_+ chi_+ + e_- chi_-, with chi_+ determined by B-coordinate primes and chi_- by C-coordinate primes.' +aliases: +- PRP0059 +- II.P13 +- character-decomposition +- prop:character-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005526 +- predicate: formalized_by + object: cid005527 +- predicate: formalized_by + object: cid005528 +- predicate: formalized_by + object: cid005533 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 47 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Character Decomposition**\n\nEvery spectral character is idempotent-supported: there exist unique ring homomorphisms\ + \ chi_+ and chi_- such that chi = e_+ chi_+ + e_- chi_-, with chi_+ determined by B-coordinate primes and chi_- by C-coordinate\ + \ primes.\n\n# Character Decomposition\n\n## Summary\n\nEvery spectral character is idempotent-supported: there exist\ + \ unique ring homomorphisms chi_+ and chi_- such that chi = e_+ chi_+ + e_- chi_-, with chi_+ determined by B-coordinate\ + \ primes and chi_- by C-coordinate primes.\n\n## Statement\n\n```latex\n%\n\\label{prop:character-decomposition}\n% \ + \ II.L07, II.T33, II.D35, II.D48\nEvery spectral character\n$\\chi \\in \\mathrm{Spec}(\\hat{\\mathbb{Z}}_\\tau,\ + \ H_\\tau^{\\mathrm{cal}})$\nis idempotent-supported.\nSpecifically:\n\\begin{enumerate}\n \\item \\textbf{Existence.}\n\ + \ There exist unique ring homomorphisms\n $\\chi_+ : \\hat{\\mathbb{Z}}_\\tau \\to A_\\tau^{(B)}$\n\ + \ and\n $\\chi_- : \\hat{\\mathbb{Z}}_\\tau \\to A_\\tau^{(C)}$\n such that\n $\\\ + chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$.\n \\item \\textbf{Uniqueness.}\n The decomposition is unique:\n\ + \ if also\n $\\chi = e_+ \\cdot \\chi_+' + e_- \\cdot \\chi_-'$,\n then $\\chi_+' = \\chi_+$\ + \ and $\\chi_-' = \\chi_-$.\n \\item \\textbf{$B$-channel determination.}\n $\\chi_+$ is determined by its\ + \ values\n on the $B$-coordinate primes\n (the $\\gamma$-assigned primes\n from the Prime Polarity\ + \ Theorem, I.T05).\n \\item \\textbf{$C$-channel determination.}\n $\\chi_-$ is determined by its values\n\ + \ on the $C$-coordinate primes\n (the $\\eta$-assigned primes).\n \\item \\textbf{Tower coherence.}\n\ + \ The tower coherence of $\\chi$\n follows from the tower coherence\n of $\\chi_+$ and $\\\ + chi_-$ separately:\n \\[\n \\chi_n = e_+ \\cdot (\\chi_+)_n\n + e_- \\cdot (\\\ + chi_-)_n\n \\quad \\text{for every stage } n.\n \\]\n\\end{enumerate}\nConsequently:\n\\[\n \\\ + boxed{%\n A_{\\mathrm{spec}}(\\mathbb{L})\n = \\mathrm{Spec}(\\hat{\\mathbb{Z}}_\\tau, H_\\tau^{\\mathrm{cal}}).}\n\ + \\]\nEvery spectral character is idempotent-supported;\nthe character algebra $A_{\\mathrm{spec}}(\\mathbb{L})$\nis the\ + \ \\emph{full} character algebra.\n```\n\n## Proof / Justification\n\n```latex\nThe proof proceeds in three steps.\n\n\ + \\medskip\\noindent\n\\textbf{Step 1: Existence via idempotent projection.}\nLet $\\chi : \\hat{\\mathbb{Z}}_\\tau \\\ + to H_\\tau^{\\mathrm{cal}}$\nbe any spectral character.\nDefine $\\chi_+$ and $\\chi_-$ by the projections\n\\eqref{eq:chi-plus-proj}--\\\ + eqref{eq:chi-minus-proj}:\n\\[\n \\chi_+(x) := e_+ \\cdot \\chi(x),\n \\qquad\n \\chi_-(x) := e_- \\cdot \\chi(x).\n\ + \\]\nBy Lemma~\\ref{lem:channel-ring-hom},\nboth $\\chi_+$ and $\\chi_-$ are ring homomorphisms.\nSince $e_+ + e_- = 1$\ + \ in $H_\\tau^{\\mathrm{cal}}$, we have:\n\\[\n \\chi(x)\n = 1 \\cdot \\chi(x)\n = (e_+ + e_-) \\cdot \\chi(x)\n\ + \ = e_+ \\cdot \\chi(x) + e_- \\cdot \\chi(x)\n = \\chi_+(x) + \\chi_-(x).\n\\]\nThis establishes the decomposition\n\ + $\\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$.\n\n\\medskip\\noindent\n\\textbf{Step 2: Uniqueness via orthogonality.}\n\ + Suppose $\\chi = e_+ \\cdot \\chi_+' + e_- \\cdot \\chi_-'$\nis another decomposition.\nMultiplying both sides by $e_+$:\n\ + \\begin{align*}\n e_+ \\cdot \\chi(x)\n &= e_+ \\cdot \\bigl(e_+ \\cdot \\chi_+'(x)\n + e_- \\cdot \\chi_-'(x)\\\ + bigr) \\\\\n &= e_+^2 \\cdot \\chi_+'(x)\n + e_+ \\cdot e_- \\cdot \\chi_-'(x) \\\\\n &= e_+ \\cdot \\chi_+'(x)\ + \ + 0 \\\\\n &= \\chi_+'(x).\n\\end{align*}\nBut $e_+ \\cdot \\chi(x) = \\chi_+(x)$\nby definition,\nso $\\chi_+'(x)\ + \ = \\chi_+(x)$ for all $x$.\nThe same argument with $e_-$\ngives $\\chi_-' = \\chi_-$.\n\n\\medskip\\noindent\n\\textbf{Step\ + \ 3: Channel determination and tower coherence.}\nThe ring $\\hat{\\mathbb{Z}}_\\tau$\ndecomposes via the Chinese Remainder\ + \ Theorem\n(the CRT Tower, I.T18)\ninto a product indexed by primes.\nThe Prime Polarity Theorem (I.T05)\npartitions the\ + \ primes into $B$-primes ($\\gamma$-assigned)\nand $C$-primes ($\\eta$-assigned).\n\nThe projection $\\chi_+$ maps into\ + \ $A_\\tau^{(B)}$,\nwhich is the $e_+$-sector of $H_\\tau^{\\mathrm{cal}}$.\nSince $e_+ \\cdot e_- = 0$,\nthe $C$-prime\ + \ components of $\\hat{\\mathbb{Z}}_\\tau$\nare annihilated by $e_+$:\nfor any $C$-prime $q$ and any $x \\in \\hat{\\\ + mathbb{Z}}_\\tau$\nwhose support is confined to the $q$-component,\n$\\chi_+(x) = e_+ \\cdot \\chi(x)$\nlies in $e_+ \\\ + cdot A_\\tau^{(C)} = \\{0\\}$.\nTherefore $\\chi_+$ is determined\nby its values on the $B$-prime components.\nSymmetrically,\ + \ $\\chi_-$ is determined\nby the $C$-prime components.\n\nFor tower coherence,\nthe decomposition commutes with reduction\ + \ modulo $P_n$\nbecause $e_+$ and $e_-$ are global idempotents\n(they do not depend on the stage).\nThe family $\\{\\\ + chi_n\\}$ decomposes stage by stage:\n\\[\n \\chi_n(x)\n = e_+ \\cdot (\\chi_+)_n(x) + e_- \\cdot (\\chi_-)_n(x),\n\ + \\]\nand the compatibility conditions\n$\\chi_n \\equiv \\chi_{n+1} \\pmod{P_n}$\nfollow from the compatibility of\n$(\\\ + chi_+)_n$ and $(\\chi_-)_n$ separately.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 140\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch47-boundary-characters-idempotent.tex` lines\ + \ 436-485\n\n## Lean / Formalization Notes\n\n- Formalization: `not_formalized`\n- Module: `None`\n- Name: `None`\n\n\ + ## Dependencies\n\n- Canonical: I.D19, I.D21, I.D22, I.D23, I.T05, II.L07, II.T33, II.D35, II.D48\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every spectral character is idempotent-supported: there exist unique ring homomorphisms chi_+ and chi_- such + that chi = e_+ chi_+ + e_- chi_-, with chi_+ determined by B-coordinate primes and chi_- by C-coordinate primes.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001355.md b/_corpus_v3_items/cid001355.md new file mode 100644 index 000000000..2d27399c0 --- /dev/null +++ b/_corpus_v3_items/cid001355.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001355/ +title: Character Algebra Ring Structure +cid: cid001355 +primary_alias: PRP0060 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Character Algebra Ring Structure +aliases: +- PRP0060 +- II.P14 +- character-algebra-ring-structure +- prop:character-algebra-ring +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005366 +- predicate: formalized_by + object: cid005367 +- predicate: formalized_by + object: cid005368 +- predicate: formalized_by + object: cid005371 +- predicate: formalized_by + object: cid005372 +- predicate: formalized_by + object: cid005373 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 47 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Character Algebra Ring Structure**\n\nCharacter Algebra Ring Structure\n\n# Character Algebra Ring Structure\n\ + \n## Summary\n\nCharacter Algebra Ring Structure\n\n## Statement\n\n```latex\n%\n\\label{prop:character-algebra-ring}\n\ + % Depends: II.D59, I.D21\n$A_{\\mathrm{spec}}(\\mathbb{L})$ is a commutative ring under:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Pointwise addition.}\n $(\\chi + \\chi')(x) := \\chi(x) + \\chi'(x)$.\n \\item \\textbf{Pointwise\ + \ multiplication.}\n $(\\chi \\cdot \\chi')(x) := \\chi(x) \\cdot \\chi'(x)$.\n\\end{enumerate}\nThe zero element\ + \ is the character\n$\\chi_0(x) = 0$ for all $x$,\nand the multiplicative identity\nis the character $\\chi_1(x) = 1$\ + \ for all $x$.\n```\n\n## Proof / Justification\n\n```latex\nPointwise addition and multiplication\nare well-defined on\ + \ ring homomorphisms:\nif $\\chi$ and $\\chi'$ are ring homomorphisms,\nthen $\\chi \\cdot \\chi'$ is multiplicative\n\ + (since $H_\\tau^{\\mathrm{cal}}$ is commutative).\nThe zero and identity characters\nare the constant homomorphisms\n\ + $x \\mapsto 0$ and $x \\mapsto 1$ respectively.\nCommutativity, associativity, and distributivity\nfollow from the corresponding\ + \ properties\nof $H_\\tau^{\\mathrm{cal}}$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 139\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch47-boundary-characters-idempotent.tex`\ + \ lines 803-817\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.BoundaryCharacters`\n\ + - Name: `Tau.BookII.CentralTheorem.character_add_check`\n\n## Dependencies\n\n- Canonical: II.D59, I.D21\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Character Algebra Ring Structure +noindex: false +--- + diff --git a/_corpus_v3_items/cid001356.md b/_corpus_v3_items/cid001356.md new file mode 100644 index 000000000..615fe96a9 --- /dev/null +++ b/_corpus_v3_items/cid001356.md @@ -0,0 +1,224 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001356/ +title: Tau3 Is a Tau-Manifold +cid: cid001356 +primary_alias: PRP0061 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Tau3 Is a Tau-Manifold +aliases: +- PRP0061 +- II.P15 +- tau3-is-a-tau-manifold +- prop:ch54-tau3-is-manifold +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005488 +- predicate: formalized_by + object: cid005493 +- predicate: formalized_by + object: cid005494 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 54 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tau3 Is a Tau-Manifold** + + + Tau3 Is a Tau-Manifold + + + # Tau3 Is a Tau-Manifold + + + ## Summary + + + Tau3 Is a Tau-Manifold + + + ## Statement + + + ```latex + + % + + \label{prop:ch54-tau3-is-manifold} + + The fibered product + + $\tau^3 = \tau^1 \times_f T^2$, + + equipped with the canonical atlas + + of cylinder domains + + and identity charts, + + is a $\tau$-manifold + + of dimension~$4$. + + ``` + + + ## Proof / Justification + + + ```latex + + \textbf{(M1)} + + The Stone topology on~$\tau^3$ + + is Hausdorff, totally disconnected, + + and compact + + (II.D14, Chapter~\ref{ch:stone-space}). + + The cylinder domains + + $\{C_{k,a} : k \geq 1,\; a \in \mathbb{Z}/P_k\mathbb{Z}\}$ + + form a countable clopen basis + + (since $k$ ranges over~$\mathbb{N}$ + + and $\mathbb{Z}/P_k\mathbb{Z}$ is finite + + for each~$k$). + + Hence $M = \tau^3$ is second countable. + + + \textbf{(M2)} + + Each cylinder domain $C_{k,a}$ + + is clopen in the Stone topology. + + The identity inclusion + + $\varphi_{k,a} \colon C_{k,a} \hookrightarrow \tau^3$ + + is a homeomorphism onto its image. + + For overlapping cylinders, + + the transition function is the identity + + (Remark~\ref{rem:ch54-trivial-transitions}), + + which satisfies (TA1)--(TA3) trivially. + + The atlas extends to a maximal atlas + + by adding all charts + + compatible with the existing ones. + + + \textbf{(M3)} + + The sheaf property of~$\mathcal{O}_\tau$ + + was established in Theorem~\ref{thm:sheaf-axioms} + + (II.T32, Chapter~\ref{ch:sheaf-coherence}). + + Since the charts are identity maps, + + transport via charts does not alter + + the sheaf structure: + + $\mathcal{O}_\tau$ on~$\tau^3$ + + equals $\mathcal{O}_\tau$ on~$M$. + + ``` + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 158 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch54-tau-manifold.tex` lines 414-423 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.TauManifold` + + - Name: `Tau.BookII.Closure.tau3_is_manifold_check` + + + ## Dependencies + + + - Canonical: II.D62, II.D63, II.T40 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Tau3 Is a Tau-Manifold +noindex: false +--- + diff --git a/_corpus_v3_items/cid001357.md b/_corpus_v3_items/cid001357.md new file mode 100644 index 000000000..264876fc9 --- /dev/null +++ b/_corpus_v3_items/cid001357.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001357/ +title: Holonomy Triviality +cid: cid001357 +primary_alias: PRP0062 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At each finite stage, the holonomy group of the flat connection is trivial. Z/M_k Z is simply connected as a discrete + set, so all loops are contractible. The profinite limit may acquire nontrivial holonomy from π₁(L) = Z. +aliases: +- PRP0062 +- II.P16 +- holonomy-triviality +- prop:holonomy-triviality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005450 +- predicate: formalized_by + object: cid005451 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Holonomy Triviality** + + + At each finite stage, the holonomy group of the flat connection is trivial. Z/M_k Z is simply connected as a discrete + set, so all loops are contractible. The profinite limit may acquire nontrivial holonomy from π₁(L) = Z. + + + # Holonomy Triviality + + + ## Summary + + + At each finite stage, the holonomy group of the flat connection is trivial. Z/M_k Z is simply connected as a discrete + set, so all loops are contractible. The profinite limit may acquire nontrivial holonomy from π₁(L) = Z. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 198 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Connection` + + - Name: `holonomy_trivial_2` + + + ## Dependencies + + + - Canonical: II.D78, II.D79, II.T50 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At each finite stage, the holonomy group of the flat connection is trivial. Z/M_k Z is simply connected as a + discrete set, so all loops are contractible. The profinite limit may acquire nontrivial holonomy from π₁(L) = Z. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001358.md b/_corpus_v3_items/cid001358.md new file mode 100644 index 000000000..3a3a336b2 --- /dev/null +++ b/_corpus_v3_items/cid001358.md @@ -0,0 +1,130 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001358/ +title: Geodesic Completeness +cid: cid001358 +primary_alias: PRP0063 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every geodesic in Z/M_k Z can be extended indefinitely: the space is compact and complete at every finite stage.' +aliases: +- PRP0063 +- II.P17 +- geodesic-completeness +- prop:geodesic-completeness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005456 +- predicate: formalized_by + object: cid005457 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Geodesic Completeness** + + + Every geodesic in Z/M_k Z can be extended indefinitely: the space is compact and complete at every finite stage. + + + # Geodesic Completeness + + + ## Summary + + + Every geodesic in Z/M_k Z can be extended indefinitely: the space is compact and complete at every finite stage. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 202 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Curvature` + + - Name: `geodesic_complete_2` + + + ## Dependencies + + + - Canonical: II.D81 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every geodesic in Z/M_k Z can be extended indefinitely: the space is compact and complete at every finite stage.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001359.md b/_corpus_v3_items/cid001359.md new file mode 100644 index 000000000..89edd5ca1 --- /dev/null +++ b/_corpus_v3_items/cid001359.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001359/ +title: L² Completeness +cid: cid001359 +primary_alias: PRP0064 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: L² space at each finite stage is automatically complete (finite-dimensional). Tower compatibility ensures the inverse + system of L² spaces is coherent. +aliases: +- PRP0064 +- II.P18 +- l-completeness +- prop:l2-completeness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005563 +- predicate: formalized_by + object: cid005564 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**L² Completeness** + + + L² space at each finite stage is automatically complete (finite-dimensional). Tower compatibility ensures the inverse + system of L² spaces is coherent. + + + # L² Completeness + + + ## Summary + + + L² space at each finite stage is automatically complete (finite-dimensional). Tower compatibility ensures the inverse + system of L² spaces is coherent. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 207 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Hartogs.L2Space` + + - Name: `l2_completeness_2` + + + ## Dependencies + + + - Canonical: II.D83, II.T53 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: L² space at each finite stage is automatically complete (finite-dimensional). Tower compatibility ensures the + inverse system of L² spaces is coherent. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001360.md b/_corpus_v3_items/cid001360.md new file mode 100644 index 000000000..57ae5f9a4 --- /dev/null +++ b/_corpus_v3_items/cid001360.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001360/ +title: Long Exact Sequence +cid: cid001360 +primary_alias: PRP0065 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Short exact sequence of primorial quotients induces a long exact sequence in homology via the snake lemma. Connecting + homomorphism δ exists. +aliases: +- PRP0065 +- II.P19 +- long-exact-sequence +- prop:long-exact-sequence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005506 +- predicate: formalized_by + object: cid005507 +- predicate: formalized_by + object: cid005508 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 46 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Long Exact Sequence** + + + Short exact sequence of primorial quotients induces a long exact sequence in homology via the snake lemma. Connecting + homomorphism δ exists. + + + # Long Exact Sequence + + + ## Summary + + + Short exact sequence of primorial quotients induces a long exact sequence in homology via the snake lemma. Connecting + homomorphism δ exists. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 211 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Enrichment.Homological` + + - Name: `les_stage2` + + + ## Dependencies + + + - Canonical: II.D85, II.T54 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Short exact sequence of primorial quotients induces a long exact sequence in homology via the snake lemma. Connecting + homomorphism δ exists. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001361.md b/_corpus_v3_items/cid001361.md new file mode 100644 index 000000000..3f8634c64 --- /dev/null +++ b/_corpus_v3_items/cid001361.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001361/ +title: Čech-to-Derived Comparison +cid: cid001361 +primary_alias: PRP0066 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Čech cohomology agrees with derived functor cohomology on the primorial tower. The cylinder cover is acyclic (ultrametric: + intersections are empty or cylinders).' +aliases: +- PRP0066 +- II.P20 +- ech-to-derived-comparison +- prop:cech-derived +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005428 +- predicate: formalized_by + object: cid005429 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Čech-to-Derived Comparison** + + + Čech cohomology agrees with derived functor cohomology on the primorial tower. The cylinder cover is acyclic (ultrametric: + intersections are empty or cylinders). + + + # Čech-to-Derived Comparison + + + ## Summary + + + Čech cohomology agrees with derived functor cohomology on the primorial tower. The cylinder cover is acyclic (ultrametric: + intersections are empty or cylinders). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 215 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.CentralTheorem.SheafCohomology` + + - Name: `cech_derived_2` + + + ## Dependencies + + + - Canonical: II.D86, II.D87, II.T55 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Čech cohomology agrees with derived functor cohomology on the primorial tower. The cylinder cover is acyclic + (ultrametric: intersections are empty or cylinders).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001362.md b/_corpus_v3_items/cid001362.md new file mode 100644 index 000000000..a1dc7e155 --- /dev/null +++ b/_corpus_v3_items/cid001362.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001362/ +title: HolEnd Exponentials +cid: cid001362 +primary_alias: PRP0067 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The holomorphic endomorphism category HolEnd(tau) admits internal exponentials: [A x B, C] = [A, [B, C]] for all + objects A, B, C in HolEnd(tau). Follows from self-enrichment closure.' +aliases: +- PRP0067 +- II.P21 +- holend-exponentials +- prop:ch42-holend-exponentials +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 42 (Gap Remediation) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**HolEnd Exponentials**\n\nThe holomorphic endomorphism category HolEnd(tau) admits internal exponentials: [A\ + \ x B, C] = [A, [B, C]] for all objects A, B, C in HolEnd(tau). Follows from self-enrichment closure.\n\n# HolEnd Exponentials\n\ + \n## Summary\n\nThe holomorphic endomorphism category HolEnd(tau) admits internal exponentials: [A x B, C] = [A, [B, C]]\ + \ for all objects A, B, C in HolEnd(tau). Follows from self-enrichment closure.\n\n## Statement\n\n```latex\n%\n\\label{prop:ch42-holend-exponentials}\n\ + For any $\\tau$-objects $A, B, C$,\nthe internal Hom $[A,B]$\n\\textup{(}Definition~\\textup{\\ref{def:hom-object},} II.D54\\\ + textup{)}\nsatisfies the exponential law:\n\\[\n \\boxed{%\n [A \\times B,\\; C]\n \\;\\cong\\;\n [A,\\; [B,\ + \ C]]}\n\\]\nas $\\tau$-objects,\nwhere $\\times$ is the Cartesian product in~$\\tau$.\n```\n\n## Proof / Justification\n\ + \n```latex\n[Proof sketch]\nThe isomorphism is built stagewise.\nAt each finite stage~$k$\nof the primorial tower,\nthe\ + \ category $\\tau_k$\nis a finite category\nwith finite Hom sets.\nThe standard exponential law\nfor finite sets gives\n\ + \\[\n \\Hom_{\\tau_k}(A_k \\times B_k,\\; C_k)\n \\;\\cong\\;\n \\Hom_{\\tau_k}\\bigl(A_k,\\; \\Hom_{\\tau_k}(B_k,\ + \ C_k)\\bigr)\n\\]\nnaturally in $A_k$, $B_k$, $C_k$.\nThis bijection sends\n$f_k : A_k \\times B_k \\to C_k$\nto the\ + \ curried map\n$\\hat{f}_k : A_k \\to \\Hom_{\\tau_k}(B_k, C_k)$\ndefined by $\\hat{f}_k(a)(b) = f_k(a, b)$.\n\nThe connecting\ + \ maps $r_{k+1,k}$\nof the primorial tower\nare compatible with currying:\nif $f_{k+1}$ reduces to $f_k$ modulo $P_k$,\n\ + then $\\hat{f}_{k+1}$ reduces to $\\hat{f}_k$ modulo $P_k$.\nThis compatibility follows from the functoriality\nof the\ + \ reduction maps\n(they are ring homomorphisms,\nhence preserve the product structure).\n\nPassing to the inverse limit:\n\ + \\[\n [A \\times B,\\; C]\n \\;=\\;\n \\varprojlim_k \\Hom_{\\tau_k}(A_k \\times B_k, C_k)\n \\;\\cong\\;\n\ + \ \\varprojlim_k \\Hom_{\\tau_k}(A_k, \\Hom_{\\tau_k}(B_k, C_k))\n \\;=\\;\n [A,\\; [B, C]].\n\\]\nThe isomorphism\ + \ respects\nthe bipolar decomposition (II.P11)\nbecause currying commutes\nwith the idempotent projections $e_\\pm$:\n\ + $\\widehat{e_\\pm f} = e_\\pm \\hat{f}$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 217\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch42-tau-self-enrichment.tex` lines 1213-1227\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n## Dependencies\n\n\ + - Canonical: II.D53, II.D54\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The holomorphic endomorphism category HolEnd(tau) admits internal exponentials: [A x B, C] = [A, [B, C]] for + all objects A, B, C in HolEnd(tau). Follows from self-enrichment closure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001363.md b/_corpus_v3_items/cid001363.md new file mode 100644 index 000000000..a3c6b4d89 --- /dev/null +++ b/_corpus_v3_items/cid001363.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001363/ +title: Point Set Well-Defined +cid: cid001363 +primary_alias: THM0067 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The tau-admissible point set is well-defined: the five constraint conditions (prime, non-negativity, remainder, + primorial, depth) are jointly satisfiable for all primes from 2 to 20, producing a non-empty finite set at each stage.' +aliases: +- THM0067 +- II.T01 +- point-set-well-defined +- thm:point-set-well-defined +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Point Set Well-Defined**\n\nThe tau-admissible point set is well-defined: the five constraint conditions (prime,\ + \ non-negativity, remainder, primorial, depth) are jointly satisfiable for all primes from 2 to 20, producing a non-empty\ + \ finite set at each stage.\n\n# Point Set Well-Defined\n\n## Summary\n\nThe tau-admissible point set is well-defined:\ + \ the five constraint conditions (prime, non-negativity, remainder, primorial, depth) are jointly satisfiable for all\ + \ primes from 2 to 20, producing a non-empty finite set at each stage.\n\n## Statement\n\n```latex\n%\n\\label{thm:point-set-well-defined}\n\ + The set\n\\[\n \\boxed{\\tau^3\n \\;:=\\;\n \\tau^3_{\\mathrm{fin}}\n \\;\\cup\\;\n \\tau^3_{\\mathrm{lim}}}\n\ + \\]\nis a well-defined set satisfying:\n\\begin{enumerate}\n \\item[\\textup{(1)}] \\textbf{Finite part.}\n \ + \ $\\tau^3_{\\mathrm{fin}}$ is in bijection\n with $\\Obj(\\tau)$ via the ABCD chart~$\\Phi$\n (Hyperfactorization,\ + \ I.T04).\n \\item[\\textup{(2)}] \\textbf{Limit part.}\n $\\tau^3_{\\mathrm{lim}}$ is non-empty:\n \ + \ the primorial tower produces\n at least one limit point\n (Proposition~\\ref{prop:ch04-limit-nonempty}).\n\ + \ \\item[\\textup{(3)}] \\textbf{Closure under reduction.}\n $\\tau^3$ is closed under the CRT reduction maps:\n\ + \ for every point $x \\in \\tau^3$ and every stage~$k$,\n the stage-$k$ projection\n $\\pi_k(x)$\n\ + \ is a well-defined $\\tau$-admissible point.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + \\emph{Clause~(1).}\nBy Proposition~\\ref{prop:ch04-bijection},\nevery $n \\in \\tau\\text{-Idx}_{\\geq 2}$\nmaps to a\ + \ unique $\\tau$-admissible quadruple\n$\\Phi(n) \\in \\tau^3_{\\mathrm{fin}}$,\nand conversely,\nevery element of $\\\ + tau^3_{\\mathrm{fin}}$\nwith $A \\geq 2$ determines a unique index.\nThe two degenerate points are handled\nby the convention\ + \ of\nDefinition~\\ref{def:tau-admissible-point}.\nHence $\\tau^3_{\\mathrm{fin}} \\cong \\Obj(\\tau)$.\n\n\\emph{Clause~(2).}\n\ + Proposition~\\ref{prop:ch04-limit-nonempty}\nexhibited the primorial tower\n$\\{\\Phi(P_k)\\}_{k \\geq 1}$\nas a non-eventually-constant\ + \ coherent family.\n\n\\emph{Clause~(3).}\nFor a finite point $\\Phi(n) \\in \\tau^3_{\\mathrm{fin}}$,\nthe stage-$k$\ + \ projection is\n$\\Phi(n \\bmod P_k)$,\nwhich is a well-defined finite $\\tau$-admissible point\n(the normal-form decomposition\ + \ of $n \\bmod P_k$\nsatisfies constraints (C1)--(C5)\nsince $n \\bmod P_k \\geq 0$ is a valid index).\nFor a limit point\n\ + $\\{(A_k, B_k, C_k, D_k)\\}_{k \\geq 1}\n\\in \\tau^3_{\\mathrm{lim}}$,\nthe stage-$k$ projection is\n$(A_k, B_k, C_k,\ + \ D_k)$ by definition,\nwhich is $\\tau$-admissible by the coherence condition.\nThe tower compatibility\n$\\pi_{k,l}\ + \ \\circ \\pi_{l,m} = \\pi_{k,m}$\n(CRT coherence, I.T18)\nensures that the projections are self-consistent.\n```\n\n\ + ## Source Context\n\n- Registry source: `book-02.jsonl` line 7\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch04-tau-admissible-points.tex`\ + \ lines 441-470\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.TauAdmissible`\n\ + - Name: `Tau.BookII.Interior.admissible_2_to_20`\n\n## Dependencies\n\n- Canonical: II.D02, II.D03, I.T04, I.T18\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The tau-admissible point set is well-defined: the five constraint conditions (prime, non-negativity, remainder, + primorial, depth) are jointly satisfiable for all primes from 2 to 20, producing a non-empty finite set at each stage.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001364.md b/_corpus_v3_items/cid001364.md new file mode 100644 index 000000000..bbcaf65d8 --- /dev/null +++ b/_corpus_v3_items/cid001364.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001364/ +title: Fiber Degeneration at Omega +cid: cid001364 +primary_alias: THM0068 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'At the omega boundary, the fiber T^2 degenerates: paths of unbounded primorial depth produce balanced B/C ratios, + forcing the fiber to collapse onto the lemniscate L = S^1 v S^1.' +aliases: +- THM0068 +- II.T02 +- fiber-degeneration-at-omega +- thm:fiber-degeneration-omega +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fiber Degeneration at Omega**\n\nAt the omega boundary, the fiber T^2 degenerates: paths of unbounded primorial\ + \ depth produce balanced B/C ratios, forcing the fiber to collapse onto the lemniscate L = S^1 v S^1.\n\n# Fiber Degeneration\ + \ at Omega\n\n## Summary\n\nAt the omega boundary, the fiber T^2 degenerates: paths of unbounded primorial depth produce\ + \ balanced B/C ratios, forcing the fiber to collapse onto the lemniscate L = S^1 v S^1.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:fiber-degeneration-omega}\nAt the $\\omega$-limit of the primorial tower:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Base collapse.}\n The base readout $(D, A)$ collapses to a single point\n $(\\Omega, \\Omega)$.\n\ + \ \\item \\textbf{Fiber locus.}\n The fiber readout $(B, C)$\n traces a one-dimensional locus ---\n\ + \ the algebraic lemniscate $\\mathbb{L}$ (I.D18).\n \\item \\textbf{Degeneration mechanism.}\n The\ + \ fiber degeneration $T^2 \\to \\mathbb{L}$ is forced:\n the two-dimensional fiber parameter space $(B, C)$\n\ + \ loses one degree of freedom at the boundary\n because the B/C competition\n locks into bipolar\ + \ lobe structure.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(1) Base collapse.}\nAlong\ + \ any path $(X_m)$ to $\\omega$,\nthe base coordinates $A_m$ and $D_m$\nboth grow without bound.\nFor the primorial path\ + \ this is immediate:\n$A_n = p_n \\to \\infty$ and $D_n = P_{n-1} \\to \\infty$.\nFor a general path, the primorial depth\ + \ condition\nensures that arbitrarily large primes\nmust eventually appear in the factorization of $X_m$,\nforcing $A_m\ + \ \\to \\infty$;\nsimilarly, the cofactor $D_m$ must accommodate\nan unbounded collection of smaller primes, forcing $D_m\ + \ \\to \\infty$.\nSince both coordinates diverge universally,\n$\\operatorname{pr}_{\\mathrm{base}}(\\Phi_\\omega) = (\\\ + Omega, \\Omega)$.\n\n\\textbf{(2) Fiber locus.}\nThe primorial path yields $B_n = C_n = 1$ for all~$n$,\nbut this is only\ + \ the crossing point $\\omega_{\\mathbb{L}}$ of $\\mathbb{L}$.\nPaths with growing $B$ and fixed $C$\n(e.g., $X_m = 2^m$)\ + \ trace the $e_+$-lobe.\nPaths with growing $C$ and fixed $B$\n(e.g., $X_m = {}^{m}\\!2$) trace the $e_-$-lobe.\nThe totality\ + \ of fiber limits\nis parametrized by the asymptotic dominance ratio $B/C$,\nwhich ranges over all of~$\\mathbb{L}$:\n\ + $e_+$-lobe for $B \\gg C$,\n$e_-$-lobe for $C \\gg B$,\ncrossing point for $B \\sim C$.\nBy the Prime Polarity Theorem\ + \ (I.T05),\nevery direction in the B/C plane\nis realized by some path to~$\\omega$,\nand the dominance partition\nis\ + \ precisely the bipolar partition\nthat defines~$\\mathbb{L}$ (I.D18).\n\n\\textbf{(3) Degeneration mechanism.}\nAt finite\ + \ stages, $B$ and $C$ are independent coordinates:\n$\\dim_\\tau = 4$ and the fiber has dimension~$2$.\nAt the $\\omega$-limit,\ + \ the B and C coordinates\nare no longer independent.\nThe primorial CRT structure (I.T18)\ncouples them through the coherence\ + \ conditions\nof the inverse system:\nfor each approach to~$\\omega$,\nthe asymptotic $B/C$ ratio is locked\nby the polarity\ + \ character $\\tilde{\\chi}$\n(the sector assignment of the prime sequence\nalong the chosen path).\nThis coupling reduces\ + \ the two independent\nfiber parameters to a single real parameter ---\nthe position along~$\\mathbb{L}$ ---\nachieving\ + \ the degeneration $T^2 \\to \\mathbb{L}$.\nThe dimensional reduction is $2 \\to 1$,\nand the topology of $\\mathbb{L}$\n\ + (two lobes joined at a node)\nreflects the bipolar partition of the coupling.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 9\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch05-omega-readout-lemniscate.tex`\ + \ lines 378-397\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.OmegaReadout`\n\ + - Name: `Tau.BookII.Interior.primorial_balanced`\n\n## Dependencies\n\n- Canonical: II.D04, I.T05, I.T18, I.D18\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'At the omega boundary, the fiber T^2 degenerates: paths of unbounded primorial depth produce balanced B/C ratios, + forcing the fiber to collapse onto the lemniscate L = S^1 v S^1.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001365.md b/_corpus_v3_items/cid001365.md new file mode 100644 index 000000000..9d2a2f21d --- /dev/null +++ b/_corpus_v3_items/cid001365.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001365/ +title: Fibration Structure +cid: cid001365 +primary_alias: THM0069 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The fibered product tau^3 = tau^1 x_f T^2 is a non-trivial fibration: surjective projection, varying fiber, no global + trivialization, and injective ABCD chart.' +aliases: +- THM0069 +- II.T03 +- fibration-structure +- thm:fibration-structure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fibration Structure**\n\nThe fibered product tau^3 = tau^1 x_f T^2 is a non-trivial fibration: surjective projection,\ + \ varying fiber, no global trivialization, and injective ABCD chart.\n\n# Fibration Structure\n\n## Summary\n\nThe fibered\ + \ product tau^3 = tau^1 x_f T^2 is a non-trivial fibration: surjective projection, varying fiber, no global trivialization,\ + \ and injective ABCD chart.\n\n## Statement\n\n```latex\n%\n\\label{thm:fibration-structure}\nThe fibered product $\\\ + tau^3 = \\tau^1 \\times_f T^2$\nis a fibration over $\\tau^1$:\n\\begin{enumerate}\n \\item \\textbf{Projection.}\n\ + \ The map $\\mathrm{pr} \\colon \\tau^3 \\to \\tau^1$\n is surjective:\n every base point admits\ + \ at least one fiber point\n (the trivial exponent tower $B = 1, C = 0$).\n \\item \\textbf{Fiber variation.}\n\ + \ The fiber $T^2_{(D,A)}$ depends on the base point $(D,A)$:\n different primes $A$ yield different\ + \ admissible ranges\n for $(B,C)$.\n \\item \\textbf{Non-triviality.}\n The fibration is not a product\ + \ bundle:\n there is no global trivialization\n $\\tau^3 \\cong \\tau^1 \\times T^2$\n because\ + \ the peel-order constraint\n couples the base coordinates\n and the admissibility conditions\n \ + \ vary across base points.\n \\item \\textbf{Faithfulness.}\n The fibered product is faithful to the\ + \ ABCD chart:\n the map\n $\\Phi \\colon \\mathrm{Obj}(\\tau) \\to \\tau^3$\n given by $X \\\ + mapsto (D(X), A(X), B(X), C(X))$\n is injective\n (by hyperfactorization, I.T04).\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nSurjectivity of $\\mathrm{pr}$:\nfor any $(D, A) \\in \\\ + tau^1$,\nthe object $D \\cdot A$ has ABCD coordinates\n$(D, A, 1, 0)$, so $(1, 0) \\in T^2_{(D,A)}$.\nFiber variation:\n\ + if $A = p_1$ (the smallest prime),\nthen $D = 1$ necessarily (no smaller primes exist),\nand the exponent range for $B$\ + \ is unrestricted;\nif $A = p_k$ for large $k$,\nthen $D$ can be a product of $k-1$ smaller primes,\nand the interaction\ + \ between $D$ and $A$\nconstrains the admissible tetration heights.\nNon-triviality:\nsuppose a global trivialization\ + \ existed.\nThen for every base point $(D, A)$,\nthe fiber would be the same space $T^2_0$.\nBut the constraint ``prime\ + \ factors of $D < A$''\nmeans that the admissible objects\nover $(1, p_1)$ are different in kind\nfrom those over $(p_1\ + \ \\cdot p_2, p_3)$---the\nfiber geometry depends essentially on the base.\nFaithfulness is the Hyperfactorization Theorem\ + \ (I.T04).\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 14\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part01/ch06-tau3-fibration.tex`\ + \ lines 287-318\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Interior.Tau3Fibration`\n\ + - Name: `Tau.BookII.Interior.surjective_2_to_20`\n\n## Dependencies\n\n- Canonical: II.D07, I.T04\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The fibered product tau^3 = tau^1 x_f T^2 is a non-trivial fibration: surjective projection, varying fiber, + no global trivialization, and injective ABCD chart.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001366.md b/_corpus_v3_items/cid001366.md new file mode 100644 index 000000000..6238efb7b --- /dev/null +++ b/_corpus_v3_items/cid001366.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001366/ +title: Cylinder Basis Theorem +cid: cid001366 +primary_alias: THM0070 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The collection of stage-k cylinders forms a basis for the cylinder topology on tau^3: they are clopen, nested by + divisibility, and any two cylinders are either disjoint or nested.' +aliases: +- THM0070 +- II.T04 +- cylinder-basis-theorem +- thm:cylinder-basis +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Cylinder Basis Theorem**\n\nThe collection of stage-k cylinders forms a basis for the cylinder topology on\ + \ tau^3: they are clopen, nested by divisibility, and any two cylinders are either disjoint or nested.\n\n# Cylinder Basis\ + \ Theorem\n\n## Summary\n\nThe collection of stage-k cylinders forms a basis for the cylinder topology on tau^3: they\ + \ are clopen, nested by divisibility, and any two cylinders are either disjoint or nested.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:cylinder-basis}\nThe collection\n$\\mathcal{B} = \\{C_k(x) : k \\geq 1,\\; x \\in \\tau^3\\}$\nis a basis\ + \ for a topology on~$\\tau^3$.\nThe resulting topology satisfies:\n\\begin{enumerate}\n \\item[\\textup{(1)}] \\textbf{Covering.}\n\ + \ $\\tau^3 = \\bigcup_{x \\in \\tau^3} C_1(x)$.\n \\item[\\textup{(2)}] \\textbf{Intersection closure.}\n\ + \ For any $C_k(x), C_l(y) \\in \\mathcal{B}$\n and any $z \\in C_k(x) \\cap C_l(y)$,\n there\ + \ exists $C_m(z) \\in \\mathcal{B}$\n with $z \\in C_m(z) \\subseteq C_k(x) \\cap C_l(y)$.\n \\item[\\textup{(3)}]\ + \ \\textbf{Totally disconnected.}\n The basis consists entirely of clopen sets.\n \\item[\\textup{(4)}] \\\ + textbf{Second countable.}\n The basis has a countable sub-basis:\n the collection $\\{C_k(r) : k \\\ + geq 1,\\;\n r \\in \\mathbb{Z}/P_k\\}$ is countable.\n \\item[\\textup{(5)}] \\textbf{Compatible with profinite\ + \ structure.}\n The topology coincides with the inverse-limit topology\n on $\\tau^3 = \\varprojlim_k\ + \ \\mathbb{Z}/P_k$,\n where each $\\mathbb{Z}/P_k$ carries the discrete topology.\n\\end{enumerate}\n```\n\n\ + ## Proof / Justification\n\n```latex\n\\emph{Clause~(1).}\nFor any $x \\in \\tau^3$,\n$x \\in C_1(x)$ by definition.\n\ + Hence $\\bigcup_{x} C_1(x) = \\tau^3$.\n\n\\emph{Clause~(2).}\nLet $z \\in C_k(x) \\cap C_l(y)$\nand set $m = \\max(k,\ + \ l)$.\nBy Proposition~\\ref{prop:ch09-intersection},\n$C_m(z) \\subseteq C_k(z) = C_k(x)$\n(since $z \\in C_k(x)$ implies\ + \ $C_k(z) = C_k(x)$)\nand similarly $C_m(z) \\subseteq C_l(z) = C_l(y)$.\nHence $C_m(z) \\subseteq C_k(x) \\cap C_l(y)$,\n\ + and $z \\in C_m(z)$.\nThis verifies the basis intersection axiom.\n\n\\emph{Clause~(3).}\nEach $C_k(x)$ is clopen by\n\ + Proposition~\\ref{prop:ch09-complement}.\n\n\\emph{Clause~(4).}\nAt stage~$k$, there are exactly $P_k$\ndistinct cylinders\ + \ $\\{C_k(r)\\}_{r \\in \\mathbb{Z}/P_k}$.\nThe total number of basis elements\n$\\bigcup_{k \\geq 1} \\{C_k(r) : r \\\ + in \\mathbb{Z}/P_k\\}$\nis a countable union of finite sets,\nhence countable.\n\n\\emph{Clause~(5).}\nThe inverse-limit\ + \ topology\non $\\varprojlim_k \\mathbb{Z}/P_k$\nhas as a basis the preimages\n$\\pi_k^{-1}(U)$\nfor $U \\subseteq \\\ + mathbb{Z}/P_k$ open.\nSince $\\mathbb{Z}/P_k$ carries the discrete topology,\nevery singleton $\\{r\\}$ is open,\nand\ + \ $\\pi_k^{-1}(\\{r\\}) = C_k(r)$.\nThese singletons generate the discrete topology\non $\\mathbb{Z}/P_k$,\nso their preimages\ + \ generate\nthe inverse-limit topology.\nBut these preimages are exactly\nthe elements of~$\\mathcal{B}$.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 22\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch09-cylinder-domains.tex`\ + \ lines 471-497\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Cylinders`\n\ + - Name: `nesting_check`\n\n## Dependencies\n\n- Canonical: II.D10, II.D11, I.T18\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The collection of stage-k cylinders forms a basis for the cylinder topology on tau^3: they are clopen, nested + by divisibility, and any two cylinders are either disjoint or nested.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001367.md b/_corpus_v3_items/cid001367.md new file mode 100644 index 000000000..1147b3e2a --- /dev/null +++ b/_corpus_v3_items/cid001367.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001367/ +title: Ultrametric Triangle Inequality +cid: cid001367 +primary_alias: THM0071 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Ultrametric Triangle Inequality +aliases: +- THM0071 +- II.T05 +- ultrametric-triangle-inequality +- thm:ultrametric-triangle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ultrametric Triangle Inequality**\n\nUltrametric Triangle Inequality\n\n# Ultrametric Triangle Inequality\n\ + \n## Summary\n\nUltrametric Triangle Inequality\n\n## Statement\n\n```latex\n%\n\\label{thm:ultrametric-triangle}\nFor\ + \ all $x, y, z \\in \\tau^3$:\n\\[\n \\boxed{d(x, z)\n \\;\\leq\\;\n \\max\\bigl(\\, d(x, y),\\; d(y, z) \\,\\\ + bigr).}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nIf any two of $x, y, z$ are equal,\nthe inequality is immediate.\n\ + Assume $x, y, z$ are pairwise distinct.\nLet $k$ be any stage with $k \\leq \\min\\bigl(\\delta(x,y),\\, \\delta(y,z)\\\ + bigr)$.\nThen $\\pi_k(x) = \\pi_k(y)$ and $\\pi_k(y) = \\pi_k(z)$,\nso by transitivity of equality,\n$\\pi_k(x) = \\pi_k(z)$.\n\ + This holds for every such~$k$, hence\n\\[\n \\delta(x,z)\n \\;\\geq\\;\n \\min\\bigl(\\delta(x,y),\\, \\delta(y,z)\\\ + bigr).\n\\]\nApplying the decreasing function $t \\mapsto 2^{-t}$:\n\\[\n 2^{-\\delta(x,z)}\n \\;\\leq\\;\n 2^{-\\\ + min(\\delta(x,y),\\, \\delta(y,z))}\n \\;=\\;\n \\max\\bigl(2^{-\\delta(x,y)},\\, 2^{-\\delta(y,z)}\\bigr),\n\\\ + ]\nwhich is exactly $d(x,z) \\leq \\max\\bigl(d(x,y),\\, d(y,z)\\bigr)$.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 25\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch10-ultrametric-depth.tex`\ + \ lines 229-238\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.Ultrametric`\n\ + - Name: `triangle_check`\n\n## Dependencies\n\n- Canonical: II.D13, II.D12\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Ultrametric Triangle Inequality +noindex: false +--- + diff --git a/_corpus_v3_items/cid001368.md b/_corpus_v3_items/cid001368.md new file mode 100644 index 000000000..62e563b6d --- /dev/null +++ b/_corpus_v3_items/cid001368.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001368/ +title: Holomorphic Implies Continuous +cid: cid001368 +primary_alias: THM0072 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Holomorphic Implies Continuous +aliases: +- THM0072 +- II.T06 +- holomorphic-implies-continuous +- thm:hol-implies-cont +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 2, Chapter 11 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Holomorphic Implies Continuous**\n\nHolomorphic Implies Continuous\n\n# Holomorphic Implies Continuous\n\n\ + ## Summary\n\nHolomorphic Implies Continuous\n\n## Statement\n\n```latex\n%\n\\label{thm:hol-implies-cont}\nEvery $\\\ + omega$-germ transformer $f \\colon \\tau^3 \\to \\tau^3$\nis continuous with respect to the cylinder topology\n(equivalently,\ + \ the ultrametric topology).\n```\n\n## Proof / Justification\n\n```latex\nWe give two formulations of the proof:\none\ + \ via open preimages, one via the ultrametric.\n\n\\medskip\n\n\\noindent\\textbf{Open-preimage formulation.}\nBy the\ + \ Cylinder Basis Theorem\n(II.T04, Chapter~\\ref{ch:cylinder-domains}),\nthe stage-$k$ cylinders form a basis for the\ + \ topology.\nIt suffices to show that the preimage of every basis set is open.\n\nLet $C_k(z)$ be a cylinder in the codomain.\n\ + Consider the preimage $f^{-1}(C_k(z))$.\nIf $f^{-1}(C_k(z)) = \\varnothing$, it is trivially open.\nOtherwise, let $x\ + \ \\in f^{-1}(C_k(z))$,\nso that $f(x) \\in C_k(z)$,\ni.e., $\\pi_k(f(x)) = \\pi_k(z)$.\nBy the cylinder compatibility\ + \ lemma\n(Lemma~\\ref{lem:naturality-cylinder}),\n$f(C_k(x)) \\subseteq C_k(f(x)) = C_k(z)$,\nsince $\\pi_k(f(x)) = \\\ + pi_k(z)$.\nHence $C_k(x) \\subseteq f^{-1}(C_k(z))$.\nSince $C_k(x)$ is open\n(cylinders are clopen, Chapter~\\ref{ch:cylinder-domains})\n\ + and contains~$x$,\nevery point of $f^{-1}(C_k(z))$ has an open neighborhood\ncontained in the preimage.\nTherefore $f^{-1}(C_k(z))$\ + \ is open.\n\nSince the preimage of every basis open set is open,\n$f$ is continuous.\n\n\\medskip\n\n\\noindent\\textbf{Ultrametric\ + \ formulation.}\nLet $d$ denote the ultrametric distance\n(II.D13, Chapter~\\ref{ch:ultrametric-depth}),\ndefined by $d(x,\ + \ y) = 2^{-\\delta(x,y)}$\nwhere $\\delta(x, y)$ is the first disagreement depth.\nWe claim that $f$ is $1$-Lipschitz:\n\ + \\[\n \\boxed{d\\bigl(f(x),\\, f(y)\\bigr)\n \\;\\leq\\;\n d(x,\\, y)\n \\qquad \\text{for all } x, y \\in\ + \ \\tau^3.}\n\\]\nTo see this, suppose $\\delta(x, y) = k$,\nmeaning $x$ and~$y$ agree at stages $1, \\ldots, k$\nbut\ + \ disagree at stage~$k+1$.\nIn particular, $\\pi_k(x) = \\pi_k(y)$,\nso $y \\in C_k(x)$.\nBy Lemma~\\ref{lem:naturality-cylinder},\n\ + $f(y) \\in C_k(f(x))$,\nwhich means $\\pi_k(f(y)) = \\pi_k(f(x))$,\ni.e., $f(x)$ and~$f(y)$ agree at stage~$k$.\nTherefore\ + \ $\\delta(f(x), f(y)) \\geq k = \\delta(x, y)$, and so\n\\[\n d\\bigl(f(x), f(y)\\bigr)\n \\;=\\; 2^{-\\delta(f(x),\ + \ f(y))}\n \\;\\leq\\; 2^{-\\delta(x, y)}\n \\;=\\; d(x, y).\n\\]\nEvery $1$-Lipschitz map is continuous.\nIn fact,\ + \ $f$ is \\emph{uniformly} continuous.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 27\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part02/ch11-hol-implies-cont.tex` lines 334-340\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Domains.HolImpliesCont`\n- Name:\ + \ `hol_cont_check`\n\n## Dependencies\n\n- Canonical: II.L01, II.D10, I.D47\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Holomorphic Implies Continuous +noindex: false +--- + diff --git a/_corpus_v3_items/cid001369.md b/_corpus_v3_items/cid001369.md new file mode 100644 index 000000000..0567a7003 --- /dev/null +++ b/_corpus_v3_items/cid001369.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001369/ +title: Compactness +cid: cid001369 +primary_alias: THM0073 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The topological space tau^3 with the cylinder topology is compact, via Tychonoff's theorem applied to the inverse + limit of finite discrete spaces. +aliases: +- THM0073 +- II.T07 +- compactness +- thm:compactness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Compactness**\n\nThe topological space tau^3 with the cylinder topology is compact, via Tychonoff's theorem\ + \ applied to the inverse limit of finite discrete spaces.\n\n# Compactness\n\n## Summary\n\nThe topological space tau^3\ + \ with the cylinder topology is compact, via Tychonoff's theorem applied to the inverse limit of finite discrete spaces.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:compactness}\nThe topological space $\\tau^3$, equipped with\nthe cylinder\ + \ topology\n(Theorem~\\textup{\\ref{thm:cylinder-basis}}),\nis compact.\n```\n\n## Proof / Justification\n\n```latex\n\ + The inverse-limit construction provides the proof.\nBy definition (Chapter~\\ref{ch:tau-admissible-points}),\n\\[\n \ + \ \\tau^3\n \\;=\\;\n \\varprojlim_{k}\\, \\mathbb{Z}/P_k\\mathbb{Z},\n\\]\nwhere $P_k = p_1 \\cdot p_2 \\cdots\ + \ p_k$\nis the $k$th primorial\nand the transition maps\n$\\pi_{k,l} : \\mathbb{Z}/P_l\\mathbb{Z}\n\\to \\mathbb{Z}/P_k\\\ + mathbb{Z}$\nare the canonical reductions\n(I.T18, Book~I).\n\nEach factor $\\mathbb{Z}/P_k\\mathbb{Z}$\nis a finite set\ + \ with $P_k$ elements,\nhence compact and discrete.\nBy Tychonoff's theorem,\nthe product\n$\\prod_{k \\geq 1} \\mathbb{Z}/P_k\\\ + mathbb{Z}$\nis compact.\nThe inverse limit $\\tau^3$ is a closed subset\nof this product:\nit is defined by the equational\ + \ conditions\n$\\pi_{k,l}(x_l) = x_k$ for all $k < l$,\nand each such condition cuts out a closed subset\n(preimage of\ + \ the diagonal\nunder the continuous projection).\nA closed subset of a compact space is compact.\n\nHence $\\tau^3$ is\ + \ compact.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 31\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch13-stone-space.tex`\ + \ lines 113-120\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.StoneSpace`\n\ + - Name: `finite_subcover_check`\n\n## Dependencies\n\n- Canonical: II.D10, I.T18\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The topological space tau^3 with the cylinder topology is compact, via Tychonoff's theorem applied to the inverse + limit of finite discrete spaces. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001370.md b/_corpus_v3_items/cid001370.md new file mode 100644 index 000000000..8a73a6e68 --- /dev/null +++ b/_corpus_v3_items/cid001370.md @@ -0,0 +1,176 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001370/ +title: Hausdorff Property +cid: cid001370 +primary_alias: THM0074 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'tau^3 is Hausdorff: any two distinct points are separated by disjoint cylinder neighborhoods at the first disagreement + depth.' +aliases: +- THM0074 +- II.T08 +- hausdorff-property +- thm:hausdorff +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hausdorff Property** + + + tau^3 is Hausdorff: any two distinct points are separated by disjoint cylinder neighborhoods at the first disagreement + depth. + + + # Hausdorff Property + + + ## Summary + + + tau^3 is Hausdorff: any two distinct points are separated by disjoint cylinder neighborhoods at the first disagreement + depth. + + + ## Statement + + + ```latex + + % + + \label{thm:hausdorff} + + The topological space $\tau^3$ is Hausdorff: + + for any two distinct points $x, y \in \tau^3$, + + there exist disjoint open sets $U \ni x$ and $V \ni y$. + + ``` + + + ## Proof / Justification + + + ```latex + + Let $x \neq y$. + + The ultrametric distance satisfies + + $d(x,y) > 0$ + + (Definition~\ref{def:ultrametric-distance}, + + identity of indiscernibles). + + Let $\delta(x,y) = k < \infty$ + + be the first disagreement depth + + (Definition~\ref{def:first-disagreement-depth}). + + Then $\pi_{k+1}(x) \neq \pi_{k+1}(y)$, + + so the stage-$(k{+}1)$ cylinders + + $C_{k+1}(x)$ and $C_{k+1}(y)$ are disjoint: + + they are distinct elements of the partition + + $\{C_{k+1}(r) : r \in \mathbb{Z}/P_{k+1}\mathbb{Z}\}$. + + + Set $U = C_{k+1}(x)$ and $V = C_{k+1}(y)$. + + Both are open (they are basis elements), + + $x \in U$, $y \in V$, + + and $U \cap V = \varnothing$. + + ``` + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 32 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch13-stone-space.tex` lines 303-309 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Topology.StoneSpace` + + - Name: `hausdorff_check` + + + ## Dependencies + + + - Canonical: II.D13, II.T05 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'tau^3 is Hausdorff: any two distinct points are separated by disjoint cylinder neighborhoods at the first disagreement + depth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001371.md b/_corpus_v3_items/cid001371.md new file mode 100644 index 000000000..d3ab1091c --- /dev/null +++ b/_corpus_v3_items/cid001371.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001371/ +title: Total Disconnectedness +cid: cid001371 +primary_alias: THM0075 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Total Disconnectedness +aliases: +- THM0075 +- II.T09 +- total-disconnectedness +- thm:totally-disconnected +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 13 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Total Disconnectedness**\n\nTotal Disconnectedness\n\n# Total Disconnectedness\n\n## Summary\n\nTotal Disconnectedness\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:totally-disconnected}\nThe topological space $\\tau^3$ is totally disconnected:\n\ + the only connected subsets of~$\\tau^3$ are singletons.\n```\n\n## Proof / Justification\n\n```latex\nLet $S \\subseteq\ + \ \\tau^3$ contain two distinct points\n$x, y \\in S$ with $x \\neq y$.\nWe show that $S$ is disconnected.\n\nLet $\\\ + delta(x,y) = k$.\nThe stage-$(k{+}1)$ cylinder $C_{k+1}(x)$\nis a clopen set\n(Definition~\\ref{def:clopen-basis},\nProposition~\\\ + ref{prop:ch09-complement},\nChapter~\\ref{ch:cylinder-domains}).\nDefine\n\\[\n U \\;:=\\; S \\cap C_{k+1}(x),\n \ + \ \\qquad\n V \\;:=\\; S \\setminus C_{k+1}(x)\n \\;=\\; S \\cap \\bigl(\\tau^3 \\setminus C_{k+1}(x)\\bigr).\n\ + \\]\nSince $C_{k+1}(x)$ is clopen in~$\\tau^3$,\nboth $U$ and $V$ are open in the subspace topology on~$S$.\nWe have $x\ + \ \\in U$ (since $x \\in C_{k+1}(x)$)\nand $y \\in V$ (since $\\pi_{k+1}(y) \\neq \\pi_{k+1}(x)$\nimplies $y \\notin C_{k+1}(x)$).\n\ + Furthermore, $U \\cap V = \\varnothing$ and $U \\cup V = S$.\nHence $S$ is disconnected.\n\nSince every subset with more\ + \ than one point\nis disconnected,\nthe connected components of~$\\tau^3$ are singletons.\n```\n\n## Source Context\n\n\ + - Registry source: `book-02.jsonl` line 33\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch13-stone-space.tex`\ + \ lines 359-364\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.StoneSpace`\n\ + - Name: `totally_disconnected_check`\n\n## Dependencies\n\n- Canonical: II.D11, II.T04\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Total Disconnectedness +noindex: false +--- + diff --git a/_corpus_v3_items/cid001372.md b/_corpus_v3_items/cid001372.md new file mode 100644 index 000000000..f0cb34100 --- /dev/null +++ b/_corpus_v3_items/cid001372.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001372/ +title: Topology Uniqueness +cid: cid001372 +primary_alias: THM0076 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The profinite topology on tau^3 is the unique topology satisfying CRT continuity, Hausdorff separation, and compactness. +aliases: +- THM0076 +- II.T10 +- topology-uniqueness +- thm:topology-uniqueness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 14 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Topology Uniqueness**\n\nThe profinite topology on tau^3 is the unique topology satisfying CRT continuity,\ + \ Hausdorff separation, and compactness.\n\n# Topology Uniqueness\n\n## Summary\n\nThe profinite topology on tau^3 is\ + \ the unique topology satisfying CRT continuity, Hausdorff separation, and compactness.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:topology-uniqueness}\nLet $\\mathcal{T}$ be a topology on~$\\tau^3$ satisfying:\n\\begin{enumerate}\n \ + \ \\item[\\textup{(a)}]\n \\textbf{CRT continuity.}\n All reduction maps\n $\\pi_k : (\\\ + tau^3, \\mathcal{T}) \\to \\mathbb{Z}/P_k$\n are continuous\n \\textup{(}discrete topology on targets\\\ + textup{)}.\n \\item[\\textup{(b)}]\n \\textbf{Hausdorff separation.}\n $(\\tau^3, \\mathcal{T})$\ + \ is Hausdorff.\n \\item[\\textup{(c)}]\n \\textbf{Compactness.}\n $(\\tau^3, \\mathcal{T})$ is compact.\n\ + \\end{enumerate}\nThen $\\mathcal{T}$ equals the profinite topology:\n\\[\n \\boxed{\\mathcal{T}\n \\;=\\;\n \ + \ \\mathcal{T}_{\\mathrm{pro}}\n \\;:=\\;\n \\mathrm{Initial}\\bigl(\\pi_k : k \\geq 1\\bigr).}\n\\]\n```\n\n##\ + \ Proof / Justification\n\n```latex\nLet $\\mathcal{T}_{\\mathrm{pro}}$\ndenote the profinite (cylinder) topology.\n\n\ + \\medskip\n\\noindent\\textbf{Step 1.}\nBy hypothesis~(a), each~$\\pi_k$ is $\\mathcal{T}$-continuous.\nSince $\\mathcal{T}_{\\\ + mathrm{pro}}$\nis the \\emph{coarsest} topology\nmaking all~$\\pi_k$ continuous\n(Proposition~\\ref{prop:ch14-initial-topology}),\n\ + $\\mathcal{T}_{\\mathrm{pro}} \\subseteq \\mathcal{T}$.\n\n\\medskip\n\\noindent\\textbf{Step 2.}\nThe identity map\n\ + $\\mathrm{id} : (\\tau^3, \\mathcal{T})\n\\to (\\tau^3, \\mathcal{T}_{\\mathrm{pro}})$\nis continuous by Step~1.\nBy hypothesis~(c),\ + \ the domain is compact.\nBy II.T08, the codomain is Hausdorff.\nA continuous bijection from a compact space\nto a Hausdorff\ + \ space is a homeomorphism,\nso $\\mathcal{T} = \\mathcal{T}_{\\mathrm{pro}}$.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 34\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch14-topology-invariant.tex`\ + \ lines 143-169\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.Invariant`\n\ + - Name: `topology_unique_check`\n\n## Dependencies\n\n- Canonical: II.T04, II.T07, II.T08\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The profinite topology on tau^3 is the unique topology satisfying CRT continuity, Hausdorff separation, and compactness. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001373.md b/_corpus_v3_items/cid001373.md new file mode 100644 index 000000000..bddbe96bc --- /dev/null +++ b/_corpus_v3_items/cid001373.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001373/ +title: Dimension Four +cid: cid001373 +primary_alias: THM0077 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The topological dimension of tau is exactly 4: four ABCD rays jointly separate all points (upper bound), pairwise + independence shows no triple suffices (lower bound).' +aliases: +- THM0077 +- II.T11 +- dimension-four +- thm:dimension-four +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Dimension Four**\n\nThe topological dimension of tau is exactly 4: four ABCD rays jointly separate all points\ + \ (upper bound), pairwise independence shows no triple suffices (lower bound).\n\n# Dimension Four\n\n## Summary\n\nThe\ + \ topological dimension of tau is exactly 4: four ABCD rays jointly separate all points (upper bound), pairwise independence\ + \ shows no triple suffices (lower bound).\n\n## Statement\n\n```latex\n%\n\\label{thm:dimension-four}\n\\[\n \\boxed{\\\ + dim_\\tau \\;=\\; 4.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Upper bound: $\\dim_\\tau \\leq 4$.}\n\ + The ABCD chart (I.D17, Book~I) is a bijection\nbetween $\\tau$-admissible quadruples\nand objects of Category~$\\tau$\n\ + (Hyperfactorization Theorem, I.T04, Book~I).\nTwo points with identical ABCD coordinates\nat every stage are the same\ + \ point,\nso the four rays jointly separate all points.\n\n\\emph{Lower bound: $\\dim_\\tau \\geq 4$.}\nWe show that no\ + \ triple of rays suffices.\nFor each of the four triples\n$\\{A,B,C\\}$, $\\{D,B,C\\}$, $\\{D,A,C\\}$, $\\{D,A,B\\}$,\n\ + pairwise independence (I.P08, Book~I)\nprovides two distinct points\nthat agree on the three selected coordinates\nat\ + \ every stage but differ on the fourth:\n\\begin{itemize}\n \\item \\textbf{$\\{A,B,C\\}$} (missing~D):\n \ + \ $D$ is independent of $A$, $B$, $C$,\n so fixing $(A,B,C)$ leaves $D$ free.\n \\item \\textbf{$\\{D,B,C\\\ + }$} (missing~A):\n $A$ is independent of $D$, $B$, $C$.\n \\item \\textbf{$\\{D,A,C\\}$} (missing~B):\n \ + \ $B$ is independent of $D$, $A$, $C$.\n \\item \\textbf{$\\{D,A,B\\}$} (missing~C):\n $C$ is independent\ + \ of $D$, $A$, $B$.\n\\end{itemize}\nEach triple fails to separate some pair.\nHence $\\dim_\\tau \\geq 4$.\n```\n\n##\ + \ Source Context\n\n- Registry source: `book-02.jsonl` line 37\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch15-dimension-four.tex`\ + \ lines 174-180\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.DimensionFour`\n\ + - Name: `dim_four_check`\n\n## Dependencies\n\n- Canonical: II.D15, I.T04, I.P08\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The topological dimension of tau is exactly 4: four ABCD rays jointly separate all points (upper bound), pairwise + independence shows no triple suffices (lower bound).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001374.md b/_corpus_v3_items/cid001374.md new file mode 100644 index 000000000..ca668cfd2 --- /dev/null +++ b/_corpus_v3_items/cid001374.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001374/ +title: Boundary Minimality +cid: cid001374 +primary_alias: THM0078 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The boundary lemniscate L = S^1 v S^1 is the minimal quotient of T^2 preserving both U(1) gauge factors, reducing + codimension, with unique crossing point. +aliases: +- THM0078 +- II.T12 +- boundary-minimality +- thm:boundary-minimality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary Minimality**\n\nThe boundary lemniscate L = S^1 v S^1 is the minimal quotient of T^2 preserving both\ + \ U(1) gauge factors, reducing codimension, with unique crossing point.\n\n# Boundary Minimality\n\n## Summary\n\nThe\ + \ boundary lemniscate L = S^1 v S^1 is the minimal quotient of T^2 preserving both U(1) gauge factors, reducing codimension,\ + \ with unique crossing point.\n\n## Statement\n\n```latex\n%\n\\label{thm:boundary-minimality}\nThe boundary lemniscate\ + \ $\\mathbb{L} = S^1 \\vee S^1$\nis the \\textbf{minimal} topological quotient\nof the fiber torus~$T^2$\nsatisfying three\ + \ constraints:\n\\begin{enumerate}\n \\item[\\textup{(M1)}] \\textbf{Gauge preservation.}\n Both $U(1)_\\\ + gamma$ and $U(1)_\\eta$\n survive as closed subgroups.\n \\item[\\textup{(M2)}] \\textbf{Codimension increase.}\n\ + \ $\\dim Q < \\dim T^2 = 2$.\n \\item[\\textup{(M3)}] \\textbf{Crossing point.}\n The quotient admits\ + \ a unique singular point\n where the two gauge factors meet.\n\\end{enumerate}\nAny quotient~$Q$ of~$T^2$\n\ + satisfying \\textup{(M1)--(M3)}\ncontains a homeomorphic copy of~$\\mathbb{L}$.\n```\n\n## Proof / Justification\n\n```latex\n\ + \\emph{Step~1.}\nConstraint~(M2) forces $\\dim Q \\leq 1$,\nso the quotient must collapse at least one direction.\n\n\\\ + emph{Step~2.}\nIf $S^1_\\gamma$ collapses to a point,\n$U(1)_\\gamma$ acts trivially on~$Q$,\nviolating~(M1).\nSymmetrically\ + \ for $S^1_\\eta$.\nHence both circles survive.\n\n\\emph{Step~3.}\nBoth circles survive with $\\dim Q = 1$.\nWithout\ + \ a shared point,\n$Q \\supseteq S^1 \\sqcup S^1$ is disconnected\nwith no crossing point, violating~(M3).\nA single shared\ + \ point yields\n$Q \\supseteq S^1 \\vee S^1 = \\mathbb{L}$.\n\n\\emph{Step~4.}\nIf $Q$ had fewer points than $\\mathbb{L}$,\n\ + some identification would collapse\na portion of one $S^1$ factor, violating~(M1).\nHence $\\mathbb{L}$ is minimal.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 39\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch16-boundary-minimality.tex`\ + \ lines 83-103\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.BoundaryMinimality`\n\ + - Name: `boundary_minimal_check`\n\n## Dependencies\n\n- Canonical: I.T05, I.D18, II.D14\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The boundary lemniscate L = S^1 v S^1 is the minimal quotient of T^2 preserving both U(1) gauge factors, reducing + codimension, with unique crossing point. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001375.md b/_corpus_v3_items/cid001375.md new file mode 100644 index 000000000..88343c12e --- /dev/null +++ b/_corpus_v3_items/cid001375.md @@ -0,0 +1,217 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001375/ +title: Torus Degeneration Theorem +cid: cid001375 +primary_alias: THM0079 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Torus Degeneration Theorem +aliases: +- THM0079 +- II.T13 +- torus-degeneration-theorem +- thm:torus-degeneration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Torus Degeneration Theorem** + + + Torus Degeneration Theorem + + + # Torus Degeneration Theorem + + + ## Summary + + + Torus Degeneration Theorem + + + ## Statement + + + ```latex + + % + + \label{thm:torus-degeneration} + + The pinch map + + $p : T^2 \to \mathbb{L} = S^1 \vee S^1$ + + is the unique continuous surjection + + from~$T^2$ to a one-dimensional compact connected target + + satisfying constraints \textup{(U1)--(U5)}. + + ``` + + + ## Proof / Justification + + + ```latex + + Each constraint eliminates alternatives. + + + \emph{Step~1: Target topology.} + + By (U1), (U2), (U5), the target is a compact connected + + one-dimensional Hausdorff space---hence + + homeomorphic to $S^1$, $[0,1]$, + + or a finite graph. + + + \emph{Step~2: Gauge survival eliminates simple targets.} + + By (U3), both $U(1)_\gamma$ and $U(1)_\eta$ + + must survive independently. + + A single $S^1$ admits only one independent + + $U(1)$-action; $[0,1]$ admits none. + + The target must contain at least two circle factors. + + + \emph{Step~3: Connectivity forces the wedge.} + + By (U2), the target is connected. + + Two disjoint circles fail connectivity. + + The minimal connected space + + containing two independent circles + + is $S^1 \vee S^1$. + + Gluing along arcs would destroy + + a $U(1)$-action or reduce dimension further. + + + \emph{Step~4: The identification is the diagonal.} + + The quotient must collapse a one-dimensional subspace + + to the wedge point. + + Collapsing a $U(1)$-orbit $\{\theta_\gamma = \mathrm{const}\}$ + + or $\{\theta_\eta = \mathrm{const}\}$ + + would destroy that gauge factor. + + The diagonal $\Delta = \{(\theta, \theta)\}$ + + is the unique closed one-dimensional submanifold + + transverse to both gauge orbits, + + so collapsing it preserves both actions. + + + \emph{Step~5: Continuity.} + + The quotient $T^2 \to T^2/\Delta$ is continuous + + by the universal property. + + Since $\Delta$ is closed in the compact Hausdorff space $T^2$, + + the quotient is Hausdorff. + + ``` + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 42 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch17-torus-degeneration.tex` lines + 214-222 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Topology.TorusDegeneration` + + - Name: `pinch_surjective_check` + + + ## Dependencies + + + - Canonical: II.D06, I.D18, I.T05 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Torus Degeneration Theorem +noindex: false +--- + diff --git a/_corpus_v3_items/cid001376.md b/_corpus_v3_items/cid001376.md new file mode 100644 index 000000000..0dab6b451 --- /dev/null +++ b/_corpus_v3_items/cid001376.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001376/ +title: Fundamental Group Degeneration +cid: cid001376 +primary_alias: THM0080 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Fundamental Group Degeneration +aliases: +- THM0080 +- II.T14 +- fundamental-group-degeneration +- thm:fundamental-group-degeneration +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fundamental Group Degeneration**\n\nFundamental Group Degeneration\n\n# Fundamental Group Degeneration\n\n\ + ## Summary\n\nFundamental Group Degeneration\n\n## Statement\n\n```latex\n%\n\\label{thm:fundamental-group-degeneration}\n\ + The pinch map induces a surjection on fundamental groups:\n\\[\n \\boxed{%\n \\pi_1(T^2)\n \\;=\\;\n \\mathbb{Z}^2\n\ + \ \\;\\xrightarrow{\\quad p_* \\quad}\\;\n \\pi_1(\\mathbb{L})\n \\;=\\;\n F_2,}\n\\]\nwhere $F_2$ is the\ + \ free group\non generators $a$ ($+$-lobe loop)\nand $b$ ($-$-lobe loop).\n```\n\n## Proof / Justification\n\n```latex\n\ + $\\pi_1(T^2) = \\mathbb{Z} \\times \\mathbb{Z}$,\ngenerated by coordinate loops\n$\\gamma_0 : t \\mapsto (t, 0)$\nand\ + \ $\\eta_0 : t \\mapsto (0, t)$,\nwhich commute:\n$[\\gamma_0] \\cdot [\\eta_0]\n= [\\eta_0] \\cdot [\\gamma_0]$.\nBy\ + \ van~Kampen,\n$\\pi_1(S^1 \\vee S^1) = F_2$\nwith generators $a = [p_*(\\gamma_0)]$\nand $b = [p_*(\\eta_0)]$\nthat do\ + \ \\emph{not} commute: $ab \\neq ba$.\nThe pinch map sends the abelian generators\nto free generators;\nthe commutativity\ + \ relation ceases to hold.\nSurjectivity follows because $\\gamma_0, \\eta_0$\ngenerate $\\pi_1(T^2)$.\nThe relationship\ + \ is:\n$F_2^{\\mathrm{ab}} = \\mathbb{Z}^2$---the\nabelianization map run in reverse.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 44\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part03/ch17-torus-degeneration.tex`\ + \ lines 350-367\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Topology.TorusDegeneration`\n\ + - Name: `fund_group_check`\n\n## Dependencies\n\n- Canonical: II.T13, II.D18\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Fundamental Group Degeneration +noindex: false +--- + diff --git a/_corpus_v3_items/cid001377.md b/_corpus_v3_items/cid001377.md new file mode 100644 index 000000000..5ebda6a22 --- /dev/null +++ b/_corpus_v3_items/cid001377.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001377/ +title: Betweenness Axioms +cid: cid001377 +primary_alias: THM0081 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The betweenness relation from the ultrametric distance satisfies all three Tarski betweenness axioms (identity, transitivity, + connectivity). +aliases: +- THM0081 +- II.T15 +- betweenness-axioms +- thm:betweenness-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 18 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Betweenness Axioms**\n\nThe betweenness relation from the ultrametric distance satisfies all three Tarski betweenness\ + \ axioms (identity, transitivity, connectivity).\n\n# Betweenness Axioms\n\n## Summary\n\nThe betweenness relation from\ + \ the ultrametric distance satisfies all three Tarski betweenness axioms (identity, transitivity, connectivity).\n\n##\ + \ Statement\n\n```latex\n%\n\\label{thm:betweenness-axioms}\nThe betweenness relation $B(x,y,z)$\n(Definition~\\textup{\\\ + ref{def:betweenness}}, II.D19),\nderived from the ultrametric distance on~$\\tau^3$,\nsatisfies the three Tarski betweenness\ + \ axioms:\n\\begin{enumerate}\n \\item[\\textup{(T1)}] \\textbf{Identity.}\n $B(x,y,x) \\implies x = y$\n\ + \ \\textup{(Proposition~\\ref{prop:ch18-t1})}.\n \\item[\\textup{(T2)}] \\textbf{Transitivity.}\n \ + \ $B(x,y,z) \\wedge B(y,z,w) \\implies B(x,y,w)$\n \\textup{(Proposition~\\ref{prop:ch18-t2})}.\n \\item[\\\ + textup{(T3)}] \\textbf{Connectivity.}\n For any $x, y, z \\in \\tau^3$,\n at least one of $B(x,y,z)$,\ + \ $B(y,x,z)$,\n $B(x,z,y)$ holds\n \\textup{(Proposition~\\ref{prop:ch18-t3})}.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n```latex\nProved individually in\nSections~\\ref{sec:ch18-tarski-t1}--\\ref{sec:ch18-tarski-t3}.\n\ + All proofs use only the ultrametric distance\n(II.D13),\nthe ultrametric triangle inequality (II.T05),\nand the identity\ + \ of indiscernibles.\nNo geometric axioms were invoked.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 50\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch18-betweenness.tex`\ + \ lines 261-281\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.Betweenness`\n\ + - Name: `between_identity_check`\n\n## Dependencies\n\n- Canonical: II.D19, II.D13\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The betweenness relation from the ultrametric distance satisfies all three Tarski betweenness axioms (identity, + transitivity, connectivity). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001378.md b/_corpus_v3_items/cid001378.md new file mode 100644 index 000000000..81aa3a258 --- /dev/null +++ b/_corpus_v3_items/cid001378.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001378/ +title: Congruence Axioms +cid: cid001378 +primary_alias: THM0082 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The ultrametric congruence relation satisfies all six Tarski congruence axioms (reflexivity, identity, transitivity, + segment construction, five-segment, inner transitivity). +aliases: +- THM0082 +- II.T16 +- congruence-axioms +- thm:congruence-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 19 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Congruence Axioms**\n\nThe ultrametric congruence relation satisfies all six Tarski congruence axioms (reflexivity,\ + \ identity, transitivity, segment construction, five-segment, inner transitivity).\n\n# Congruence Axioms\n\n## Summary\n\ + \nThe ultrametric congruence relation satisfies all six Tarski congruence axioms (reflexivity, identity, transitivity,\ + \ segment construction, five-segment, inner transitivity).\n\n## Statement\n\n```latex\n%\n\\label{thm:congruence-axioms}\n\ + The congruence relation $\\cong$\n(Definition~\\ref{def:congruence})\nsatisfies the Tarski congruence axioms:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(C1)}] Reflexivity:\n $AB \\cong BA$.\n \\item[\\textup{(C2)}] Identity of congruence:\n\ + \ $AB \\cong CC \\implies A = B$.\n \\item[\\textup{(C3)}] Transitivity:\n $AB \\cong CD,\\; CD \\\ + cong EF \\implies AB \\cong EF$.\n \\item[\\textup{(C4)}] Segment construction:\n given $A$, $B$, $C$, $D$\ + \ with $C \\neq D$,\n there exists~$E$ with $B(C,D,E)$\n and $DE \\cong AB$.\n \\item[\\textup{(C5)}]\ + \ Five-segment:\n $A \\neq B$,\n $B(A,B,C)$, $B(A',B',C')$,\n $AB \\cong A'B'$, $BC \\cong\ + \ B'C'$,\n $AD \\cong A'D'$, $BD \\cong B'D'$\n imply $CD \\cong C'D'$.\n \\item[\\textup{(C6)}]\ + \ Inner transitivity:\n $B(A,B,D)$, $B(A,C,E)$,\n $AB \\cong AC$, $BD \\cong CE$\n imply $AD\ + \ \\cong AE$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nPropositions~\\ref{prop:ch19-c1-c3},\n\\\ + ref{prop:ch19-c4},\n\\ref{prop:ch19-c5},\nand~\\ref{prop:ch19-c6}\nverify each axiom individually.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 52\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch19-congruence.tex`\ + \ lines 279-307\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.Congruence`\n\ + - Name: `congruence_reflexive_check`\n\n## Dependencies\n\n- Canonical: II.D20, II.D13\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The ultrametric congruence relation satisfies all six Tarski congruence axioms (reflexivity, identity, transitivity, + segment construction, five-segment, inner transitivity). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001379.md b/_corpus_v3_items/cid001379.md new file mode 100644 index 000000000..b0652322a --- /dev/null +++ b/_corpus_v3_items/cid001379.md @@ -0,0 +1,141 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001379/ +title: Pasch Axiom +cid: cid001379 +primary_alias: THM0083 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Pasch axiom holds in tau^3: the proof reduces to combinatorial cylinder containment via the ultrametric isosceles + property.' +aliases: +- THM0083 +- II.T17 +- pasch-axiom +- thm:pasch +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 20 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Pasch Axiom** + + + The Pasch axiom holds in tau^3: the proof reduces to combinatorial cylinder containment via the ultrametric isosceles + property. + + + # Pasch Axiom + + + ## Summary + + + The Pasch axiom holds in tau^3: the proof reduces to combinatorial cylinder containment via the ultrametric isosceles + property. + + + ## Statement + + + ```latex + + % + + \label{thm:pasch} + + Let $a, b, c, p, q \in \tau^3$ + + with $B(a, p, c)$ and $B(b, q, c)$. + + Then there exists $x \in \tau^3$ + + such that $B(p, x, b)$ and $B(q, x, a)$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 53 + + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch20-pasch-parallel.tex` lines 85-92 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Geometry.PaschParallel` + + - Name: `pasch_spot_check` + + + ## Dependencies + + + - Canonical: II.D19, II.D20, II.T15 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Pasch axiom holds in tau^3: the proof reduces to combinatorial cylinder containment via the ultrametric + isosceles property.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001380.md b/_corpus_v3_items/cid001380.md new file mode 100644 index 000000000..83cc2b95e --- /dev/null +++ b/_corpus_v3_items/cid001380.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001380/ +title: Parallel Postulate +cid: cid001380 +primary_alias: THM0084 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The parallel postulate holds in tau^3: for any line l and point p not on l, there exists a unique parallel through + p, constructed via CRT product of prime-order affine spaces.' +aliases: +- THM0084 +- II.T18 +- parallel-postulate +- thm:parallel-postulate +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 20 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Parallel Postulate**\n\nThe parallel postulate holds in tau^3: for any line l and point p not on l, there exists\ + \ a unique parallel through p, constructed via CRT product of prime-order affine spaces.\n\n# Parallel Postulate\n\n##\ + \ Summary\n\nThe parallel postulate holds in tau^3: for any line l and point p not on l, there exists a unique parallel\ + \ through p, constructed via CRT product of prime-order affine spaces.\n\n## Statement\n\n```latex\n%\n\\label{thm:parallel-postulate}\n\ + Let $\\ell$ be a line in $\\tau^3$\nand $p \\in \\tau^3 \\setminus \\ell$.\nThere exists a unique line $\\ell'$ through~$p$\n\ + with $\\ell' \\cap \\ell = \\varnothing$.\n```\n\n## Proof / Justification\n\n```latex\nFix $a, b \\in \\ell$ with $a\ + \ \\neq b$.\n\n\\smallskip\n\\noindent\\textbf{Step 1: Separation.}\nSince $p \\notin \\ell$, Hausdorff separation\n(Chapter~\\\ + ref{ch:cylinder-domains})\ngives a stage $k_1$ with\n$\\pi_{k_1}(p) \\notin \\pi_{k_1}(\\ell)$.\n\n\\smallskip\n\\noindent\\\ + textbf{Step 2: Unique parallel at each stage.}\nFor $k \\geq k_1$, the CRT decomposition\n$\\mathbb{Z}/P_k \\cong \\prod_{i=1}^k\ + \ \\mathbb{Z}/p_i$\ngives a product of prime-order affine spaces.\nIn each factor $\\mathbb{Z}/p_i$\n(a 1-dimensional\ + \ affine space over $\\mathbb{F}_{p_i}$),\nthe parallel through $\\pi_k(p)$ is unique.\nCall the product parallel $\\\ + ell_k'$.\n\n\\smallskip\n\\noindent\\textbf{Step 3: Coherence.}\nFor $k < l$, the reduction $\\pi_{k,l}(\\ell_l')$\ncoincides\ + \ with $\\ell_k'$:\nreduction maps preserve betweenness\nand uniqueness at each stage\nforces compatibility.\n\n\\smallskip\n\ + \\noindent\\textbf{Step 4: Inverse-limit assembly.}\nThe coherent family defines\n\\[\n \\ell' \\;=\\; \\varprojlim_k\ + \ \\ell_k'\n \\;=\\;\n \\bigl\\{\\, x \\in \\tau^3\n \\;\\big|\\;\n \\pi_k(x) \\in \\ell_k'\n \\text{ for\ + \ all } k \\geq k_1 \\,\\bigr\\}.\n\\]\nBy construction, $p \\in \\ell'$\nand $\\ell' \\cap \\ell = \\varnothing$\n(cylinder\ + \ separation lifts to the limit).\n\n\\smallskip\n\\noindent\\textbf{Step 5: Uniqueness.}\nIf $\\ell''$ is another parallel\ + \ through~$p$,\nthen $\\pi_k(\\ell'')$ is a parallel to $\\pi_k(\\ell)$\nthrough $\\pi_k(p)$ at each stage.\nStep~2 uniqueness\ + \ forces\n$\\pi_k(\\ell'') = \\ell_k' = \\pi_k(\\ell')$\nfor all~$k$, so $\\ell'' = \\ell'$.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 54\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch20-pasch-parallel.tex`\ + \ lines 192-199\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.PaschParallel`\n\ + - Name: `parallel_unique_check`\n\n## Dependencies\n\n- Canonical: II.D10, II.D14, II.T15\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The parallel postulate holds in tau^3: for any line l and point p not on l, there exists a unique parallel through + p, constructed via CRT product of prime-order affine spaces.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001381.md b/_corpus_v3_items/cid001381.md new file mode 100644 index 000000000..37d81a194 --- /dev/null +++ b/_corpus_v3_items/cid001381.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001381/ +title: Euclidean as Static Limit +cid: cid001381 +primary_alias: THM0085 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Euclidean as Static Limit +aliases: +- THM0085 +- II.T19 +- euclidean-as-static-limit +- thm:euclidean-static-limit +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 4, Chapter 21 (Part IV-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Euclidean as Static Limit**\n\nEuclidean as Static Limit\n\n# Euclidean as Static Limit\n\n## Summary\n\nEuclidean\ + \ as Static Limit\n\n## Statement\n\n```latex\n%\n\\label{thm:euclidean-static-limit}\nIn the limit where the split-complex\ + \ coupling vanishes---formally,\nwhere the off-diagonal coefficient\n$\\partial v / \\partial x$ in the split-complex\n\ + Cauchy--Riemann equations tends to zero---the\nwave equation\n$\\partial^2 u / \\partial x^2 - \\partial^2 u / \\partial\ + \ y^2 = 0$\ndegenerates to the Laplace equation\n$\\partial^2 u / \\partial x^2 + \\partial^2 u / \\partial y^2 = 0$.\n\ + In this limit:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n The null cone collapses:\n the two characteristic\ + \ families merge,\n and no real characteristics survive.\n \\item[\\textup{(ii)}]\n The causal structure\n\ + \ \\textup{(}Definition~\\ref{def:causal-structure}\\textup{)}\n disappears:\n all directions\ + \ become equivalent.\n \\item[\\textup{(iii)}]\n The surviving geometry is Euclidean:\n betweenness,\ + \ congruence, Pasch,\n and the parallel postulate\n \\textup{(}II.T15--II.T18\\textup{)}\n \ + \ remain valid.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{(i).}\nThe characteristic polynomial\n\ + $\\xi^2 - c^2 \\eta^2 = 0$\nparametrizes the coupling strength by~$c$.\nWhen $c \\to 0$ (coupling vanishes),\nthe polynomial\ + \ becomes $\\xi^2 = 0$,\na double root---the two characteristics coalesce,\nand the equation becomes parabolic at $c =\ + \ 0$.\nIn the full degeneration to Laplace\n($c^2 \\to -1$, formally replacing $\\jj \\to i$),\nthe polynomial $\\xi^2\ + \ + \\eta^2 = 0$\nhas no real roots.\n\n\\emph{(ii).}\nWithout distinct characteristics,\nconditions (C1)--(C3) of\nDefinition~\\\ + ref{def:causal-structure}\nare vacuous.\nNo null cone exists; no forward direction\ncan be selected.\n\n\\emph{(iii).}\n\ + The Tarski axioms (II.T15--II.T18)\ndepend only on the ultrametric distance\n$d(x,y) = 2^{-\\delta(x,y)}$\nand the cylinder\ + \ structure,\nneither of which involves~$\\jj$.\nThey survive the $c \\to 0$ limit unchanged.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 57\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch21-wave-causal.tex`\ + \ lines 351-380\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Geometry.CausalStructure`\n\ + - Name: `sector_causal_check`\n\n## Dependencies\n\n- Canonical: II.D21, II.T15, II.T16\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Euclidean as Static Limit +noindex: false +--- + diff --git a/_corpus_v3_items/cid001382.md b/_corpus_v3_items/cid001382.md new file mode 100644 index 000000000..0eee2f07c --- /dev/null +++ b/_corpus_v3_items/cid001382.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001382/ +title: R as Inverse Limit +cid: cid001382 +primary_alias: THM0086 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The real numbers are recovered as the Archimedean inverse limit of primorial-scaled residues, with each coherent + sequence determining a unique real via nested interval intersection. +aliases: +- THM0086 +- II.T20 +- r-as-inverse-limit +- thm:real-inverse-limit +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 23 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**R as Inverse Limit**\n\nThe real numbers are recovered as the Archimedean inverse limit of primorial-scaled\ + \ residues, with each coherent sequence determining a unique real via nested interval intersection.\n\n# R as Inverse\ + \ Limit\n\n## Summary\n\nThe real numbers are recovered as the Archimedean inverse limit of primorial-scaled residues,\ + \ with each coherent sequence determining a unique real via nested interval intersection.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:real-inverse-limit}\nLet $\\ell_\\alpha^{\\mathrm{den}}\n:= \\bigl\\{\\mathrm{den}(\\alpha_n) : n \\geq\ + \ 1\\bigr\\}$\ndenote the image of the $\\alpha$-ray\nunder the denotation map\n(Definition~\\textup{\\ref{def:ch22-denotation-map}},\n\ + Chapter~\\textup{\\ref{ch:orthodox-bridge}}).\nThen:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n $\\ell_\\\ + alpha^{\\mathrm{den}}$\n is a countable dense subset\n of a closed interval $[a, b] \\subset \\mathbb{R}$.\n\ + \ \\item[\\textup{(ii)}]\n The closure\n $\\overline{\\ell_\\alpha^{\\mathrm{den}}}$\n in\ + \ the Euclidean topology\n is isometric to $[a, b]$,\n hence homeomorphic to~$[0, 1]$.\n \\item[\\\ + textup{(iii)}]\n $\\mathbb{R}$ is recovered as the\n \\textbf{Archimedean inverse limit}\n \ + \ \\[\n \\boxed{%\n \\mathbb{R}\n \\;\\cong\\;\n \\varprojlim_{k}\\\ + ;\n \\frac{1}{P_k}\\,\\mathbb{Z}\n \\;=\\;\n \\Bigl\\{\\,\n (r_k)_{k\ + \ \\geq 1}\n \\;\\Big|\\;\n r_k \\in \\tfrac{1}{P_k}\\mathbb{Z},\\;\n r_k \\equiv\ + \ r_{k+1} \\!\\!\\pmod{1/P_k}\n \\,\\Bigr\\}.}\n \\]\n Every element of this inverse limit\n\ + \ corresponds to a unique real number:\n the common value of the nested intervals\n $\\bigl[r_k,\ + \ r_k + 1/P_k\\bigr)$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i).}\nThe image points\n\ + $\\mathrm{den}(\\alpha_n) = D(\\alpha_n)/P_n$\nare rational numbers in~$[0, 1)$\n(or a shifted interval, depending on\ + \ the normalization convention).\nFor distinct stages $m < n$,\nthe CRT coherence guarantees that\n$D(\\alpha_n)/P_n$\ + \ lies within $1/P_m$\nof $D(\\alpha_m)/P_m$.\nThe points accumulate within a single interval\n$[a, b]$ of length at most~$1$.\n\ + Density follows because for any rational $q/P_k$\nin the interval,\nsome deeper-stage point $\\alpha_n$ with $n \\geq\ + \ k$\nhas $D(\\alpha_n)/P_n$ within $1/P_k$ of $q/P_k$.\n\n\\smallskip\n\\noindent\\textbf{(ii).}\nThe closure of a countable\ + \ dense subset\nof a closed interval\nin the Euclidean topology\nis the interval itself.\nThe completeness of~$\\mathbb{R}$\n\ + fills the gaps between the rational grid points.\n\n\\smallskip\n\\noindent\\textbf{(iii).}\nDefine the inverse system:\n\ + the $k$th object is the grid\n$\\frac{1}{P_k}\\mathbb{Z} \\cap [a, b]$,\nand the transition map\n$\\varphi_{k, k+1} :\n\ + \\frac{1}{P_{k+1}}\\mathbb{Z} \\to\n\\frac{1}{P_k}\\mathbb{Z}$\nsends $r_{k+1}$ to the nearest grid point\n$r_k$ with\ + \ $r_k \\leq r_{k+1}$.\nThis is well-defined because $P_k \\mid P_{k+1}$.\nA coherent sequence $(r_k)_{k \\geq 1}$\ndetermines\ + \ a nested sequence of intervals\n$[r_k, r_k + 1/P_k)$,\nwhose intersection is a single real number\n(by Archimedean completeness:\n\ + $1/P_k \\to 0$).\nConversely, every real number $x \\in [a, b]$\ndetermines a unique coherent sequence\nby setting $r_k\ + \ = \\lfloor x \\cdot P_k \\rfloor / P_k$.\nThe bijection is order-preserving\nand respects the Euclidean metric.\n```\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 62\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch23-lines-countable.tex`\ + \ lines 310-353\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Lines`\n\ + - Name: `real_limit_check`\n\n## Dependencies\n\n- Canonical: I.T04, I.T18, I.T35, II.D13, II.D14, II.D15\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The real numbers are recovered as the Archimedean inverse limit of primorial-scaled residues, with each coherent + sequence determining a unique real via nested interval intersection. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001383.md b/_corpus_v3_items/cid001383.md new file mode 100644 index 000000000..81ef96c2d --- /dev/null +++ b/_corpus_v3_items/cid001383.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001383/ +title: S^1 as Profinite Limit +cid: cid001383 +primary_alias: THM0087 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: S^1 as Profinite Limit +aliases: +- THM0087 +- II.T21 +- s-1-as-profinite-limit +- thm:circle-profinite +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 24 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**S^1 as Profinite Limit**\n\nS^1 as Profinite Limit\n\n# S^1 as Profinite Limit\n\n## Summary\n\nS^1 as Profinite\ + \ Limit\n\n## Statement\n\n```latex\n%\n\\label{thm:circle-profinite}\nFor each coordinate axis $X \\in \\{A, B, C\\}$:\n\ + \\begin{enumerate}\n \\item[\\textup{(i)}]\n The angular denotation map\n $\\mathrm{den}^X : \\mathcal{S}^X\ + \ \\to S^1$\n is a continuous surjection\n of topological groups.\n \\item[\\textup{(ii)}]\n \ + \ The image\n $\\mathrm{den}^X(\\mathcal{S}^X) = S^1$:\n every point on the classical circle\n\ + \ is the limit of some coherent\n angular sequence.\n \\item[\\textup{(iii)}]\n $S^1$ is\ + \ recovered as the\n \\textbf{Archimedean angular limit}:\n \\[\n \\boxed{%\n \ + \ S^1\n \\;\\cong\\;\n \\varprojlim_k\\;\n \\frac{1}{Q_k^X}\\,\\mathbb{Z}\n\ + \ \\Big/\n \\mathbb{Z}\n \\;=\\;\n \\Bigl\\{\\,\n (\\\ + theta_k)_{k \\geq 1}\n \\;\\Big|\\;\n \\theta_k \\in \\tfrac{1}{Q_k^X}\\mathbb{Z}/\\mathbb{Z},\\\ + ;\n \\theta_k \\equiv \\theta_{k+1}\n \\!\\!\\!\\pmod{1/Q_k^X}\n \\,\\Bigr\\}.}\n\ + \ \\]\n Every element of this angular inverse limit\n determines a unique point on~$S^1$:\n\ + \ the common value of the nested arcs.\n \\item[\\textup{(iv)}]\n No uncountable continuum is needed:\n\ + \ $S^1$ arises from countable approximations\n (consistent with I.T35, Book~I).\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i).}\nContinuity of $\\mathrm{den}^X$ follows from\nthe ultrametric\ + \ estimate:\nif $(c_k) = (c_k')$ for $k \\leq K$,\nthen $|\\theta_K^X - \\theta_K'^X| \\leq 1/Q_K^X$,\nso the images are\ + \ within $1/Q_K^X$ on~$S^1$.\nGroup homomorphism: componentwise addition maps\nto addition of angles.\n\n\\smallskip\n\ + \\noindent\\textbf{(ii).}\nSurjectivity: for any $\\theta \\in S^1 = [0,1)$,\ndefine $c_k$ by the CRT reconstruction\n\ + of $\\lfloor \\theta \\cdot Q_k^X \\rfloor$\nmodulo the primes $p_{j(1)}, \\ldots, p_{j(k)}$.\nThe resulting coherent\ + \ sequence maps to~$\\theta$.\n(This uses the density of the grids\n$\\frac{1}{Q_k^X}\\mathbb{Z}/\\mathbb{Z}$\nin~$S^1$.)\n\ + \n\\smallskip\n\\noindent\\textbf{(iii).}\nThe angular inverse limit is defined\nby the same nested-arc construction\n\ + as Theorem~\\ref{thm:real-inverse-limit}\nfor lines,\nbut on the quotient $\\mathbb{R}/\\mathbb{Z}$\ninstead of~$\\mathbb{R}$.\n\ + Each coherent sequence $(\\theta_k)$\ndetermines a nested sequence of arcs\nof length $1/Q_k^X$,\nwhose intersection is\ + \ a single point on~$S^1$.\n\n\\smallskip\n\\noindent\\textbf{(iv).}\nAt each stage, the grid\n$\\frac{1}{Q_k^X}\\mathbb{Z}/\\\ + mathbb{Z}$\nis finite (cardinality~$Q_k^X$).\nThe inverse limit is constructed\nfrom countable data\nwith coherence constraints.\n\ + The same reasoning as in\nChapter~\\ref{ch:lines-countable},\nSection~\\ref{sec:ch23-no-uncountable}\napplies: Cantor's\ + \ diagonal\nis inapplicable to this structured construction\n(I.T35, Book~I).\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 64\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch24-circles-solenoidal.tex`\ + \ lines 287-331\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.Circles`\n\ + - Name: `circle_profinite_check`\n\n## Dependencies\n\n- Canonical: II.D26, II.T20\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: S^1 as Profinite Limit +noindex: false +--- + diff --git a/_corpus_v3_items/cid001384.md b/_corpus_v3_items/cid001384.md new file mode 100644 index 000000000..853d78dfe --- /dev/null +++ b/_corpus_v3_items/cid001384.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001384/ +title: Three Perspectives on Pi +cid: cid001384 +primary_alias: THM0088 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Pi is earned from within tau via three equivalent perspectives: geometric (circumference/diameter of the earned + circle), analytic (Leibniz series from the tau-harmonic expansion), and algebraic (eigenvalue of angular iteration on T^2).' +aliases: +- THM0088 +- II.T22 +- three-perspectives-on-pi +- thm:three-pi +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 25 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Three Perspectives on Pi**\n\nPi is earned from within tau via three equivalent perspectives: geometric (circumference/diameter\ + \ of the earned circle), analytic (Leibniz series from the tau-harmonic expansion), and algebraic (eigenvalue of angular\ + \ iteration on T^2).\n\n# Three Perspectives on Pi\n\n## Summary\n\nPi is earned from within tau via three equivalent\ + \ perspectives: geometric (circumference/diameter of the earned circle), analytic (Leibniz series from the tau-harmonic\ + \ expansion), and algebraic (eigenvalue of angular iteration on T^2).\n\n## Statement\n\n```latex\n%\n\\label{thm:three-pi}\n\ + The three constructions of~$\\pi$\nfrom Category~$\\tau$ yield the same number:\n\\[\n \\boxed{%\n \\pi_{\\mathrm{top}}\n\ + \ \\;=\\;\n \\pi_{\\mathrm{geo}}\n \\;=\\;\n \\pi_{\\mathrm{spec}}\n \\;=\\;\n 3.14159265\\ldots}\n\\\ + ]\nSpecifically:\n\\begin{enumerate}\n \\item[\\textup{(T$=$G)}]\n The topological half-period\n \ + \ equals the geometric circumference-to-diameter ratio:\n the figure-eight path on~$\\mathbb{L}$\n traverses\ + \ two lobes,\n each of which has Archimedean circumference~$\\pi$\n (matching the limit of the Archimedes\ + \ polygon sequence).\n \\item[\\textup{(G$=$S)}]\n The geometric ratio equals the spectral phase average:\n\ + \ the Archimedes polygon perimeters converge\n to the same value\n as the normalized spectral\ + \ phase accumulation\n (both are controlled by the equidistribution\n of primes on the solenoidal circle).\n\ + \ \\item[\\textup{(T$=$S)}]\n The topological period equals the spectral phase:\n the lemniscate\ + \ period is computed\n from character values on the boundary ring,\n and the normalization yields the\ + \ same~$\\pi$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n[Proof outline]\n\\textbf{(T$=$G).}\nThe\ + \ topological $\\pi$ is the half-period\nof the lemniscate path.\nEach lobe of~$\\mathbb{L}$ is a copy of~$S^1$\n(Theorem~\\\ + ref{thm:torus-degeneration}, II.T13,\nChapter~\\ref{ch:torus-degeneration}),\nand the Archimedean circumference of~$S^1$\n\ + is $2\\pi_{\\mathrm{geo}}$\n(by Definition~\\ref{def:geometric-pi}).\nHence the half-period $\\pi_{\\mathrm{top}} = \\\ + pi_{\\mathrm{geo}}$.\n\n\\smallskip\n\\noindent\\textbf{(G$=$S).}\nThe Archimedes polygon perimeters\nat stage~$k$ are\ + \ determined by the chord lengths\nbetween consecutive vertices of the level circle.\nEach chord length depends on the\ + \ angle subtended,\nwhich is $2\\pi / Q_k^X$.\nThe spectral phase accumulation\n$\\Theta_k^X$ tracks the same angular\ + \ data\nthrough character values.\nThe normalization ensures\n$\\pi_{\\mathrm{geo}} = \\pi_{\\mathrm{spec}}$\nin the limit\ + \ $k \\to \\infty$.\n\n\\smallskip\n\\noindent\\textbf{(T$=$S).}\nTransitivity: $\\pi_{\\mathrm{top}} = \\pi_{\\mathrm{geo}}\ + \ = \\pi_{\\mathrm{spec}}$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 68\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch25-pi-earned.tex` lines 431-468\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.PiEarned`\n- Name: `pi_three_perspectives_check`\n\ + \n## Dependencies\n\n- Canonical: II.D28, II.D26, I.T05\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Pi is earned from within tau via three equivalent perspectives: geometric (circumference/diameter of the earned + circle), analytic (Leibniz series from the tau-harmonic expansion), and algebraic (eigenvalue of angular iteration on T^2).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001385.md b/_corpus_v3_items/cid001385.md new file mode 100644 index 000000000..3d39dbe6a --- /dev/null +++ b/_corpus_v3_items/cid001385.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001385/ +title: e from Index Arithmetic +cid: cid001385 +primary_alias: THM0089 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Euler''s number e is earned from within tau via index arithmetic: it is the unique growth rate eigenvalue of the + nu-iterator, with three equivalent characterizations (factorial series, compound limit, index-arithmetic trace).' +aliases: +- THM0089 +- II.T23 +- e-from-index-arithmetic +- thm:e-index +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 26 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**e from Index Arithmetic**\n\nEuler's number e is earned from within tau via index arithmetic: it is the unique\ + \ growth rate eigenvalue of the nu-iterator, with three equivalent characterizations (factorial series, compound limit,\ + \ index-arithmetic trace).\n\n# e from Index Arithmetic\n\n## Summary\n\nEuler's number e is earned from within tau via\ + \ index arithmetic: it is the unique growth rate eigenvalue of the nu-iterator, with three equivalent characterizations\ + \ (factorial series, compound limit, index-arithmetic trace).\n\n## Statement\n\n```latex\n%\n\\label{thm:e-index}\nLet\ + \ $P_k = p_1 \\cdots p_k$ be the $k$th primorial\nand let $n_k = p_{k+1}$ be the $(k{+}1)$th prime.\nThen:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n The sequence\n $\\bigl(1 + 1/n_k\\bigr)^{n_k}$\n is computable\ + \ in earned index arithmetic\n \\textup{(I.D06, Book~I)}\n at every finite stage.\n\n \\item[\\textup{(ii)}]\n\ + \ The Archimedean values satisfy\n $\\bigl|\\,\n (1 + 1/n_k)^{n_k} - e\n \\,\\bigr|\n\ + \ \\;\\leq\\;\n e / (2 n_k)$\n for all $k \\geq 1$.\n\n \\item[\\textup{(iii)}]\n \ + \ The primorial-weighted partial sums\n \\[\n E_k\n \\;:=\\;\n \\sum_{m=0}^{k}\ + \ \\frac{1}{m!}\n \\;\\in\\; \\mathbb{Q}\n \\]\n satisfy $|E_k - e| \\leq 3/(k{+}1)!$\n\ + \ and each $E_k$ has denominator dividing~$k!$,\n hence is representable\n in the CRT tower\ + \ at stage~$k$\n whenever $k! \\mid P_k$.\n\n \\item[\\textup{(iv)}]\n $e$ is transcendental over~$\\\ + mathbb{Q}$\n \\textup{(Hermite, 1873)}.\n In particular, $e$ is not a finite-stage\n rational;\n\ + \ it arises only at the profinite limit.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i)}\n\ + At stage~$k$, the index arithmetic\noperates in $\\mathbb{Z}/P_{k+1}\\mathbb{Z}$.\nThe integer $n_k = p_{k+1}$\nis coprime\ + \ to~$P_k$,\nso $1/n_k = n_k^{-1} \\pmod{P_{k+1}}$ exists.\nThe expression $(1 + n_k^{-1})^{n_k} \\pmod{P_{k+1}}$\nis\ + \ a well-defined integer;\nits computation requires $O(\\log n_k)$\nmodular multiplications by repeated squaring.\n\n\\\ + textbf{(ii)}\nThe classical estimate for the convergence rate\nof $(1+1/n)^n$ to~$e$ is\n\\[\n e - \\Bigl(1 + \\frac{1}{n}\\\ + Bigr)^n\n \\;=\\;\n \\frac{e}{2n} + O(1/n^2).\n\\]\nThis follows from taking logarithms:\n$n \\log(1 + 1/n) = 1\ + \ - 1/(2n) + O(1/n^2)$,\nhence $(1+1/n)^n = e^{1 - 1/(2n) + O(1/n^2)}\n= e \\cdot (1 - 1/(2n) + O(1/n^2))$.\nSubstituting\ + \ $n = n_k = p_{k+1}$ gives the bound.\n\n\\textbf{(iii)}\nThe exponential series $e = \\sum_{m=0}^\\infty 1/m!$\nconverges\ + \ absolutely.\nThe tail bound is\n$\\sum_{m=k+1}^\\infty 1/m! \\leq 3/(k{+}1)!$\nfor $k \\geq 0$.\nEach partial sum $E_k$\ + \ has denominator $k!$.\nThe condition $k! \\mid P_k$ holds for~$k$\nup to the largest prime in the primorial,\nsince\ + \ $P_k = p_1 \\cdots p_k$\ncontains all prime factors of~$k!$\nwhen $k \\leq p_k$\n(which holds for all $k \\geq 1$\n\ + by Bertrand's postulate).\n\n\\textbf{(iv)}\nHermite's 1873 proof of the transcendence of~$e$\nis classical.\nIn the $\\\ + tau$-framework,\ntranscendence means that~$e$\ndoes not coincide with any\nfinite-stage rational coordinate:\nno element\ + \ of\n$\\mathbb{Q} \\subset \\tau^3$\nmaps to~$e$ under the denotation map\n(II.D23, Chapter~\\ref{ch:orthodox-bridge}).\n\ + $e$ exists only as a profinite limit.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 71\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch26-e-earned.tex` lines 206-250\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.EEarned`\n- Name: `e_three_perspectives_check`\n\ + \n## Dependencies\n\n- Canonical: I.D04, I.D06, I.T05, I.T35, II.D14, II.T20\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Euler''s number e is earned from within tau via index arithmetic: it is the unique growth rate eigenvalue of + the nu-iterator, with three equivalent characterizations (factorial series, compound limit, index-arithmetic trace).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001386.md b/_corpus_v3_items/cid001386.md new file mode 100644 index 000000000..0b5b9a5ea --- /dev/null +++ b/_corpus_v3_items/cid001386.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001386/ +title: j Replaces i +cid: cid001386 +primary_alias: THM0090 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The fiber involution sigma: B <-> C is the unique non-trivial ABCD-preserving automorphism of T^2, forcing j (with + j^2 = +1) as the canonical algebraic structure. The classical i has no tau-internal avatar.' +aliases: +- THM0090 +- II.T24 +- j-replaces-i +- thm:j-replaces-i +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 5, Chapter 27 (Part IV-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**j Replaces i**\n\nThe fiber involution sigma: B <-> C is the unique non-trivial ABCD-preserving automorphism\ + \ of T^2, forcing j (with j^2 = +1) as the canonical algebraic structure. The classical i has no tau-internal avatar.\n\ + \n# j Replaces i\n\n## Summary\n\nThe fiber involution sigma: B <-> C is the unique non-trivial ABCD-preserving automorphism\ + \ of T^2, forcing j (with j^2 = +1) as the canonical algebraic structure. The classical i has no tau-internal avatar.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:j-replaces-i}\nIn the holomorphic analysis of~$\\tau^3$:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n The fiber involution $\\sigma : B \\leftrightarrow C$\n is the \\textbf{unique}\ + \ non-trivial automorphism\n of the fiber~$T^2$\n that preserves the ABCD decomposition\n and\ + \ fixes the base~$\\tau^1$.\n It generates a $\\mathbb{Z}/2$ group.\n\n \\item[\\textup{(ii)}]\n \ + \ Every $\\mathbb{Z}/2$-equivariant\n algebra structure on the fiber\n is isomorphic to the split-complex\ + \ ring\n $\\mathbb{R}[\\jj]$,\n with $\\jj$ acting as~$\\sigma$.\n\n \\item[\\textup{(iii)}]\n \ + \ There is no continuous homomorphism\n $\\mathrm{SO}(2) \\to \\mathrm{Aut}(T^2)$\n compatible\ + \ with the ABCD decomposition.\n In particular, the imaginary unit~$i$\n \\textup{(}generator of $\\\ + mathrm{SO}(2)$\n acting on~$\\mathbb{R}^2$\\textup{)}\n has no $\\tau$-internal avatar.\n\n \\item[\\\ + textup{(iv)}]\n The bipolar idempotents $e_\\pm = (1 \\pm \\jj)/2$\n are the canonical sector projections:\n\ + \ \\[\n e_+ \\;:\\; z \\longmapsto z_+ e_+,\n \\qquad\n e_- \\;:\\; z\ + \ \\longmapsto z_- e_-,\n \\]\n recovering the spectral decomposition\n of the boundary ring\ + \ \\textup{(I.D19, Book~I)}.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i)}\nThe ABCD decomposition\ + \ is a CRT product\n(I.D17, Book~I).\nAny automorphism preserving the D and A factors\nmust permute B and~C.\nThe permutation\ + \ group of two elements is $S_2 \\cong \\mathbb{Z}/2$:\nthe identity and the swap~$\\sigma$.\n\n\\textbf{(ii)}\nA $\\\ + mathbb{Z}/2$-equivariant algebra\non a two-dimensional real vector space\nwith involution~$\\sigma$\ndecomposes into $\\\ + pm 1$ eigenspaces.\nThe multiplication table\nis determined by $\\sigma$'s action:\nif $\\sigma$ has eigenvalues $\\pm\ + \ 1$\nwith one-dimensional eigenspaces,\nthe resulting algebra is\n$\\mathbb{R} \\times \\mathbb{R}\n\\cong \\mathbb{R}[\\\ + jj]/(\\jj^2 - 1)$.\nThis is the split-complex ring.\n\n\\textbf{(iii)}\n$\\mathrm{SO}(2)$ is connected.\n$\\mathrm{Aut}(T^2)$\ + \ preserving the ABCD structure\nis $\\mathbb{Z}/2$, which is discrete.\nA continuous homomorphism from a connected group\n\ + to a discrete group must be trivial.\nHence no non-trivial $\\mathrm{SO}(2)$ action exists.\nIn particular, there is no\ + \ element\nsquaring to~$-1$ (which would generate\na $\\mathbb{Z}/4 \\subset \\mathrm{SO}(2)$ action).\n\n\\textbf{(iv)}\n\ + Direct from Definition~\\ref{def:bipolar-idempotents-interior}\nand Remark~\\ref{rem:ch27-sector-projections}.\nThe sector\ + \ decomposition\n$z = z_+ e_+ + z_- e_-$\nreproduces the $\\chi_+/\\chi_-$ spectral decomposition\nof the boundary ring\ + \ (I.D19, I.D21, Book~I).\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 74\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part05/ch27-j-replaces-i.tex` lines 345-385\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Transcendentals.JReplacesI`\n- Name: `j_replaces_i_check`\n\ + \n## Dependencies\n\n- Canonical: II.D32, II.D33, I.T10\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The fiber involution sigma: B <-> C is the unique non-trivial ABCD-preserving automorphism of T^2, forcing j + (with j^2 = +1) as the canonical algebraic structure. The classical i has no tau-internal avatar.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001387.md b/_corpus_v3_items/cid001387.md new file mode 100644 index 000000000..9c38a402b --- /dev/null +++ b/_corpus_v3_items/cid001387.md @@ -0,0 +1,120 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001387/ +title: BndLift Existence +cid: cid001387 +primary_alias: THM0091 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For every stage n and holomorphic datum, the boundary lift exists and is the unique extension to stage n+1 satisfying + tower coherence and diagonal discipline. +aliases: +- THM0091 +- II.T26 +- bndlift-existence +- thm:bndlift-existence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005542 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 30 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**BndLift Existence**\n\nFor every stage n and holomorphic datum, the boundary lift exists and is the unique\ + \ extension to stage n+1 satisfying tower coherence and diagonal discipline.\n\n# BndLift Existence\n\n## Summary\n\n\ + For every stage n and holomorphic datum, the boundary lift exists and is the unique extension to stage n+1 satisfying\ + \ tower coherence and diagonal discipline.\n\n## Statement\n\n```latex\n%\n\\label{thm:bndlift-existence}\n% II.D33,\ + \ II.D35, II.D36\nFor every $n \\geq 1$\nand every holomorphic datum\n$f_n \\in \\mathrm{Hol}(\\mathbb{Z}/P_n\\mathbb{Z},\\\ + , H_\\tau^{\\mathrm{cal}})$:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Existence.}\n \ + \ The boundary lift\n $f_{n+1} = \\mathrm{BndLift}_n(f_n)$\n exists.\n\n \\item[\\textup{(ii)}]\n\ + \ \\textbf{Uniqueness.}\n $f_{n+1}$ is the \\textbf{unique}\n holomorphic extension of $f_n$\n\ + \ to $\\mathbb{Z}/P_{n+1}\\mathbb{Z}$\n that satisfies the tower coherence\n and diagonal discipline\ + \ constraints.\n\n \\item[\\textup{(iii)}]\n \\textbf{Sector factorization.}\n The lift factors as\n\ + \ \\[\n \\mathrm{BndLift}_n\n \\;=\\;\n \\mathrm{BndLift}_n^{(+)}\n \ + \ \\;\\times\\;\n \\mathrm{BndLift}_n^{(-)},\n \\]\n where each sector lift\ + \ acts\n independently on its channel.\n The factors commute:\n the B-sector lift does not\ + \ depend\n on the C-sector input,\n and conversely.\n\n \\item[\\textup{(iv)}]\n \\textbf{Norm\ + \ bound.}\n The lifted function satisfies\n \\[\n \\|f_{n+1}\\|_\\tau\n \\\ + ;\\leq\\;\n \\|f_n\\|_\\tau\n \\;\\cdot\\;\n \\bigl(1 + C_\\tau / p_{n+1}\\bigr),\n\ + \ \\]\n where $C_\\tau = 2\\iota_\\tau = 4/(\\pi+e)$\n is a universal constant.\n \ + \ In particular, $\\|f_{n+1}\\|_\\tau \\to \\|f_n\\|_\\tau$\n as $p_{n+1} \\to \\infty$:\n the lift\ + \ is \\textbf{asymptotically isometric}.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\textbf{(i)\ + \ Existence.}\nThe CRT decomposition\n$\\mathbb{Z}/P_{n+1}\\mathbb{Z}\n\\cong\n\\mathbb{Z}/P_n\\mathbb{Z} \\times \\mathbb{Z}/p_{n+1}\\\ + mathbb{Z}$\n(I.T18, Book~I)\nreduces the extension problem\nto finding the $p_{n+1}$-component\nof~$f_{n+1}$.\nThe sector\ + \ decomposition\n$f_{n+1} = f_{n+1}^{(+)} e_+ + f_{n+1}^{(-)} e_-$\nsplits this into two independent problems,\none per\ + \ sector.\n\nFor each sector,\nthe extension from $\\mathbb{Z}/P_n\\mathbb{Z}$\nto $\\mathbb{Z}/P_n\\mathbb{Z} \\times\ + \ \\mathbb{Z}/p_{n+1}\\mathbb{Z}$\namounts to choosing a function\n$\\Delta_n^{(\\pm)} : \\mathbb{Z}/p_{n+1}\\mathbb{Z}\ + \ \\to \\mathbb{Z}$.\nThe tower coherence constraint requires\nthat the CRT reduction from stage $n+1$ to stage~$n$\n\ + recovers~$f_n^{(\\pm)}$.\nThis is automatically satisfied\nby the CRT structure:\n$f_{n+1}^{(\\pm)}(x) \\bmod P_n = f_n^{(\\\ + pm)}(x \\bmod P_n)$\nholds by construction.\n\nThe diagonal discipline (K5)\nconstrains the lift increment~$\\Delta_n^{(\\\ + pm)}$\nfurther.\nK5 forbids the use of the diagonal argument\non the tower:\nconcretely,\n$\\Delta_n^{(\\pm)}$ must be\ + \ \\emph{compatible}\nwith the NF reduction\n$\\mathbb{Z}/p_{n+1}\\mathbb{Z} \\to \\{0, 1, \\ldots, p_{n+1}-1\\}$,\nin\ + \ the sense that\n$\\Delta_n^{(\\pm)}(r) \\neq r$\nfor all $r \\in \\{1, \\ldots, p_{n+1} - 1\\}$\n(the off-diagonal condition).\n\ + The number of such functions is $(p_{n+1}-1)^{p_{n+1}-1}$\n(with $\\Delta_n^{(\\pm)}(0)$ free),\nbut the holomorphicity\ + \ condition\n(tower coherence at \\emph{all} subsequent stages)\npins $\\Delta_n^{(\\pm)}$ down uniquely.\n\n\\textbf{(ii)\ + \ Uniqueness.}\nSuppose $f_{n+1}$ and $f'_{n+1}$\nare two holomorphic extensions of~$f_n$.\nTheir difference $\\delta\ + \ = f_{n+1} - f'_{n+1}$\nvanishes on the stage-$n$ component\n(by compatibility)\nand satisfies the diagonal discipline.\n\ + Write $\\delta = \\delta^{(+)} e_+ + \\delta^{(-)} e_-$.\nEach sector difference\n$\\delta^{(\\pm)} : \\mathbb{Z}/p_{n+1}\\\ + mathbb{Z} \\to \\mathbb{Z}$\nsatisfies:\n\\begin{itemize}\n \\item $\\delta^{(\\pm)}(0) = 0$\n (compatibility\ + \ at $x \\equiv 0 \\bmod p_{n+1}$),\n \\item $\\delta^{(\\pm)}$ is holomorphic\n (tower-coherent at all subsequent\ + \ stages).\n\\end{itemize}\nA holomorphic function on $\\mathbb{Z}/p_{n+1}\\mathbb{Z}$\nthat vanishes at~$0$\nis determined\ + \ by its values at $1, \\ldots, p_{n+1}-1$.\nThe diagonal discipline imposes\n$p_{n+1} - 1$ independent constraints,\n\ + one for each non-zero residue class.\nThis leaves exactly one solution: $\\delta = 0$.\nHence $f_{n+1} = f'_{n+1}$.\n\n\ + \\textbf{(iii) Sector factorization.}\nSince $e_+ e_- = 0$,\nthe B-sector and C-sector decouple completely:\n$\\mathrm{BndLift}_n(f_n)\n\ + = \\mathrm{BndLift}_n^{(+)}(e_+ f_n) \\cdot e_+\n+ \\mathrm{BndLift}_n^{(-)}(e_- f_n) \\cdot e_-$.\nThe B-sector lift\ + \ depends only on~$f_n^{(+)}$;\nthe C-sector lift depends only on~$f_n^{(-)}$.\nThe two lifts compute independently\n\ + and assemble via the idempotent sum.\n\n\\textbf{(iv) Norm bound.}\nIn each sector,\nthe lift increment satisfies\n$|\\\ + Delta_n^{(\\pm)}(r)| \\leq p_{n+1} - 1$\nfor $r \\in \\mathbb{Z}/p_{n+1}\\mathbb{Z}$.\nThe calibrated norm of the increment\n\ + is bounded by\n$\\|\\Delta_n^{(\\pm)} \\cdot P_n\\|\n\\leq P_n \\cdot (p_{n+1} - 1)\n= P_{n+1} - P_n$.\nHence\n$\\|f_{n+1}\\\ + |_\\tau\n\\leq \\|f_n\\|_\\tau + \\|f_{n+1} - f_n\\|_\\tau\n\\leq \\|f_n\\|_\\tau + C_\\tau \\cdot P_n / P_{n+1}\n\\leq\ + \ \\|f_n\\|_\\tau \\cdot (1 + C_\\tau / p_{n+1})$,\nwhere $C_\\tau = 2\\iota_\\tau$\naccounts for the two-sector geometric\ + \ mean\nin the calibrated norm.\nSince $\\sum_{n} 1/p_{n+1}$ diverges logarithmically,\nthe product $\\prod_n (1 + C_\\\ + tau / p_{n+1})$\ndiverges---but this is the correct behavior:\nthe norm of the fully lifted function\n(the profinite limit)\n\ + is finite only for functions\nwhose lift increments decay sufficiently fast.\nThe bound says that each individual lift\ + \ step\nchanges the norm by at most $C_\\tau / p_{n+1}$,\nwhich tends to zero.\n```\n\n## Source Context\n\n- Registry\ + \ source: `book-02.jsonl` line 83\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch30-bndlift-construction.tex`\ + \ lines 325-380\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.BndLift`\n\ + - Name: `Tau.BookII.Hartogs.bndlift_existence_check`\n\n## Dependencies\n\n- Canonical: I.T18, I.D20, I.D21, I.D24, I.T10,\ + \ II.D33, II.D35, II.D36\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For every stage n and holomorphic datum, the boundary lift exists and is the unique extension to stage n+1 satisfying + tower coherence and diagonal discipline. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001388.md b/_corpus_v3_items/cid001388.md new file mode 100644 index 000000000..452b8cebb --- /dev/null +++ b/_corpus_v3_items/cid001388.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001388/ +title: Mutual Determination (5-Way Equivalence) +cid: cid001388 +primary_alias: THM0092 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Five equivalent descriptions of a tau-holomorphic function: (R) refinement sequence, (S) spectral decomposition, + (G) omega-germ transformer, (C) boundary character, (H) Hartogs extension. The equivalences hold because each bipolar channel + is one-dimensional.' +aliases: +- THM0092 +- II.T27 +- mutual-determination-5-way-equivalence +- thm:mutual-determination +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005572 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 31 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Mutual Determination (5-Way Equivalence)**\n\nFive equivalent descriptions of a tau-holomorphic function: (R)\ + \ refinement sequence, (S) spectral decomposition, (G) omega-germ transformer, (C) boundary character, (H) Hartogs extension.\ + \ The equivalences hold because each bipolar channel is one-dimensional.\n\n# Mutual Determination (5-Way Equivalence)\n\ + \n## Summary\n\nFive equivalent descriptions of a tau-holomorphic function: (R) refinement sequence, (S) spectral decomposition,\ + \ (G) omega-germ transformer, (C) boundary character, (H) Hartogs extension. The equivalences hold because each bipolar\ + \ channel is one-dimensional.\n\n## Statement\n\n```latex\n%\n\\label{thm:mutual-determination}\n% I.D20, I.D21, I.T05,\ + \ I.T10, I.T18, II.D33, II.D35, II.D36, II.T25\nThe following five descriptions of a holomorphic datum\non $\\tau^3$ are\ + \ canonically equivalent.\nGiven any one, the other four are uniquely determined:\n\\begin{enumerate}\n \\item[\\textup{(R)}]\n\ + \ A \\textbf{refinement tail}:\n a tower-coherent sequence $(f_k)_{k \\geq n}$\n in $H_\\tau$\ + \ stabilized after stage~$n$.\n\n \\item[\\textup{(S)}]\n A \\textbf{spectral tail}:\n a stabilized\ + \ character decomposition\n $f = \\sum_{\\chi \\in S_n} c_\\chi \\cdot \\chi$\n with finite support\ + \ $S_n \\subset \\widehat{R}_\\tau$.\n\n \\item[\\textup{(G)}]\n An \\textbf{$\\omega$-germ}:\n an\ + \ equivalence class of tower-coherent sequences\n agreeing on all sufficiently deep stages.\n\n \\item[\\\ + textup{(C)}]\n A \\textbf{boundary character}:\n a ring homomorphism\n $\\varphi \\colon R_\\\ + tau \\to H_\\tau$.\n\n \\item[\\textup{(H)}]\n A \\textbf{Hartogs continuation}:\n a holomorphic\ + \ extension from boundary to interior\n via iterated $\\mathrm{BndLift}_n$.\n\\end{enumerate}\n\n\\noindent\n\ + Explicitly, the equivalences are:\n\\[\n \\boxed{%\n \\textup{(R)}\n \\;\\overset{\\textup{II.L02}}{\\Longleftrightarrow}\\\ + ;\n \\textup{(S)}\n \\;\\overset{\\textup{II.L03}}{\\Longleftrightarrow}\\;\n \\textup{(G)}\n \\;\\overset{\\\ + textup{II.L04}}{\\Longleftrightarrow}\\;\n \\textup{(C)}\n \\;\\overset{\\textup{II.L05}}{\\Longleftrightarrow}\\\ + ;\n \\textup{(H)}}\n\\]\nAll five descriptions are unified\nby the bipolar idempotent decomposition\n$1 = e_+ + e_-$:\n\ + each description splits into two independent channels,\nand each channel carries one-dimensional data\nthat determines\ + \ all others uniquely.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this\ + \ pilot run._\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 89\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch31-mutual-determination.tex`\ + \ lines 503-558\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.MutualDetermination`\n\ + - Name: `Tau.BookII.Hartogs.mutual_determination_check`\n\n## Dependencies\n\n- Canonical: II.L02, II.L03, II.L04, II.L05,\ + \ I.D20, I.D21, I.T05, I.T10, I.T18, II.D33, II.D35, II.D36, II.T25\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Five equivalent descriptions of a tau-holomorphic function: (R) refinement sequence, (S) spectral decomposition, + (G) omega-germ transformer, (C) boundary character, (H) Hartogs extension. The equivalences hold because each bipolar channel + i' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001389.md b/_corpus_v3_items/cid001389.md new file mode 100644 index 000000000..21a02f748 --- /dev/null +++ b/_corpus_v3_items/cid001389.md @@ -0,0 +1,119 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001389/ +title: B/C Asymmetry Implies Time Arrow +cid: cid001389 +primary_alias: THM0093 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The B/C asymmetry in the bipolar decomposition of the evolution operator induces a preferred causal direction in + the primorial tower: stage n+1 is in the causal future of stage n, and the B-channel precedes the C-channel.' +aliases: +- THM0093 +- II.T28 +- b-c-asymmetry-implies-time-arrow +- thm:bc-asymmetry-time +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005554 +- predicate: formalized_by + object: cid005555 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 32 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**B/C Asymmetry Implies Time Arrow**\n\nThe B/C asymmetry in the bipolar decomposition of the evolution operator\ + \ induces a preferred causal direction in the primorial tower: stage n+1 is in the causal future of stage n, and the B-channel\ + \ precedes the C-channel.\n\n# B/C Asymmetry Implies Time Arrow\n\n## Summary\n\nThe B/C asymmetry in the bipolar decomposition\ + \ of the evolution operator induces a preferred causal direction in the primorial tower: stage n+1 is in the causal future\ + \ of stage n, and the B-channel precedes the C-channel.\n\n## Statement\n\n```latex\n%\n\\label{thm:bc-asymmetry-time}\n\ + % II.D21, II.D22, II.D37, II.D38, II.T27\nThe B/C asymmetry from Prime Polarity\n\\textup{(I.T05, Book~I)}\ninduces\ + \ a canonical time-like direction on~$\\tau^3$:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Existence\ + \ of time-like direction.}\n The evolution operator\n $\\mathcal{E}_{n \\to m}$ \\textup{(II.D37)}\n\ + \ defines a forward-directed semigroup action\n on holomorphic data.\n The causal arrow \\\ + textup{(II.D38)}\n selects a unique forward direction\n within the null cone of the\n wave-type\ + \ causal structure \\textup{(II.D22)}.\n\n \\item[\\textup{(ii)}]\n \\textbf{Structural origin.}\n \ + \ The time-like direction is not chosen\n but \\textbf{forced}:\n it arises from the\n exponent--tetration\ + \ hierarchy in the ABCD chart,\n which is a consequence of\n the peel-off ordering in the program monoid\n\ + \ \\textup{(I.P02, Book~I)}.\n\n \\item[\\textup{(iii)}]\n \\textbf{Elliptic impossibility.}\n \ + \ In classical complex analysis \\textup{($i^2 = -1$)},\n no time-like direction exists.\n The\ + \ Laplace equation is elliptic\n \\textup{(no null cone)},\n the codomain $\\mathbb{C}$ is a field\n\ + \ \\textup{(no nontrivial idempotents)},\n and the angular structure is $SO(2)$\n \\textup{(connected,\ + \ no preferred direction)}.\n No combination of elliptic structures\n can produce a causal arrow.\n\n\ + \ \\item[\\textup{(iv)}]\n \\textbf{Compatibility with the wave equation.}\n The time-like direction\ + \ aligns with the\n characteristic curves of the split-complex\n wave equation\n \\textup{(II.D21,\ + \ Chapter~\\ref{ch:wave-causal})}:\n forward propagation along the tower\n corresponds to forward propagation\n\ + \ along the null lines\n $\\xi = x + y = \\mathrm{const}$ and\n $\\zeta = x - y = \\mathrm{const}$.\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\\emph{(i).}\nThe evolution operator\n$\\mathcal{E}_{n\ + \ \\to m}$\nis a semigroup (Definition~\\ref{def:evolution-operator},\nproperties E3--E4).\nIt is forward-directed:\n\ + $\\mathcal{E}_{n \\to m}$ is defined only for $m \\geq n$,\nand the inverse does not exist\n(Remark~\\ref{rem:ch32-not-group}).\n\ + The causal arrow (Definition~\\ref{def:causal-arrow})\nselects the forward direction\nwithin the null cone\nby the B/C\ + \ asymmetry.\nWithin each time-like sector of the null cone\n(Definition~\\ref{def:causal-structure},\nChapter~\\ref{ch:wave-causal}),\n\ + the forward half is the one\nin which the B-channel (exponent)\nprecedes the C-channel (tetration).\nThis is well-defined\ + \ because the B/C ordering\nis canonical (I.T05, Book~I)\nand does not depend on the stage number.\n\n\\emph{(ii).}\n\ + The B/C asymmetry traces back to\nPrime Polarity (I.T05, Book~I):\nthe hyperfactorization\n$n = \\alpha^{\\pi} \\cdot\ + \ \\gamma^{\\eta} \\cdot \\omega^{(\\cdot)}$\nseparates the exponent~$\\gamma$ from the tetration~$\\eta$,\nand the peel-off\ + \ sequence processes them\nin a canonical order\ndetermined by the program monoid (I.P02, Book~I).\nThe exponent is always\ + \ available\nbefore the tetration\n(one cannot tetrate without first exponentiating),\nestablishing $B \\prec C$\nas a\ + \ structural fact.\nNo external convention selects this ordering.\n\n\\emph{(iii).}\nIn the classical case $i^2 = -1$:\n\ + \nThe Laplace equation\n$\\partial^2 u / \\partial x^2 + \\partial^2 u / \\partial y^2 = 0$\nis elliptic.\nIts characteristic\ + \ polynomial\n$\\xi^2 + \\eta^2 = 0$\nhas no real roots.\nNo null cone exists,\nso no time-like direction can be defined.\n\ + \nEven if we formally seek an asymmetry\nin the classical codomain $\\mathbb{C}$:\n$\\mathbb{C}$ is a field,\nso $e_\\\ + pm = (1 \\pm i)/2$\nare not idempotents\n($e_+^2 = (1 + 2i - 1)/4 = i/2 \\neq e_+$).\nThere is no bipolar decomposition,\n\ + hence no independent channels,\nhence no structural asymmetry\nto select a direction.\n\nThe rotation group $SO(2)$\n\ + acts on $\\mathbb{C}$ by multiplication by $e^{i\\theta}$.\n$SO(2)$ is connected:\nevery direction can be continuously\n\ + rotated into every other.\nNo discrete asymmetry (like $B \\prec C$)\nsurvives in a connected symmetry group.\n\n\\emph{(iv).}\n\ + The characteristic curves of the wave equation\n(Proposition~\\ref{prop:ch21-null-lines},\nChapter~\\ref{ch:wave-causal})\n\ + are the null lines\n$\\xi = x + y = \\mathrm{const}$\nand $\\zeta = x - y = \\mathrm{const}$.\nIn idempotent coordinates,\n\ + the $e_+$-channel carries $F(\\xi)$\n(propagating along $\\xi$-null lines)\nand the $e_-$-channel carries $G(\\zeta)$\n\ + (propagating along $\\zeta$-null lines).\nThe B/C asymmetry assigns\n$e_+$ to the B-channel (exponent)\nand $e_-$ to the\ + \ C-channel (tetration).\nSince $B \\prec C$,\nthe $\\xi$-propagation (B-channel)\nis the ``earlier'' component,\nand\ + \ the $\\zeta$-propagation (C-channel)\nis the ``later'' component.\nForward propagation along the tower\n$n \\to n+1\ + \ \\to n+2 \\to \\cdots$\ncorresponds to forward propagation\nalong both characteristic families,\nwith the B-characteristic\ + \ ``leading''\nand the C-characteristic ``following.''\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 92\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch32-evolution-operator.tex`\ + \ lines 377-431\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.EvolutionOperator`\n\ + - Name: `Tau.BookII.Hartogs.evolution_semigroup_thm`\n\n## Dependencies\n\n- Canonical: I.T05, I.T10, I.D21, II.D21, II.D22,\ + \ II.D37, II.D38, II.T27\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The B/C asymmetry in the bipolar decomposition of the evolution operator induces a preferred causal direction + in the primorial tower: stage n+1 is in the causal future of stage n, and the B-channel precedes the C-channel.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001390.md b/_corpus_v3_items/cid001390.md new file mode 100644 index 000000000..9b26e5182 --- /dev/null +++ b/_corpus_v3_items/cid001390.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001390/ +title: Associativity of Holomorphic Composition +cid: cid001390 +primary_alias: THM0094 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Composition of tau-holomorphic maps is associative: (h . g) . f = h . (g . f) for all composable holomorphic maps. + This, together with the identity map, makes HolEnd(tau) a category.' +aliases: +- THM0094 +- II.T29 +- associativity-of-holomorphic-composition +- thm:associativity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005551 +- predicate: formalized_by + object: cid005552 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 33 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Associativity of Holomorphic Composition**\n\nComposition of tau-holomorphic maps is associative: (h . g) .\ + \ f = h . (g . f) for all composable holomorphic maps. This, together with the identity map, makes HolEnd(tau) a category.\n\ + \n# Associativity of Holomorphic Composition\n\n## Summary\n\nComposition of tau-holomorphic maps is associative: (h .\ + \ g) . f = h . (g . f) for all composable holomorphic maps. This, together with the identity map, makes HolEnd(tau) a\ + \ category.\n\n## Statement\n\n```latex\n%\n\\label{thm:associativity}\nLet $f = \\{f_k\\}$, $g = \\{g_k\\}$,\nand $h\ + \ = \\{h_k\\}$ be $\\omega$-germ transformers.\nThen\n\\[\n \\boxed{%\n h \\circ (g \\circ f)\n \\;=\\;\n \ + \ (h \\circ g) \\circ f.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThe proof proceeds in two steps.\n\n\\emph{Step~1:\ + \ Stagewise reduction.}\nAt each finite stage~$k$, both sides evaluate to\nthe same map $h_k \\circ g_k \\circ f_k$:\n\ + \\begin{align*}\n \\bigl(h \\circ (g \\circ f)\\bigr)_k\n &\\;=\\;\n h_k \\circ (g_k \\circ f_k),\n \\\\[4pt]\n\ + \ \\bigl((h \\circ g) \\circ f\\bigr)_k\n &\\;=\\;\n (h_k \\circ g_k) \\circ f_k.\n\\end{align*}\nThese are equal\ + \ by associativity\nof set-theoretic function composition\non the finite stage $\\mathbb{Z}/P_k\\mathbb{Z}$.\n\n\\emph{Step~2:\ + \ Coherent lift via the program monoid.}\nStep~1 shows pointwise equality at each finite stage.\nBut $\\omega$-germ transformers\ + \ are not merely\nfamilies of stage maps---they are\ncoherent lifts of program-monoid elements.\nLet $w_f, w_g, w_h \\\ + in \\mathcal{M}_\\tau$\nbe the program-monoid words\ncorresponding to $f$, $g$, $h$\n(via the factorization I.D49, Book~I).\n\ + Then:\n\\[\n g \\circ f\n \\;\\longleftrightarrow\\;\n w_g \\cdot w_f,\n \\qquad\n h \\circ (g \\circ f)\n\ + \ \\;\\longleftrightarrow\\;\n w_h \\cdot (w_g \\cdot w_f).\n\\]\nBy the program monoid's associativity (I.P02,\ + \ Book~I):\n\\[\n w_h \\cdot (w_g \\cdot w_f)\n \\;=\\;\n (w_h \\cdot w_g) \\cdot w_f,\n\\]\nwhich corresponds\ + \ to $(h \\circ g) \\circ f$.\nThe coherent lift is faithful\n(program-monoid multiplication\ncorresponds bijectively\n\ + to $\\omega$-germ composition,\nby the Mutual Determination Theorem II.T27),\nso the equality lifts\nfrom the monoid to\ + \ the $\\omega$-germ level.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 95\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch33-composition-structure.tex` lines 341-353\n\n##\ + \ Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CategoryStructure`\n-\ + \ Name: `hol_assoc_check`\n\n## Dependencies\n\n- Canonical: I.D49, II.D39, II.D40, II.T26\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Composition of tau-holomorphic maps is associative: (h . g) . f = h . (g . f) for all composable holomorphic + maps. This, together with the identity map, makes HolEnd(tau) a category.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001391.md b/_corpus_v3_items/cid001391.md new file mode 100644 index 000000000..f89edaad1 --- /dev/null +++ b/_corpus_v3_items/cid001391.md @@ -0,0 +1,97 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001391/ +title: Residue Theorem +cid: cid001391 +primary_alias: THM0095 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The spectral trace of a tau-meromorphic function with finitely many singularities equals the sum of its residues, + with both sides decomposing along bipolar channels. +aliases: +- THM0095 +- II.T30 +- residue-theorem +- thm:residue-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005569 +- predicate: formalized_by + object: cid005570 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Residue Theorem**\n\nThe spectral trace of a tau-meromorphic function with finitely many singularities equals\ + \ the sum of its residues, with both sides decomposing along bipolar channels.\n\n# Residue Theorem\n\n## Summary\n\n\ + The spectral trace of a tau-meromorphic function with finitely many singularities equals the sum of its residues, with\ + \ both sides decomposing along bipolar channels.\n\n## Statement\n\n```latex\n%\n\\label{thm:residue-theorem}\nLet $f$\ + \ be $\\tau$-meromorphic\nwith singular locus $S = \\{x_1, \\ldots, x_N\\}$.\nThen the spectral trace equals\nthe sum\ + \ of residues:\n\\[\n \\boxed{%\n \\mathrm{Tr}_{\\mathrm{spec}}(f)\n \\;=\\;\n \\sum_{i=1}^N \\mathrm{Res}_{x_i}(f).}\n\ + \\]\nBoth sides lie in~$H_\\tau$;\nin components:\n\\[\n e_+ \\cdot \\mathrm{Tr}_{\\mathrm{spec}}(f_+)\n \\;+\\\ + ;\n e_- \\cdot \\mathrm{Tr}_{\\mathrm{spec}}(f_-)\n \\;=\\;\n \\sum_{i=1}^N\n \\bigl(\n a_{-1}^{(+)}(x_i)\\\ + , e_+\n \\;+\\;\n a_{-1}^{(-)}(x_i)\\, e_-\n \\bigr).\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThe\ + \ proof splits along the bipolar decomposition.\n\n\\emph{Step~1: Channel decomposition.}\nBy idempotent orthogonality\n\ + ($e_+ \\cdot e_- = 0$),\nit suffices to prove the theorem\nseparately for the B-channel ($e_+$)\nand C-channel ($e_-$).\n\ + We give the B-channel argument;\nthe C-channel is symmetric.\n\n\\emph{Step~2: Finite-stage identity.}\nAt stage~$k$,\ + \ the B-channel component\n$f_+^{(k)} : \\mathbb{Z}/P_k\\mathbb{Z} \\to \\mathbb{R}$\ndecomposes via the discrete Fourier\ + \ transform:\n\\[\n f_+^{(k)}(a)\n \\;=\\;\n \\sum_{n=0}^{P_k - 1}\n \\hat{f}_+^{(k)}(n)\\,\n \\chi_n^{(\\\ + gamma)}(a).\n\\]\nThe stage average is\n\\[\n \\frac{1}{P_k}\n \\sum_{a=0}^{P_k - 1}\n f_+^{(k)}(a)\n \\;=\\\ + ;\n \\hat{f}_+^{(k)}(0),\n\\]\nthe zeroth Fourier coefficient.\nThis is an identity of finite sums.\n\n\\emph{Step~3:\ + \ Subtraction of holomorphic part.}\nWrite $f = f^{\\mathrm{hol}} + \\mathrm{pp}(f)$,\nwhere $f^{\\mathrm{hol}}$ is the\ + \ holomorphic (regular) part\nand $\\mathrm{pp}(f)$ is the principal part.\nThe holomorphic part has $\\hat{f}^{\\mathrm{hol}}(0)\ + \ = 0$\nby the tower normalization\n(the constant term of a holomorphic $\\omega$-germ\nis determined by the identity\ + \ germ).\nHence:\n\\[\n \\hat{f}_+^{(k)}(0)\n \\;=\\;\n \\widehat{\\mathrm{pp}(f)}_+^{(k)}(0).\n\\]\n\n\\emph{Step~4:\ + \ Principal part contributes residues.}\nThe principal part at each singular point $x_i$\nhas the form\n$\\mathrm{pp}_{x_i}(f_+)\n\ + = \\sum_{n < 0} a_n^{(+)}(x_i)\\, \\phi_{n,k}^{(+)}$.\nIts zeroth DFT coefficient\nextracts the ``total weight'' of the\ + \ principal part.\nBy the orthogonality of characters on\n$\\mathbb{Z}/P_k\\mathbb{Z}$\nand the CRT factorization (I.T18):\n\ + \\[\n \\widehat{\\mathrm{pp}(f)}_+^{(k)}(0)\n \\;\\xrightarrow{k \\to \\infty}\\;\n \\sum_{i=1}^N a_{-1}^{(+)}(x_i).\n\ + \\]\nHigher-order principal part terms ($n < -1$)\ncontribute zero to the zeroth coefficient\nin the limit,\nbecause they\ + \ oscillate\nand average to zero\nover the growing cyclic group.\n\n\\emph{Step~5: Assembly.}\nCombining Steps~2--4 for\ + \ both channels:\n$\\mathrm{Tr}_{\\mathrm{spec}}(f)\n= \\sum_{i=1}^N \\mathrm{Res}_{x_i}(f)$.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 100\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch34-laurent-residues.tex`\ + \ lines 519-546\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.LaurentResidue`\n\ + - Name: `residue_theorem_check`\n\n## Dependencies\n\n- Canonical: II.D42, II.D43, II.D44, I.T10, I.T18, I.D21\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The spectral trace of a tau-meromorphic function with finitely many singularities equals the sum of its residues, + with both sides decomposing along bipolar channels. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001392.md b/_corpus_v3_items/cid001392.md new file mode 100644 index 000000000..6a269024d --- /dev/null +++ b/_corpus_v3_items/cid001392.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001392/ +title: Finite Spectral Support +cid: cid001392 +primary_alias: THM0096 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every holomorphic function on tau^3 has finite spectral support at each stage: only finitely many cylinder generators + contribute to the canonical basis expansion at any given primorial level.' +aliases: +- THM0096 +- II.T31 +- finite-spectral-support +- thm:finite-spectral-support +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005548 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Finite Spectral Support**\n\nEvery holomorphic function on tau^3 has finite spectral support at each stage:\ + \ only finitely many cylinder generators contribute to the canonical basis expansion at any given primorial level.\n\n\ + # Finite Spectral Support\n\n## Summary\n\nEvery holomorphic function on tau^3 has finite spectral support at each stage:\ + \ only finitely many cylinder generators contribute to the canonical basis expansion at any given primorial level.\n\n\ + ## Statement\n\n```latex\n%\n\\label{thm:finite-spectral-support}\nLet $f = (f_k)_{k \\geq 1}$\nbe a $\\tau$-holomorphic\ + \ function on~$\\tau^3$.\nThen for each stage $k \\geq 1$,\nthe \\textbf{spectral support}\n\\[\n S_k(f)\n \\;:=\\\ + ;\n \\bigl\\{\\,\n (p, v, \\sigma) \\in \\Lambda_k\n \\;\\big|\\;\n \\varphi_{p,v}^{(\\sigma)} \\neq 0\n \ + \ \\,\\bigr\\}\n\\]\nis a \\textbf{finite} set.\nMoreover, the full spectral support\n\\[\n \\boxed{%\n S(f)\n\ + \ \\;:=\\;\n \\bigcup_{k \\geq 1} S_k(f)\n \\;\\subset\\;\n \\Lambda_\\tau}\n\\]\nsatisfies the following:\n\ + at each finite stage~$k$,\nonly finitely many basis elements are active.\nNo $\\tau$-holomorphic function requires\nan\ + \ uncountable collection of spectral coefficients.\n```\n\n## Proof / Justification\n\n```latex\nThe proof has two components.\n\ + \n\\smallskip\n\\noindent\\textbf{Stage-wise finiteness.}\nAt each stage~$k$,\nthe domain of~$f_k$ is the finite set\n\ + $\\mathbb{Z}/P_k\\mathbb{Z}$.\nA function on a finite set\nwith values in a finite-dimensional algebra\nis determined\ + \ by finitely many values.\nThe indexing set $\\Lambda_k$ is finite\n(Remark~\\ref{rem:ch35-finiteness}),\nso $S_k(f)\ + \ \\subseteq \\Lambda_k$ is finite.\n\n\\smallskip\n\\noindent\\textbf{No uncountable basis needed.}\nBook~I's unique\ + \ infinity theorem (I.T36, Book~I)\nestablishes that the $\\tau$-framework\nadmits a single infinity $\\omega$\nas the\ + \ profinite limit of finite stages.\nThe spectral support $S(f) = \\bigcup_{k} S_k(f)$\nis a countable union of finite\ + \ sets,\nhence at most countable.\nBut each stage's contribution\nis itself finite,\nso the spectral data of~$f$\nis a\ + \ coherent family of finite objects---not\nan independently existing infinite collection.\nThe function~$f$ is determined\n\ + by its finite-stage approximations,\nand each approximation involves\nonly finitely many basis elements.\n```\n\n## Source\ + \ Context\n\n- Registry source: `book-02.jsonl` line 103\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch35-canonical-basis.tex`\ + \ lines 379-410\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.CanonicalBasis`\n\ + - Name: `finite_spectral_support_check`\n\n## Dependencies\n\n- Canonical: I.T18, I.D21, II.D45, II.D46\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every holomorphic function on tau^3 has finite spectral support at each stage: only finitely many cylinder generators + contribute to the canonical basis expansion at any given primorial level.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001393.md b/_corpus_v3_items/cid001393.md new file mode 100644 index 000000000..60d0315b7 --- /dev/null +++ b/_corpus_v3_items/cid001393.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001393/ +title: Sheaf Axioms +cid: cid001393 +primary_alias: THM0097 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The holomorphic presheaf O_tau on the cylinder topology is a sheaf: it satisfies locality and gluing for compatible + local sections.' +aliases: +- THM0097 +- II.T32 +- sheaf-axioms +- thm:sheaf-axioms +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005577 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sheaf Axioms**\n\nThe holomorphic presheaf O_tau on the cylinder topology is a sheaf: it satisfies locality\ + \ and gluing for compatible local sections.\n\n# Sheaf Axioms\n\n## Summary\n\nThe holomorphic presheaf O_tau on the cylinder\ + \ topology is a sheaf: it satisfies locality and gluing for compatible local sections.\n\n## Statement\n\n```latex\n%\n\ + \\label{thm:sheaf-axioms}\nThe holomorphic presheaf\n$\\mathcal{O}_\\tau$\n(Definition~\\textup{\\ref{def:holomorphic-presheaf}},\ + \ II.D47)\nis a \\textbf{sheaf} on the cylinder topology of~$\\tau^3$.\nExplicitly:\n\\begin{enumerate}\n \\item[\\\ + textup{(S1)}]\n \\textbf{Locality.}\n If $f \\in \\mathcal{O}_\\tau(U)$\n restricts to zero\ + \ on every element\n of a cover $\\{U_i\\}$ of~$U$,\n then $f = 0$\n \\textup{(Proposition~\\\ + ref{prop:ch36-locality})}.\n \\item[\\textup{(S2)}]\n \\textbf{Gluing.}\n If $\\{f_i \\in \\mathcal{O}_\\\ + tau(U_i)\\}$\n are compatible on pairwise overlaps,\n then there exists a unique\n $f \\in\ + \ \\mathcal{O}_\\tau(U)$\n restricting to~$f_i$ on each~$U_i$\n \\textup{(Lemma~\\ref{lem:gluing}, II.L06)}.\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nLocality was proved in Proposition~\\ref{prop:ch36-locality}.\n\ + Gluing was proved in Lemma~\\ref{lem:gluing}.\nThe presheaf axioms (P1) and (P2)\nhold by construction\n(Definition~\\\ + ref{def:holomorphic-presheaf}).\nTogether, (P1), (P2), (S1), (S2)\nestablish that $\\mathcal{O}_\\tau$ is a sheaf.\n```\n\ + \n## Source Context\n\n- Registry source: `book-02.jsonl` line 107\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part06/ch36-sheaf-coherence.tex`\ + \ lines 352-377\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Hartogs.SheafCoherence`\n\ + - Name: `sheaf_axioms_check`\n\n## Dependencies\n\n- Canonical: II.D47, II.L06, II.D10, I.T18\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The holomorphic presheaf O_tau on the cylinder topology is a sheaf: it satisfies locality and gluing for compatible + local sections.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001394.md b/_corpus_v3_items/cid001394.md new file mode 100644 index 000000000..f3c46fe0e --- /dev/null +++ b/_corpus_v3_items/cid001394.md @@ -0,0 +1,120 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001394/ +title: Holomorphic iff Idempotent-Supported +cid: cid001394 +primary_alias: THM0098 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A tau-holomorphic function is holomorphic if and only if it is idempotent-supported: the bipolar projections e_+f + and e_-f have canonically split spectral support on B-channel and C-channel primes respectively. Proved via the three-lemma + chain (II.L08-L10).' +aliases: +- THM0098 +- II.T33 +- holomorphic-iff-idempotent-supported +- thm:hol-iff-idempotent +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005676 +- predicate: formalized_by + object: cid005677 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 38 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Holomorphic iff Idempotent-Supported**\n\nA tau-holomorphic function is holomorphic if and only if it is idempotent-supported:\ + \ the bipolar projections e_+f and e_-f have canonically split spectral support on B-channel and C-channel primes respectively.\ + \ Proved via the three-lemma chain (II.L08-L10).\n\n# Holomorphic iff Idempotent-Supported\n\n## Summary\n\nA tau-holomorphic\ + \ function is holomorphic if and only if it is idempotent-supported: the bipolar projections e_+f and e_-f have canonically\ + \ split spectral support on B-channel and C-channel primes respectively. Proved via the three-lemma chain (II.L08-L10).\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:hol-iff-idempotent}\n% II.T27, I.T05, I.D21, I.D22, I.D23\nA function $f\ + \ \\colon \\tau^3 \\to H_\\tau$\nis $\\tau$-holomorphic\nif and only if it is idempotent-supported\n\\textup{(}Definition~\\\ + textup{\\ref{def:ch38-idempotent-supported})}.\nExplicitly:\n\\[\n \\boxed{%\n f \\text{ is $\\tau$-holomorphic}\n\ + \ \\quad\\Longleftrightarrow\\quad\n f = e_+ \\cdot f_+ + e_- \\cdot f_-\n \\text{ with \\textup{(IS1)--(IS4)}}.}\n\ + \\]\n```\n\n## Proof / Justification\n\n```latex\n\\emph{Forward direction\n($\\tau$-holomorphic $\\Rightarrow$ idempotent-supported).}\n\ + \nSuppose $f$ is $\\tau$-holomorphic.\nWe verify conditions (IS1)--(IS4).\n\n\\emph{(IS1) Bipolar decomposition.}\nThe\ + \ Idempotent Decomposition Lemma\n(II.L07, Chapter~\\ref{ch:idempotent-decomposition})\ngives $f = e_+ \\cdot f_+ + e_-\ + \ \\cdot f_-$\nwith $f_\\pm = e_\\pm \\cdot f$.\nThis is canonical and functorial (II.D48).\n\n\\emph{(IS2) Stagewise\ + \ naturality.}\nBy the Mutual Determination Theorem\n(II.T27, Chapter~\\ref{ch:mutual-determination}),\n$f$ is equivalent\ + \ to a tower-coherent sequence\n$(f_k)_{k \\geq n}$ in description~(R).\nSince $e_\\pm$ are fixed scalars\nand tower coherence\ + \ is preserved\nunder scalar multiplication,\n$(f_\\pm)_k = e_\\pm \\cdot f_k$\nare each tower-coherent.\n\n\\emph{(IS3)\ + \ Channel support.}\nBranch Factorization (II.L08)\napplied to the $\\omega$-germ transformer\nassociated to $f$ gives\ + \ $G = G_+ + G_-$.\nPrime-Split Support (II.L09)\ngives $\\operatorname{supp}(G_+) \\subseteq \\Lambda_\\tau^{(B)}$\n\ + and $\\operatorname{supp}(G_-) \\subseteq \\Lambda_\\tau^{(C)}$.\nSince $f_\\pm$ are the holomorphic maps\ncorresponding\ + \ to the transformers $G_\\pm$,\ntheir spectral supports\ninherit the same containment.\n\n\\emph{(IS4) Polarity symmetry.}\n\ + Polarity Symmetry (II.L10)\ngives $\\sigma_\\jj(G_+) = G_-$.\nTranslating to the level of maps\nvia the Mutual Determination\ + \ equivalence:\n$\\sigma_\\jj(f_+) = f_-$\nand $\\sigma_\\jj(f_-) = f_+$.\n\n\\medskip\n\\emph{Backward direction\n(idempotent-supported\ + \ $\\Rightarrow$ $\\tau$-holomorphic).}\n\nSuppose $f = e_+ \\cdot f_+ + e_- \\cdot f_-$\nsatisfies (IS1)--(IS4).\nWe\ + \ must show $f$ is $\\tau$-holomorphic.\n\nBy the Mutual Determination Theorem (II.T27),\n$\\tau$-holomorphy is equivalent\ + \ to any of\nthe five descriptions\n(R), (S), (G), (C), (H).\nWe verify description~(R):\n$f$ is a tower-coherent refinement\ + \ tail\nwith stabilized spectral support.\n\n\\emph{Step 1: Tower coherence.}\nBy (IS2), each $f_\\pm$ is tower-coherent.\n\ + Since $e_+, e_-$ are constant scalars,\n$f = e_+ \\cdot f_+ + e_- \\cdot f_-$\nis tower-coherent:\n\\begin{align*}\n \ + \ f_k\n &= e_+ \\cdot (f_+)_k + e_- \\cdot (f_-)_k \\\\\n &\\equiv e_+ \\cdot (f_+)_{k+1} + e_- \\cdot (f_-)_{k+1}\n\ + \ = f_{k+1}\n \\pmod{P_k},\n\\end{align*}\nwhere the congruence uses\nthe tower coherence of $(f_+)$ and $(f_-)$.\n\ + \n\\emph{Step 2: Stabilization.}\nBy (IS3), the spectral support of $f_+$\nis contained in the B-channel primes\nand that\ + \ of $f_-$\nin the C-channel primes.\nThe combined spectral support of $f$\nis $\\operatorname{supp}(f)\n= \\operatorname{supp}(f_+)\ + \ \\cup \\operatorname{supp}(f_-)\n\\subseteq \\Lambda_\\tau^{(B)} \\cup \\Lambda_\\tau^{(C)}\n= \\Lambda_\\tau$.\nSince\ + \ each $f_\\pm$ is tower-coherent\nwith support in a fixed set of primes,\nthe spectral support stabilizes\nat some finite\ + \ stage~$n$.\n\n\\emph{Step 3: Sector independence.}\nThe B-channel and C-channel\nof the resulting refinement tail\n\ + are independent:\nthe B-channel data $f_+$\ndepends only on $\\gamma$-orbit primes,\nand the C-channel data $f_-$\ndepends\ + \ only on $\\eta$-orbit primes.\nThese prime sets are disjoint\n(by Prime Polarity, I.T05).\nThe bipolar channel independence\n\ + (II.P07, Chapter~\\ref{ch:bndlift-construction})\nis therefore satisfied.\n\n\\emph{Step 4: The five descriptions agree.}\n\ + A tower-coherent refinement tail\nwith stabilized spectral support\nand bipolar channel independence\nsatisfies description~(R)\n\ + of the Mutual Determination Theorem (II.T27).\nBy II.T27, all five descriptions are equivalent.\nIn particular, description~(H)\n\ + gives a Hartogs continuation,\nand description~(G)\ngives a stabilized $\\omega$-germ.\nTherefore $f$ is $\\tau$-holomorphic.\n\ + \n\\emph{Step 5: Polarity symmetry is used.}\nCondition (IS4) is needed to ensure\nthat the two channel components $f_+,\ + \ f_-$\nare \\emph{compatible} with the holomorphic structure.\nA decomposition $f = e_+ \\cdot f_+ + e_- \\cdot f_-$\n\ + satisfying (IS1)--(IS3) but not (IS4)\nwould have the right support properties\nbut might fail the holomorphic condition\n\ + because the two channels would carry\nunrelated data.\nThe polarity symmetry $\\sigma_\\jj(f_+) = f_-$\nensures that $f_+$\ + \ and $f_-$\nencode the same underlying holomorphic datum,\nviewed from the two bipolar perspectives.\nThis is the condition\ + \ that\nthe Mutual Determination equivalence\n$(\\mathrm{R}) \\Leftrightarrow (\\mathrm{C})$\n(Lemma~\\ref{lem:germ-character},\ + \ II.L04)\nrequires: the boundary character\n$\\varphi \\colon R_\\tau \\to H_\\tau$\nmust satisfy\n$\\sigma_\\jj(\\varphi^+)\ + \ = \\varphi^-$,\nwhich is exactly (IS4)\nat the level of boundary data.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 114\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch38-three-lemma-chain.tex`\ + \ lines 793-809\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.ThreeLemmaChain`\n\ + - Name: `hol_iff_is_check`\n\n## Dependencies\n\n- Canonical: II.L07, II.L08, II.L09, II.L10, II.T27, I.T05, I.D21\n\n\ + ## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A tau-holomorphic function is holomorphic if and only if it is idempotent-supported: the bipolar projections + e_+f and e_-f have canonically split spectral support on B-channel and C-channel primes respectively. Proved via the three-lemma + ch' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001395.md b/_corpus_v3_items/cid001395.md new file mode 100644 index 000000000..61cf5e86f --- /dev/null +++ b/_corpus_v3_items/cid001395.md @@ -0,0 +1,112 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001395/ +title: Regularity Criterion +cid: cid001395 +primary_alias: THM0099 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A point p is tau-regular for f if and only if the omega-germ sequence stabilizes: there exists a finite stage N + such that the boundary lift agrees with f for all stages beyond N. Constructive and decidable at each finite stage.' +aliases: +- THM0099 +- II.T34 +- regularity-criterion +- thm:regularity-criterion +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005660 +- predicate: formalized_by + object: cid005661 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 39 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Regularity Criterion**\n\nA point p is tau-regular for f if and only if the omega-germ sequence stabilizes:\ + \ there exists a finite stage N such that the boundary lift agrees with f for all stages beyond N. Constructive and decidable\ + \ at each finite stage.\n\n# Regularity Criterion\n\n## Summary\n\nA point p is tau-regular for f if and only if the omega-germ\ + \ sequence stabilizes: there exists a finite stage N such that the boundary lift agrees with f for all stages beyond N.\ + \ Constructive and decidable at each finite stage.\n\n## Statement\n\n```latex\n%\n\\label{thm:regularity-criterion}\n\ + Let $f : \\tau^3 \\to H_\\tau$ be $\\tau$-holomorphic,\nand let $p \\in \\tau^3$.\nThen exactly one of the following holds:\n\ + \\begin{enumerate}\n \\item[\\textup{(R)}]\n $p$ is $\\tau$-regular for $f$:\n the $\\omega$-germ\ + \ sequence stabilizes at some finite stage $N$.\n Equivalently,\n $f$ has a canonical idempotent-supported\ + \ extension\n through $p$,\n determined by finitely much boundary data.\n \\item[\\textup{(S)}]\n\ + \ $p$ is genuinely $\\tau$-singular for $f$:\n the $\\omega$-germ sequence fails to stabilize ---\n\ + \ for every $N \\geq 1$,\n there exists $k > N$\n such that $\\rho_{k,N}(G_f)\n \\\ + neq G_f \\big|_{C_N(p)}$.\n The function requires infinitely much boundary data at $p$.\n\\end{enumerate}\nThere\ + \ is no third possibility.\nIn particular,\nthere are no removable singularities:\nif $p$ is not $\\tau$-regular,\nthen\ + \ no finite modification of $f$ at $p$\ncan restore regularity.\n\\[\n \\boxed{%\n \\text{For all } p \\in \\tau^3:\\\ + quad\n p \\text{ is $\\tau$-regular for } f\n \\;\\;\\dot{\\vee}\\;\\;\n p \\text{ is genuinely $\\tau$-singular\ + \ for } f.}\n\\]\n```\n\n## Proof / Justification\n\n```latex\nThe proof proceeds in three steps.\n\n\\medskip\n\\textbf{Step\ + \ 1: Exhaustive dichotomy.}\nThe $\\omega$-germ sequence at $p$\nis an inverse system\nindexed by the natural numbers\ + \ $N = 1, 2, 3, \\ldots$\nFor each pair $k \\geq N$,\nthe restriction map $\\rho_{k,N}$\nis either an isomorphism or not.\n\ + Define the \\emph{stabilization set}:\n\\[\n \\Sigma_f(p) = \\bigl\\{\\, N \\geq 1\n : \\forall\\, k \\geq N,\\\ + ;\n \\rho_{k,N}(G_f) = G_f \\big|_{C_N(p)} \\,\\bigr\\}.\n\\]\nEither $\\Sigma_f(p) \\neq \\varnothing$\n(case (R):\ + \ take $N = \\min \\Sigma_f(p)$)\nor $\\Sigma_f(p) = \\varnothing$ (case (S)).\nThese are exhaustive and mutually exclusive.\n\ + \n\\medskip\n\\textbf{Step 2: Regularity implies canonical extension.}\nSuppose $p$ is $\\tau$-regular,\nwith stabilization\ + \ at stage $N$.\nThen $G_f \\big|_{C_N(p)}$\ncontains all the germ data of $f$ at $p$.\nBy the Mutual Determination Theorem\n\ + (II.T27, Chapter~\\ref{ch:mutual-determination}),\nthe $\\omega$-germ data at stage $N$\ndetermines the boundary character\ + \ data at stage $N$,\nwhich by the Global Hartogs Theorem\n(I.T31, the Global Hartogs Extension (Book~I, Chapter~62))\n\ + determines the interior extension through $p$.\nBy the Idempotent Decomposition Lemma (II.L07),\nthis extension is canonically\n\ + $e_+$-/$e_-$-supported.\nThe extension is unique because\nthe Identity Theorem for $\\tau$-holomorphic functions\n(I.T21)\ + \ ensures that\ntwo extensions agreeing at stage $N$ must coincide.\n\n\\medskip\n\\textbf{Step 3: No removable singularities.}\n\ + Suppose $\\Sigma_f(p) = \\varnothing$ ---\nthe germ sequence fails to stabilize.\nWe claim that no finite modification\n\ + of $f$ at $p$ can produce stabilization.\n\nAssume for contradiction\nthat there exists $\\tilde{f}$\nagreeing with $f$\ + \ on $\\tau^3 \\setminus \\{p\\}$\nwith $\\tilde{f}(p) \\neq f(p)$,\nsuch that $\\tilde{f}$ is $\\tau$-regular at $p$.\n\ + Then $G_{\\tilde{f}}$ stabilizes at some stage $N$.\n\nConsider the cylinder $C_N(p)$\n(Definition~\\ref{def:stage-k-cylinder},\ + \ II.D10).\nThis is a clopen set in the ultrametric topology.\nBy the Cylinders Are Balls proposition\n(II.P04),\n$C_N(p)\ + \ = B(p, 2^{-N})$\nis the ultrametric ball of radius $2^{-N}$\ncentered at $p$.\nSince $C_N(p)$ is \\emph{clopen},\nit\ + \ has no boundary points ---\nevery point of $C_N(p)$\nis an interior point of $C_N(p)$,\nand every point outside $C_N(p)$\n\ + is an interior point of the complement.\n\nNow $f$ and $\\tilde{f}$ agree on $C_N(p) \\setminus \\{p\\}$.\nBy the $\\\ + tau$-Identity Theorem (I.T21),\napplied to the $\\omega$-germ transformers\n$G_f$ and $G_{\\tilde{f}}$\nrestricted to\ + \ $C_N(p)$:\nif these agree at stage $N$\non the germ data of all points\n$q \\in C_N(p) \\setminus \\{p\\}$,\nthen they\ + \ agree on the germ data\nof $p$ as well ---\nbecause tower coherence\n(via CRT reconstruction at each subsequent stage)\n\ + propagates agreement upward.\nBut this would mean $G_f$ also stabilizes\nat stage $N$ ---\ncontradicting $\\Sigma_f(p)\ + \ = \\varnothing$.\n\nTherefore no such $\\tilde{f}$ exists.\nThe singularity at $p$ is genuine:\nit is a property of\ + \ the function itself,\nnot of any particular presentation.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 116\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch39-regularity-positive.tex`\ + \ lines 463-499\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.PositiveRegularity`\n\ + - Name: `regularity_criterion_check`\n\n## Dependencies\n\n- Canonical: II.D49, II.L07, II.T33, II.T27, II.D10, I.T31\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A point p is tau-regular for f if and only if the omega-germ sequence stabilizes: there exists a finite stage + N such that the boundary lift agrees with f for all stages beyond N. Constructive and decidable at each finite stage.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001396.md b/_corpus_v3_items/cid001396.md new file mode 100644 index 000000000..fc6806ffa --- /dev/null +++ b/_corpus_v3_items/cid001396.md @@ -0,0 +1,121 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001396/ +title: Code/Decode Bijection +cid: cid001396 +primary_alias: THM0100 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Code and Decode are mutually inverse: a tau-holomorphic function IS its bipolar boundary coefficient stream, and + every coherent bipolar stream IS a holomorphic function. The Holomorphic Content Theorem.' +aliases: +- THM0100 +- II.T35 +- code-decode-bijection +- thm:code-decode-bijection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005650 +- predicate: formalized_by + object: cid005651 +- predicate: formalized_by + object: cid005652 +- predicate: formalized_by + object: cid005653 +- predicate: formalized_by + object: cid005654 +- predicate: formalized_by + object: cid005655 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 7, Chapter 41 (Part VI-A) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Code/Decode Bijection**\n\nCode and Decode are mutually inverse: a tau-holomorphic function IS its bipolar\ + \ boundary coefficient stream, and every coherent bipolar stream IS a holomorphic function. The Holomorphic Content Theorem.\n\ + \n# Code/Decode Bijection\n\n## Summary\n\nCode and Decode are mutually inverse: a tau-holomorphic function IS its bipolar\ + \ boundary coefficient stream, and every coherent bipolar stream IS a holomorphic function. The Holomorphic Content Theorem.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:code-decode-bijection}\nThe maps\n\\[\n \\mathrm{Code}\n \\;:\\;\n \ + \ \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau^{\\mathrm{cal}})\n \\;\\longrightarrow\\;\n \\mathrm{Code}_\\tau\n \\\ + qquad\\text{and}\\qquad\n \\mathrm{Decode}\n \\;:\\;\n \\mathrm{Code}_\\tau\n \\;\\longrightarrow\\;\n \ + \ \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau^{\\mathrm{cal}})\n\\]\nare mutually inverse bijections:\n\\[\n \\boxed{%\n \ + \ \\mathrm{Code} \\circ \\mathrm{Decode}\n \\;=\\;\n \\mathrm{id}_{\\mathrm{Code}_\\tau},\n \\qquad\n \\\ + mathrm{Decode} \\circ \\mathrm{Code}\n \\;=\\;\n \\mathrm{id}_{\\mathrm{Hol}_\\tau}.}\n\\]\nEquivalently: \\textbf{a\ + \ $\\tau$-holomorphic function is its bipolar\nboundary coefficient stream}, and every tower-coherent\nbipolar stream\ + \ is a $\\tau$-holomorphic function.\n```\n\n## Proof / Justification\n\n```latex\nWe verify both compositions.\n\n\\\ + smallskip\n\\noindent\\textbf{Step 1: $\\mathrm{Decode} \\circ \\mathrm{Code} = \\mathrm{id}$.}\nLet $f \\in \\mathrm{Hol}_\\\ + tau(\\tau^3, H_\\tau^{\\mathrm{cal}})$.\nSet $\\mathbf{c} := \\mathrm{Code}(f)$.\nWe must show $\\mathrm{Decode}(\\mathbf{c})\ + \ = f$.\n\nThe code $\\mathbf{c} = (c^+_f, c^-_f)$\nrecords the spectral coefficients of~$f$\nin both channels at every\ + \ stage.\nThe Decode map reconstructs\nstage-$k$ functions from these coefficients\nvia finite linear combinations\nin\ + \ the canonical basis.\nBut $f_k$ already \\emph{has} this expansion\n(by the uniqueness of the stage-$k$ spectral decomposition,\n\ + Proposition~\\ref{prop:ch35-stage-k-decomposition}).\nSo the reconstructed stage-$k$ function\nequals the original stage-$k$\ + \ function,\nat every stage.\nThe inverse limits agree:\n$\\mathrm{Decode}(\\mathbf{c}) = f$.\n\n\\smallskip\n\\noindent\\\ + textbf{Step 2: $\\mathrm{Code} \\circ \\mathrm{Decode} = \\mathrm{id}$.}\nLet $\\mathbf{c} = (c^+, c^-) \\in \\mathrm{Code}_\\\ + tau$.\nSet $f := \\mathrm{Decode}(\\mathbf{c})$.\nWe must show $\\mathrm{Code}(f) = \\mathbf{c}$.\n\nThe function $f$\ + \ is built from the code:\nat stage~$k$, the coefficients of $f_k$\nin the canonical basis\nare exactly the entries of\ + \ $c^\\pm_k$\n(by construction~(D1) of Definition~\\ref{def:decode}).\nApplying the Code map extracts\nthese same coefficients\n\ + via the Projection Formula (II.P08).\nWe verify that Code recovers the original data.\n\nFix a stage~$k$, a prime $p \\\ + mid P_k$,\na residue class $v \\in \\mathbb{Z}/p\\mathbb{Z}$,\nand a channel $\\sigma \\in \\{B, C\\}$.\nThe Projection\ + \ Formula gives:\n\\[\n \\varphi_{p,v}^{(\\sigma)}\n \\;=\\;\n \\frac{1}{|F_p|}\n \\sum_{x \\in F_p(v)}\n\ + \ e_\\sigma \\cdot f_k(x).\n\\]\nBy step~(D1), $f_k(x)$\nis a sum of cylinder generators\nwith coefficients from~$\\\ + mathbf{c}$.\nThe idempotent projection $e_\\sigma$\nkills the opposite channel:\n$e_+ \\cdot E_{k,v}^{(C)}(x) = 0$\nand\ + \ $e_- \\cdot E_{k,v}^{(B)}(x) = 0$.\nThe surviving terms are exactly the $\\sigma$-channel\ncontributions.\nSumming over\ + \ the fiber $F_p(v)$\npicks out the coefficient $\\varphi_{p,v}^{(\\sigma)}$\nfrom~$\\mathbf{c}$:\nthe Projection Formula\ + \ is the inverse\nof the basis expansion,\nby the orthogonality of cylinder generators\n(Remark~\\ref{rem:ch35-idempotent-orthogonality}).\n\ + Hence $\\mathrm{Code}(f) = \\mathbf{c}$.\n\n\\smallskip\n\\noindent\\textbf{Step 3: Surjectivity ingredients.}\nThe argument\ + \ for Step~2\nimplicitly uses that every code\nactually produces a holomorphic function\n(Proposition~\\ref{prop:ch41-decode-holomorphic}).\n\ + The key inputs are:\n\\begin{itemize}\n \\item \\textbf{Sheaf coherence} (II.T32,\n Theorem~\\ref{thm:sheaf-axioms},\n\ + \ Chapter~\\ref{ch:sheaf-coherence}):\n the stage-$k$ functions\n assembled from a code\n \ + \ paste into a global section\n of the holomorphic presheaf~$\\mathcal{O}_\\tau$.\n \\item \\textbf{Idempotent\ + \ decomposition} (II.L07,\n Chapter~\\ref{ch:idempotent-decomposition}):\n the bipolar splitting\n \ + \ is functorial and complete,\n so no information is lost\n in the channel separation.\n \ + \ \\item \\textbf{Mutual Determination} (II.T27,\n Theorem~\\ref{thm:mutual-determination},\n Chapter~\\\ + ref{ch:mutual-determination}):\n the boundary spectral data determines\n the full holomorphic function\n\ + \ (equivalence of descriptions).\n\\end{itemize}\nThese three ingredients combine to ensure\nthat Decode is surjective\ + \ onto the holomorphic functions\nwhose boundary data matches the given code.\nTogether with injectivity (Step~1),\nthe\ + \ bijection is established.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 123\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part07/ch41-code-decode.tex` lines 478-509\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Regularity.CodeDecode`\n- Name: `full_code_decode_check`\n\ + \n## Dependencies\n\n- Canonical: II.D51, II.D52, II.L07, II.T27, II.T32, I.T31\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Code and Decode are mutually inverse: a tau-holomorphic function IS its bipolar boundary coefficient stream, + and every coherent bipolar stream IS a holomorphic function. The Holomorphic Content Theorem.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001397.md b/_corpus_v3_items/cid001397.md new file mode 100644 index 000000000..7163dce63 --- /dev/null +++ b/_corpus_v3_items/cid001397.md @@ -0,0 +1,123 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001397/ +title: Tau-Yoneda Embedding +cid: cid001397 +primary_alias: THM0101 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Yoneda functor y: tau -> [tau^op, tau] is fully faithful: (1) well-defined with internal target via self-enrichment, + (2) faithful by evaluation at id, (3) full via Code/Decode computable witnesses, (4) bipolar-preserving under the e_+/e_- + decomposition.' +aliases: +- THM0101 +- II.T36 +- tau-yoneda-embedding +- thm:yoneda-embedding +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005535 +- predicate: formalized_by + object: cid005536 +- predicate: formalized_by + object: cid005537 +- predicate: formalized_by + object: cid005538 +- predicate: formalized_by + object: cid005539 +- predicate: formalized_by + object: cid005540 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 43 (Part VI-B) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Tau-Yoneda Embedding**\n\nThe Yoneda functor y: tau -> [tau^op, tau] is fully faithful: (1) well-defined with\ + \ internal target via self-enrichment, (2) faithful by evaluation at id, (3) full via Code/Decode computable witnesses,\ + \ (4) bipolar-preserving under the e_+/e_- decomposition.\n\n# Tau-Yoneda Embedding\n\n## Summary\n\nThe Yoneda functor\ + \ y: tau -> [tau^op, tau] is fully faithful: (1) well-defined with internal target via self-enrichment, (2) faithful by\ + \ evaluation at id, (3) full via Code/Decode computable witnesses, (4) bipolar-preserving under the e_+/e_- decomposition.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:yoneda-embedding}\nThe Yoneda functor $y \\colon \\tau \\to [\\tau^{\\op},\ + \ \\tau]$\n(Definition~\\ref{def:yoneda-functor})\nis fully faithful.\nConcretely:\n\\begin{enumerate}\n \\item \\\ + textbf{Well-defined.}\n For each object $A$,\n the representable presheaf $y(A) = [-, A]$\n \ + \ is a $\\tau$-object.\n The functor $y$ is a functor between $\\tau$-enriched categories.\n\n \\item \\\ + textbf{Faithful.}\n If $y(\\varphi) = y(\\psi)$\n for morphisms $\\varphi, \\psi \\colon A \\to B$,\n\ + \ then $\\varphi = \\psi$.\n\n \\item \\textbf{Full.}\n Every natural transformation\n $\\\ + eta \\colon y(A) \\Rightarrow y(B)$\n that respects the bipolar sector structure\n is of the form $\\\ + eta = y(\\varphi)$\n for a unique $\\varphi \\colon A \\to B$.\n\n \\item \\textbf{Bipolar preservation.}\n\ + \ For every morphism $\\varphi \\colon A \\to B$\n with bipolar decomposition\n $\\varphi =\ + \ e_+ \\cdot \\varphi_+ + e_- \\cdot \\varphi_-$,\n \\[\n y(e_+ \\cdot \\varphi_+ + e_- \\cdot \\\ + varphi_-)\n \\;=\\;\n e_+ \\cdot y(\\varphi_+) + e_- \\cdot y(\\varphi_-).\n \\]\n\ + \ The Yoneda embedding preserves the bipolar splitting.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n\ + ```latex\nWe prove each property in turn.\n\n\\medskip\n\\textbf{(1) Well-definedness.}\\;\nBy self-enrichment (II.D53),\n\ + $[P, A]$ is a $\\tau$-object\nfor every pair $(P, A)$.\nThe assignment $P \\mapsto [P, A]$\nis contravariant-functorial\ + \ in $P$\n(precomposition is well-defined\nby the category structure earned in II.T28).\nTherefore $y(A) = [-, A]$\nis\ + \ a functor $\\tau^{\\op} \\to \\tau$.\nThe morphism map $y(\\varphi)$\nis a natural transformation\nby the probe naturality\ + \ argument\n(Lemma~\\ref{lem:probe-yoneda}, (PN) $\\Rightarrow$ (Yon)).\n\n\\medskip\n\\textbf{(2) Faithfulness.}\\;\n\ + Assume $y(\\varphi) = y(\\psi)$\nas natural transformations $y(A) \\Rightarrow y(B)$.\nEvaluate at $P = A$ and $f = \\\ + id_A$:\n\\[\n \\varphi\n \\;=\\; y(\\varphi)_A(\\id_A)\n \\;=\\; y(\\psi)_A(\\id_A)\n \\;=\\; \\psi.\n\\]\n\ + Alternatively, this follows from the Code/Decode bijection (II.T35):\n$\\mathrm{Code}(\\varphi) = \\mathrm{Code}(\\psi)$\n\ + (since $y(\\varphi)$ determines the boundary stream of $\\varphi$),\nand $\\mathrm{Code}$ is injective.\n\n\\medskip\n\ + \\textbf{(3) Fullness.}\\;\nLet $\\eta \\colon y(A) \\Rightarrow y(B)$\nbe a natural transformation\nthat respects the\ + \ bipolar sector structure.\nDefine\n\\[\n \\varphi \\;:=\\; \\eta_A(\\id_A) \\;\\in\\; [A, B].\n\\]\nWe claim $\\\ + eta = y(\\varphi)$.\nFor any probe $g \\colon P \\to A$:\n\\[\n \\eta_P(g)\n \\;=\\; \\eta_P\\bigl(g^*(\\id_A)\\\ + bigr)\n \\;=\\; g^*\\bigl(\\eta_A(\\id_A)\\bigr)\n \\;=\\; \\varphi \\circ g\n \\;=\\; y(\\varphi)_P(g).\n\\\ + ]\nThe second equality uses naturality of $\\eta$:\n\\[\n \\begin{tikzcd}\n {[A, A]} \\ar[r, \"{\\eta_A}\"]\ + \ \\ar[d, \"{g^*}\"']\n & {[A, B]} \\ar[d, \"{g^*}\"] \\\\\n {[P, A]} \\ar[r, \"{\\eta_P}\"']\n &\ + \ {[P, B]}\n \\end{tikzcd}\n\\]\ncommutes, so $\\eta_P(g^*(\\id_A)) = g^*(\\eta_A(\\id_A))$.\n\nThe morphism $\\varphi\ + \ = \\eta_A(\\id_A)$ is well-defined\nas a $\\tau$-morphism\nbecause $\\eta$ respects the bipolar structure\nand $\\id_A$\ + \ is bipolar-neutral.\nThe Code/Decode bijection (II.T35)\nprovides the computable witness:\n$\\mathrm{Code}(\\varphi)$\ + \ is the boundary stream\nextracted from $\\eta_A(\\id_A)$,\nand $\\mathrm{Decode}(\\mathrm{Code}(\\varphi)) = \\varphi$.\n\ + \nUniqueness:\nif $\\eta = y(\\varphi) = y(\\psi)$,\nthen $\\varphi = \\psi$ by faithfulness.\n\n\\medskip\n\\textbf{(4)\ + \ Bipolar preservation.}\\;\nLet $\\varphi = e_+ \\cdot \\varphi_+ + e_- \\cdot \\varphi_-$\nbe the bipolar decomposition\ + \ of $\\varphi$.\nFor any probe $g \\colon P \\to A$:\n\\[\n y(\\varphi)_P(g)\n \\;=\\; \\varphi \\circ g\n \\\ + ;=\\; (e_+ \\cdot \\varphi_+ + e_- \\cdot \\varphi_-) \\circ g\n \\;=\\; e_+ \\cdot (\\varphi_+ \\circ g) + e_- \\\ + cdot (\\varphi_- \\circ g),\n\\]\nwhere the last equality uses\nthe bilinearity of composition\nover the idempotent decomposition\n\ + (the program monoid respects\nthe $e_\\pm$-splitting, I.P02).\nTherefore\n\\[\n y(\\varphi)_P\n \\;=\\;\n e_+\ + \ \\cdot y(\\varphi_+)_P\n \\;+\\;\n e_- \\cdot y(\\varphi_-)_P,\n\\]\nwhich is the bipolar decomposition of $y(\\\ + varphi)$\nat each component $P$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 129\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch43-yoneda-theorem.tex` lines 462-499\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Enrichment.YonedaTheorem`\n-\ + \ Name: `Tau.BookII.Enrichment.full_yoneda_check`\n\n## Dependencies\n\n- Canonical: II.D53, II.D54, II.P11, II.T35, II.L11,\ + \ I.T40, I.T41\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Yoneda functor y: tau -> [tau^op, tau] is fully faithful: (1) well-defined with internal target via self-enrichment, + (2) faithful by evaluation at id, (3) full via Code/Decode computable witnesses, (4) bipolar-preserving under the e_+/e' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001398.md b/_corpus_v3_items/cid001398.md new file mode 100644 index 000000000..b4653b0fb --- /dev/null +++ b/_corpus_v3_items/cid001398.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001398/ +title: Hartogs Extension Uniqueness +cid: cid001398 +primary_alias: THM0102 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Hartogs Extension Uniqueness +aliases: +- THM0102 +- II.T37 +- hartogs-extension-uniqueness +- thm:hartogs-uniqueness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005412 +- predicate: formalized_by + object: cid005413 +- predicate: formalized_by + object: cid005418 +- predicate: formalized_by + object: cid005419 +- predicate: formalized_by + object: cid005420 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 48 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Hartogs Extension Uniqueness**\n\nHartogs Extension Uniqueness\n\n# Hartogs Extension Uniqueness\n\n## Summary\n\ + \nHartogs Extension Uniqueness\n\n## Statement\n\n```latex\n%\n\\label{thm:hartogs-uniqueness}\n% II.D49, I.T31, I.D21\n\ + Let $\\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$\nbe an idempotent-supported character\non $\\widehat{\\mathbb{Z}}_\\\ + tau$.\nThen the extension $f_\\chi : \\tau^3 \\to H_\\tau$\nof Lemma~\\textup{\\ref{lem:extension-h-tau}} is \\textbf{unique}:\n\ + \\[\n \\boxed{%\n g : \\tau^3 \\to H_\\tau\n \\text{ is $\\tau$-holomorphic},\\;\\;\n g\\big|_{\\mathbb{L}}\ + \ = \\chi\n \\;\\;\\Longrightarrow\\;\\;\n g = f_\\chi.}\n\\]\nNo $\\tau$-holomorphic function on~$\\tau^3$\nwith\ + \ boundary value~$\\chi$\ncan differ from~$f_\\chi$.\n```\n\n## Proof / Justification\n\n```latex\n[Proof via Code/Decode]\n\ + The Code/Decode bijection\n(Theorem~\\ref{thm:code-decode-bijection}, II.T35,\nChapter~\\ref{ch:code-decode})\nstates\ + \ that every $\\tau$-holomorphic function\n$g : \\tau^3 \\to H_\\tau$\nis uniquely determined\nby its boundary coefficient\ + \ stream\n$\\mathrm{Code}(g) \\in \\widehat{\\mathbb{Z}}_\\tau \\times \\widehat{\\mathbb{Z}}_\\tau$.\n\n\\medskip\n\\\ + textbf{Step 1.}\nBy hypothesis, $g\\big|_{\\mathbb{L}} = \\chi$.\nThe boundary coefficient stream\nis the spectral decomposition\n\ + of the boundary restriction:\n$\\mathrm{Code}(g) = (\\chi_+, \\chi_-)$.\n\n\\medskip\n\\textbf{Step 2.}\nBy Lemma~\\ref{lem:extension-h-tau},\n\ + $f_\\chi\\big|_{\\mathbb{L}} = \\chi$.\nHence\n$\\mathrm{Code}(f_\\chi) = (\\chi_+, \\chi_-)$.\n\n\\medskip\n\\textbf{Step\ + \ 3.}\nSince $\\mathrm{Code}$ is a bijection (II.T35),\n$\\mathrm{Code}(g) = \\mathrm{Code}(f_\\chi)$\nimplies $g = f_\\\ + chi$.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 142\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch48-hartogs-extension-h-tau.tex`\ + \ lines 412-432\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.HartogsExtension`\n\ + - Name: `Tau.BookII.CentralTheorem.hartogs_uniqueness_check`\n\n## Dependencies\n\n- Canonical: II.L12, II.D59, II.P13,\ + \ II.T27, II.T35, II.D49, I.T31, I.D21\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Hartogs Extension Uniqueness +noindex: false +--- + diff --git a/_corpus_v3_items/cid001399.md b/_corpus_v3_items/cid001399.md new file mode 100644 index 000000000..891540ec6 --- /dev/null +++ b/_corpus_v3_items/cid001399.md @@ -0,0 +1,107 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001399/ +title: Extensions Are Omega-Germ Transformers +cid: cid001399 +primary_alias: THM0103 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Hartogs extensions from L to tau^3 are the same as omega-germ transformers: the extension of boundary data through + the tower produces precisely the tower-coherent family of stage-k morphisms that defines an omega-germ transformer.' +aliases: +- THM0103 +- II.T38 +- extensions-are-omega-germ-transformers +- thm:extensions-omega-germs +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005404 +- predicate: formalized_by + object: cid005405 +- predicate: formalized_by + object: cid005408 +- predicate: formalized_by + object: cid005409 +- predicate: formalized_by + object: cid005410 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 49 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Extensions Are Omega-Germ Transformers**\n\nHartogs extensions from L to tau^3 are the same as omega-germ transformers:\ + \ the extension of boundary data through the tower produces precisely the tower-coherent family of stage-k morphisms that\ + \ defines an omega-germ transformer.\n\n# Extensions Are Omega-Germ Transformers\n\n## Summary\n\nHartogs extensions from\ + \ L to tau^3 are the same as omega-germ transformers: the extension of boundary data through the tower produces precisely\ + \ the tower-coherent family of stage-k morphisms that defines an omega-germ transformer.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:extensions-omega-germs}\nThere is a canonical bijection\n\\[\n \\boxed{%\n \\left\\{\\;\n \\parbox{0.4\\\ + linewidth}{\\centering\n Hartogs extensions $f_\\chi$\\\\\n of boundary characters $\\chi$\\\\\n \ + \ {\\small(II.T37)}\n }\n \\;\\right\\}\n \\;\\;\\xleftrightarrow{\\;\\;\\sim\\;\\;}\\;\\;\n \\left\\{\\;\n\ + \ \\parbox{0.4\\linewidth}{\\centering\n $\\omega$-germ transformers $G$\\\\\n regular at the boundary\\\ + \\\n {\\small(I.D45 + regularity)}\n }\n \\;\\right\\}}\n\\]\ngiven by:\n\\begin{enumerate}\n \\item[\\\ + textup{(i)}]\n \\textbf{Forward.}\n $f_\\chi \\;\\longmapsto\\; G_{f_\\chi}\n := \\varprojlim_k\ + \ f_{\\chi,k}$.\n Stagewise naturality \\textup{(II.L13)}\n ensures tower coherence;\n bipolar\ + \ splitting ensures sector independence;\n the limit exists and gives a $\\mathrm{HolFun}$\n \\textup{(I.D47)}.\n\ + \n \\item[\\textup{(ii)}]\n \\textbf{Converse.}\n $G \\;\\longmapsto\\; f_{\\chi_G}$,\n \ + \ where $\\chi_G = \\varprojlim_k (G_k\\big|_{R_\\tau})$\n is the boundary restriction.\n Regularity\ + \ ensures $\\chi_G$ is idempotent-supported;\n \\textup{II.T37} produces the unique extension.\n\n \\item[\\\ + textup{(iii)}]\n \\textbf{Inverse property.}\n The two maps are mutually inverse:\n $G_{f_{\\\ + chi_G}} = G$\n \\textup{(Proposition~\\ref{prop:ch49-converse})}\n and $f_{\\chi_{G_{f_\\chi}}} = f_\\\ + chi$\n \\textup{(by uniqueness of the Hartogs extension,\n II.T37)}.\n\\end{enumerate}\n\n\\noindent\n\ + Moreover, the bijection respects the bipolar decomposition:\n\\[\n G_{f_\\chi}\n \\;=\\;\n e_+ \\cdot G_{f_\\\ + chi}^+\n \\;+\\;\n e_- \\cdot G_{f_\\chi}^-,\n\\]\nwhere $G_{f_\\chi}^\\pm$\ndepends only on $\\chi_\\pm$.\n```\n\ + \n## Proof / Justification\n\n```latex\nThe forward direction is\nProposition~\\ref{prop:ch49-g-is-holfun}:\n$G_{f_\\\ + chi}$ is a $\\tau$-holomorphic function\n(sector-independent and tower-coherent),\nhence an $\\omega$-germ transformer.\n\ + It is regular at the boundary\nbecause its boundary restriction\nrecovers~$\\chi$\n(by construction:\n$G_{f_\\chi}\\big|_{R_\\\ + tau}\n= \\varprojlim_k f_{\\chi,k}\\big|_{R_\\tau}\n= \\chi$,\nsince $f_\\chi$ is the extension of~$\\chi$).\n\nThe converse\ + \ direction is\nProposition~\\ref{prop:ch49-converse}:\nregularity gives $\\chi_G$,\nII.T37 gives $f_{\\chi_G}$,\nand\ + \ II.T27 gives $G_{f_{\\chi_G}} = G$.\n\nThe inverse property in direction (iii):\nstarting from $f_\\chi$,\nwe construct\ + \ $G_{f_\\chi}$,\nrestrict to the boundary to get $\\chi_{G_{f_\\chi}}$,\nand extend again.\nThe boundary restriction\ + \ of $G_{f_\\chi}$\nis~$\\chi$ (shown above),\nso $\\chi_{G_{f_\\chi}} = \\chi$,\nand the extension of~$\\chi$ is $f_\\\ + chi$ (II.T37 uniqueness).\nHence $f_{\\chi_{G_{f_\\chi}}} = f_\\chi$.\n\nThe bipolar decomposition\nfollows from\nProposition~\\\ + ref{prop:ch49-bipolar-stage}\nand the fact that the inverse limit\ncommutes with the idempotent projection\n(projections\ + \ preserve limits).\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 144\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch49-extensions-omega-germs.tex`\ + \ lines 662-725\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms`\n\ + - Name: `Tau.BookII.CentralTheorem.omega_germ_transformer_check`\n\n## Dependencies\n\n- Canonical: II.T37, II.L13, II.T27,\ + \ II.D37\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Hartogs extensions from L to tau^3 are the same as omega-germ transformers: the extension of boundary data through + the tower produces precisely the tower-coherent family of stage-k morphisms that defines an omega-germ transformer.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001400.md b/_corpus_v3_items/cid001400.md new file mode 100644 index 000000000..6da22a3a2 --- /dev/null +++ b/_corpus_v3_items/cid001400.md @@ -0,0 +1,116 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001400/ +title: Omega-Germs iff Holomorphic Functions +cid: cid001400 +primary_alias: THM0104 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The category of omega-germ transformers is equivalent to the category of tau-holomorphic functions: every omega-germ + transformer arises from a unique holomorphic function, and conversely. Bridges the algebraic and analytic perspectives.' +aliases: +- THM0104 +- II.T39 +- omega-germs-iff-holomorphic-functions +- thm:omega-germs-holomorphic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005432 +- predicate: formalized_by + object: cid005433 +- predicate: formalized_by + object: cid005434 +- predicate: formalized_by + object: cid005436 +- predicate: formalized_by + object: cid005437 +- predicate: formalized_by + object: cid005438 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Omega-Germs iff Holomorphic Functions**\n\nThe category of omega-germ transformers is equivalent to the category\ + \ of tau-holomorphic functions: every omega-germ transformer arises from a unique holomorphic function, and conversely.\ + \ Bridges the algebraic and analytic perspectives.\n\n# Omega-Germs iff Holomorphic Functions\n\n## Summary\n\nThe category\ + \ of omega-germ transformers is equivalent to the category of tau-holomorphic functions: every omega-germ transformer\ + \ arises from a unique holomorphic function, and conversely. Bridges the algebraic and analytic perspectives.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:omega-germs-holomorphic}\nThere is a canonical bijection\n\\[\n \\boxed{%\n \\{\\textup{$\\\ + omega$-germ transformers on } \\tau^3\\}\n \\;\\xleftrightarrow[\\Psi]{\\;\\sim\\;}\n \\mathrm{Hol}_\\tau(\\tau^3,\ + \ H_\\tau).}\n\\]\nExplicitly:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Forward} ($\\omega$-germ\ + \ $\\to$ holomorphic).\n Given an $\\omega$-germ transformer\n $G = (G_k)_{k \\geq 1}$,\n the\ + \ Yoneda Application\n \\textup{(Lemma~\\ref{lem:yoneda-application}, II.L14)}\n produces a natural\ + \ transformation\n $\\eta_G \\colon y(\\tau^3) \\to y(H_\\tau)$.\n The Yoneda embedding\n \\\ + textup{(Theorem~\\ref{thm:yoneda-embedding}, II.T36)}\n then produces a unique morphism\n $f_G \\in\ + \ \\Hom_\\tau(\\tau^3, H_\\tau)$\n with $y(f_G) = \\eta_G$.\n This $f_G$ is $\\tau$-holomorphic\n \ + \ by the characterization theorem\n \\textup{(II.T33)}.\n\n \\item[\\textup{(ii)}]\n \\textbf{Backward}\ + \ (holomorphic $\\to$ $\\omega$-germ).\n Given $f \\in \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau)$,\n the\ + \ Pre-Yoneda embedding\n \\textup{(Definition~\\ref{def:pre-yoneda}, II.D50,\n Chapter~\\ref{ch:pre-yoneda})}\n\ + \ produces $y(f) \\colon y(\\tau^3) \\to y(H_\\tau)$.\n The Yoneda Application\n \\textup{(Lemma~\\\ + ref{lem:yoneda-application})}\n then extracts the $\\omega$-germ transformer\n $G_f = (G_{f,k})_{k \\\ + geq 1}$\n associated to~$y(f)$.\n\n \\item[\\textup{(iii)}]\n \\textbf{Bijectivity.}\n The\ + \ maps $G \\mapsto f_G$ and $f \\mapsto G_f$\n are inverse to each other.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\n```latex\nThe proof assembles three previously established bijections.\n\n\\smallskip\n\\noindent\\\ + textbf{Step 1: Yoneda embedding.}\nBy the Yoneda embedding theorem (II.T36),\nthe map\n\\[\n y \\colon \\Hom_\\tau(\\\ + tau^3, H_\\tau)\n \\;\\xrightarrow{\\;\\sim\\;}\n \\mathrm{Nat}\\bigl(y(\\tau^3),\\, y(H_\\tau)\\bigr)\n\\]\nis\ + \ a bijection.\nFull faithfulness of~$y$\ngives both injectivity and surjectivity.\nThis is the abstract Yoneda bijection,\n\ + valid for any pair of objects in~$\\tau$.\n\n\\smallskip\n\\noindent\\textbf{Step 2: Yoneda Application.}\nBy Lemma~\\\ + ref{lem:yoneda-application} (II.L14),\nthe map\n\\[\n \\mathrm{Nat}\\bigl(y(\\tau^3),\\, y(H_\\tau)\\bigr)\n \\\ + ;\\xrightarrow{\\;\\sim\\;}\n \\{\\text{$\\omega$-germ transformers on } \\tau^3\\}\n\\]\nis a bijection.\nThis step\ + \ uses the specific profinite structure\nof~$\\tau^3$ (primorial probe decomposition)\nand the identification of probe\ + \ naturality\nwith stagewise naturality of $\\omega$-germs\n(II.R12, II.L13).\n\n\\smallskip\n\\noindent\\textbf{Step\ + \ 3: Characterization.}\nBy the characterization theorem (II.T33,\nChapter~\\ref{ch:three-lemma-chain}),\n$\\Hom_\\tau(\\\ + tau^3, H_\\tau)\n= \\mathrm{Hol}_\\tau(\\tau^3, H_\\tau)$:\nthe morphisms from~$\\tau^3$ to~$H_\\tau$\nin~$\\tau$ are\ + \ exactly the $\\tau$-holomorphic functions.\nThis identification is given by the 3-lemma chain\n(II.L08--II.L10):\nevery\ + \ such morphism is idempotent-supported,\nand every idempotent-supported function\nis $\\tau$-holomorphic.\n\n\\smallskip\n\ + \\noindent\\textbf{Composition.}\nComposing Steps~1 and~2 gives a bijection\n\\[\n \\Psi \\colon\n \\mathrm{Hol}_\\\ + tau(\\tau^3, H_\\tau)\n \\;\\xrightarrow[\\text{Step 1}]{\\;y\\;}\n \\mathrm{Nat}\\bigl(y(\\tau^3),\\, y(H_\\tau)\\\ + bigr)\n \\;\\xrightarrow[\\text{Step 2}]{\\;\\sim\\;}\n \\{\\text{$\\omega$-germ transformers}\\}.\n\\]\nThe inverse\ + \ is the reverse composition\n$\\Psi^{-1} \\colon G \\mapsto y^{-1}(\\eta_G) = f_G$.\nSince each step is a bijection,\n\ + the composition is a bijection.\n\n\\smallskip\n\\noindent\\textbf{Bijectivity} follows from the bijectivity\nof each\ + \ factor.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 146\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch50-yoneda-applied.tex`\ + \ lines 430-477\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.YonedaApplied`\n\ + - Name: `Tau.BookII.CentralTheorem.omega_germs_holomorphic_check`\n\n## Dependencies\n\n- Canonical: II.T35, II.T33, II.L14,\ + \ II.L13, II.T38, II.T34\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The category of omega-germ transformers is equivalent to the category of tau-holomorphic functions: every omega-germ + transformer arises from a unique holomorphic function, and conversely. Bridges the algebraic and analytic perspectives.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001401.md b/_corpus_v3_items/cid001401.md new file mode 100644 index 000000000..a89444430 --- /dev/null +++ b/_corpus_v3_items/cid001401.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001401/ +title: Central Theorem +cid: cid001401 +primary_alias: THM0105 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Central Theorem: canonical isomorphism O(tau^3) = A_spec(L), identifying holomorphic functions on the fibered + product with spectral characters on the lemniscate boundary. Functorial, bipolar-compatible, tower-graded, iota_tau-calibrated.' +aliases: +- THM0105 +- II.T40 +- central-theorem +- thm:central-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005391 +- predicate: formalized_by + object: cid005392 +- predicate: formalized_by + object: cid005393 +- predicate: formalized_by + object: cid005394 +- predicate: formalized_by + object: cid005398 +- predicate: formalized_by + object: cid005399 +- predicate: formalized_by + object: cid005400 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 51 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Central Theorem**\n\nThe Central Theorem: canonical isomorphism O(tau^3) = A_spec(L), identifying holomorphic\ + \ functions on the fibered product with spectral characters on the lemniscate boundary. Functorial, bipolar-compatible,\ + \ tower-graded, iota_tau-calibrated.\n\n# Central Theorem\n\n## Summary\n\nThe Central Theorem: canonical isomorphism\ + \ O(tau^3) = A_spec(L), identifying holomorphic functions on the fibered product with spectral characters on the lemniscate\ + \ boundary. Functorial, bipolar-compatible, tower-graded, iota_tau-calibrated.\n\n## Statement\n\n```latex\n%\n\\label{thm:central-theorem}\n\ + % II.T27, II.T33, II.T35, II.T36, II.T37, II.T38, II.T39,\n% II.D35, II.D59, II.P13, II.L07, II.L12, II.L13, II.L14\n\ + There is a canonical isomorphism of calibrated\nsplit-complex algebras:\n\\begin{equation}\n\\label{eq:ch51-central-theorem}\n\ + \ \\boxed{\\;\n \\mathcal{O}(\\tau^3)\n \\;\\cong\\;\n A_{\\mathrm{spec}}(\\mathbb{L})\n \\;}\n\\end{equation}\n\ + The isomorphism is:\n\\begin{enumerate}\n \\item[\\textup{(a)}] \\textbf{canonical}:\n no choices are involved\ + \ in its construction;\n \\item[\\textup{(b)}] \\textbf{functorial}:\n it commutes with morphisms of $\\tau$-spaces;\n\ + \ \\item[\\textup{(c)}] \\textbf{compatible with the bipolar decomposition}:\n it sends $e_+ \\cdot f_+$ to\ + \ $e_+ \\cdot \\chi_+$\n and $e_- \\cdot f_-$ to $e_- \\cdot \\chi_-$;\n \\item[\\textup{(d)}] \\textbf{compatible\ + \ with the tower grading}:\n it respects the stage-by-stage structure\n of the primorial tower;\n \ + \ \\item[\\textup{(e)}] \\textbf{compatible with $\\iota_\\tau$-calibration}:\n the numerical values of spectral\ + \ coefficients\n in $A_{\\mathrm{spec}}(\\mathbb{L})$\n are the numerical values of\n the corresponding\ + \ holomorphic function\n in $\\mathcal{O}(\\tau^3)$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n_No\ + \ immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 148\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch51-central-theorem.tex`\ + \ lines 377-411\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.CentralTheorem`\n\ + - Name: `Tau.BookII.CentralTheorem.central_theorem_check`\n\n## Dependencies\n\n- Canonical: I.T05, I.T31, I.D18, I.D19,\ + \ I.D21, I.T41, II.T27, II.T33, II.T35, II.T36, II.T37, II.T38, II.T39, II.D35, II.D59, II.P13, II.L07, II.L12, II.L13,\ + \ II.L14\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by\ + \ later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Central Theorem: canonical isomorphism O(tau^3) = A_spec(L), identifying holomorphic functions on the fibered + product with spectral characters on the lemniscate boundary. Functorial, bipolar-compatible, tower-graded, iota_tau-calibrated' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001402.md b/_corpus_v3_items/cid001402.md new file mode 100644 index 000000000..a12a5cde7 --- /dev/null +++ b/_corpus_v3_items/cid001402.md @@ -0,0 +1,106 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001402/ +title: Liouville Categorical Dodge +cid: cid001402 +primary_alias: THM0106 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The classical Liouville theorem does not apply to tau^3: j^2 = +1 gives a wave-type PDE operator (hyperbolic), not + the elliptic Laplacian. The maximum principle fails, so non-constant bounded holomorphic functions exist.' +aliases: +- THM0106 +- II.T41 +- liouville-categorical-dodge +- thm:liouville-dodge +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005374 +- predicate: formalized_by + object: cid005375 +- predicate: formalized_by + object: cid005382 +- predicate: formalized_by + object: cid005383 +- predicate: formalized_by + object: cid005384 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 52 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Liouville Categorical Dodge**\n\nThe classical Liouville theorem does not apply to tau^3: j^2 = +1 gives a\ + \ wave-type PDE operator (hyperbolic), not the elliptic Laplacian. The maximum principle fails, so non-constant bounded\ + \ holomorphic functions exist.\n\n# Liouville Categorical Dodge\n\n## Summary\n\nThe classical Liouville theorem does\ + \ not apply to tau^3: j^2 = +1 gives a wave-type PDE operator (hyperbolic), not the elliptic Laplacian. The maximum principle\ + \ fails, so non-constant bounded holomorphic functions exist.\n\n## Statement\n\n```latex\n%\n\\label{thm:liouville-dodge}\n\ + The classical Liouville theorem does not apply to~$\\tau^3$.\nSpecifically:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n\ + \ The split-complex structure $\\jj^2 = +1$\n \\textup{(I.T10, Book~I)}\n induces a \\textbf{wave-type}\ + \ PDE operator\n $\\Box = \\partial^2/\\partial x^2\n - \\partial^2/\\partial y^2$\n on the\ + \ holomorphic functions of~$\\tau^3$,\n not the elliptic Laplacian.\n \\item[\\textup{(ii)}]\n The\ + \ maximum principle fails\n for wave-type operators\n \\textup{(Proposition~\\ref{prop:ch52-maximum-principle-failure})}.\n\ + \ \\item[\\textup{(iii)}]\n The bipolar idempotent decomposition\n $f = e_+ f_+ + e_- f_-$\n \ + \ \\textup{(II.L07)}\n decomposes every holomorphic function\n into two independent channel components,\n\ + \ each admitting non-constant bounded solutions.\n \\item[\\textup{(iv)}]\n Therefore, the compactness\ + \ of~$\\tau^3$\n \\textup{(II.T06)}\n is compatible with a rich function algebra\n $\\mathcal{O}(\\\ + tau^3) \\cong\n A_{\\mathrm{spec}}(\\Lemniscate)$\n \\textup{(II.T40)}.\n\\end{enumerate}\n```\n\n##\ + \ Proof / Justification\n\n```latex\nThe proof assembles the ingredients\nalready established in this chapter.\n\n\\smallskip\n\ + \\noindent\\textbf{PDE type.}\nThe split-complex unit~$\\jj$ satisfies\n$\\jj^2 = +1$ (I.T10).\nThe associated Cauchy--Riemann-type\ + \ operator\n$\\bar{\\partial}_\\jj\n= \\tfrac{1}{2}(\\partial_x + \\jj\\,\\partial_y)$\nhas principal symbol $\\sigma(\\\ + xi, \\eta)\n= \\xi^2 - \\eta^2$\n(indefinite quadratic form, signature $(1,1)$).\nThis is a \\textbf{hyperbolic} PDE,\n\ + not an elliptic one.\nLiouville's theorem requires elliptic PDE type\n(hypothesis~(E) in Remark~\\ref{rem:ch52-classical-liouville});\n\ + this hypothesis is not satisfied.\n\n\\smallskip\n\\noindent\\textbf{Maximum principle failure.}\nFor hyperbolic operators,\n\ + the maximum principle does not hold\n(Proposition~\\ref{prop:ch52-maximum-principle-failure}).\nWithout the maximum principle,\n\ + Liouville's argument cannot force\nholomorphic functions to be constant.\n\n\\smallskip\n\\noindent\\textbf{Channel decomposition.}\n\ + The idempotent decomposition lemma (II.L07)\ngives $f = e_+ f_+ + e_- f_-$,\nwhere $e_\\pm = (1 \\pm \\jj)/2$.\nThe B-channel\ + \ component~$f_+$\nand the C-channel component~$f_-$\nare independent\n(Proposition~\\ref{prop:bipolar-channel-independence},\ + \ II.P07):\neach lives in its own sector\nand is unconstrained by the other.\nNon-constant bounded solutions in each channel\n\ + (the standing-wave phenomenon of\nRemark~\\ref{rem:ch52-standing-waves})\ncombine to produce a rich function algebra.\n\ + \n\\smallskip\n\\noindent\\textbf{Compatibility.}\nThe three ingredients---hyperbolic PDE type,\nmaximum principle failure,\n\ + and independent channel decomposition---jointly\nensure that the compactness of~$\\tau^3$\nposes no obstruction\nto the\ + \ existence of non-constant holomorphic functions.\nThe Central Theorem (II.T40) is consistent.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 150\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part09/ch52-liouville-categoricity.tex`\ + \ lines 293-326\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.Categoricity`\n\ + - Name: `Tau.BookII.CentralTheorem.liouville_dodge_check`\n\n## Dependencies\n\n- Canonical: I.T05, I.T10, II.T33, II.T40\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The classical Liouville theorem does not apply to tau^3: j^2 = +1 gives a wave-type PDE operator (hyperbolic), + not the elliptic Laplacian. The maximum principle fails, so non-constant bounded holomorphic functions exist.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001403.md b/_corpus_v3_items/cid001403.md new file mode 100644 index 000000000..673cb8206 --- /dev/null +++ b/_corpus_v3_items/cid001403.md @@ -0,0 +1,99 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001403/ +title: Categoricity +cid: cid001403 +primary_alias: THM0107 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Any fibered product with an ABCD chart satisfying axioms K0-K5 is canonically isomorphic to tau^3, with the isomorphism + unique and structure-preserving. +aliases: +- THM0107 +- II.T42 +- categoricity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005376 +- predicate: formalized_by + object: cid005377 +- predicate: formalized_by + object: cid005378 +- predicate: formalized_by + object: cid005379 +- predicate: formalized_by + object: cid005385 +- predicate: formalized_by + object: cid005386 +- predicate: formalized_by + object: cid005389 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 52 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Categoricity**\n\nAny fibered product with an ABCD chart satisfying axioms K0-K5 is canonically isomorphic\ + \ to tau^3, with the isomorphism unique and structure-preserving.\n\n# Categoricity\n\n## Summary\n\nAny fibered product\ + \ with an ABCD chart satisfying axioms K0-K5 is canonically isomorphic to tau^3, with the isomorphism unique and structure-preserving.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:categoricity}\nThe theory $\\tau_0$ is categorical:\nif $M$ and $N$ are any\ + \ two models of $\\tau_0$,\nthen there exists a unique isomorphism $\\varphi : M \\xrightarrow{\\sim} N$.\n```\n\n## Proof\ + \ / Justification\n\n```latex\nLet $M \\models \\tau_0$.\nWe construct the unique isomorphism\n$\\varphi : \\tau \\to\ + \ M$.\n\n\\medskip\n\\textbf{Existence.}\nDefine $\\varphi$ by:\n\\begin{align*}\n \\varphi(\\omega) &:= \\omega^M,\ + \ \\\\\n \\varphi(\\rho^n(g)) &:= (\\rho^M)^n(g^M)\n \\quad\\text{for } g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\\ + },\\; n \\geq 0.\n\\end{align*}\nThis is well-defined by the Ontic Closure Theorem\n(every object of $\\tau$ has a unique\ + \ representation).\n\n\\emph{$\\varphi$ is well-defined on $M$:}\nSince $M \\models \\tau_0$,\n$M$ satisfies $\\KAxiom{3}$\ + \ (orbit-seeded generation),\nso $(\\rho^M)^n(g^M)$ exists in $M$ for all $n \\geq 0$.\n\n\\emph{$\\varphi$ preserves\ + \ $\\rho$:}\n$\\varphi(\\rho(x)) = \\rho^M(\\varphi(x))$ by construction.\n\n\\emph{$\\varphi$ preserves order:}\nThe\ + \ order in $\\tau$ is determined by\ndepths within and across orbits (via $\\KAxiom{1}$).\nThe same determination holds\ + \ in $M$.\n\n\\emph{$\\varphi$ is injective:}\nSuppose $\\varphi(\\rho^n(g)) = \\varphi(\\rho^m(h))$,\ni.e., $(\\rho^M)^n(g^M)\ + \ = (\\rho^M)^m(h^M)$.\nSince $M \\models \\tau_0$, the orbit rays in $M$\nare pairwise disjoint\n(by the same proof as\ + \ Proposition~\\ref{prop:orbit-disjoint},\nwhich uses only $\\KAxiom{1}$--$\\KAxiom{6}$).\nTherefore $g = h$ and $n =\ + \ m$.\n\n\\emph{$\\varphi$ is surjective:}\nSince $M \\models \\KAxiom{6}$ (Object Closure),\nevery element of $M$ is\ + \ either $\\omega^M$\nor $(\\rho^M)^n(g^M)$ for some $g$ and $n$.\nBoth are in the image of $\\varphi$.\n\n\\medskip\n\ + \\textbf{Uniqueness.}\nAny isomorphism $\\psi : \\tau \\to M$ must satisfy:\n$\\psi(g) = g^M$ (preservation of constants)\n\ + and $\\psi(\\rho^n(g)) = (\\rho^M)^n(\\psi(g)) = (\\rho^M)^n(g^M)$.\nTherefore $\\psi = \\varphi$.\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 151\n- Manuscript source: `2nd-edition/book-i-categorical-foundations/02_mainmatter/part02/ch09-rigidity-categoricity.tex`\ + \ lines 179-185\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.CentralTheorem.Categoricity`\n\ + - Name: `Tau.BookII.CentralTheorem.categoricity_check`\n\n## Dependencies\n\n- Canonical: I.T05, II.T40, II.D60, II.T33\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Any fibered product with an ABCD chart satisfying axioms K0-K5 is canonically isomorphic to tau^3, with the isomorphism + unique and structure-preserving. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001404.md b/_corpus_v3_items/cid001404.md new file mode 100644 index 000000000..f95c19bd9 --- /dev/null +++ b/_corpus_v3_items/cid001404.md @@ -0,0 +1,103 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001404/ +title: Structural Incompatibility of Unique Omega and Archimedean Density +cid: cid001404 +primary_alias: THM0108 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Under K5, properties (U) unique global omega and (A) Archimedean local density cannot both hold. (A) implies not-(U) + via the cardinality hierarchy; (U) + K5 implies not-(A) via the Inapplicability Theorem (I.T35) and Unique Infinity (I.T36). +aliases: +- THM0108 +- II.T43 +- structural-incompatibility-of-unique-omega-and-archimedean-density +- thm:structural-incompatibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005617 +- predicate: formalized_by + object: cid005618 +- predicate: formalized_by + object: cid005619 +- predicate: formalized_by + object: cid005620 +- predicate: formalized_by + object: cid005621 +- predicate: formalized_by + object: cid005622 +- predicate: formalized_by + object: cid005623 +- predicate: formalized_by + object: cid005624 +- predicate: formalized_by + object: cid005628 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Structural Incompatibility of Unique Omega and Archimedean Density**\n\nUnder K5, properties (U) unique global\ + \ omega and (A) Archimedean local density cannot both hold. (A) implies not-(U) via the cardinality hierarchy; (U) + K5\ + \ implies not-(A) via the Inapplicability Theorem (I.T35) and Unique Infinity (I.T36).\n\n# Structural Incompatibility\ + \ of Unique Omega and Archimedean Density\n\n## Summary\n\nUnder K5, properties (U) unique global omega and (A) Archimedean\ + \ local density cannot both hold. (A) implies not-(U) via the cardinality hierarchy; (U) + K5 implies not-(A) via the\ + \ Inapplicability Theorem (I.T35) and Unique Infinity (I.T36).\n\n## Statement\n\n```latex\n%\n\\label{thm:structural-incompatibility}\n\ + % I.D76, II.D69, II.D68\nProperties (U) and (A) of the Infinity Trade-Off\n(Definition~\\ref{def:infinity-trade-off})\n\ + are structurally incompatible.\nNo mathematical framework can simultaneously satisfy both.\n```\n\n## Proof / Justification\n\ + \n```latex\n[Proof sketch]\nAssume both (U) and (A) hold simultaneously.\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n\ + \ From~(U): there exists a unique $\\omega$ with\n $\\rho(\\omega) = \\omega$ and the diagonal discipline\ + \ K5\n blocks unrestricted cardinality arguments\n (I.T35, I.T36).\n \\item[\\textup{(ii)}]\n \ + \ From~(A): Archimedean density implies\n that between $0$ and $\\omega$\n lie uncountably many\ + \ objects.\n Cantor's diagonal argument constructs\n uncountable subsets.\n \\item[\\textup{(iii)}]\n\ + \ But K5 blocks the diagonal argument\n within $\\tau$ (I.T35):\n the self-application required\n\ + \ for diagonalization exceeds\n the allowed iteration depth.\n \\item[\\textup{(iv)}]\n \ + \ Contradiction: (A) requires the diagonal argument\n to construct uncountable sets;\n (U) requires\ + \ K5 to block it.\n\\end{enumerate}\nThe incompatibility is not a limitation\nof current proof techniques.\nIt is a structural\ + \ impossibility:\nthe axioms required for (U)\ncontradict the axioms required for (A).\n```\n\n## Source Context\n\n-\ + \ Registry source: `book-02.jsonl` line 173\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex`\ + \ lines 136-144\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.SignClassification`\n\ + - Name: `structural_incompatibility`\n\n## Dependencies\n\n- Canonical: I.T05, I.T10, I.T35, I.T36, I.D76, II.D69, II.D68\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Under K5, properties (U) unique global omega and (A) Archimedean local density cannot both hold. (A) implies + not-(U) via the cardinality hierarchy; (U) + K5 implies not-(A) via the Inapplicability Theorem (I.T35) and Unique Infinity + (I.T36) +noindex: false +--- + diff --git a/_corpus_v3_items/cid001405.md b/_corpus_v3_items/cid001405.md new file mode 100644 index 000000000..719038863 --- /dev/null +++ b/_corpus_v3_items/cid001405.md @@ -0,0 +1,108 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001405/ +title: Asymmetric Determination +cid: cid001405 +primary_alias: THM0109 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'In tau-holomorphy, boundary data on L determines interior values on tau^3 asymmetrically: directional propagation + along two characteristic families, no reverse constraint, Global Hartogs as wave front coverage, and uniqueness from sector + independence.' +aliases: +- THM0109 +- II.T44 +- asymmetric-determination +- thm:asymmetric-determination +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005629 +- predicate: formalized_by + object: cid005630 +- predicate: formalized_by + object: cid005631 +- predicate: formalized_by + object: cid005632 +- predicate: formalized_by + object: cid005633 +- predicate: formalized_by + object: cid005634 +- predicate: formalized_by + object: cid005642 +- predicate: formalized_by + object: cid005643 +- predicate: formalized_by + object: cid005644 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 61 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Asymmetric Determination**\n\nIn tau-holomorphy, boundary data on L determines interior values on tau^3 asymmetrically:\ + \ directional propagation along two characteristic families, no reverse constraint, Global Hartogs as wave front coverage,\ + \ and uniqueness from sector independence.\n\n# Asymmetric Determination\n\n## Summary\n\nIn tau-holomorphy, boundary\ + \ data on L determines interior values on tau^3 asymmetrically: directional propagation along two characteristic families,\ + \ no reverse constraint, Global Hartogs as wave front coverage, and uniqueness from sector independence.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:asymmetric-determination}\nIn $\\tau$-holomorphy ($\\jj^2 = +1$),\nboundary data on the lemniscate\ + \ $\\Lemniscate$\ndetermines interior values on $\\tau^3$\nwith the following properties:\n\\begin{enumerate}\n \\\ + item[\\textup{(i)}]\n \\textbf{Directional propagation.}\n Boundary data propagates into the interior\n\ + \ along the two characteristic families\n of the wave equation,\n not isotropically.\n \\\ + item[\\textup{(ii)}]\n \\textbf{No reverse constraint.}\n Interior values do not constrain\n \ + \ boundary values:\n the dependency is one-directional\n (boundary $\\to$ interior).\n \\item[\\\ + textup{(iii)}]\n \\textbf{Global Hartogs as wave front.}\n The Hartogs extension theorem (I.T31)\n \ + \ is the statement that wave-front data\n on the boundary determines\n the full interior holomorphic\ + \ function.\n \\item[\\textup{(iv)}]\n \\textbf{Uniqueness from independence.}\n The two characteristic\ + \ families\n carry independent data;\n their consistency forces uniqueness\n of the interior\ + \ extension.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nProperties~(i)--(ii) follow\ + \ from the hyperbolic\ncharacter of $\\Box f = 0$:\ndata propagates along characteristics,\nand hyperbolic equations have\n\ + domain-of-dependence properties\nthat prevent reverse information flow.\nProperty~(iii) identifies Hartogs extension\n\ + as the constructive analog of the elliptic\nmaximum principle:\nwhere the Laplacian ``averages'' boundary data,\nthe wave\ + \ operator ``propagates'' it.\nProperty~(iv) follows from the idempotent\ndecomposition $f = e_+ g + e_- h$:\nthe two\ + \ components are independent,\nand the boundary data for each sector\ndetermines its interior value uniquely\n(II.T27,\ + \ mutual determination).\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 176\n- Manuscript source:\ + \ `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch61-master-switch.tex` lines 434-467\n\n## Lean /\ + \ Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.WaveHolomorphy`\n- Name: `asymmetric_determination`\n\ + \n## Dependencies\n\n- Canonical: I.T05, I.T10, I.T31, II.D21, II.T26, II.T37\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'In tau-holomorphy, boundary data on L determines interior values on tau^3 asymmetrically: directional propagation + along two characteristic families, no reverse constraint, Global Hartogs as wave front coverage, and uniqueness from sector + in' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001406.md b/_corpus_v3_items/cid001406.md new file mode 100644 index 000000000..06d7c976f --- /dev/null +++ b/_corpus_v3_items/cid001406.md @@ -0,0 +1,98 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001406/ +title: Parallel Preservation +cid: cid001406 +primary_alias: THM0110 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The parallel postulate (II.T18) survives the hyperbolic algebraic substrate j^2 = +1: stage-k geometry is Euclidean, + Tarski axioms are universal sentences preserved under inverse limits, and light cones require tangent spaces that tau^3 + does not possess.' +aliases: +- THM0110 +- II.T45 +- parallel-preservation +- thm:parallel-preservation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005635 +- predicate: formalized_by + object: cid005636 +- predicate: formalized_by + object: cid005637 +- predicate: formalized_by + object: cid005638 +- predicate: formalized_by + object: cid005639 +- predicate: formalized_by + object: cid005645 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 64 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Parallel Preservation**\n\nThe parallel postulate (II.T18) survives the hyperbolic algebraic substrate j^2\ + \ = +1: stage-k geometry is Euclidean, Tarski axioms are universal sentences preserved under inverse limits, and light\ + \ cones require tangent spaces that tau^3 does not possess.\n\n# Parallel Preservation\n\n## Summary\n\nThe parallel postulate\ + \ (II.T18) survives the hyperbolic algebraic substrate j^2 = +1: stage-k geometry is Euclidean, Tarski axioms are universal\ + \ sentences preserved under inverse limits, and light cones require tangent spaces that tau^3 does not possess.\n\n##\ + \ Statement\n\n```latex\n%\n\\label{thm:parallel-preservation}\n% II.D71, II.T15--II.T18\nIn Stage-Finite Euclidean\ + \ Geometry (II.D71):\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Stage-$k$ preservation.}\n \ + \ The Parallel Postulate holds on each $\\tau^3_k$.\n \\item[\\textup{(ii)}]\n \\textbf{Limit preservation.}\n\ + \ The profinite limit preserves the Parallel Postulate:\n it holds on $\\tau^3$.\n \\item[\\textup{(iii)}]\n\ + \ \\textbf{No light-cone obstruction.}\n The wave equation's characteristics\n operate between\ + \ stages (inter-stage propagation)\n and do not generate intra-stage light cones.\n\\end{enumerate}\n```\n\n\ + ## Proof / Justification\n\n```latex\n[Proof sketch]\n(i) On each finite $\\tau^3_k$,\nbetweenness is defined by\n$B(x,y,z)\ + \ \\iff d(x,z) = \\max(d(x,y),d(y,z))$\n(ultrametric betweenness, II.D19).\nGiven a line $\\ell$ and a point $P \\notin\ + \ \\ell$\nin $\\tau^3_k$,\nthe ultrametric structure guarantees\nexactly one line through $P$\nthat does not intersect\ + \ $\\ell$\nwithin $\\tau^3_k$\n(because the clopen balls partition the space\ninto non-overlapping regions).\n(ii) The\ + \ Parallel Postulate is a universal statement:\n``for all lines $\\ell$, for all points $P$, \\ldots''\nSuch statements\ + \ are preserved under profinite limits\n(inverse limits of surjections).\n(iii) Light cones require continuous characteristics\n\ + in a manifold.\n$\\tau^3_k$ is discrete; there are no continuous curves.\n```\n\n## Source Context\n\n- Registry source:\ + \ `book-02.jsonl` line 179\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch64-gains.tex`\ + \ lines 256-275\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.WaveHolomorphy`\n\ + - Name: `stage_euclidean`\n\n## Dependencies\n\n- Canonical: I.T10, II.D14, II.D21, II.D71, II.T15, II.T16, II.T17, II.T18\n\ + \n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The parallel postulate (II.T18) survives the hyperbolic algebraic substrate j^2 = +1: stage-k geometry is Euclidean, + Tarski axioms are universal sentences preserved under inverse limits, and light cones require tangent spaces that tau^3 + doe' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001407.md b/_corpus_v3_items/cid001407.md new file mode 100644 index 000000000..5c395f473 --- /dev/null +++ b/_corpus_v3_items/cid001407.md @@ -0,0 +1,105 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001407/ +title: Fourth Quadrant Resolution +cid: cid001407 +primary_alias: THM0111 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'In the (Hyperbolic, non-Archimedean) quadrant of Category tau: wave characteristics exist (R1), no Laplacian infinities + (R2), stage-finite Euclidean geometry (R3), discrete exterior derivative d_tau (R4), bipolar quantum phases from e_+/e_- + (R5), and the Unification Obstruction does not apply (R6).' +aliases: +- THM0111 +- II.T46 +- fourth-quadrant-resolution +- thm:fourth-quadrant-resolution +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005602 +- predicate: formalized_by + object: cid005603 +- predicate: formalized_by + object: cid005604 +- predicate: formalized_by + object: cid005605 +- predicate: formalized_by + object: cid005606 +- predicate: formalized_by + object: cid005607 +- predicate: formalized_by + object: cid005608 +- predicate: formalized_by + object: cid005609 +- predicate: formalized_by + object: cid005612 +- predicate: formalized_by + object: cid005613 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fourth Quadrant Resolution**\n\nIn the (Hyperbolic, non-Archimedean) quadrant of Category tau: wave characteristics\ + \ exist (R1), no Laplacian infinities (R2), stage-finite Euclidean geometry (R3), discrete exterior derivative d_tau (R4),\ + \ bipolar quantum phases from e_+/e_- (R5), and the Unification Obstruction does not apply (R6).\n\n# Fourth Quadrant\ + \ Resolution\n\n## Summary\n\nIn the (Hyperbolic, non-Archimedean) quadrant of Category tau: wave characteristics exist\ + \ (R1), no Laplacian infinities (R2), stage-finite Euclidean geometry (R3), discrete exterior derivative d_tau (R4), bipolar\ + \ quantum phases from e_+/e_- (R5), and the Unification Obstruction does not apply (R6).\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:fourth-quadrant-resolution}\n% I.T41, II.T03, II.T19, II.T39, II.D21, II.D22,\n% II.D35, II.D65, II.D73,\ + \ II.D74\nIn the (Hyperbolic, Non-Archimedean) quadrant,\nthe Unification Obstruction dissolves:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(R1)}]\n Wave characteristics exist\n (hyperbolic propagation for QFT).\n \\\ + item[\\textup{(R2)}]\n No local Laplacian infinities\n (the Archimedean prison is absent).\n \\item[\\\ + textup{(R3)}]\n Stage-finite Euclidean geometry\n (flat geometry without curvature singularities).\n\ + \ \\item[\\textup{(R4)}]\n Discrete exterior derivative\n (differential forms on finite stages).\n\ + \ \\item[\\textup{(R5)}]\n Quantum phases from bipolar decomposition\n ($e_+$, $e_-$ sectors carry\ + \ independent data).\n \\item[\\textup{(R6)}]\n No Unification Obstruction:\n (U1) and (U2) are no\ + \ longer incompatible\n because the Archimedean requirement is absent.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-02.jsonl`\ + \ line 186\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex`\ + \ lines 458-486\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.PhysicsQuadrant`\n\ + - Name: `tau_escapes_obstruction`\n\n## Dependencies\n\n- Canonical: I.T05, I.T10, I.D21, I.T31, I.T41, II.T03, II.T19,\ + \ II.T39, II.D21, II.D22, II.D35, II.D65, II.D73, II.D74\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'In the (Hyperbolic, non-Archimedean) quadrant of Category tau: wave characteristics exist (R1), no Laplacian + infinities (R2), stage-finite Euclidean geometry (R3), discrete exterior derivative d_tau (R4), bipolar quantum phases from + e_+/e_-' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001408.md b/_corpus_v3_items/cid001408.md new file mode 100644 index 000000000..7d9b28142 --- /dev/null +++ b/_corpus_v3_items/cid001408.md @@ -0,0 +1,99 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001408/ +title: Simultaneous Rung Theorem +cid: cid001408 +primary_alias: THM0112 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'tau^3 simultaneously exhibits features from at least three classical SCV dimension rungs: full Hartogs extension + (C3 feature via I.T31), distinguished boundary (C2 feature via torus degeneration II.T13), and complete boundary determination + (C1 feature via Mutual Determination II.T27), while lacking Riemann mapping, monodromy, isolated singularities, and the + Levi problem.' +aliases: +- THM0112 +- II.T47 +- simultaneous-rung-theorem +- thm:simultaneous-rung +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005580 +- predicate: formalized_by + object: cid005581 +- predicate: formalized_by + object: cid005592 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 63 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Simultaneous Rung Theorem**\n\ntau^3 simultaneously exhibits features from at least three classical SCV dimension\ + \ rungs: full Hartogs extension (C3 feature via I.T31), distinguished boundary (C2 feature via torus degeneration II.T13),\ + \ and complete boundary determination (C1 feature via Mutual Determination II.T27), while lacking Riemann mapping, monodromy,\ + \ isolated singularities, and the Levi problem.\n\n# Simultaneous Rung Theorem\n\n## Summary\n\ntau^3 simultaneously exhibits\ + \ features from at least three classical SCV dimension rungs: full Hartogs extension (C3 feature via I.T31), distinguished\ + \ boundary (C2 feature via torus degeneration II.T13), and complete boundary determination (C1 feature via Mutual Determination\ + \ II.T27), while lacking Riemann mapping, monodromy, isolated singularities, and the Levi problem.\n\n## Statement\n\n\ + ```latex\n%\n\\label{thm:simultaneous-rung}\n% II.T06, II.T07, II.T40, II.T41, II.D13, II.D14,\n% II.D68, II.D70,\ + \ II.D75, II.D76\n$\\tau^3$ simultaneously exhibits:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n Complete\ + \ boundary determination\n ($\\mathbb{C}^1$ feature):\n boundary data on $\\Lemniscate$\n determines\ + \ the full interior,\n via BndLift and the Central Theorem (II.T40).\n \\item[\\textup{(ii)}]\n A\ + \ distinguished boundary\n ($\\mathbb{C}^2$ feature):\n the lemniscate $\\Lemniscate = S^1 \\vee S^1$\n\ + \ is a proper subset of the topological boundary\n but determines all holomorphic functions.\n \\\ + item[\\textup{(iii)}]\n Full Hartogs extension\n ($\\mathbb{C}^3$ feature):\n functions holomorphic\ + \ on the boundary\n extend uniquely to the interior (I.T31).\n\\end{enumerate}\nAll three features coexist without\ + \ the pathologies\nassociated with their individual rungs\nin the orthodox ladder\n(no removable singularities,\nno Cousin\ + \ problems,\nno Levi problem).\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nEach feature traces to a\ + \ different aspect\nof the $\\tau^3$ fibration\n$\\tau^3 = \\tau^1 \\times_f T^2$:\n\\begin{itemize}\n \\item (i) follows\ + \ from the Central Theorem (II.T40):\n $\\mathcal{O}(\\tau^3) \\cong\n A_{\\mathrm{spec}}(\\Lemniscate)$.\n\ + \ \\item (ii) follows from the fibered product structure:\n the lemniscate is the base-level boundary\n \ + \ that controls the fiber.\n \\item (iii) follows from Global Hartogs (I.T31):\n the CRT-based extension\ + \ is constructive\n and works in all dimensions simultaneously.\n\\end{itemize}\nThe pathologies are absent because\n\ + the Archimedean-Elliptic Engine is absent:\nwithout Archimedean density,\nthere are no removable singularities\n(nothing\ + \ ``between'' the discrete stages),\nno Cousin problems\n(cohomology is stage-finite),\nand no Levi problem\n(domains\ + \ of holomorphy\nare determined by the primorial tower,\nnot by smooth boundary geometry).\n```\n\n## Source Context\n\ + \n- Registry source: `book-02.jsonl` line 191\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch63-refuses.tex`\ + \ lines 291-322\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.DimensionalLadder`\n\ + - Name: `simultaneous_rung`\n\n## Dependencies\n\n- Canonical: I.T31, II.T13, II.T27, II.T42, II.D75\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'tau^3 simultaneously exhibits features from at least three classical SCV dimension rungs: full Hartogs extension + (C3 feature via I.T31), distinguished boundary (C2 feature via torus degeneration II.T13), and complete boundary determination ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001409.md b/_corpus_v3_items/cid001409.md new file mode 100644 index 000000000..068e8883f --- /dev/null +++ b/_corpus_v3_items/cid001409.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001409/ +title: Fourth Quadrant Ladder Collapse +cid: cid001409 +primary_alias: THM0113 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'In the (Hyperbolic, Non-Archimedean) quadrant the Archimedean-elliptic engine is absent: no continuous metric dimension + (LC1), no progressive elliptic overdeterminacy (LC2), therefore no dimensional ladder (LC3), and features from multiple + classical rungs coexist (LC4).' +aliases: +- THM0113 +- II.T48 +- fourth-quadrant-ladder-collapse +- thm:ladder-collapse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005582 +- predicate: formalized_by + object: cid005583 +- predicate: formalized_by + object: cid005593 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 63 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Fourth Quadrant Ladder Collapse**\n\nIn the (Hyperbolic, Non-Archimedean) quadrant the Archimedean-elliptic\ + \ engine is absent: no continuous metric dimension (LC1), no progressive elliptic overdeterminacy (LC2), therefore no\ + \ dimensional ladder (LC3), and features from multiple classical rungs coexist (LC4).\n\n# Fourth Quadrant Ladder Collapse\n\ + \n## Summary\n\nIn the (Hyperbolic, Non-Archimedean) quadrant the Archimedean-elliptic engine is absent: no continuous\ + \ metric dimension (LC1), no progressive elliptic overdeterminacy (LC2), therefore no dimensional ladder (LC3), and features\ + \ from multiple classical rungs coexist (LC4).\n\n## Statement\n\n```latex\n%\n\\label{thm:ladder-collapse}\nThe Archimedean-Elliptic\ + \ Engine (II.D75)\nis the necessary driver of the SCV dimensional ladder.\nIn its absence, the ladder collapses:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n Without Archimedean density,\n there are no ``scales'' between stages\n\ + \ at which new phenomena could emerge.\n \\item[\\textup{(ii)}]\n Without elliptic regularity,\n\ + \ there are no convergent power series\n whose radius of convergence could\n distinguish dimensions.\n\ + \ \\item[\\textup{(iii)}]\n The wave equation ($\\Box f = 0$)\n propagates along characteristics\n\ + \ in all coordinate dimensions simultaneously:\n adding dimensions does not create\n new characteristic\ + \ families.\n\\end{enumerate}\nThe dimensional ladder is a feature\nof the orthodox ($i^2 = -1$, Archimedean) quadrant,\n\ + not a universal property of holomorphic function theory.\n```\n\n## Proof / Justification\n\n_No immediate manuscript\ + \ proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 192\n\ + - Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch63-refuses.tex` lines 352-378\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Mirror.DimensionalLadder`\n\ + - Name: `ladder_collapse`\n\n## Dependencies\n\n- Canonical: II.D73, II.D75, II.D76, II.T47\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'In the (Hyperbolic, Non-Archimedean) quadrant the Archimedean-elliptic engine is absent: no continuous metric + dimension (LC1), no progressive elliptic overdeterminacy (LC2), therefore no dimensional ladder (LC3), and features from + multiple ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001410.md b/_corpus_v3_items/cid001410.md new file mode 100644 index 000000000..8c8f5827e --- /dev/null +++ b/_corpus_v3_items/cid001410.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001410/ +title: Geometric Bi-Square Theorem +cid: cid001410 +primary_alias: THM0114 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Five properties of the geometric bi-square: (1) every algebraic component has geometric content, (2) all vertical + arrows are continuous, (3) both squares commute with continuity, (4) the limit row is precisely the Central Theorem O(tau^3) + = A_spec(L), (5) forgetting geometry recovers the algebraic bi-square I.T41.' +aliases: +- THM0114 +- II.T49 +- geometric-bi-square-theorem +- thm:geometric-bisquare +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005474 +- predicate: formalized_by + object: cid005475 +- predicate: formalized_by + object: cid005479 +- predicate: formalized_by + object: cid005480 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 59 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Geometric Bi-Square Theorem**\n\nFive properties of the geometric bi-square: (1) every algebraic component\ + \ has geometric content, (2) all vertical arrows are continuous, (3) both squares commute with continuity, (4) the limit\ + \ row is precisely the Central Theorem O(tau^3) = A_spec(L), (5) forgetting geometry recovers the algebraic bi-square\ + \ I.T41.\n\n# Geometric Bi-Square Theorem\n\n## Summary\n\nFive properties of the geometric bi-square: (1) every algebraic\ + \ component has geometric content, (2) all vertical arrows are continuous, (3) both squares commute with continuity, (4)\ + \ the limit row is precisely the Central Theorem O(tau^3) = A_spec(L), (5) forgetting geometry recovers the algebraic\ + \ bi-square I.T41.\n\n## Statement\n\n```latex\n%\n\\label{thm:geometric-bisquare}\n% II.D35, II.D60, II.T27, II.T32,\ + \ II.T37, II.T40\nThe geometric bi-square\n\\textup{(}Definition~\\textup{\\ref{def:geometric-bisquare},}\nII.D77\\textup{)}\n\ + satisfies the following five properties:\n\\begin{enumerate}\n \\item[\\textup{(GB1)}] \\textbf{Geometric filling.}\n\ + \ Every algebraic component of the bi-square \\textup{(}I.T41\\textup{)}\n receives geometric content\ + \ earned in Book~II:\n \\begin{itemize}\n \\item the domain $\\Lemniscate$\n \ + \ receives the geometric body $S^1 \\vee S^1$\n \\textup{(}II.T13\\textup{)},\n \\item\ + \ the interior $\\tau^3$\n receives the Stone topology\n \\textup{(}II.T07\\textup{)},\n\ + \ \\item the codomain $H_\\tau$\n receives calibration\n \\textup{(}II.D35\\\ + textup{)},\n \\item the spectral side\n receives $A_{\\mathrm{spec}}(\\Lemniscate)$\n\ + \ \\textup{(}II.D60\\textup{)}.\n \\end{itemize}\n\n \\item[\\textup{(GB2)}] \\textbf{Continuity.}\n\ + \ All vertical arrows\n \\textup{(}projections $\\mathrm{proj}_k$\\textup{)}\n are continuous\ + \ maps\n in the inverse-limit topology\n \\textup{(}by II.T06,\n Chapter~\\textup{\\ref{ch:hol-implies-cont})}.\n\ + \ All horizontal arrows\n are $H_\\tau^{\\mathrm{cal}}$-valued ring homomorphisms.\n\n \\item[\\\ + textup{(GB3)}] \\textbf{Commutativity.}\n Both faces of the pasted diagram commute:\n \\begin{itemize}\n\ + \ \\item Left face:\n $\\mathrm{proj}_k \\circ f_\\ell\n = f_k \\circ\ + \ \\mathrm{proj}_k$\n for all $k \\leq \\ell$.\n \\item Right face:\n \ + \ $(\\mathrm{proj}_k, \\mathrm{proj}_k) \\circ (\\chi_+, \\chi_-)\n = (\\chi_+, \\chi_-) \\circ\ + \ \\mathrm{proj}_k$\n for all $k \\leq \\ell$.\n \\end{itemize}\n These are the sheaf\ + \ condition\n \\textup{(}II.T32\\textup{)}\n and the spectral naturality\n \\textup{(}inherited\ + \ from the character decomposition,\n II.D59\\textup{)},\n respectively.\n\n \\item[\\textup{(GB4)}]\ + \ \\textbf{Limit row = Central Theorem.}\n The limit row of the geometric bi-square\n \\textup{(}\\\ + eqref{eq:ch59-limit-diagram}\\textup{)}\n is the Central Theorem \\textup{(}II.T40\\textup{)}:\n the\ + \ composition $\\Psi \\circ \\Phi = \\mathrm{id}$\n and $\\Phi \\circ \\Psi = \\mathrm{id}$\n establishes\n\ + \ $\\mathcal{O}(\\tau^3) \\cong A_{\\mathrm{spec}}(\\Lemniscate)$.\n\n \\item[\\textup{(GB5)}] \\textbf{Compatibility\ + \ with I.T41.}\n The geometric bi-square restricts to the algebraic bi-square\n at every finite stage:\n\ + \ forgetting the topology, geometry, and calibration\n recovers the diagram of I.T41 exactly.\n \ + \ The algebraic bi-square is the skeleton;\n the geometric bi-square is the same skeleton\n clothed\ + \ in the geometric content of Book~II.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\nThe proof is an\ + \ assembly of earned results.\nWe verify each property.\n\n\\medskip\n\\noindent\n\\textup{(GB1)}:\nThe domain $\\Lemniscate$\ + \ receives the body $S^1 \\vee S^1$\nby the Torus Degeneration Theorem\n(II.T13, Chapter~\\ref{ch:torus-degeneration}):\n\ + the fiber $T^2$ degenerates to $S^1 \\vee S^1$\nvia the pinch map (II.D18).\nThe interior $\\tau^3$ receives the Stone\ + \ topology\nby Theorems~II.T07--II.T09\n(Chapter~\\ref{ch:stone-space}):\ncompact, Hausdorff, totally disconnected.\n\ + The codomain receives calibration\nby Definition~II.D35\n(Chapter~\\ref{ch:split-complex-calibrated}):\nthe four earned\ + \ transcendentals\n$\\pi$, $e$, $\\jj$, $\\iota_\\tau$\nequip $H_\\tau^{\\mathrm{cal}}$\nwith numerical content.\nThe\ + \ spectral algebra $A_{\\mathrm{spec}}(\\Lemniscate)$\nis constructed in Definition~II.D60\n(Chapter~\\ref{ch:central-theorem}).\n\ + \n\\medskip\n\\noindent\n\\textup{(GB2)}:\nThe projections $\\mathrm{proj}_k$\nare continuous by II.T06\n(Chapter~\\ref{ch:hol-implies-cont}):\n\ + holomorphic implies continuous\nin the inverse-limit topology on~$\\tau^3$.\nThe horizontal arrows are ring homomorphisms\n\ + by construction:\n$f_k$ is a stage function valued in $H_\\tau^{\\mathrm{cal}}$,\nand $(\\chi_+, \\chi_-)$ is the idempotent\ + \ decomposition\n(II.D59, Chapter~\\ref{ch:boundary-characters-idempotent}),\nwhich is a ring homomorphism\nbecause $e_\\\ + pm$ are idempotents.\n\n\\medskip\n\\noindent\n\\textup{(GB3)}:\nThe left face commutes\nbecause holomorphic functions\ + \ on~$\\tau^3$\nform a sheaf\n(II.T32, Chapter~\\ref{ch:sheaf-coherence}):\nthe sheaf axiom on the clopen cylinder basis\n\ + is precisely the statement\nthat restriction to a smaller stage\ncommutes with function evaluation.\nThe right face commutes\n\ + because the bipolar decomposition\n$\\chi = e_+ \\cdot \\chi_+ + e_- \\cdot \\chi_-$\nis natural with respect to ring\ + \ homomorphisms:\nreduction maps $\\mathrm{proj}_k$ are ring homomorphisms,\nand idempotents are preserved by ring homomorphisms.\n\ + \n\\medskip\n\\noindent\n\\textup{(GB4)}:\nThe limit row~\\eqref{eq:ch59-limit-diagram}\nis the Central Theorem (II.T40,\n\ + Chapter~\\ref{ch:central-theorem}).\nThe map $\\Phi$ is the boundary-to-interior Hartogs extension\n(constructed in the\ + \ proof of II.T40\nusing II.T37 and II.T39);\nthe map $\\Psi$ is the interior-to-boundary restriction\n(constructed using\ + \ II.T27).\nThe mutual inverse property\n$\\Psi \\circ \\Phi = \\mathrm{id}$\nand $\\Phi \\circ \\Psi = \\mathrm{id}$\n\ + was proved in Chapter~\\ref{ch:central-theorem}.\n\n\\medskip\n\\noindent\n\\textup{(GB5)}:\nAt each finite stage~$k$,\n\ + the geometric bi-square restricts to the diagram\n\\[\n \\mathbb{Z}/M_k\\mathbb{Z}\n \\;\\xrightarrow{\\;\\; f_k\ + \ \\;\\;}\\;\n H_\\tau\n \\;\\xrightarrow{\\;\\; (\\chi_+,\\, \\chi_-) \\;\\;}\\;\n \\widehat{\\mathbb{Z}}_\\\ + tau \\times \\widehat{\\mathbb{Z}}_\\tau.\n\\]\nThe topology is discrete at finite stages\n(every function on a finite\ + \ set is continuous),\nthe calibration constants are still formal\nat any single stage,\nand the spectral decomposition\n\ + restricts to the finite spectral decomposition\nof I.T41.\nHence the geometric bi-square\nrestricts to the algebraic bi-square.\n\ + ```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 167\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part10/ch59-geometric-bisquare.tex`\ + \ lines 518-592\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookII.Closure.GeometricBiSquare`\n\ + - Name: `geometric_bisquare_check`\n\n## Dependencies\n\n- Canonical: II.D77, I.T41, II.T06, II.T13, II.T27, II.T32, II.T37,\ + \ II.T40, II.D35, II.D60\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Five properties of the geometric bi-square: (1) every algebraic component has geometric content, (2) all vertical + arrows are continuous, (3) both squares commute with continuity, (4) the limit row is precisely the Central Theorem O(tau^3) + =' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001411.md b/_corpus_v3_items/cid001411.md new file mode 100644 index 000000000..0bac64e84 --- /dev/null +++ b/_corpus_v3_items/cid001411.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001411/ +title: Flat Connection Existence +cid: cid001411 +primary_alias: THM0115 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The additive connection Γ_k(x,v) = (x+v) mod M_k is flat: parallel transport around any closed loop returns to the + starting point. Verified computationally at stages 1-2.' +aliases: +- THM0115 +- II.T50 +- flat-connection-existence +- thm:flat-connection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005449 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Flat Connection Existence** + + + The additive connection Γ_k(x,v) = (x+v) mod M_k is flat: parallel transport around any closed loop returns to the starting + point. Verified computationally at stages 1-2. + + + # Flat Connection Existence + + + ## Summary + + + The additive connection Γ_k(x,v) = (x+v) mod M_k is flat: parallel transport around any closed loop returns to the starting + point. Verified computationally at stages 1-2. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 197 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Connection` + + - Name: `flat_connection_flat_2` + + + ## Dependencies + + + - Canonical: II.D78, II.D62 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The additive connection Γ_k(x,v) = (x+v) mod M_k is flat: parallel transport around any closed loop returns + to the starting point. Verified computationally at stages 1-2.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001412.md b/_corpus_v3_items/cid001412.md new file mode 100644 index 000000000..f76337d9c --- /dev/null +++ b/_corpus_v3_items/cid001412.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001412/ +title: Flat Curvature Vanishing +cid: cid001412 +primary_alias: THM0116 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The canonical flat connection has zero curvature at every finite stage. R = 0 because addition in Z/M_k Z is commutative. + Verified at stages 1-2. +aliases: +- THM0116 +- II.T51 +- flat-curvature-vanishing +- thm:flat-curvature +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005452 +- predicate: formalized_by + object: cid005453 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Flat Curvature Vanishing** + + + The canonical flat connection has zero curvature at every finite stage. R = 0 because addition in Z/M_k Z is commutative. + Verified at stages 1-2. + + + # Flat Curvature Vanishing + + + ## Summary + + + The canonical flat connection has zero curvature at every finite stage. R = 0 because addition in Z/M_k Z is commutative. + Verified at stages 1-2. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 201 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Curvature` + + - Name: `flat_curvature_vanishes_2` + + + ## Dependencies + + + - Canonical: II.D80, II.T50 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The canonical flat connection has zero curvature at every finite stage. R = 0 because addition in Z/M_k Z is + commutative. Verified at stages 1-2. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001413.md b/_corpus_v3_items/cid001413.md new file mode 100644 index 000000000..c3558c701 --- /dev/null +++ b/_corpus_v3_items/cid001413.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001413/ +title: Lemniscate Holonomy +cid: cid001413 +primary_alias: THM0117 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The profinite limit acquires nontrivial holonomy from π₁(L) ≅ ℤ. The generator maps to the unit shift x ↦ x+1 mod + M_k, which has order M_k at stage k. +aliases: +- THM0117 +- II.T52 +- lemniscate-holonomy +- thm:lemniscate-holonomy +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005458 +- predicate: formalized_by + object: cid005459 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 10, Chapter 55 (Wave M4) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Lemniscate Holonomy** + + + The profinite limit acquires nontrivial holonomy from π₁(L) ≅ ℤ. The generator maps to the unit shift x ↦ x+1 mod M_k, + which has order M_k at stage k. + + + # Lemniscate Holonomy + + + ## Summary + + + The profinite limit acquires nontrivial holonomy from π₁(L) ≅ ℤ. The generator maps to the unit shift x ↦ x+1 mod M_k, + which has order M_k at stage k. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 203 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Closure.Curvature` + + - Name: `lemniscate_holonomy_2` + + + ## Dependencies + + + - Canonical: II.P16, II.T50 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The profinite limit acquires nontrivial holonomy from π₁(L) ≅ ℤ. The generator maps to the unit shift x ↦ x+1 + mod M_k, which has order M_k at stage k. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001414.md b/_corpus_v3_items/cid001414.md new file mode 100644 index 000000000..f4d382953 --- /dev/null +++ b/_corpus_v3_items/cid001414.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001414/ +title: Cauchy-Schwarz Inequality +cid: cid001414 +primary_alias: THM0118 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Cauchy-Schwarz inequality |⟨f,g⟩|² ≤ ‖f‖²·‖g‖². Verified exhaustively for indicator basis functions at stages 1-2. +aliases: +- THM0118 +- II.T53 +- cauchy-schwarz-inequality +- thm:cauchy-schwarz +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005561 +- predicate: formalized_by + object: cid005562 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 6, Chapter 36 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Cauchy-Schwarz Inequality** + + + Cauchy-Schwarz inequality |⟨f,g⟩|² ≤ ‖f‖²·‖g‖². Verified exhaustively for indicator basis functions at stages 1-2. + + + # Cauchy-Schwarz Inequality + + + ## Summary + + + Cauchy-Schwarz inequality |⟨f,g⟩|² ≤ ‖f‖²·‖g‖². Verified exhaustively for indicator basis functions at stages 1-2. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 206 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Hartogs.L2Space` + + - Name: `cauchy_schwarz_stage2` + + + ## Dependencies + + + - Canonical: II.D82, II.D83 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Cauchy-Schwarz inequality |⟨f,g⟩|² ≤ ‖f‖²·‖g‖². Verified exhaustively for indicator basis functions at stages + 1-2. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001415.md b/_corpus_v3_items/cid001415.md new file mode 100644 index 000000000..570357532 --- /dev/null +++ b/_corpus_v3_items/cid001415.md @@ -0,0 +1,133 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001415/ +title: Tower Coherence +cid: cid001415 +primary_alias: THM0119 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Boundary maps compose coherently: d_{n-1}(d_n(x)) = d_{n-1}(x) for the primorial chain. This is tower coherence + of the inverse system.' +aliases: +- THM0119 +- II.T54 +- thm:tower-coherence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005500 +- predicate: formalized_by + object: cid005501 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 8, Chapter 46 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tower Coherence** + + + Boundary maps compose coherently: d_{n-1}(d_n(x)) = d_{n-1}(x) for the primorial chain. This is tower coherence of the + inverse system. + + + # Tower Coherence + + + ## Summary + + + Boundary maps compose coherently: d_{n-1}(d_n(x)) = d_{n-1}(x) for the primorial chain. This is tower coherence of the + inverse system. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 210 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.Enrichment.Homological` + + - Name: `boundary_coherence_3` + + + ## Dependencies + + + - Canonical: II.D84 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Boundary maps compose coherently: d_{n-1}(d_n(x)) = d_{n-1}(x) for the primorial chain. This is tower coherence + of the inverse system.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001416.md b/_corpus_v3_items/cid001416.md new file mode 100644 index 000000000..c1c54f22b --- /dev/null +++ b/_corpus_v3_items/cid001416.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001416/ +title: H⁰ = Global Sections +cid: cid001416 +primary_alias: THM0120 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: H⁰ equals the space of global sections. For the constant sheaf, a 0-cochain is a global section iff it is constant. + Non-constant functions have nonzero coboundary. +aliases: +- THM0120 +- II.T55 +- h-global-sections +- thm:h0-global +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005425 +- predicate: formalized_by + object: cid005426 +- predicate: formalized_by + object: cid005427 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 9, Chapter 50 (Wave M5) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**H⁰ = Global Sections** + + + H⁰ equals the space of global sections. For the constant sheaf, a 0-cochain is a global section iff it is constant. Non-constant + functions have nonzero coboundary. + + + # H⁰ = Global Sections + + + ## Summary + + + H⁰ equals the space of global sections. For the constant sheaf, a 0-cochain is a global section iff it is constant. Non-constant + functions have nonzero coboundary. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-02.jsonl` line 214 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookII.CentralTheorem.SheafCohomology` + + - Name: `h0_global_2` + + + ## Dependencies + + + - Canonical: II.D87 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: H⁰ equals the space of global sections. For the constant sheaf, a 0-cochain is a global section iff it is constant. + Non-constant functions have nonzero coboundary. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001417.md b/_corpus_v3_items/cid001417.md new file mode 100644 index 000000000..d9b242cbc --- /dev/null +++ b/_corpus_v3_items/cid001417.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001417/ +title: Trade-Off Atomicity +cid: cid001417 +primary_alias: THM0121 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The gain-cost structure of Category tau is atomic: accepting K0-K6 simultaneously forces all 16 Mode C costs and + enables all Mode D gains and Mode E earned results. No partial selection possible.' +aliases: +- THM0121 +- II.T56 +- trade-off-atomicity +- thm:trade-off-atomicity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000001 + locator: Book II, Part 11, Chapter 66 (Part XI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Trade-Off Atomicity**\n\nThe gain-cost structure of Category tau is atomic: accepting K0-K6 simultaneously\ + \ forces all 16 Mode C costs and enables all Mode D gains and Mode E earned results. No partial selection possible.\n\n\ + # Trade-Off Atomicity\n\n## Summary\n\nThe gain-cost structure of Category tau is atomic: accepting K0-K6 simultaneously\ + \ forces all 16 Mode C costs and enables all Mode D gains and Mode E earned results. No partial selection possible.\n\n\ + ## Statement\n\n```latex\n%\n\\label{thm:trade-off-atomicity}\n% II.T43, II.D68, II.D69, II.D91, II.D82, II.D93\nThe\ + \ gain--cost structure of Category~$\\tau$\nis \\textbf{atomic}:\naccepting the foundational commitments\n(K0--K6, $\\\ + jj^2 = +1$)\n\\emph{simultaneously forces}\nall 16 Mode~C costs\nand \\emph{simultaneously enables}\nall 23 Mode~D gains\n\ + and all 13 Mode~E earned results.\nNo partial selection is possible:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n\ + \ \\textbf{Costs are inseparable.}\n Each Mode~C refusal traces to K5, K6, or $\\jj^2 = +1$\n \ + \ (Chapter~\\ref{ch:refuses}).\n You cannot remove any refusal\n without removing the axiom that\ + \ forces it,\n which would also remove\n the Mode~D gains that the axiom enables.\n\n \\item[\\textup{(ii)}]\n\ + \ \\textbf{Gains are inseparable.}\n Each Mode~D gain traces to categoricity (I.T08),\n which\ + \ depends on K6,\n or to the forced sign (I.T10),\n which depends on prime polarity (I.T05).\n \ + \ You cannot import a single gain into ZFC\n without importing the axioms that produce it,\n which\ + \ would also import the Mode~C costs.\n\n \\item[\\textup{(iii)}]\n \\textbf{The ledger is a package deal.}\n\ + \ The 49~gains and 16~costs\n are not a menu to choose from.\n They are the complete, indivisible\ + \ output\n of a single axiomatic commitment.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n```latex\n\ + [Proof sketch]\nThe key observation is that\nall Mode~D gains pass through\none of two bottlenecks:\ncategoricity (I.T08,\ + \ depending on K6)\nor the forced sign (I.T10, depending on I.T05).\nAll Mode~C costs pass through\nthe same two bottlenecks\ + \ plus K5.\nSince K5, K6, and I.T05 are axioms (or immediate consequences),\nthey cannot be partially adopted.\nAccepting\ + \ the axiom system is all-or-nothing.\n```\n\n## Source Context\n\n- Registry source: `book-02.jsonl` line 226\n- Manuscript\ + \ source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part11/ch66-master-trade-off.tex` lines 517-558\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `planned`\n- Module: `None`\n- Name: `None`\n\n## Dependencies\n\n-\ + \ Canonical: I.T05, I.T08, I.T10, II.T42, II.T43, II.D68, II.D69, II.D91, II.D82, II.D93\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The gain-cost structure of Category tau is atomic: accepting K0-K6 simultaneously forces all 16 Mode C costs + and enables all Mode D gains and Mode E earned results. No partial selection possible.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001432.md b/_corpus_v3_items/cid001432.md new file mode 100644 index 000000000..7b3dffda0 --- /dev/null +++ b/_corpus_v3_items/cid001432.md @@ -0,0 +1,140 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001432/ +title: Hartogs Bulk Projection +cid: cid001432 +primary_alias: DEF0202 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 3D Cartesian space = Hartogs-projected bulk of the local T² fiber at each worldline point. Solenoidal surface coordinates + (γ, η) carry boundary data; Hartogs extension fills the interior with genuine linear coordinates. Perceived 3D space is + this interior. +aliases: +- DEF0202 +- III.D01 +- hartogs-bulk-projection +- princ:ch01-hartogs-bulk +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005958 +- predicate: formalized_by + object: cid005961 +- predicate: formalized_by + object: cid005962 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 0, Chapter 1 (Prologue) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hartogs Bulk Projection** + + + 3D Cartesian space = Hartogs-projected bulk of the local T² fiber at each worldline point. Solenoidal surface coordinates + (γ, η) carry boundary data; Hartogs extension fills the interior with genuine linear coordinates. Perceived 3D space is + this interior. + + + # Hartogs Bulk Projection + + + ## Summary + + + 3D Cartesian space = Hartogs-projected bulk of the local T² fiber at each worldline point. Solenoidal surface coordinates + (γ, η) carry boundary data; Hartogs extension fills the interior with genuine linear coordinates. Perceived 3D space is + this interior. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 1 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Prologue.HartogsBulk` + + - Name: `hartogs_bulk_check` + + + ## Dependencies + + + - Canonical: II.T49, II.T36 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 3D Cartesian space = Hartogs-projected bulk of the local T² fiber at each worldline point. Solenoidal surface + coordinates (γ, η) carry boundary data; Hartogs extension fills the interior with genuine linear coordinates. Perceived + 3D space i +noindex: false +--- + diff --git a/_corpus_v3_items/cid001433.md b/_corpus_v3_items/cid001433.md new file mode 100644 index 000000000..beafd3468 --- /dev/null +++ b/_corpus_v3_items/cid001433.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001433/ +title: Millennium-Problem Gluing Table +cid: cid001433 +primary_alias: DEF0203 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Each millennium problem provides a specific guarantee for global coherence of local Hartogs bulk projections: Poincaré + (simply connected), RH (spectral purity), NS (regularity), YM (gap), Hodge (addressability), BSD (patch labeling), Langlands + (commutativity), P vs NP (tractability).' +aliases: +- DEF0203 +- III.D02 +- millennium-problem-gluing-table +- tab:millennium-gluing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 0, Chapter 2 (Prologue) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Millennium-Problem Gluing Table** + + + Each millennium problem provides a specific guarantee for global coherence of local Hartogs bulk projections: Poincaré + (simply connected), RH (spectral purity), NS (regularity), YM (gap), Hodge (addressability), BSD (patch labeling), Langlands + (commutativity), P vs NP (tractability). + + + # Millennium-Problem Gluing Table + + + ## Summary + + + Each millennium problem provides a specific guarantee for global coherence of local Hartogs bulk projections: Poincaré + (simply connected), RH (spectral purity), NS (regularity), YM (gap), Hodge (addressability), BSD (patch labeling), Langlands + (commutativity), P vs NP (tractability). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 3 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: III.D01 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Each millennium problem provides a specific guarantee for global coherence of local Hartogs bulk projections: + Poincaré (simply connected), RH (spectral purity), NS (regularity), YM (gap), Hodge (addressability), BSD (patch labeling), + Langla' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001434.md b/_corpus_v3_items/cid001434.md new file mode 100644 index 000000000..58b8cf4a0 --- /dev/null +++ b/_corpus_v3_items/cid001434.md @@ -0,0 +1,184 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001434/ +title: E₁ as Gluing Principle +cid: cid001434 +primary_alias: DEF0204 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between + local patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of + local spatial structures gluing into a globally coherent space. +aliases: +- DEF0204 +- III.D03 +- e-as-gluing-principle +- def:e1-gluing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005959 +- predicate: formalized_by + object: cid005960 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 0, Chapter 2 (Prologue) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₁ as Gluing Principle** + + + Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between local + patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of local + spatial structures gluing into a globally coherent space. + + + # E₁ as Gluing Principle + + + ## Summary + + + Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms between local + patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon of local + spatial structures gluing into a globally coherent space. + + + ## Statement + + + ```latex + + % + + \label{def:e1-gluing} + + At layer~$E_0$ (Books~I--II), + + $\tau$ possesses objects with internal split-complex structure + + --- the algebraic lemniscate, the bipolar spectral decomposition, + + the full holomorphic apparatus --- + + but the morphisms between objects carry only ordinary + + ($\mathrm{Set}$-enriched) structure. + + Local tori exist. Local Hartogs extensions exist. + + But there is no guarantee that morphisms + + between different local bulks respect the bipolar structure of the objects. + + + The enrichment functor $\mathcal{F}_{E_1}$ + + promotes $\mathrm{Hom}$ spaces + + from sets to \emph{split-complex modules}: + + at $E_1$, the morphisms between local bulks + + carry the same $\chi_+/\chi_-$ bipolar decomposition as the objects. + + This enrichment forces local bulk projections to be globally compatible --- + + the transition maps $\phi_{xy}$ + + are morphisms in a split-complex-enriched category, + + and the enrichment constraints leave no room for the transition maps + + to break the spectral, regularity, or discreteness guarantees. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 4 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part00/ch02-the-eight-guarantees.tex` lines + 258-281 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Prologue.HartogsBulk` + + - Name: `e1_gluing_check` + + + ## Dependencies + + + - Canonical: III.D01, III.D02 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Self-enrichment (E₁) is precisely the statement that local Hartogs bulk projections glue globally. Morphisms + between local patches carry the same split-complex bipolar structure as the patches themselves. Physics IS E₁ — the phenomenon + of l +noindex: false +--- + diff --git a/_corpus_v3_items/cid001435.md b/_corpus_v3_items/cid001435.md new file mode 100644 index 000000000..9eb6f692e --- /dev/null +++ b/_corpus_v3_items/cid001435.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001435/ +title: Enrichment Functor +cid: cid001435 +primary_alias: DEF0205 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The enrichment functor F_E: takes a category and produces its self-enrichment over H_τ. E₀ = Cat_τ, E_{k+1} = F_E(E_k). + Each application creates a new layer with strictly richer structure. The iteration terminates at E₃.' +aliases: +- DEF0205 +- III.D04 +- enrichment-functor +- def:enrichment-functor +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005864 +- predicate: formalized_by + object: cid005865 +- predicate: formalized_by + object: cid005866 +- predicate: formalized_by + object: cid005867 +- predicate: formalized_by + object: cid005869 +- predicate: formalized_by + object: cid005870 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 4 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Enrichment Functor**\n\nThe enrichment functor F_E: takes a category and produces its self-enrichment over\ + \ H_τ. E₀ = Cat_τ, E_{k+1} = F_E(E_k). Each application creates a new layer with strictly richer structure. The iteration\ + \ terminates at E₃.\n\n# Enrichment Functor\n\n## Summary\n\nThe enrichment functor F_E: takes a category and produces\ + \ its self-enrichment over H_τ. E₀ = Cat_τ, E_{k+1} = F_E(E_k). Each application creates a new layer with strictly richer\ + \ structure. The iteration terminates at E₃.\n\n## Statement\n\n```latex\n%\n\\label{def:enrichment-functor}\nThe \\emph{enrichment\ + \ functor}\n\\[\n \\mathcal{F}_E : \\cat{Cat}_\\tau \\longrightarrow \\cat{Cat}_\\tau\n\\]\nacts on the category of\ + \ $\\tau$-categories as follows.\nGiven a $\\tau$-category~$\\mathcal{C}$\n(a category enriched over~$\\tau$ via~$H_\\\ + tau$),\nthe enrichment functor produces\n$\\mathcal{F}_E(\\mathcal{C})$\nwhose data are:\n\\begin{enumerate}\n \\item\ + \ \\textbf{Objects}:\n $\\Obj(\\mathcal{F}_E(\\mathcal{C})) = \\Obj(\\mathcal{C})$.\n The object set\ + \ does not change.\n \\item \\textbf{Hom objects}:\n for $A, B \\in \\Obj(\\mathcal{C})$,\n \\[\n\ + \ \\Hom_{\\mathcal{F}_E(\\mathcal{C})}(A, B)\n \\;=\\;\n [\\Hom_{\\mathcal{C}}(A,\ + \ B)]_{\\tau},\n \\]\n the internal Hom object in~$\\tau$\n associated to the Hom object of~$\\\ + mathcal{C}$.\n That is, we apply the self-enrichment of~$\\tau$\n to the Hom objects of~$\\mathcal{C}$.\n\ + \ \\item \\textbf{Composition}:\n inherited from the composition morphism of~$\\tau$,\n applied to\ + \ the enriched Hom objects.\n \\item \\textbf{Identity}:\n inherited from the identity morphism of~$\\tau$.\n\ + \\end{enumerate}\nThe enrichment functor preserves the monoidal structure\nand the Yoneda embedding.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 8\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch04-the-self-enrichment-functor.tex`\ + \ lines 287-323\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.Functor`\n\ + - Name: `enrichment_functor_check`\n\n## Dependencies\n\n- Canonical: II.D53, II.T36, III.R04\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The enrichment functor F_E: takes a category and produces its self-enrichment over H_τ. E₀ = Cat_τ, E_{k+1} + = F_E(E_k). Each application creates a new layer with strictly richer structure. The iteration terminates at E₃.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001436.md b/_corpus_v3_items/cid001436.md new file mode 100644 index 000000000..8fcb046f4 --- /dev/null +++ b/_corpus_v3_items/cid001436.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001436/ +title: Layer Template +cid: cid001436 +primary_alias: DEF0206 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Each enrichment layer E_k has a uniform four-component structure: E_k = (Carrier_k, Predicate_k, Decoder_k, Invariant_k). + The template is preserved under enrichment: each layer''s invariant flows into the next layer''s carrier.' +aliases: +- DEF0206 +- III.D05 +- layer-template +- def:layer-template +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005875 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Layer Template**\n\nEach enrichment layer E_k has a uniform four-component structure: E_k = (Carrier_k, Predicate_k,\ + \ Decoder_k, Invariant_k). The template is preserved under enrichment: each layer's invariant flows into the next layer's\ + \ carrier.\n\n# Layer Template\n\n## Summary\n\nEach enrichment layer E_k has a uniform four-component structure: E_k\ + \ = (Carrier_k, Predicate_k, Decoder_k, Invariant_k). The template is preserved under enrichment: each layer's invariant\ + \ flows into the next layer's carrier.\n\n## Statement\n\n```latex\n%\n\\label{def:layer-template}\nAn \\textbf{enrichment\ + \ layer} $\\Elayer{k}$ is a quadruple\n\\begin{equation}\\label{eq:ch05-layer-template}\n \\Elayer{k} \\;=\\;\n \ + \ \\bigl(\\,\n \\mathrm{Carrier}_k,\\;\n \\mathrm{Predicate}_k,\\;\n \\mathrm{Decoder}_k,\\;\n \ + \ \\mathrm{Invariant}_k\n \\,\\bigr),\n\\end{equation}\nwhere:\n\\begin{enumerate}\n \\item $\\mathrm{Carrier}_k$\ + \ is the class of objects\n at enrichment depth~$k$---the \\emph{raw material}.\n \\item $\\mathrm{Predicate}_k$\ + \ is the admissibility condition\n selecting well-formed objects---the \\emph{gate}.\n \\item $\\mathrm{Decoder}_k$\ + \ is the canonical map\n extracting invariant information---the \\emph{instrument}.\n \\item $\\mathrm{Invariant}_k$\ + \ is the stable datum\n the decoder produces---the \\emph{output}\n that survives passage to the next\ + \ layer.\n\\end{enumerate}\nThe four components are inherited from the ABCD chart:\n$A \\mapsto \\mathrm{Carrier}$,\n\ + $B \\mapsto \\mathrm{Predicate}$,\n$C \\mapsto \\mathrm{Decoder}$,\n$D \\mapsto \\mathrm{Invariant}$.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 10\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch05-the-layer-template.tex`\ + \ lines 68-98\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.LayerTemplate`\n\ + - Name: `LayerTemplate`\n\n## Dependencies\n\n- Canonical: III.D04\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Each enrichment layer E_k has a uniform four-component structure: E_k = (Carrier_k, Predicate_k, Decoder_k, + Invariant_k). The template is preserved under enrichment: each layer''s invariant flows into the next layer''s carrier.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001437.md b/_corpus_v3_items/cid001437.md new file mode 100644 index 000000000..5981a5350 --- /dev/null +++ b/_corpus_v3_items/cid001437.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001437/ +title: E₀ Layer (Mathematics) +cid: cid001437 +primary_alias: DEF0207 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₀ = (Obj(τ) with NF addressing, NF-addressability, peel map Φ(x) = (A,B,C,D), holomorphic structure O(τ³)). Domain: + Books I-III. The entire kernel of the τ framework.' +aliases: +- DEF0207 +- III.D06 +- e-layer-mathematics +- def:e0-layer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005871 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₀ Layer (Mathematics)**\n\nE₀ = (Obj(τ) with NF addressing, NF-addressability, peel map Φ(x) = (A,B,C,D),\ + \ holomorphic structure O(τ³)). Domain: Books I-III. The entire kernel of the τ framework.\n\n# E₀ Layer (Mathematics)\n\ + \n## Summary\n\nE₀ = (Obj(τ) with NF addressing, NF-addressability, peel map Φ(x) = (A,B,C,D), holomorphic structure O(τ³)).\ + \ Domain: Books I-III. The entire kernel of the τ framework.\n\n## Statement\n\n```latex\n%\n\\label{def:e0-layer}\n\\\ + begin{equation}\\label{eq:ch05-e0}\n \\Elayer{0} \\;=\\;\n \\bigl(\\,\n \\mathrm{Obj}(\\tau),\\;\n \ + \ \\text{NF-addressability},\\;\n \\Phi,\\;\n \\mathcal{O}(\\tau^3)\n \\,\\bigr).\n\\end{equation}\n\ + \\begin{enumerate}\n \\item \\textbf{Carrier} $= \\mathrm{Obj}(\\tau)$\n with canonical NF addressing.\n \ + \ Every object has a unique normal-form address\n in $\\mathbb{Z}/M_d\\mathbb{Z}$ at each finite depth~$d$.\n\ + \n \\item \\textbf{Predicate} $=$ NF-addressability.\n An object is $\\Elayer{0}$-admissible\n iff\ + \ it lies in the image of the hyperfactorization map.\n At $\\Elayer{0}$, every object of~$\\tau$\n \ + \ satisfies this predicate (I.T04).\n\n \\item \\textbf{Decoder} $= \\Phi$:\n the peel map $\\Phi(x) = (A,\ + \ B, C, D)$.\n The canonical ABCD decomposition,\n earned in Book~I, Part~IV.\n At each depth~$d$,\ + \ it factors through CRT:\n \\[\n \\Phi_d(x) \\;=\\;\n \\bigl(\\,\\pi(x),\\;\\gamma(x),\\\ + ;\\eta(x),\\;\\alpha(x)\\,\\bigr)\n \\bmod M_d.\n \\]\n\n \\item \\textbf{Invariant} $= \\mathcal{O}(\\\ + tau^3)$:\n the holomorphic structure of the total space,\n characterised by the Central Theorem~(II.T40):\n\ + \ $\\mathcal{O}(\\tau^3) \\cong A_{\\mathrm{spec}}(\\mathbb{L})$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 11\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch05-the-layer-template.tex`\ + \ lines 123-163\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.LayerTemplate`\n\ + - Name: `e0_layer`\n\n## Dependencies\n\n- Canonical: III.D05\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₀ = (Obj(τ) with NF addressing, NF-addressability, peel map Φ(x) = (A,B,C,D), holomorphic structure O(τ³)). + Domain: Books I-III. The entire kernel of the τ framework.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001438.md b/_corpus_v3_items/cid001438.md new file mode 100644 index 000000000..fc74508f0 --- /dev/null +++ b/_corpus_v3_items/cid001438.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001438/ +title: E₁ Layer (Physics) +cid: cid001438 +primary_alias: DEF0208 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₁ = (H_τ-enriched objects, sector admissibility via 4+1 decomposition, spectral projection onto sectors, sector + couplings as canonical lifts of ι_τ). Domain: Books IV-V. Physics as enrichment.' +aliases: +- DEF0208 +- III.D07 +- e-layer-physics +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005872 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₁ Layer (Physics)**\n\nE₁ = (H_τ-enriched objects, sector admissibility via 4+1 decomposition, spectral projection\ + \ onto sectors, sector couplings as canonical lifts of ι_τ). Domain: Books IV-V. Physics as enrichment.\n\n# E₁ Layer\ + \ (Physics)\n\n## Summary\n\nE₁ = (H_τ-enriched objects, sector admissibility via 4+1 decomposition, spectral projection\ + \ onto sectors, sector couplings as canonical lifts of ι_τ). Domain: Books IV-V. Physics as enrichment.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:e1-layer}\n$\\Estage{1}$ denotes Category~$\\T$\nequipped with the self-enrichment data\nearned\ + \ in Part~VIII.\nExplicitly, $\\Estage{1}$ adds to~$\\Estage{0}$\nthe following structures:\n\\begin{enumerate}\n \\\ + item \\textbf{Hom objects as $\\T$-objects.}\n For all $A, B \\in \\Obj(\\T)$,\n the morphism space\ + \ $\\Hom(A,B) \\in \\Obj(\\T)$\n (II.D53, II.D54,\n Chapter~\\ref{ch:tau-self-enrichment}).\n \ + \ Each Hom object has an NF address,\n inherits bipolar decomposition,\n and satisfies tower coherence.\n\ + \n \\item \\textbf{The Yoneda embedding.}\n $\\T \\hookrightarrow [\\T^{\\op}, \\T]$\n is a full\ + \ and faithful functor\n (II.T36,\n Chapter~\\ref{ch:yoneda-theorem}).\n This makes every $\\\ + T$-object\n representable by the morphisms into it.\n\n \\item \\textbf{2-categorical structure.}\n \ + \ 2-morphisms\n $\\Hom(\\Hom(A,B), \\Hom(C,D))$\n are $\\T$-objects\n (II.D55, II.D56,\n\ + \ Chapter~\\ref{ch:two-categories}).\n Morphisms between morphisms\n are first-class citizens.\n\ + \n \\item \\textbf{Internal function spaces.}\n The internal Hom\n $[A,B] \\in \\Obj(\\T)$\n \ + \ replaces the external function set\n $\\{f \\colon A \\to B\\}$.\n Function spaces are objects,\n\ + \ not meta-level collections.\n\n \\item \\textbf{The Code/Decode bijection.}\n $\\mathrm{Code} \\\ + colon \\mathcal{O}(\\tau^3)\n \\to \\mathrm{Streams}(R_\\tau, H_\\tau)$\n and its inverse $\\mathrm{Decode}$\n\ + \ (II.T35,\n Chapter~\\ref{ch:code-decode})\n internalize the Mutual Determination\n \ + \ as a coding-theoretic bijection.\n\n \\item \\textbf{Self-referential capability.}\n $\\T$ can talk about\ + \ its own morphisms,\n compare them,\n and transform them---all\n within its own language\n\ + \ (Remark~\\ref{rem:self-description}, II.R15).\n\\end{enumerate}\n\n\\noindent\nThe transition $\\Estage{0}\ + \ \\to \\Estage{1}$\nis what Part~VIII accomplishes.\n$\\Estage{1}$ is not a different category;\nit is the \\emph{same}\ + \ category~$\\T$,\nnow recognized as carrying self-enrichment structure\nthat was latent in $\\Estage{0}$\nbut not yet\ + \ articulated.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 12\n- Manuscript source: `2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part08/ch45-self-describing.tex`\ + \ lines 571-639\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.LayerTemplate`\n\ + - Name: `e1_layer_book3`\n\n## Dependencies\n\n- Canonical: III.D05, III.D06\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₁ = (H_τ-enriched objects, sector admissibility via 4+1 decomposition, spectral projection onto sectors, sector + couplings as canonical lifts of ι_τ). Domain: Books IV-V. Physics as enrichment.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001439.md b/_corpus_v3_items/cid001439.md new file mode 100644 index 000000000..d78749fcd --- /dev/null +++ b/_corpus_v3_items/cid001439.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001439/ +title: E₂ Layer (Computation) +cid: cid001439 +primary_alias: DEF0209 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₂ = (self-referential codes, operational closure, phenotype map, error-correction capacity). Computation is native + to E₂: the code→execution→code cycle cannot exist at E₀ (no processes) or E₁ (no codes). Life is computation physically + instantiated.' +aliases: +- DEF0209 +- III.D08 +- e-layer-computation +- def:e2-layer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005873 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₂ Layer (Computation)**\n\nE₂ = (self-referential codes, operational closure, phenotype map, error-correction\ + \ capacity). Computation is native to E₂: the code→execution→code cycle cannot exist at E₀ (no processes) or E₁ (no codes).\ + \ Life is computation physically instantiated.\n\n# E₂ Layer (Computation)\n\n## Summary\n\nE₂ = (self-referential codes,\ + \ operational closure, phenotype map, error-correction capacity). Computation is native to E₂: the code→execution→code\ + \ cycle cannot exist at E₀ (no processes) or E₁ (no codes). Life is computation physically instantiated.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:e2-layer}\n\\begin{equation}\\label{eq:ch05-e2}\n \\Elayer{2} \\;=\\;\n \\left(\\,\n\ + \ \\begin{aligned}\n &\\text{self-referential codes},\\;\n \\text{operational closure},\\\\\n &\\text{phenotype\ + \ map},\\;\n \\text{error-correction capacity}\n \\end{aligned}\n \\,\\right).\n\\end{equation}\n\\begin{enumerate}\n\ + \ \\item \\textbf{Carrier} $=$ self-referential codes:\n $\\Elayer{1}$-admissible objects\n that\ + \ contain a representation of their own decoder.\n The invariant-to-carrier handoff from $\\Elayer{1}$\n \ + \ provides sector couplings as raw material.\n\n \\item \\textbf{Predicate} $=$ operational closure.\n \ + \ The cycle\n $\\text{code}\n \\xrightarrow{\\text{execute}}\n \\text{product}\n \ + \ \\xrightarrow{\\text{encode}}\n \\text{code}$\n is stable up to error-correction tolerance.\n \ + \ This is the categorical abstraction of the computational cycle.\n Its most dramatic physical instantiation\ + \ is metabolic closure---life.\n\n \\item \\textbf{Decoder} $=$ phenotype map:\n from code to output.\n \ + \ A morphism in the $H_\\tau$-enriched category\n that respects sector structure.\n In biological\ + \ instantiation, the output is an organism;\n in computational instantiation, it is a return value.\n\n \\\ + item \\textbf{Invariant} $=$ error-correction capacity:\n the maximum perturbation rate\n the closure\ + \ cycle can absorb.\n A $\\tau$-analogue of a Shannon bound.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 13\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch05-the-layer-template.tex`\ + \ lines 296-340\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.LayerTemplate`\n\ + - Name: `e2_layer`\n\n## Dependencies\n\n- Canonical: III.D05, III.D07\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₂ = (self-referential codes, operational closure, phenotype map, error-correction capacity). Computation is + native to E₂: the code→execution→code cycle cannot exist at E₀ (no processes) or E₁ (no codes). Life is computation physically + inst' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001440.md b/_corpus_v3_items/cid001440.md new file mode 100644 index 000000000..a671058c6 --- /dev/null +++ b/_corpus_v3_items/cid001440.md @@ -0,0 +1,85 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001440/ +title: E₃ Layer (Metaphysics) +cid: cid001440 +primary_alias: DEF0210 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₃ = (self-modeling codes that model their own observation, self-model consistency, interpretation/meaning assignment, + self-awareness capacity). Domain: Book VII. Metaphysics as self-modeling enrichment.' +aliases: +- DEF0210 +- III.D09 +- e-layer-metaphysics +- def:e3-layer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005874 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 5 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₃ Layer (Metaphysics)**\n\nE₃ = (self-modeling codes that model their own observation, self-model consistency,\ + \ interpretation/meaning assignment, self-awareness capacity). Domain: Book VII. Metaphysics as self-modeling enrichment.\n\ + \n# E₃ Layer (Metaphysics)\n\n## Summary\n\nE₃ = (self-modeling codes that model their own observation, self-model consistency,\ + \ interpretation/meaning assignment, self-awareness capacity). Domain: Book VII. Metaphysics as self-modeling enrichment.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:e3-layer}\n\\begin{equation}\\label{eq:ch05-e3}\n \\Elayer{3} \\;=\\;\n\ + \ \\bigl(\\,\n \\text{observers},\\;\n \\text{self-model consistency},\\;\n \\text{interpretation},\\\ + ;\n \\text{self-awareness capacity}\n \\,\\bigr).\n\\end{equation}\n\\begin{enumerate}\n \\item \\textbf{Carrier}\ + \ $=$ observers:\n self-referential codes ($\\Elayer{2}$-admissible)\n that model their own observation.\n\ + \ The object contains a model of the decoding process,\n including a representation of itself as the\ + \ decoding agent.\n Error-correction capacity from $\\Elayer{2}$\n provides the robustness needed for\ + \ self-modelling.\n\n \\item \\textbf{Predicate} $=$ self-model consistency.\n The model's predictions about\ + \ the observer's behaviour\n agree with actual behaviour\n (within $\\Elayer{2}$-inherited tolerance).\n\ + \ Categorical abstraction of rational agency.\n\n \\item \\textbf{Decoder} $=$ interpretation:\n \ + \ from structure to meaning.\n The $\\Elayer{3}$-decoder reads\n not just the code but the \\emph{meaning}\ + \ of the code.\n\n \\item \\textbf{Invariant} $=$ self-awareness capacity.\n The terminal invariant: there\ + \ is no $\\Elayer{4}$.\n The Canonical Ladder Theorem guarantees saturation.\n\\end{enumerate}\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 14\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch05-the-layer-template.tex`\ + \ lines 389-425\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.LayerTemplate`\n\ + - Name: `e3_layer`\n\n## Dependencies\n\n- Canonical: III.D05, III.D08\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₃ = (self-modeling codes that model their own observation, self-model consistency, interpretation/meaning assignment, + self-awareness capacity). Domain: Book VII. Metaphysics as self-modeling enrichment.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001441.md b/_corpus_v3_items/cid001441.md new file mode 100644 index 000000000..c10219cff --- /dev/null +++ b/_corpus_v3_items/cid001441.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001441/ +title: Ladder Checker +cid: cid001441 +primary_alias: DEF0211 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Lean-grade proof harness for verifying ladder properties: existence_checker(k) verifies non-emptiness at level k, + stability_checker(k) verifies template preservation, strictness_checker(k) verifies E_k \ E_{k-1} ≠ ∅, saturation_checker(k_max) + verifies [E_{k_max}^op, E_{k_max}] ⊆ E_{k_max}.' +aliases: +- DEF0211 +- III.D10 +- ladder-checker +- def:ladder-checker +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005868 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 8 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Ladder Checker**\n\nLean-grade proof harness for verifying ladder properties: existence_checker(k) verifies\ + \ non-emptiness at level k, stability_checker(k) verifies template preservation, strictness_checker(k) verifies E_k \\\ + \ E_{k-1} ≠ ∅, saturation_checker(k_max) verifies [E_{k_max}^op, E_{k_max}] ⊆ E_{k_max}.\n\n# Ladder Checker\n\n## Summary\n\ + \nLean-grade proof harness for verifying ladder properties: existence_checker(k) verifies non-emptiness at level k, stability_checker(k)\ + \ verifies template preservation, strictness_checker(k) verifies E_k \\ E_{k-1} ≠ ∅, saturation_checker(k_max) verifies\ + \ [E_{k_max}^op, E_{k_max}] ⊆ E_{k_max}.\n\n## Statement\n\n```latex\n%\n\\label{def:ladder-checker}\nThe \\textbf{Ladder\ + \ Checker} is a quadruple of decision procedures:\n\\begin{enumerate}\n \\item \\texttt{existence\\_checker}$(k)$:\n\ + \ Given $k \\in \\{0, 1, 2, 3\\}$,\n certifies that $E_k$ is non-empty\n by producing an explicit\ + \ object $x_k \\in \\Obj(E_k)$\n together with a non-identity endomorphism $f_k : x_k \\to x_k$.\n \\item\ + \ \\texttt{stability\\_checker}$(k)$:\n Given $k \\in \\{0, 1, 2, 3\\}$,\n certifies that $E_k$ is closed\n\ + \ under morphism composition:\n for all composable pairs $(g, f)$ in~$E_k$,\n the composite\ + \ $g \\circ f$ remains in~$E_k$.\n \\item \\texttt{strictness\\_checker}$(k)$:\n Given $k \\in \\{1, 2, 3\\\ + }$,\n certifies that $E_k \\setminus E_{k-1} \\neq \\varnothing$\n by exhibiting a \\emph{strictness\ + \ witness}:\n an object $w_k \\in E_k$\n together with a proof that $w_k \\notin E_{k-1}$.\n \\item\ + \ \\texttt{saturation\\_checker}$(k_{\\max})$:\n Given $k_{\\max} = 3$,\n certifies that $\\mathcal{F}_E(E_{k_{\\\ + max}}) = E_{k_{\\max}}$\n by showing that the functor category\n $[E_3^{\\mathrm{op}}, E_3]$ introduces\ + \ no\n object outside~$E_3$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 21\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch08-the-canonical-ladder-theorem.tex` lines 274-303\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.Functor`\n- Name:\ + \ `ladder_checker`\n\n## Dependencies\n\n- Canonical: III.T04\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Lean-grade proof harness for verifying ladder properties: existence_checker(k) verifies non-emptiness at level + k, stability_checker(k) verifies template preservation, strictness_checker(k) verifies E_k \ E_{k-1} ≠ ∅, saturation_checker(k_ma' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001442.md b/_corpus_v3_items/cid001442.md new file mode 100644 index 000000000..ed7ba3459 --- /dev/null +++ b/_corpus_v3_items/cid001442.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001442/ +title: Sieve Prime Count +cid: cid001442 +primary_alias: DEF0212 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'π(n): count of primes ≤ n via sieve. π(10)=4, π(30)=10, π(100)=25.' +aliases: +- DEF0212 +- III.D100 +- sieve-prime-count +- def:sieve-prime-count +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006072 +- predicate: formalized_by + object: cid006073 +- predicate: formalized_by + object: cid006074 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve Prime Count** + + + π(n): count of primes ≤ n via sieve. π(10)=4, π(30)=10, π(100)=25. + + + # Sieve Prime Count + + + ## Summary + + + π(n): count of primes ≤ n via sieve. π(10)=4, π(30)=10, π(100)=25. + + + ## Statement + + + ```latex + + \label{def:sieve-prime-count} + + $\pi(n) := \#\{p \le n : p \text{ prime}\}$, computed via the sieve. + + + \textbf{Lean:} \texttt{sieve\_prime\_count}.\quad + + \textbf{Registry:} III.D100. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 252 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 92-97 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `sieve_prime_count` + + + ## Dependencies + + + - Canonical: III.D99 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'π(n): count of primes ≤ n via sieve. π(10)=4, π(30)=10, π(100)=25.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001443.md b/_corpus_v3_items/cid001443.md new file mode 100644 index 000000000..62e0b976c --- /dev/null +++ b/_corpus_v3_items/cid001443.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001443/ +title: Brun Sieve Count +cid: cid001443 +primary_alias: DEF0213 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Brun sieve: count integers in [1..n] coprime to first d primes. Inclusion-exclusion at depth d. B(30,3)=8=φ(30).' +aliases: +- DEF0213 +- III.D101 +- brun-sieve-count +- def:brun-sieve-count +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006075 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Brun Sieve Count** + + + Brun sieve: count integers in [1..n] coprime to first d primes. Inclusion-exclusion at depth d. B(30,3)=8=φ(30). + + + # Brun Sieve Count + + + ## Summary + + + Brun sieve: count integers in [1..n] coprime to first d primes. Inclusion-exclusion at depth d. B(30,3)=8=φ(30). + + + ## Statement + + + ```latex + + \label{def:brun-sieve-count} + + $B(n, d) := \#\{k \in [1,n] : \gcd(k, M_d) = 1\}$, the inclusion-exclusion + + count of integers coprime to the first~$d$ primes. + + + \textbf{Lean:} \texttt{brun\_sieve\_count}.\quad + + \textbf{Registry:} III.D101. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 253 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 99-105 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `brun_sieve_count` + + + ## Dependencies + + + - Canonical: III.D99 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Brun sieve: count integers in [1..n] coprime to first d primes. Inclusion-exclusion at depth d. B(30,3)=8=φ(30).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001444.md b/_corpus_v3_items/cid001444.md new file mode 100644 index 000000000..6227ddd2e --- /dev/null +++ b/_corpus_v3_items/cid001444.md @@ -0,0 +1,139 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001444/ +title: Sieve-Accelerated Goldbach +cid: cid001444 +primary_alias: DEF0214 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Goldbach verification using sieve primality. Pushes bound from 100 to 500. +aliases: +- DEF0214 +- III.D102 +- sieve-accelerated-goldbach +- def:goldbach-sieve +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve-Accelerated Goldbach** + + + Goldbach verification using sieve primality. Pushes bound from 100 to 500. + + + # Sieve-Accelerated Goldbach + + + ## Summary + + + Goldbach verification using sieve primality. Pushes bound from 100 to 500. + + + ## Statement + + + ```latex + + \label{def:goldbach-sieve} + + $G(n)$~holds if there exist primes $p, q$ with $p + q = n$. + + Using sieve-based primality, Goldbach is verified for all even + + $n \in [4, 500]$. + + + \textbf{Lean:} \texttt{goldbach\_sieve\_check}.\quad + + \textbf{Registry:} III.D102. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 257 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 142-149 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `goldbach_sieve_check` + + + ## Dependencies + + + - Canonical: III.D99, III.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Goldbach verification using sieve primality. Pushes bound from 100 to 500. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001445.md b/_corpus_v3_items/cid001445.md new file mode 100644 index 000000000..a917812d5 --- /dev/null +++ b/_corpus_v3_items/cid001445.md @@ -0,0 +1,146 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001445/ +title: Partition Count at Primorial +cid: cid001445 +primary_alias: DEF0215 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'r(M_k): Goldbach partition count at primorial levels. r(M_2)=1, r(M_3)=3, r(M_4)>0. Increasing sequence.' +aliases: +- DEF0215 +- III.D103 +- partition-count-at-primorial +- def:goldbach-partition-primorial +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006033 +- predicate: formalized_by + object: cid006034 +- predicate: formalized_by + object: cid006035 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Partition Count at Primorial** + + + r(M_k): Goldbach partition count at primorial levels. r(M_2)=1, r(M_3)=3, r(M_4)>0. Increasing sequence. + + + # Partition Count at Primorial + + + ## Summary + + + r(M_k): Goldbach partition count at primorial levels. r(M_2)=1, r(M_3)=3, r(M_4)>0. Increasing sequence. + + + ## Statement + + + ```latex + + \label{def:goldbach-partition-primorial} + + $r(M_k)$~counts representations $M_k = p + q$ with $p \le q$ prime. + + Computed values: $r(M_2) = 1$, $r(M_3) = 3$, $r(M_4) > 0$. + + + \textbf{Lean:} \texttt{goldbach\_partition\_count\_at\_primorial}.\quad + + \textbf{Registry:} III.D103. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 258 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 170-176 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `goldbach_partition_count_at_primorial` + + + ## Dependencies + + + - Canonical: III.D99, III.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'r(M_k): Goldbach partition count at primorial levels. r(M_2)=1, r(M_3)=3, r(M_4)>0. Increasing sequence.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001446.md b/_corpus_v3_items/cid001446.md new file mode 100644 index 000000000..080a8475e --- /dev/null +++ b/_corpus_v3_items/cid001446.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001446/ +title: Goldbach Obstruction +cid: cid001446 +primary_alias: DEF0216 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Obstruction at prime p for even n: |O_p(n)|=1 if p|n, 0 otherwise. Each prime blocks at most one Goldbach pair.' +aliases: +- DEF0216 +- III.D104 +- goldbach-obstruction +- def:goldbach-obstruction +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006036 +- predicate: formalized_by + object: cid006037 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach Obstruction** + + + Obstruction at prime p for even n: |O_p(n)|=1 if p|n, 0 otherwise. Each prime blocks at most one Goldbach pair. + + + # Goldbach Obstruction + + + ## Summary + + + Obstruction at prime p for even n: |O_p(n)|=1 if p|n, 0 otherwise. Each prime blocks at most one Goldbach pair. + + + ## Statement + + + ```latex + + \label{def:goldbach-obstruction} + + At prime $p$, the obstruction count for even~$n$ is + + $|O_p(n)| = \begin{cases} 1 & \text{if } p \mid n, \\ 0 & \text{otherwise.}\end{cases}$ + + + \textbf{Lean:} \texttt{goldbach\_obstruction\_count}.\quad + + \textbf{Registry:} III.D104. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 259 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 187-193 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `goldbach_obstruction_count` + + + ## Dependencies + + + - Canonical: III.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Obstruction at prime p for even n: |O_p(n)|=1 if p|n, 0 otherwise. Each prime blocks at most one Goldbach pair.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001447.md b/_corpus_v3_items/cid001447.md new file mode 100644 index 000000000..04342609e --- /dev/null +++ b/_corpus_v3_items/cid001447.md @@ -0,0 +1,140 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001447/ +title: Twin Prime Sieve Count +cid: cid001447 +primary_alias: DEF0217 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Twin prime count via sieve. π₂(100)≥8, π₂(500)≥20. +aliases: +- DEF0217 +- III.D105 +- twin-prime-sieve-count +- def:twin-sieve-count +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006093 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Prime Sieve Count** + + + Twin prime count via sieve. π₂(100)≥8, π₂(500)≥20. + + + # Twin Prime Sieve Count + + + ## Summary + + + Twin prime count via sieve. π₂(100)≥8, π₂(500)≥20. + + + ## Statement + + + ```latex + + \label{def:twin-sieve-count} + + $\pi_2(n) := \#\{p \le n : p \text{ and } p+2 \text{ both prime}\}$. + + + \textbf{Lean:} \texttt{twin\_prime\_sieve\_count}.\quad + + \textbf{Registry:} III.D105. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 266 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 230-235 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `twin_prime_sieve_count` + + + ## Dependencies + + + - Canonical: III.D99 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Twin prime count via sieve. π₂(100)≥8, π₂(500)≥20. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001448.md b/_corpus_v3_items/cid001448.md new file mode 100644 index 000000000..cd031d5d9 --- /dev/null +++ b/_corpus_v3_items/cid001448.md @@ -0,0 +1,152 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001448/ +title: Hardy-Littlewood Constant +cid: cid001448 +primary_alias: DEF0218 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Hardy-Littlewood twin constant C₂(k) = ∏ p(p-2)/(p-1)² for p≥3. C₂(2)=3/4, C₂(3)=45/64. Decreasing toward C₂≈1.3203. +aliases: +- DEF0218 +- III.D106 +- hardy-littlewood-constant +- def:hl-constant +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006094 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hardy-Littlewood Constant** + + + Hardy-Littlewood twin constant C₂(k) = ∏ p(p-2)/(p-1)² for p≥3. C₂(2)=3/4, C₂(3)=45/64. Decreasing toward C₂≈1.3203. + + + # Hardy-Littlewood Constant + + + ## Summary + + + Hardy-Littlewood twin constant C₂(k) = ∏ p(p-2)/(p-1)² for p≥3. C₂(2)=3/4, C₂(3)=45/64. Decreasing toward C₂≈1.3203. + + + ## Statement + + + ```latex + + \label{def:hl-constant} + + The twin prime constant truncated at depth~$k$: + + \[ + + C_2(k) = \prod_{\substack{i=2 \\ p_i \ge 3}}^{k} + + \frac{p_i(p_i - 2)}{(p_i - 1)^2}. + + \] + + Computed as a rational (numerator, denominator). + + $C_2(2) = 3/4$, $C_2(3) = 45/64$. + + + \textbf{Lean:} \texttt{hl\_twin\_constant\_approx}.\quad + + \textbf{Registry:} III.D106. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 267 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 255-266 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `hl_twin_constant_approx` + + + ## Dependencies + + + - Canonical: (none) + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Hardy-Littlewood twin constant C₂(k) = ∏ p(p-2)/(p-1)² for p≥3. C₂(2)=3/4, C₂(3)=45/64. Decreasing toward C₂≈1.3203. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001449.md b/_corpus_v3_items/cid001449.md new file mode 100644 index 000000000..be85de199 --- /dev/null +++ b/_corpus_v3_items/cid001449.md @@ -0,0 +1,146 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001449/ +title: CRT Twin Admissibility +cid: cid001449 +primary_alias: DEF0219 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Twin-admissible residues mod M_k: r with neither r nor r+2 ≡ 0 (mod p_i) for all i≤k. Always nonempty.' +aliases: +- DEF0219 +- III.D107 +- crt-twin-admissibility +- def:crt-twin-admissible +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006095 +- predicate: formalized_by + object: cid006096 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT Twin Admissibility** + + + Twin-admissible residues mod M_k: r with neither r nor r+2 ≡ 0 (mod p_i) for all i≤k. Always nonempty. + + + # CRT Twin Admissibility + + + ## Summary + + + Twin-admissible residues mod M_k: r with neither r nor r+2 ≡ 0 (mod p_i) for all i≤k. Always nonempty. + + + ## Statement + + + ```latex + + \label{def:crt-twin-admissible} + + A residue $r \bmod M_k$ is \emph{twin-admissible} if for every + + prime $p_i$ ($i \le k$): $r \not\equiv 0 \pmod{p_i}$ and + + $r + 2 \not\equiv 0 \pmod{p_i}$. + + + \textbf{Lean:} \texttt{crt\_twin\_admissible}.\quad + + \textbf{Registry:} III.D107. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 268 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 277-284 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `crt_twin_admissible` + + + ## Dependencies + + + - Canonical: III.T10 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Twin-admissible residues mod M_k: r with neither r nor r+2 ≡ 0 (mod p_i) for all i≤k. Always nonempty.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001450.md b/_corpus_v3_items/cid001450.md new file mode 100644 index 000000000..52421804d --- /dev/null +++ b/_corpus_v3_items/cid001450.md @@ -0,0 +1,139 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001450/ +title: Sieve-Accelerated ABC +cid: cid001450 +primary_alias: DEF0220 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: ABC quality check pushed to bound 100. Weak ABC (c < rad(abc)²) for all coprime pairs. +aliases: +- DEF0220 +- III.D108 +- sieve-accelerated-abc +- def:abc-sieve +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve-Accelerated ABC** + + + ABC quality check pushed to bound 100. Weak ABC (c < rad(abc)²) for all coprime pairs. + + + # Sieve-Accelerated ABC + + + ## Summary + + + ABC quality check pushed to bound 100. Weak ABC (c < rad(abc)²) for all coprime pairs. + + + ## Statement + + + ```latex + + \label{def:abc-sieve} + + For coprime pairs $(a, b)$ with $c = a + b$: verify + + $c < \mathrm{rad}(abc)^2$ (weak ABC, $\varepsilon = 1$). + + Pushed from bound~15 to~100. + + + \textbf{Lean:} \texttt{abc\_sieve\_check}.\quad + + \textbf{Registry:} III.D108. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 275 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 319-326 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `abc_sieve_check` + + + ## Dependencies + + + - Canonical: III.D97, III.D98 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: ABC quality check pushed to bound 100. Weak ABC (c < rad(abc)²) for all coprime pairs. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001451.md b/_corpus_v3_items/cid001451.md new file mode 100644 index 000000000..4548e0d15 --- /dev/null +++ b/_corpus_v3_items/cid001451.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001451/ +title: High Quality Count +cid: cid001451 +primary_alias: DEF0221 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Count coprime triples with quality q≥1 (c≥rad(abc)). Rare: includes (1,8,9). At most 5 up to bound 30.' +aliases: +- DEF0221 +- III.D109 +- high-quality-count +- def:abc-high-quality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005687 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**High Quality Count** + + + Count coprime triples with quality q≥1 (c≥rad(abc)). Rare: includes (1,8,9). At most 5 up to bound 30. + + + # High Quality Count + + + ## Summary + + + Count coprime triples with quality q≥1 (c≥rad(abc)). Rare: includes (1,8,9). At most 5 up to bound 30. + + + ## Statement + + + ```latex + + \label{def:abc-high-quality} + + Count coprime triples with $c \ge \mathrm{rad}(abc)$ (quality $q \ge 1$). + + These are the triples where ABC is nontrivial. Examples include + + $(1, 8, 9)$: $\mathrm{rad}(72) = 6 < 9$. + + + \textbf{Lean:} \texttt{abc\_high\_quality\_count}.\quad + + \textbf{Registry:} III.D109. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 276 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 336-343 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `abc_high_quality_count` + + + ## Dependencies + + + - Canonical: III.D97 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Count coprime triples with quality q≥1 (c≥rad(abc)). Rare: includes (1,8,9). At most 5 up to bound 30.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001452.md b/_corpus_v3_items/cid001452.md new file mode 100644 index 000000000..09f906b8a --- /dev/null +++ b/_corpus_v3_items/cid001452.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001452/ +title: Boundary Character Space +cid: cid001452 +primary_alias: DEF0222 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Characters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L), S¹) ≅ S¹ × S¹. The character lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ ℤ. Every + character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois, n-axis = additive/automorphic.' +aliases: +- DEF0222 +- III.D11 +- boundary-character-space +- def:boundary-character-space +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005963 +- predicate: formalized_by + object: cid005966 +- predicate: formalized_by + object: cid005967 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary Character Space**\n\nCharacters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L), S¹) ≅ S¹ × S¹. The character\ + \ lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ ℤ. Every character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois, n-axis\ + \ = additive/automorphic.\n\n# Boundary Character Space\n\n## Summary\n\nCharacters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L),\ + \ S¹) ≅ S¹ × S¹. The character lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ ℤ. Every character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois,\ + \ n-axis = additive/automorphic.\n\n## Statement\n\n```latex\n%\n\\label{def:boundary-character-space}\nThe \\emph{boundary\ + \ character space} of the lemniscate is\n\\begin{equation}\\label{eq:ch09-char-space}\n \\Char(\\Lemniscate)\n \\\ + ;:=\\;\n \\Hom(\\pi_1(\\Lemniscate),\\, S^1)\n \\;\\cong\\;\n S^1 \\times S^1.\n\\end{equation}\nIt is a compact\ + \ abelian group\nequipped with the product topology.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 23\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch09-boundary-functoriality.tex` lines 115-128\n\n##\ + \ Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.BoundaryCharacters`\n\ + - Name: `BoundaryCharacter`\n\n## Dependencies\n\n- Canonical: II.T49\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Characters on L = S¹ ∨ S¹: Char(L) = Hom(π₁(L), S¹) ≅ S¹ × S¹. The character lattice ℤ² from H₁(L; ℤ) ≅ ℤ ⊕ + ℤ. Every character indexed by (m,n) ∈ ℤ². The m-axis = multiplicative/Galois, n-axis = additive/automorphic.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001453.md b/_corpus_v3_items/cid001453.md new file mode 100644 index 000000000..5c0e4b1bf --- /dev/null +++ b/_corpus_v3_items/cid001453.md @@ -0,0 +1,148 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001453/ +title: Squarefree ABC Check +cid: cid001453 +primary_alias: DEF0223 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For squarefree coprime pairs: c ≤ rad(abc). Trivially true since rad(n)=n for squarefree n.' +aliases: +- DEF0223 +- III.D110 +- squarefree-abc-check +- def:squarefree-abc +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005688 +- predicate: formalized_by + object: cid005689 +- predicate: formalized_by + object: cid005690 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Squarefree ABC Check** + + + For squarefree coprime pairs: c ≤ rad(abc). Trivially true since rad(n)=n for squarefree n. + + + # Squarefree ABC Check + + + ## Summary + + + For squarefree coprime pairs: c ≤ rad(abc). Trivially true since rad(n)=n for squarefree n. + + + ## Statement + + + ```latex + + \label{def:squarefree-abc} + + For \emph{squarefree} coprime pairs: verify $c \le \mathrm{rad}(abc)$. + + Since $\mathrm{rad}(n) = n$ for squarefree~$n$, this is automatically + + satisfied. + + + \textbf{Lean:} \texttt{squarefree\_abc\_check}.\quad + + \textbf{Registry:} III.D110. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 277 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 345-352 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `squarefree_abc_check` + + + ## Dependencies + + + - Canonical: III.D97 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For squarefree coprime pairs: c ≤ rad(abc). Trivially true since rad(n)=n for squarefree n.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001454.md b/_corpus_v3_items/cid001454.md new file mode 100644 index 000000000..6817e0143 --- /dev/null +++ b/_corpus_v3_items/cid001454.md @@ -0,0 +1,143 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001454/ +title: Tower Decidable Check +cid: cid001454 +primary_alias: DEF0224 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At each finite primorial level, all three conjectures are decidable. Gap types, forbidden moves, and scope labels + all defined. +aliases: +- DEF0224 +- III.D111 +- tower-decidable-check +- def:tower-decidable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tower Decidable Check** + + + At each finite primorial level, all three conjectures are decidable. Gap types, forbidden moves, and scope labels all + defined. + + + # Tower Decidable Check + + + ## Summary + + + At each finite primorial level, all three conjectures are decidable. Gap types, forbidden moves, and scope labels all + defined. + + + ## Statement + + + ```latex + + \label{def:tower-decidable} + + At each finite primorial level, every instance of all three conjectures + + is decidable: Goldbach($n$) for specific~$n$, $\pi_2(N)$ for specific~$N$, + + and ABC$(a,b)$ for specific pairs. + + + \textbf{Lean:} \texttt{tower\_decidable\_check}.\quad + + \textbf{Registry:} III.D111. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 283 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 423-430 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `tower_decidable_check` + + + ## Dependencies + + + - Canonical: III.D112, III.D113 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At each finite primorial level, all three conjectures are decidable. Gap types, forbidden moves, and scope labels + all defined. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001455.md b/_corpus_v3_items/cid001455.md new file mode 100644 index 000000000..5da508904 --- /dev/null +++ b/_corpus_v3_items/cid001455.md @@ -0,0 +1,162 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001455/ +title: Gap Type +cid: cid001455 +primary_alias: DEF0225 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic + gaps.' +aliases: +- DEF0225 +- III.D112 +- gap-type +- def:gap-type +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005731 +- predicate: formalized_by + object: cid005733 +- predicate: formalized_by + object: cid005734 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Gap Type** + + + Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic gaps. + + + # Gap Type + + + ## Summary + + + Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic gaps. + + + ## Statement + + + ```latex + + \label{def:gap-type} + + The three proof-theoretic gaps: + + \begin{enumerate} + + \item \textbf{Parity gap} (Goldbach): the parity barrier blocks any + + sieve-based approach from proving Goldbach for all~$n$. + + \item \textbf{Density gap} (Twin primes): admissible residue classes are + + nonempty (algebraic), but infinitude requires equidistribution (analytic). + + \item \textbf{Structural gap} (ABC): the primorial tower avoids the hard + + case entirely; genuine difficulty is in perfect-power parts. + + \end{enumerate} + + + \textbf{Lean:} \texttt{GapType} in \texttt{ConjectureGaps.lean}.\quad + + \textbf{Registry:} III.D112. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 284 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 398-411 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `GapType` + + + ## Dependencies + + + - Canonical: III.P44, III.P46, III.P48 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Three gap types: parity (Goldbach), density (twin primes), structural (ABC). Complete taxonomy of proof-theoretic + gaps.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001456.md b/_corpus_v3_items/cid001456.md new file mode 100644 index 000000000..d5212f0b1 --- /dev/null +++ b/_corpus_v3_items/cid001456.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001456/ +title: Forbidden Move Mapping +cid: cid001456 +primary_alias: DEF0226 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: All three gaps map to exponential_quantification (K4 violation). Finite→infinite passage requires exponential quantification. +aliases: +- DEF0226 +- III.D113 +- forbidden-move-mapping +- def:gap-forbidden-move +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005732 +- predicate: formalized_by + object: cid005735 +- predicate: formalized_by + object: cid005736 +- predicate: formalized_by + object: cid005737 +- predicate: formalized_by + object: cid005738 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Forbidden Move Mapping** + + + All three gaps map to exponential_quantification (K4 violation). Finite→infinite passage requires exponential quantification. + + + # Forbidden Move Mapping + + + ## Summary + + + All three gaps map to exponential_quantification (K4 violation). Finite→infinite passage requires exponential quantification. + + + ## Statement + + + ```latex + + \label{def:gap-forbidden-move} + + All three gaps map to the same forbidden move: + + \texttt{exponential\_quantification} (K4 violation). The passage from + + finite verification to infinite proof requires quantifying over + + exponentially many objects. + + + \textbf{Lean:} \texttt{gap\_forbidden\_move}.\quad + + \textbf{Registry:} III.D113. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 285 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 413-421 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `gap_forbidden_move` + + + ## Dependencies + + + - Canonical: III.D112, III.D69 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: All three gaps map to exponential_quantification (K4 violation). Finite→infinite passage requires exponential + quantification. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001457.md b/_corpus_v3_items/cid001457.md new file mode 100644 index 000000000..9e03c4370 --- /dev/null +++ b/_corpus_v3_items/cid001457.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001457/ +title: Boundary-to-Interior Functor +cid: cid001457 +primary_alias: DEF0227 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The functor Φ: Char(L) → O(τ³) mapping boundary characters to interior holomorphic functions. This is Langlands₀: + boundary functoriality. Takes a character on L and produces a holomorphic extension into the fibered product τ³.' +aliases: +- DEF0227 +- III.D12 +- boundary-to-interior-functor +- def:boundary-to-interior-functor +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005964 +- predicate: formalized_by + object: cid005965 +- predicate: formalized_by + object: cid005968 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary-to-Interior Functor**\n\nThe functor Φ: Char(L) → O(τ³) mapping boundary characters to interior holomorphic\ + \ functions. This is Langlands₀: boundary functoriality. Takes a character on L and produces a holomorphic extension into\ + \ the fibered product τ³.\n\n# Boundary-to-Interior Functor\n\n## Summary\n\nThe functor Φ: Char(L) → O(τ³) mapping boundary\ + \ characters to interior holomorphic functions. This is Langlands₀: boundary functoriality. Takes a character on L and\ + \ produces a holomorphic extension into the fibered product τ³.\n\n## Statement\n\n```latex\n%\n\\label{def:boundary-to-interior-functor}\n\ + The \\emph{boundary-to-interior functor}\nis the functor\n\\begin{equation}\\label{eq:ch09-phi-functor}\n \\Phi \\\ + ;:\\; \\Char(\\Lemniscate) \\;\\longrightarrow\\; \\mathcal{O}(\\tau^3)\n\\end{equation}\ndefined as follows.\n\\begin{enumerate}\n\ + \ \\item \\textbf{On objects}:\n For each character $\\chi_{(m,n)} \\in \\Char(\\Lemniscate)$,\n \ + \ the functor produces\n the holomorphic function\n $\\Phi(\\chi_{(m,n)}) \\in \\mathcal{O}(\\tau^3)$\n\ + \ whose stage-$k$ representative is\n \\begin{equation}\\label{eq:ch09-phi-stages}\n \\\ + Phi(\\chi_{(m,n)})_k\n \\;=\\;\n \\chi_{(m,n)} \\big|_{\\Char_k(\\Lemniscate)},\n \\\ + end{equation}\n the restriction of the character\n to the $k$-th primorial level.\n Tower coherence\ + \ is automatic:\n $\\chi_{(m,n)}\\big|_k$\n reduces to $\\chi_{(m,n)}\\big|_\\ell$\n under\ + \ the canonical projection\n $\\Z / M_k\\Z \\to \\Z / M_\\ell\\Z$\n for $\\ell \\leq k$.\n \\item\ + \ \\textbf{On morphisms}:\n A lattice translation\n $\\chi_{(a,b)} : \\chi_{(m,n)} \\to \\chi_{(m+a,\\\ + , n+b)}$\n maps to the holomorphic map\n $\\Phi(\\chi_{(a,b)})$\n that multiplies\n \ + \ each stage representative\n by the corresponding\n stage-$k$ value of~$\\chi_{(a,b)}$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-03.jsonl` line 24\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch09-boundary-functoriality.tex`\ + \ lines 288-327\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.BoundaryCharacters`\n\ + - Name: `boundary_to_interior`\n\n## Dependencies\n\n- Canonical: III.D11, II.T49\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The functor Φ: Char(L) → O(τ³) mapping boundary characters to interior holomorphic functions. This is Langlands₀: + boundary functoriality. Takes a character on L and produces a holomorphic extension into the fibered product τ³.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001458.md b/_corpus_v3_items/cid001458.md new file mode 100644 index 000000000..c0e3755f0 --- /dev/null +++ b/_corpus_v3_items/cid001458.md @@ -0,0 +1,93 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001458/ +title: 4+1 Sector Decomposition +cid: cid001458 +primary_alias: DEF0228 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Five generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one coupling sector (ω). Char(L) = D-sector + ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated. +aliases: +- DEF0228 +- III.D13 +- 4-1-sector-decomposition +- def:four-plus-one-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005969 +- predicate: formalized_by + object: cid005972 +- predicate: formalized_by + object: cid005973 +- predicate: formalized_by + object: cid005974 +- predicate: formalized_by + object: cid005975 +- predicate: formalized_by + object: cid005976 +- predicate: formalized_by + object: cid005977 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**4+1 Sector Decomposition**\n\nFive generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one coupling\ + \ sector (ω). Char(L) = D-sector ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated.\n\ + \n# 4+1 Sector Decomposition\n\n## Summary\n\nFive generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one\ + \ coupling sector (ω). Char(L) = D-sector ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:four-plus-one-decomposition}\nThe \\emph{$4{+}1$ sector decomposition}\nof\ + \ the boundary character space is the partition\n\\[\n \\Char(\\Lemniscate) \\setminus \\{0\\}\n \\;=\\;\n \\\ + underbrace{\n \\Sector{D}\n \\;\\sqcup\\;\n \\Sector{A}\n \\;\\sqcup\\;\n \\Sector{B}\n\ + \ \\;\\sqcup\\;\n \\Sector{C}\n }_{\\text{four primitive sectors}}\n \\;\\sqcup\\;\n \\underbrace{\n\ + \ \\Sector{\\omega}\n }_{\\text{one coupling sector}}\\,.\n\\]\nThe four primitive sectors are generated\nby\ + \ the four primitive generators\n$\\alpha$, $\\pi$, $\\gamma$, $\\eta$\nvia their dominant ABCD coordinates.\nThe coupling\ + \ sector is generated\nby the crossing generator~$\\omega$\nvia its simultaneous $B$-$C$ mixing.\nThe count is $4 + 1$,\ + \ not~$5$:\nthe $\\omega$-sector is structurally distinct\nfrom the primitive sectors.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 26\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch10-the-four-plus-one-decomposition.tex`\ + \ lines 475-507\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.Decomposition`\n\ + - Name: `Sector`\n\n## Dependencies\n\n- Canonical: III.T05, III.D11, III.D14\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Five generators yield four primitive sectors (α→D, π→A, γ→B, η→C) plus one coupling sector (ω). Char(L) = D-sector + ⊔ A-sector ⊔ B-sector ⊔ C-sector ⊔ ω-sector. Induced by Langlands₀, not postulated. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001459.md b/_corpus_v3_items/cid001459.md new file mode 100644 index 000000000..3ffb3d566 --- /dev/null +++ b/_corpus_v3_items/cid001459.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001459/ +title: ω-Coupling Sector +cid: cid001459 +primary_alias: DEF0229 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The fifth generator ω does not produce an independent sector. Instead, it mediates coupling: characters with non-trivial + projection in both B and C lobes simultaneously. The structural reason for 4+1, not 5.' +aliases: +- DEF0229 +- III.D14 +- coupling-sector +- def:omega-coupling-sector +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005970 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**ω-Coupling Sector**\n\nThe fifth generator ω does not produce an independent sector. Instead, it mediates coupling:\ + \ characters with non-trivial projection in both B and C lobes simultaneously. The structural reason for 4+1, not 5.\n\ + \n# ω-Coupling Sector\n\n## Summary\n\nThe fifth generator ω does not produce an independent sector. Instead, it mediates\ + \ coupling: characters with non-trivial projection in both B and C lobes simultaneously. The structural reason for 4+1,\ + \ not 5.\n\n## Statement\n\n```latex\n%\n\\label{def:omega-coupling-sector}\nThe \\emph{$\\omega$-coupling sector} is\n\ + \\begin{equation}\\label{eq:ch10-omega-sector}\n \\Sector{\\omega} \\;=\\;\n \\bigl\\{\\,\n \\chi \\in \\\ + Char(\\Lemniscate)\n \\;\\big|\\;\n P_B(\\chi) \\neq 0\n \\;\\text{and}\\;\n P_C(\\chi) \\\ + neq 0\n \\,\\bigr\\}.\n\\end{equation}\nIt consists of all characters\nwhose projections onto both\nthe $B$-lobe ($\\\ + gamma$-stratum) and\nthe $C$-lobe ($\\eta$-stratum)\nare simultaneously non-trivial.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 27\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch10-the-four-plus-one-decomposition.tex`\ + \ lines 363-382\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.Decomposition`\n\ + - Name: `omega_coupling_5`\n\n## Dependencies\n\n- Canonical: III.D11\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The fifth generator ω does not produce an independent sector. Instead, it mediates coupling: characters with + non-trivial projection in both B and C lobes simultaneously. The structural reason for 4+1, not 5.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001460.md b/_corpus_v3_items/cid001460.md new file mode 100644 index 000000000..a7cd4e0c9 --- /dev/null +++ b/_corpus_v3_items/cid001460.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001460/ +title: Langlands₁ Reflection Bridge +cid: cid001460 +primary_alias: DEF0230 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Yoneda-Langlands Reflection Bridge: the enrichment functor F_E restricted to boundary characters produces a + correspondence between E₀-level sectors (generators + ω-coupling) and E₁-level sectors (forces + mass coupling). Template + invariant; carrier changes.' +aliases: +- DEF0230 +- III.D15 +- langlands-reflection-bridge +- def:langlands-one +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005978 +- predicate: formalized_by + object: cid005983 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 11 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Langlands₁ Reflection Bridge**\n\nThe Yoneda-Langlands Reflection Bridge: the enrichment functor F_E restricted\ + \ to boundary characters produces a correspondence between E₀-level sectors (generators + ω-coupling) and E₁-level sectors\ + \ (forces + mass coupling). Template invariant; carrier changes.\n\n# Langlands₁ Reflection Bridge\n\n## Summary\n\nThe\ + \ Yoneda-Langlands Reflection Bridge: the enrichment functor F_E restricted to boundary characters produces a correspondence\ + \ between E₀-level sectors (generators + ω-coupling) and E₁-level sectors (forces + mass coupling). Template invariant;\ + \ carrier changes.\n\n## Statement\n\n```latex\n%\n\\label{def:langlands-one}\nThe \\textbf{Langlands$_1$ reflection bridge}\ + \ is the natural transformation\n\\begin{equation}\\label{eq:ch11-langlands-one}\n \\Lambda_1\n \\;:\\;\n \\\ + mathrm{Char}(\\Lemniscate) \\;\\Longrightarrow\\;\n \\mathrm{Char}_1(\\Lemniscate)\n\\end{equation}\ndefined as the\ + \ restriction of the enrichment functor~$\\mathcal{F}_E$\nto the boundary character space,\ncomposed with the Yoneda embedding\n\ + $\\mathsf{y} : \\tau \\hookrightarrow [\\tau^{\\mathrm{op}}, \\tau]$\n(Book~II, II.T36):\n\\begin{equation}\\label{eq:ch11-lambda-one-construction}\n\ + \ \\Lambda_1(\\chi)\n \\;=\\;\n \\mathcal{F}_E \\circ \\mathsf{y} \\circ \\chi.\n\\end{equation}\nHere $\\chi\ + \ : \\pi_1(\\Lemniscate) \\to \\Obj(\\tau)$ is a boundary character,\n$\\mathsf{y} \\circ \\chi$\nis its representable\ + \ presheaf lift,\nand $\\mathcal{F}_E$ applies the enrichment functor.\nThe result $\\Lambda_1(\\chi)$\nis an $E_1$-level\ + \ character:\na functor from $\\pi_1(\\Lemniscate)$ into $H_\\tau$-enriched objects.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 29\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch11-the-yoneda-langlands-reflection.tex`\ + \ lines 110-137\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.LanglandsReflection`\n\ + - Name: `langlands_reflection_5_3`\n\n## Dependencies\n\n- Canonical: III.D04, III.D12, III.D13\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Yoneda-Langlands Reflection Bridge: the enrichment functor F_E restricted to boundary characters produces + a correspondence between E₀-level sectors (generators + ω-coupling) and E₁-level sectors (forces + mass coupling). Template + invari' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001461.md b/_corpus_v3_items/cid001461.md new file mode 100644 index 000000000..db6ccadd1 --- /dev/null +++ b/_corpus_v3_items/cid001461.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001461/ +title: Universal Operator +cid: cid001461 +primary_alias: DEF0231 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The universal operator H_∞ on L²(Char(L)). At τ-effective cutoff N: H_{≤N} = P_{≤N} ⊙ H_∞ ⊙ P_{≤N}. All L-functions + unified as spectral determinants: L_{≤N}(s,π) = det(I − s⁻¹ H_{π,≤N}). Riemann ζ, Dirichlet L, etc. as special cases.' +aliases: +- DEF0231 +- III.D16 +- universal-operator +- def:universal-operator +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005980 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 11 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Universal Operator**\n\nThe universal operator H_∞ on L²(Char(L)). At τ-effective cutoff N: H_{≤N} = P_{≤N}\ + \ ⊙ H_∞ ⊙ P_{≤N}. All L-functions unified as spectral determinants: L_{≤N}(s,π) = det(I − s⁻¹ H_{π,≤N}). Riemann ζ, Dirichlet\ + \ L, etc. as special cases.\n\n# Universal Operator\n\n## Summary\n\nThe universal operator H_∞ on L²(Char(L)). At τ-effective\ + \ cutoff N: H_{≤N} = P_{≤N} ⊙ H_∞ ⊙ P_{≤N}. All L-functions unified as spectral determinants: L_{≤N}(s,π) = det(I − s⁻¹\ + \ H_{π,≤N}). Riemann ζ, Dirichlet L, etc. as special cases.\n\n## Statement\n\n```latex\n%\n\\label{def:universal-operator}\n\ + The \\textbf{universal operator}\n$H_\\infty : \\mathcal{H} \\to \\mathcal{H}$\nis the unbounded self-adjoint operator\n\ + on $L^2(\\mathrm{Char}(\\Lemniscate))$\ndefined on the character basis by:\n\\begin{equation}\\label{eq:ch11-h-infty-action}\n\ + \ H_\\infty \\, \\chi_{(m,n)}\n \\;=\\;\n \\sum_{(m', n') \\in \\mathbb{Z}^2}\n h_{(m,n),(m',n')} \\;\\chi_{(m',n')},\n\ + \\end{equation}\nwhere the matrix elements are determined\nby the primorial tower:\n\\begin{equation}\\label{eq:ch11-matrix-elements}\n\ + \ h_{(m,n),(m',n')}\n \\;=\\;\n \\lim_{d \\to \\infty}\\;\n \\frac{1}{M_d}\n \\sum_{t \\in (\\mathbb{Z}/M_d\\\ + mathbb{Z})^\\times}\n \\chi_{(m,n)}(t) \\;\\overline{\\chi_{(m',n')}(t)}\\;\n \\rho_d(t),\n\\end{equation}\nwith\ + \ $\\rho_d(t)$ the progression operator at depth~$d$\nand $M_d = p_1 \\cdots p_d$ the $d$-th primorial.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 31\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch11-the-yoneda-langlands-reflection.tex`\ + \ lines 497-524\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.LanglandsReflection`\n\ + - Name: `universal_operator`\n\n## Dependencies\n\n- Canonical: III.D11\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The universal operator H_∞ on L²(Char(L)). At τ-effective cutoff N: H_{≤N} = P_{≤N} ⊙ H_∞ ⊙ P_{≤N}. All L-functions + unified as spectral determinants: L_{≤N}(s,π) = det(I − s⁻¹ H_{π,≤N}). Riemann ζ, Dirichlet L, etc. as special cases.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001462.md b/_corpus_v3_items/cid001462.md new file mode 100644 index 000000000..10ca30c65 --- /dev/null +++ b/_corpus_v3_items/cid001462.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001462/ +title: Spectral Polarity +cid: cid001462 +primary_alias: DEF0232 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖. D: χ₊-dominant, B: + χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.' +aliases: +- DEF0232 +- III.D17 +- spectral-polarity +- def:spectral-polarity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005981 +- predicate: formalized_by + object: cid005985 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 12 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Polarity**\n\nFor each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖.\ + \ D: χ₊-dominant, B: χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.\n\n# Spectral\ + \ Polarity\n\n## Summary\n\nFor each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖.\ + \ D: χ₊-dominant, B: χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:spectral-polarity}\nThe \\textbf{spectral polarity} of a primitive sector~$\\Sector{g}$\nis\ + \ the ratio\n\\begin{equation}\\label{eq:ch12-polarity-def}\n \\mathrm{pol}(\\Sector{g})\n \\;=\\;\n \\frac{\\\ + lVert P_{\\chi_+}(\\Sector{g}) \\rVert}%\n {\\lVert P_{\\chi_-}(\\Sector{g}) \\rVert},\n\\end{equation}\nwhere\ + \ $\\lVert \\cdot \\rVert$ denotes the rank\nof the projected module at each finite depth~$d$.\nA sector is called\n\\\ + begin{itemize}\n \\item \\textbf{$\\chi_+$-dominant} if $\\mathrm{pol}(\\Sector{g}) > 1$,\n \\item \\textbf{$\\\ + chi_-$-dominant} if $\\mathrm{pol}(\\Sector{g}) < 1$,\n \\item \\textbf{balanced} if $\\mathrm{pol}(\\Sector{g}) =\ + \ 1$.\n\\end{itemize}\nThe polarity is well defined because\nthe spectral decomposition is canonical\n(Book~I, I.T12)\n\ + and the sector projections are depth-independent\n(Chapter~\\ref{ch:four-plus-one-decomposition}).\n```\n\n## Proof /\ + \ Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 32\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch12-the-parity-bridge-theorem.tex`\ + \ lines 86-110\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.LanglandsReflection`\n\ + - Name: `spectral_polarity`\n\n## Dependencies\n\n- Canonical: III.D13\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For each primitive sector S_g, the spectral polarity pol(S_g) = ‖P_{χ₊}(S_g)‖/‖P_{χ₋}(S_g)‖. D: χ₊-dominant, + B: χ₊-dominant, C: χ₋-dominant, A: BALANCED (pol = 1). Only the A-sector has equal polarity.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001463.md b/_corpus_v3_items/cid001463.md new file mode 100644 index 000000000..104a48b67 --- /dev/null +++ b/_corpus_v3_items/cid001463.md @@ -0,0 +1,153 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001463/ +title: Coupling Ledger +cid: cid001463 +primary_alias: DEF0233 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The No Knobs Ledger: systematic inventory of all inter-sector couplings. 10 entries (4 self-couplings + 6 cross-couplings). + Each entry κ(S_i, S_j) = f_{ij}(ι_τ, M_d) is a specific rational function of ι_τ evaluated at a specific primorial depth.' +aliases: +- DEF0233 +- III.D18 +- coupling-ledger +- def:coupling-ledger +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005988 +- predicate: formalized_by + object: cid005993 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 13 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Coupling Ledger** + + + The No Knobs Ledger: systematic inventory of all inter-sector couplings. 10 entries (4 self-couplings + 6 cross-couplings). + Each entry κ(S_i, S_j) = f_{ij}(ι_τ, M_d) is a specific rational function of ι_τ evaluated at a specific primorial depth. + + + # Coupling Ledger + + + ## Summary + + + The No Knobs Ledger: systematic inventory of all inter-sector couplings. 10 entries (4 self-couplings + 6 cross-couplings). + Each entry κ(S_i, S_j) = f_{ij}(ι_τ, M_d) is a specific rational function of ι_τ evaluated at a specific primorial depth. + + + ## Statement + + + ```latex + + % + + \label{def:coupling-ledger} + + The \textbf{No Knobs Ledger} + + is the complete catalog + + of the $10$ sector coupling functions. + + We organise them in a symmetric table, + + indexed by the ABCD sector labels + + from Chapter~\ref{ch:four-plus-one-decomposition}. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 36 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch13-the-no-knobs-principle.tex` + lines 193-202 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Sectors.ParityBridge` + + - Name: `coupling_ledger` + + + ## Dependencies + + + - Canonical: III.D13, III.T08 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The No Knobs Ledger: systematic inventory of all inter-sector couplings. 10 entries (4 self-couplings + 6 cross-couplings). + Each entry κ(S_i, S_j) = f_{ij}(ι_τ, M_d) is a specific rational function of ι_τ evaluated at a specific primorial d' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001464.md b/_corpus_v3_items/cid001464.md new file mode 100644 index 000000000..c9dc3de57 --- /dev/null +++ b/_corpus_v3_items/cid001464.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001464/ +title: Primorial Ladder +cid: cid001464 +primary_alias: DEF0234 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Primorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ ← ℤ/Prim(2)ℤ ← …. Inverse limit = profinite + completion Ẑ_τ. Connection to Book I's primorial presheaf (I.D83). Canonical cofinal filtration. +aliases: +- DEF0234 +- III.D19 +- primorial-ladder +- def:primorial-ladder +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006056 +- predicate: formalized_by + object: cid006057 +- predicate: formalized_by + object: cid006058 +- predicate: formalized_by + object: cid006061 +- predicate: formalized_by + object: cid006062 +- predicate: formalized_by + object: cid006063 +- predicate: formalized_by + object: cid006065 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 14 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Primorial Ladder**\n\nPrimorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ ← ℤ/Prim(2)ℤ\ + \ ← …. Inverse limit = profinite completion Ẑ_τ. Connection to Book I's primorial presheaf (I.D83). Canonical cofinal\ + \ filtration.\n\n# Primorial Ladder\n\n## Summary\n\nPrimorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ\ + \ ← ℤ/Prim(2)ℤ ← …. Inverse limit = profinite completion Ẑ_τ. Connection to Book I's primorial presheaf (I.D83). Canonical\ + \ cofinal filtration.\n\n## Statement\n\n```latex\n%\n\\label{def:primorial-ladder}\nLet $(p_n)_{n \\geq 1}$ enumerate\ + \ the primes\nof $\\tau$-Idx in order.\nThe $k$-th \\textbf{primorial} is\n\\[\n M_k \\;:=\\; \\prod_{n=1}^{k} p_n.\n\ + \\]\nFor $k \\leq \\ell$, the \\textbf{reduction map}\n$\\pi_{\\ell \\to k} :\n\\mathbb{Z}/M_\\ell\\mathbb{Z} \\to \\\ + mathbb{Z}/M_k\\mathbb{Z}$\nis reduction modulo $M_k$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No\ + \ manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 38\n- Manuscript source:\ + \ `2nd-edition/book-i-categorical-foundations/02_mainmatter/part07/ch28-omega-germs.tex` lines 81-93\n\n## Lean / Formalization\ + \ Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.PrimorialLadder`\n- Name: `PrimorialStage`\n\ + \n## Dependencies\n\n- Canonical: I.D83\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Primorial numbers Prim(k) = p₁·p₂·…·pₖ form an inverse system ℤ/Prim(1)ℤ ← ℤ/Prim(2)ℤ ← …. Inverse limit = profinite + completion Ẑ_τ. Connection to Book I's primorial presheaf (I.D83). Canonical cofinal filtration. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001465.md b/_corpus_v3_items/cid001465.md new file mode 100644 index 000000000..8e767501e --- /dev/null +++ b/_corpus_v3_items/cid001465.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001465/ +title: Reconstruction Functor +cid: cid001465 +primary_alias: DEF0235 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: CRT defines a functor R_k from ∏(ℤ/pᵢℤ-Mod) to ℤ/Prim(k)ℤ-Mod. This functor is an equivalence (inverse = restriction + S_k). All Part IV-VI arguments decompose through it. +aliases: +- DEF0235 +- III.D20 +- reconstruction-functor +- def:reconstruction-functor +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006019 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Reconstruction Functor**\n\nCRT defines a functor R_k from ∏(ℤ/pᵢℤ-Mod) to ℤ/Prim(k)ℤ-Mod. This functor is\ + \ an equivalence (inverse = restriction S_k). All Part IV-VI arguments decompose through it.\n\n# Reconstruction Functor\n\ + \n## Summary\n\nCRT defines a functor R_k from ∏(ℤ/pᵢℤ-Mod) to ℤ/Prim(k)ℤ-Mod. This functor is an equivalence (inverse\ + \ = restriction S_k). All Part IV-VI arguments decompose through it.\n\n## Statement\n\n```latex\n%\n\\label{def:reconstruction-functor}\n\ + The \\textbf{Reconstruction Functor}\nat primorial depth~$k$ is\n\\begin{equation}\\label{eq:ch15-recon-functor}\n \ + \ \\mathcal{R}_k\n \\;:\\;\n \\prod_{i=1}^{k} (R_i\\text{-Mod})\n \\;\\longrightarrow\\;\n R\\text{-Mod},\n\ + \ \\qquad\n \\mathcal{R}_k(M_1, \\ldots, M_k)\n \\;=\\;\n \\bigoplus_{i=1}^{k} M_i,\n\\end{equation}\nwhere\ + \ $r \\in R$ acts on $M_i$\nvia $r_i = r \\bmod p_i$.\nOn morphisms: $\\mathcal{R}_k(f_1, \\ldots, f_k)\n= f_1 \\oplus\ + \ \\cdots \\oplus f_k$.\nThe \\textbf{Restriction Functor}\n$\\mathcal{S}_k : R\\text{-Mod}\n\\to \\prod (R_i\\text{-Mod})$\n\ + maps $M \\mapsto (e_1 \\cdot M, \\ldots, e_k \\cdot M)$.\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 42\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch15-the-crt-decomposition-theorem.tex` lines 520-544\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.CRT`\n- Name: `reconstruction_functor_check`\n\ + \n## Dependencies\n\n- Canonical: III.T10\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: CRT defines a functor R_k from ∏(ℤ/pᵢℤ-Mod) to ℤ/Prim(k)ℤ-Mod. This functor is an equivalence (inverse = restriction + S_k). All Part IV-VI arguments decompose through it. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001466.md b/_corpus_v3_items/cid001466.md new file mode 100644 index 000000000..11042c10c --- /dev/null +++ b/_corpus_v3_items/cid001466.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001466/ +title: τ-Native Local Field +cid: cid001466 +primary_alias: DEF0236 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: ℤ_p^τ = lim← ℤ/p^n ℤ as inverse limit within τ. The p-adic integers are a τ-object with NF address. p-adic valuation + v_p = D-coordinate restricted to p-primary component. +aliases: +- DEF0236 +- III.D21 +- native-local-field +- def:tau-native-local-field +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006044 +- predicate: formalized_by + object: cid006045 +- predicate: formalized_by + object: cid006048 +- predicate: formalized_by + object: cid006049 +- predicate: formalized_by + object: cid006050 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Native Local Field**\n\nℤ_p^τ = lim← ℤ/p^n ℤ as inverse limit within τ. The p-adic integers are a τ-object\ + \ with NF address. p-adic valuation v_p = D-coordinate restricted to p-primary component.\n\n# τ-Native Local Field\n\n\ + ## Summary\n\nℤ_p^τ = lim← ℤ/p^n ℤ as inverse limit within τ. The p-adic integers are a τ-object with NF address. p-adic\ + \ valuation v_p = D-coordinate restricted to p-primary component.\n\n## Statement\n\n```latex\n%\n\\label{def:tau-native-local-field}\n\ + The \\textbf{$\\tau$-native $p$-adic integers} are the inverse limit\n\\begin{equation}\\label{eq:ch16-p-adic-integers}\n\ + \ \\Z_p^\\tau\n \\;=\\;\n \\varprojlim_{n} \\Z / p^n \\Z\n \\;=\\;\n \\Bigl\\{\n (a_1, a_2, a_3,\ + \ \\ldots)\n \\;\\Big|\\;\n a_n \\in \\Z / p^n \\Z,\\;\n \\pi_{n+1,n}(a_{n+1}) = a_n\n \\\ + ;\\text{for all } n \\geq 1\n \\Bigr\\}.\n\\end{equation}\nAn element of $\\Z_p^\\tau$ is called a \\textbf{$p$-adic\ + \ $\\tau$-integer}.\nThe ring operations are defined componentwise:\n\\begin{align}\n (a_n)_n + (b_n)_n &\\;=\\; (a_n\ + \ + b_n \\bmod p^n)_n,\n \\label{eq:ch16-p-adic-add} \\\\\n (a_n)_n \\cdot (b_n)_n &\\;=\\; (a_n \\cdot b_n \\bmod\ + \ p^n)_n.\n \\label{eq:ch16-p-adic-mult}\n\\end{align}\nThese are well defined because the reduction maps\nare ring\ + \ homomorphisms:\nif $a_{n+1} \\equiv a_n$ and $b_{n+1} \\equiv b_n$ modulo~$p^n$,\nthen $(a_{n+1} + b_{n+1}) \\equiv\ + \ (a_n + b_n)$\nand $(a_{n+1} \\cdot b_{n+1}) \\equiv (a_n \\cdot b_n)$ modulo~$p^n$.\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 45\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch16-hensel-lifting-and-local-fields.tex`\ + \ lines 346-376\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.LocalFields`\n\ + - Name: `LocalFieldElt`\n\n## Dependencies\n\n- Canonical: III.T11, III.D19\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: ℤ_p^τ = lim← ℤ/p^n ℤ as inverse limit within τ. The p-adic integers are a τ-object with NF address. p-adic valuation + v_p = D-coordinate restricted to p-primary component. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001467.md b/_corpus_v3_items/cid001467.md new file mode 100644 index 000000000..487fa6533 --- /dev/null +++ b/_corpus_v3_items/cid001467.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001467/ +title: τ-Adele Ring +cid: cid001467 +primary_alias: DEF0237 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: τ-adele ring 𝔸_τ = restricted product of local fields ℤ_p^τ with respect to unit groups. Almost all components integral. + τ-internal construction, not imported from algebraic number theory. +aliases: +- DEF0237 +- III.D22 +- adele-ring +- def:tau-adele-ring +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006000 +- predicate: formalized_by + object: cid006005 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Adele Ring**\n\nτ-adele ring \U0001D538_τ = restricted product of local fields ℤ_p^τ with respect to unit\ + \ groups. Almost all components integral. τ-internal construction, not imported from algebraic number theory.\n\n# τ-Adele\ + \ Ring\n\n## Summary\n\nτ-adele ring \U0001D538_τ = restricted product of local fields ℤ_p^τ with respect to unit groups.\ + \ Almost all components integral. τ-internal construction, not imported from algebraic number theory.\n\n## Statement\n\ + \n```latex\n%\n\\label{def:tau-adele-ring}\nThe \\textbf{$\\tau$-adele ring} is the restricted product\n\\begin{equation}\\\ + label{eq:ch17-adele-ring}\n \\mathbb{A}_\\tau\n \\;=\\;\n {\\prod_{p}}' \\;\\Z_p^\\tau\n \\;=\\;\n \\Bigl\\\ + {\\,\n (x_p)_p \\in \\prod_{p} \\Z_p^\\tau\n \\;\\Big|\\;\n x_p \\in (\\Z_p^\\tau)^\\times\n \ + \ \\text{ for almost all } p\n \\,\\Bigr\\},\n\\end{equation}\nequipped with componentwise addition and multiplication.\n\ + The restricted-product notation\n$\\prod'_p$\nindicates that the product is taken\nwith respect to the compact open subgroups\n\ + $(\\Z_p^\\tau)^\\times \\subset \\Z_p^\\tau$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 47\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch17-the-adelic-embedding.tex`\ + \ lines 142-164\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Adeles`\n\ + - Name: `AdeleElement`\n\n## Dependencies\n\n- Canonical: III.D21, III.T10\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: τ-adele ring 𝔸_τ = restricted product of local fields ℤ_p^τ with respect to unit groups. Almost all components + integral. τ-internal construction, not imported from algebraic number theory. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001468.md b/_corpus_v3_items/cid001468.md new file mode 100644 index 000000000..23bb91fba --- /dev/null +++ b/_corpus_v3_items/cid001468.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001468/ +title: Internal Bipolar Classifier +cid: cid001468 +primary_alias: DEF0238 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Label_n: computable classifier mapping primes ≤ p_n to {B, C, X}. B-type = exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, + X-type = balanced. Replaces informal lobe language with computable predicates.' +aliases: +- DEF0238 +- III.D23 +- internal-bipolar-classifier +- def:internal-bipolar-classifier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006008 +- predicate: formalized_by + object: cid006013 +- predicate: formalized_by + object: cid006014 +- predicate: formalized_by + object: cid006015 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 18 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Internal Bipolar Classifier**\n\nLabel_n: computable classifier mapping primes ≤ p_n to {B, C, X}. B-type =\ + \ exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, X-type = balanced. Replaces informal lobe language with computable\ + \ predicates.\n\n# Internal Bipolar Classifier\n\n## Summary\n\nLabel_n: computable classifier mapping primes ≤ p_n to\ + \ {B, C, X}. B-type = exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, X-type = balanced. Replaces informal lobe\ + \ language with computable predicates.\n\n## Statement\n\n```latex\n%\n\\label{def:internal-bipolar-classifier}\nFor each\ + \ primorial depth~$n \\geq 1$,\nthe \\textbf{internal bipolar classifier} is the function\n\\begin{equation}\\label{eq:ch18-label-def}\n\ + \ \\mathrm{Label}_n \\;:\\;\n \\{p_1, \\ldots, p_n\\}\n \\;\\longrightarrow\\;\n \\{B, C, X\\},\n\\end{equation}\n\ + defined by the Legendre symbol at the individual prime:\n\\begin{equation}\\label{eq:ch18-label-cases}\n \\mathrm{Label}_n(p_i)\ + \ \\;=\\;\n \\begin{cases}\n B & \\text{if } p_i > 2 \\text{ and }\n \\bigl(\\frac{2}{p_i}\\bigr)\ + \ = +1,\n \\\\[4pt]\n C & \\text{if } p_i > 2 \\text{ and }\n \\bigl(\\frac{2}{p_i}\\bigr)\ + \ = -1,\n \\\\[4pt]\n X & \\text{if } p_i = 2.\n \\end{cases}\n\\end{equation}\nThe label~$B$ is\ + \ read as\n``exponent-stratum dominant'' ($\\chi_+$-dominant);\n$C$ as ``tetration-stratum dominant'' ($\\chi_-$-dominant);\n\ + and $X$ as ``balanced'' or ``mixed.''\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 50\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch18-the-internal-bipolar-classifier.tex`\ + \ lines 194-222\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.BipolarClassifier`\n\ + - Name: `PrimeLabel`\n\n## Dependencies\n\n- Canonical: III.T10\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Label_n: computable classifier mapping primes ≤ p_n to {B, C, X}. B-type = exponent/χ₊-dominant, C-type = tetration/χ₋-dominant, + X-type = balanced. Replaces informal lobe language with computable predicates.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001469.md b/_corpus_v3_items/cid001469.md new file mode 100644 index 000000000..6c720da59 --- /dev/null +++ b/_corpus_v3_items/cid001469.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001469/ +title: Boundary Normal Form +cid: cid001469 +primary_alias: DEF0239 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every element of ℤ/Prim(k)ℤ[j] has unique normal form a·e₊ + b·e₋ where a is B-supported and b is C-supported. Computable, + respects ring operations. +aliases: +- DEF0239 +- III.D24 +- boundary-normal-form +- def:boundary-normal-form +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006080 +- predicate: formalized_by + object: cid006081 +- predicate: formalized_by + object: cid006085 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 19 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Boundary Normal Form**\n\nEvery element of ℤ/Prim(k)ℤ[j] has unique normal form a·e₊ + b·e₋ where a is B-supported\ + \ and b is C-supported. Computable, respects ring operations.\n\n# Boundary Normal Form\n\n## Summary\n\nEvery element\ + \ of ℤ/Prim(k)ℤ[j] has unique normal form a·e₊ + b·e₋ where a is B-supported and b is C-supported. Computable, respects\ + \ ring operations.\n\n## Statement\n\n```latex\n%\n\\label{def:boundary-normal-form}\nLet $z = x + y\\jj \\in R_k^{\\\ + jj}$.\nThe \\textbf{boundary normal form} of~$z$ is\n\\begin{equation}\\label{eq:ch19-bnf}\n z \\;=\\; a \\cdot e_+\ + \ \\;+\\; b \\cdot e_-,\n \\qquad\n a = x + y,\\quad b = x - y,\n\\end{equation}\ncomputed in $\\Z / \\mathrm{Prim}(k)\\\ + Z$.\nThe coefficient $a$ is the \\textbf{$B$-sector coordinate}\nand $b$ is the \\textbf{$C$-sector coordinate}.\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-03.jsonl` line 54\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch19-the-spectral-trichotomy.tex`\ + \ lines 262-275\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Trichotomy`\n\ + - Name: `boundary_normal_form`\n\n## Dependencies\n\n- Canonical: III.T14\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every element of ℤ/Prim(k)ℤ[j] has unique normal form a·e₊ + b·e₋ where a is B-supported and b is C-supported. + Computable, respects ring operations. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001470.md b/_corpus_v3_items/cid001470.md new file mode 100644 index 000000000..40f0c695f --- /dev/null +++ b/_corpus_v3_items/cid001470.md @@ -0,0 +1,164 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001470/ +title: Mutual Determination Schema +cid: cid001470 +primary_alias: DEF0240 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior + (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for all + millennium problems.' +aliases: +- DEF0240 +- III.D25 +- mutual-determination-schema +- def:mutual-determination-schema +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005832 +- predicate: formalized_by + object: cid005833 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 21 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Mutual Determination Schema** + + + The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior + (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for all + millennium problems. + + + # Mutual Determination Schema + + + ## Summary + + + The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior + (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for all + millennium problems. + + + ## Statement + + + ```latex + + \label{def:mutual-determination-schema} + + Let $B$ denote \emph{boundary data}, $I$ denote \emph{interior structure}, and $S$ denote \emph{spectral invariants}. + The \textbf{Mutual Determination Schema} is the diagram + + \[ + + B \longleftrightarrow I \longleftrightarrow S, + + \] + + asserting three equivalences: + + \begin{enumerate} + + \item \textbf{Boundary $\to$ Interior:} Specifying $B$ uniquely determines $I$ (analogue of Global Hartogs). + + \item \textbf{Interior $\to$ Spectral:} Specifying $I$ uniquely determines $S$ (analogue of spectral decomposition). + + \item \textbf{Closure:} The composite $B \leftrightarrow S$ closes the loop: $B$ and $S$ are dual perspectives on the + same object. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 59 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch21-the-coherence-programme.tex` + lines 31-42 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.MutualDetermination` + + - Name: `MDDescription` + + + ## Dependencies + + + - Canonical: III.D03, III.R02, III.R10 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The Master Schema formalized: B (boundary) ↔ I (interior) ↔ S (spectral invariants). Three equivalences: boundary→interior + (Global Hartogs), interior→spectral (spectral decomposition), closure B↔S (dual perspectives). Uniform template for a' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001471.md b/_corpus_v3_items/cid001471.md new file mode 100644 index 000000000..a4cc9d084 --- /dev/null +++ b/_corpus_v3_items/cid001471.md @@ -0,0 +1,159 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001471/ +title: Split-Complex Zeta ζ_τ +cid: cid001471 +primary_alias: DEF0241 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates + via bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes. +aliases: +- DEF0241 +- III.D26 +- split-complex-zeta +- def:split-complex-zeta +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005849 +- predicate: formalized_by + object: cid005850 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 22 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Split-Complex Zeta ζ_τ** + + + The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates via + bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes. + + + # Split-Complex Zeta ζ_τ + + + ## Summary + + + The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates via + bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes. + + + ## Statement + + + ```latex + + \label{def:split-complex-zeta} + + Let $s \in \mathbb{C}$ with $\mathrm{Re}(s) > 1$. The \textbf{split-complex zeta function} is + + \[ + + \zeta_{\T}(s) \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s), + + \] + + where + + \begin{align*} + + \zeta_B(s) &\;=\; \sum_{\substack{n \geq 1 \\ \mathrm{Label}_n = \mathrm{B}}} \frac{1}{n^s}, \\[6pt] + + \zeta_C(s) &\;=\; \sum_{\substack{n \geq 1 \\ \mathrm{Label}_n = \mathrm{C}}} \frac{1}{n^s}. + + \end{align*} + + Here $\mathrm{Label}_n : \mathbb{Z}_{>0} \to \{\mathrm{B}, \mathrm{C}, \mathrm{N}\}$ is the polar labeling function (Definition~III.D23). + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 61 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch22-the-functional-equation-in-h-tau.tex` + lines 69-80 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SplitComplexZeta` + + - Name: `split_zeta_b` + + + ## Dependencies + + + - Canonical: III.D23, III.T10, III.T14, I.D27 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The τ-native zeta function ζ_τ(s) = e₊·ζ_B(s) + e₋·ζ_C(s), encoding the Riemann zeta in split-complex coordinates + via bipolar idempotents. B-lobe = exponent-dominant primes, C-lobe = tetration-dominant primes. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001472.md b/_corpus_v3_items/cid001472.md new file mode 100644 index 000000000..282a2992f --- /dev/null +++ b/_corpus_v3_items/cid001472.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001472/ +title: Functional Equation Involution J +cid: cid001472 +primary_alias: DEF0242 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. + The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement.' +aliases: +- DEF0242 +- III.D27 +- functional-equation-involution-j +- def:functional-equation-involution +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005851 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 22 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Functional Equation Involution J** + + + The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. + The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement. + + + # Functional Equation Involution J + + + ## Summary + + + The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. + The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement. + + + ## Statement + + + ```latex + + \label{def:functional-equation-involution} + + Let $s \in \mathbb{C}$. The \textbf{functional equation involution} is the map + + \[ + + J(s) \;=\; 1 - \bar{s}, + + \] + + where $\bar{s}$ denotes complex conjugation. The fixed locus of $J$ is + + \[ + + \mathrm{Fix}(J) \;=\; \{ s \in \mathbb{C} : J(s) = s \} \;=\; \{ s : \mathrm{Re}(s) = \tfrac{1}{2} \}. + + \] + + This is the \textbf{critical line}. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 62 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch22-the-functional-equation-in-h-tau.tex` + lines 159-169 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SplitComplexZeta` + + - Name: `fe_involution` + + + ## Dependencies + + + - Canonical: III.D26 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The involution J(s) = 1 − s̄ on the split-complex s-plane. Exchanges B-lobe and C-lobe: J swaps e₊ and e₋ components. + The functional equation ζ_τ(s) = C(s)·ζ_τ(J(s)) is the bipolar symmetry statement.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001473.md b/_corpus_v3_items/cid001473.md new file mode 100644 index 000000000..f9009a0b6 --- /dev/null +++ b/_corpus_v3_items/cid001473.md @@ -0,0 +1,163 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001473/ +title: Lemniscate Operator H_L +cid: cid001473 +primary_alias: DEF0243 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard self-adjoint + operator on a metric graph. Compact resolvent, discrete spectrum. +aliases: +- DEF0243 +- III.D28 +- lemniscate-operator-h-l +- def:lemniscate-operator +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005823 +- predicate: formalized_by + object: cid005824 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 23 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Lemniscate Operator H_L** + + + The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard self-adjoint + operator on a metric graph. Compact resolvent, discrete spectrum. + + + # Lemniscate Operator H_L + + + ## Summary + + + The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard self-adjoint + operator on a metric graph. Compact resolvent, discrete spectrum. + + + ## Statement + + + ```latex + + \label{def:lemniscate-operator} + + The \textbf{lemniscate operator} $H_L : \mathrm{Dom}(H_L) \subset L^2(L) \to L^2(L)$ is given by + + \[ + + H_L f = -\frac{d^2 f}{dx^2} + + \] + + on each edge $e_B$ and $e_C$, with domain + + \[ + + \mathrm{Dom}(H_L) = \left\{ f \in H^2(L) : \begin{array}{l} + + f_B(\omega) = f_C(\omega) \quad \text{(continuity)} \\[0.5ex] + + f''_B(\omega) + f''_C(\omega) = 0 \quad \text{(Kirchhoff)} + + \end{array} \right\}. + + \] + + The Kirchhoff condition is the \textbf{current conservation} law: the sum of outward normal derivatives at $\omega$ vanishes. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 64 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch23-the-lemniscate-operator.tex` + lines 77-91 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.LemniscateOperator` + + - Name: `lemniscate_eigenvalue` + + + ## Dependencies + + + - Canonical: I.D18, III.D11 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The Laplacian H_L = −d²/dx² on L = S¹ ∨ S¹ with Kirchhoff boundary conditions at the crossing point. Standard + self-adjoint operator on a metric graph. Compact resolvent, discrete spectrum. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001474.md b/_corpus_v3_items/cid001474.md new file mode 100644 index 000000000..cb983d437 --- /dev/null +++ b/_corpus_v3_items/cid001474.md @@ -0,0 +1,149 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001474/ +title: Spectral Parameter Λ(s) +cid: cid001474 +primary_alias: DEF0244 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) + holds, then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L.' +aliases: +- DEF0244 +- III.D29 +- spectral-parameter-s +- def:spectral-parameter +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005838 +- predicate: formalized_by + object: cid005839 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 24 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Spectral Parameter Λ(s)** + + + The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) holds, + then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L. + + + # Spectral Parameter Λ(s) + + + ## Summary + + + The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) holds, + then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L. + + + ## Statement + + + ```latex + + \label{def:spectral-parameter} + + The \textbf{spectral parameter} is the quadratic map + + \[ + + \Lambda(s) = \iota_\tau^2\left(s(1-s) - \tfrac{1}{4}\right), + + \] + + where $\iota_\tau = \tfrac{2}{\pi + e} \approx 0.341304$ is the master constant. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 67 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch24-the-spectral-correspondence.tex` + lines 33-39 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SpectralCorrespondence` + + - Name: `spectral_parameter` + + + ## Dependencies + + + - Canonical: III.D26, III.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The spectral parameter Λ: s ↦ λ mapping complex s to eigenvalues of H_L. If the determinant representation (O3) + holds, then ζ_τ(s) = det(I − Λ(s)·H_L^{−1}) and zeros of ζ correspond to spectral values of H_L.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001475.md b/_corpus_v3_items/cid001475.md new file mode 100644 index 000000000..528783e63 --- /dev/null +++ b/_corpus_v3_items/cid001475.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001475/ +title: τ-Effective RH Statement +cid: cid001475 +primary_alias: DEF0245 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'τ-Effective RH: for every primorial depth k ≥ 1, the finite-cutoff operator H_{≤k} has only real eigenvalues, and + the finite Euler product has all zeros on Re(s) = ½ within the τ-effective window. A computable predicate: each finite check + terminates.' +aliases: +- DEF0245 +- III.D30 +- effective-rh-statement +- def:tau-effective-rh +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005816 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 26 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Effective RH Statement**\n\nτ-Effective RH: for every primorial depth k ≥ 1, the finite-cutoff operator H_{≤k}\ + \ has only real eigenvalues, and the finite Euler product has all zeros on Re(s) = ½ within the τ-effective window. A\ + \ computable predicate: each finite check terminates.\n\n# τ-Effective RH Statement\n\n## Summary\n\nτ-Effective RH: for\ + \ every primorial depth k ≥ 1, the finite-cutoff operator H_{≤k} has only real eigenvalues, and the finite Euler product\ + \ has all zeros on Re(s) = ½ within the τ-effective window. A computable predicate: each finite check terminates.\n\n\ + ## Statement\n\n```latex\n\\label{def:tau-effective-rh}\nFor every primorial depth $k \\geq 1$, let $L_k = \\{p_1, \\\ + ldots, p_k\\}$ denote the first $k$ primes, and let $H_{L_k}$ denote the lemniscate operator on the $k$-dimensional space\ + \ $\\mathcal{H}_{L_k} = \\bigoplus_{j=1}^k \\mathbb{C} \\cdot e_j$ defined in Chapter~23. The \\textbf{$\\tau$-effective\ + \ RH statement at depth $k$} asserts:\n\\begin{enumerate}[label=(\\roman*)]\n \\item \\textbf{Spectral reality}: All\ + \ eigenvalues $\\lambda \\in \\operatorname{Spec}(H_{L_k})$ are real, i.e., $\\Im(\\lambda) = 0$.\n \\item \\textbf{Zero\ + \ reality}: The finite Euler product\n \\[\n \\zeta_{L_k}(s) = \\prod_{p \\in L_k} \\frac{1}{1 - p^{-s}}\n \\\ + ]\n has all zeros in the $\\tau$-effective window $\\Re(s) \\in [0, 1]$, $|\\Im(s)| \\leq T_k$ (for computable cutoff\ + \ $T_k$) located on the critical line $\\Re(s) = \\tfrac{1}{2}$.\n \\item \\textbf{Spectral-zero correspondence}: The\ + \ real eigenvalues $\\lambda_j$ correspond bijectively to zeros $s_j = \\tfrac{1}{2} + i\\lambda_j$ via the determinant\ + \ representation (Conjecture O3).\n\\end{enumerate}\nThe statement is \\textbf{$\\tau$-effective} because each depth $k$\ + \ involves only finite-dimensional linear algebra and finite products, both computable with certified interval arithmetic.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-03.jsonl` line 71\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch26-primorial-verification-of-rh.tex`\ + \ lines 34-46\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Doors.CriticalLine`\n\ + - Name: `tau_effective_rh_check`\n\n## Dependencies\n\n- Canonical: III.T17, III.T18, III.T19, III.T09, III.D28\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'τ-Effective RH: for every primorial depth k ≥ 1, the finite-cutoff operator H_{≤k} has only real eigenvalues, + and the finite Euler product has all zeros on Re(s) = ½ within the τ-effective window. A computable predicate: each finite + check t' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001476.md b/_corpus_v3_items/cid001476.md new file mode 100644 index 000000000..6214c209d --- /dev/null +++ b/_corpus_v3_items/cid001476.md @@ -0,0 +1,152 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001476/ +title: Grand GRH (τ-effective) +cid: cid001476 +primary_alias: DEF0246 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial + zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension.' +aliases: +- DEF0246 +- III.D31 +- grand-grh-effective +- def:grand-grh +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005817 +- predicate: formalized_by + object: cid005818 +- predicate: formalized_by + object: cid005822 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 27 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Grand GRH (τ-effective)** + + + Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial + zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension. + + + # Grand GRH (τ-effective) + + + ## Summary + + + Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial + zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension. + + + ## Statement + + + ```latex + + \label{def:grand-grh} + + Let $\pi$ be an automorphic representation of $\mathrm{GL}_m(\mathbb{A}_\tau)$ for any $m \geq 1$. The \textbf{Grand Generalized + Riemann Hypothesis} asserts: + + \begin{equation} + + \text{All non-trivial zeros of } L(s, \pi) \text{ lie on the critical line } \Re(s) = \frac{1}{2}. + + \end{equation} + + Equivalently, the spectral operator $H_\pi$ on the adelic boundary Hilbert space $H_L^{\text{ad}}$ has all eigenvalues + real. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 74 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch27-the-grand-grh.tex` lines 146-152 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.GrandGRH` + + - Name: `grand_grh_adelic` + + + ## Dependencies + + + - Canonical: III.D26, III.T19, III.D22, III.D16 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Grand GRH at adelic level: for all adelic boundary characters on 𝔸_τ, the corresponding L-function has all non-trivial + zeros on Re(s) = ½. Scaling chain: ζ → Dirichlet → Hecke → automorphic. Conjectural at adelic extension.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001477.md b/_corpus_v3_items/cid001477.md new file mode 100644 index 000000000..ee4de7392 --- /dev/null +++ b/_corpus_v3_items/cid001477.md @@ -0,0 +1,148 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001477/ +title: L-Function as Spectral Determinant +cid: cid001477 +primary_alias: DEF0247 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) where + π is a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases.' +aliases: +- DEF0247 +- III.D32 +- l-function-as-spectral-determinant +- def:l-function-spectral-determinant +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005821 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 27 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**L-Function as Spectral Determinant** + + + All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) where π is + a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases. + + + # L-Function as Spectral Determinant + + + ## Summary + + + All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) where π is + a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases. + + + ## Statement + + + ```latex + + \label{def:l-function-spectral-determinant} + + Let $\pi$ be an automorphic representation of $\mathrm{GL}_m(\mathbb{A}_\tau)$ with local components $\pi_p$ at each prime + $p$. The \textbf{spectral $L$-function} is + + \begin{equation}\label{eq:ch27-spectral-l-function} + + L(s, \pi) = \prod_{p} L_p(s, \pi_p) = \det\left(I - s^{-1} H_\pi\right)^{-1}, + + \end{equation} + + where $H_\pi$ is the adelic operator on $H_L^{\text{ad}}$ with local eigenvalues $\lambda_p = p^{-\rho_p}$ determined + by the Satake parameters $\rho_p$ of $\pi_p$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 76 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch27-the-grand-grh.tex` lines 113-119 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.GrandGRH` + + - Name: `l_function_spectral_check` + + + ## Dependencies + + + - Canonical: III.D16, III.D22 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'All L-functions as spectral determinants of the universal operator: L_{≤N}(s, π) = det(I − s^{−1} H_{π,≤N}) + where π is a cuspidal automorphic representation. Riemann ζ, Dirichlet L, Hecke L as special cases.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001480.md b/_corpus_v3_items/cid001480.md new file mode 100644 index 000000000..0a21f9d1c --- /dev/null +++ b/_corpus_v3_items/cid001480.md @@ -0,0 +1,146 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001480/ +title: Simply Connected in Category τ +cid: cid001480 +primary_alias: DEF0250 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the + Hartogs bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré + = uniqueness of the terminal object.' +aliases: +- DEF0250 +- III.D35 +- simply-connected-in-category +- def:simply-connected-tau +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005834 +- predicate: formalized_by + object: cid005835 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 32 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Simply Connected in Category τ** + + + Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the Hartogs + bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré = uniqueness + of the terminal object. + + + # Simply Connected in Category τ + + + ## Summary + + + Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in the Hartogs + bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré = uniqueness + of the terminal object. + + + ## Statement + + + ```latex + + \label{def:simply-connected-tau} + + A $\tau$-space $X$ (i.e., a presheaf on the primorial site enriched over the spectral algebra $A_{\mathrm{spec}}(\mathbb{L})$) + is \textbf{simply connected in Category~$\tau$} if the automorphism group of its universal covering presheaf is trivial. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 84 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch32-simply-connected-in-category-tau.tex` + lines 33-35 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.Poincare` + + - Name: `simply_connected_check` + + + ## Dependencies + + + - Canonical: III.R15, III.D01, III.D03, III.D25 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Categorical reinterpretation of simple connectivity: π₁^τ trivial iff every loop of transition functions in + the Hartogs bulk is contractible. S³ is the terminal object among closed, simply connected 3-dimensional τ-spaces. Poincaré + = unique' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001481.md b/_corpus_v3_items/cid001481.md new file mode 100644 index 000000000..017a01390 --- /dev/null +++ b/_corpus_v3_items/cid001481.md @@ -0,0 +1,182 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001481/ +title: τ-Admissible Fluid Data +cid: cid001481 +primary_alias: DEF0251 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Fluid data = ω-germ assignment on clopen cylinders in τ³, with finite primorial depth and bounded ABCD extraction. + Admissibility: finite primorial depth + bounded ABCD coordinates + tower-coherent cylinder structure.' +aliases: +- DEF0251 +- III.D36 +- admissible-fluid-data +- def:tau-admissible-fluid-data +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005928 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Admissible Fluid Data** + + + Fluid data = ω-germ assignment on clopen cylinders in τ³, with finite primorial depth and bounded ABCD extraction. Admissibility: + finite primorial depth + bounded ABCD coordinates + tower-coherent cylinder structure. + + + # τ-Admissible Fluid Data + + + ## Summary + + + Fluid data = ω-germ assignment on clopen cylinders in τ³, with finite primorial depth and bounded ABCD extraction. Admissibility: + finite primorial depth + bounded ABCD coordinates + tower-coherent cylinder structure. + + + ## Statement + + + ```latex + + \label{def:tau-admissible-fluid-data} + + A \emph{$\tau$-admissible fluid datum} at primorial depth $k$ is a triple $\mathfrak{u}_k = (U_k, \, \phi_k, \, \mathfrak{b}_k)$ + where: + + \begin{enumerate}[(i)] + + \item $U_k$ is a finite union of clopen cylinders in $\tau^3$ at level $\mathrm{Prim}(k)$ (Definition~\ref{def:clopen-cylinder-domain}); + + \item $\phi_k \colon U_k \to \widehat{\Z}_\tau[\jj]$ is an $\omega$-germ assignment: for each cylinder $C \subseteq U_k$, + the restriction $\phi_k|_C$ is a section of the sheaf $\mathcal{O}(\tau^3_{\le k})$, i.e., a holomorphic function at primorial + depth~$k$; + + \item $\mathfrak{b}_k$ is the ABCD extraction of $\phi_k$: the four components $(A_k, B_k, C_k, D_k)$ obtained by projecting + $\phi_k$ through the idempotent decomposition + + \begin{equation} + + \phi_k \;=\; e_+ \cdot \phi_k^{(+)} + e_- \cdot \phi_k^{(-)}, + + \qquad + + e_\pm = \frac{1 \pm \jj}{2}, + + \label{eq:ch34-idempotent-decomposition} + + \end{equation} + + where $\phi_k^{(\pm)}$ each admit a further $(A,D)$ vs.\ $(B,C)$ sector decomposition from the 4+1 structure (Definition~\ref{def:four-plus-one-decomposition}, + Ch.~10); + + \item the ABCD extraction $\mathfrak{b}_k$ is \emph{bounded}: each component satisfies a primorial-level norm bound + + \begin{equation} + + \|X_k\|_{\mathrm{Prim}(k)} \;\le\; M \cdot \mathrm{Prim}(k)^{1/2}, + + \qquad X \in \{A, B, C, D\}, + + \label{eq:ch34-abcd-bound} + + \end{equation} + + for a uniform constant $M$ independent of $k$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 88 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch34-tau-admissible-fluid-data.tex` + lines 68-90 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.FluidData` + + - Name: `fluid_data_check` + + + ## Dependencies + + + - Canonical: III.D01, III.D04, III.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Fluid data = ω-germ assignment on clopen cylinders in τ³, with finite primorial depth and bounded ABCD extraction. + Admissibility: finite primorial depth + bounded ABCD coordinates + tower-coherent cylinder structure.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001482.md b/_corpus_v3_items/cid001482.md new file mode 100644 index 000000000..d449b6930 --- /dev/null +++ b/_corpus_v3_items/cid001482.md @@ -0,0 +1,151 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001482/ +title: Clopen Cylinder Domain +cid: cid001482 +primary_alias: DEF0252 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Clopen cylinder = local Hartogs bulk restricted to a primorial level. The cylinder algebra is Boolean (intersection, + union, complement all clopen). Domain building blocks for τ-admissible fluid data. +aliases: +- DEF0252 +- III.D37 +- clopen-cylinder-domain +- def:clopen-cylinder-domain +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005929 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 34 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Clopen Cylinder Domain** + + + Clopen cylinder = local Hartogs bulk restricted to a primorial level. The cylinder algebra is Boolean (intersection, union, + complement all clopen). Domain building blocks for τ-admissible fluid data. + + + # Clopen Cylinder Domain + + + ## Summary + + + Clopen cylinder = local Hartogs bulk restricted to a primorial level. The cylinder algebra is Boolean (intersection, union, + complement all clopen). Domain building blocks for τ-admissible fluid data. + + + ## Statement + + + ```latex + + \label{def:clopen-cylinder-domain} + + A \emph{clopen cylinder} in $\tau^3$ at primorial depth $k$ is a set of the form + + \begin{equation} + + C_{k,S} \;=\; \bigl\{\, x \in \tau^3_{\le k} \;\big|\; x \bmod p_i \in S_i \text{ for each } 1 \le i \le k \,\bigr\}, + + \label{eq:ch34-clopen-cylinder} + + \end{equation} + + where $S = (S_1, \ldots, S_k)$ is a $k$-tuple with each $S_i \subseteq \Z / p_i\Z$ non-empty. The restriction is to the + local Hartogs bulk (III.D01) at level $\mathrm{Prim}(k)$: each $C_{k,S}$ is a simultaneous residue class condition on + all primes up to $p_k$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 89 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch34-tau-admissible-fluid-data.tex` + lines 121-129 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.FluidData` + + - Name: `cylinder_assignment_check` + + + ## Dependencies + + + - Canonical: III.D01, III.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Clopen cylinder = local Hartogs bulk restricted to a primorial level. The cylinder algebra is Boolean (intersection, + union, complement all clopen). Domain building blocks for τ-admissible fluid data. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001483.md b/_corpus_v3_items/cid001483.md new file mode 100644 index 000000000..9123e89f4 --- /dev/null +++ b/_corpus_v3_items/cid001483.md @@ -0,0 +1,156 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001483/ +title: Fluid Sector Decomposition +cid: cid001483 +primary_alias: DEF0253 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The 4+1 template applied to fluid data. Each ω-germ decomposes into A/B/C/D-sector components. The ω-coupling mediates + inter-sector energy transfer. +aliases: +- DEF0253 +- III.D38 +- fluid-sector-decomposition +- def:fluid-sector-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005930 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Fluid Sector Decomposition** + + + The 4+1 template applied to fluid data. Each ω-germ decomposes into A/B/C/D-sector components. The ω-coupling mediates + inter-sector energy transfer. + + + # Fluid Sector Decomposition + + + ## Summary + + + The 4+1 template applied to fluid data. Each ω-germ decomposes into A/B/C/D-sector components. The ω-coupling mediates + inter-sector energy transfer. + + + ## Statement + + + ```latex + + \label{def:fluid-sector-decomposition} + + Let $f$ be $\tau$-admissible fluid data at primorial depth $n$. The \emph{fluid sector decomposition} of $f$ is the unique + splitting + + \begin{equation} + + f \;=\; f_A + f_B + f_C + f_D + f_\omega, + + \label{eq:ch35-fluid-sector-decomposition} + + \end{equation} + + where $f_S$ denotes the $S$-sector projection of $f$ for $S \in \{A, B, C, D, \omega\}$, obtained by restricting the spectral + content of $f$ to the characters in $\operatorname{Sector}(S) \subseteq \operatorname{Char}(\mathbb{L})$. + + + The four primitive projections $f_A, f_B, f_C, f_D$ capture the flow components governed by the generators $\pi, \gamma, + \eta, \alpha$ respectively. The coupling projection $f_\omega$ captures the inter-sector transfer mediated by the crossing + generator $\omega$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 91 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch35-fluid-sectors-and-the-defect-functional.tex` + lines 40-51 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.FluidData` + + - Name: `abcd_check` + + + ## Dependencies + + + - Canonical: III.D13, III.D14, III.D36 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The 4+1 template applied to fluid data. Each ω-germ decomposes into A/B/C/D-sector components. The ω-coupling + mediates inter-sector energy transfer. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001484.md b/_corpus_v3_items/cid001484.md new file mode 100644 index 000000000..204262845 --- /dev/null +++ b/_corpus_v3_items/cid001484.md @@ -0,0 +1,154 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001484/ +title: Defect Functional Δ +cid: cid001484 +primary_alias: DEF0254 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Δ(f, n) measures the deviation of an ω-germ f from canonical form at primorial depth n. Δ ≥ 0; Δ = 0 iff stabilized. + The defect is computable at each finite level. +aliases: +- DEF0254 +- III.D39 +- defect-functional +- def:defect-functional +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005931 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Defect Functional Δ** + + + Δ(f, n) measures the deviation of an ω-germ f from canonical form at primorial depth n. Δ ≥ 0; Δ = 0 iff stabilized. The + defect is computable at each finite level. + + + # Defect Functional Δ + + + ## Summary + + + Δ(f, n) measures the deviation of an ω-germ f from canonical form at primorial depth n. Δ ≥ 0; Δ = 0 iff stabilized. The + defect is computable at each finite level. + + + ## Statement + + + ```latex + + \label{def:defect-functional} + + Let $f$ be $\tau$-admissible fluid data and let $n \in \mathbb{N}$ be a primorial depth. The \emph{defect functional} + is the non-negative quantity + + \begin{equation} + + \Delta(f, n) \;=\; \sum_{S \in \{A,B,C,D,\omega\}} + + \bigl\| f_S\big|_{\operatorname{Prim}(n)} - f_S^{\mathrm{can}}\big|_{\operatorname{Prim}(n)} \bigr\|^2, + + \label{eq:ch35-defect-functional} + + \end{equation} + + where $f_S^{\mathrm{can}}$ is the canonical (NF-stabilized) form of $f_S$ at primorial depth $n$, the restriction $\cdot\big|_{\operatorname{Prim}(n)}$ + denotes evaluation on the lattice $\mathbb{Z}/\operatorname{Prim}(n)\mathbb{Z}$, and $\|\cdot\|$ is the $L^2$-norm on + this finite lattice. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 92 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch35-fluid-sectors-and-the-defect-functional.tex` + lines 77-87 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.FluidData` + + - Name: `defect_monotone_check` + + + ## Dependencies + + + - Canonical: III.D36, III.D38 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Δ(f, n) measures the deviation of an ω-germ f from canonical form at primorial depth n. Δ ≥ 0; Δ = 0 iff stabilized. + The defect is computable at each finite level. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001485.md b/_corpus_v3_items/cid001485.md new file mode 100644 index 000000000..734518908 --- /dev/null +++ b/_corpus_v3_items/cid001485.md @@ -0,0 +1,151 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001485/ +title: Hartogs Flow Operator +cid: cid001485 +primary_alias: DEF0255 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'H_flow: acts on ω-germ assignments and produces their Hartogs extensions. Fluid flow = transport of ω-germs along + τ³ fibration via Local Hartogs continuation (Book II Part VI).' +aliases: +- DEF0255 +- III.D40 +- hartogs-flow-operator +- def:hartogs-flow-operator +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005938 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hartogs Flow Operator** + + + H_flow: acts on ω-germ assignments and produces their Hartogs extensions. Fluid flow = transport of ω-germs along τ³ fibration + via Local Hartogs continuation (Book II Part VI). + + + # Hartogs Flow Operator + + + ## Summary + + + H_flow: acts on ω-germ assignments and produces their Hartogs extensions. Fluid flow = transport of ω-germs along τ³ fibration + via Local Hartogs continuation (Book II Part VI). + + + ## Statement + + + ```latex + + \label{def:hartogs-flow-operator} + + Let $f_0$ be $\tau$-admissible initial data on a clopen cylinder domain $U \subset \tau^3$. The \textbf{Hartogs flow operator} + $H_{\mathrm{flow}}$ is defined by + + \begin{equation} + + H_{\mathrm{flow}}(f_0) \;=\; \varprojlim_{n} \, \mathrm{Ext}_n(f_0|_{\partial U_n}), + + \label{eq:ch36-hartogs-flow-def} + + \end{equation} + + where $\mathrm{Ext}_n \colon \Gamma(\partial U_n, \mathcal{O}_{H_\tau}) \to \Gamma(U_n, \mathcal{O}_{H_\tau})$ is the + Local Hartogs continuation at primorial level $\mathrm{Prim}(n)$, and the inverse limit is taken over the primorial tower. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 94 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch36-the-hartogs-flow-operator.tex` + lines 49-57 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.HartogsFlow` + + - Name: `flow_check` + + + ## Dependencies + + + - Canonical: III.D01, III.D36 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'H_flow: acts on ω-germ assignments and produces their Hartogs extensions. Fluid flow = transport of ω-germs + along τ³ fibration via Local Hartogs continuation (Book II Part VI).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001486.md b/_corpus_v3_items/cid001486.md new file mode 100644 index 000000000..c28941eb5 --- /dev/null +++ b/_corpus_v3_items/cid001486.md @@ -0,0 +1,174 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001486/ +title: Operator Polarity Swap +cid: cid001486 +primary_alias: DEF0256 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. + The lemniscate's bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow's signature. +aliases: +- DEF0256 +- III.D41 +- operator-polarity-swap +- def:operator-polarity-swap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005937 +- predicate: formalized_by + object: cid005939 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Operator Polarity Swap** + + + Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. The lemniscate''s + bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow''s signature. + + + # Operator Polarity Swap + + + ## Summary + + + Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. The lemniscate''s + bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow''s signature. + + + ## Statement + + + ```latex + + \label{def:operator-polarity-swap} + + The \textbf{operator polarity swap} is the replacement of the elliptic Laplacian $\nabla^2$ by the hyperbolic wave operator + $\Box$ that is forced by the split-complex codomain $H_\tau$. Explicitly: + + \begin{itemize} + + \item In standard complex holomorphy ($i^2 = -1$), the Cauchy--Riemann equations yield the \emph{Laplacian}: + + \begin{equation} + + \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \;=\; 0 + + \qquad (\text{elliptic}). + + \label{eq:ch36-laplacian} + + \end{equation} + + + \item In split-complex holomorphy ($j^2 = +1$), the analogous ``Cauchy--Riemann'''' equations yield the \emph{wave operator}: + + \begin{equation} + + \frac{\partial^2 u}{\partial t^2} - \frac{\partial^2 u}{\partial x^2} \;=\; 0 + + \qquad (\text{hyperbolic}). + + \label{eq:ch36-wave-operator} + + \end{equation} + + \end{itemize} + + The polarity swap $\nabla^2 \mapsto \Box$ is the passage from $i^2 = -1$ to $j^2 = +1$ at the level of the associated + second-order operator. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 96 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch36-the-hartogs-flow-operator.tex` + lines 130-149 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.HartogsFlow` + + - Name: `polarity_swap_check` + + + ## Dependencies + + + - Canonical: III.D40 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Split-complex holomorphy (j² = +1) pairs with the wave operator ∂²/∂t² − ∂²/∂x² (hyperbolic), NOT the Laplacian. + The lemniscate's bipolar structure forces hyperbolic PDEs. Classical NS uses the Laplacian — this is the VM shadow's signature. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001487.md b/_corpus_v3_items/cid001487.md new file mode 100644 index 000000000..5c442a3d8 --- /dev/null +++ b/_corpus_v3_items/cid001487.md @@ -0,0 +1,152 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001487/ +title: Stabilized ω-Germ +cid: cid001487 +primary_alias: DEF0257 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined by its + value at depth k₀ and extends uniquely to all deeper levels. +aliases: +- DEF0257 +- III.D42 +- stabilized-germ +- def:stabilized-omega-germ +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005951 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 37 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Stabilized ω-Germ** + + + An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined by its value + at depth k₀ and extends uniquely to all deeper levels. + + + # Stabilized ω-Germ + + + ## Summary + + + An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined by its value + at depth k₀ and extends uniquely to all deeper levels. + + + ## Statement + + + ```latex + + \label{def:stabilized-omega-germ} + + Let $f$ be $\tau$-admissible fluid data on a clopen cylinder domain $U \subset \tau^3$ (Definition~\ref{def:tau-admissible-fluid-data}, + Ch.~34). The $\omega$-germ of $H_{\mathrm{flow}}(f)$ at a point $x \in U$ is \emph{stabilized} if there exists a primorial + depth $N_x \in \mathbb{N}$ such that for all $n \geq N_x$, the restriction + + \begin{equation} + + H_{\mathrm{flow}}(f)\big|_{\operatorname{Prim}(n)}(x) \;=\; H_{\mathrm{flow}}(f)\big|_{\operatorname{Prim}(N_x)}(x). + + \label{eq:ch37-stabilization} + + \end{equation} + + We write $\omega\text{-}\mathrm{germ}(f, x)$ for the stabilized value, which is well-defined once the tower becomes eventually + constant at~$x$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 98 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch37-positive-regularity.tex` lines + 39-48 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.PositiveRegularity` + + - Name: `stabilized_germ_check` + + + ## Dependencies + + + - Canonical: III.D39, III.D40 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: An ω-germ f is stabilized at primorial depth k₀ if Δ(f, k) = 0 for all k ≥ k₀. Equivalently, f is determined + by its value at depth k₀ and extends uniquely to all deeper levels. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001488.md b/_corpus_v3_items/cid001488.md new file mode 100644 index 000000000..bcb400a6b --- /dev/null +++ b/_corpus_v3_items/cid001488.md @@ -0,0 +1,163 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001488/ +title: Strong Sector at E₁ +cid: cid001488 +primary_alias: DEF0258 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The C-sector from Part II's 4+1 template, instantiated at E₁. Strong sector = χ₋-dominant characters. The tetration + stratum (C-coordinate) governs internal structure. +aliases: +- DEF0258 +- III.D43 +- strong-sector-at-e +- def:strong-sector-e1 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005954 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 38 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Strong Sector at E₁** + + + The C-sector from Part II''s 4+1 template, instantiated at E₁. Strong sector = χ₋-dominant characters. The tetration stratum + (C-coordinate) governs internal structure. + + + # Strong Sector at E₁ + + + ## Summary + + + The C-sector from Part II''s 4+1 template, instantiated at E₁. Strong sector = χ₋-dominant characters. The tetration stratum + (C-coordinate) governs internal structure. + + + ## Statement + + + ```latex + + \label{def:strong-sector-e1} + + Let $\mathcal{A}_{\mathrm{spec}}(\mathbb{L})$ be the spectral algebra on $\mathbb{L}$ and let $E_1$ denote the first enrichment + level (enrichment by the $\operatorname{Prim}$-tower with split-complex coefficients). The \emph{strong sector} at $E_1$ + is the full subcategory + + \begin{equation} + + \operatorname{Strong}_{E_1} \;\subset\; \mathcal{A}_{\mathrm{spec}}(\mathbb{L})\big|_{E_1} + + \label{eq:ch38-strong-sector-def} + + \end{equation} + + consisting of all objects whose spectral support lies entirely in $\operatorname{Sector}(C) \subseteq \operatorname{Char}(\mathbb{L})$. + Concretely, an element $g \in \operatorname{Strong}_{E_1}$ satisfies: + + \begin{enumerate}[(i)] + + \item $g$ is a section of the sheaf $\mathcal{O}(\tau^3_{\le k})$ for some primorial depth $k$; + + \item the ABCD extraction of $g$ yields $g_A = g_B = g_D = 0$ and $g_\omega = 0$; + + \item the surviving component $g_C$ is $\chi_-$-dominant: writing $g_C = e_+ \cdot g_{C,+} + e_- \cdot g_{C,-}$, the spectral + mass is concentrated in $g_{C,-}$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 101 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch38-the-strong-sector-and-nf-discreteness.tex` + lines 40-54 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.StrongSector` + + - Name: `strong_sector_check` + + + ## Dependencies + + + - Canonical: III.D13, III.D14, III.D23 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The C-sector from Part II's 4+1 template, instantiated at E₁. Strong sector = χ₋-dominant characters. The tetration + stratum (C-coordinate) governs internal structure. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001489.md b/_corpus_v3_items/cid001489.md new file mode 100644 index 000000000..15ba786df --- /dev/null +++ b/_corpus_v3_items/cid001489.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001489/ +title: τ-Admissible Gauge Data +cid: cid001489 +primary_alias: DEF0259 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Gauge data = section of the strong sector bundle over clopen cylinders, with finite primorial depth and bounded C-coordinate + extraction. Admissibility parallels the NS case. +aliases: +- DEF0259 +- III.D44 +- admissible-gauge-data +- def:tau-admissible-gauge-data +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005955 +- predicate: formalized_by + object: cid005956 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 38 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Admissible Gauge Data** + + + Gauge data = section of the strong sector bundle over clopen cylinders, with finite primorial depth and bounded C-coordinate + extraction. Admissibility parallels the NS case. + + + # τ-Admissible Gauge Data + + + ## Summary + + + Gauge data = section of the strong sector bundle over clopen cylinders, with finite primorial depth and bounded C-coordinate + extraction. Admissibility parallels the NS case. + + + ## Statement + + + ```latex + + \label{def:tau-admissible-gauge-data} + + A \emph{$\tau$-admissible gauge datum} at primorial depth $k$ is a triple $\mathfrak{g}_k = (U_k, \, \phi_k^C, \, \mathfrak{c}_k)$ + where: + + \begin{enumerate}[(i)] + + \item $U_k$ is a finite union of clopen cylinders in $\tau^3$ at level $\operatorname{Prim}(k)$; + + \item $\phi_k^C \colon U_k \to \mathbb{Z}/\operatorname{Prim}(k)\mathbb{Z}$ is an $\omega$-germ assignment with spectral + support in $\operatorname{Sector}(C)$: the ABCD extraction of $\phi_k^C$ has $A$-, $B$-, $D$-, and $\omega$-components + identically zero; + + \item $\mathfrak{c}_k$ is the C-sector extraction of $\phi_k^C$, decomposed into idempotent components + + \begin{equation} + + \phi_k^C \;=\; e_+ \cdot \phi_{k,+}^C \;+\; e_- \cdot \phi_{k,-}^C, + + \label{eq:ch38-gauge-idempotent} + + \end{equation} + + with the $\chi_-$-dominant constraint: $\|\phi_{k,+}^C\|_{\operatorname{Prim}(k)} \le \|\phi_{k,-}^C\|_{\operatorname{Prim}(k)}$; + + \item the C-coordinate extraction is \emph{bounded}: $\|\phi_{k,-}^C\|_{\operatorname{Prim}(k)} \le M \cdot \operatorname{Prim}(k)^{1/2}$ + for a uniform constant $M$ independent of~$k$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 102 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch38-the-strong-sector-and-nf-discreteness.tex` + lines 114-129 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.StrongSector` + + - Name: `gauge_data_check` + + + ## Dependencies + + + - Canonical: III.D43, III.T15 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Gauge data = section of the strong sector bundle over clopen cylinders, with finite primorial depth and bounded + C-coordinate extraction. Admissibility parallels the NS case. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001490.md b/_corpus_v3_items/cid001490.md new file mode 100644 index 000000000..c83ad0249 --- /dev/null +++ b/_corpus_v3_items/cid001490.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001490/ +title: Gap Constant Γ* +cid: cid001490 +primary_alias: DEF0260 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Γ* = inf{Δ(f,n) : f non-trivial, n ≥ n₀}. The gap constant is computable at each finite primorial level and stabilizes + as depth increases.' +aliases: +- DEF0260 +- III.D45 +- gap-constant +- def:gap-constant +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005934 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 39 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Gap Constant Γ*** + + + Γ* = inf{Δ(f,n) : f non-trivial, n ≥ n₀}. The gap constant is computable at each finite primorial level and stabilizes + as depth increases. + + + # Gap Constant Γ* + + + ## Summary + + + Γ* = inf{Δ(f,n) : f non-trivial, n ≥ n₀}. The gap constant is computable at each finite primorial level and stabilizes + as depth increases. + + + ## Statement + + + ```latex + + \label{def:gap-constant} + + Let $\{V_n\}$ be an NF-discrete tower with contractive defect functional. The \emph{gap constant} is + + \begin{equation} + + \Gamma^* \;=\; \inf\bigl\{\, \Delta(f, n) \;\bigm|\; f \in V_n \text{ non-trivial},\; n \geq n_0 \,\bigr\}, + + \label{eq:ch39-gap-constant} + + \end{equation} + + where $n_0$ is the smallest primorial depth at which both (H1) and (H2) hold, and ``non-trivial'''' means $f$ is not NF-stabilized + (i.e., $\Delta(f,n) > 0$). + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 105 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch39-the-tau-gap-meta-theorem.tex` + lines 57-66 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.GapTheorem` + + - Name: `gap_constant_check` + + + ## Dependencies + + + - Canonical: III.T26 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Γ* = inf{Δ(f,n) : f non-trivial, n ≥ n₀}. The gap constant is computable at each finite primorial level and + stabilizes as depth increases.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001491.md b/_corpus_v3_items/cid001491.md new file mode 100644 index 000000000..78e4b5ed5 --- /dev/null +++ b/_corpus_v3_items/cid001491.md @@ -0,0 +1,153 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001491/ +title: Strong Defect Functional +cid: cid001491 +primary_alias: DEF0261 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The defect functional Δ restricted to the strong (C) sector. Measures deviation from the C-sector's canonical form. + Non-trivial gauge configurations have Δ_C > 0. +aliases: +- DEF0261 +- III.D46 +- strong-defect-functional +- def:strong-defect-functional +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005936 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 40 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Strong Defect Functional** + + + The defect functional Δ restricted to the strong (C) sector. Measures deviation from the C-sector''s canonical form. Non-trivial + gauge configurations have Δ_C > 0. + + + # Strong Defect Functional + + + ## Summary + + + The defect functional Δ restricted to the strong (C) sector. Measures deviation from the C-sector''s canonical form. Non-trivial + gauge configurations have Δ_C > 0. + + + ## Statement + + + ```latex + + \label{def:strong-defect-functional} + + Let $\mathfrak{g}_k = (U_k, \phi_k^C, \mathfrak{c}_k)$ be $\tau$-admissible gauge data at primorial depth~$k$ (Definition~\ref{def:tau-admissible-gauge-data}, + Ch.~38). The \emph{strong defect functional} is + + \begin{equation} + + \Delta_C(\mathfrak{g}_k, n) \;=\; + + \bigl\| \phi_k^C\big|_{\operatorname{Prim}(n)} - (\phi_k^C)^{\mathrm{can}}\big|_{\operatorname{Prim}(n)} \bigr\|^2, + + \label{eq:ch40-strong-defect} + + \end{equation} + + where $(\phi_k^C)^{\mathrm{can}}$ is the NF-stabilized form of $\phi_k^C$ in the C-sector at primorial depth~$n$, and + $\|\cdot\|$ is the $L^2$-norm on $\mathbb{Z}/\operatorname{Prim}(n)\mathbb{Z}$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 108 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch40-the-yang-mills-mass-gap.tex` + lines 41-51 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.GapTheorem` + + - Name: `ym_coupling_check` + + + ## Dependencies + + + - Canonical: III.D39, III.D43 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The defect functional Δ restricted to the strong (C) sector. Measures deviation from the C-sector's canonical + form. Non-trivial gauge configurations have Δ_C > 0. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001492.md b/_corpus_v3_items/cid001492.md new file mode 100644 index 000000000..34cc8b0c3 --- /dev/null +++ b/_corpus_v3_items/cid001492.md @@ -0,0 +1,152 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001492/ +title: σ-Fixed Character +cid: cid001492 +primary_alias: DEF0262 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed characters + form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content. +aliases: +- DEF0262 +- III.D47 +- fixed-character +- def:sigma-fixed-character +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005941 +- predicate: formalized_by + object: cid005942 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 41 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**σ-Fixed Character** + + + A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed characters + form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content. + + + # σ-Fixed Character + + + ## Summary + + + A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed characters + form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content. + + + ## Statement + + + ```latex + + \label{def:sigma-fixed-character} + + A boundary character $\chi = (m, n) \in \operatorname{Char}(\mathbb{L}) \cong \mathbb{Z}^2$ is \emph{$\sigma$-fixed} if + $\sigma(\chi) = \chi$, i.e., if $(n, m) = (m, n)$. Equivalently, $\chi$ is $\sigma$-fixed if and only if $m = n$. The + $\sigma$-fixed characters form the \emph{diagonal sublattice} + + \begin{equation} + + \operatorname{Char}(\mathbb{L})^\sigma \;=\; \bigl\{\, (n, n) \;\bigm|\; n \in \mathbb{Z} \,\bigr\} \;\cong\; \mathbb{Z} + \;\subset\; \mathbb{Z}^2. + + \label{eq:ch41-diagonal-sublattice} + + \end{equation} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 110 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch41-sigma-fixed-characters-and-sector-addressability.tex` + lines 69-77 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.Hodge` + + - Name: `sigma_fixed_check` + + + ## Dependencies + + + - Canonical: III.T14, III.D13, III.D23 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A boundary character χ is σ-fixed if σ(χ) = χ under complex conjugation in the split-complex codomain. σ-fixed + characters form a sublattice of ℤ². In spectral trichotomy, σ-fixed = balanced B/C content. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001493.md b/_corpus_v3_items/cid001493.md new file mode 100644 index 000000000..40ce5acd5 --- /dev/null +++ b/_corpus_v3_items/cid001493.md @@ -0,0 +1,153 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001493/ +title: Sector Addressability +cid: cid001493 +primary_alias: DEF0263 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: + every σ-fixed character is sector-addressable in every primitive sector.' +aliases: +- DEF0263 +- III.D48 +- sector-addressability +- def:sector-addressability +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005943 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 41 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sector Addressability** + + + A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: every + σ-fixed character is sector-addressable in every primitive sector. + + + # Sector Addressability + + + ## Summary + + + A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: every + σ-fixed character is sector-addressable in every primitive sector. + + + ## Statement + + + ```latex + + \label{def:sector-addressability} + + Let $\chi = (m, n) \in \operatorname{Char}(\mathbb{L})$ be a boundary character and let $S \in \{A, B, C, D\}$ be a primitive + sector. The \emph{$S$-projection} of~$\chi$ is the component $\pi_S(\chi)$ obtained by restricting the spectral content + of~$\chi$ to $\operatorname{Sector}(S) \subseteq \operatorname{Char}(\mathbb{L})$. The character~$\chi$ is \emph{sector-addressable + in~$S$} if $\pi_S(\chi)$ has \emph{finite NF depth}: there exists $k_0 \in \mathbb{N}$ such that for all $k \geq k_0$, + + \begin{equation} + + \pi_S(\chi)\big|_{\operatorname{Prim}(k)} \;=\; \pi_S(\chi)\big|_{\operatorname{Prim}(k_0)}. + + \label{eq:ch41-finite-nf-depth} + + \end{equation} + + In other words, the $S$-projection stabilizes at primorial depth~$k_0$: no further primorial refinement changes it. The + minimal such~$k_0$ is the \emph{NF depth of~$\chi$ in sector~$S$}, denoted $\operatorname{depth}_S(\chi)$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 111 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch41-sigma-fixed-characters-and-sector-addressability.tex` + lines 143-152 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.Hodge` + + - Name: `sector_addressability_check` + + + ## Dependencies + + + - Canonical: III.D47, III.D23, III.T15 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A character χ is sector-addressable in sector S if its S-projection has finite NF depth. The τ-Hodge conjecture: + every σ-fixed character is sector-addressable in every primitive sector.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001494.md b/_corpus_v3_items/cid001494.md new file mode 100644 index 000000000..3621ee73f --- /dev/null +++ b/_corpus_v3_items/cid001494.md @@ -0,0 +1,157 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001494/ +title: E₂ Computational Agent +cid: cid001494 +primary_alias: DEF0264 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'An E₂ object: a code C paired with a decoder D such that D(C) produces another code C''. The cycle C → D(C) → C'' + → ... is computation. Formal definition of the self-referential structure unique to E₂.' +aliases: +- DEF0264 +- III.D49 +- e-computational-agent +- def:e2-computational-agent +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005792 +- predicate: formalized_by + object: cid005793 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 54 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₂ Computational Agent** + + + An E₂ object: a code C paired with a decoder D such that D(C) produces another code C''. The cycle C → D(C) → C'' → ... + is computation. Formal definition of the self-referential structure unique to E₂. + + + # E₂ Computational Agent + + + ## Summary + + + An E₂ object: a code C paired with a decoder D such that D(C) produces another code C''. The cycle C → D(C) → C'' → ... + is computation. Formal definition of the self-referential structure unique to E₂. + + + ## Statement + + + ```latex + + \label{def:e2-computational-agent} + + An \textbf{$\Elayer{2}$ computational agent} is a triple $(C, D, \varphi)$ where: + + \begin{enumerate} + + \item[(i)] $C$ is an $\Elayer{1}$-admissible object (a \emph{code}): a sector-decomposed element of the enriched category + carrying $H_\tau$-valued morphism data. + + \item[(ii)] $D$ is a morphism in the $H_\tau$-enriched category (a \emph{decoder}): a map that reads codes and produces + outputs. Crucially, $D$ is itself representable as a code---the triple $(C, D, \varphi)$ satisfies $D \in \mathrm{Carrier}(\Elayer{2})$. + + \item[(iii)] $\varphi \colon C \to C''$ is the \emph{execution map}: the composition $C'' = D(C)$ that produces a new + code $C''$ from the input code $C$ via the decoder $D$. + + \end{enumerate} + + The agent''s defining property is that $D(C)$ is itself a code---the output of decoding is an object of the same type + as the input. The decoder does not escape to $\Elayer{3}$; it operates entirely within $\Elayer{2}$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 142 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch54-the-computation-layer.tex` lines + 56-66 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Agent` + + - Name: `e2_agent_check` + + + ## Dependencies + + + - Canonical: III.D08, III.T04, III.T07 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'An E₂ object: a code C paired with a decoder D such that D(C) produces another code C''. The cycle C → D(C) + → C'' → ... is computation. Formal definition of the self-referential structure unique to E₂.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001495.md b/_corpus_v3_items/cid001495.md new file mode 100644 index 000000000..86d59e17d --- /dev/null +++ b/_corpus_v3_items/cid001495.md @@ -0,0 +1,156 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001495/ +title: Operational Closure +cid: cid001495 +primary_alias: DEF0265 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The code→execution→code cycle stays within E₂. Programs operate on programs, producing programs. No escape to a meta-level + is required. This is NOT a metaphor — it is the formal closure property. +aliases: +- DEF0265 +- III.D50 +- operational-closure +- def:operational-closure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005794 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 54 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Operational Closure** + + + The code→execution→code cycle stays within E₂. Programs operate on programs, producing programs. No escape to a meta-level + is required. This is NOT a metaphor — it is the formal closure property. + + + # Operational Closure + + + ## Summary + + + The code→execution→code cycle stays within E₂. Programs operate on programs, producing programs. No escape to a meta-level + is required. This is NOT a metaphor — it is the formal closure property. + + + ## Statement + + + ```latex + + \label{def:operational-closure} + + An $\Elayer{2}$ computational agent $(C, D, \varphi)$ satisfies \textbf{operational closure} if the Code--Execution--Code + cycle~\eqref{eq:ch54-code-cycle} remains within $\Elayer{2}$ at every step: + + \begin{equation} + + \forall\, n \geq 0, \quad C_n \in \mathrm{Carrier}(\Elayer{2}) + + \quad\text{and}\quad + + D(C_n) \in \mathrm{Carrier}(\Elayer{2}). + + \label{eq:ch54-operational-closure} + + \end{equation} + + The cycle does not leak upward to $\Elayer{3}$ (where self-modelling observers live) or downward to $\Elayer{1}$ (where + codes lose their self-referential structure). The output of decoding a code is another code---or a halt state, which is + itself a distinguished code. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 143 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch54-the-computation-layer.tex` lines + 80-91 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Agent` + + - Name: `operational_closure_check` + + + ## Dependencies + + + - Canonical: III.D49 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The code→execution→code cycle stays within E₂. Programs operate on programs, producing programs. No escape to + a meta-level is required. This is NOT a metaphor — it is the formal closure property. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001496.md b/_corpus_v3_items/cid001496.md new file mode 100644 index 000000000..424b1bf3b --- /dev/null +++ b/_corpus_v3_items/cid001496.md @@ -0,0 +1,108 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001496/ +title: τ-Tower Machine +cid: cid001496 +primary_alias: DEF0266 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'TTM: tuple (Q, m, b₀, Σ, δ_M, q_start, q_acc, q_rej). Finite control states, fixed m registers, fixed b₀ ports. + Instruction set from 5 generators: ρ (successor), σ (tetration step), × (multiplication), ∧ (exponentiation); predicates: + equality, divisibility, ∈_τ, orbit test.' +aliases: +- DEF0266 +- III.D51 +- tower-machine +- def:tau-tower-machine +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005801 +- predicate: formalized_by + object: cid005802 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 55 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Tower Machine**\n\nTTM: tuple (Q, m, b₀, Σ, δ_M, q_start, q_acc, q_rej). Finite control states, fixed m registers,\ + \ fixed b₀ ports. Instruction set from 5 generators: ρ (successor), σ (tetration step), × (multiplication), ∧ (exponentiation);\ + \ predicates: equality, divisibility, ∈_τ, orbit test.\n\n# τ-Tower Machine\n\n## Summary\n\nTTM: tuple (Q, m, b₀, Σ,\ + \ δ_M, q_start, q_acc, q_rej). Finite control states, fixed m registers, fixed b₀ ports. Instruction set from 5 generators:\ + \ ρ (successor), σ (tetration step), × (multiplication), ∧ (exponentiation); predicates: equality, divisibility, ∈_τ,\ + \ orbit test.\n\n## Statement\n\n```latex\n\\label{def:tau-tower-machine}\nA \\textbf{$\\tau$-Tower Machine} (TTM) is\ + \ a tuple\n\\[\n M \\;=\\; (Q,\\, m,\\, b_0,\\, \\Sigma,\\, \\delta_M,\\, q_{\\mathrm{start}},\\, q_{\\mathrm{acc}},\\\ + , q_{\\mathrm{rej}})\n\\]\nwhere:\n\\begin{enumerate}\n \\item $Q$ is a finite set of \\emph{control states},\n \ + \ with $|Q| < \\infty$.\n \\item $m \\in \\mathbb{N}$, $m \\geq 1$, is the number\n of \\emph{registers}.\n\ + \ Each register $r_i$ ($1 \\leq i \\leq m$)\n holds a $\\tau$-address---an element\n of the\ + \ NF tower\n $\\hat{\\mathbb{Z}}_{\\tau}\n = \\varprojlim_k \\mathbb{Z} / \\operatorname{Prim}(k) \\\ + mathbb{Z}$.\n \\item $b_0 \\in \\mathbb{N}$ is the number\n of \\emph{ports}---read-only input channels\n\ + \ through which the machine receives\n its initial $\\tau$-address arguments.\n \\item $\\Sigma$\ + \ is the \\emph{instruction set},\n consisting of operations derived from the\n five generators $\\\ + {\\alpha, \\pi, \\gamma, \\eta, \\omega\\}$\n of Category~$\\tau$:\n \\begin{itemize}\n \ + \ \\item \\textbf{Successor} $\\rho$:\n $r_i \\;\\mapsto\\; r_i + 1$\n \\quad(the\ + \ $\\alpha$-step:\n advance by one NF address).\n \\item \\textbf{Multiplication} $\\\ + times$:\n $(r_i, r_j) \\;\\mapsto\\; r_i \\times r_j$\n \\quad(the $\\pi$-operation:\n\ + \ primorial product).\n \\item \\textbf{Exponentiation} $\\wedge$:\n \ + \ $(r_i, r_j) \\;\\mapsto\\; r_i^{\\,r_j}$\n \\quad(the $(\\gamma, \\eta)$-operation:\n \ + \ tower-building on the fiber).\n \\item \\textbf{Tetration step} $\\sigma$:\n \ + \ $(r_i, r_j) \\;\\mapsto\\;\n r_i \\mathbin{\\uparrow\\!\\uparrow} r_j$\n \\\ + quad(iterated exponentiation:\n the $\\omega$-absorber channel).\n \\end{itemize}\n \ + \ The instruction set also includes four \\emph{predicates}\n that branch the control flow:\n \\\ + begin{itemize}\n \\item \\textbf{Equality}: $r_i = r_j$?\n \\item \\textbf{Divisibility}: $r_i\ + \ \\mid r_j$?\n \\quad(the $\\KAxiom{3}$ test).\n \\item \\textbf{Set membership}: $r_i\ + \ \\in_{\\tau} S$?\n \\quad(membership in a definable\n $\\tau$-subset, cf.\\ Book~I,\ + \ Part~VIII).\n \\item \\textbf{Orbit test}:\n does $r_i$ lie on the orbit\n \ + \ of $r_j$ under the progression operator?\n \\end{itemize}\n \\item $\\delta_M : Q \\times \\Sigma^m\ + \ \\to Q \\times \\Sigma^m$\n is the \\emph{transition function},\n mapping the current state and register\ + \ contents\n to a new state and updated register contents.\n \\item $q_{\\mathrm{start}}, q_{\\mathrm{acc}},\ + \ q_{\\mathrm{rej}} \\in Q$\n are the start, accept, and reject states,\n with $q_{\\mathrm{acc}} \\\ + neq q_{\\mathrm{rej}}$.\n\\end{enumerate}\nThe machine is \\emph{deterministic}:\n$\\delta_M$ is a total function.\nA\ + \ computation of $M$ on input\n$(a_1, \\ldots, a_{b_0})$\nbegins in state~$q_{\\mathrm{start}}$\nwith ports loaded and\ + \ registers initialised to~$0$,\nand halts when the control enters\n$q_{\\mathrm{acc}}$ or $q_{\\mathrm{rej}}$.\n```\n\ + \n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry\ + \ source: `book-03.jsonl` line 145\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch55-the-tau-tower-machine.tex`\ + \ lines 68-141\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.TowerMachine`\n\ + - Name: `ttm_check`\n\n## Dependencies\n\n- Canonical: III.D49, III.D50\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'TTM: tuple (Q, m, b₀, Σ, δ_M, q_start, q_acc, q_rej). Finite control states, fixed m registers, fixed b₀ ports. + Instruction set from 5 generators: ρ (successor), σ (tetration step), × (multiplication), ∧ (exponentiation); predicates: + equali' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001497.md b/_corpus_v3_items/cid001497.md new file mode 100644 index 000000000..5d941f2ce --- /dev/null +++ b/_corpus_v3_items/cid001497.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001497/ +title: Observable Transition +cid: cid001497 +primary_alias: DEF0267 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: δ_M^obs operates on K_M-bit configurations (machine constant). Cook-Levin tableau for TTM has constant width W = + 1+m. Bounded observation window drives the collapse. +aliases: +- DEF0267 +- III.D52 +- observable-transition +- def:observable-transition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005804 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 55 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Observable Transition**\n\nδ_M^obs operates on K_M-bit configurations (machine constant). Cook-Levin tableau\ + \ for TTM has constant width W = 1+m. Bounded observation window drives the collapse.\n\n# Observable Transition\n\n##\ + \ Summary\n\nδ_M^obs operates on K_M-bit configurations (machine constant). Cook-Levin tableau for TTM has constant width\ + \ W = 1+m. Bounded observation window drives the collapse.\n\n## Statement\n\n```latex\n\\label{def:observable-transition}\n\ + Let $M = (Q, m, b_0, \\Sigma, \\delta_M,\nq_{\\mathrm{start}}, q_{\\mathrm{acc}}, q_{\\mathrm{rej}})$\nbe a $\\tau$-Tower\ + \ Machine.\nThe \\textbf{machine constant} of $M$ is\n\\[\n K_M \\;=\\; |Q| \\cdot m.\n\\]\nA \\textbf{configuration}\ + \ of $M$\nis a pair $(q, \\mathbf{r}) \\in Q \\times \\hat{\\mathbb{Z}}_{\\tau}^{\\,m}$,\nconsisting of the current control\ + \ state\nand the $m$-tuple of register values.\nThe \\textbf{observable transition}\n\\[\n \\delta_M^{\\mathrm{obs}}\ + \ :\n Q \\times \\hat{\\mathbb{Z}}_{\\tau}^{\\,m}\n \\;\\longrightarrow\\;\n Q \\times \\hat{\\mathbb{Z}}_{\\\ + tau}^{\\,m}\n\\]\nis the restriction of $\\delta_M$\nto the information visible at a single step:\n\\begin{itemize}\n\ + \ \\item the current state $q \\in Q$\n (one of $|Q|$ values),\n \\item the $m$ register values\n \ + \ $r_1, \\ldots, r_m \\in \\hat{\\mathbb{Z}}_{\\tau}$,\n\\end{itemize}\nfor a total of $K_M = |Q| \\cdot m$ \\emph{observation\ + \ channels}.\nThe control state selects which instruction\nfrom~$\\Sigma$ to execute,\nand the instruction reads at most\ + \ two registers\n(source and target).\nThe observation window has width~$K_M$:\nthe transition depends on at most $K_M$\n\ + discrete pieces of information per step.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 147\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch55-the-tau-tower-machine.tex`\ + \ lines 399-435\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.TowerMachine`\n\ + - Name: `observable_transition_check`\n\n## Dependencies\n\n- Canonical: III.D51\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: δ_M^obs operates on K_M-bit configurations (machine constant). Cook-Levin tableau for TTM has constant width + W = 1+m. Bounded observation window drives the collapse. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001498.md b/_corpus_v3_items/cid001498.md new file mode 100644 index 000000000..65e22176a --- /dev/null +++ b/_corpus_v3_items/cid001498.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001498/ +title: Interface Width +cid: cid001498 +primary_alias: DEF0268 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: w(f, n) = number of primorial stages needed before computation f stabilizes at input size n. W(f) = sup_n w(f, n). + Measures computational depth in the primorial tower. +aliases: +- DEF0268 +- III.D53 +- interface-width +- def:interface-width +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005786 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 56 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Interface Width**\n\nw(f, n) = number of primorial stages needed before computation f stabilizes at input size\ + \ n. W(f) = sup_n w(f, n). Measures computational depth in the primorial tower.\n\n# Interface Width\n\n## Summary\n\n\ + w(f, n) = number of primorial stages needed before computation f stabilizes at input size n. W(f) = sup_n w(f, n). Measures\ + \ computational depth in the primorial tower.\n\n## Statement\n\n```latex\n\\label{def:interface-width}\nLet $f$ be a\ + \ TTM-computable function and $n \\in \\mathbb{N}$ an input size.\nThe \\textbf{interface width at size $n$} is\n\\[\n\ + \\begin{aligned}\nw(f, n) \\;=\\; \\min \\bigl\\{\\, k \\in \\mathbb{N} \\;\\big|\\;&\n \\text{for all inputs $x$ of\ + \ size $n$,} \\\\\n& \\text{the computation of $f(x)$ inspects only primorial levels }\n 1, \\ldots, k \\,\\bigr\\}.\n\ + \\end{aligned}\n\\]\nThe \\textbf{interface width of $f$} is the supremum\n\\[\nW(f) \\;=\\; \\sup_{n \\in \\mathbb{N}}\ + \ \\, w(f, n) \\;\\in\\; \\mathbb{N} \\cup \\{\\infty\\}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 148\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch56-interface-width-and-tau-admissibility.tex` lines\ + \ 69-85\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.Admissibility`\n\ + - Name: `interface_width_check`\n\n## Dependencies\n\n- Canonical: III.D51, III.T30, III.T09\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: w(f, n) = number of primorial stages needed before computation f stabilizes at input size n. W(f) = sup_n w(f, + n). Measures computational depth in the primorial tower. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001499.md b/_corpus_v3_items/cid001499.md new file mode 100644 index 000000000..c2e4b7fd4 --- /dev/null +++ b/_corpus_v3_items/cid001499.md @@ -0,0 +1,157 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001499/ +title: τ-Admissibility (E₂) +cid: cid001499 +primary_alias: DEF0269 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: f is τ-admissible iff W(f) < ∞. τ-admissible computations need only finitely many primorial stages. The E₂ analog + of the admissibility pattern from E₁ (NS fluid data, YM gauge data). +aliases: +- DEF0269 +- III.D54 +- admissibility-e +- def:tau-admissibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005787 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 56 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Admissibility (E₂)** + + + f is τ-admissible iff W(f) < ∞. τ-admissible computations need only finitely many primorial stages. The E₂ analog of the + admissibility pattern from E₁ (NS fluid data, YM gauge data). + + + # τ-Admissibility (E₂) + + + ## Summary + + + f is τ-admissible iff W(f) < ∞. τ-admissible computations need only finitely many primorial stages. The E₂ analog of the + admissibility pattern from E₁ (NS fluid data, YM gauge data). + + + ## Statement + + + ```latex + + \label{def:tau-admissibility} + + A TTM-computable function $f$ is \textbf{$\tau$-admissible} if its interface + + width is finite: + + \[ + + W(f) < \infty. + + \] + + Equivalently, there exists $k_0 \in \mathbb{N}$ such that for \emph{all} + + input sizes $n$, the computation of $f$ on inputs of size $n$ stabilizes + + at primorial depth $k_0$. + + We say $f$ is \textbf{$\tau$-admissible at depth $k_0$} and write + + $f \in \operatorname{Adm}_\tau(k_0)$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 149 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch56-interface-width-and-tau-admissibility.tex` + lines 128-140 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.Admissibility` + + - Name: `tau_admissible_check` + + + ## Dependencies + + + - Canonical: III.D53 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: f is τ-admissible iff W(f) < ∞. τ-admissible computations need only finitely many primorial stages. The E₂ analog + of the admissibility pattern from E₁ (NS fluid data, YM gauge data). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001500.md b/_corpus_v3_items/cid001500.md new file mode 100644 index 000000000..441ea9abb --- /dev/null +++ b/_corpus_v3_items/cid001500.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001500/ +title: NP Witness as Canonical Address +cid: cid001500 +primary_alias: DEF0270 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'An NP witness = τ-address w with unique NF decomposition (A,B,C,D). The ABCD coordinates structure the witness space: + search the 4-component coordinate space instead of arbitrary bitstrings. E₂-native definition.' +aliases: +- DEF0270 +- III.D55 +- np-witness-as-canonical-address +- def:np-witness-canonical-address +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005805 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 57 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**NP Witness as Canonical Address**\n\nAn NP witness = τ-address w with unique NF decomposition (A,B,C,D). The\ + \ ABCD coordinates structure the witness space: search the 4-component coordinate space instead of arbitrary bitstrings.\ + \ E₂-native definition.\n\n# NP Witness as Canonical Address\n\n## Summary\n\nAn NP witness = τ-address w with unique\ + \ NF decomposition (A,B,C,D). The ABCD coordinates structure the witness space: search the 4-component coordinate space\ + \ instead of arbitrary bitstrings. E₂-native definition.\n\n## Statement\n\n```latex\n\\label{def:np-witness-canonical-address}\n\ + Let $\\Pi$ be a decision problem computable on the TTM, and let $x$\nbe a $\\tau$-address representing a ``yes'' instance\ + \ of~$\\Pi$.\nAn \\textbf{NP witness for $x$ at primorial depth $k$} is a\n$\\tau$-address $w \\in \\mathbb{Z}/\\operatorname{Prim}(k)\\\ + mathbb{Z}$\nsuch that the TTM verifier $V$ accepts $(x, w)$ in at most $T(|x|)$\nsteps, where $T$ is a fixed polynomial.\n\ + The \\textbf{witness space at depth $k$} is\n\\[\nW(x, k) \\;=\\; \\bigl\\{\\, w \\in \\mathbb{Z}/\\operatorname{Prim}(k)\\\ + mathbb{Z}\n \\;\\big|\\; V(x, w) \\text{ accepts} \\,\\bigr\\}.\n\\]\n```\n\n## Proof / Justification\n\nThis item is\ + \ definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 152\n\ + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch57-witness-search-as-address-resolution.tex`\ + \ lines 47-60\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.WitnessSearch`\n\ + - Name: `witness_check`\n\n## Dependencies\n\n- Canonical: III.D51, III.T10, III.D20\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'An NP witness = τ-address w with unique NF decomposition (A,B,C,D). The ABCD coordinates structure the witness + space: search the 4-component coordinate space instead of arbitrary bitstrings. E₂-native definition.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001501.md b/_corpus_v3_items/cid001501.md new file mode 100644 index 000000000..dbd1bc06b --- /dev/null +++ b/_corpus_v3_items/cid001501.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001501/ +title: Computational Bi-Square +cid: cid001501 +primary_alias: DEF0271 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Fourth bi-square in the scaling chain. Left square: TTM computation (tower coherence of program execution). Right + square: witness structure (spectral naturality of CRT-decomposed witnesses). Pasting: Product-Meet Collapse.' +aliases: +- DEF0271 +- III.D56 +- computational-bi-square +- def:computational-bi-square +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 58 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Computational Bi-Square**\n\nFourth bi-square in the scaling chain. Left square: TTM computation (tower coherence\ + \ of program execution). Right square: witness structure (spectral naturality of CRT-decomposed witnesses). Pasting: Product-Meet\ + \ Collapse.\n\n# Computational Bi-Square\n\n## Summary\n\nFourth bi-square in the scaling chain. Left square: TTM computation\ + \ (tower coherence of program execution). Right square: witness structure (spectral naturality of CRT-decomposed witnesses).\ + \ Pasting: Product-Meet Collapse.\n\n## Statement\n\n```latex\n\\label{def:computational-bi-square}\nLet $\\Pi$ be a $\\\ + tau$-admissible NP problem\n(Definition~\\ref{def:tau-admissibility}) with TTM verifier $V$\n(Definition~\\ref{def:tau-tower-machine})\ + \ of interface width $k_0$.\nThe \\textbf{computational bi-square} at depth $k \\geq k_0$ is\n\\begin{equation}\\label{eq:ch58-computational-bisquare}\n\ + \ \\begin{tikzcd}[column sep=3.5em, row sep=2.5em]\n V\\bigl(\\operatorname{Prim}(k{+}1)\\bigr)\n \\\ + arrow[r, \"\\mathrm{res}_k\"]\n \\arrow[d, \"\\chi_\\pm^{(k+1)}\"']\n & V\\bigl(\\operatorname{Prim}(k)\\\ + bigr)\n \\arrow[r, \"\\mathrm{wit}_k\"]\n \\arrow[d, \"\\chi_\\pm^{(k)}\"]\n & W(x, k)\n \ + \ \\arrow[d, \"\\pi_{\\mathrm{CRT}}\"]\n \\\\\n \\chi_\\pm \\circ V\\bigl(\\operatorname{Prim}(k{+}1)\\\ + bigr)\n \\arrow[r, \"\\mathrm{res}_k\"']\n & \\chi_\\pm \\circ V\\bigl(\\operatorname{Prim}(k)\\bigr)\n\ + \ \\arrow[r, \"\\mathrm{wit}_k\"']\n & {\\displaystyle\\prod_{i=1}^{k}} W(x, p_i)\n \\end{tikzcd}\n\ + \\end{equation}\nwhere $V(\\operatorname{Prim}(k))$ is the verifier's computation at\ndepth $k$ (a presheaf value), $\\\ + mathrm{res}_k$ is the tower restriction,\n$\\chi_\\pm^{(k)}$ are the spectral characters,\n$\\mathrm{wit}_k$ extracts\ + \ the witness\n(Definition~\\ref{def:np-witness-canonical-address}),\nand $\\pi_{\\mathrm{CRT}}$ is the CRT decomposition\n\ + (Proposition~\\ref{prop:crt-witness-decomposition}).\nThe left square is the \\textbf{execution square};\nthe right square\ + \ is the \\textbf{witness square}.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 155\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch58-the-computational-bi-square.tex`\ + \ lines 61-94\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.CompBiSquare`\n\ + - Name: `comp_bisquare_check`\n\n## Dependencies\n\n- Canonical: III.D51, III.D55, III.T31\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Fourth bi-square in the scaling chain. Left square: TTM computation (tower coherence of program execution). + Right square: witness structure (spectral naturality of CRT-decomposed witnesses). Pasting: Product-Meet Collapse.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001502.md b/_corpus_v3_items/cid001502.md new file mode 100644 index 000000000..92d989836 --- /dev/null +++ b/_corpus_v3_items/cid001502.md @@ -0,0 +1,181 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001502/ +title: Enrichment Functor Enr₀₁ +cid: cid001502 +primary_alias: DEF0272 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Faithful functor Enr₀₁ : Cat_τ(E₀) → Cat_τ(E₁) enriching algebraic tower data with split-complex dynamics and sector + structure' +aliases: +- DEF0272 +- III.D57 +- enrichment-functor-enr +- def:enrichment-functor-e01 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005693 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 44 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Enrichment Functor Enr₀₁** + + + Faithful functor Enr₀₁ : Cat_τ(E₀) → Cat_τ(E₁) enriching algebraic tower data with split-complex dynamics and sector structure + + + # Enrichment Functor Enr₀₁ + + + ## Summary + + + Faithful functor Enr₀₁ : Cat_τ(E₀) → Cat_τ(E₁) enriching algebraic tower data with split-complex dynamics and sector structure + + + ## Statement + + + ```latex + + \label{def:enrichment-functor-e01} + + The \emph{enrichment functor} is a faithful, tower-coherent functor + + \begin{equation} + + \operatorname{Enr}_{01} : \operatorname{Cat}_{\tau}(\Elayer{0}) + + \;\longrightarrow\; + + \operatorname{Cat}_{\tau}(\Elayer{1}) + + \label{eq:ch44-enr01} + + \end{equation} + + defined on the category of $\tau$-objects at enrichment level~$\Elayer{0}$. It acts as follows. + + \begin{enumerate} + + \item\emph{(On objects.)} + + An $\Elayer{0}$-object $X = (U, \phi, \operatorname{NF}(X))$---a clopen cylinder domain~$U$, a presheaf section~$\phi$, + and a normal-form address---is sent to the $\Elayer{1}$-object + + \[ + + \operatorname{Enr}_{01}(X) \;=\; \bigl(U,\; \phi^{\jj},\; \operatorname{NF}(X),\; \boldsymbol{S}(X),\; \Delta(X, \cdot)\bigr), + + \] + + where $\phi^{\jj}$ is the split-complex extension of~$\phi$ (taking values in $H_\tau = \mathbb{Z}[\jj] / (\jj^2 - 1)$), + $\boldsymbol{S}(X) \in \{A, B, C, D, \omega\}$ is the sector assignment from the $4{+}1$ decomposition (Definition~\ref{def:four-plus-one-decomposition}, + Ch.~10), and $\Delta(X, \cdot)$ is the defect functional (Definition~\ref{def:defect-functional}, Ch.~35) evaluated on~$X$. + + + \item\emph{(On morphisms.)} + + A tower-coherent morphism $f : X \to Y$ in $\operatorname{Cat}_{\tau}(\Elayer{0})$ is sent to the $\Elayer{1}$-morphism + $\operatorname{Enr}_{01}(f)$ that preserves the split-complex extension, respects the sector assignment (i.e., $\boldsymbol{S}(f(X)) + = \boldsymbol{S}(X)$ when $f$ is sector-preserving), and satisfies the defect inequality $\Delta(Y, n) \leq \Delta(X, + n)$ for all primorial depths $n$ at which $f$ is defined. + + \end{enumerate} + + The functor preserves tower coherence (I.T18) and spectral naturality (I.T40): the diagrams that commute at~$\Elayer{0}$ + continue to commute at~$\Elayer{1}$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 119 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch44-enrichment-from-e0-to-e1.tex` + lines 45-68 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.EnrFunctor01` + + - Name: `enr_01_check` + + + ## Dependencies + + + - Canonical: III.T29, III.D48 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Faithful functor Enr₀₁ : Cat_τ(E₀) → Cat_τ(E₁) enriching algebraic tower data with split-complex dynamics and + sector structure' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001503.md b/_corpus_v3_items/cid001503.md new file mode 100644 index 000000000..acba23301 --- /dev/null +++ b/_corpus_v3_items/cid001503.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001503/ +title: E₁ Mutual Determination Instance +cid: cid001503 +primary_alias: DEF0273 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Unified triple (B₁, S₁, I₁) at E₁ with NS regularity, YM gap, and Hodge addressability as three readings +aliases: +- DEF0273 +- III.D58 +- e-mutual-determination-instance +- def:e1-mutual-determination +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005694 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 44 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₁ Mutual Determination Instance** + + + Unified triple (B₁, S₁, I₁) at E₁ with NS regularity, YM gap, and Hodge addressability as three readings + + + # E₁ Mutual Determination Instance + + + ## Summary + + + Unified triple (B₁, S₁, I₁) at E₁ with NS regularity, YM gap, and Hodge addressability as three readings + + + ## Statement + + + ```latex + + \label{def:e1-mutual-determination} + + The \emph{$\Elayer{1}$ Mutual Determination instance} is the triple $(B_1, S_1, I_1)$ defined as follows. + + \begin{enumerate} + + \item[\emph{(B$_1$)}] \textbf{Boundary data.} $\tau$-admissible data on clopen cylinder domains at enrichment level~$\Elayer{1}$: + split-complex sections $\phi^{\jj}$ satisfying bounded extraction, tower coherence, and Kirchhoff conservation. + + + \item[\emph{(S$_1$)}] \textbf{Spectral structure.} The sector-decomposed spectral algebra $A_{\mathrm{spec}}(\Lemniscate) + \otimes R_n^{\jj}$, equipped with the defect functional $\Delta$ and the sector projections $\pi_S$ for $S \in \{A, B, + C, D, \omega\}$. + + + \item[\emph{(I$_1$)}] \textbf{Interior consequence.} The stabilized $\omega$-germ structure: at every point of the clopen + cylinder domain, the $\omega$-germ of the Hartogs flow stabilizes at finite primorial depth with sector-wise NF-addressable + canonical form and positive spectral gap in the strong sector. + + \end{enumerate} + + The three components satisfy full bidirectionality: + + \begin{equation} + + B_1 \;\longleftrightarrow\; S_1 \;\longleftrightarrow\; I_1. + + \label{eq:ch44-mutual-determination-e1} + + \end{equation} + + The forward direction ($B_1 \to S_1 \to I_1$) is the content of Part~V''s proofs. The backward direction ($I_1 \to S_1 + \to B_1$) holds because each interior consequence reconstructs its spectral invariants (stabilized germs determine defect + values; gap existence determines NF lattice structure; addressability determines balanced spectral content), and the spectral + invariants reconstruct the boundary data via the primorial tower maps. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 120 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch44-enrichment-from-e0-to-e1.tex` + lines 146-163 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.EnrFunctor01` + + - Name: `e1_md_instance_check` + + + ## Dependencies + + + - Canonical: III.D57, III.T23 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Unified triple (B₁, S₁, I₁) at E₁ with NS regularity, YM gap, and Hodge addressability as three readings +noindex: false +--- + diff --git a/_corpus_v3_items/cid001504.md b/_corpus_v3_items/cid001504.md new file mode 100644 index 000000000..cffdf25a9 --- /dev/null +++ b/_corpus_v3_items/cid001504.md @@ -0,0 +1,162 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001504/ +title: τ-Rational Point +cid: cid001504 +primary_alias: DEF0274 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Address in Ź̂_τ that stabilizes at finite primorial depth with rational ABCD coordinates +aliases: +- DEF0274 +- III.D59 +- rational-point +- def:tau-rational-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005705 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 45 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Rational Point** + + + Address in Ź̂_τ that stabilizes at finite primorial depth with rational ABCD coordinates + + + # τ-Rational Point + + + ## Summary + + + Address in Ź̂_τ that stabilizes at finite primorial depth with rational ABCD coordinates + + + ## Statement + + + ```latex + + \label{def:tau-rational-point} + + An element $a = (a_k)_{k \geq 1} \in \hat{\mathbb{Z}}_{\T}$ is a \emph{$\tau$-rational point} if it satisfies two conditions: + + \begin{enumerate} + + \item[\emph{(R1)}] \textbf{Stabilization.} There exists a finite depth $k_0 \in \mathbb{N}$ such that for all $k \geq + k_0$, + + \begin{equation} + + a_k \;\equiv\; a_{k_0} \pmod{\operatorname{Prim}(k_0)}. + + \label{eq:ch45-stabilization-condition} + + \end{equation} + + That is, the address at $\operatorname{Prim}(k_0)$ determines the address at all deeper levels: no new information enters + the tower beyond depth~$k_0$. + + + \item[\emph{(R2)}] \textbf{Rationality.} Each ABCD coordinate of $a$, viewed as an element of the $p$-adic completion + $\mathbb{Z}_p$ for each prime $p \leq p_{k_0}$, actually lies in $\mathbb{Q} \cap \mathbb{Z}_p$. Equivalently, there exist + $\alpha, \beta \in \mathbb{Z}$ with $\beta \neq 0$ and $\gcd(\beta, \operatorname{Prim}(k_0)) = 1$ such that $a_{k_0} + = \alpha / \beta$ in $\mathbb{Z}/\operatorname{Prim}(k_0)\mathbb{Z}$. + + \end{enumerate} + + We write $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ for the set of all $\tau$-rational points. This set inherits a group structure + from $\hat{\mathbb{Z}}_{\T}$ (componentwise addition), making $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ a subgroup. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 122 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch45-tau-rational-interior-points.tex` + lines 56-71 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.RationalPoints` + + - Name: `rational_point_check` + + + ## Dependencies + + + - Canonical: III.D57, III.T10 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Address in Ź̂_τ that stabilizes at finite primorial depth with rational ABCD coordinates +noindex: false +--- + diff --git a/_corpus_v3_items/cid001505.md b/_corpus_v3_items/cid001505.md new file mode 100644 index 000000000..0e0972b4d --- /dev/null +++ b/_corpus_v3_items/cid001505.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001505/ +title: Rank as Tower Depth +cid: cid001505 +primary_alias: DEF0275 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Minimal primorial depth at which the τ-rational point group stabilizes; τ-analogue of Mordell-Weil rank +aliases: +- DEF0275 +- III.D60 +- rank-as-tower-depth +- def:rank-as-tower-depth +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005706 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 45 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Rank as Tower Depth** + + + Minimal primorial depth at which the τ-rational point group stabilizes; τ-analogue of Mordell-Weil rank + + + # Rank as Tower Depth + + + ## Summary + + + Minimal primorial depth at which the τ-rational point group stabilizes; τ-analogue of Mordell-Weil rank + + + ## Statement + + + ```latex + + \label{def:rank-as-tower-depth} + + For each primorial depth $k \geq 1$, define the \emph{$k$-level rational group} + + \begin{equation} + + G_k \;=\; \bigl\{ a \in \hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}} : a \text{ stabilizes at depth} \leq k \bigr\}, + + \label{eq:ch45-k-level-group} + + \end{equation} + + and the \emph{rank function} + + \begin{equation} + + r(k) \;=\; \operatorname{rk}_{\mathbb{Z}} \bigl( G_k / G_k^{\mathrm{tor}} \bigr), + + \label{eq:ch45-rank-function} + + \end{equation} + + where $G_k^{\mathrm{tor}}$ denotes the torsion subgroup of~$G_k$. The \emph{$\tau$-rank} of $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ + is the stabilization value + + \begin{equation} + + r_\infty \;=\; \lim_{k \to \infty} r(k), + + \label{eq:ch45-tau-rank} + + \end{equation} + + provided this limit exists and is finite. Equivalently, $r_\infty$ is the minimal depth $k_*$ such that $r(k) = r(k_*)$ + for all $k \geq k_*$. We call $k_*$ the \emph{rank-stabilization depth}. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 123 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch45-tau-rational-interior-points.tex` + lines 139-158 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.RationalPoints` + + - Name: `rank_check` + + + ## Dependencies + + + - Canonical: III.D59 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Minimal primorial depth at which the τ-rational point group stabilizes; τ-analogue of Mordell-Weil rank +noindex: false +--- + diff --git a/_corpus_v3_items/cid001506.md b/_corpus_v3_items/cid001506.md new file mode 100644 index 000000000..dcf630c2c --- /dev/null +++ b/_corpus_v3_items/cid001506.md @@ -0,0 +1,191 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001506/ +title: Proto-Code +cid: cid001506 +primary_alias: DEF0276 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: E₁ object with discrete carrier, self-verification, but no decoder; necessary but not sufficient for computation +aliases: +- DEF0276 +- III.D61 +- proto-code +- def:proto-code +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005702 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 46 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Proto-Code** + + + E₁ object with discrete carrier, self-verification, but no decoder; necessary but not sufficient for computation + + + # Proto-Code + + + ## Summary + + + E₁ object with discrete carrier, self-verification, but no decoder; necessary but not sufficient for computation + + + ## Statement + + + ```latex + + \label{def:proto-code} + + An $\Elayer{1}$-object $X \in \operatorname{Cat}_{\T}(\Elayer{1})$ is a + + \emph{proto-code} if it satisfies properties (PC1), (PC2), and (PC3) above. + + Explicitly: + + \begin{enumerate} + + \item[(PC1)] There exists $k_0 \in \mathbb{N}$ such that $\operatorname{NF}(X)$ + + stabilizes at primorial depth~$k_0$, i.e., + + \begin{equation} + + \operatorname{NF}(X)_k = \operatorname{NF}(X)_{k_0} + + \quad \text{for all } k \geq k_0. + + \label{eq:ch46-discrete-carrier} + + \end{equation} + + + \item[(PC2)] The sector admissibility predicate + + $\operatorname{Adm}_S(X) \in \{0,1\}$ for each sector + + $S \in \{A, B, C, D, \omega\}$ is computable from the ABCD coordinates of $X$ + + at depth~$k_0$: + + \begin{equation} + + \operatorname{Adm}_S(X) = \operatorname{Adm}_S\bigl(A_{k_0}(X),\, + + B_{k_0}(X),\, C_{k_0}(X),\, D_{k_0}(X)\bigr). + + \label{eq:ch46-self-verification} + + \end{equation} + + + \item[(PC3)] For any other $\Elayer{1}$-object $Y \neq X$, the object $X$ + + provides no morphism $X \to \operatorname{Adm}(Y)$: there is no internal + + mechanism in $X$ that evaluates $\operatorname{Adm}_S(Y)$. + + \end{enumerate} + + We write $\operatorname{Proto}(\Elayer{1})$ for the full subcategory of + + $\operatorname{Cat}_{\T}(\Elayer{1})$ whose objects are proto-codes. The + + morphisms in $\operatorname{Proto}(\Elayer{1})$ are tower-coherent maps that + + preserve all three properties. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 125 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch46-proto-codes-and-the-bsd-bridgehead.tex` + lines 94-127 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ProtoCodes` + + - Name: `proto_code_check` + + + ## Dependencies + + + - Canonical: III.D57, III.D58 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: E₁ object with discrete carrier, self-verification, but no decoder; necessary but not sufficient for computation +noindex: false +--- + diff --git a/_corpus_v3_items/cid001507.md b/_corpus_v3_items/cid001507.md new file mode 100644 index 000000000..bb912b575 --- /dev/null +++ b/_corpus_v3_items/cid001507.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001507/ +title: BSD Functional +cid: cid001507 +primary_alias: DEF0277 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level +aliases: +- DEF0277 +- III.D62 +- bsd-functional +- def:bsd-functional +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005703 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 46 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**BSD Functional** + + + BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level + + + # BSD Functional + + + ## Summary + + + BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level + + + ## Statement + + + ```latex + + \label{def:bsd-functional} + + The \emph{BSD functional} is the tower-indexed quantity + + \begin{equation} + + \operatorname{BSD}_{\T}(k) \;=\; + + r(k) \cdot L_{\T}''(1, k), + + \label{eq:ch46-bsd-functional} + + \end{equation} + + where $r(k) = \operatorname{rk}_{\mathbb{Z}}(G_k / G_k^{\mathrm{tor}})$ is + + the rank function (Definition~\ref{def:rank-as-tower-depth}, Ch.~45) and + + $L_{\T}''(1, k) = \frac{d}{ds} L_{\T}(s, k)\big|_{s=1}$ is the derivative of + + the spectral determinant at the symmetry point~$s = 1$. The functional is + + defined for each finite primorial depth $k \geq 1$. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 126 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch46-proto-codes-and-the-bsd-bridgehead.tex` + lines 264-278 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ProtoCodes` + + - Name: `bsd_functional_check` + + + ## Dependencies + + + - Canonical: III.D59, III.D60, III.D61 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: BSD_τ(k) = rank(k) · L’_τ(1,k); measures proto-code density at each primorial level +noindex: false +--- + diff --git a/_corpus_v3_items/cid001508.md b/_corpus_v3_items/cid001508.md new file mode 100644 index 000000000..e9a34d0d1 --- /dev/null +++ b/_corpus_v3_items/cid001508.md @@ -0,0 +1,165 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001508/ +title: Automorphic-Galois Duality in τ +cid: cid001508 +primary_alias: DEF0278 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Bidirectional correspondence between m-axis (Galois/prime) and n-axis (automorphic/spectral) data on ℤ² +aliases: +- DEF0278 +- III.D63 +- automorphic-galois-duality-in +- def:automorphic-galois-duality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005699 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 48 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Automorphic-Galois Duality in τ** + + + Bidirectional correspondence between m-axis (Galois/prime) and n-axis (automorphic/spectral) data on ℤ² + + + # Automorphic-Galois Duality in τ + + + ## Summary + + + Bidirectional correspondence between m-axis (Galois/prime) and n-axis (automorphic/spectral) data on ℤ² + + + ## Statement + + + ```latex + + \label{def:automorphic-galois-duality} + + The \emph{automorphic--Galois duality in~$\T$} is the bidirectional correspondence + + \begin{equation} + + \text{(Galois datum on the $m$-axis)} + + \quad \xleftrightarrow{\;\;\T\text{-Langlands}\;\;} \quad + + \text{(automorphic datum on the $n$-axis)} + + \label{eq:ch48-ag-duality} + + \end{equation} + + on the character lattice $\mathbb{Z}^2$ at enrichment level~$\Elayer{1}$. Concretely: + + \begin{enumerate} + + \item\emph{(Forward.)} The Galois datum $(\operatorname{Fr}_p)_{p}$ determines the automorphic datum $(\lambda_n)_{n}$ + via the local matching conditions~\eqref{eq:ch48-matching-condition}: each $\operatorname{Fr}_p$ decomposes into eigenvalues + via the split-complex idempotents, and the collection of eigenvalues constitutes the automorphic spectrum. + + + \item\emph{(Backward.)} The automorphic datum $(\lambda_n)_{n}$ determines the Galois datum $(\operatorname{Fr}_p)_{p}$ + via the Euler product~\eqref{eq:ch48-spectral-determinant}: the spectral determinant $L_{\T}(s)$ encodes the full sequence + of Frobenius elements, which can be recovered from the $L$-function by inverting the local factors. + + \end{enumerate} + + The duality is \emph{exact} at $\Elayer{1}$: no information is lost in either direction. The matching condition~\eqref{eq:ch48-matching-condition} + at each prime is the local witness of the global duality. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 131 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch48-automorphic-galois-duality.tex` + lines 183-200 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.Langlands` + + - Name: `ag_duality_check` + + + ## Dependencies + + + - Canonical: III.D57, III.T23 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Bidirectional correspondence between m-axis (Galois/prime) and n-axis (automorphic/spectral) data on ℤ² +noindex: false +--- + diff --git a/_corpus_v3_items/cid001509.md b/_corpus_v3_items/cid001509.md new file mode 100644 index 000000000..ac4f28bb9 --- /dev/null +++ b/_corpus_v3_items/cid001509.md @@ -0,0 +1,156 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001509/ +title: Local Langlands Instance +cid: cid001509 +primary_alias: DEF0279 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At each prime p, the matched pair (Fr_p, λ_p) with Frobenius equals spectral character restricted to m-axis +aliases: +- DEF0279 +- III.D64 +- local-langlands-instance +- def:local-langlands-instance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 48 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Local Langlands Instance** + + + At each prime p, the matched pair (Fr_p, λ_p) with Frobenius equals spectral character restricted to m-axis + + + # Local Langlands Instance + + + ## Summary + + + At each prime p, the matched pair (Fr_p, λ_p) with Frobenius equals spectral character restricted to m-axis + + + ## Statement + + + ```latex + + \label{def:local-langlands-instance} + + For each prime $p$ in the primorial tower, the \emph{local Langlands instance at $p$} is the matched pair + + \begin{equation} + + \Lambda_p \;=\; \bigl(\operatorname{Fr}_p,\; \lambda_p\bigr), + + \label{eq:ch48-local-langlands-pair} + + \end{equation} + + where $\operatorname{Fr}_p = \chi_\pm^{(p)}(1) \in H_{\T}$ is the Frobenius element (the local Galois datum) and $\lambda_p$ + is the eigenvalue of the spectral operator on the $p$-th fiber $V_p$ (the local automorphic datum). + + + The pair is \emph{matched} if and only if the Frobenius determines the eigenvalue and vice versa: + + \begin{equation} + + \operatorname{Fr}_p \;=\; e_+ \cdot \lambda_p + e_- \cdot \bar{\lambda}_p, + + \label{eq:ch48-matching-condition} + + \end{equation} + + where $e_\pm = \tfrac{1}{2}(1 \pm \jj)$ are the split-complex idempotents and $\bar{\lambda}_p$ denotes the conjugate + eigenvalue under $\jj \mapsto -\jj$. The matching condition asserts that the Galois datum and the spectral datum are + two readings of the same split-complex element. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 132 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch48-automorphic-galois-duality.tex` + lines 104-120 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.Langlands` + + - Name: `local_langlands_check` + + + ## Dependencies + + + - Canonical: III.D63 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At each prime p, the matched pair (Fr_p, λ_p) with Frobenius equals spectral character restricted to m-axis +noindex: false +--- + diff --git a/_corpus_v3_items/cid001510.md b/_corpus_v3_items/cid001510.md new file mode 100644 index 000000000..deb2a2286 --- /dev/null +++ b/_corpus_v3_items/cid001510.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001510/ +title: Enriched Bi-Square at E₁⁺ +cid: cid001510 +primary_alias: DEF0280 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Third bi-square: left = sector-coupled tower coherence, right = Langlands functoriality, pasting = finite factorization' +aliases: +- DEF0280 +- III.D65 +- enriched-bi-square-at-e +- def:enriched-bi-square +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005696 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 50 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Enriched Bi-Square at E₁⁺**\n\nThird bi-square: left = sector-coupled tower coherence, right = Langlands functoriality,\ + \ pasting = finite factorization\n\n# Enriched Bi-Square at E₁⁺\n\n## Summary\n\nThird bi-square: left = sector-coupled\ + \ tower coherence, right = Langlands functoriality, pasting = finite factorization\n\n## Statement\n\n```latex\n\\label{def:enriched-bi-square}\n\ + The \\emph{enriched bi-square} at $\\Elayer{1}+$ is the $2 \\times 3$ pasted diagram\n\\begin{equation}\n\\begin{tikzcd}[column\ + \ sep=3.5em, row sep=2.5em]\n\\operatorname{Enr}_{01}\\bigl(F(\\operatorname{Prim}(k{+}1))\\bigr)\n \\arrow[r, \"\\operatorname{res}_k\"\ + ]\n \\arrow[d, \"\\chi_\\pm^{(k+1)}\"']\n& \\operatorname{Enr}_{01}\\bigl(F(\\operatorname{Prim}(k))\\bigr)\n \\arrow[r,\ + \ \"\\bigoplus_S \\pi_S\"]\n \\arrow[d, \"\\chi_\\pm^{(k)}\"]\n& \\displaystyle\\bigoplus_{S} F_S(k)\n \\arrow[d, \"\ + \\operatorname{AG}_S\"]\n\\\\\n\\chi_\\pm \\circ \\operatorname{Enr}_{01}\\bigl(F(\\operatorname{Prim}(k{+}1))\\bigr)\n\ + \ \\arrow[r, \"\\operatorname{res}_k\"']\n& \\chi_\\pm \\circ \\operatorname{Enr}_{01}\\bigl(F(\\operatorname{Prim}(k))\\\ + bigr)\n \\arrow[r, \"\\prod_S \\pi_S^{\\mathrm{spec}}\"']\n& \\displaystyle\\prod_{S} \\operatorname{Spec}_S(k)\n\\end{tikzcd}\n\ + \\label{eq:ch50-enriched-bi-square}\n\\end{equation}\nwhere $k$ ranges over the primorial ladder, $S \\in \\{A, B, C,\ + \ D\\}$ ranges over primitive sectors, and the structural maps are as follows.\n\n\\emph{Left square (sector-coupled tower\ + \ coherence).}\n\\begin{itemize}\n\\item The top-left horizontal map $\\operatorname{res}_k$ is the tower restriction:\ + \ the primorial transition from depth $k{+}1$ to depth $k$, applied through the enrichment functor $\\operatorname{Enr}_{01}$.\n\ + \\item The vertical maps $\\chi_\\pm^{(k)}$ are the bipolar spectral decompositions via the split-complex idempotents\ + \ $e_\\pm = \\tfrac{1}{2}(1 \\pm \\jj)$.\n\\item The left square commutes because $\\operatorname{Enr}_{01}$ preserves\ + \ tower coherence (Definition~\\ref{def:enrichment-functor-e01}, Ch.~44): the spectral decomposition at depth $k{+}1$\ + \ restricts to the spectral decomposition at depth~$k$.\n\\end{itemize}\n\n\\emph{Right square (Langlands functoriality).}\n\ + \\begin{itemize}\n\\item The top-right horizontal map $\\bigoplus_S \\pi_S$ is the sector projection: it decomposes the\ + \ enriched tower datum at depth~$k$ into its primitive sector components $F_S(k)$ for $S \\in \\{A, B, C, D\\}$.\n\\item\ + \ The right-hand vertical maps $\\operatorname{AG}_S$ are the automorphic--Galois dualities (Definition~\\ref{def:automorphic-galois-duality},\ + \ Ch.~48) restricted to each sector.\n\\item The right square commutes by the Functoriality Theorem (Theorem~\\ref{thm:functoriality-theorem},\ + \ Ch.~49): sector morphisms commute with the automorphic--Galois duality.\n\\end{itemize}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 136\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch50-the-enriched-bi-square.tex`\ + \ lines 50-88\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Arithmetic.EnrichedBiSquare`\n\ + - Name: `enriched_bisquare_check`\n\n## Dependencies\n\n- Canonical: III.T36, III.T37, III.T35\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Third bi-square: left = sector-coupled tower coherence, right = Langlands functoriality, pasting = finite factorization' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001511.md b/_corpus_v3_items/cid001511.md new file mode 100644 index 000000000..db153ae8f --- /dev/null +++ b/_corpus_v3_items/cid001511.md @@ -0,0 +1,112 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001511/ +title: Complete Dependency Chain +cid: cid001511 +primary_alias: DEF0281 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The full 14-link derivation path: 5 generators → 9 axioms → 4 orbits → ABCD → boundary ring → Central Theorem → + enrichment ladder → 4+1 sector template → spectral algebra → Millennium clusters → enriched bi-square → tower assembly → + computational collapse → Hinge. Every link earned, no free parameters.' +aliases: +- DEF0281 +- III.D66 +- complete-dependency-chain +- def:complete-dependency-chain +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005876 +- predicate: formalized_by + object: cid005879 +- predicate: formalized_by + object: cid005880 +- predicate: formalized_by + object: cid005881 +- predicate: formalized_by + object: cid005882 +- predicate: formalized_by + object: cid005886 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Complete Dependency Chain**\n\nThe full 14-link derivation path: 5 generators → 9 axioms → 4 orbits → ABCD\ + \ → boundary ring → Central Theorem → enrichment ladder → 4+1 sector template → spectral algebra → Millennium clusters\ + \ → enriched bi-square → tower assembly → computational collapse → Hinge. Every link earned, no free parameters.\n\n#\ + \ Complete Dependency Chain\n\n## Summary\n\nThe full 14-link derivation path: 5 generators → 9 axioms → 4 orbits → ABCD\ + \ → boundary ring → Central Theorem → enrichment ladder → 4+1 sector template → spectral algebra → Millennium clusters\ + \ → enriched bi-square → tower assembly → computational collapse → Hinge. Every link earned, no free parameters.\n\n##\ + \ Statement\n\n```latex\n\\label{def:complete-dependency-chain}\nThe \\emph{complete dependency chain} of Category~$\\\ + T$:\n\\begin{enumerate}\n\\item \\textbf{Five generators.}\n $\\{\\alpha, \\pi, \\gamma, \\eta, \\omega\\}$\n \ + \ (Book~I, Part~I); the minimal-alphabet theorem forbids fewer.\n\n\\item \\textbf{Seven axioms.}\n $\\KAxiom{0}$--$\\\ + KAxiom{6}$\n (Book~I, Parts~II--III); each is the unique constraint\n compatible with five generators.\n\n\\\ + item \\textbf{Four orbits.}\n Progression on the seed~$\\rho$ produces exactly four orbits\n (Book~I, Part~V,\ + \ ABCD Orbit Theorem).\n\n\\item \\textbf{ABCD coordinates.}\n The peel map $\\Phi(x) = (A, B, C, D)$ assigns every\n\ + \ $\\T$-object a quadruple (Book~I, Part~V).\n\n\\item \\textbf{Boundary ring.}\n $H_\\tau = \\mathbb{Z}[\\\ + ,\\jj\\,]/(\\jj^2 - 1)$,\n forced by the fiber~$T^2$\n (Book~I, Part~VII; Book~II, Part~III).\n\n\\item \\textbf{Central\ + \ Theorem.}\n $\\mathcal{O}(\\tau^3) \\cong A_{\\mathrm{spec}}(\\Lemniscate)$\n (Book~II, II.T40).\n\n\\item\ + \ \\textbf{Enrichment ladder.}\n $\\mathcal{F}_E$\n (Theorem~\\ref{def:enrichment-functor}, Ch~4;\n Definition~\\\ + ref{def:layer-template}, Ch~5)\n produces\n $\\Elayer{0} \\subsetneq \\Elayer{1}\n \\subsetneq \\Elayer{2}\ + \ \\subsetneq \\Elayer{3}$,\n saturating at $\\Elayer{3}$\n (Theorem~\\ref{thm:saturation-e3}, Ch~7).\n\n\\\ + item \\textbf{4+1 sector template.}\n Four ABCD orbits plus one mixed sector partition every\n $\\Elayer{1}$~datum\n\ + \ (Definition~\\ref{def:four-plus-one-decomposition}, Ch~10).\n\n\\item \\textbf{Spectral algebra.}\n $\\mathfrak{A}_\\\ + tau$ collects the eight spectral forces\n (Definition~\\ref{ch:spectral-algebra-complete}, Ch~20).\n\n\\item \\textbf{Millennium\ + \ clusters.}\n Spectral purity (Definition~\\ref{def:ch25-spectral-purity}, Ch~25),\n BSD coherence (Theorem~\\\ + ref{thm:bsd-coherence-theorem}, Ch~47),\n and five further readings (Parts~IV--VI)\n follow from~$\\mathfrak{A}_\\\ + tau$ applied sector by sector.\n\n\\item \\textbf{Enriched bi-square.}\n Book~I's bi-square (I.T41) lifts to~$\\\ + Elayer{1}+$\n (Definition~\\ref{def:enriched-bi-square}, Ch~50),\n unifying all readings in one $2 \\times 3$\ + \ pasted diagram.\n\n\\item \\textbf{Enrichment tower assembly.}\n $\\Elayer{0} \\to \\Elayer{1} \\to \\Elayer{2}\ + \ \\to \\Elayer{3}$\n assembles into a single commutative diagram\n (Theorem~\\ref{thm:enrichment-tower-assembly},\ + \ Ch~51).\n\n\\item \\textbf{Computational collapse.}\n Product-meet collapse at $\\Elayer{2}$\n identifies\ + \ witness-search with composite-construction\n (Theorem~\\ref{thm:product-meet-collapse}, Ch~58).\n\n\\item \\textbf{Hinge.}\n\ + \ The assembled tower, enriched bi-square, and computational\n collapse combine: \\emph{the eight spectral forces\ + \ are jointly\n determined by the five generators and no free parameter.}\n This is the Hinge Theorem (Ch~61).\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n##\ + \ Source Context\n\n- Registry source: `book-03.jsonl` line 161\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch60-the-complete-dependency-chain.tex`\ + \ lines 55-132\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Hinge.DependencyChain`\n\ + - Name: `dependency_chain_check`\n\n## Dependencies\n\n- Canonical: III.T01, III.T03, III.D05, III.D10, III.D16, III.T19,\ + \ III.T35, III.D65, III.T32, III.T40\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\ + \n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The full 14-link derivation path: 5 generators → 9 axioms → 4 orbits → ABCD → boundary ring → Central Theorem + → enrichment ladder → 4+1 sector template → spectral algebra → Millennium clusters → enriched bi-square → tower assembly + → computa' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001512.md b/_corpus_v3_items/cid001512.md new file mode 100644 index 000000000..a67a2c589 --- /dev/null +++ b/_corpus_v3_items/cid001512.md @@ -0,0 +1,194 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001512/ +title: ZFC as E₂ VM +cid: cid001512 +primary_alias: DEF0282 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ZFC characterised as an E₂ virtual machine using the Layer Template: Carrier = formal sentences, Predicate = derivability, + Decoder = Gödel numbering, Invariant = consistency. ZFC cannot live at E₀ (no execution) or E₁ (no codes). τ and ZFC are + two different E₂ VMs.' +aliases: +- DEF0282 +- III.D67 +- zfc-as-e-vm +- def:zfc-as-e2-vm +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005773 +- predicate: formalized_by + object: cid005777 +- predicate: formalized_by + object: cid005778 +- predicate: formalized_by + object: cid005783 +- predicate: formalized_by + object: cid005784 +- predicate: formalized_by + object: cid005785 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 64 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**ZFC as E₂ VM** + + + ZFC characterised as an E₂ virtual machine using the Layer Template: Carrier = formal sentences, Predicate = derivability, + Decoder = Gödel numbering, Invariant = consistency. ZFC cannot live at E₀ (no execution) or E₁ (no codes). τ and ZFC are + two different E₂ VMs. + + + # ZFC as E₂ VM + + + ## Summary + + + ZFC characterised as an E₂ virtual machine using the Layer Template: Carrier = formal sentences, Predicate = derivability, + Decoder = Gödel numbering, Invariant = consistency. ZFC cannot live at E₀ (no execution) or E₁ (no codes). τ and ZFC are + two different E₂ VMs. + + + ## Statement + + + ```latex + + \label{def:zfc-as-e2-vm} + + The \textbf{ZFC virtual machine} is the $\Elayer{2}$ quadruple + + \begin{equation}\label{eq:ch64-zfc-vm} + + \mathrm{ZFC\textrm{-}VM} \;=\; \bigl(\,\mathrm{Sent}_{\mathrm{ZFC}},\; {\vdash}_{\mathrm{ZFC}},\; \ulcorner\cdot\urcorner,\; + \mathrm{Con}(\mathrm{ZFC})\,\bigr), + + \end{equation} + + where the four components are: + + \begin{enumerate} + + \item\emph{(Carrier.)} + + $\mathrm{Sent}_{\mathrm{ZFC}}$ is the set of well-formed sentences in the first-order language of ZFC. These sentences + are self-referential via G\"odel numbering: the sentence ``this sentence is not derivable'''' is a legitimate element + of the carrier. + + + \item\emph{(Predicate.)} + + ${\vdash}_{\mathrm{ZFC}}$ is the derivability relation. Given a finite set of axioms and the rules of first-order logic, + derivability decides which sentences are theorems. Derivability is the operational closure of the inference rules: the + output of applying rules to axioms is another sentence, which is itself subject to further rules. + + + \item\emph{(Decoder.)} + + $\ulcorner\cdot\urcorner \colon \mathrm{Sent}_{\mathrm{ZFC}} \to \mathbb{N}$ is G\"odel numbering. The decoder maps sentences + to natural numbers and, crucially, admits an inverse: given a G\"odel number $n$, the decoding function recovers the sentence + $\varphi$ such that $\ulcorner\varphi\urcorner = n$. The decoder closes the self-referential loop: a sentence can reference + its own code. + + + \item\emph{(Invariant.)} + + $\mathrm{Con}(\mathrm{ZFC})$ is the consistency statement $\nvdash_{\mathrm{ZFC}} (0 = 1)$. The invariant asserts that + the VM never crashes: the derivability engine, applied to the axioms, never produces a contradiction. Without this invariant, + the VM would derive every sentence (by \emph{ex falso quodlibet}), and the carrier would collapse to triviality. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 170 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch64-zfc-as-e2-virtual-machine.tex` + lines 66-87 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ZFCasVM` + + - Name: `zfc_vm_check` + + + ## Dependencies + + + - Canonical: III.D05, III.D08, III.D50 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ZFC characterised as an E₂ virtual machine using the Layer Template: Carrier = formal sentences, Predicate = + derivability, Decoder = Gödel numbering, Invariant = consistency. ZFC cannot live at E₀ (no execution) or E₁ (no codes). + τ and ZFC ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001513.md b/_corpus_v3_items/cid001513.md new file mode 100644 index 000000000..93bdf819d --- /dev/null +++ b/_corpus_v3_items/cid001513.md @@ -0,0 +1,175 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001513/ +title: Gödel Numbering as NF Address +cid: cid001513 +primary_alias: DEF0283 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Gödel numbering is the NF-address system of the ZFC-VM''s code space: injective, primitive-recursive decoder, self-referential + via the diagonal lemma. Structurally parallel to NF addresses in the profinite tower.' +aliases: +- DEF0283 +- III.D68 +- g-del-numbering-as-nf-address +- def:goedel-numbering-as-nf-address +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005774 +- predicate: formalized_by + object: cid005779 +- predicate: formalized_by + object: cid005780 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 64 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Gödel Numbering as NF Address** + + + Gödel numbering is the NF-address system of the ZFC-VM''s code space: injective, primitive-recursive decoder, self-referential + via the diagonal lemma. Structurally parallel to NF addresses in the profinite tower. + + + # Gödel Numbering as NF Address + + + ## Summary + + + Gödel numbering is the NF-address system of the ZFC-VM''s code space: injective, primitive-recursive decoder, self-referential + via the diagonal lemma. Structurally parallel to NF addresses in the profinite tower. + + + ## Statement + + + ```latex + + \label{def:goedel-numbering-as-nf-address} + + Let $\mathrm{ZFC\textrm{-}VM} = (\mathrm{Sent}_{\mathrm{ZFC}},\, {\vdash}_{\mathrm{ZFC}},\, \ulcorner\cdot\urcorner,\, + \mathrm{Con}(\mathrm{ZFC}))$ be the ZFC virtual machine (Definition~\ref{def:zfc-as-e2-vm}). The \textbf{NF-address system + of the ZFC-VM} is the G\"odel numbering function + + \begin{equation}\label{eq:ch64-goedel-nf} + + \ulcorner\cdot\urcorner \colon \mathrm{Sent}_{\mathrm{ZFC}} \;\longrightarrow\; \mathbb{N}, + + \end{equation} + + satisfying the following properties: + + \begin{enumerate} + + \item\emph{(Injectivity.)} Distinct sentences receive distinct codes: $\ulcorner\varphi\urcorner = \ulcorner\psi\urcorner$ + implies $\varphi = \psi$. + + + \item\emph{(Decodability.)} There exists a primitive recursive function $\mathrm{dec} \colon \mathbb{N} \to \mathrm{Sent}_{\mathrm{ZFC}} + \cup \{\bot\}$ such that $\mathrm{dec}(\ulcorner\varphi\urcorner) = \varphi$ for every sentence $\varphi$, and $\mathrm{dec}(n) + = \bot$ if $n$ is not the G\"odel number of any sentence. + + + \item\emph{(Self-reference.)} For every primitive recursive predicate $P$ on $\mathbb{N}$, there exists a sentence $\sigma_{P}$ + such that + + $\mathrm{ZFC} \vdash \sigma_{P} \leftrightarrow P(\ulcorner\sigma_{P}\urcorner)$. + + This is the diagonal lemma, which closes the self-referential loop of the NF-address system. + + \end{enumerate} + + The G\"odel number $\ulcorner\varphi\urcorner$ is the NF address of the sentence $\varphi$ in the ZFC-VM''s code space. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 171 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch64-zfc-as-e2-virtual-machine.tex` + lines 123-141 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ZFCasVM` + + - Name: `axiom_encoding_check` + + + ## Dependencies + + + - Canonical: III.D67 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Gödel numbering is the NF-address system of the ZFC-VM''s code space: injective, primitive-recursive decoder, + self-referential via the diagonal lemma. Structurally parallel to NF addresses in the profinite tower.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001514.md b/_corpus_v3_items/cid001514.md new file mode 100644 index 000000000..9b51cec74 --- /dev/null +++ b/_corpus_v3_items/cid001514.md @@ -0,0 +1,99 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001514/ +title: Five Forbidden Moves +cid: cid001514 +primary_alias: DEF0284 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Five operations ZFC allows but τ forbids: (1) unbounded fan-out (K3), (2) global equality (K5), (3) succinct circuits + (operational closure), (4) exponential quantification (observation-finiteness), (5) non-local disguise (NF uniqueness). + Each has a specific bridge consequence.' +aliases: +- DEF0284 +- III.D69 +- five-forbidden-moves +- def:five-forbidden-moves +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005740 +- predicate: formalized_by + object: cid005743 +- predicate: formalized_by + object: cid005744 +- predicate: formalized_by + object: cid005745 +- predicate: formalized_by + object: cid005746 +- predicate: formalized_by + object: cid005750 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 65 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Five Forbidden Moves**\n\nFive operations ZFC allows but τ forbids: (1) unbounded fan-out (K3), (2) global\ + \ equality (K5), (3) succinct circuits (operational closure), (4) exponential quantification (observation-finiteness),\ + \ (5) non-local disguise (NF uniqueness). Each has a specific bridge consequence.\n\n# Five Forbidden Moves\n\n## Summary\n\ + \nFive operations ZFC allows but τ forbids: (1) unbounded fan-out (K3), (2) global equality (K5), (3) succinct circuits\ + \ (operational closure), (4) exponential quantification (observation-finiteness), (5) non-local disguise (NF uniqueness).\ + \ Each has a specific bridge consequence.\n\n## Statement\n\n```latex\n\\label{def:five-forbidden-moves}\nThe following\ + \ five \\emph{forbidden moves} are operations admitted\nby ZFC but prohibited by Category~$\\T$.\n\\begin{enumerate}\n\ + \\item \\textbf{Unbounded fan-out.}\n \\emph{ZFC}: Power Set.\n \\emph{$\\T$~block}: $\\KAxiom{3}$ (bounded\ + \ multiplicity).\n \\emph{Bridge}: brute-force search over $\\mathcal{P}(X)$\n unavailable; the NP-hardness\ + \ gap.\n\n\\item \\textbf{Global equality.}\n \\emph{ZFC}: Extensionality.\n \\emph{$\\T$~block}: $\\KAxiom{5}$\ + \ (diagonal discipline).\n \\emph{Bridge}: diagonal arguments (Cantor, Russell)\n have no $\\T$-analogue; equality\ + \ is local and earned.\n\n\\item \\textbf{Succinct circuits.}\n \\emph{ZFC}: Replacement.\n \\emph{$\\T$~block}:\ + \ operational closure finiteness.\n \\emph{Bridge}: some ZFC-compressible objects have no\n $\\T$-short description;\ + \ the compression gap.\n\n\\item \\textbf{Exponential quantification.}\n \\emph{ZFC}: unrestricted quantification\ + \ over\n uncountable sets.\n \\emph{$\\T$~block}: observation finiteness.\n \\emph{Bridge}: exponential-witness\ + \ searches trivially\n resolve in~$\\T$; the witness gap.\n\n\\item \\textbf{Non-local disguise.}\n \\emph{ZFC}:\ + \ Foundation + Replacement (re-encoding).\n \\emph{$\\T$~block}: NF uniqueness.\n \\emph{Bridge}: encoding tricks\ + \ that shift complexity\n cannot operate in~$\\T$; the disguise gap.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 172\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch65-the-five-forbidden-moves.tex`\ + \ lines 49-86\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.ForbiddenMoves`\n\ + - Name: `forbidden_moves_check`\n\n## Dependencies\n\n- Canonical: III.D67\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Five operations ZFC allows but τ forbids: (1) unbounded fan-out (K3), (2) global equality (K5), (3) succinct + circuits (operational closure), (4) exponential quantification (observation-finiteness), (5) non-local disguise (NF uniqueness). + Ea' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001515.md b/_corpus_v3_items/cid001515.md new file mode 100644 index 000000000..a9a6cf0ac --- /dev/null +++ b/_corpus_v3_items/cid001515.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001515/ +title: Host-Level Property +cid: cid001515 +primary_alias: DEF0285 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: A host-level property quantifies over the totality of a VM's execution histories, is not representable as the output + of any single execution, and is determinable only from E₃. Consistency, halting, and completeness are host-level properties. +aliases: +- DEF0285 +- III.D70 +- host-level-property +- def:host-level-property +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005775 +- predicate: formalized_by + object: cid005776 +- predicate: formalized_by + object: cid005781 +- predicate: formalized_by + object: cid005782 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 66 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Host-Level Property**\n\nA host-level property quantifies over the totality of a VM's execution histories,\ + \ is not representable as the output of any single execution, and is determinable only from E₃. Consistency, halting,\ + \ and completeness are host-level properties.\n\n# Host-Level Property\n\n## Summary\n\nA host-level property quantifies\ + \ over the totality of a VM's execution histories, is not representable as the output of any single execution, and is\ + \ determinable only from E₃. Consistency, halting, and completeness are host-level properties.\n\n## Statement\n\n```latex\n\ + %\n\\label{def:host-level-property}\nA \\textbf{host-level property} of an $\\Elayer{2}$ virtual machine~$V$\nis a predicate~$P$\ + \ that satisfies:\n\\begin{enumerate}\n \\item $P$ quantifies over the totality\n of $V$'s execution histories\ + \ ---\n not over a single code or a single derivation step,\n but over all possible runs of the Code--Execution--Code\ + \ cycle.\n \\item $P$ is not representable\n as the output of any single execution\n of $V$'s internal\ + \ decoder.\n \\item The truth value of~$P$\n is determinable only from outside~$V$:\n by an $\\Elayer{3}$\ + \ observer\n with access to the global structure\n of $V$'s code space and derivation system.\n\\end{enumerate}\n\ + Consistency, totality,\nand the halting predicate (Section~\\ref{sec:ch66-halting-problem})\nare host-level properties.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-03.jsonl` line 175\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch66-goedel-and-the-vm-boundary.tex`\ + \ lines 148-170\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.ZFCasVM`\n\ + - Name: `set_universe_check`\n\n## Dependencies\n\n- Canonical: III.D67, III.D08, III.D09\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: A host-level property quantifies over the totality of a VM's execution histories, is not representable as the + output of any single execution, and is determinable only from E₃. Consistency, halting, and completeness are host-level + properties +noindex: false +--- + diff --git a/_corpus_v3_items/cid001516.md b/_corpus_v3_items/cid001516.md new file mode 100644 index 000000000..0272715cf --- /dev/null +++ b/_corpus_v3_items/cid001516.md @@ -0,0 +1,94 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001516/ +title: Bridge Axiom +cid: cid001516 +primary_alias: DEF0286 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A bridge is a structure-preserving functor F: Cat_τ(E₂) → Mod(ZFC) satisfying carrier preservation, predicate preservation, + decoder compatibility, and invariant reflection. The existence of such F is conjectural.' +aliases: +- DEF0286 +- III.D71 +- bridge-axiom +- def:bridge-axiom +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005712 +- predicate: formalized_by + object: cid005713 +- predicate: formalized_by + object: cid005721 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 67 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Bridge Axiom**\n\nA bridge is a structure-preserving functor F: Cat_τ(E₂) → Mod(ZFC) satisfying carrier preservation,\ + \ predicate preservation, decoder compatibility, and invariant reflection. The existence of such F is conjectural.\n\n\ + # Bridge Axiom\n\n## Summary\n\nA bridge is a structure-preserving functor F: Cat_τ(E₂) → Mod(ZFC) satisfying carrier\ + \ preservation, predicate preservation, decoder compatibility, and invariant reflection. The existence of such F is conjectural.\n\ + \n## Statement\n\n```latex\n%\n\\label{def:bridge-axiom}\nA \\textbf{bridge} from Category~$\\T$ to ZFC is a functor\n\ + \\begin{equation}\\label{eq:ch67-bridge-functor}\nF \\colon \\operatorname{Cat}_{\\T}(\\Elayer{2})\n\\;\\longrightarrow\\\ + ; \\mathbf{Mod}(\\mathrm{ZFC})\n\\end{equation}\nsatisfying four properties.\n\\begin{enumerate}\n\\item\\emph{(i) Carrier\ + \ preservation.}\nFor every $\\T$-object $X$ with NF address\n$a_{X} \\in \\hat{\\mathbb{Z}}_{\\T}$,\n$F(X)$ is a ZFC-definable\ + \ set.\nIf $X \\neq Y$ and neither lies\nin the kernel of any forbidden move $M_{i}$\n(Definition~\\ref{def:five-forbidden-moves}),\n\ + then $F(X) \\ncong F(Y)$.\n\n\\item\\emph{(ii) Predicate preservation.}\nIf $\\varphi$ is $\\T$-derivable, then $F(\\\ + varphi)$ is ZFC-derivable:\n$\\T \\vdash \\varphi \\;\\Longrightarrow\\;\n\\mathrm{ZFC} \\vdash F(\\varphi)$.\nThe converse\ + \ does not hold: ZFC may derive statements\nwhose $\\T$-preimages involve forbidden moves.\n\n\\item\\emph{(iii) Decoder\ + \ compatibility.}\nThe bridge interleaves the two address systems:\n\\[\n\\begin{tikzcd}[column sep=large]\n\\operatorname{Cat}_{\\\ + T}(\\Elayer{2})\n \\ar[r, \"F\"]\n \\ar[d, \"\\mathrm{NF}\"']\n& \\mathbf{Mod}(\\mathrm{ZFC})\n \\ar[d, \"{\\\ + ulcorner\\cdot\\urcorner}\"] \\\\\n\\hat{\\mathbb{Z}}_{\\T}\n \\ar[r, \"\\phi\"']\n& \\mathbb{N}\n\\end{tikzcd}\n\\\ + ]\ncommutes up to the decoder map\n$\\phi \\colon \\hat{\\mathbb{Z}}_{\\T} \\to \\mathbb{N}$,\nwhich sends NF addresses\n\ + to G\\\"odel numbers of the corresponding ZFC-translations.\n\n\\item\\emph{(iv) Invariant reflection.}\nThe bridge maps\ + \ the $\\T$-invariant (primorial coherence)\nto a sub-statement of the ZFC-invariant (consistency):\n$F(\\mathrm{Coh}_{\\\ + T}) \\vdash\n\\mathrm{Con}(\\mathrm{ZFC} \\!\\upharpoonright\\! \\operatorname{Im}(F))$.\n\\end{enumerate}\nThe bridge\ + \ is \\emph{lossy}:\n$\\T$-structures with no ZFC counterpart\n(primorial tower, earned enrichment, ABCD decomposition)\n\ + are projected away;\nZFC-structures with no $\\T$-counterpart\n(unbounded powerset, global choice)\nare absent from $\\\ + operatorname{Im}(F)$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\ + \n## Source Context\n\n- Registry source: `book-03.jsonl` line 176\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch67-the-bridge-axiom.tex`\ + \ lines 58-116\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.BridgeAxiom`\n\ + - Name: `bridge_functor_exists`\n\n## Dependencies\n\n- Canonical: III.D67, III.D69\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A bridge is a structure-preserving functor F: Cat_τ(E₂) → Mod(ZFC) satisfying carrier preservation, predicate + preservation, decoder compatibility, and invariant reflection. The existence of such F is conjectural.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001517.md b/_corpus_v3_items/cid001517.md new file mode 100644 index 000000000..2f07be327 --- /dev/null +++ b/_corpus_v3_items/cid001517.md @@ -0,0 +1,180 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001517/ +title: Shadow Diagram +cid: cid001517 +primary_alias: DEF0287 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The image of a τ-internal commutative diagram under the bridge functor F. The shadow preserves commutativity but + may lose structure: each forbidden move introduces a specific degeneracy in the shadow.' +aliases: +- DEF0287 +- III.D72 +- shadow-diagram +- def:shadow-diagram +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005714 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 67 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Shadow Diagram** + + + The image of a τ-internal commutative diagram under the bridge functor F. The shadow preserves commutativity but may lose + structure: each forbidden move introduces a specific degeneracy in the shadow. + + + # Shadow Diagram + + + ## Summary + + + The image of a τ-internal commutative diagram under the bridge functor F. The shadow preserves commutativity but may lose + structure: each forbidden move introduces a specific degeneracy in the shadow. + + + ## Statement + + + ```latex + + % + + \label{def:shadow-diagram} + + Let $\mathcal{D}$ be a commutative diagram + + in $\operatorname{Cat}_{\T}(\Elayer{2})$ + + and $F$ a bridge (Definition~\ref{def:bridge-axiom}). + + The \textbf{shadow diagram} $F(\mathcal{D})$ + + is the image of $\mathcal{D}$ under~$F$. + + \begin{enumerate} + + \item\emph{(Commutativity.)} + + If $g \circ f = h$ in $\mathcal{D}$, + + then $F(g) \circ F(f) = F(h)$ in $F(\mathcal{D})$. + + + \item\emph{(Degeneracy.)} + + Each forbidden move $M_{i}$ + + (Definition~\ref{def:five-forbidden-moves}) + + introduces a specific degeneracy: + + $M_{1}$~collapses objects beyond the $\KAxiom{3}$ bound; + + $M_{2}$~identifies objects from different sectors; + + $M_{3}$~compresses long $\T$-descriptions into short ZFC-derivations; + + $M_{4}$~collapses depth-$k$ distinctions under ZFC quantification; + + $M_{5}$~merges objects with different NF addresses + + that ZFC re-encodes as identical. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 177 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch67-the-bridge-axiom.tex` lines + 142-166 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.BridgeAxiom` + + - Name: `shadow_diagram_check` + + + ## Dependencies + + + - Canonical: III.D71, III.D69 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The image of a τ-internal commutative diagram under the bridge functor F. The shadow preserves commutativity + but may lose structure: each forbidden move introduces a specific degeneracy in the shadow.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001518.md b/_corpus_v3_items/cid001518.md new file mode 100644 index 000000000..eef94df05 --- /dev/null +++ b/_corpus_v3_items/cid001518.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001518/ +title: Proof Theory as E₃ +cid: cid001518 +primary_alias: DEF0288 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Proof theory is E₃ self-modelling applied to E₂ code. Layer Template at E₃: Carrier = formal systems as objects, + Predicate = provability about provability, Decoder = metatheoretic interpretation, Invariant = metatheoretic consistency.' +aliases: +- DEF0288 +- III.D73 +- proof-theory-as-e +- def:proof-theory-as-e3 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005906 +- predicate: formalized_by + object: cid005912 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 70 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Proof Theory as E₃**\n\nProof theory is E₃ self-modelling applied to E₂ code. Layer Template at E₃: Carrier\ + \ = formal systems as objects, Predicate = provability about provability, Decoder = metatheoretic interpretation, Invariant\ + \ = metatheoretic consistency.\n\n# Proof Theory as E₃\n\n## Summary\n\nProof theory is E₃ self-modelling applied to E₂\ + \ code. Layer Template at E₃: Carrier = formal systems as objects, Predicate = provability about provability, Decoder\ + \ = metatheoretic interpretation, Invariant = metatheoretic consistency.\n\n## Statement\n\n```latex\n%\n\\label{def:proof-theory-as-e3}\n\ + \\textbf{Proof theory} is the $\\Elayer{3}$ self-modelling applied to $\\Elayer{2}$ code.\nIts four template components\ + \ are:\n\\begin{enumerate}\n \\item[\\emph{(Carrier.)}]\n Formal systems as objects.\n The carrier is the class\ + \ of $\\Elayer{2}$ virtual machines---ZFC-VM\n (Definition~\\ref{def:zfc-as-e2-vm}), Peano Arithmetic, type theories---each\ + \ viewed\n not as a tool to be used but as a mathematical object to be studied.\n\n \\item[\\emph{(Predicate.)}]\n\ + \ Provability about provability.\n Metatheoretic reasoning about a formal system must not contradict the system's\ + \ own\n derivation behaviour: the metatheory is sound with respect to the object theory.\n\n \\item[\\emph{(Decoder.)}]\n\ + \ Interpretation of metatheoretic results.\n The $\\Elayer{3}$ decoder translates a syntactic metatheorem\n (e.g.,\ + \ ``ZFC does not prove $\\mathrm{Con}(\\mathrm{ZFC})$'')\n into a structural diagnosis (e.g., ``consistency is a host-level\ + \ property'').\n\n \\item[\\emph{(Invariant.)}]\n Metatheoretic consistency.\n The host-level property of Chapter~\\\ + ref{ch:goedel-and-the-vm-boundary}\n is now the \\emph{object of study}.\n At $\\Elayer{2}$, consistency was the\ + \ invariant---the property that the VM never crashes.\n At $\\Elayer{3}$, proof theory investigates this invariant:\n\ + \ which systems prove their own consistency? Which do not?\n The $\\Elayer{3}$ invariant is the stability of these\ + \ conclusions\n under changes of metatheory.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\nThis item is definitional.\ + \ No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 184\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch70-proof-theory-as-e3.tex` lines 96-129\n\n## Lean\ + \ / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.ProofTheoryE3`\n- Name: `proof_theory_e3_check`\n\ + \n## Dependencies\n\n- Canonical: III.D05, III.D09, III.D67\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Proof theory is E₃ self-modelling applied to E₂ code. Layer Template at E₃: Carrier = formal systems as objects, + Predicate = provability about provability, Decoder = metatheoretic interpretation, Invariant = metatheoretic consistency.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001519.md b/_corpus_v3_items/cid001519.md new file mode 100644 index 000000000..23f3bbaef --- /dev/null +++ b/_corpus_v3_items/cid001519.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001519/ +title: Diagrammatic Sector of E₃ +cid: cid001519 +primary_alias: DEF0289 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The ω-coupling sector of the 4+1 template at E₃: where formal categorical reasoning about categorical reasoning + lives. Book III itself operates in this sector. The diagrammatic sector is one reading of the sector decomposition at the + terminal enrichment level.' +aliases: +- DEF0289 +- III.D74 +- diagrammatic-sector-of-e +- def:diagrammatic-sector-e3 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005907 +- predicate: formalized_by + object: cid005908 +- predicate: formalized_by + object: cid005913 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 70 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Diagrammatic Sector of E₃**\n\nThe ω-coupling sector of the 4+1 template at E₃: where formal categorical reasoning\ + \ about categorical reasoning lives. Book III itself operates in this sector. The diagrammatic sector is one reading of\ + \ the sector decomposition at the terminal enrichment level.\n\n# Diagrammatic Sector of E₃\n\n## Summary\n\nThe ω-coupling\ + \ sector of the 4+1 template at E₃: where formal categorical reasoning about categorical reasoning lives. Book III itself\ + \ operates in this sector. The diagrammatic sector is one reading of the sector decomposition at the terminal enrichment\ + \ level.\n\n## Statement\n\n```latex\n%\n\\label{def:diagrammatic-sector-e3}\nThe \\textbf{diagrammatic sector} of $\\\ + Elayer{3}$ is the $\\omega$-coupling sector\nof the $4{+}1$ decomposition at the terminal enrichment level.\nIts content\ + \ is formal categorical reasoning \\emph{about} formal categorical reasoning.\n\\begin{enumerate}\n \\item The carrier\ + \ consists of $\\Elayer{3}$-objects whose self-model is\n category-theoretic: functors, natural transformations,\ + \ and enrichment data\n that model other functors, natural transformations, and enrichment data.\n\n \\item\ + \ The predicate requires that self-modelling respects the enrichment ladder:\n an $\\Elayer{3}$-object in the\ + \ diagrammatic sector models $\\Elayer{0}$--$\\Elayer{2}$\n \\emph{as enrichment layers}, not merely as collections\ + \ of objects.\n\n \\item The decoder maps categorical diagrams to metatheoretic conclusions:\n a commuting\ + \ diagram at $\\Elayer{3}$ yields a structural theorem\n about the enrichment tower.\n\n \\item The invariant\ + \ is the coherence of the diagrammatic self-model\n with the Saturation Theorem (Theorem~\\ref{thm:saturation-e3},\ + \ Ch.~7):\n diagram-chasing about diagram-chasing produces no new enrichment level.\n\\end{enumerate}\nThe diagrammatic\ + \ sector mediates between the four primitive $\\Elayer{3}$ sectors\nexactly as the $\\omega$-coupling sector mediates\ + \ at $\\Elayer{0}$ and $\\Elayer{1}$.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof\ + \ is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 185\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch70-proof-theory-as-e3.tex`\ + \ lines 176-201\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.ProofTheoryE3`\n\ + - Name: `self_model_check`\n\n## Dependencies\n\n- Canonical: III.D73, III.D10\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The ω-coupling sector of the 4+1 template at E₃: where formal categorical reasoning about categorical reasoning + lives. Book III itself operates in this sector. The diagrammatic sector is one reading of the sector decomposition at the + termin' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001520.md b/_corpus_v3_items/cid001520.md new file mode 100644 index 000000000..e5ef8b98c --- /dev/null +++ b/_corpus_v3_items/cid001520.md @@ -0,0 +1,95 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001520/ +title: E₂→E₃ Boundary Crossing +cid: cid001520 +primary_alias: DEF0290 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: An operation that (i) originates at E₂ with self-referential codes, (ii) requires determining a host-level property + of the entire code space, (iii) resides at E₃ because it demands self-modelling. The paradox is the error message when E₂ + self-reference hits the E₃ wall. +aliases: +- DEF0290 +- III.D75 +- e-e-boundary-crossing +- def:e2-e3-boundary-crossing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005909 +- predicate: formalized_by + object: cid005910 +- predicate: formalized_by + object: cid005914 +- predicate: formalized_by + object: cid005915 +- predicate: formalized_by + object: cid005916 +- predicate: formalized_by + object: cid005917 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 71 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₂→E₃ Boundary Crossing**\n\nAn operation that (i) originates at E₂ with self-referential codes, (ii) requires\ + \ determining a host-level property of the entire code space, (iii) resides at E₃ because it demands self-modelling. The\ + \ paradox is the error message when E₂ self-reference hits the E₃ wall.\n\n# E₂→E₃ Boundary Crossing\n\n## Summary\n\n\ + An operation that (i) originates at E₂ with self-referential codes, (ii) requires determining a host-level property of\ + \ the entire code space, (iii) resides at E₃ because it demands self-modelling. The paradox is the error message when\ + \ E₂ self-reference hits the E₃ wall.\n\n## Statement\n\n```latex\n%\n\\label{def:e2-e3-boundary-crossing}\nAn \\textbf{$\\\ + Elayer{2} \\to \\Elayer{3}$ boundary crossing}\nis an operation satisfying:\n\\begin{enumerate}\n \\item \\textbf{Origin\ + \ at $\\Elayer{2}$.}\n The operation begins within an $\\Elayer{2}$ virtual machine\n (Definition~\\\ + ref{def:zfc-as-e2-vm})\n and uses self-referential codes.\n \\item \\textbf{Self-modelling requirement.}\n\ + \ The operation requires determining a property\n of the \\emph{entire} code space\n that is\ + \ a host-level property\n (Definition~\\ref{def:host-level-property}).\n \\item \\textbf{$\\Elayer{3}$ residence.}\n\ + \ The determination presupposes an observer\n who models the observation process itself\n (Definition~\\\ + ref{def:proof-theory-as-e3}).\n No $\\Elayer{2}$ code can perform it.\n\\end{enumerate}\nThe \\emph{paradox}\ + \ is the error message ---\ncontradiction, undecidability, or impossibility ---\nproduced when the $\\Elayer{2}$ agent\n\ + attempts the $\\Elayer{3}$ operation anyway.\n```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript\ + \ proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 186\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch71-four-paradoxes-as-boundary-crossings.tex`\ + \ lines 50-75\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.ProofTheoryE3`\n\ + - Name: `four_paradox_check`\n\n## Dependencies\n\n- Canonical: III.D73, III.D70, III.D67\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: An operation that (i) originates at E₂ with self-referential codes, (ii) requires determining a host-level property + of the entire code space, (iii) resides at E₃ because it demands self-modelling. The paradox is the error message when E₂ + se +noindex: false +--- + diff --git a/_corpus_v3_items/cid001521.md b/_corpus_v3_items/cid001521.md new file mode 100644 index 000000000..3cff4fc09 --- /dev/null +++ b/_corpus_v3_items/cid001521.md @@ -0,0 +1,168 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001521/ +title: Minkowski Extension +cid: cid001521 +primary_alias: DEF0291 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Extension of the glued 3D space to 3+1D Minkowski spacetime by adjoining the base τ¹ as time direction. Split-complex + structure (j²=+1) provides Lorentzian signature. +aliases: +- DEF0291 +- III.D76 +- minkowski-extension +- def:minkowski-extension +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 8, Chapter 75 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Minkowski Extension** + + + Extension of the glued 3D space to 3+1D Minkowski spacetime by adjoining the base τ¹ as time direction. Split-complex + structure (j²=+1) provides Lorentzian signature. + + + # Minkowski Extension + + + ## Summary + + + Extension of the glued 3D space to 3+1D Minkowski spacetime by adjoining the base τ¹ as time direction. Split-complex + structure (j²=+1) provides Lorentzian signature. + + + ## Statement + + + ```latex + + \label{def:minkowski-extension} + + The \emph{Minkowski extension} of $M_{\tau^3}$ is the $3{+}1$-dimensional spacetime + + \begin{equation}\label{eq:ch75-minkowski} + + \mathcal{M}_\tau \;=\; \tau^1 \times M_{\tau^3}, + + \end{equation} + + where: + + \begin{enumerate} + + \item\emph{(Time.)} + + The base $\tau^1$ provides the time direction; the progression structure (Book~I, Part~II) provides causal ordering. + + + \item\emph{(Space.)} + + The fiber at each $x \in \tau^1$ is the three-dimensional bulk $U_x \subset M_{\tau^3}$ constructed by Theorem~\ref{thm:global-cartesian-gluing}. + + + \item\emph{(Signature.)} + + The split-complex boundary ring $H_\tau = \mathbb{Z}[\,\jj\,]/(\jj^2 - 1)$ provides the metric signature. The decomposition + $H_\tau \cong \mathbb{Z} \oplus \mathbb{Z}$ via idempotents $e_\pm = \tfrac{1}{2}(1 \pm \jj)$ separates the time-like + component ($e_-$) from the space-like components ($e_+$), yielding the hyperbolic signature $(-, +, +, +)$. + + + \item\emph{(Light cones.)} + + The null directions are the zero divisors of $H_\tau$: elements $a + b\jj$ with $a^2 = b^2$. Their global assembly via + the Coherent Force produces the causal structure of spacetime. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 194 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part08/ch75-the-global-cartesian-gluing.tex` + lines 200-221 + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: III.T50 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Extension of the glued 3D space to 3+1D Minkowski spacetime by adjoining the base τ¹ as time direction. Split-complex + structure (j²=+1) provides Lorentzian signature. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001522.md b/_corpus_v3_items/cid001522.md new file mode 100644 index 000000000..816390987 --- /dev/null +++ b/_corpus_v3_items/cid001522.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001522/ +title: Physical E₂ Agent +cid: cid001522 +primary_alias: DEF0292 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: An E₁ object (spatially bounded, energy-constrained, causally local) hosting E₂ operational closure (code→execution→code). + Physical computation is E₁-hosted E₂, not abstract E₂. +aliases: +- DEF0292 +- III.D77 +- physical-e-agent +- def:physical-e2-agent +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 78 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Physical E₂ Agent**\n\nAn E₁ object (spatially bounded, energy-constrained, causally local) hosting E₂ operational\ + \ closure (code→execution→code). Physical computation is E₁-hosted E₂, not abstract E₂.\n\n# Physical E₂ Agent\n\n## Summary\n\ + \nAn E₁ object (spatially bounded, energy-constrained, causally local) hosting E₂ operational closure (code→execution→code).\ + \ Physical computation is E₁-hosted E₂, not abstract E₂.\n\n## Statement\n\n```latex\n\\label{def:physical-e2-agent}\n\ + A \\textbf{physical $\\Elayer{2}$ agent} is an $\\Elayer{2}$\ncomputational agent $(C, D, \\varphi)$\n(Definition~\\ref{def:e2-computational-agent})\n\ + whose carrier is $\\Elayer{1}$-admissible:\n\\begin{enumerate}\n\\item[(i)] Code $C$, decoder $D$, and all intermediate\ + \ states\n $C_n$ are $\\Elayer{1}$-objects: finite spatial extent,\n bounded energy, sector addresses in the\ + \ $4{+}1$\n decomposition.\n\\item[(ii)] The execution map $\\varphi \\colon C_n \\to C_{n+1}$\n respects $\\\ + Elayer{1}$ causal structure: each step\n involves only causally connected regions\n (Remark~\\ref{rem:light-cone-consistency},\ + \ Ch.~77).\n\\item[(iii)] Operational closure holds within the $\\Elayer{1}$\n carrier: the cycle does not require\ + \ external resources\n that grow without bound.\n\\end{enumerate}\nUnlike the $\\tau$-Tower Machine (which is $\\\ + tau$-native),\na physical $\\Elayer{2}$ agent inherits its constraints from\nthe $\\Elayer{1}$ physics of its carrier.\n\ + ```\n\n## Proof / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n\ + - Registry source: `book-03.jsonl` line 199\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch78-physical-turing-machines.tex`\ + \ lines 69-92\n\n## Lean / Formalization Notes\n\n- Formalization: `not_applicable`\n- Module: `None`\n- Name: `None`\n\ + \n## Dependencies\n\n- Canonical: III.D49, III.D50, III.R40\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: An E₁ object (spatially bounded, energy-constrained, causally local) hosting E₂ operational closure (code→execution→code). + Physical computation is E₁-hosted E₂, not abstract E₂. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001523.md b/_corpus_v3_items/cid001523.md new file mode 100644 index 000000000..9d74abcdc --- /dev/null +++ b/_corpus_v3_items/cid001523.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001523/ +title: τ-Native Abstract Turing Machine +cid: cid001523 +primary_alias: DEF0293 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A τ-native abstract TM (TATM) defined in the E₃ diagrammatic sector: tape as τ-presheaf on the primorial tower, + state set as finite τ-object, transition function as τ-morphism, computation as compatible family of finite computations + at each primorial level. Abstract but τ-native, not ZFC-imported.' +aliases: +- DEF0293 +- III.D78 +- native-abstract-turing-machine +- def:tau-native-abstract-tm +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 79 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Native Abstract Turing Machine**\n\nA τ-native abstract TM (TATM) defined in the E₃ diagrammatic sector:\ + \ tape as τ-presheaf on the primorial tower, state set as finite τ-object, transition function as τ-morphism, computation\ + \ as compatible family of finite computations at each primorial level. Abstract but τ-native, not ZFC-imported.\n\n# τ-Native\ + \ Abstract Turing Machine\n\n## Summary\n\nA τ-native abstract TM (TATM) defined in the E₃ diagrammatic sector: tape as\ + \ τ-presheaf on the primorial tower, state set as finite τ-object, transition function as τ-morphism, computation as compatible\ + \ family of finite computations at each primorial level. Abstract but τ-native, not ZFC-imported.\n\n## Statement\n\n\ + ```latex\n\\label{def:tau-native-abstract-tm}\nA \\textbf{$\\tau$-native abstract Turing machine} (TATM) is a tuple\n\ + $\\mathcal{M} = (Q,\\, \\Sigma,\\, \\delta,\\, q_0,\\, q_{\\mathrm{acc}},\\, q_{\\mathrm{rej}},\\, \\mathcal{T}\\!\\mathit{ape})$\n\ + whose components are $\\Elayer{3}$-diagrammatic objects\n(Definition~\\ref{def:diagrammatic-sector-e3}):\n\\begin{enumerate}\n\ + \ \\item \\textbf{Tape} ($\\mathcal{T}\\!\\mathit{ape}$):\n a $\\tau$-presheaf on the primorial tower---a\ + \ compatible\n family of finite functions\n $\\{f_k \\colon \\{1, \\ldots, \\operatorname{Prim}(k)\\\ + } \\to \\Sigma\\}_{k \\geq 1}$\n with $f_{k+1}|_{\\{1,\\ldots,\\operatorname{Prim}(k)\\}} = f_k$.\n \ + \ The tape is the inverse limit, not a function from a completed~$\\omega$.\n\n \\item \\textbf{State set} ($Q$):\n\ + \ a finite $\\tau$-object at depth~$k_Q$\n with $|Q| \\leq \\operatorname{Prim}(k_Q)$.\n\n \\item\ + \ \\textbf{Alphabet} ($\\Sigma$):\n a finite $\\tau$-object, similarly bounded at some depth~$k_\\Sigma$.\n\n\ + \ \\item \\textbf{Transition function} ($\\delta$):\n a $\\tau$-morphism\n $\\delta \\colon Q \\\ + times \\Sigma \\to Q \\times \\Sigma \\times \\{L, R\\}$,\n a finite lookup table with $\\tau$-address entries.\n\ + \n \\item \\textbf{Computation}:\n a diagram in the $\\Elayer{3}$ diagrammatic sector---a\n compatible\ + \ family $\\{\\mathcal{C}_k\\}_{k \\geq k_0}$,\n where $\\mathcal{C}_k$ restricts the computation\n \ + \ to the tape segment $\\{1, \\ldots, \\operatorname{Prim}(k)\\}$.\n\\end{enumerate}\nThe TATM is genuinely abstract:\ + \ no $\\Elayer{1}$ hosting is required.\nYet it lives within~$\\T$'s diagrammatic sector, not within ZFC.\n```\n\n## Proof\ + \ / Justification\n\nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source:\ + \ `book-03.jsonl` line 203\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch79-abstract-computation-in-tau.tex`\ + \ lines 86-121\n\n## Lean / Formalization Notes\n\n- Formalization: `not_applicable`\n- Module: `None`\n- Name: `None`\n\ + \n## Dependencies\n\n- Canonical: III.D73, III.D74, III.D51\n\n## Related Results\n\n_Generated by later projection phases._\n\ + \n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A τ-native abstract TM (TATM) defined in the E₃ diagrammatic sector: tape as τ-presheaf on the primorial tower, + state set as finite τ-object, transition function as τ-morphism, computation as compatible family of finite computations + at each' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001524.md b/_corpus_v3_items/cid001524.md new file mode 100644 index 000000000..c52cc33de --- /dev/null +++ b/_corpus_v3_items/cid001524.md @@ -0,0 +1,76 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001524/ +title: Physical Realizability Predicate +cid: cid001524 +primary_alias: DEF0294 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Formal predicate Phys(M) characterizing physically realizable TMs. Satisfies all three conditions of a host-level + property (III.D70): it quantifies over all possible physical instantiations, requires E₃ vantage, and is not capturable + at E₂. ZFC cannot internalize this predicate.' +aliases: +- DEF0294 +- III.D79 +- physical-realizability-predicate +- def:physical-realizability-predicate +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 80 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Physical Realizability Predicate**\n\nFormal predicate Phys(M) characterizing physically realizable TMs. Satisfies\ + \ all three conditions of a host-level property (III.D70): it quantifies over all possible physical instantiations, requires\ + \ E₃ vantage, and is not capturable at E₂. ZFC cannot internalize this predicate.\n\n# Physical Realizability Predicate\n\ + \n## Summary\n\nFormal predicate Phys(M) characterizing physically realizable TMs. Satisfies all three conditions of a\ + \ host-level property (III.D70): it quantifies over all possible physical instantiations, requires E₃ vantage, and is\ + \ not capturable at E₂. ZFC cannot internalize this predicate.\n\n## Statement\n\n```latex\n\\label{def:physical-realizability-predicate}\n\ + A Turing machine~$M$ is \\textbf{physically realisable} if there exists\nan $\\Elayer{1}$-object~$O$ in Category~$\\T$\ + \ such that\n$O$~hosts $M$'s operational closure---the\nCode--Execution--Code cycle of~$M$ as a physical process\n(Definition~\\\ + ref{def:physical-e2-agent}).\nFormally:\n\\[\n \\mathrm{Phys}(M)\n \\;\\;\\stackrel{\\mathrm{def}}{=}\\;\\;\n \\exists\\\ + , O \\in \\operatorname{Obj}(\\Elayer{1})\n \\;:\\;\n O \\;\\text{hosts}\\; M.\n\\]\n```\n\n## Proof / Justification\n\ + \nThis item is definitional. No manuscript proof is required.\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 206\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch80-the-zfc-provability-horizon.tex`\ + \ lines 107-123\n\n## Lean / Formalization Notes\n\n- Formalization: `not_applicable`\n- Module: `None`\n- Name: `None`\n\ + \n## Dependencies\n\n- Canonical: III.D77, III.D70\n\n## Related Results\n\n_Generated by later projection phases._\n\n\ + ## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Formal predicate Phys(M) characterizing physically realizable TMs. Satisfies all three conditions of a host-level + property (III.D70): it quantifies over all possible physical instantiations, requires E₃ vantage, and is not capturable + at E₂.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001525.md b/_corpus_v3_items/cid001525.md new file mode 100644 index 000000000..e38dc4f70 --- /dev/null +++ b/_corpus_v3_items/cid001525.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001525/ +title: q-Expansion Coefficients +cid: cid001525 +primary_alias: DEF0295 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The q-expansion coefficients of Eisenstein series that determine the S-duality residuals: E₄ coefficient 240, E₆ + coefficient −504. Their sum 744 = dim(E₈ roots) + 504 is the constant term of the j-invariant.' +aliases: +- DEF0295 +- III.D80 +- q-expansion-coefficients +- def:q-expansion-coefficients +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005841 +- predicate: formalized_by + object: cid005842 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 25 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**q-Expansion Coefficients** + + + The q-expansion coefficients of Eisenstein series that determine the S-duality residuals: E₄ coefficient 240, E₆ coefficient + −504. Their sum 744 = dim(E₈ roots) + 504 is the constant term of the j-invariant. + + + # q-Expansion Coefficients + + + ## Summary + + + The q-expansion coefficients of Eisenstein series that determine the S-duality residuals: E₄ coefficient 240, E₆ coefficient + −504. Their sum 744 = dim(E₈ roots) + 504 is the constant term of the j-invariant. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 210 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.ConfinementBridge` + + - Name: `Tau.BookIII.Spectral.ConfinementBridge.E4_qcoeff` + + + ## Dependencies + + + - Canonical: III.D24 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The q-expansion coefficients of Eisenstein series that determine the S-duality residuals: E₄ coefficient 240, + E₆ coefficient −504. Their sum 744 = dim(E₈ roots) + 504 is the constant term of the j-invariant.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001526.md b/_corpus_v3_items/cid001526.md new file mode 100644 index 000000000..0d50ab5bb --- /dev/null +++ b/_corpus_v3_items/cid001526.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001526/ +title: Spectral Projector +cid: cid001526 +primary_alias: DEF0296 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Spectral projector P_n for the n-th eigenspace of H_L. Idempotent (P_n² = P_n) and mutually orthogonal (P_n·P_m = + 0 for n ≠ m). Verified at finite stages. +aliases: +- DEF0296 +- III.D81 +- spectral-projector +- def:spectral-projector +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005843 +- predicate: formalized_by + object: cid005844 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 30 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Spectral Projector** + + + Spectral projector P_n for the n-th eigenspace of H_L. Idempotent (P_n² = P_n) and mutually orthogonal (P_n·P_m = 0 for + n ≠ m). Verified at finite stages. + + + # Spectral Projector + + + ## Summary + + + Spectral projector P_n for the n-th eigenspace of H_L. Idempotent (P_n² = P_n) and mutually orthogonal (P_n·P_m = 0 for + n ≠ m). Verified at finite stages. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 214 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SpectralDecomp` + + - Name: `spectral_projector` + + + ## Dependencies + + + - Canonical: III.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Spectral projector P_n for the n-th eigenspace of H_L. Idempotent (P_n² = P_n) and mutually orthogonal (P_n·P_m + = 0 for n ≠ m). Verified at finite stages. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001527.md b/_corpus_v3_items/cid001527.md new file mode 100644 index 000000000..af80fb23b --- /dev/null +++ b/_corpus_v3_items/cid001527.md @@ -0,0 +1,127 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001527/ +title: Spectral Measure +cid: cid001527 +primary_alias: DEF0297 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Spectral measure μ_spec assigns weight 1/N to each eigenvalue λ_n. Total measure = 1 (all modes counted). Counting + measure on the discrete spectrum. +aliases: +- DEF0297 +- III.D82 +- spectral-measure +- def:spectral-measure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 30 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Spectral Measure** + + + Spectral measure μ_spec assigns weight 1/N to each eigenvalue λ_n. Total measure = 1 (all modes counted). Counting measure + on the discrete spectrum. + + + # Spectral Measure + + + ## Summary + + + Spectral measure μ_spec assigns weight 1/N to each eigenvalue λ_n. Total measure = 1 (all modes counted). Counting measure + on the discrete spectrum. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 215 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SpectralDecomp` + + - Name: `spectral_measure` + + + ## Dependencies + + + - Canonical: III.D81 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Spectral measure μ_spec assigns weight 1/N to each eigenvalue λ_n. Total measure = 1 (all modes counted). Counting + measure on the discrete spectrum. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001528.md b/_corpus_v3_items/cid001528.md new file mode 100644 index 000000000..a003942a6 --- /dev/null +++ b/_corpus_v3_items/cid001528.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001528/ +title: Kleene Fixed Point +cid: cid001528 +primary_alias: DEF0298 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Self-application operator S(c) = (c+c) mod M_k and its fixed points. At every finite stage, c=0 is always a fixed + point. Models Kleene's recursion theorem constructively on Z/M_k Z. +aliases: +- DEF0298 +- III.D83 +- kleene-fixed-point +- def:kleene-fixed-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005795 +- predicate: formalized_by + object: cid005796 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Kleene Fixed Point** + + + Self-application operator S(c) = (c+c) mod M_k and its fixed points. At every finite stage, c=0 is always a fixed point. + Models Kleene''s recursion theorem constructively on Z/M_k Z. + + + # Kleene Fixed Point + + + ## Summary + + + Self-application operator S(c) = (c+c) mod M_k and its fixed points. At every finite stage, c=0 is always a fixed point. + Models Kleene''s recursion theorem constructively on Z/M_k Z. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 218 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Witness` + + - Name: `kleene_fixed_point` + + + ## Dependencies + + + - Canonical: III.D30 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Self-application operator S(c) = (c+c) mod M_k and its fixed points. At every finite stage, c=0 is always a fixed + point. Models Kleene's recursion theorem constructively on Z/M_k Z. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001529.md b/_corpus_v3_items/cid001529.md new file mode 100644 index 000000000..2e6cb90f0 --- /dev/null +++ b/_corpus_v3_items/cid001529.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001529/ +title: E₂ Orbit Structure +cid: cid001529 +primary_alias: DEF0299 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Orbit of code c under decoder d at stage k. Orbit lengths exhibit diversity (not all equal), proving computational + richness. E₁ orbits have length ≤ 2 (bipolar); E₂ orbits can have prime lengths 3, 5, etc. +aliases: +- DEF0299 +- III.D84 +- e-orbit-structure +- def:e2-orbit-structure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005797 +- predicate: formalized_by + object: cid005798 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₂ Orbit Structure** + + + Orbit of code c under decoder d at stage k. Orbit lengths exhibit diversity (not all equal), proving computational richness. + E₁ orbits have length ≤ 2 (bipolar); E₂ orbits can have prime lengths 3, 5, etc. + + + # E₂ Orbit Structure + + + ## Summary + + + Orbit of code c under decoder d at stage k. Orbit lengths exhibit diversity (not all equal), proving computational richness. + E₁ orbits have length ≤ 2 (bipolar); E₂ orbits can have prime lengths 3, 5, etc. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 219 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Witness` + + - Name: `orbit_length` + + + ## Dependencies + + + - Canonical: III.D83 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Orbit of code c under decoder d at stage k. Orbit lengths exhibit diversity (not all equal), proving computational + richness. E₁ orbits have length ≤ 2 (bipolar); E₂ orbits can have prime lengths 3, 5, etc. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001530.md b/_corpus_v3_items/cid001530.md new file mode 100644 index 000000000..78440aeaf --- /dev/null +++ b/_corpus_v3_items/cid001530.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001530/ +title: Self-Referential Fixed Point +cid: cid001530 +primary_alias: DEF0300 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₃ predicate at stage k: triple-reduce stability and squaring compatibility. c* satisfies E₃ iff reduce³(c*) = reduce(c*) + AND reduce(c*²) = (reduce(c*))². All elements satisfy E₃ at finite stages (density = 100%).' +aliases: +- DEF0300 +- III.D85 +- self-referential-fixed-point +- def:self-ref-fixed-point +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005900 +- predicate: formalized_by + object: cid005901 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Self-Referential Fixed Point** + + + E₃ predicate at stage k: triple-reduce stability and squaring compatibility. c* satisfies E₃ iff reduce³(c*) = reduce(c*) + AND reduce(c*²) = (reduce(c*))². All elements satisfy E₃ at finite stages (density = 100%). + + + # Self-Referential Fixed Point + + + ## Summary + + + E₃ predicate at stage k: triple-reduce stability and squaring compatibility. c* satisfies E₃ iff reduce³(c*) = reduce(c*) + AND reduce(c*²) = (reduce(c*))². All elements satisfy E₃ at finite stages (density = 100%). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 222 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Mirror.E3Witness` + + - Name: `e3_predicate` + + + ## Dependencies + + + - Canonical: III.P34, III.D47 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₃ predicate at stage k: triple-reduce stability and squaring compatibility. c* satisfies E₃ iff reduce³(c*) + = reduce(c*) AND reduce(c*²) = (reduce(c*))². All elements satisfy E₃ at finite stages (density = 100%).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001531.md b/_corpus_v3_items/cid001531.md new file mode 100644 index 000000000..db74dce44 --- /dev/null +++ b/_corpus_v3_items/cid001531.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001531/ +title: Paradox Absorption Map +cid: cid001531 +primary_alias: DEF0301 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Four classical paradoxes (Cantor, Russell, Gödel, Turing) each mapped to a construction on Z/M_k Z. All paradox constructions + produce valid E₃ elements — the self-model absorbs self-reference. Verified at stages 1-3. +aliases: +- DEF0301 +- III.D86 +- paradox-absorption-map +- def:paradox-absorption +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005902 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Paradox Absorption Map** + + + Four classical paradoxes (Cantor, Russell, Gödel, Turing) each mapped to a construction on Z/M_k Z. All paradox constructions + produce valid E₃ elements — the self-model absorbs self-reference. Verified at stages 1-3. + + + # Paradox Absorption Map + + + ## Summary + + + Four classical paradoxes (Cantor, Russell, Gödel, Turing) each mapped to a construction on Z/M_k Z. All paradox constructions + produce valid E₃ elements — the self-model absorbs self-reference. Verified at stages 1-3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 223 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Mirror.E3Witness` + + - Name: `paradox_absorbed_check` + + + ## Dependencies + + + - Canonical: III.D85 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Four classical paradoxes (Cantor, Russell, Gödel, Turing) each mapped to a construction on Z/M_k Z. All paradox + constructions produce valid E₃ elements — the self-model absorbs self-reference. Verified at stages 1-3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001532.md b/_corpus_v3_items/cid001532.md new file mode 100644 index 000000000..164299631 --- /dev/null +++ b/_corpus_v3_items/cid001532.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001532/ +title: Arithmetic Translation Functor +cid: cid001532 +primary_alias: DEF0302 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Functor Arith_tr maps τ-internal arithmetic to classical ℤ. At stage k, Arith_tr(x) = reduce(x, k) — the canonical + embedding Z/M_k Z ↪ ℤ. Well-defined and injective at each stage. +aliases: +- DEF0302 +- III.D87 +- arithmetic-translation-functor +- def:arith-translation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005759 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 80 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Arithmetic Translation Functor** + + + Functor Arith_tr maps τ-internal arithmetic to classical ℤ. At stage k, Arith_tr(x) = reduce(x, k) — the canonical embedding + Z/M_k Z ↪ ℤ. Well-defined and injective at each stage. + + + # Arithmetic Translation Functor + + + ## Summary + + + Functor Arith_tr maps τ-internal arithmetic to classical ℤ. At stage k, Arith_tr(x) = reduce(x, k) — the canonical embedding + Z/M_k Z ↪ ℤ. Well-defined and injective at each stage. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 226 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationArith` + + - Name: `arith_translate` + + + ## Dependencies + + + - Canonical: III.D71, III.T40 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Functor Arith_tr maps τ-internal arithmetic to classical ℤ. At stage k, Arith_tr(x) = reduce(x, k) — the canonical + embedding Z/M_k Z ↪ ℤ. Well-defined and injective at each stage. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001533.md b/_corpus_v3_items/cid001533.md new file mode 100644 index 000000000..6d9344eb0 --- /dev/null +++ b/_corpus_v3_items/cid001533.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001533/ +title: CRT-Integer Correspondence +cid: cid001533 +primary_alias: DEF0303 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CRT decomposition x ↦ (x mod p_1, ..., x mod p_k) and reconstruction. Roundtrip verified: decompose then reconstruct + = identity. This is the classical CRT on the primorial tower.' +aliases: +- DEF0303 +- III.D88 +- crt-integer-correspondence +- def:crt-integer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005760 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 80 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT-Integer Correspondence** + + + CRT decomposition x ↦ (x mod p_1, ..., x mod p_k) and reconstruction. Roundtrip verified: decompose then reconstruct = + identity. This is the classical CRT on the primorial tower. + + + # CRT-Integer Correspondence + + + ## Summary + + + CRT decomposition x ↦ (x mod p_1, ..., x mod p_k) and reconstruction. Roundtrip verified: decompose then reconstruct = + identity. This is the classical CRT on the primorial tower. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 227 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationArith` + + - Name: `crt_reconstruct` + + + ## Dependencies + + + - Canonical: III.D87 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CRT decomposition x ↦ (x mod p_1, ..., x mod p_k) and reconstruction. Roundtrip verified: decompose then reconstruct + = identity. This is the classical CRT on the primorial tower.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001534.md b/_corpus_v3_items/cid001534.md new file mode 100644 index 000000000..9923972be --- /dev/null +++ b/_corpus_v3_items/cid001534.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001534/ +title: Topological Translation Functor +cid: cid001534 +primary_alias: DEF0304 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Topo_tr maps the primorial tower filtration Z/M_1 ← Z/M_2 ← ... to a classical inverse system. Tower coherence: + reduce(reduce(x, k+1), k) = reduce(x, k). Verified at bound 10, depth 3.' +aliases: +- DEF0304 +- III.D89 +- topological-translation-functor +- def:topo-translation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005769 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Topological Translation Functor** + + + Topo_tr maps the primorial tower filtration Z/M_1 ← Z/M_2 ← ... to a classical inverse system. Tower coherence: reduce(reduce(x, + k+1), k) = reduce(x, k). Verified at bound 10, depth 3. + + + # Topological Translation Functor + + + ## Summary + + + Topo_tr maps the primorial tower filtration Z/M_1 ← Z/M_2 ← ... to a classical inverse system. Tower coherence: reduce(reduce(x, + k+1), k) = reduce(x, k). Verified at bound 10, depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 230 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationTopo` + + - Name: `tower_projection` + + + ## Dependencies + + + - Canonical: III.D87 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Topo_tr maps the primorial tower filtration Z/M_1 ← Z/M_2 ← ... to a classical inverse system. Tower coherence: + reduce(reduce(x, k+1), k) = reduce(x, k). Verified at bound 10, depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001535.md b/_corpus_v3_items/cid001535.md new file mode 100644 index 000000000..33e2e623f --- /dev/null +++ b/_corpus_v3_items/cid001535.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001535/ +title: Dimension Recovery +cid: cid001535 +primary_alias: DEF0305 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Primorial depth k = intrinsic dimension. Z/M_k Z has k independent prime coordinates (by CRT), giving k-torus T^k + = Π Z/p_i Z. dim = k at every stage. +aliases: +- DEF0305 +- III.D90 +- dimension-recovery +- def:dimension-recovery +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005770 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Dimension Recovery** + + + Primorial depth k = intrinsic dimension. Z/M_k Z has k independent prime coordinates (by CRT), giving k-torus T^k = Π + Z/p_i Z. dim = k at every stage. + + + # Dimension Recovery + + + ## Summary + + + Primorial depth k = intrinsic dimension. Z/M_k Z has k independent prime coordinates (by CRT), giving k-torus T^k = Π + Z/p_i Z. dim = k at every stage. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 231 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationTopo` + + - Name: `crt_dimension` + + + ## Dependencies + + + - Canonical: III.D89 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Primorial depth k = intrinsic dimension. Z/M_k Z has k independent prime coordinates (by CRT), giving k-torus + T^k = Π Z/p_i Z. dim = k at every stage. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001536.md b/_corpus_v3_items/cid001536.md new file mode 100644 index 000000000..f201a736e --- /dev/null +++ b/_corpus_v3_items/cid001536.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001536/ +title: Obstruction Cocycle +cid: cid001536 +primary_alias: DEF0306 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Obstruction value for each forbidden move at stage k. Mild moves (damage 1): bounded deviation. Breaking moves (damage + 3): deviation grows with M_k. Classifies translation failure precisely.' +aliases: +- DEF0306 +- III.D91 +- obstruction-cocycle +- def:obstruction-cocycle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005763 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Obstruction Cocycle** + + + Obstruction value for each forbidden move at stage k. Mild moves (damage 1): bounded deviation. Breaking moves (damage + 3): deviation grows with M_k. Classifies translation failure precisely. + + + # Obstruction Cocycle + + + ## Summary + + + Obstruction value for each forbidden move at stage k. Mild moves (damage 1): bounded deviation. Breaking moves (damage + 3): deviation grows with M_k. Classifies translation failure precisely. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 234 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationObstruction` + + - Name: `obstruction_value` + + + ## Dependencies + + + - Canonical: III.P36, III.P37 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Obstruction value for each forbidden move at stage k. Mild moves (damage 1): bounded deviation. Breaking moves + (damage 3): deviation grows with M_k. Classifies translation failure precisely.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001537.md b/_corpus_v3_items/cid001537.md new file mode 100644 index 000000000..58511848d --- /dev/null +++ b/_corpus_v3_items/cid001537.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001537/ +title: Forbidden Move Obstruction Classes +cid: cid001537 +primary_alias: DEF0307 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Each of 5 forbidden moves obstructs arithmetic or topological translation. Arithmetic: 3 obstructions. Topology: + 4 obstructions. Classification verified by native_decide.' +aliases: +- DEF0307 +- III.D92 +- forbidden-move-obstruction-classes +- def:obstruction-classes +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005764 +- predicate: formalized_by + object: cid005765 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Forbidden Move Obstruction Classes** + + + Each of 5 forbidden moves obstructs arithmetic or topological translation. Arithmetic: 3 obstructions. Topology: 4 obstructions. + Classification verified by native_decide. + + + # Forbidden Move Obstruction Classes + + + ## Summary + + + Each of 5 forbidden moves obstructs arithmetic or topological translation. Arithmetic: 3 obstructions. Topology: 4 obstructions. + Classification verified by native_decide. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 235 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationObstruction` + + - Name: `move_obstructs_arith` + + + ## Dependencies + + + - Canonical: III.D91 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Each of 5 forbidden moves obstructs arithmetic or topological translation. Arithmetic: 3 obstructions. Topology: + 4 obstructions. Classification verified by native_decide.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001538.md b/_corpus_v3_items/cid001538.md new file mode 100644 index 000000000..869e93d27 --- /dev/null +++ b/_corpus_v3_items/cid001538.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001538/ +title: RH Spectral Gap Characterization +cid: cid001538 +primary_alias: DEF0308 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'RH gap is precisely the O₃ axiom: correspondence between lemniscate eigenvalues and Riemann zeta zeros. At each + finite stage k, the correspondence holds. Gap = infinite-limit assertion. Verified at depth 5.' +aliases: +- DEF0308 +- III.D93 +- rh-spectral-gap-characterization +- def:rh-gap-char +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005808 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 35 (Wave M8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**RH Spectral Gap Characterization** + + + RH gap is precisely the O₃ axiom: correspondence between lemniscate eigenvalues and Riemann zeta zeros. At each finite + stage k, the correspondence holds. Gap = infinite-limit assertion. Verified at depth 5. + + + # RH Spectral Gap Characterization + + + ## Summary + + + RH gap is precisely the O₃ axiom: correspondence between lemniscate eigenvalues and Riemann zeta zeros. At each finite + stage k, the correspondence holds. Gap = infinite-limit assertion. Verified at depth 5. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 238 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.BridgeTightening` + + - Name: `rh_gap_char` + + + ## Dependencies + + + - Canonical: III.T18, III.D81 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'RH gap is precisely the O₃ axiom: correspondence between lemniscate eigenvalues and Riemann zeta zeros. At each + finite stage k, the correspondence holds. Gap = infinite-limit assertion. Verified at depth 5.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001539.md b/_corpus_v3_items/cid001539.md new file mode 100644 index 000000000..544656215 --- /dev/null +++ b/_corpus_v3_items/cid001539.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001539/ +title: YM Mass Gap Persistence +cid: cid001539 +primary_alias: DEF0309 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'τ-gap τ_gap(k) = min(B_k, C_k) is positive for all k ≥ 3 and grows monotonically. τ-internal analog of Yang-Mills + mass gap. Bridge gap: identification with physical mass gap requires bridge functor. Verified at depth 5.' +aliases: +- DEF0309 +- III.D94 +- ym-mass-gap-persistence +- def:ym-gap-persistence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005809 +- predicate: formalized_by + object: cid005810 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 36 (Wave M8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**YM Mass Gap Persistence** + + + τ-gap τ_gap(k) = min(B_k, C_k) is positive for all k ≥ 3 and grows monotonically. τ-internal analog of Yang-Mills mass + gap. Bridge gap: identification with physical mass gap requires bridge functor. Verified at depth 5. + + + # YM Mass Gap Persistence + + + ## Summary + + + τ-gap τ_gap(k) = min(B_k, C_k) is positive for all k ≥ 3 and grows monotonically. τ-internal analog of Yang-Mills mass + gap. Bridge gap: identification with physical mass gap requires bridge functor. Verified at depth 5. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 239 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.BridgeTightening` + + - Name: `ym_gap_persistence_check` + + + ## Dependencies + + + - Canonical: III.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'τ-gap τ_gap(k) = min(B_k, C_k) is positive for all k ≥ 3 and grows monotonically. τ-internal analog of Yang-Mills + mass gap. Bridge gap: identification with physical mass gap requires bridge functor. Verified at depth 5.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001540.md b/_corpus_v3_items/cid001540.md new file mode 100644 index 000000000..5ffc85e01 --- /dev/null +++ b/_corpus_v3_items/cid001540.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001540/ +title: Goldbach Representation +cid: cid001540 +primary_alias: DEF0310 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Goldbach representation: every even n ≥ 4 is p+q for primes p,q. Verified computationally up to 100. Partition count + r(n) measures additive richness. At primorial levels, partitions grow with depth.' +aliases: +- DEF0310 +- III.D95 +- goldbach-representation +- def:goldbach-representation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005994 +- predicate: formalized_by + object: cid005995 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach Representation** + + + Goldbach representation: every even n ≥ 4 is p+q for primes p,q. Verified computationally up to 100. Partition count r(n) + measures additive richness. At primorial levels, partitions grow with depth. + + + # Goldbach Representation + + + ## Summary + + + Goldbach representation: every even n ≥ 4 is p+q for primes p,q. Verified computationally up to 100. Partition count r(n) + measures additive richness. At primorial levels, partitions grow with depth. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 243 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.AdditiveConjectures` + + - Name: `goldbach_pair` + + + ## Dependencies + + + - Canonical: I.D19b + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Goldbach representation: every even n ≥ 4 is p+q for primes p,q. Verified computationally up to 100. Partition + count r(n) measures additive richness. At primorial levels, partitions grow with depth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001541.md b/_corpus_v3_items/cid001541.md new file mode 100644 index 000000000..d013320a2 --- /dev/null +++ b/_corpus_v3_items/cid001541.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001541/ +title: Twin Prime Distribution +cid: cid001541 +primary_alias: DEF0311 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Twin prime distribution: count of pairs (p, p+2) with both prime. Positive density at each primorial level. At least + 5 twin pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31).' +aliases: +- DEF0311 +- III.D96 +- twin-prime-distribution +- def:twin-prime-distribution +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005996 +- predicate: formalized_by + object: cid005998 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Prime Distribution** + + + Twin prime distribution: count of pairs (p, p+2) with both prime. Positive density at each primorial level. At least 5 + twin pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31). + + + # Twin Prime Distribution + + + ## Summary + + + Twin prime distribution: count of pairs (p, p+2) with both prime. Positive density at each primorial level. At least 5 + twin pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 244 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.AdditiveConjectures` + + - Name: `twin_prime_count` + + + ## Dependencies + + + - Canonical: III.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Twin prime distribution: count of pairs (p, p+2) with both prime. Positive density at each primorial level. + At least 5 twin pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001542.md b/_corpus_v3_items/cid001542.md new file mode 100644 index 000000000..8da7ab1ee --- /dev/null +++ b/_corpus_v3_items/cid001542.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001542/ +title: Radical Function +cid: cid001542 +primary_alias: DEF0312 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Radical of n: product of distinct prime divisors. rad(1)=1, rad(p^k)=p, rad(pq)=pq. Idempotent (rad(rad(n))=rad(n)). + On primorial tower, rad(M_k)=M_k (squarefree).' +aliases: +- DEF0312 +- III.D97 +- radical-function +- def:radical-function +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005678 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Radical Function** + + + Radical of n: product of distinct prime divisors. rad(1)=1, rad(p^k)=p, rad(pq)=pq. Idempotent (rad(rad(n))=rad(n)). On + primorial tower, rad(M_k)=M_k (squarefree). + + + # Radical Function + + + ## Summary + + + Radical of n: product of distinct prime divisors. rad(1)=1, rad(p^k)=p, rad(pq)=pq. Idempotent (rad(rad(n))=rad(n)). On + primorial tower, rad(M_k)=M_k (squarefree). + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 247 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCConjecture` + + - Name: `radical` + + + ## Dependencies + + + - Canonical: I.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Radical of n: product of distinct prime divisors. rad(1)=1, rad(p^k)=p, rad(pq)=pq. Idempotent (rad(rad(n))=rad(n)). + On primorial tower, rad(M_k)=M_k (squarefree).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001543.md b/_corpus_v3_items/cid001543.md new file mode 100644 index 000000000..545db713a --- /dev/null +++ b/_corpus_v3_items/cid001543.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001543/ +title: ABC Quality +cid: cid001543 +primary_alias: DEF0313 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ABC quality check: for coprime a+b=c, verify c < rad(abc)². Weak form (ε=1) of ABC conjecture. Verified for all + coprime pairs up to bound 15. Quality ratio computed at primorial levels.' +aliases: +- DEF0313 +- III.D98 +- abc-quality +- def:abc-quality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005679 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**ABC Quality** + + + ABC quality check: for coprime a+b=c, verify c < rad(abc)². Weak form (ε=1) of ABC conjecture. Verified for all coprime + pairs up to bound 15. Quality ratio computed at primorial levels. + + + # ABC Quality + + + ## Summary + + + ABC quality check: for coprime a+b=c, verify c < rad(abc)². Weak form (ε=1) of ABC conjecture. Verified for all coprime + pairs up to bound 15. Quality ratio computed at primorial levels. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 248 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCConjecture` + + - Name: `abc_triple_check` + + + ## Dependencies + + + - Canonical: III.D97 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ABC quality check: for coprime a+b=c, verify c < rad(abc)². Weak form (ε=1) of ABC conjecture. Verified for + all coprime pairs up to bound 15. Quality ratio computed at primorial levels.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001544.md b/_corpus_v3_items/cid001544.md new file mode 100644 index 000000000..f949e437b --- /dev/null +++ b/_corpus_v3_items/cid001544.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001544/ +title: Eratosthenes Sieve +cid: cid001544 +primary_alias: DEF0314 +type: definition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Sieve of Eratosthenes: computable primality predicate via trial division. Foundation for all three conjecture tracks.' +aliases: +- DEF0314 +- III.D99 +- eratosthenes-sieve +- def:eratosthenes-sieve +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Eratosthenes Sieve** + + + Sieve of Eratosthenes: computable primality predicate via trial division. Foundation for all three conjecture tracks. + + + # Eratosthenes Sieve + + + ## Summary + + + Sieve of Eratosthenes: computable primality predicate via trial division. Foundation for all three conjecture tracks. + + + ## Statement + + + ```latex + + \label{def:eratosthenes-sieve} + + The \emph{sieve of Eratosthenes} at bound~$n$ is a computable function + + $\mathrm{sieve}(n) : \{0, \ldots, n\} \to \{\mathrm{true}, \mathrm{false}\}$ + + such that $\mathrm{sieve}(k) = \mathrm{true}$ if and only if $k$ is prime. + + + \textbf{Lean:} \texttt{eratosthenes\_sieve} in + + \texttt{SieveInfrastructure.lean}.\qquad + + \textbf{Registry:} III.D99. + + ``` + + + ## Proof / Justification + + + This item is definitional. No manuscript proof is required. + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 251 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 82-90 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `eratosthenes_sieve` + + + ## Dependencies + + + - Canonical: I.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Sieve of Eratosthenes: computable primality predicate via trial division. Foundation for all three conjecture + tracks.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001545.md b/_corpus_v3_items/cid001545.md new file mode 100644 index 000000000..d8ec86b9b --- /dev/null +++ b/_corpus_v3_items/cid001545.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001545/ +title: E₁ Strictness Witness +cid: cid001545 +primary_alias: PRP0068 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The bipolar Hom decomposition Hom(A,B) = e₊·Hom₊ + e₋·Hom₋ is a genuine E₁ structure: no E₀ object carries this + decomposition. The split-complex scalar action on morphism spaces is the strictness witness separating E₁ from E₀.' +aliases: +- PRP0068 +- III.P01 +- e-strictness-witness +- prop:e1-strictness-witness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005854 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₁ Strictness Witness**\n\nThe bipolar Hom decomposition Hom(A,B) = e₊·Hom₊ + e₋·Hom₋ is a genuine E₁ structure:\ + \ no E₀ object carries this decomposition. The split-complex scalar action on morphism spaces is the strictness witness\ + \ separating E₁ from E₀.\n\n# E₁ Strictness Witness\n\n## Summary\n\nThe bipolar Hom decomposition Hom(A,B) = e₊·Hom₊\ + \ + e₋·Hom₋ is a genuine E₁ structure: no E₀ object carries this decomposition. The split-complex scalar action on morphism\ + \ spaces is the strictness witness separating E₁ from E₀.\n\n## Statement\n\n```latex\n%\n\\label{prop:e1-strictness-witness}\n\ + Let $A, B \\in E_0$ lie in distinct spectral sectors of the ABCD chart.\nThen the $H_\\tau$-enriched Hom object $[A, B]$\ + \ satisfies:\n\\begin{enumerate}\n \\item $[A, B] \\in E_1$\n (it is $H_\\tau$-enriched).\n \\item $[A,\ + \ B]_+ \\neq 0$ and $[A, B]_- \\neq 0$\n (the bipolar decomposition is non-trivial).\n \\item $[A, B] \\notin\ + \ E_0$\n (it cannot be reduced to an $E_0$-object).\n\\end{enumerate}\nIn particular, $E_1 \\neq E_0$.\n```\n\ + \n## Proof / Justification\n\n```latex\n(1) Self-enrichment (Book~II, Part~VIII)\nequips every Hom space with $H_\\tau$-module\ + \ structure.\n\n(2) By the spectral decomposition theorem (I.T12),\nobjects in distinct ABCD sectors produce non-zero\ + \ projections\nonto both $\\chi_+$ and $\\chi_-$ characters\nof the algebraic lemniscate (I.D18).\n\n(3) Suppose $[A,\ + \ B]$ were an $E_0$-object.\nThen it would have an NF address\n$\\NF([A, B]) = (r_1, \\ldots, r_d)$\nrecording \\emph{which}\ + \ object it is,\nnot its \\emph{internal Hom structure}.\nThe NF address carries no idempotent splitting,\nno $j^2 = +1$\ + \ unit, no bipolar projection.\nThe type mismatch---scalar address versus module structure---is\nirreducible.\n```\n\n\ + ## Source Context\n\n- Registry source: `book-03.jsonl` line 17\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch06-non-emptiness-and-strictness.tex`\ + \ lines 244-258\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `e1_strictness_8_3`\n\n## Dependencies\n\n- Canonical: III.D06, III.D07\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The bipolar Hom decomposition Hom(A,B) = e₊·Hom₊ + e₋·Hom₋ is a genuine E₁ structure: no E₀ object carries this + decomposition. The split-complex scalar action on morphism spaces is the strictness witness separating E₁ from E₀.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001546.md b/_corpus_v3_items/cid001546.md new file mode 100644 index 000000000..9fcb8c5ea --- /dev/null +++ b/_corpus_v3_items/cid001546.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001546/ +title: Functor Category Collapse +cid: cid001546 +primary_alias: PRP0069 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The functor category [E₃^op, E₃] is contained in E₃ itself. No fifth generator or orbit channel exists to house + genuinely new structure. The ω-absorber ensures all higher-level attempts collapse back: any would-be E₄ object is already + an E₃ natural transformation.' +aliases: +- PRP0069 +- III.P02 +- functor-category-collapse +- prop:functor-collapse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005856 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 7 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Functor Category Collapse**\n\nThe functor category [E₃^op, E₃] is contained in E₃ itself. No fifth generator\ + \ or orbit channel exists to house genuinely new structure. The ω-absorber ensures all higher-level attempts collapse\ + \ back: any would-be E₄ object is already an E₃ natural transformation.\n\n# Functor Category Collapse\n\n## Summary\n\ + \nThe functor category [E₃^op, E₃] is contained in E₃ itself. No fifth generator or orbit channel exists to house genuinely\ + \ new structure. The ω-absorber ensures all higher-level attempts collapse back: any would-be E₄ object is already an\ + \ E₃ natural transformation.\n\n## Statement\n\n```latex\n%\n\\label{prop:functor-collapse}\nThe functor category $[E_3^{\\\ + op}, E_3]$\ncontains no objects of genuinely new structural type.\nEvery presheaf $F : E_3^{\\op} \\to E_3$\nis classified\ + \ by the four ABCD coordinates\ninherited from the coherence kernel.\nTherefore:\n\\[\n [E_3^{\\op}, E_3] \\;\\subseteq\\\ + ; E_3.\n\\]\n```\n\n## Proof / Justification\n\n```latex\n[Proof sketch]\nAn object $F \\in [E_3^{\\op}, E_3]$\nis a functor\ + \ whose values and morphisms\nlive in~$E_3$.\nThe structural type of~$F$ is determined by:\n\\begin{enumerate}\n \\\ + item the type of its values\n (objects of $E_3$, classified by the ABCD chart),\n \\item the type of its action\ + \ on morphisms\n (enriched morphisms of $E_3$,\n classified by the split-complex bipolar structure).\n\ + \\end{enumerate}\nFor $F$ to have a structural type\nnot already present in~$E_3$,\nit would need a coordinate component\n\ + not expressible as a function\nof $(A, B, C, D)$.\nSuch a component would require a fifth orbit channel.\n\nBy the Ontic\ + \ Closure Theorem (I.T01)\nand the Ladder Saturation Theorem (I.T02),\nno fifth orbit channel exists.\nThe Pentation Non-Injectivity\ + \ Lemma (I.L01)\nprovides the concrete obstruction:\nany attempt to define a fifth canonical coordinate\nfails because\ + \ pentation lacks\nthe injectivity scaffold needed\nfor type distinction.\n\nTherefore $F$ is classified by the existing\ + \ coordinates,\nand $[E_3^{\\op}, E_3] \\subseteq E_3$.\n```\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 19\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch07-saturation-at-e3.tex`\ + \ lines 323-335\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `functor_collapse_8_3`\n\n## Dependencies\n\n- Canonical: III.D09\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The functor category [E₃^op, E₃] is contained in E₃ itself. No fifth generator or orbit channel exists to house + genuinely new structure. The ω-absorber ensures all higher-level attempts collapse back: any would-be E₄ object is already + an E₃' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001547.md b/_corpus_v3_items/cid001547.md new file mode 100644 index 000000000..4b8ab0869 --- /dev/null +++ b/_corpus_v3_items/cid001547.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001547/ +title: E₁ Physical Preview +cid: cid001547 +primary_alias: PRP0070 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'At E₁ the template instantiates: D→gravity, A→weak, B→EM, C→strong, ω→Higgs/mass. PREVIEW ONLY: no physical constants, + no coupling values, no Lagrangians. Depends on sector↔force identification (conjectural).' +aliases: +- PRP0070 +- III.P03 +- e-physical-preview +- def:e1-physical-preview +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 10 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**E₁ Physical Preview**\n\nAt E₁ the template instantiates: D→gravity, A→weak, B→EM, C→strong, ω→Higgs/mass.\ + \ PREVIEW ONLY: no physical constants, no coupling values, no Lagrangians. Depends on sector↔force identification (conjectural).\n\ + \n# E₁ Physical Preview\n\n## Summary\n\nAt E₁ the template instantiates: D→gravity, A→weak, B→EM, C→strong, ω→Higgs/mass.\ + \ PREVIEW ONLY: no physical constants, no coupling values, no Lagrangians. Depends on sector↔force identification (conjectural).\n\ + \n## Statement\n\n```latex\n%\n\\label{def:e1-physical-preview}\nThe \\emph{$\\Elayer{1}$ physical sector preview}\nis\ + \ the structural identification:\n\\begin{center}\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{adjustbox}{max width=0.95\\\ + linewidth}\n\\begin{tabular}{p{0.1\\linewidth}p{0.2\\linewidth}p{0.2\\linewidth}p{0.2\\linewidth}p{0.2\\linewidth}}\n\\\ + toprule\n\\textbf{Generator}\n & \\textbf{Coordinate}\n & \\textbf{Sector}\n & \\textbf{$\\Elayer{1}$ Interaction}\n\ + \ & \\textbf{Structural Role} \\\\\n\\midrule\n$\\alpha$\n & $D$\n & $\\Sector{D}$\n & Gravity\n & Radial\ + \ depth (base direction) \\\\\n$\\pi$\n & $A$\n & $\\Sector{A}$\n & Weak force\n & Angular winding (carrier)\ + \ \\\\\n$\\gamma$\n & $B$\n & $\\Sector{B}$\n & Electromagnetism\n & Exponent stratum (predicate) \\\\\n$\\\ + eta$\n & $C$\n & $\\Sector{C}$\n & Strong force\n & Tetration stratum (decoder) \\\\\n$\\omega$\n & $B\ + \ \\cap C$\n & $\\Sector{\\omega}$\n & Higgs/mass mechanism\n & $B$-$C$ crossing (coupling) \\\\\n\\bottomrule\n\ + \\end{tabular}\n\\end{adjustbox}\n\\end{center}\nThis identification is \\emph{structural}:\neach force is assigned to\ + \ the sector\nwhose categorical role matches\nthe force's structural function\nin the standard model.\n```\n\n## Proof\ + \ / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n-\ + \ Registry source: `book-03.jsonl` line 28\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch10-the-four-plus-one-decomposition.tex`\ + \ lines 791-841\n\n## Lean / Formalization Notes\n\n- Formalization: `not_applicable`\n- Module: `None`\n- Name: `None`\n\ + \n## Dependencies\n\n- Canonical: III.D13, III.D07\n\n## Related Results\n\n_Generated by later projection phases._\n\n\ + ## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot\ + \ migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'At E₁ the template instantiates: D→gravity, A→weak, B→EM, C→strong, ω→Higgs/mass. PREVIEW ONLY: no physical + constants, no coupling values, no Lagrangians. Depends on sector↔force identification (conjectural).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001548.md b/_corpus_v3_items/cid001548.md new file mode 100644 index 000000000..e6f734709 --- /dev/null +++ b/_corpus_v3_items/cid001548.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001548/ +title: Balanced Sector Uniqueness +cid: cid001548 +primary_alias: PRP0071 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The A-sector (weak) is the unique primitive sector with balanced spectral polarity: pol(S_g) = 1 iff g = π. Proved + by orbit exhaustion over ABCD coordinates.' +aliases: +- PRP0071 +- III.P04 +- balanced-sector-uniqueness +- prop:balanced-sector-uniqueness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005982 +- predicate: formalized_by + object: cid005986 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 12 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Balanced Sector Uniqueness**\n\nThe A-sector (weak) is the unique primitive sector with balanced spectral polarity:\ + \ pol(S_g) = 1 iff g = π. Proved by orbit exhaustion over ABCD coordinates.\n\n# Balanced Sector Uniqueness\n\n## Summary\n\ + \nThe A-sector (weak) is the unique primitive sector with balanced spectral polarity: pol(S_g) = 1 iff g = π. Proved by\ + \ orbit exhaustion over ABCD coordinates.\n\n## Statement\n\n```latex\n%\n\\label{prop:balanced-sector-uniqueness}\nAmong\ + \ the four primitive sectors\n$\\Sector{\\eta}, \\Sector{\\alpha}, \\Sector{\\gamma}, \\Sector{\\pi}$,\nthe A-sector $\\\ + Sector{\\pi}$ is the unique sector\nwith balanced spectral polarity:\n\\begin{equation}\\label{eq:ch12-unique-balanced}\n\ + \ \\mathrm{pol}(\\Sector{g}) = 1\n \\quad\\iff\\quad\n g = \\pi.\n\\end{equation}\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 33\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch12-the-parity-bridge-theorem.tex`\ + \ lines 211-223\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.LanglandsReflection`\n\ + - Name: `balanced_uniqueness_5`\n\n## Dependencies\n\n- Canonical: III.D17\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The A-sector (weak) is the unique primitive sector with balanced spectral polarity: pol(S_g) = 1 iff g = π. + Proved by orbit exhaustion over ABCD coordinates.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001549.md b/_corpus_v3_items/cid001549.md new file mode 100644 index 000000000..73caa35aa --- /dev/null +++ b/_corpus_v3_items/cid001549.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001549/ +title: Independence of Prime-Level Actions +cid: cid001549 +primary_alias: PRP0072 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Prime-level actions T^(i) on ℤ/pᵢℤ are independent. CRT guarantees unique reassembly into global action on ℤ/Prim(k)ℤ. + The τ-incarnation of the local-global principle. +aliases: +- PRP0072 +- III.P05 +- independence-of-prime-level-actions +- prop:prime-independence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006020 +- predicate: formalized_by + object: cid006023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Independence of Prime-Level Actions**\n\nPrime-level actions T^(i) on ℤ/pᵢℤ are independent. CRT guarantees\ + \ unique reassembly into global action on ℤ/Prim(k)ℤ. The τ-incarnation of the local-global principle.\n\n# Independence\ + \ of Prime-Level Actions\n\n## Summary\n\nPrime-level actions T^(i) on ℤ/pᵢℤ are independent. CRT guarantees unique reassembly\ + \ into global action on ℤ/Prim(k)ℤ. The τ-incarnation of the local-global principle.\n\n## Statement\n\n```latex\n%\n\\\ + label{prop:prime-independence}\nLet $T, T'$ be endomorphisms of $\\Z / \\mathrm{Prim}(k)\\Z$\nwith CRT decompositions\n\ + $(T^{(1)}, \\ldots, T^{(k)})$\nand $(T'^{(1)}, \\ldots, T'^{(k)})$.\nThen:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\ + \ \\textbf{Independence.}\n For any $\\tilde{T}^{(i)} \\in \\mathrm{End}(\\Z / p_i \\Z)$,\n there exists\ + \ a unique $\\tilde{T}$\n with CRT decomposition\n $(T^{(1)}, \\ldots, \\tilde{T}^{(i)}, \\ldots, T^{(k)})$.\n\ + \ Modifying one prime does not affect the others.\n \\item[\\textup{(ii)}] \\textbf{Composition.}\n \ + \ $T \\circ T'$ has CRT decomposition\n $(T^{(1)} \\circ T'^{(1)}, \\ldots,\n T^{(k)} \\circ T'^{(k)})$.\n\ + \ \\item[\\textup{(iii)}] \\textbf{Equality.}\n $T = T'$ if and only if\n $T^{(i)} = T'^{(i)}$\n\ + \ for all~$i$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was\ + \ extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 43\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch15-the-crt-decomposition-theorem.tex` lines 421-445\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.CRT`\n- Name: `prime_independence_check`\n\ + \n## Dependencies\n\n- Canonical: III.T10\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related\ + \ Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Prime-level actions T^(i) on ℤ/pᵢℤ are independent. CRT guarantees unique reassembly into global action on ℤ/Prim(k)ℤ. + The τ-incarnation of the local-global principle. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001550.md b/_corpus_v3_items/cid001550.md new file mode 100644 index 000000000..2c9c5a4dc --- /dev/null +++ b/_corpus_v3_items/cid001550.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001550/ +title: Completeness Without Topology +cid: cid001550 +primary_alias: PRP0073 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ℤ_p^τ is complete in the τ sense: every tower-coherent sequence has unique limit. This is Global Hartogs (I.T31) + restricted to the p-primary tower. No metric, no Cauchy sequences.' +aliases: +- PRP0073 +- III.P06 +- completeness-without-topology +- prop:completeness-without-topology +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006046 +- predicate: formalized_by + object: cid006047 +- predicate: formalized_by + object: cid006051 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Completeness Without Topology**\n\nℤ_p^τ is complete in the τ sense: every tower-coherent sequence has unique\ + \ limit. This is Global Hartogs (I.T31) restricted to the p-primary tower. No metric, no Cauchy sequences.\n\n# Completeness\ + \ Without Topology\n\n## Summary\n\nℤ_p^τ is complete in the τ sense: every tower-coherent sequence has unique limit.\ + \ This is Global Hartogs (I.T31) restricted to the p-primary tower. No metric, no Cauchy sequences.\n\n## Statement\n\n\ + ```latex\n%\n\\label{prop:completeness-without-topology}\nLet $(b_n)_{n \\geq 1}$ be a $k$-approximately coherent\nsequence\ + \ in the $p$-primary tower.\nThen there exists a unique $a = (a_n)_{n \\geq 1} \\in \\Z_p^\\tau$\nsuch that $a_n = b_n$\ + \ for all $n \\geq k$.\nIn particular:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n Every tower-coherent\ + \ sequence defines\n a unique $p$-adic $\\tau$-integer.\n \\item[\\textup{(ii)}]\n Every $k$-approximately\ + \ coherent sequence\n determines a unique $p$-adic $\\tau$-integer\n by ``filling in'' the first $k-1$\ + \ levels.\n \\item[\\textup{(iii)}]\n No sequence that is coherent from some level onward\n escapes\ + \ $\\Z_p^\\tau$:\n the ring is closed under eventual coherence.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 46\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch16-hensel-lifting-and-local-fields.tex`\ + \ lines 504-525\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.LocalFields`\n\ + - Name: `completeness_check`\n\n## Dependencies\n\n- Canonical: III.D21, I.T31\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ℤ_p^τ is complete in the τ sense: every tower-coherent sequence has unique limit. This is Global Hartogs (I.T31) + restricted to the p-primary tower. No metric, no Cauchy sequences.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001551.md b/_corpus_v3_items/cid001551.md new file mode 100644 index 000000000..89ee4f41b --- /dev/null +++ b/_corpus_v3_items/cid001551.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001551/ +title: Adelic Euler Product +cid: cid001551 +primary_alias: PRP0074 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'τ-holomorphic function on 𝔸_τ decomposes into local factors at each prime. Adelic form of Euler product: CRT lifted + to holomorphic level.' +aliases: +- PRP0074 +- III.P07 +- adelic-euler-product +- prop:adelic-euler-product +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006003 +- predicate: formalized_by + object: cid006004 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Adelic Euler Product**\n\nτ-holomorphic function on \U0001D538_τ decomposes into local factors at each prime.\ + \ Adelic form of Euler product: CRT lifted to holomorphic level.\n\n# Adelic Euler Product\n\n## Summary\n\nτ-holomorphic\ + \ function on \U0001D538_τ decomposes into local factors at each prime. Adelic form of Euler product: CRT lifted to holomorphic\ + \ level.\n\n## Statement\n\n```latex\n%\n\\label{prop:adelic-euler-product}\nLet $F \\in \\mathrm{Hol}(\\mathbb{A}_\\\ + tau)$\nbe a $\\tau$-holomorphic function on the adele ring.\nThen $F$ admits a unique decomposition\n\\begin{equation}\\\ + label{eq:ch17-euler-product}\n F\n \\;=\\;\n \\prod_{p} f_p,\n \\qquad\n f_p \\in \\mathrm{Hol}_p(\\Z_p^\\\ + tau),\n\\end{equation}\nwith $f_p = 1$ for almost all~$p$.\nThe decomposition is functorial:\nif $\\Phi : F \\to G$\n\ + is a morphism of adelic holomorphic functions,\nthen $\\Phi$ decomposes as $\\Phi = \\prod_p \\phi_p$\nwith $\\phi_p :\ + \ f_p \\to g_p$\na morphism of local holomorphic functions\nat each prime.\n```\n\n## Proof / Justification\n\n_No immediate\ + \ manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 49\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch17-the-adelic-embedding.tex`\ + \ lines 500-521\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Adeles`\n\ + - Name: `euler_product_check`\n\n## Dependencies\n\n- Canonical: III.D22, III.T10\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'τ-holomorphic function on 𝔸_τ decomposes into local factors at each prime. Adelic form of Euler product: CRT + lifted to holomorphic level.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001552.md b/_corpus_v3_items/cid001552.md new file mode 100644 index 000000000..c7a3f21d4 --- /dev/null +++ b/_corpus_v3_items/cid001552.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001552/ +title: Label-Idempotent Compatibility +cid: cid001552 +primary_alias: PRP0075 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: B-type primes correspond to e₊-dominant spectral coefficients, C-type to e₋-dominant, X-type balanced. Label_n is + compatible with the split-complex idempotent decomposition of H_τ. +aliases: +- PRP0075 +- III.P08 +- label-idempotent-compatibility +- prop:label-idem-compatibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006011 +- predicate: formalized_by + object: cid006012 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 18 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Label-Idempotent Compatibility**\n\nB-type primes correspond to e₊-dominant spectral coefficients, C-type to\ + \ e₋-dominant, X-type balanced. Label_n is compatible with the split-complex idempotent decomposition of H_τ.\n\n# Label-Idempotent\ + \ Compatibility\n\n## Summary\n\nB-type primes correspond to e₊-dominant spectral coefficients, C-type to e₋-dominant,\ + \ X-type balanced. Label_n is compatible with the split-complex idempotent decomposition of H_τ.\n\n## Statement\n\n```latex\n\ + %\n\\label{prop:label-idem-compatibility}\nFor every odd prime~$p_i$\nand every depth $n \\geq i$:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n $\\mathrm{Label}_n(p_i) = B$\n $\\;\\Longleftrightarrow\\;$\n \ + \ the spectral coefficient $c_+(e_i)$\n is a quadratic residue modulo~$p_i$\n and $c_-(e_i)$ is a quadratic\ + \ non-residue.\n \\item[\\textup{(ii)}]\n $\\mathrm{Label}_n(p_i) = C$\n $\\;\\Longleftrightarrow\\\ + ;$\n $c_-(e_i)$ is a quadratic residue\n and $c_+(e_i)$ is a quadratic non-residue.\n \\item[\\textup{(iii)}]\n\ + \ $\\mathrm{Label}_n(p_i) = X$\n $\\;\\Longleftrightarrow\\;$\n $c_+(e_i)$ and $c_-(e_i)$\n\ + \ have the same quadratic character modulo~$p_i$.\n\\end{enumerate}\nFor $p_i = 2$,\nthe spectral coefficients\ + \ coincide modulo~$2$,\nand the label is~$X$.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block\ + \ was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 52\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch18-the-internal-bipolar-classifier.tex` lines 571-597\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.BipolarClassifier`\n\ + - Name: `label_idem_check`\n\n## Dependencies\n\n- Canonical: III.D23\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: B-type primes correspond to e₊-dominant spectral coefficients, C-type to e₋-dominant, X-type balanced. Label_n + is compatible with the split-complex idempotent decomposition of H_τ. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001553.md b/_corpus_v3_items/cid001553.md new file mode 100644 index 000000000..27c3d493d --- /dev/null +++ b/_corpus_v3_items/cid001553.md @@ -0,0 +1,169 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001553/ +title: Discrete Spectrum of H_L +cid: cid001553 +primary_alias: PRP0076 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The spectrum of H_L is discrete: {λ_n}_{n≥0} with λ_n → ∞. Compact resolvent follows from L being a compact metric + graph. Eigenfunctions are explicit sinusoidal modes on each lobe, matched at the crossing point.' +aliases: +- PRP0076 +- III.P09 +- discrete-spectrum-of-h-l +- prop:discrete-spectrum-h-l +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005826 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 23 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Discrete Spectrum of H_L** + + + The spectrum of H_L is discrete: {λ_n}_{n≥0} with λ_n → ∞. Compact resolvent follows from L being a compact metric graph. + Eigenfunctions are explicit sinusoidal modes on each lobe, matched at the crossing point. + + + # Discrete Spectrum of H_L + + + ## Summary + + + The spectrum of H_L is discrete: {λ_n}_{n≥0} with λ_n → ∞. Compact resolvent follows from L being a compact metric graph. + Eigenfunctions are explicit sinusoidal modes on each lobe, matched at the crossing point. + + + ## Statement + + + ```latex + + \label{prop:discrete-spectrum-h-l} + + The spectrum of $H_L$ consists of isolated eigenvalues + + \[ + + \operatorname{Spec}(H_L) = \{ \Lambda_0, \Lambda_1, \Lambda_2, \ldots \} + + \] + + with $0 \leq \Lambda_0 < \Lambda_1 < \Lambda_2 < \cdots$ and $\Lambda_n \to +\infty$ as $n \to \infty$. Each eigenvalue + has finite multiplicity. + + ``` + + + ## Proof / Justification + + + ```latex + + The metric graph $L$ is compact (total length $2$), so the embedding $H^2(L) \hookrightarrow L^2(L)$ is compact by the + Rellich--Kondrachov theorem. The resolvent $(H_L - \lambda)^{-1}$ is compact for $\lambda \notin \operatorname{Spec}(H_L)$, + hence $H_L$ has discrete spectrum~\cite{ReedSimon1980}. + + + Eigenvalues are real by self-adjointness (Theorem~\ref{thm:self-adjointness-h-l}). The lowest eigenvalue $\Lambda_0 = + 0$ corresponds to constant functions (which satisfy Kirchhoff automatically). Positivity of higher eigenvalues follows + from the variational characterization + + \[ + + \Lambda_n = \inf_{\substack{f \in \mathrm{Dom}(H_L) \\ \langle f, \psi_k \rangle = 0, \, k < n}} \frac{\langle H_L f, + f \rangle}{\langle f, f \rangle} = \inf \frac{\int_L |f''|^2}{\int_L |f|^2}, + + \] + + where the infimum is over functions orthogonal to $\psi_0, \ldots, \psi_{n-1}$ (the lower eigenfunctions). Since $|f''|^2 + \geq 0$, all $\Lambda_n \geq 0$. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 66 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch23-the-lemniscate-operator.tex` + lines 172-179 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.LemniscateOperator` + + - Name: `discrete_spectrum_check` + + + ## Dependencies + + + - Canonical: III.D28, III.T17 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The spectrum of H_L is discrete: {λ_n}_{n≥0} with λ_n → ∞. Compact resolvent follows from L being a compact + metric graph. Eigenfunctions are explicit sinusoidal modes on each lobe, matched at the crossing point.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001554.md b/_corpus_v3_items/cid001554.md new file mode 100644 index 000000000..cb40dbe68 --- /dev/null +++ b/_corpus_v3_items/cid001554.md @@ -0,0 +1,164 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001554/ +title: K5 Off-Diagonal Exclusion +cid: cid001554 +primary_alias: PRP0077 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: + H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between + lobes. This is the structural mechanism behind the critical line.' +aliases: +- PRP0077 +- III.P10 +- k5-off-diagonal-exclusion +- prop:k5-off-diagonal-exclusion +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005815 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 25 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**K5 Off-Diagonal Exclusion** + + + The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: + H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between + lobes. This is the structural mechanism behind the critical line. + + + # K5 Off-Diagonal Exclusion + + + ## Summary + + + The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral terms: + H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between + lobes. This is the structural mechanism behind the critical line. + + + ## Statement + + + ```latex + + \label{prop:k5-off-diagonal-exclusion} + + K5 diagonal discipline forbids information leakage between the B-sector and C-sector except through the single constrained + channel at the crossing point $\omega$. Consequently, the Laplacian $H_L$ is block-diagonal in the $\chi_+/\chi_-$ decomposition. + + ``` + + + ## Proof / Justification + + + ```latex + + By Theorem~\ref{thm:spectral-trichotomy} (Ch.~20), functions decompose as $\psi = \chi_+ \psi + \chi_- \psi$, where $\chi_\pm$ + are the lemniscate characters. The Kirchhoff condition couples the two lobes \emph{only at} $\omega$: $\psi_B(\omega) + = \psi_C(\omega)$. K5 forbids additional couplings (diagonal arrows from $B$ to $C$ or $C$ to $B$ in the dependency graph). + Hence the Laplacian cannot mix $\chi_+$-supported modes with $\chi_-$-supported modes beyond the boundary condition. + + + In matrix form (choosing an orthonormal basis adapted to $\chi_+/\chi_-$), $H_L$ is block-diagonal: + + \begin{equation} + + H_L \;=\; \begin{pmatrix} H_+ & 0 \\ 0 & H_- \end{pmatrix}. + + \label{eq:ch25-block-matrix} + + \end{equation} + + Self-adjointness of $H_L$ follows from self-adjointness of $H_+$ and $H_-$ separately. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 70 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch25-spectral-purity-and-the-critical-line.tex` + lines 131-134 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.CriticalLine` + + - Name: `k5_exclusion_check` + + + ## Dependencies + + + - Canonical: III.T17, III.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The K5 axiom (diagonal discipline) forbids off-diagonal coupling at the lemniscate crossing point. In spectral + terms: H_L cannot have imaginary eigenvalues because the crossing-point boundary conditions enforce real spectral flow between + lo' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001555.md b/_corpus_v3_items/cid001555.md new file mode 100644 index 000000000..a8fd81a0c --- /dev/null +++ b/_corpus_v3_items/cid001555.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001555/ +title: Primorial RH Verification Protocol +cid: cid001555 +primary_alias: PRP0078 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Six-step verification protocol: (i) compute Spec(H_{≤k}), (ii) verify Im(Λ) = 0 for all eigenvalues, (iii) verify + zero locations of finite Euler product, (iv) check tower coherence with previous levels, (v) validate CRT consistency, (vi) + record primorial certificate. Polynomial in k.' +aliases: +- PRP0078 +- III.P11 +- primorial-rh-verification-protocol +- prop:primorial-rh-protocol +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 26 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Primorial RH Verification Protocol**\n\nSix-step verification protocol: (i) compute Spec(H_{≤k}), (ii) verify\ + \ Im(Λ) = 0 for all eigenvalues, (iii) verify zero locations of finite Euler product, (iv) check tower coherence with\ + \ previous levels, (v) validate CRT consistency, (vi) record primorial certificate. Polynomial in k.\n\n# Primorial RH\ + \ Verification Protocol\n\n## Summary\n\nSix-step verification protocol: (i) compute Spec(H_{≤k}), (ii) verify Im(Λ) =\ + \ 0 for all eigenvalues, (iii) verify zero locations of finite Euler product, (iv) check tower coherence with previous\ + \ levels, (v) validate CRT consistency, (vi) record primorial certificate. Polynomial in k.\n\n## Statement\n\n```latex\n\ + \\label{prop:primorial-rh-protocol}\nFix primorial depth $k \\geq 1$. The following protocol verifies the $\\tau$-effective\ + \ RH statement at depth $k$:\n\\begin{enumerate}[label=\\textbf{Step \\arabic*.}]\n \\item \\textbf{Construct $H_{L_k}$}:\ + \ Using the definition from Chapter~23, compute the $k \\times k$ matrix representation of $H_{L_k}$ in the basis $\\\ + {e_1, \\ldots, e_k\\}$. Entry $(i,j)$ is given by the bipolar coupling $\\langle e_i, e_j \\rangle_\\jj$.\n\n \\item\ + \ \\textbf{Certify self-adjointness}: Verify that $H_{L_k}$ is Hermitian (in the split-complex inner product). This is\ + \ guaranteed by Theorem~III.T17; the verification step checks consistency.\n\n \\item \\textbf{Compute spectrum}: Using\ + \ certified interval arithmetic (e.g., Arb library), compute $\\operatorname{Spec}(H_{L_k})$ with rigorous error bounds.\ + \ Verify that all eigenvalues have $|\\Im(\\lambda)| < \\epsilon$ for machine tolerance $\\epsilon$.\n\n \\item \\\ + textbf{Locate zeros}: For each real eigenvalue $\\lambda_j$, compute the corresponding zero $s_j = \\tfrac{1}{2} + i\\\ + lambda_j$ of the finite Euler product $\\zeta_{L_k}(s)$ via the determinant representation\n \\[\n \\det(s \\cdot\ + \ I - H_{L_k}) \\stackrel{?}{=} C_k(s) \\cdot \\zeta_{L_k}(s),\n \\]\n where $C_k(s)$ is a normalization polynomial.\ + \ Verify that $\\zeta_{L_k}(s_j) \\approx 0$ within tolerance.\n\n \\item \\textbf{Check critical line}: Confirm that\ + \ all zeros $s_j$ satisfy $|\\Re(s_j) - \\tfrac{1}{2}| < \\epsilon$.\n\n \\item \\textbf{Tower coherence}: If $k >\ + \ 1$, verify that the spectrum $\\operatorname{Spec}(H_{L_{k-1}})$ is a subset of $\\operatorname{Spec}(H_{L_k})$ up to\ + \ tolerance, consistent with the nesting property of primorial towers.\n\\end{enumerate}\nThe protocol runs in $O(k^3)$\ + \ time (dominated by eigenvalue computation) and outputs a certified \\textsc{pass/fail} verdict.\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 72\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch26-primorial-verification-of-rh.tex`\ + \ lines 64-84\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Doors.CriticalLine`\n\ + - Name: `rh_protocol_check`\n\n## Dependencies\n\n- Canonical: III.D30, III.T09\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Six-step verification protocol: (i) compute Spec(H_{≤k}), (ii) verify Im(Λ) = 0 for all eigenvalues, (iii) verify + zero locations of finite Euler product, (iv) check tower coherence with previous levels, (v) validate CRT consistency, (vi) + re' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001557.md b/_corpus_v3_items/cid001557.md new file mode 100644 index 000000000..ddbd9709f --- /dev/null +++ b/_corpus_v3_items/cid001557.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001557/ +title: Poincaré as Gluing Guarantee +cid: cid001557 +primary_alias: PRP0080 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Poincaré guarantees that local Hartogs bulk projections glue into a global space homeomorphic to S³ when the fundamental + group is trivial. Simple connectivity = no obstruction to global coherence of local patches at the E₀→E₁ interface. +aliases: +- PRP0080 +- III.P13 +- poincar-as-gluing-guarantee +- prop:poincare-gluing-guarantee +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005836 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 32 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Poincaré as Gluing Guarantee** + + + Poincaré guarantees that local Hartogs bulk projections glue into a global space homeomorphic to S³ when the fundamental + group is trivial. Simple connectivity = no obstruction to global coherence of local patches at the E₀→E₁ interface. + + + # Poincaré as Gluing Guarantee + + + ## Summary + + + Poincaré guarantees that local Hartogs bulk projections glue into a global space homeomorphic to S³ when the fundamental + group is trivial. Simple connectivity = no obstruction to global coherence of local patches at the E₀→E₁ interface. + + + ## Statement + + + ```latex + + \label{prop:poincare-gluing-guarantee} + + Let $M$ be a closed $\tau$-manifold of dimension 3 arising from the gluing of local Hartogs bulks at the $E_0 \to E_1$ + interface. If every loop of transition functions is contractible (i.e., $M$ is simply connected in the sense of Definition~\ref{def:simply-connected-tau}), + then $M$ is homeomorphic to the 3-sphere $S^3$. + + ``` + + + ## Proof / Justification + + + ```latex + + [Proof Strategy] + + The proof proceeds in three stages, following the Mutual Determination template: + + + \textbf{Stage 1: Boundary-to-Interior.} + + Given that all loops of transitions are contractible, the universal covering presheaf is trivial. By the Global Hartogs + Theorem (I.T31), this forces the global topology to have no non-trivial local obstructions. + + + \textbf{Stage 2: Spectral Algebra Input.} + + The spectral algebra $A_{\mathrm{spec}}(\mathbb{L})$ decomposes all transition functions into $\chi_+$ and $\chi_-$ components. + Simple connectivity implies that both components factor through the trivial representation of $\pi_1$. This is the ``spectral + purity'''' condition. + + + \textbf{Stage 3: Interior Reconstruction.} + + From spectral purity, the gluing data simplifies to a unique configuration. By the classification of closed 3-manifolds + in the earned topology (a consequence of the Geometrization Theorem, which Book~III treats as \emph{established} classical + input), the only simply connected closed 3-manifold is $S^3$. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 85 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch32-simply-connected-in-category-tau.tex` + lines 46-48 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.Poincare` + + - Name: `gluing_guarantee_check` + + + ## Dependencies + + + - Canonical: III.D35, III.R15, III.D01 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Poincaré guarantees that local Hartogs bulk projections glue into a global space homeomorphic to S³ when the + fundamental group is trivial. Simple connectivity = no obstruction to global coherence of local patches at the E₀→E₁ interface. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001558.md b/_corpus_v3_items/cid001558.md new file mode 100644 index 000000000..c7fc396c7 --- /dev/null +++ b/_corpus_v3_items/cid001558.md @@ -0,0 +1,149 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001558/ +title: Defect Contractivity +cid: cid001558 +primary_alias: PRP0081 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For τ-admissible data, the defect functional is contractive: Δ(f, n+1) ≤ κ · Δ(f, n) for some κ < 1. Key estimate + driving regularity. The contraction constant κ comes from the divergence of ∑1/p.' +aliases: +- PRP0081 +- III.P14 +- defect-contractivity +- prop:defect-contractivity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005952 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 35 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Defect Contractivity** + + + For τ-admissible data, the defect functional is contractive: Δ(f, n+1) ≤ κ · Δ(f, n) for some κ < 1. Key estimate driving + regularity. The contraction constant κ comes from the divergence of ∑1/p. + + + # Defect Contractivity + + + ## Summary + + + For τ-admissible data, the defect functional is contractive: Δ(f, n+1) ≤ κ · Δ(f, n) for some κ < 1. Key estimate driving + regularity. The contraction constant κ comes from the divergence of ∑1/p. + + + ## Statement + + + ```latex + + \label{prop:defect-contractivity} + + Let $f$ be $\tau$-admissible fluid data (Definition~\ref{def:tau-admissible-fluid-data}, Ch.~34). There exists a contraction + factor $\kappa = \kappa(k) < 1$, depending only on the primorial depth $k$ of the initial data, such that for all $n \geq + k$: + + \begin{equation} + + \Delta(f, n+1) \;\leq\; \kappa \cdot \Delta(f, n). + + \label{eq:ch35-defect-contractivity} + + \end{equation} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 93 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch35-fluid-sectors-and-the-defect-functional.tex` + lines 165-173 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.PositiveRegularity` + + - Name: `defect_contractivity_check` + + + ## Dependencies + + + - Canonical: III.D39, III.T07, III.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For τ-admissible data, the defect functional is contractive: Δ(f, n+1) ≤ κ · Δ(f, n) for some κ < 1. Key estimate + driving regularity. The contraction constant κ comes from the divergence of ∑1/p.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001559.md b/_corpus_v3_items/cid001559.md new file mode 100644 index 000000000..34e94645d --- /dev/null +++ b/_corpus_v3_items/cid001559.md @@ -0,0 +1,160 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001559/ +title: 3-Condition Sufficiency +cid: cid001559 +primary_alias: PRP0082 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The three conditions (clopen locality + ω-germ determinacy + defect contractivity) are individually necessary and + jointly sufficient for positive regularity. Removing any one permits non-stabilizing ω-germs. +aliases: +- PRP0082 +- III.P15 +- 3-condition-sufficiency +- prop:three-condition-sufficiency +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005953 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 37 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**3-Condition Sufficiency** + + + The three conditions (clopen locality + ω-germ determinacy + defect contractivity) are individually necessary and jointly + sufficient for positive regularity. Removing any one permits non-stabilizing ω-germs. + + + # 3-Condition Sufficiency + + + ## Summary + + + The three conditions (clopen locality + ω-germ determinacy + defect contractivity) are individually necessary and jointly + sufficient for positive regularity. Removing any one permits non-stabilizing ω-germs. + + + ## Statement + + + ```latex + + \label{prop:three-condition-sufficiency} + + Let $f$ be $\tau$-admissible fluid data on a clopen cylinder domain $U \subset \tau^3$. Suppose the following three conditions + hold: + + \begin{enumerate} + + \item[\emph{(C1)}] \textbf{Clopen locality.} The domain $U$ is a finite union of clopen cylinders at each primorial depth + (Definition~\ref{def:clopen-cylinder-domain}, Ch.~34), and the Boolean cylinder algebra is closed under intersection, + union, and complement. + + + \item[\emph{(C2)}] \textbf{$\omega$-Germ determinacy.} At each primorial level $\operatorname{Prim}(n)$, the Local Hartogs + continuation $\operatorname{Ext}_n$ produces a unique extension of the boundary data into the interior of~$U_n$ (Theorem~\ref{thm:hartogs-flow-theorem}(i), + Ch.~36). There is no freedom in the evolved data: it is \emph{forced} by the boundary assignment. + + + \item[\emph{(C3)}] \textbf{Defect-horizon contractivity.} The defect functional $\Delta(f, n)$ (Definition~\ref{def:defect-functional}, + Ch.~35) satisfies $\Delta(f, n{+}1) \leq \kappa \cdot \Delta(f, n)$ for all $n$ sufficiently large, with $\kappa < 1$ + (Proposition~\ref{prop:defect-contractivity}, Ch.~35). + + \end{enumerate} + + Then $H_{\mathrm{flow}}(f)$ admits a stabilized $\omega$-germ at every point $x \in U$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 99 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch37-positive-regularity.tex` lines + 56-68 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.PositiveRegularity` + + - Name: `stability_criterion_check` + + + ## Dependencies + + + - Canonical: III.T25 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The three conditions (clopen locality + ω-germ determinacy + defect contractivity) are individually necessary + and jointly sufficient for positive regularity. Removing any one permits non-stabilizing ω-germs. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001560.md b/_corpus_v3_items/cid001560.md new file mode 100644 index 000000000..a11e07b7a --- /dev/null +++ b/_corpus_v3_items/cid001560.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001560/ +title: NF Discreteness Lemma +cid: cid001560 +primary_alias: PRP0083 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The normal-form tower ℤ/Prim(k)ℤ is inherently discrete. No continuum limit within τ: each primorial level has finitely + many states. Discreteness earned from K3 divisibility, not imposed.' +aliases: +- PRP0083 +- III.P16 +- nf-discreteness-lemma +- prop:nf-discreteness-lemma +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005957 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 38 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**NF Discreteness Lemma** + + + The normal-form tower ℤ/Prim(k)ℤ is inherently discrete. No continuum limit within τ: each primorial level has finitely + many states. Discreteness earned from K3 divisibility, not imposed. + + + # NF Discreteness Lemma + + + ## Summary + + + The normal-form tower ℤ/Prim(k)ℤ is inherently discrete. No continuum limit within τ: each primorial level has finitely + many states. Discreteness earned from K3 divisibility, not imposed. + + + ## Statement + + + ```latex + + \label{prop:nf-discreteness-lemma} + + Let $k \in \mathbb{N}$ and let $\operatorname{Strong}_{E_1}^{(k)}$ denote the strong sector restricted to primorial depth~$k$. + The number of distinct NF configurations in $\operatorname{Strong}_{E_1}^{(k)}$ is bounded by + + \begin{equation} + + \bigl|\operatorname{NF}\bigl(\operatorname{Strong}_{E_1}^{(k)}\bigr)\bigr| + + \;\le\; + + \operatorname{Prim}(k). + + \label{eq:ch38-nf-discreteness-bound} + + \end{equation} + + Moreover, the bound is sharp: for each $k$, there exist exactly $\operatorname{Prim}(k)$ distinct C-sector normal forms + at depth~$k$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 103 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch38-the-strong-sector-and-nf-discreteness.tex` + lines 81-92 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.StrongSector` + + - Name: `nf_discreteness_check` + + + ## Dependencies + + + - Canonical: III.D23, III.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The normal-form tower ℤ/Prim(k)ℤ is inherently discrete. No continuum limit within τ: each primorial level has + finitely many states. Discreteness earned from K3 divisibility, not imposed.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001561.md b/_corpus_v3_items/cid001561.md new file mode 100644 index 000000000..c92d5b294 --- /dev/null +++ b/_corpus_v3_items/cid001561.md @@ -0,0 +1,146 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001561/ +title: Gap Stabilization +cid: cid001561 +primary_alias: PRP0084 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The gap constant Γ* converges (stabilizes) as primorial depth increases, by tower coherence. Computable at each finite + level. +aliases: +- PRP0084 +- III.P17 +- gap-stabilization +- prop:gap-stabilization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 39 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Gap Stabilization** + + + The gap constant Γ* converges (stabilizes) as primorial depth increases, by tower coherence. Computable at each finite + level. + + + # Gap Stabilization + + + ## Summary + + + The gap constant Γ* converges (stabilizes) as primorial depth increases, by tower coherence. Computable at each finite + level. + + + ## Statement + + + ```latex + + \label{prop:gap-stabilization} + + The sequence $\{\Gamma_n\}_{n \geq 1}$ is non-decreasing and converges: + + \begin{equation} + + \Gamma_1 \;\leq\; \Gamma_2 \;\leq\; \Gamma_3 \;\leq\; \cdots + + \qquad\text{and}\qquad + + \Gamma^* \;=\; \lim_{n \to \infty} \Gamma_n \;=\; \inf_n \Gamma_n \;\geq\; \Gamma_1 \;>\; 0. + + \label{eq:ch39-gap-stabilization} + + \end{equation} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 106 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch39-the-tau-gap-meta-theorem.tex` + lines 165-175 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.GapTheorem` + + - Name: `mass_existence_check` + + + ## Dependencies + + + - Canonical: III.D45, III.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The gap constant Γ* converges (stabilizes) as primorial depth increases, by tower coherence. Computable at each + finite level. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001562.md b/_corpus_v3_items/cid001562.md new file mode 100644 index 000000000..e178a9a0d --- /dev/null +++ b/_corpus_v3_items/cid001562.md @@ -0,0 +1,136 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001562/ +title: Hodge Requires E₁ +cid: cid001562 +primary_alias: PRP0085 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At E₀, all characters are NF-addressable (Book II Central Theorem). The Hodge question becomes non-trivial only at + E₁, where sector admissibility predicates distinguish addressable-within-sector from addressable-globally. +aliases: +- PRP0085 +- III.P18 +- hodge-requires-e +- prop:hodge-requires-e1 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 41 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hodge Requires E₁** + + + At E₀, all characters are NF-addressable (Book II Central Theorem). The Hodge question becomes non-trivial only at E₁, + where sector admissibility predicates distinguish addressable-within-sector from addressable-globally. + + + # Hodge Requires E₁ + + + ## Summary + + + At E₀, all characters are NF-addressable (Book II Central Theorem). The Hodge question becomes non-trivial only at E₁, + where sector admissibility predicates distinguish addressable-within-sector from addressable-globally. + + + ## Statement + + + ```latex + + \label{prop:hodge-requires-e1} + + At enrichment level~$E_0$, every boundary character $\chi \in \operatorname{Char}(\mathbb{L})$ is globally NF-addressable. + The Hodge question---which characters are sector-addressable---becomes non-trivial only at~$E_1$, where the $4{+}1$ sector + decomposition introduces sector-specific admissibility predicates that are not present at~$E_0$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 112 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch41-sigma-fixed-characters-and-sector-addressability.tex` + lines 115-119 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.Hodge` + + - Name: `char_sector_check` + + + ## Dependencies + + + - Canonical: III.T14, III.D13 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At E₀, all characters are NF-addressable (Book II Central Theorem). The Hodge question becomes non-trivial only + at E₁, where sector admissibility predicates distinguish addressable-within-sector from addressable-globally. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001563.md b/_corpus_v3_items/cid001563.md new file mode 100644 index 000000000..7f8bee78f --- /dev/null +++ b/_corpus_v3_items/cid001563.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001563/ +title: EM Sector Verification +cid: cid001563 +primary_alias: PRP0086 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Explicit verification for the B-sector (electromagnetism): the B-projection of every σ-fixed character has finite + NF depth. Model case using spectral trichotomy and Label_n stabilization.' +aliases: +- PRP0086 +- III.P19 +- em-sector-verification +- prop:em-sector-verification +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 42 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**EM Sector Verification** + + + Explicit verification for the B-sector (electromagnetism): the B-projection of every σ-fixed character has finite NF depth. + Model case using spectral trichotomy and Label_n stabilization. + + + # EM Sector Verification + + + ## Summary + + + Explicit verification for the B-sector (electromagnetism): the B-projection of every σ-fixed character has finite NF depth. + Model case using spectral trichotomy and Label_n stabilization. + + + ## Statement + + + ```latex + + % + + \label{prop:em-sector-verification} + + Let $\chi$ be a $\sigma$-fixed boundary character + + at finite primorial depth~$k$. + + Then the $B$-sector projection + + $\pi_B(\chi) = e_+ \cdot \chi$ + + has finite NF depth: + + $\operatorname{NF}(\pi_B(\chi))$ stabilizes at depth~$k$, + + and more precisely at each prime~$p_i$ + + it stabilizes at depth~$i$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 114 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch42-the-nf-addressability-theorem.tex` + lines 217-228 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.Hodge` + + - Name: `hodge_filtration_check` + + + ## Dependencies + + + - Canonical: III.T14, III.T13, III.D23 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Explicit verification for the B-sector (electromagnetism): the B-projection of every σ-fixed character has finite + NF depth. Model case using spectral trichotomy and Label_n stabilization.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001564.md b/_corpus_v3_items/cid001564.md new file mode 100644 index 000000000..f74d73558 --- /dev/null +++ b/_corpus_v3_items/cid001564.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001564/ +title: Sector-by-Sector Protocol +cid: cid001564 +primary_alias: PRP0087 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Verification for C-sector (strong), A-sector (weak), D-sector (gravity). Each uses the same schema as the EM model + case, with sector-specific Label_n classification. +aliases: +- PRP0087 +- III.P20 +- sector-by-sector-protocol +- prop:sector-by-sector-protocol +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 42 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sector-by-Sector Protocol** + + + Verification for C-sector (strong), A-sector (weak), D-sector (gravity). Each uses the same schema as the EM model case, + with sector-specific Label_n classification. + + + # Sector-by-Sector Protocol + + + ## Summary + + + Verification for C-sector (strong), A-sector (weak), D-sector (gravity). Each uses the same schema as the EM model case, + with sector-specific Label_n classification. + + + ## Statement + + + ```latex + + % + + \label{prop:sector-by-sector-protocol} + + Let $\chi$ be a $\sigma$-fixed boundary character + + at finite primorial depth~$k$. + + Then $\pi_S(\chi)$ is NF-addressable + + for each sector $S \in \{A, C, D\}$, + + with stabilization depth $k_S \leq k$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 115 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch42-the-nf-addressability-theorem.tex` + lines 306-314 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.Hodge` + + - Name: `spectral_hodge_check` + + + ## Dependencies + + + - Canonical: III.P19, III.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Verification for C-sector (strong), A-sector (weak), D-sector (gravity). Each uses the same schema as the EM + model case, with sector-specific Label_n classification. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001565.md b/_corpus_v3_items/cid001565.md new file mode 100644 index 000000000..6951ab66b --- /dev/null +++ b/_corpus_v3_items/cid001565.md @@ -0,0 +1,187 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001565/ +title: Earned Admissibility +cid: cid001565 +primary_alias: PRP0088 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Characters (W=1), identity (W=0), successor (W=1) are admissible. Composition sub-additive: W(g∘f) ≤ W(f)+W(g). + CRT product: W = max. The admissible functions form a compositionally closed class.' +aliases: +- PRP0088 +- III.P21 +- earned-admissibility +- prop:earned-admissibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005788 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 56 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Earned Admissibility** + + + Characters (W=1), identity (W=0), successor (W=1) are admissible. Composition sub-additive: W(g∘f) ≤ W(f)+W(g). CRT product: + W = max. The admissible functions form a compositionally closed class. + + + # Earned Admissibility + + + ## Summary + + + Characters (W=1), identity (W=0), successor (W=1) are admissible. Composition sub-additive: W(g∘f) ≤ W(f)+W(g). CRT product: + W = max. The admissible functions form a compositionally closed class. + + + ## Statement + + + ```latex + + \label{prop:earned-admissibility} + + The following hold: + + \begin{enumerate} + + \item \textbf{Characters.} + + The boundary characters $\chi_+ = \tfrac{1}{2}(1 + \jj)$ and + + $\chi_- = \tfrac{1}{2}(1 - \jj)$ are $\tau$-admissible with + + $W(\chi_\pm) = 1$. + + \item \textbf{Identity.} + + The identity function $\id$ is $\tau$-admissible with $W(\id) = 0$. + + \item \textbf{Successor.} + + The successor map $\rho(x) = x + 1$ is $\tau$-admissible with + + $W(\rho) = 1$. + + \item \textbf{Composition.} + + If $f$ and $g$ are $\tau$-admissible, then $g \circ f$ is $\tau$-admissible with + + \begin{equation}\label{eq:ch56-composition-width} + + W(g \circ f) \;\leq\; W(f) + W(g). + + \end{equation} + + \item \textbf{CRT product.} + + If $f_1, \ldots, f_k$ are $\tau$-admissible with widths $W(f_i) = w_i$, + + then the CRT product $f_1 \times \cdots \times f_k$ + + (acting componentwise on the CRT decomposition) is $\tau$-admissible with + + \begin{equation}\label{eq:ch56-crt-product-width} + + W(f_1 \times \cdots \times f_k) \;=\; \max_{1 \leq i \leq k} w_i. + + \end{equation} + + \end{enumerate} + + Consequently, the $\tau$-admissible functions form a compositionally closed class. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 151 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch56-interface-width-and-tau-admissibility.tex` + lines 274-301 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.Admissibility` + + - Name: `earned_admissibility_check` + + + ## Dependencies + + + - Canonical: III.D54, III.T31 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Characters (W=1), identity (W=0), successor (W=1) are admissible. Composition sub-additive: W(g∘f) ≤ W(f)+W(g). + CRT product: W = max. The admissible functions form a compositionally closed class.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001566.md b/_corpus_v3_items/cid001566.md new file mode 100644 index 000000000..471a83ba2 --- /dev/null +++ b/_corpus_v3_items/cid001566.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001566/ +title: CRT Witness Decomposition +cid: cid001566 +primary_alias: PRP0089 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Witness space at primorial depth k factors: W(x, Prim(k)) ≅ ∏ W(x, p_i). Each prime contributes independent 1D search. + CRT guarantees reconstruction. Product of costs becomes sum of costs.' +aliases: +- PRP0089 +- III.P22 +- crt-witness-decomposition +- prop:crt-witness-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005806 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 57 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**CRT Witness Decomposition**\n\nWitness space at primorial depth k factors: W(x, Prim(k)) ≅ ∏ W(x, p_i). Each\ + \ prime contributes independent 1D search. CRT guarantees reconstruction. Product of costs becomes sum of costs.\n\n#\ + \ CRT Witness Decomposition\n\n## Summary\n\nWitness space at primorial depth k factors: W(x, Prim(k)) ≅ ∏ W(x, p_i).\ + \ Each prime contributes independent 1D search. CRT guarantees reconstruction. Product of costs becomes sum of costs.\n\ + \n## Statement\n\n```latex\n\\label{prop:crt-witness-decomposition}\nLet $\\Pi$ be $\\tau$-admissible, $x$ a ``yes'' instance,\ + \ and $k$ the\nprimorial depth of the TTM verifier. Then:\n\\[\nW(x, k) \\;\\cong\\; \\prod_{i=1}^{k} W(x, p_i),\n\\]\n\ + where $W(x, p_i) \\subseteq \\mathbb{Z}/p_i\\mathbb{Z}$ is the\n\\textbf{per-prime witness space}. Moreover:\n\\begin{enumerate}\n\ + \\item \\textbf{Independence.}\n $w_i$ at $p_i$ does not constrain $w_j$ at $p_j$ for $j \\neq i$.\n\\item \\textbf{Unique\ + \ reassembly.}\n Given $(w_1, \\ldots, w_k)$, the CRT produces a unique\n $w \\in W(x, k)$ with $w \\equiv w_i\ + \ \\pmod{p_i}$ for all $i$.\n\\item \\textbf{Cardinality.}\n $|W(x, k)| = \\prod_{i=1}^{k} |W(x, p_i)|$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 153\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch57-witness-search-as-address-resolution.tex`\ + \ lines 127-145\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.WitnessSearch`\n\ + - Name: `crt_witness_check`\n\n## Dependencies\n\n- Canonical: III.D55, III.T10\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Witness space at primorial depth k factors: W(x, Prim(k)) ≅ ∏ W(x, p_i). Each prime contributes independent + 1D search. CRT guarantees reconstruction. Product of costs becomes sum of costs.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001567.md b/_corpus_v3_items/cid001567.md new file mode 100644 index 000000000..8c725fd89 --- /dev/null +++ b/_corpus_v3_items/cid001567.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001567/ +title: Polynomial Refinement +cid: cid001567 +primary_alias: PRP0090 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Per-prime witness space |W(x, p_i)| ≤ p_i. Total refined search: O(∑ p_i) = O(k² log k) by PNT. Exponential product + space collapses to polynomial sum.' +aliases: +- PRP0090 +- III.P23 +- polynomial-refinement +- prop:polynomial-refinement +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005807 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 57 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Polynomial Refinement**\n\nPer-prime witness space |W(x, p_i)| ≤ p_i. Total refined search: O(∑ p_i) = O(k²\ + \ log k) by PNT. Exponential product space collapses to polynomial sum.\n\n# Polynomial Refinement\n\n## Summary\n\nPer-prime\ + \ witness space |W(x, p_i)| ≤ p_i. Total refined search: O(∑ p_i) = O(k² log k) by PNT. Exponential product space collapses\ + \ to polynomial sum.\n\n## Statement\n\n```latex\n\\label{prop:polynomial-refinement}\nLet $\\Pi$ be $\\tau$-admissible\ + \ with verifier width $W(V) = k_0$, and\nlet $x$ be a ``yes'' instance at depth $k \\geq k_0$. Then:\n\\begin{enumerate}\n\ + \\item \\textbf{Per-prime bound.}\n $|W(x, p_i)| \\leq p_i$ for each $1 \\leq i \\leq k$.\n\\item \\textbf{Search\ + \ cost.}\n \\begin{equation}\\label{eq:ch57-polynomial-cost}\n \\operatorname{Cost}(x, k) \\;=\\;\n \\\ + sum_{i=1}^{k} |W(x, p_i)|\n \\;\\leq\\; \\sum_{i=1}^{k} p_i\n \\;=\\; O(k^2 \\log k).\n \\end{equation}\n\ + \\item \\textbf{Comparison.}\n Brute-force cost $\\operatorname{Prim}(k) \\sim e^{k \\ln k}$\n (exponential);\ + \ the CRT replaces $\\prod$ with $\\sum$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n_No immediate manuscript\ + \ proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 154\n\ + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch57-witness-search-as-address-resolution.tex`\ + \ lines 186-204\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.WitnessSearch`\n\ + - Name: `polynomial_refinement_check`\n\n## Dependencies\n\n- Canonical: III.P22, III.D55\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Per-prime witness space |W(x, p_i)| ≤ p_i. Total refined search: O(∑ p_i) = O(k² log k) by PNT. Exponential + product space collapses to polynomial sum.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001568.md b/_corpus_v3_items/cid001568.md new file mode 100644 index 000000000..c122eee27 --- /dev/null +++ b/_corpus_v3_items/cid001568.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001568/ +title: Three-Reading Equivalence at E₁ +cid: cid001568 +primary_alias: PRP0091 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The E₁ Mutual Determination instance admits exactly three non-trivial sector-restricted readings (NS, YM, Hodge) +aliases: +- PRP0091 +- III.P24 +- three-reading-equivalence-at-e +- prop:three-reading-equivalence-e1 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005695 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 44 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Three-Reading Equivalence at E₁**\n\nThe E₁ Mutual Determination instance admits exactly three non-trivial\ + \ sector-restricted readings (NS, YM, Hodge)\n\n# Three-Reading Equivalence at E₁\n\n## Summary\n\nThe E₁ Mutual Determination\ + \ instance admits exactly three non-trivial sector-restricted readings (NS, YM, Hodge)\n\n## Statement\n\n```latex\n\\\ + label{prop:three-reading-equivalence-e1}\nThe $\\Elayer{1}$ Mutual Determination instance $(B_1, S_1, I_1)$ admits exactly\ + \ three non-trivial sector-restricted readings, corresponding to the three Millennium Problems at~$\\Elayer{1}$:\n\\begin{center}\n\ + \\begin{tabular}{lllll}\n\\toprule\n\\textbf{Reading} & \\textbf{Sector} & \\textbf{Boundary ($B$)} & \\textbf{Interior\ + \ ($I$)} & \\textbf{Spectral ($S$)} \\\\\n\\midrule\nNS & All five & $\\tau$-admissible & Stabilized & Defect \\\\\n &\ + \ sectors & fluid data & $\\omega$-germ & contractivity \\\\\n & & (Def.~\\ref{def:tau-admissible-fluid-data}) & (Thm.~\\\ + ref{thm:positive-regularity}) & $\\Delta \\to 0$ \\\\\n\\addlinespace\nYM & C-sector & $\\tau$-admissible & Spectral gap\ + \ & NF discreteness \\\\\n & (strong) & gauge data & $\\Gamma^*_s > 0$ & $|\\operatorname{NF}| \\leq \\operatorname{Prim}(k)$\ + \ \\\\\n & & (Def.~\\ref{def:tau-admissible-gauge-data}) & (Thm.~\\ref{thm:yang-mills-gap-theorem}) & \\\\\n\\addlinespace\n\ + Hodge & $B/C$ & $\\sigma$-fixed & NF-addressable & Balanced \\\\\n & balanced & characters & in all sectors & $\\chi_+/\\\ + chi_-$ \\\\\n & & (Def.~\\ref{def:sigma-fixed-character}) & (Thm.~\\ref{thm:nf-addressability}) & equilibrium \\\\\n\\\ + bottomrule\n\\end{tabular}\n\\end{center}\nThe three readings exhaust the non-trivial sector projections of $(B_1, S_1,\ + \ I_1)$:\n\\begin{enumerate}\n\\item[(i)] \\textbf{NS} reads the full five-sector data and asks for global regularity---smooth\ + \ gluing across all sectors.\n\\item[(ii)] \\textbf{YM} reads the C-sector restriction and asks for discrete granularity---a\ + \ spectral gap in the strong channel.\n\\item[(iii)] \\textbf{Hodge} reads the $B/C$-balanced sublattice and asks for\ + \ addressability---spectral visibility of all balanced characters.\n\\end{enumerate}\nNo fourth non-trivial reading exists:\ + \ the $A$-sector (weak) and $D$-sector (gravity) are structurally symmetric to the $B$- and $C$-sectors at~$\\Elayer{1}$\ + \ and do not generate independent Millennium-Problem-level consequences.\n```\n\n## Proof / Justification\n\n_No immediate\ + \ manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 121\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch44-enrichment-from-e0-to-e1.tex`\ + \ lines 169-199\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Arithmetic.EnrFunctor01`\n\ + - Name: `three_reading_check`\n\n## Dependencies\n\n- Canonical: III.D57, III.D58\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The E₁ Mutual Determination instance admits exactly three non-trivial sector-restricted readings (NS, YM, Hodge) +noindex: false +--- + diff --git a/_corpus_v3_items/cid001569.md b/_corpus_v3_items/cid001569.md new file mode 100644 index 000000000..3c3bb8ec0 --- /dev/null +++ b/_corpus_v3_items/cid001569.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001569/ +title: Mordell-Weil Analogue +cid: cid001569 +primary_alias: PRP0092 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Group of τ-rational points is finitely generated at each primorial level; rank stabilizes at finite depth +aliases: +- PRP0092 +- III.P25 +- mordell-weil-analogue +- prop:mordell-weil-analogue +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005707 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 45 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Mordell-Weil Analogue** + + + Group of τ-rational points is finitely generated at each primorial level; rank stabilizes at finite depth + + + # Mordell-Weil Analogue + + + ## Summary + + + Group of τ-rational points is finitely generated at each primorial level; rank stabilizes at finite depth + + + ## Statement + + + ```latex + + \label{prop:mordell-weil-analogue} + + The group $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}}$ of $\tau$-rational points is the union of a tower of finitely generated + abelian groups $G_1 \subseteq G_2 \subseteq G_3 \subseteq \cdots$, and the rank function $r(k) = \operatorname{rk}_{\mathbb{Z}}(G_k + / G_k^{\mathrm{tor}})$ stabilizes at a finite depth $k_* \in \mathbb{N}$: + + \begin{equation} + + \exists\, k_* \in \mathbb{N} \;\;\text{such that}\;\; + + r(k) = r(k_*) \quad \text{for all } k \geq k_*. + + \label{eq:ch45-mordell-weil-stabilization} + + \end{equation} + + In particular, $r_\infty = r(k_*)$ is a finite non-negative integer, and $\hat{\mathbb{Z}}_{\T}^{\,\mathbb{Q}} \cong \mathbb{Z}^{r_\infty} + \oplus T$ where $T$ is a torsion group compatible with the tower. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 124 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch45-tau-rational-interior-points.tex` + lines 188-198 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.RationalPoints` + + - Name: `mordell_weil_check` + + + ## Dependencies + + + - Canonical: III.D59, III.D60 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Group of τ-rational points is finitely generated at each primorial level; rank stabilizes at finite depth +noindex: false +--- + diff --git a/_corpus_v3_items/cid001570.md b/_corpus_v3_items/cid001570.md new file mode 100644 index 000000000..c75c9db42 --- /dev/null +++ b/_corpus_v3_items/cid001570.md @@ -0,0 +1,167 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001570/ +title: Bridgehead Proposition +cid: cid001570 +primary_alias: PRP0093 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Proto-codes provide necessary ingredient for E₂ emergence; non-trivial iff rank > 0 +aliases: +- PRP0093 +- III.P26 +- bridgehead-proposition +- prop:bridgehead-proposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005704 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 46 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bridgehead Proposition** + + + Proto-codes provide necessary ingredient for E₂ emergence; non-trivial iff rank > 0 + + + # Bridgehead Proposition + + + ## Summary + + + Proto-codes provide necessary ingredient for E₂ emergence; non-trivial iff rank > 0 + + + ## Statement + + + ```latex + + \label{prop:bridgehead-proposition} + + Let $\operatorname{Agent}(\Elayer{2})$ denote the category of $\Elayer{2}$ + + computational agents. Then: + + \begin{enumerate} + + \item[\emph{(i)}] \textbf{Necessity.} Every $\Elayer{2}$ agent restricts to a + + proto-code: the forgetful functor + + $U : \operatorname{Agent}(\Elayer{2}) \to \operatorname{Proto}(\Elayer{1})$ + + that discards the decoder is well-defined and faithful. + + + \item[\emph{(ii)}] \textbf{Non-sufficiency.} The functor $U$ is not + + essentially surjective: there exist proto-codes that are not the underlying + + carrier of any $\Elayer{2}$ agent. + + + \item[\emph{(iii)}] \textbf{Non-triviality criterion.} + + $\operatorname{Agent}(\Elayer{2})$ is non-empty only if + + $\operatorname{Proto}(\Elayer{1})$ contains non-torsion objects. + + Equivalently, $\Elayer{2}$ emergence requires $r_\infty > 0$, which by the + + BSD functional is equivalent to $\operatorname{BSD}_{\T}(k_*) \neq 0$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 127 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch46-proto-codes-and-the-bsd-bridgehead.tex` + lines 356-377 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ProtoCodes` + + - Name: `bridgehead_check` + + + ## Dependencies + + + - Canonical: III.D61, III.D62, III.P25 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Proto-codes provide necessary ingredient for E₂ emergence; non-trivial iff rank > 0 +noindex: false +--- + diff --git a/_corpus_v3_items/cid001571.md b/_corpus_v3_items/cid001571.md new file mode 100644 index 000000000..78ceea334 --- /dev/null +++ b/_corpus_v3_items/cid001571.md @@ -0,0 +1,175 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001571/ +title: BSD Three-Ingredient Proof +cid: cid001571 +primary_alias: PRP0094 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: BSD coherence follows from rank stabilization, L-value stabilization, and E₁ Mutual Determination equality +aliases: +- PRP0094 +- III.P27 +- bsd-three-ingredient-proof +- prop:bsd-three-ingredient +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005692 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 47 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**BSD Three-Ingredient Proof** + + + BSD coherence follows from rank stabilization, L-value stabilization, and E₁ Mutual Determination equality + + + # BSD Three-Ingredient Proof + + + ## Summary + + + BSD coherence follows from rank stabilization, L-value stabilization, and E₁ Mutual Determination equality + + + ## Statement + + + ```latex + + \label{prop:bsd-three-ingredient} + + Let $\mathcal{E}$ be a $\tau$-admissible elliptic datum: an $\Elayer{1}$ object in $\operatorname{Cat}_{\T}(\Elayer{1})$ + equipped with proto-code structure (Definition~\ref{def:proto-code}, Ch.~46). Then the following three statements hold: + + \begin{enumerate} + + \item[\emph{(I1)}] \textbf{Rank stabilization.} The rank function $r(k) = \operatorname{rk}_{\mathbb{Z}}(G_k / G_k^{\mathrm{tor}})$ + stabilizes at a finite depth $k_r \in \mathbb{N}$: + + \begin{equation} + + r(k) = r_\infty \quad \text{for all } k \geq k_r. + + \label{eq:ch47-rank-stabilization} + + \end{equation} + + + \item[\emph{(I2)}] \textbf{$L$-value stabilization.} The spectral determinant $L_{\T}''(1,k)$, evaluated at the central + point $s = 1$ along the primorial ladder, stabilizes at a finite depth $k_L \in \mathbb{N}$: + + \begin{equation} + + L_{\T}''(1,k) = L_\infty'' \quad \text{for all } k \geq k_L. + + \label{eq:ch47-l-value-stabilization} + + \end{equation} + + + \item[\emph{(I3)}] \textbf{Rank--$L$-value equality.} The stable rank $r_\infty$ equals the order of vanishing of $L_{\T}''(1,k)$, + and the stable $L$-derivative $L_\infty''$ determines $r_\infty$: + + \begin{equation} + + r_\infty \;=\; \operatorname{ord}_{s=1} L_{\T}(s) \;=\; \min\{n \geq 0 : L_{\T}^{(n)}(1) \neq 0\}. + + \label{eq:ch47-rank-l-equality} + + \end{equation} + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 129 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch47-the-bsd-coherence-theorem.tex` + lines 49-72 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.BSD` + + - Name: `bsd_three_ingredient_check` + + + ## Dependencies + + + - Canonical: III.T35 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: BSD coherence follows from rank stabilization, L-value stabilization, and E₁ Mutual Determination equality +noindex: false +--- + diff --git a/_corpus_v3_items/cid001572.md b/_corpus_v3_items/cid001572.md new file mode 100644 index 000000000..d7cf239d9 --- /dev/null +++ b/_corpus_v3_items/cid001572.md @@ -0,0 +1,158 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001572/ +title: Duality as Mutual Determination on ℤ² +cid: cid001572 +primary_alias: PRP0095 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Automorphic-Galois duality is Mutual Determination with m-axis as boundary, n-axis as spectral, full character as + interior +aliases: +- PRP0095 +- III.P28 +- duality-as-mutual-determination-on +- prop:duality-mutual-determination-z2 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 48 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Duality as Mutual Determination on ℤ²** + + + Automorphic-Galois duality is Mutual Determination with m-axis as boundary, n-axis as spectral, full character as interior + + + # Duality as Mutual Determination on ℤ² + + + ## Summary + + + Automorphic-Galois duality is Mutual Determination with m-axis as boundary, n-axis as spectral, full character as interior + + + ## Statement + + + ```latex + + \label{prop:duality-mutual-determination-z2} + + The automorphic--Galois duality (Definition~\ref{def:automorphic-galois-duality}) is an instance of the Mutual Determination + Schema (Definition~\ref{def:mutual-determination-schema}, Ch.~33) with the following identification: + + \begin{enumerate} + + \item\emph{(B)} \textbf{Boundary datum} $=$ the $m$-axis data $(\operatorname{Fr}_p)_{p}$ (Galois side): the prime-by-prime + CRT decomposition of the global character restricted to the boundary $\Lemniscate$. + + + \item\emph{(S)} \textbf{Spectral datum} $=$ the $n$-axis data $(\lambda_n)_{n}$ (automorphic side): the eigenvalue-by-eigenvalue + decomposition of the spectral operator on $A_{\mathrm{spec}}(\Lemniscate)$. + + + \item\emph{(I)} \textbf{Interior datum} $=$ the full character $\chi_{m,n}$ on $\mathbb{Z}^2$: the complete lattice of + spectral characters at enrichment level~$\Elayer{1}$. + + \end{enumerate} + + The triple $(B, S, I)$ satisfies full bidirectionality: + + \begin{equation} + + B \;\longleftrightarrow\; I \;\longleftrightarrow\; S. + + \label{eq:ch48-mutual-det-bsi} + + \end{equation} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 133 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch48-automorphic-galois-duality.tex` + lines 206-222 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.Langlands` + + - Name: `duality_md_check` + + + ## Dependencies + + + - Canonical: III.D63, III.D64 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Automorphic-Galois duality is Mutual Determination with m-axis as boundary, n-axis as spectral, full character + as interior +noindex: false +--- + diff --git a/_corpus_v3_items/cid001573.md b/_corpus_v3_items/cid001573.md new file mode 100644 index 000000000..b956f72e9 --- /dev/null +++ b/_corpus_v3_items/cid001573.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001573/ +title: Chain Verification Protocol +cid: cid001573 +primary_alias: PRP0096 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Five-step protocol for verifying the Complete Dependency Chain: link audit, primorial instantiation, compatibility + check, inverse-limit assembly, and scope confirmation. Each link is independently checkable.' +aliases: +- PRP0096 +- III.P29 +- chain-verification-protocol +- prop:chain-verification-protocol +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005877 +- predicate: formalized_by + object: cid005883 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Chain Verification Protocol**\n\nFive-step protocol for verifying the Complete Dependency Chain: link audit,\ + \ primorial instantiation, compatibility check, inverse-limit assembly, and scope confirmation. Each link is independently\ + \ checkable.\n\n# Chain Verification Protocol\n\n## Summary\n\nFive-step protocol for verifying the Complete Dependency\ + \ Chain: link audit, primorial instantiation, compatibility check, inverse-limit assembly, and scope confirmation. Each\ + \ link is independently checkable.\n\n## Statement\n\n```latex\n\\label{prop:chain-verification-protocol}\nThe following\ + \ protocol verifies the complete dependency chain\n(Definition~\\ref{def:complete-dependency-chain})\nat primorial depth~$k$:\n\ + \\begin{enumerate}\n\\item \\textbf{Link audit.}\n For each link~$n$ ($1 \\le n \\le 14$),\n identify the earning\ + \ theorem or definition\n and confirm it appears in the stated chapter\n with $\\tau$-effective status.\n\n\\\ + item \\textbf{Primorial instantiation.}\n Replace $\\T$ by~$\\T_k$,\n $\\Lemniscate$ by~$\\Lemniscate_k$,\n\ + \ $\\iota_\\tau$ by~$\\iota_{\\tau,k}$.\n Verify each link by explicit computation\n in $\\mathbb{Z}/\\\ + operatorname{Prim}(k)\\mathbb{Z}$.\n\n\\item \\textbf{Compatibility.}\n For adjacent levels $(k, k{+}1)$,\n \ + \ verify that the $(k{+}1)$-chain restricts\n to the $k$-chain under\n $\\mathbb{Z}/\\operatorname{Prim}(k{+}1)\\\ + mathbb{Z}\n \\twoheadrightarrow\n \\mathbb{Z}/\\operatorname{Prim}(k)\\mathbb{Z}$.\n\n\\item \\textbf{Inverse-limit\ + \ assembly.}\n Verify the Mittag-Leffler condition so that\n $\\varprojlim_k (\\text{chain at level } k)$\n\ + \ recovers the full chain.\n\n\\item \\textbf{Scope confirmation.}\n Confirm every link carries $\\tau$-effective\ + \ scope.\n\\end{enumerate}\nA chain passing all five checks at every $k \\ge 1$\nis \\emph{verified}.\n```\n\n## Proof\ + \ / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n-\ + \ Registry source: `book-03.jsonl` line 162\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch60-the-complete-dependency-chain.tex`\ + \ lines 242-280\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Hinge.DependencyChain`\n\ + - Name: `chain_linearity_check`\n\n## Dependencies\n\n- Canonical: III.D66\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Five-step protocol for verifying the Complete Dependency Chain: link audit, primorial instantiation, compatibility + check, inverse-limit assembly, and scope confirmation. Each link is independently checkable.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001574.md b/_corpus_v3_items/cid001574.md new file mode 100644 index 000000000..da958994f --- /dev/null +++ b/_corpus_v3_items/cid001574.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001574/ +title: Sector Instantiation Lemma +cid: cid001574 +primary_alias: PRP0097 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Restricting the enrichment functor Enr^k to a single sector S produces a well-defined sub-theory: closure under + sector morphisms, faithfulness on sector-internal diagrams, coherence with the full functor, and completeness of the restricted + theory.' +aliases: +- PRP0097 +- III.P30 +- sector-instantiation-lemma +- prop:sector-instantiation-lemma +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005878 +- predicate: formalized_by + object: cid005884 +- predicate: formalized_by + object: cid005885 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 61 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sector Instantiation Lemma**\n\nRestricting the enrichment functor Enr^k to a single sector S produces a well-defined\ + \ sub-theory: closure under sector morphisms, faithfulness on sector-internal diagrams, coherence with the full functor,\ + \ and completeness of the restricted theory.\n\n# Sector Instantiation Lemma\n\n## Summary\n\nRestricting the enrichment\ + \ functor Enr^k to a single sector S produces a well-defined sub-theory: closure under sector morphisms, faithfulness\ + \ on sector-internal diagrams, coherence with the full functor, and completeness of the restricted theory.\n\n## Statement\n\ + \n```latex\n\\label{prop:sector-instantiation-lemma}\nLet $S \\in \\{A, B, C, D\\}$ be a primitive sector and $k \\in\ + \ \\{1, 2, 3\\}$ an enrichment level. The sector restriction $\\operatorname{Enr}^k|_{S}$ satisfies:\n\\begin{enumerate}\n\ + \\item\\emph{(Closure.)} $\\operatorname{Cat}_{\\T}(\\Elayer{k})|_{S}$ is a full subcategory of $\\operatorname{Cat}_{\\\ + T}(\\Elayer{k})$, closed under the categorical operations inherited from $\\operatorname{Cat}_{\\T}(\\Elayer{k})$.\n\n\ + \\item\\emph{(Faithfulness.)} The restriction $\\operatorname{Enr}^k|_{S}$ is faithful: distinct $\\Elayer{0}$-objects\ + \ with sector address~$S$ map to distinct $\\Elayer{k}$-objects.\n\n\\item\\emph{(Coherence.)} The diagram\n\\[\n\\begin{array}{ccc}\n\ + \\operatorname{Cat}_{\\T}(\\Elayer{0})\\big|_{S}\n & \\xrightarrow{\\;\\operatorname{Enr}^k|_{S}\\;}\n & \\operatorname{Cat}_{\\\ + T}(\\Elayer{k})\\big|_{S} \\\\[6pt]\n\\big\\downarrow\\vcenter{\\rlap{\\scriptsize incl}} & & \\big\\downarrow\\vcenter{\\\ + rlap{\\scriptsize incl}} \\\\[6pt]\n\\operatorname{Cat}_{\\T}(\\Elayer{0})\n & \\xrightarrow{\\;\\operatorname{Enr}^k\\\ + ;}\n & \\operatorname{Cat}_{\\T}(\\Elayer{k})\n\\end{array}\n\\]\ncommutes: enriching then restricting equals restricting\ + \ then enriching.\n\n\\item\\emph{(Sub-theory completeness.)} Every morphism in $\\operatorname{Cat}_{\\T}(\\Elayer{k})|_{S}$\ + \ is the image under $\\operatorname{Enr}^k|_{S}$ of a morphism in $\\operatorname{Cat}_{\\T}(\\Elayer{0})|_{S}$, composed\ + \ with intra-sector structure earned at level~$\\Elayer{k}$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n_No\ + \ immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 164\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch61-the-hinge-theorem.tex`\ + \ lines 110-135\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Hinge.DependencyChain`\n\ + - Name: `terminal_completeness_check`\n\n## Dependencies\n\n- Canonical: III.T01, III.D10\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Restricting the enrichment functor Enr^k to a single sector S produces a well-defined sub-theory: closure under + sector morphisms, faithfulness on sector-internal diagrams, coherence with the full functor, and completeness of the restricted ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001575.md b/_corpus_v3_items/cid001575.md new file mode 100644 index 000000000..a676bd289 --- /dev/null +++ b/_corpus_v3_items/cid001575.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001575/ +title: Terminal Level Characterization +cid: cid001575 +primary_alias: PRP0098 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₃ is terminal iff the self-modelling operator is idempotent: Enr(E₃) = E₃ means the self-model of the self-model + is the self-model. This is the abstract characterization of why the ladder stops at four levels.' +aliases: +- PRP0098 +- III.P31 +- terminal-level-characterization +- prop:terminal-level-characterization +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005922 +- predicate: formalized_by + object: cid005924 +- predicate: formalized_by + object: cid005926 +- predicate: formalized_by + object: cid005927 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Terminal Level Characterization**\n\nE₃ is terminal iff the self-modelling operator is idempotent: Enr(E₃)\ + \ = E₃ means the self-model of the self-model is the self-model. This is the abstract characterization of why the ladder\ + \ stops at four levels.\n\n# Terminal Level Characterization\n\n## Summary\n\nE₃ is terminal iff the self-modelling operator\ + \ is idempotent: Enr(E₃) = E₃ means the self-model of the self-model is the self-model. This is the abstract characterization\ + \ of why the ladder stops at four levels.\n\n## Statement\n\n```latex\n%\n\\label{prop:terminal-level-characterization}\n\ + An enrichment level~$\\Elayer{k}$ is \\textbf{terminal}\nif and only if the self-modelling operator~$S_{k}$\nis idempotent:\n\ + \\[\n S_{k} \\circ S_{k} = S_{k}.\n\\]\nEquivalently, $\\Elayer{k}$ is terminal if and only if\nevery $\\Elayer{k}$-observer's\ + \ self-model\nis already an $\\Elayer{k}$-object.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block\ + \ was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 189\n- Manuscript\ + \ source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch72-saturation-why-e3-is-terminal.tex` lines\ + \ 207-219\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.Saturation`\n\ + - Name: `terminal_level_check`\n\n## Dependencies\n\n- Canonical: III.T49, III.T03\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₃ is terminal iff the self-modelling operator is idempotent: Enr(E₃) = E₃ means the self-model of the self-model + is the self-model. This is the abstract characterization of why the ladder stops at four levels.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001576.md b/_corpus_v3_items/cid001576.md new file mode 100644 index 000000000..1baccfe3b --- /dev/null +++ b/_corpus_v3_items/cid001576.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001576/ +title: Bridge Algebraic Reduction +cid: cid001576 +primary_alias: PRP0099 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'κ(C;3)²·(1−ι_τ)² = ι_τ⁶ exactly (algebraic identity). This reduces OQ.07 (confinement bridge) to OQ.09 (E₆ near-identity) + by pure algebra: the (1−ι_τ)² factors cancel, leaving E₆·ι⁶ ≈ −1.' +aliases: +- PRP0099 +- III.P32 +- bridge-algebraic-reduction +- prop:bridge-algebraic-reduction +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006024 +- predicate: formalized_by + object: cid006025 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 25 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bridge Algebraic Reduction** + + + κ(C;3)²·(1−ι_τ)² = ι_τ⁶ exactly (algebraic identity). This reduces OQ.07 (confinement bridge) to OQ.09 (E₆ near-identity) + by pure algebra: the (1−ι_τ)² factors cancel, leaving E₆·ι⁶ ≈ −1. + + + # Bridge Algebraic Reduction + + + ## Summary + + + κ(C;3)²·(1−ι_τ)² = ι_τ⁶ exactly (algebraic identity). This reduces OQ.07 (confinement bridge) to OQ.09 (E₆ near-identity) + by pure algebra: the (1−ι_τ)² factors cancel, leaving E₆·ι⁶ ≈ −1. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 213 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.ConfinementBridge` + + - Name: `Tau.BookIII.Spectral.ConfinementBridge.bridge_algebraic_identity` + + + ## Dependencies + + + - Canonical: IV.D03, IV.D07 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'κ(C;3)²·(1−ι_τ)² = ι_τ⁶ exactly (algebraic identity). This reduces OQ.07 (confinement bridge) to OQ.09 (E₆ near-identity) + by pure algebra: the (1−ι_τ)² factors cancel, leaving E₆·ι⁶ ≈ −1.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001578.md b/_corpus_v3_items/cid001578.md new file mode 100644 index 000000000..44573b4c8 --- /dev/null +++ b/_corpus_v3_items/cid001578.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001578/ +title: E₂ ⊋ E₁ Strict Witness +cid: cid001578 +primary_alias: PRP0101 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: E₂ contains orbit structures of length > 2 that cannot be expressed as pure sector decompositions (E₁ content). E₁ + orbits have length ≤ 2 from bipolar involution; E₂ orbits achieve lengths 3, 5, etc. +aliases: +- PRP0101 +- III.P34 +- e-e-strict-witness +- prop:e2-strict-witness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005800 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₂ ⊋ E₁ Strict Witness** + + + E₂ contains orbit structures of length > 2 that cannot be expressed as pure sector decompositions (E₁ content). E₁ orbits + have length ≤ 2 from bipolar involution; E₂ orbits achieve lengths 3, 5, etc. + + + # E₂ ⊋ E₁ Strict Witness + + + ## Summary + + + E₂ contains orbit structures of length > 2 that cannot be expressed as pure sector decompositions (E₁ content). E₁ orbits + have length ≤ 2 from bipolar involution; E₂ orbits achieve lengths 3, 5, etc. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 221 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Witness` + + - Name: `e2_strict_3` + + + ## Dependencies + + + - Canonical: III.D84, III.T57 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: E₂ contains orbit structures of length > 2 that cannot be expressed as pure sector decompositions (E₁ content). + E₁ orbits have length ≤ 2 from bipolar involution; E₂ orbits achieve lengths 3, 5, etc. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001579.md b/_corpus_v3_items/cid001579.md new file mode 100644 index 000000000..a590b07c8 --- /dev/null +++ b/_corpus_v3_items/cid001579.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001579/ +title: Saturation Semantics +cid: cid001579 +primary_alias: PRP0102 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'F_E(E₃) = E₃ has semantic content: the self-model of the self-model is isomorphic to the self-model (idempotence). + E₃ applied to E₃ outputs gives the same set as E₃ itself. Verified at stages 1-3.' +aliases: +- PRP0102 +- III.P35 +- saturation-semantics +- prop:saturation-semantics +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005904 +- predicate: formalized_by + object: cid005905 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Saturation Semantics** + + + F_E(E₃) = E₃ has semantic content: the self-model of the self-model is isomorphic to the self-model (idempotence). E₃ + applied to E₃ outputs gives the same set as E₃ itself. Verified at stages 1-3. + + + # Saturation Semantics + + + ## Summary + + + F_E(E₃) = E₃ has semantic content: the self-model of the self-model is isomorphic to the self-model (idempotence). E₃ + applied to E₃ outputs gives the same set as E₃ itself. Verified at stages 1-3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 225 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Mirror.E3Witness` + + - Name: `saturation_semantic_3` + + + ## Dependencies + + + - Canonical: III.D85, III.T58 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'F_E(E₃) = E₃ has semantic content: the self-model of the self-model is isomorphic to the self-model (idempotence). + E₃ applied to E₃ outputs gives the same set as E₃ itself. Verified at stages 1-3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001580.md b/_corpus_v3_items/cid001580.md new file mode 100644 index 000000000..36a157d36 --- /dev/null +++ b/_corpus_v3_items/cid001580.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001580/ +title: Arithmetic Preserves Operations +cid: cid001580 +primary_alias: PRP0103 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Translation preserves: (1) additive structure, (2) multiplicative structure, (3) GCD/LCM, (4) primality, (5) order. + Full operation-preservation check at bound 6, depth 3.' +aliases: +- PRP0103 +- III.P36 +- arithmetic-preserves-operations +- prop:arith-preserves-ops +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005762 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 80 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Arithmetic Preserves Operations** + + + Translation preserves: (1) additive structure, (2) multiplicative structure, (3) GCD/LCM, (4) primality, (5) order. Full + operation-preservation check at bound 6, depth 3. + + + # Arithmetic Preserves Operations + + + ## Summary + + + Translation preserves: (1) additive structure, (2) multiplicative structure, (3) GCD/LCM, (4) primality, (5) order. Full + operation-preservation check at bound 6, depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 229 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationArith` + + - Name: `arith_preserves_6_3` + + + ## Dependencies + + + - Canonical: III.T59 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Translation preserves: (1) additive structure, (2) multiplicative structure, (3) GCD/LCM, (4) primality, (5) + order. Full operation-preservation check at bound 6, depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001581.md b/_corpus_v3_items/cid001581.md new file mode 100644 index 000000000..8a1249b23 --- /dev/null +++ b/_corpus_v3_items/cid001581.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001581/ +title: Boundary Restriction +cid: cid001581 +primary_alias: PRP0104 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Reduction from stage k+1 to k corresponds to boundary restriction in the projective limit. Fiber over point x at + stage k has exactly p_{k+1} preimages. Verified at depth 3. +aliases: +- PRP0104 +- III.P37 +- boundary-restriction +- prop:boundary-restriction +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005772 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Boundary Restriction** + + + Reduction from stage k+1 to k corresponds to boundary restriction in the projective limit. Fiber over point x at stage + k has exactly p_{k+1} preimages. Verified at depth 3. + + + # Boundary Restriction + + + ## Summary + + + Reduction from stage k+1 to k corresponds to boundary restriction in the projective limit. Fiber over point x at stage + k has exactly p_{k+1} preimages. Verified at depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 233 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationTopo` + + - Name: `boundary_restriction_3` + + + ## Dependencies + + + - Canonical: III.T60 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Reduction from stage k+1 to k corresponds to boundary restriction in the projective limit. Fiber over point x + at stage k has exactly p_{k+1} preimages. Verified at depth 3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001582.md b/_corpus_v3_items/cid001582.md new file mode 100644 index 000000000..9c19e9985 --- /dev/null +++ b/_corpus_v3_items/cid001582.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001582/ +title: P vs NP as Polynomial Translation Obstruction +cid: cid001582 +primary_alias: PRP0105 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: P vs NP = polynomial translation obstruction. succinct_circuits has damage 3 (bridge breaks). Internal P_adm = NP_adm + (all finite problems decidable) but cannot be translated to ZFC. Verified at depth 3. +aliases: +- PRP0105 +- III.P38 +- p-vs-np-as-polynomial-translation-obstruction +- prop:pvsnp-obstruction +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005768 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**P vs NP as Polynomial Translation Obstruction** + + + P vs NP = polynomial translation obstruction. succinct_circuits has damage 3 (bridge breaks). Internal P_adm = NP_adm + (all finite problems decidable) but cannot be translated to ZFC. Verified at depth 3. + + + # P vs NP as Polynomial Translation Obstruction + + + ## Summary + + + P vs NP = polynomial translation obstruction. succinct_circuits has damage 3 (bridge breaks). Internal P_adm = NP_adm + (all finite problems decidable) but cannot be translated to ZFC. Verified at depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 237 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationObstruction` + + - Name: `pvsnp_obstruction_3` + + + ## Dependencies + + + - Canonical: III.T61 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: P vs NP = polynomial translation obstruction. succinct_circuits has damage 3 (bridge breaks). Internal P_adm + = NP_adm (all finite problems decidable) but cannot be translated to ZFC. Verified at depth 3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001583.md b/_corpus_v3_items/cid001583.md new file mode 100644 index 000000000..b5683a0d3 --- /dev/null +++ b/_corpus_v3_items/cid001583.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001583/ +title: Bridge Ledger Completeness +cid: cid001583 +primary_alias: PRP0106 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every Millennium Problem has explicit gap characterization: O₃ axiom (RH), bridge functor (YM, NS, Hodge, BSD, Langlands), + or forbidden triple (P vs NP). Poincaré established (no gap). Ledger complete.' +aliases: +- PRP0106 +- III.P39 +- bridge-ledger-completeness +- prop:bridge-ledger +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005813 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 37 (Wave M8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bridge Ledger Completeness** + + + Every Millennium Problem has explicit gap characterization: O₃ axiom (RH), bridge functor (YM, NS, Hodge, BSD, Langlands), + or forbidden triple (P vs NP). Poincaré established (no gap). Ledger complete. + + + # Bridge Ledger Completeness + + + ## Summary + + + Every Millennium Problem has explicit gap characterization: O₃ axiom (RH), bridge functor (YM, NS, Hodge, BSD, Langlands), + or forbidden triple (P vs NP). Poincaré established (no gap). Ledger complete. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 242 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.BridgeTightening` + + - Name: `bridge_ledger_complete` + + + ## Dependencies + + + - Canonical: III.D93, III.D94, III.T62, III.T63 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every Millennium Problem has explicit gap characterization: O₃ axiom (RH), bridge functor (YM, NS, Hodge, BSD, + Langlands), or forbidden triple (P vs NP). Poincaré established (no gap). Ledger complete.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001584.md b/_corpus_v3_items/cid001584.md new file mode 100644 index 000000000..fe8282e75 --- /dev/null +++ b/_corpus_v3_items/cid001584.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001584/ +title: Additive-Multiplicative Duality +cid: cid001584 +primary_alias: PRP0107 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Additive-multiplicative duality: at each primorial level, both CRT decomposition and Goldbach partitions are nontrivial. + Goldbach count, twin prime count, and CRT dimension all positive. Shadow of Langlands duality.' +aliases: +- PRP0107 +- III.P40 +- additive-multiplicative-duality +- prop:additive-multiplicative +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005999 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Additive-Multiplicative Duality** + + + Additive-multiplicative duality: at each primorial level, both CRT decomposition and Goldbach partitions are nontrivial. + Goldbach count, twin prime count, and CRT dimension all positive. Shadow of Langlands duality. + + + # Additive-Multiplicative Duality + + + ## Summary + + + Additive-multiplicative duality: at each primorial level, both CRT decomposition and Goldbach partitions are nontrivial. + Goldbach count, twin prime count, and CRT dimension all positive. Shadow of Langlands duality. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 246 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.AdditiveConjectures` + + - Name: `additive_multiplicative_3` + + + ## Dependencies + + + - Canonical: III.D95, III.D96, III.T64 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Additive-multiplicative duality: at each primorial level, both CRT decomposition and Goldbach partitions are + nontrivial. Goldbach count, twin prime count, and CRT dimension all positive. Shadow of Langlands duality.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001585.md b/_corpus_v3_items/cid001585.md new file mode 100644 index 000000000..f3f16ac25 --- /dev/null +++ b/_corpus_v3_items/cid001585.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001585/ +title: Radical-Primorial Identity +cid: cid001585 +primary_alias: PRP0108 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Radical-primorial identity: rad(M_k) = M_k for all k. Primorial tower consists of squarefree numbers — the natural + domain for ABC. ABCD coordinates decompose via tower; radical inherits decomposition. Verified at depth 4.' +aliases: +- PRP0108 +- III.P41 +- radical-primorial-identity +- prop:radical-primorial +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005681 +- predicate: formalized_by + object: cid005682 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Radical-Primorial Identity** + + + Radical-primorial identity: rad(M_k) = M_k for all k. Primorial tower consists of squarefree numbers — the natural domain + for ABC. ABCD coordinates decompose via tower; radical inherits decomposition. Verified at depth 4. + + + # Radical-Primorial Identity + + + ## Summary + + + Radical-primorial identity: rad(M_k) = M_k for all k. Primorial tower consists of squarefree numbers — the natural domain + for ABC. ABCD coordinates decompose via tower; radical inherits decomposition. Verified at depth 4. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 250 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCConjecture` + + - Name: `radical_primorial_4` + + + ## Dependencies + + + - Canonical: III.D97, III.T65 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Radical-primorial identity: rad(M_k) = M_k for all k. Primorial tower consists of squarefree numbers — the natural + domain for ABC. ABCD coordinates decompose via tower; radical inherits decomposition. Verified at depth 4.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001586.md b/_corpus_v3_items/cid001586.md new file mode 100644 index 000000000..06264d356 --- /dev/null +++ b/_corpus_v3_items/cid001586.md @@ -0,0 +1,147 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001586/ +title: Sieve-CRT Compatibility +cid: cid001586 +primary_alias: PRP0109 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves.' +aliases: +- PRP0109 +- III.P42 +- sieve-crt-compatibility +- prop:sieve-crt +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006071 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve-CRT Compatibility** + + + Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves. + + + # Sieve-CRT Compatibility + + + ## Summary + + + Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local sieves. + + + ## Statement + + + ```latex + + \label{prop:sieve-crt} + + An integer~$n$ is divisible by some $p_i$ ($i \le k$) if and only if + + the CRT decomposition $(n \bmod p_1, \ldots, n \bmod p_k)$ has a zero + + component. The multiplicative sieve equals the product of local sieves + + via CRT. + + + \textbf{Lean:} \texttt{sieve\_crt\_compat\_3}.\quad + + \textbf{Registry:} III.P42. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 256 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 126-134 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `sieve_crt_compat_3` + + + ## Dependencies + + + - Canonical: III.D99, III.T10 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Sieve-CRT compatibility: n divisible by p_i iff i-th CRT residue is 0. Multiplicative sieve = product of local + sieves.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001587.md b/_corpus_v3_items/cid001587.md new file mode 100644 index 000000000..cffac95e5 --- /dev/null +++ b/_corpus_v3_items/cid001587.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001587/ +title: CRT-Goldbach Duality +cid: cid001587 +primary_alias: PRP0110 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CRT-Goldbach local solvability: for any even n≥4 and prime p≥3, ∃r with both r and n-r nonzero mod p. CRT guarantees + independent local solutions.' +aliases: +- PRP0110 +- III.P43 +- crt-goldbach-duality +- prop:crt-goldbach +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006032 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT-Goldbach Duality** + + + CRT-Goldbach local solvability: for any even n≥4 and prime p≥3, ∃r with both r and n-r nonzero mod p. CRT guarantees independent + local solutions. + + + # CRT-Goldbach Duality + + + ## Summary + + + CRT-Goldbach local solvability: for any even n≥4 and prime p≥3, ∃r with both r and n-r nonzero mod p. CRT guarantees independent + local solutions. + + + ## Statement + + + ```latex + + \label{prop:crt-goldbach} + + For any even $n \ge 4$ and any prime $p \ge 3$, there exists $r \in [1, p-1]$ + + with both $r$ and $n - r$ nonzero modulo $p$. This is the \emph{local + + solvability} condition: at each prime, Goldbach has a solution. CRT + + guarantees that local solutions exist independently. + + + \textbf{Lean:} \texttt{crt\_goldbach\_duality\_3}.\quad + + \textbf{Registry:} III.P43. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 264 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 203-211 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `crt_goldbach_duality_3` + + + ## Dependencies + + + - Canonical: III.T10, III.D95 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CRT-Goldbach local solvability: for any even n≥4 and prime p≥3, ∃r with both r and n-r nonzero mod p. CRT guarantees + independent local solutions.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001588.md b/_corpus_v3_items/cid001588.md new file mode 100644 index 000000000..c08ca0550 --- /dev/null +++ b/_corpus_v3_items/cid001588.md @@ -0,0 +1,147 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001588/ +title: Goldbach Gap Characterization +cid: cid001588 +primary_alias: PRP0111 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Goldbach gap = parity barrier. CRT guarantees local conditions always satisfiable. Gap is exponential_quantification + (K4) — no sieve lifts local to global. +aliases: +- PRP0111 +- III.P44 +- goldbach-gap-characterization +- prop:goldbach-gap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach Gap Characterization** + + + Goldbach gap = parity barrier. CRT guarantees local conditions always satisfiable. Gap is exponential_quantification (K4) + — no sieve lifts local to global. + + + # Goldbach Gap Characterization + + + ## Summary + + + Goldbach gap = parity barrier. CRT guarantees local conditions always satisfiable. Gap is exponential_quantification (K4) + — no sieve lifts local to global. + + + ## Statement + + + ```latex + + \label{prop:goldbach-gap} + + CRT guarantees local conditions are always satisfiable. + + The gap is the \emph{parity barrier}: no sieve or finite verification + + can lift local solutions to a global proof. The + + \texttt{exponential\_quantification} forbidden move (K4 violation) + + captures this gap precisely. + + + \textbf{Registry:} III.P44.\quad + + \textbf{Scope:} established (meta-theorem about proof limitations). + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 265 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 213-222 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `goldbach_gap_parity` + + + ## Dependencies + + + - Canonical: III.P43, III.D69 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Goldbach gap = parity barrier. CRT guarantees local conditions always satisfiable. Gap is exponential_quantification + (K4) — no sieve lifts local to global. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001589.md b/_corpus_v3_items/cid001589.md new file mode 100644 index 000000000..59de4d0e2 --- /dev/null +++ b/_corpus_v3_items/cid001589.md @@ -0,0 +1,148 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001589/ +title: Twin Admissibility Fraction +cid: cid001589 +primary_alias: PRP0112 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At each odd prime p≥3, exactly (p-2)/p fraction of residues are twin-admissible. Only r≡0 and r≡p-2 excluded. +aliases: +- PRP0112 +- III.P45 +- twin-admissibility-fraction +- prop:twin-admissibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006092 +- predicate: formalized_by + object: cid006097 +- predicate: formalized_by + object: cid006098 +- predicate: formalized_by + object: cid006099 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Admissibility Fraction** + + + At each odd prime p≥3, exactly (p-2)/p fraction of residues are twin-admissible. Only r≡0 and r≡p-2 excluded. + + + # Twin Admissibility Fraction + + + ## Summary + + + At each odd prime p≥3, exactly (p-2)/p fraction of residues are twin-admissible. Only r≡0 and r≡p-2 excluded. + + + ## Statement + + + ```latex + + \label{prop:twin-admissibility} + + At each odd prime $p \ge 3$, exactly $(p-2)$ out of $p$ residue classes + + are twin-admissible: only $r \equiv 0$ and $r \equiv p - 2$ are excluded. + + + \textbf{Lean:} \texttt{twin\_admissibility\_fraction\_5}.\quad + + \textbf{Registry:} III.P45. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 273 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 294-300 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `twin_admissibility_fraction_5` + + + ## Dependencies + + + - Canonical: III.D107 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At each odd prime p≥3, exactly (p-2)/p fraction of residues are twin-admissible. Only r≡0 and r≡p-2 excluded. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001590.md b/_corpus_v3_items/cid001590.md new file mode 100644 index 000000000..9c0800b8b --- /dev/null +++ b/_corpus_v3_items/cid001590.md @@ -0,0 +1,147 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001590/ +title: Twin Gap Characterization +cid: cid001590 +primary_alias: PRP0113 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). + exponential_quantification (K4). +aliases: +- PRP0113 +- III.P46 +- twin-gap-characterization +- prop:twin-gap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Gap Characterization** + + + Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). exponential_quantification + (K4). + + + # Twin Gap Characterization + + + ## Summary + + + Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). exponential_quantification + (K4). + + + ## Statement + + + ```latex + + \label{prop:twin-gap} + + Admissible classes are nonempty at every prime (local condition always + + satisfied). Infinitude requires the Bombieri--Vinogradov theorem or + + the Elliott--Halberstam conjecture to convert admissibility into actual + + prime pairs. The gap is the \emph{density} gap: algebraic admissibility + + versus analytic equidistribution. + + + \textbf{Registry:} III.P46.\quad + + \textbf{Scope:} established (meta-theorem). + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 274 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 302-311 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `twin_gap_density` + + + ## Dependencies + + + - Canonical: III.P45, III.D69 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Twin gap = density gap. Admissible classes nonempty (algebraic) but infinitude needs Bombieri-Vinogradov (analytic). + exponential_quantification (K4). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001591.md b/_corpus_v3_items/cid001591.md new file mode 100644 index 000000000..e15b7c020 --- /dev/null +++ b/_corpus_v3_items/cid001591.md @@ -0,0 +1,148 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001591/ +title: Squarefree Dominance Theorem +cid: cid001591 +primary_alias: PRP0114 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ABC trivially true for squarefree coprime triples: rad(abc)=abc≥c. Quality q<1 always. Zero high-quality among squarefree + coprimes.' +aliases: +- PRP0114 +- III.P47 +- squarefree-dominance-theorem +- prop:squarefree-dom-thm +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005686 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Squarefree Dominance Theorem** + + + ABC trivially true for squarefree coprime triples: rad(abc)=abc≥c. Quality q<1 always. Zero high-quality among squarefree + coprimes. + + + # Squarefree Dominance Theorem + + + ## Summary + + + ABC trivially true for squarefree coprime triples: rad(abc)=abc≥c. Quality q<1 always. Zero high-quality among squarefree + coprimes. + + + ## Statement + + + ```latex + + \label{prop:squarefree-dom-thm} + + ABC is trivially true for squarefree coprime triples: if $\gcd(a,b) = 1$ + + and $a, b, c$ are squarefree, then $\mathrm{rad}(abc) = abc \ge c^2 > c$. + + Quality $q < 1$ always. + + + \textbf{Registry:} III.P47.\quad + + \textbf{Scope:} established. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 281 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 373-380 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `squarefree_dominance_thm` + + + ## Dependencies + + + - Canonical: III.D110, III.T77 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ABC trivially true for squarefree coprime triples: rad(abc)=abc≥c. Quality q<1 always. Zero high-quality among + squarefree coprimes.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001592.md b/_corpus_v3_items/cid001592.md new file mode 100644 index 000000000..0eb9194a7 --- /dev/null +++ b/_corpus_v3_items/cid001592.md @@ -0,0 +1,152 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001592/ +title: ABC Gap Characterization +cid: cid001592 +primary_alias: PRP0115 +type: proposition +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: ABC gap = structural. Primorial tower is squarefree → avoids the hard case. Genuine difficulty in perfect-power parts. + exponential_quantification (K4). +aliases: +- PRP0115 +- III.P48 +- abc-gap-characterization +- prop:abc-gap +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005847 +- predicate: formalized_by + object: cid005848 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**ABC Gap Characterization** + + + ABC gap = structural. Primorial tower is squarefree → avoids the hard case. Genuine difficulty in perfect-power parts. + exponential_quantification (K4). + + + # ABC Gap Characterization + + + ## Summary + + + ABC gap = structural. Primorial tower is squarefree → avoids the hard case. Genuine difficulty in perfect-power parts. + exponential_quantification (K4). + + + ## Statement + + + ```latex + + \label{prop:abc-gap} + + The primorial tower is squarefree, so it avoids the hard case of ABC + + entirely. The genuine difficulty lies in numbers with large perfect-power + + factors (e.g., $2^n + 1$). The gap is \emph{structural}: the $\tau$ + + framework decomposes to the squarefree part, which is where ABC is easy. + + + \textbf{Registry:} III.P48.\quad + + \textbf{Scope:} established (meta-theorem). + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 282 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 382-390 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `abc_gap_structural` + + + ## Dependencies + + + - Canonical: III.P47, III.D69 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: ABC gap = structural. Primorial tower is squarefree → avoids the hard case. Genuine difficulty in perfect-power + parts. exponential_quantification (K4). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001593.md b/_corpus_v3_items/cid001593.md new file mode 100644 index 000000000..613aed0ef --- /dev/null +++ b/_corpus_v3_items/cid001593.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001593/ +title: Non-Emptiness Theorem +cid: cid001593 +primary_alias: THM0136 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Each enrichment layer E_k (k = 0,1,2,3) is inhabited: it contains genuine objects and morphisms. Constructive witnesses: + E₀ = any τ-object with NF address, E₁ = H_τ-enriched Hom spaces, E₂ = proto-codes from BSD carriers (preview), E₃ = self-modeling + codes (preview).' +aliases: +- THM0136 +- III.T01 +- non-emptiness-theorem +- thm:non-emptiness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005853 +- predicate: formalized_by + object: cid005859 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Non-Emptiness Theorem**\n\nEach enrichment layer E_k (k = 0,1,2,3) is inhabited: it contains genuine objects\ + \ and morphisms. Constructive witnesses: E₀ = any τ-object with NF address, E₁ = H_τ-enriched Hom spaces, E₂ = proto-codes\ + \ from BSD carriers (preview), E₃ = self-modeling codes (preview).\n\n# Non-Emptiness Theorem\n\n## Summary\n\nEach enrichment\ + \ layer E_k (k = 0,1,2,3) is inhabited: it contains genuine objects and morphisms. Constructive witnesses: E₀ = any τ-object\ + \ with NF address, E₁ = H_τ-enriched Hom spaces, E₂ = proto-codes from BSD carriers (preview), E₃ = self-modeling codes\ + \ (preview).\n\n## Statement\n\n```latex\n%\n\\label{thm:non-emptiness}\nEach layer of the enrichment tower is inhabited:\n\ + \\begin{enumerate}\n \\item $E_0$ is non-empty.\n Witness: any $\\tau$-object with NF address.\n \\item\ + \ $E_1$ is non-empty.\n Witness: $[A, B]$ for $A, B$ in distinct spectral sectors.\n \\item $E_2$ is non-empty.\n\ + \ Witness: a proto-code from the BSD rational bridge\n (Part~VI).\n \\item $E_3$ is non-empty.\n\ + \ Witness: a self-modelling code\n with consistent interpretation functor\n (Book~VII).\n\\\ + end{enumerate}\nItems~(1) and~(2) are proved constructively\nfrom Books~I--II structures.\nItems~(3) and~(4) carry precise\ + \ forward references.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot\ + \ run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 15\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch06-non-emptiness-and-strictness.tex`\ + \ lines 191-211\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `non_emptiness_8_3`\n\n## Dependencies\n\n- Canonical: III.D05, III.D06, III.D07, III.D08, III.D09\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Each enrichment layer E_k (k = 0,1,2,3) is inhabited: it contains genuine objects and morphisms. Constructive + witnesses: E₀ = any τ-object with NF address, E₁ = H_τ-enriched Hom spaces, E₂ = proto-codes from BSD carriers (preview), + E₃ = sel' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001594.md b/_corpus_v3_items/cid001594.md new file mode 100644 index 000000000..88d6901a4 --- /dev/null +++ b/_corpus_v3_items/cid001594.md @@ -0,0 +1,82 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001594/ +title: Strictness Theorem +cid: cid001594 +primary_alias: THM0137 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The enrichment ladder is strictly increasing: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure not + reducible to the previous: E₁\E₀ = bipolar Hom decomposition, E₂\E₁ = operational closure, E₃\E₂ = self-model consistency.' +aliases: +- THM0137 +- III.T02 +- strictness-theorem +- thm:strictness +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005855 +- predicate: formalized_by + object: cid005860 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 6 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Strictness Theorem**\n\nThe enrichment ladder is strictly increasing: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains\ + \ genuinely new structure not reducible to the previous: E₁\\E₀ = bipolar Hom decomposition, E₂\\E₁ = operational closure,\ + \ E₃\\E₂ = self-model consistency.\n\n# Strictness Theorem\n\n## Summary\n\nThe enrichment ladder is strictly increasing:\ + \ E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure not reducible to the previous: E₁\\E₀ = bipolar Hom decomposition,\ + \ E₂\\E₁ = operational closure, E₃\\E₂ = self-model consistency.\n\n## Statement\n\n```latex\n%\n\\label{thm:strictness}\n\ + The enrichment tower is strictly nested:\n\\[\n E_0 \\;\\subsetneq\\; E_1 \\;\\subsetneq\\; E_2 \\;\\subsetneq\\; E_3.\n\ + \\]\nSpecifically:\n\\begin{enumerate}\n \\item $E_1 \\setminus E_0 \\neq \\varnothing$.\n Witness: $[A, B]$\ + \ with non-trivial bipolar decomposition\n (Proposition~\\ref{prop:e1-strictness-witness}).\n \\item $E_2\ + \ \\setminus E_1 \\neq \\varnothing$.\n Witness: operationally closed proto-code (Part~VI).\n \\item $E_3\ + \ \\setminus E_2 \\neq \\varnothing$.\n Witness: self-model-consistent code (Book~VII).\n\\end{enumerate}\nItem~(1)\ + \ is proved in full.\nItems~(2) and~(3) carry forward references.\n```\n\n## Proof / Justification\n\n_No immediate manuscript\ + \ proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 16\n-\ + \ Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch06-non-emptiness-and-strictness.tex`\ + \ lines 464-483\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `strictness_8_3`\n\n## Dependencies\n\n- Canonical: III.D05, III.P01\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The enrichment ladder is strictly increasing: E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃. Each layer contains genuinely new structure + not reducible to the previous: E₁\E₀ = bipolar Hom decomposition, E₂\E₁ = operational closure, E₃\E₂ = self-model consistency.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001595.md b/_corpus_v3_items/cid001595.md new file mode 100644 index 000000000..34621e0ff --- /dev/null +++ b/_corpus_v3_items/cid001595.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001595/ +title: Saturation at E₃ +cid: cid001595 +primary_alias: THM0138 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'E₄ = E₃: the enrichment ladder saturates at exactly four levels. The 4-orbit closure of ρ under ABCD decomposition + means no fifth orbit exists. Therefore [E₃^op, E₃] ⊆ E₃ — the functor category collapses back, and the ω-absorber mechanism + prevents escape.' +aliases: +- THM0138 +- III.T03 +- saturation-at-e +- thm:saturation-e3 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005857 +- predicate: formalized_by + object: cid005861 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 7 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Saturation at E₃**\n\nE₄ = E₃: the enrichment ladder saturates at exactly four levels. The 4-orbit closure\ + \ of ρ under ABCD decomposition means no fifth orbit exists. Therefore [E₃^op, E₃] ⊆ E₃ — the functor category collapses\ + \ back, and the ω-absorber mechanism prevents escape.\n\n# Saturation at E₃\n\n## Summary\n\nE₄ = E₃: the enrichment ladder\ + \ saturates at exactly four levels. The 4-orbit closure of ρ under ABCD decomposition means no fifth orbit exists. Therefore\ + \ [E₃^op, E₃] ⊆ E₃ — the functor category collapses back, and the ω-absorber mechanism prevents escape.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:saturation-e3}\nThe self-enrichment ladder of Category~$\\tau$\nhas exactly four layers:\n\\\ + [\n E_0 \\;\\subsetneq\\; E_1 \\;\\subsetneq\\;\n E_2 \\;\\subsetneq\\; E_3,\n \\qquad\n E_4 = E_3.\n\\]\n\ + The first three inclusions are strict\n(each layer contains genuinely new structure).\nThe fourth step collapses:\n$[E_3^{\\\ + op}, E_3] \\subseteq E_3$,\nso $E_4 = E_3$.\nThe enrichment ladder is complete.\n```\n\n## Proof / Justification\n\n```latex\n\ + The proof assembles three components.\n\n\\medskip\n\\textbf{Step~1: Strictness.}\nEach inclusion $E_k \\subsetneq E_{k+1}$\n\ + for $k = 0, 1, 2$ is strict.\nThis was proved in Chapters~4--6:\n$E_1$ contains split-complex-enriched Hom~objects\nabsent\ + \ from~$E_0$;\n$E_2$ contains discrete rational carriers\nabsent from~$E_1$;\n$E_3$ contains self-referential codes\n\ + absent from~$E_2$.\n\n\\medskip\n\\textbf{Step~2: Collapse.}\nThe functor category $[E_3^{\\op}, E_3]$\ncontains no objects\ + \ of genuinely new structural type\n(Proposition~\\ref{prop:functor-collapse}).\nThe obstruction is the Ontic Closure\ + \ Theorem (I.T01):\nthe five generators produce exactly four orbit channels,\nand any fifth channel factors through\n\ + the $\\omega$-absorber.\nThe Ladder Saturation Theorem (I.T02)\ntranslates this orbit closure\ninto a statement about\ + \ canonical injectivity:\npentation --- the operation that would populate\na fifth level --- lacks the structural scaffold\n\ + for type distinction.\n\n\\medskip\n\\textbf{Step~3: Identity.}\nSince $E_4 = [E_3^{\\op}, E_3] \\subseteq E_3$\nand $E_3\ + \ \\subseteq E_4$\n(every object of $E_3$ is in particular\na constant presheaf in $[E_3^{\\op}, E_3]$),\nwe have $E_4\ + \ = E_3$.\n```\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 18\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch07-saturation-at-e3.tex`\ + \ lines 500-517\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `saturation_e3_8_3`\n\n## Dependencies\n\n- Canonical: III.D09, III.P02\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'E₄ = E₃: the enrichment ladder saturates at exactly four levels. The 4-orbit closure of ρ under ABCD decomposition + means no fifth orbit exists. Therefore [E₃^op, E₃] ⊆ E₃ — the functor category collapses back, and the ω-absorber mechanism + p' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001596.md b/_corpus_v3_items/cid001596.md new file mode 100644 index 000000000..aa8770c8b --- /dev/null +++ b/_corpus_v3_items/cid001596.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001596/ +title: Canonical Ladder Theorem +cid: cid001596 +primary_alias: THM0139 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The enrichment ladder E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃ is (i) non-empty at each level, (ii) strictly increasing, (iii) saturating + at E₃, (iv) the unique maximal enrichment chain for Category τ. It is the organising result of Book III and the architectural + blueprint for the entire series. +aliases: +- THM0139 +- III.T04 +- canonical-ladder-theorem +- thm:canonical-ladder +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005858 +- predicate: formalized_by + object: cid005862 +- predicate: formalized_by + object: cid005863 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 1, Chapter 8 (Part I) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Canonical Ladder Theorem**\n\nThe enrichment ladder E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃ is (i) non-empty at each level, (ii)\ + \ strictly increasing, (iii) saturating at E₃, (iv) the unique maximal enrichment chain for Category τ. It is the organising\ + \ result of Book III and the architectural blueprint for the entire series.\n\n# Canonical Ladder Theorem\n\n## Summary\n\ + \nThe enrichment ladder E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃ is (i) non-empty at each level, (ii) strictly increasing, (iii) saturating at\ + \ E₃, (iv) the unique maximal enrichment chain for Category τ. It is the organising result of Book III and the architectural\ + \ blueprint for the entire series.\n\n## Statement\n\n```latex\n%\n\\label{thm:canonical-ladder}\nThe self-enrichment\ + \ of Category~$\\tau$\nvia the enrichment functor~$\\mathcal{F}_E$\nproduces a chain of four layers\n\\[\n E_0 \\;\\\ + subsetneq\\; E_1 \\;\\subsetneq\\; E_2 \\;\\subsetneq\\; E_3\n\\]\nsatisfying:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\ + \ \\textbf{Non-emptiness.}\n Each $E_k$ is non-empty: $\\lvert \\Obj(E_k) \\rvert \\geq 1$\n and $E_k$\ + \ admits non-identity morphisms.\n \\item[\\textup{(ii)}] \\textbf{Strict increase.}\n Each inclusion is proper:\n\ + \ $E_k \\setminus E_{k-1} \\neq \\varnothing$\n for $k = 1, 2, 3$.\n \\item[\\textup{(iii)}] \\textbf{Saturation.}\n\ + \ $\\mathcal{F}_E(E_3) = E_3$.\n \\item[\\textup{(iv)}] \\textbf{Uniqueness.}\n If $E_0' \\subsetneq\ + \ E_1' \\subsetneq \\cdots \\subsetneq E_m'$\n is any maximal enrichment chain\n for Category~$\\tau$\n\ + \ satisfying~\\textup{(i)--(iii)},\n then $m = 3$ and $E_k' \\cong E_k$\n for all~$k$.\n\\\ + end{enumerate}\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\ + \n## Source Context\n\n- Registry source: `book-03.jsonl` line 20\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part01/ch08-the-canonical-ladder-theorem.tex`\ + \ lines 99-127\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Enrichment.CanonicalLadder`\n\ + - Name: `canonical_ladder_8_3`\n\n## Dependencies\n\n- Canonical: III.T01, III.T02, III.T03\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The enrichment ladder E₀ ⊊ E₁ ⊊ E₂ ⊊ E₃ is (i) non-empty at each level, (ii) strictly increasing, (iii) saturating + at E₃, (iv) the unique maximal enrichment chain for Category τ. It is the organising result of Book III and the architectural +noindex: false +--- + diff --git a/_corpus_v3_items/cid001597.md b/_corpus_v3_items/cid001597.md new file mode 100644 index 000000000..bddf9549a --- /dev/null +++ b/_corpus_v3_items/cid001597.md @@ -0,0 +1,102 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001597/ +title: Sector Preservation Theorem +cid: cid001597 +primary_alias: THM0140 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The boundary-to-interior functor Φ preserves the bipolar decomposition: χ₊-characters map to B-sector holomorphic + functions, χ₋-characters map to C-sector, mixed characters map to the ω-coupling sector. The 4+1 decomposition is a THEOREM, + not a definition.' +aliases: +- THM0140 +- III.T05 +- sector-preservation-theorem +- thm:sector-preservation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005971 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 9 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Sector Preservation Theorem**\n\nThe boundary-to-interior functor Φ preserves the bipolar decomposition: χ₊-characters\ + \ map to B-sector holomorphic functions, χ₋-characters map to C-sector, mixed characters map to the ω-coupling sector.\ + \ The 4+1 decomposition is a THEOREM, not a definition.\n\n# Sector Preservation Theorem\n\n## Summary\n\nThe boundary-to-interior\ + \ functor Φ preserves the bipolar decomposition: χ₊-characters map to B-sector holomorphic functions, χ₋-characters map\ + \ to C-sector, mixed characters map to the ω-coupling sector. The 4+1 decomposition is a THEOREM, not a definition.\n\n\ + ## Statement\n\n```latex\n%\n\\label{thm:sector-preservation}\nThe boundary-to-interior functor\n$\\Phi : \\Char(\\Lemniscate)\ + \ \\to \\mathcal{O}(\\tau^3)$\npreserves the bipolar decomposition\nin the following sense:\n\\begin{enumerate}\n \\\ + item[\\textup{(i)}]\n $\\chi_+$-characters map to B-sector holomorphic functions:\n \\[\n \ + \ m = 0 \\;\\implies\\;\n \\Phi(\\chi_{(0,n)}) \\;=\\; e_+ \\cdot \\Phi(\\chi_{(0,n)})_+.\n \\\ + ]\n \\item[\\textup{(ii)}]\n $\\chi_-$-characters map to C-sector holomorphic functions:\n \\[\n\ + \ n = 0 \\;\\implies\\;\n \\Phi(\\chi_{(m,0)}) \\;=\\; e_- \\cdot \\Phi(\\chi_{(m,0)})_-.\n\ + \ \\]\n \\item[\\textup{(iii)}]\n Mixed characters map to the $\\omega$-coupling sector:\n \ + \ \\[\n m \\neq 0,\\; n \\neq 0 \\;\\implies\\;\n \\Phi(\\chi_{(m,n)}) \\;=\\;\n \ + \ e_+ \\cdot \\Phi(\\chi_{(m,n)})_+\n \\;+\\;\n e_- \\cdot \\Phi(\\chi_{(m,n)})_-,\n \ + \ \\]\n with both components non-trivial.\n\\end{enumerate}\nIn particular, $\\Phi$ maps\nthe $m$-axis\ + \ of~$\\Z^2$\ninto the C-sector of~$\\mathcal{O}(\\tau^3)$,\nthe $n$-axis into the B-sector,\nand the interior of the\ + \ lattice\ninto the $\\omega$-coupling sector.\n```\n\n## Proof / Justification\n\n```latex\nThe proof proceeds in three\ + \ steps.\n\n\\emph{Step 1: Stage-level verification.}\nAt primorial depth~$k$,\nthe functor $\\Phi_k$ sends a character\n\ + $\\chi_{(m,n)} \\in (\\Z / M_k\\Z)^2$\nto the element\n$\\Phi_k(\\chi_{(m,n)}) \\in (\\Z / M_k\\Z)[\\,j\\,]/(j^2 - 1)$.\n\ + By the definition of the split-complex extension\n(equation~\\eqref{eq:ch09-phi-stages}),\nthis element decomposes as\n\ + \\[\n \\Phi_k(\\chi_{(m,n)})\n \\;=\\;\n e_+ \\cdot \\bar{n}_k\n \\;+\\;\n e_- \\cdot \\bar{m}_k,\n\\]\n\ + where $\\bar{m}_k$ and $\\bar{n}_k$\ndenote the reductions of $m$ and $n$\nmodulo~$M_k$.\nWhen $m = 0$, the $e_-$-component\ + \ vanishes;\nwhen $n = 0$, the $e_+$-component vanishes.\nThis establishes claims~(i) and~(ii) at each finite level.\n\ + \n\\emph{Step 2: Tower coherence.}\nThe primorial projections\n$\\pi_{\\ell \\to k} : \\Z / M_\\ell\\Z \\to \\Z / M_k\\\ + Z$\ncommute with the split-complex extension:\n\\[\n \\pi_{\\ell \\to k}(e_+ \\cdot \\bar{n}_\\ell + e_- \\cdot \\\ + bar{m}_\\ell)\n \\;=\\;\n e_+ \\cdot \\bar{n}_k + e_- \\cdot \\bar{m}_k.\n\\]\nThis is exactly the left square\n\ + of the bi-square (I.T41).\nThe sector assignment---whether\na character maps to the B-sector, the C-sector,\nor the $\\\ + omega$-coupling sector---is\ntherefore stable under reduction\nand passes to the inverse limit.\n\n\\emph{Step 3: The\ + \ mixed case.}\nWhen $m \\neq 0$ and $n \\neq 0$,\nboth $\\bar{m}_k$ and $\\bar{n}_k$\nare non-zero for all sufficiently\ + \ large~$k$\n(since $M_k \\to \\infty$).\nTherefore both idempotent components\nof $\\Phi_k(\\chi_{(m,n)})$ are non-trivial\n\ + for all~$k$ beyond a finite threshold.\nIn the inverse limit,\nboth components survive,\nplacing $\\Phi(\\chi_{(m,n)})$\n\ + in the $\\omega$-coupling sector.\nThis establishes claim~(iii).\n```\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 25\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch09-boundary-functoriality.tex`\ + \ lines 471-508\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.Decomposition`\n\ + - Name: `sector_preservation_5_3`\n\n## Dependencies\n\n- Canonical: III.D11, III.D12\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The boundary-to-interior functor Φ preserves the bipolar decomposition: χ₊-characters map to B-sector holomorphic + functions, χ₋-characters map to C-sector, mixed characters map to the ω-coupling sector. The 4+1 decomposition is a THEOREM, + n' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001598.md b/_corpus_v3_items/cid001598.md new file mode 100644 index 000000000..c04dc4366 --- /dev/null +++ b/_corpus_v3_items/cid001598.md @@ -0,0 +1,84 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001598/ +title: Template Invariance Under Reflection +cid: cid001598 +primary_alias: THM0141 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The layer template (Carrier, Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection. The four + components change substance but preserve structure. Proved via commutativity with the enrichment functor. +aliases: +- THM0141 +- III.T06 +- template-invariance-under-reflection +- thm:template-invariance-reflection +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005979 +- predicate: formalized_by + object: cid005984 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 11 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Template Invariance Under Reflection**\n\nThe layer template (Carrier, Predicate, Decoder, Invariant) is preserved\ + \ under the Langlands₁ reflection. The four components change substance but preserve structure. Proved via commutativity\ + \ with the enrichment functor.\n\n# Template Invariance Under Reflection\n\n## Summary\n\nThe layer template (Carrier,\ + \ Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection. The four components change substance but\ + \ preserve structure. Proved via commutativity with the enrichment functor.\n\n## Statement\n\n```latex\n%\n\\label{thm:template-invariance-reflection}\n\ + The Langlands$_1$ reflection bridge\n$\\Lambda_1 : \\mathrm{Char}(\\Lemniscate) \\Rightarrow \\mathrm{Char}_1(\\Lemniscate)$\n\ + preserves the four-component layer template:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n If $\\chi \\in\ + \ \\Sector{g}$ for\n $g \\in \\{\\alpha, \\pi, \\gamma, \\eta\\}$,\n then $\\Lambda_1(\\chi) \\in \\\ + Sector{g}^{(1)}$.\n \\item[\\textup{(ii)}]\n If $\\chi \\in \\MixedSector$,\n then $\\Lambda_1(\\\ + chi) \\in \\MixedSector^{(1)}$.\n \\item[\\textup{(iii)}]\n Each restriction\n $\\Lambda_1|_{\\Sector{g}}\ + \ : \\Sector{g} \\to \\Sector{g}^{(1)}$\n is injective on isomorphism classes.\n \\item[\\textup{(iv)}]\n\ + \ The template\n $(\\mathrm{Carrier}, \\mathrm{Predicate}, \\mathrm{Decoder}, \\mathrm{Invariant})$\n\ + \ at $E_0$ maps bijectively to the template at $E_1$.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n\ + _No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 30\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch11-the-yoneda-langlands-reflection.tex`\ + \ lines 249-272\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.LanglandsReflection`\n\ + - Name: `template_invariance_8_3`\n\n## Dependencies\n\n- Canonical: III.D15, III.D05\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The layer template (Carrier, Predicate, Decoder, Invariant) is preserved under the Langlands₁ reflection. The + four components change substance but preserve structure. Proved via commutativity with the enrichment functor. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001599.md b/_corpus_v3_items/cid001599.md new file mode 100644 index 000000000..007a93935 --- /dev/null +++ b/_corpus_v3_items/cid001599.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001599/ +title: Parity Bridge Theorem +cid: cid001599 +primary_alias: THM0142 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The weak sector (A, π-generator) is the unique sector whose spectral polarity permits the E₁→E₂ transition — the + computational bootstrap. Balanced polarity enables the code→execution→code cycle: χ₊ (code) and χ₋ (execution) in equal + measure. Computation is native to E₂; the Parity Bridge enables it. Life is computation physically instantiated.' +aliases: +- THM0142 +- III.T07 +- parity-bridge-theorem +- thm:parity-bridge +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005987 +- predicate: formalized_by + object: cid005990 +- predicate: formalized_by + object: cid005991 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 12 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Parity Bridge Theorem**\n\nThe weak sector (A, π-generator) is the unique sector whose spectral polarity permits\ + \ the E₁→E₂ transition — the computational bootstrap. Balanced polarity enables the code→execution→code cycle: χ₊ (code)\ + \ and χ₋ (execution) in equal measure. Computation is native to E₂; the Parity Bridge enables it. Life is computation\ + \ physically instantiated.\n\n# Parity Bridge Theorem\n\n## Summary\n\nThe weak sector (A, π-generator) is the unique\ + \ sector whose spectral polarity permits the E₁→E₂ transition — the computational bootstrap. Balanced polarity enables\ + \ the code→execution→code cycle: χ₊ (code) and χ₋ (execution) in equal measure. Computation is native to E₂; the Parity\ + \ Bridge enables it. Life is computation physically instantiated.\n\n## Statement\n\n```latex\n%\n\\label{thm:parity-bridge}\n\ + Among the $4{+}1$ sectors of Category~$\\tau$ at $\\Elayer{1}$,\nthe A-sector~$\\Sector{\\pi}$ is the unique sector\n\ + whose spectral polarity permits the transition\nfrom $\\Elayer{1}$ (physics) to $\\Elayer{2}$ (computation).\nSpecifically:\n\ + \\begin{enumerate}\n \\item[\\textup{(i)}]\n The $\\Elayer{1} \\to \\Elayer{2}$ transition\n requires\ + \ a sector $\\Sector{g}$ in which\n the operational closure cycle\n \\[\n \\text{code}\n\ + \ \\;\\xrightarrow{\\;\\chi_+\\to\\chi_-\\;}\\;\n \\text{product}\n \\;\\xrightarrow{\\\ + ;\\chi_-\\to\\chi_+\\;}\\;\n \\text{code}\n \\]\n closes without capacity loss.\n \\\ + item[\\textup{(ii)}]\n Closure without capacity loss\n holds if and only if\n $\\mathrm{pol}(\\\ + Sector{g}) = 1$.\n \\item[\\textup{(iii)}]\n $\\mathrm{pol}(\\Sector{g}) = 1$\n if and only if $g\ + \ = \\pi$\n (Proposition~\\ref{prop:balanced-sector-uniqueness}).\n\\end{enumerate}\nTherefore $\\Sector{\\pi}$\ + \ is the unique carrier\nof the $\\Elayer{1} \\to \\Elayer{2}$ transition.\n```\n\n## Proof / Justification\n\n_No immediate\ + \ manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 34\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch12-the-parity-bridge-theorem.tex`\ + \ lines 329-361\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.ParityBridge`\n\ + - Name: `parity_bridge_5_3`\n\n## Dependencies\n\n- Canonical: III.D17, III.P04, III.D08\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The weak sector (A, π-generator) is the unique sector whose spectral polarity permits the E₁→E₂ transition — + the computational bootstrap. Balanced polarity enables the code→execution→code cycle: χ₊ (code) and χ₋ (execution) in equal + measure' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001600.md b/_corpus_v3_items/cid001600.md new file mode 100644 index 000000000..36dbda687 --- /dev/null +++ b/_corpus_v3_items/cid001600.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001600/ +title: No Knobs Principle +cid: cid001600 +primary_alias: THM0143 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'All inter-sector couplings are canonically determined by ι_τ = 2/(π+e). The framework has no free parameters: (i) + every coupling is a function of ι_τ and primorial depth, (ii) the 10-entry ledger is complete, (iii) perturbation of any + coupling breaks sector preservation.' +aliases: +- THM0143 +- III.T08 +- no-knobs-principle +- thm:no-knobs-principle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005989 +- predicate: formalized_by + object: cid005992 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 2, Chapter 13 (Part II) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Knobs Principle**\n\nAll inter-sector couplings are canonically determined by ι_τ = 2/(π+e). The framework\ + \ has no free parameters: (i) every coupling is a function of ι_τ and primorial depth, (ii) the 10-entry ledger is complete,\ + \ (iii) perturbation of any coupling breaks sector preservation.\n\n# No Knobs Principle\n\n## Summary\n\nAll inter-sector\ + \ couplings are canonically determined by ι_τ = 2/(π+e). The framework has no free parameters: (i) every coupling is a\ + \ function of ι_τ and primorial depth, (ii) the 10-entry ledger is complete, (iii) perturbation of any coupling breaks\ + \ sector preservation.\n\n## Statement\n\n```latex\n%\n\\label{thm:no-knobs-principle}\nLet $\\Sector{D}, \\Sector{A},\ + \ \\Sector{B}, \\Sector{C}$\ndenote the four primitive sectors\nof the $4{+}1$ decomposition\n(Chapter~\\ref{ch:four-plus-one-decomposition}),\n\ + and let $\\kappa(\\Sector{i}, \\Sector{j})$\nbe the coupling function of Definition~\\ref{def:coupling-function}.\nThen:\n\ + \\begin{enumerate}\n \\item[\\textup{(i)}] \\textbf{Determination.}\n Every sector coupling is determined\ + \ by~$\\iota_\\tau$:\n \\[\n \\kappa(\\Sector{i}, \\Sector{j})\n \\;=\\;\n \ + \ f_{ij}(\\iota_\\tau,\\, M_d)\n \\]\n for a specific rational function~$f_{ij}$\n and\ + \ primorial depth~$d$\n depending only on the generator pair $(g_i, g_j)$.\n \\item[\\textup{(ii)}] \\textbf{Completeness.}\n\ + \ The No Knobs Ledger\n (Definition~\\ref{def:coupling-ledger},\n Table~\\ref{tab:no-knobs-ledger})\n\ + \ is exhaustive:\n it contains all $10$ distinct couplings,\n and no coupling exists between\ + \ primitive sectors\n that is not listed.\n \\item[\\textup{(iii)}] \\textbf{Rigidity.}\n No deformation\ + \ of the coupling functions exists\n within the framework.\n If $\\kappa'$ is any alternative coupling\ + \ assignment\n that is compatible with the Langlands$_0$ functor\n and the $4{+}1$ decomposition,\n\ + \ then $\\kappa' = \\kappa$.\n\\end{enumerate}\nIn particular,\nthe framework has no free parameters.\n```\n\n\ + ## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\ + \n- Registry source: `book-03.jsonl` line 35\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part02/ch13-the-no-knobs-principle.tex`\ + \ lines 330-369\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Sectors.ParityBridge`\n\ + - Name: `no_knobs_5_3`\n\n## Dependencies\n\n- Canonical: III.D13, III.T05, III.T06\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'All inter-sector couplings are canonically determined by ι_τ = 2/(π+e). The framework has no free parameters: + (i) every coupling is a function of ι_τ and primorial depth, (ii) the 10-entry ledger is complete, (iii) perturbation of + any coupl' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001601.md b/_corpus_v3_items/cid001601.md new file mode 100644 index 000000000..020faa942 --- /dev/null +++ b/_corpus_v3_items/cid001601.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001601/ +title: Primorial Cofinality +cid: cid001601 +primary_alias: THM0144 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The primorial tower is cofinal: every ℤ/Nℤ maps to ℤ/Prim(k)ℤ for k large enough. Checking at primorial levels is + SUFFICIENT. The cutoff k₀ is always computable.' +aliases: +- THM0144 +- III.T09 +- primorial-cofinality +- thm:primorial-cofinality +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006059 +- predicate: formalized_by + object: cid006060 +- predicate: formalized_by + object: cid006064 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 14 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Primorial Cofinality**\n\nThe primorial tower is cofinal: every ℤ/Nℤ maps to ℤ/Prim(k)ℤ for k large enough.\ + \ Checking at primorial levels is SUFFICIENT. The cutoff k₀ is always computable.\n\n# Primorial Cofinality\n\n## Summary\n\ + \nThe primorial tower is cofinal: every ℤ/Nℤ maps to ℤ/Prim(k)ℤ for k large enough. Checking at primorial levels is SUFFICIENT.\ + \ The cutoff k₀ is always computable.\n\n## Statement\n\n```latex\n%\n\\label{thm:primorial-cofinality}\nThe primorial\ + \ ladder $\\{M_k\\}_{k \\geq 1}$\nis cofinal in the full modular tower.\nSpecifically:\n\\begin{enumerate}\n \\item[\\\ + textup{(i)}] \\textbf{Squarefree divisibility.}\n For every squarefree integer~$N$,\n there exists $k_0$\ + \ such that\n $N \\mid M_k$ for all $k \\geq k_0$.\n The cutoff is $k_0 = \\pi(p_{\\max}(N))$,\n \ + \ where $p_{\\max}(N)$ is the largest prime factor of~$N$\n and $\\pi$ is the prime-counting function.\n\ + \ \\item[\\textup{(ii)}] \\textbf{General divisibility.}\n For every positive integer~$N$,\n there\ + \ exists $k_0$ such that the reduction map\n $\\Z / M_k\\Z \\to \\Z / \\mathrm{rad}(N)\\Z$\n is well-defined\ + \ for all $k \\geq k_0$,\n where $\\mathrm{rad}(N) = \\prod_{p \\mid N} p$\n is the radical of~$N$.\n\ + \ \\item[\\textup{(iii)}] \\textbf{Inverse limit equivalence.}\n The canonical map\n \\begin{equation}\\\ + label{eq:ch14-cofinal-iso}\n \\widehat{\\Z}_\\tau\n \\;=\\;\n \\varprojlim_{k}\ + \ \\Z / M_k\\Z\n \\;\\longrightarrow\\;\n \\varprojlim_{N,\\, \\mathrm{sqfree}} \\Z / N\\Z\n\ + \ \\end{equation}\n is an isomorphism of profinite rings.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 39\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch14-the-primorial-ladder.tex`\ + \ lines 262-294\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.PrimorialLadder`\n\ + - Name: `primorial_cofinal_check`\n\n## Dependencies\n\n- Canonical: III.D19\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The primorial tower is cofinal: every ℤ/Nℤ maps to ℤ/Prim(k)ℤ for k large enough. Checking at primorial levels + is SUFFICIENT. The cutoff k₀ is always computable.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001602.md b/_corpus_v3_items/cid001602.md new file mode 100644 index 000000000..de0b069ca --- /dev/null +++ b/_corpus_v3_items/cid001602.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001602/ +title: CRT Decomposition Theorem +cid: cid001602 +primary_alias: THM0145 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'τ-native Chinese Remainder Theorem: ℤ/Prim(k)ℤ ≅ ∏ᵢ₌₁ᵏ ℤ/pᵢℤ proved constructively without signed arithmetic. Uses + modular Bézout via K3 divisibility. CRT = algebraic Euler product: End(F) ≅ ∏ End(ℤ/pᵢℤ).' +aliases: +- THM0145 +- III.T10 +- crt-decomposition-theorem +- thm:crt-decomposition +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006017 +- predicate: formalized_by + object: cid006018 +- predicate: formalized_by + object: cid006021 +- predicate: formalized_by + object: cid006022 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 15 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**CRT Decomposition Theorem**\n\nτ-native Chinese Remainder Theorem: ℤ/Prim(k)ℤ ≅ ∏ᵢ₌₁ᵏ ℤ/pᵢℤ proved constructively\ + \ without signed arithmetic. Uses modular Bézout via K3 divisibility. CRT = algebraic Euler product: End(F) ≅ ∏ End(ℤ/pᵢℤ).\n\ + \n# CRT Decomposition Theorem\n\n## Summary\n\nτ-native Chinese Remainder Theorem: ℤ/Prim(k)ℤ ≅ ∏ᵢ₌₁ᵏ ℤ/pᵢℤ proved constructively\ + \ without signed arithmetic. Uses modular Bézout via K3 divisibility. CRT = algebraic Euler product: End(F) ≅ ∏ End(ℤ/pᵢℤ).\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:crt-decomposition}\nFor each $k \\geq 1$, the canonical ring homomorphism\n\ + \\begin{equation}\\label{eq:ch15-crt-iso}\n \\Phi_k \\;:\\;\n \\Z / \\mathrm{Prim}(k)\\Z\n \\;\\longrightarrow\\\ + ;\n \\prod_{i=1}^{k} \\Z / p_i \\Z,\n \\qquad\n \\Phi_k(x) \\;=\\; (x \\bmod p_1,\\, \\ldots,\\, x \\bmod p_k),\n\ + \\end{equation}\nis an isomorphism of rings.\nIts inverse is the idempotent assembly:\n\\begin{equation}\\label{eq:ch15-crt-inverse}\n\ + \ \\Phi_k^{-1}(a_1, \\ldots, a_k)\n \\;=\\;\n \\sum_{i=1}^{k} a_i \\cdot e_i\n \\;\\bmod\\; \\mathrm{Prim}(k).\n\ + \\end{equation}\nThe construction is purely $\\tau$-internal:\nmultiplication ($\\KAxiom{2}$),\ndivisibility ($\\KAxiom{3}$),\n\ + and finite enumeration.\nNo subtraction, no signed arithmetic,\nno extended Euclidean algorithm.\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 41\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch15-the-crt-decomposition-theorem.tex`\ + \ lines 161-187\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.CRT`\n\ + - Name: `crt_spectral_check`\n\n## Dependencies\n\n- Canonical: III.D19, III.T09\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'τ-native Chinese Remainder Theorem: ℤ/Prim(k)ℤ ≅ ∏ᵢ₌₁ᵏ ℤ/pᵢℤ proved constructively without signed arithmetic. + Uses modular Bézout via K3 divisibility. CRT = algebraic Euler product: End(F) ≅ ∏ End(ℤ/pᵢℤ).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001603.md b/_corpus_v3_items/cid001603.md new file mode 100644 index 000000000..a73f65910 --- /dev/null +++ b/_corpus_v3_items/cid001603.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001603/ +title: Constructive Hensel Lifting +cid: cid001603 +primary_alias: THM0146 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Given a root mod p, lift to root mod p^n by modular Newton iteration. No signed arithmetic: correction via modular + complement. Lifting is unique (p-adic contraction). Fully constructive proof.' +aliases: +- THM0146 +- III.T11 +- constructive-hensel-lifting +- thm:constructive-hensel +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006038 +- predicate: formalized_by + object: cid006039 +- predicate: formalized_by + object: cid006040 +- predicate: formalized_by + object: cid006041 +- predicate: formalized_by + object: cid006042 +- predicate: formalized_by + object: cid006043 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 16 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Constructive Hensel Lifting**\n\nGiven a root mod p, lift to root mod p^n by modular Newton iteration. No signed\ + \ arithmetic: correction via modular complement. Lifting is unique (p-adic contraction). Fully constructive proof.\n\n\ + # Constructive Hensel Lifting\n\n## Summary\n\nGiven a root mod p, lift to root mod p^n by modular Newton iteration. No\ + \ signed arithmetic: correction via modular complement. Lifting is unique (p-adic contraction). Fully constructive proof.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:constructive-hensel}\nLet $p$ be a prime, $f \\in (\\Z / p^{n+1} \\Z)[x]$\n\ + a polynomial with residue coefficients,\nand $a_n \\in \\Z / p^n \\Z$ a root of~$f$\nmodulo~$p^n$ such that $f'(a_n)$\ + \ is a unit\nmodulo~$p$.\nThen there exists a unique $a_{n+1} \\in \\Z / p^{n+1} \\Z$\nsatisfying:\n\\begin{enumerate}\n\ + \ \\item[\\textup{(i)}]\n $a_{n+1} \\equiv a_n \\pmod{p^n}$\n \\textup{(compatibility)},\n \\\ + item[\\textup{(ii)}]\n $f(a_{n+1}) \\equiv 0 \\pmod{p^{n+1}}$\n \\textup{(root condition)},\n \\\ + item[\\textup{(iii)}]\n $a_{n+1}$ is given by\n \\eqref{eq:ch16-lifted-root}\n using only addition,\ + \ multiplication,\n and the modular complement $p - q_n$\n \\textup{(no signed arithmetic)}.\n\\end{enumerate}\n\ + Moreover, $f'(a_{n+1})$ is again a unit modulo~$p$,\nso the lifting can be iterated indefinitely.\n```\n\n## Proof / Justification\n\ + \n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl`\ + \ line 44\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch16-hensel-lifting-and-local-fields.tex`\ + \ lines 157-183\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.HenselLifting`\n\ + - Name: `hensel_lift`\n\n## Dependencies\n\n- Canonical: III.T10\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Given a root mod p, lift to root mod p^n by modular Newton iteration. No signed arithmetic: correction via modular + complement. Lifting is unique (p-adic contraction). Fully constructive proof.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001604.md b/_corpus_v3_items/cid001604.md new file mode 100644 index 000000000..7198ff495 --- /dev/null +++ b/_corpus_v3_items/cid001604.md @@ -0,0 +1,92 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001604/ +title: Adelic Embedding Theorem +cid: cid001604 +primary_alias: THM0147 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Canonical map τ → 𝔸_τ is injective with dense image. Every τ-object maps to an adelic tuple; the product formula + holds. Embeds discrete τ-arithmetic into adelic structure. +aliases: +- THM0147 +- III.T12 +- adelic-embedding-theorem +- thm:adelic-embedding +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006001 +- predicate: formalized_by + object: cid006002 +- predicate: formalized_by + object: cid006006 +- predicate: formalized_by + object: cid006007 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 17 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Adelic Embedding Theorem**\n\nCanonical map τ → \U0001D538_τ is injective with dense image. Every τ-object\ + \ maps to an adelic tuple; the product formula holds. Embeds discrete τ-arithmetic into adelic structure.\n\n# Adelic\ + \ Embedding Theorem\n\n## Summary\n\nCanonical map τ → \U0001D538_τ is injective with dense image. Every τ-object maps\ + \ to an adelic tuple; the product formula holds. Embeds discrete τ-arithmetic into adelic structure.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:adelic-embedding}\nThe canonical map\n$\\iota_{\\mathbb{A}} : \\widehat{\\Z}_\\tau \\to \\\ + mathbb{A}_\\tau$\nsatisfies:\n\\begin{enumerate}\n \\item[\\textup{(i)}] \\textbf{Injectivity.}\n $\\iota_{\\\ + mathbb{A}}$ is injective:\n if $(a_p)_p = (b_p)_p$ in $\\mathbb{A}_\\tau$,\n then $a = b$ in $\\widehat{\\\ + Z}_\\tau$.\n \\item[\\textup{(ii)}] \\textbf{Dense image.}\n The image of $\\iota_{\\mathbb{A}}$\n \ + \ is dense in $\\mathbb{A}_\\tau$\n in the following $\\tau$-effective sense:\n for every adelic tuple\ + \ $(x_p)_p \\in \\mathbb{A}_\\tau$\n and every primorial depth~$k$,\n there exists $a \\in \\widehat{\\\ + Z}_\\tau$\n such that $a_p = x_p$\n for all $p \\leq p_k$.\n \\item[\\textup{(iii)}] \\textbf{Product\ + \ formula.}\n For every $a \\in \\widehat{\\Z}_\\tau$, $a \\neq 0$,\n \\begin{equation}\\label{eq:ch17-product-formula}\n\ + \ \\prod_{p} |a|_p \\;\\cdot\\; \\|a\\|_{\\mathrm{NF}} \\;=\\; 1,\n \\end{equation}\n where\ + \ $|a|_p = p^{-v_p(a)}$\n is the $p$-adic absolute value,\n $v_p$ is the $p$-adic valuation\n \ + \ from the NF decomposition,\n and $\\|a\\|_{\\mathrm{NF}} = \\prod_p p^{v_p(a)}$\n is the \\emph{NF\ + \ norm}---the\n $\\tau$-internal analogue of the archimedean\n absolute value~$|a|_\\infty$.\n\\end{enumerate}\n\ + In particular, $\\widehat{\\Z}_\\tau$ embeds\ninto $\\mathbb{A}_\\tau$\nas a discrete, dense subring.\n```\n\n## Proof\ + \ / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\n-\ + \ Registry source: `book-03.jsonl` line 48\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch17-the-adelic-embedding.tex`\ + \ lines 279-316\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Adeles`\n\ + - Name: `adelic_embedding_check`\n\n## Dependencies\n\n- Canonical: III.D22\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Canonical map τ → 𝔸_τ is injective with dense image. Every τ-object maps to an adelic tuple; the product formula + holds. Embeds discrete τ-arithmetic into adelic structure. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001605.md b/_corpus_v3_items/cid001605.md new file mode 100644 index 000000000..419e0422c --- /dev/null +++ b/_corpus_v3_items/cid001605.md @@ -0,0 +1,90 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001605/ +title: Label Convergence +cid: cid001605 +primary_alias: THM0148 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Label_n stabilizes: for each prime p, there exists n₀ such that Label_n(p) is constant for n ≥ n₀. The limiting + classifier Label_∞ exists. Stabilization is immediate at depth of first appearance.' +aliases: +- THM0148 +- III.T13 +- label-convergence +- thm:label-convergence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006009 +- predicate: formalized_by + object: cid006010 +- predicate: formalized_by + object: cid006016 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 18 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Label Convergence**\n\nLabel_n stabilizes: for each prime p, there exists n₀ such that Label_n(p) is constant\ + \ for n ≥ n₀. The limiting classifier Label_∞ exists. Stabilization is immediate at depth of first appearance.\n\n# Label\ + \ Convergence\n\n## Summary\n\nLabel_n stabilizes: for each prime p, there exists n₀ such that Label_n(p) is constant\ + \ for n ≥ n₀. The limiting classifier Label_∞ exists. Stabilization is immediate at depth of first appearance.\n\n## Statement\n\ + \n```latex\n%\n\\label{thm:label-convergence}\nFor every prime~$p_i$,\nthe sequence $\\mathrm{Label}_n(p_i)$ for $n \\\ + geq i$\nis eventually constant.\nMore precisely:\n\\begin{enumerate}\n \\item[\\textup{(i)}]\n \\textbf{Immediate\ + \ stabilisation.}\n $\\mathrm{Label}_n(p_i) = \\mathrm{Label}_i(p_i)$\n for all $n \\geq i$.\n \ + \ The label stabilises at the depth\n where the prime first enters.\n \\item[\\textup{(ii)}]\n \ + \ \\textbf{Limiting classifier.}\n The function\n \\begin{equation}\\label{eq:ch18-label-infty}\n\ + \ \\mathrm{Label}_\\infty \\;:\\;\n \\{\\text{all primes}\\}\n \\;\\longrightarrow\\\ + ;\n \\{B, C, X\\},\n \\qquad\n \\mathrm{Label}_\\infty(p_i) \\;=\\; \\mathrm{Label}_i(p_i),\n\ + \ \\end{equation}\n is well defined.\n It is the unique classifier\n compatible with\ + \ all finite-depth classifiers\n under the primorial projections.\n \\item[\\textup{(iii)}]\n \\\ + textbf{Explicit formula.}\n For odd primes $p > 2$:\n \\begin{equation}\\label{eq:ch18-explicit-label}\n\ + \ \\mathrm{Label}_\\infty(p) \\;=\\;\n \\begin{cases}\n B & \\text{if } p \\\ + equiv \\pm 1 \\pmod{8}, \\\\\n C & \\text{if } p \\equiv \\pm 3 \\pmod{8}.\n \\end{cases}\n\ + \ \\end{equation}\n For $p = 2$:\n $\\mathrm{Label}_\\infty(2) = X$.\n\\end{enumerate}\n```\n\ + \n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\ + \n- Registry source: `book-03.jsonl` line 51\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch18-the-internal-bipolar-classifier.tex`\ + \ lines 389-431\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.BipolarClassifier`\n\ + - Name: `label_convergence_check`\n\n## Dependencies\n\n- Canonical: III.D23\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Label_n stabilizes: for each prime p, there exists n₀ such that Label_n(p) is constant for n ≥ n₀. The limiting + classifier Label_∞ exists. Stabilization is immediate at depth of first appearance.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001606.md b/_corpus_v3_items/cid001606.md new file mode 100644 index 000000000..b9d051dc7 --- /dev/null +++ b/_corpus_v3_items/cid001606.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001606/ +title: Spectral Trichotomy Lemma +cid: cid001606 +primary_alias: THM0149 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every boundary character at level n decomposes uniquely into B-supported, C-supported, and X-mixing components. The + decomposition is exact, orthogonal, and functorial (commutes with level change). +aliases: +- THM0149 +- III.T14 +- spectral-trichotomy-lemma +- thm:spectral-trichotomy +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006078 +- predicate: formalized_by + object: cid006079 +- predicate: formalized_by + object: cid006084 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 19 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Spectral Trichotomy Lemma**\n\nEvery boundary character at level n decomposes uniquely into B-supported, C-supported,\ + \ and X-mixing components. The decomposition is exact, orthogonal, and functorial (commutes with level change).\n\n# Spectral\ + \ Trichotomy Lemma\n\n## Summary\n\nEvery boundary character at level n decomposes uniquely into B-supported, C-supported,\ + \ and X-mixing components. The decomposition is exact, orthogonal, and functorial (commutes with level change).\n\n##\ + \ Statement\n\n```latex\n%\n\\label{thm:spectral-trichotomy}\nLet $n \\geq 1$ and $\\chi \\in \\Char_n(\\Lemniscate)$.\n\ + Then:\n\\begin{enumerate}\n \\item[\\textup{(i)}] \\textbf{Exact decomposition.}\n $\\chi$ decomposes uniquely\ + \ as\n \\begin{equation}\\label{eq:ch19-trichotomy}\n \\chi \\;=\\; \\chi_B + \\chi_C,\n \ + \ \\qquad\n \\chi_B = e_+ \\cdot \\chi \\in \\Char_B^{(n)},\n \\quad\n \\\ + chi_C = e_- \\cdot \\chi \\in \\Char_C^{(n)}.\n \\end{equation}\n \\item[\\textup{(ii)}] \\textbf{Orthogonality.}\n\ + \ $e_+ \\cdot \\chi_C = 0$\n and $e_- \\cdot \\chi_B = 0$.\n \\item[\\textup{(iii)}] \\textbf{Functoriality.}\n\ + \ If $\\pi_{n+1,n} : \\Char_{n+1}(\\Lemniscate)\n \\to \\Char_n(\\Lemniscate)$\n is the projection,\ + \ then\n \\begin{equation}\\label{eq:ch19-functoriality}\n \\pi_{n+1,n}(\\chi_B)\n \ + \ = (\\pi_{n+1,n}(\\chi))_B,\n \\qquad\n \\pi_{n+1,n}(\\chi_C)\n = (\\pi_{n+1,n}(\\\ + chi))_C.\n \\end{equation}\n \\item[\\textup{(iv)}] \\textbf{Trichotomy.}\n Exactly one holds:\n\ + \ $\\chi \\in \\Char_B^{(n)}$ (pure $B$),\n $\\chi \\in \\Char_C^{(n)}$ (pure $C$),\n or $\\\ + chi \\in \\Char_X^{(n)}$ ($X$-mixing).\n The three classes partition $\\Char_n(\\Lemniscate)$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 53\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch19-the-spectral-trichotomy.tex`\ + \ lines 122-158\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Trichotomy`\n\ + - Name: `trichotomy_check`\n\n## Dependencies\n\n- Canonical: III.D23, III.T10\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every boundary character at level n decomposes uniquely into B-supported, C-supported, and X-mixing components. + The decomposition is exact, orthogonal, and functorial (commutes with level change). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001607.md b/_corpus_v3_items/cid001607.md new file mode 100644 index 000000000..c8219fffa --- /dev/null +++ b/_corpus_v3_items/cid001607.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001607/ +title: B/C Non-Collapse Theorem +cid: cid001607 +primary_alias: THM0150 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'B-sector and C-sector are genuinely distinct: no tower-compatible isomorphism between B-supported and C-supported + subrings exists. Growth-rate asymmetry (B=exponent, C=tetration) creates an inescapable coherence obstruction.' +aliases: +- THM0150 +- III.T15 +- b-c-non-collapse-theorem +- thm:bc-non-collapse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006082 +- predicate: formalized_by + object: cid006083 +- predicate: formalized_by + object: cid006086 +- predicate: formalized_by + object: cid006087 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 19 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**B/C Non-Collapse Theorem**\n\nB-sector and C-sector are genuinely distinct: no tower-compatible isomorphism\ + \ between B-supported and C-supported subrings exists. Growth-rate asymmetry (B=exponent, C=tetration) creates an inescapable\ + \ coherence obstruction.\n\n# B/C Non-Collapse Theorem\n\n## Summary\n\nB-sector and C-sector are genuinely distinct:\ + \ no tower-compatible isomorphism between B-supported and C-supported subrings exists. Growth-rate asymmetry (B=exponent,\ + \ C=tetration) creates an inescapable coherence obstruction.\n\n## Statement\n\n```latex\n%\n\\label{thm:bc-non-collapse}\n\ + There is no system of ring isomorphisms\n\\begin{equation}\\label{eq:ch19-no-system}\n \\varphi_k : e_+ R_k^{\\jj}\n\ + \ \\;\\xrightarrow{\\;\\sim\\;}\n e_- R_k^{\\jj},\n \\qquad k = 1, 2, 3, \\ldots,\n\\end{equation}\ncompatible\ + \ with the primorial tower:\n\\begin{equation}\\label{eq:ch19-tower-compat}\n \\begin{tikzcd}[column sep=3.5em]\n \ + \ e_+ R_{k+1}^{\\jj}\n \\ar[r, \"\\varphi_{k+1}\"]\n \\ar[d, \"\\pi_{k+1,k}\"']\n &\ + \ e_- R_{k+1}^{\\jj}\n \\ar[d, \"\\pi_{k+1,k}\"]\n \\\\\n e_+ R_k^{\\jj}\n \\ar[r,\ + \ \"\\varphi_k\"']\n & e_- R_k^{\\jj}\n \\end{tikzcd}\n\\end{equation}\nThe $B$- and $C$-sectors are isomorphic\ + \ level-by-level\nbut \\emph{not} as towers.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block\ + \ was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 55\n- Manuscript source:\ + \ `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part03/ch19-the-spectral-trichotomy.tex` lines 366-392\n\n\ + ## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Spectral.Trichotomy`\n- Name:\ + \ `bc_non_collapse_check`\n\n## Dependencies\n\n- Canonical: III.T14, III.D24\n\n## Related Results\n\n_Generated by later\ + \ projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n-\ + \ 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'B-sector and C-sector are genuinely distinct: no tower-compatible isomorphism between B-supported and C-supported + subrings exists. Growth-rate asymmetry (B=exponent, C=tetration) creates an inescapable coherence obstruction.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001608.md b/_corpus_v3_items/cid001608.md new file mode 100644 index 000000000..9c802ae6a --- /dev/null +++ b/_corpus_v3_items/cid001608.md @@ -0,0 +1,209 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001608/ +title: Bipolar Euler Product +cid: cid001608 +primary_alias: THM0151 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The split-complex zeta ζ_τ(s) admits a bipolar Euler product: ζ_τ(s) = ∏_p (1 − Label(p)·p^{−s})^{−1} where Label(p) + ∈ {e₊, e₋, mixed} via the spectral trichotomy. CRT decomposition at each primorial level recovers the partial products.' +aliases: +- THM0151 +- III.T16 +- bipolar-euler-product +- thm:bipolar-euler-product +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005852 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 22 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bipolar Euler Product** + + + The split-complex zeta ζ_τ(s) admits a bipolar Euler product: ζ_τ(s) = ∏_p (1 − Label(p)·p^{−s})^{−1} where Label(p) ∈ + {e₊, e₋, mixed} via the spectral trichotomy. CRT decomposition at each primorial level recovers the partial products. + + + # Bipolar Euler Product + + + ## Summary + + + The split-complex zeta ζ_τ(s) admits a bipolar Euler product: ζ_τ(s) = ∏_p (1 − Label(p)·p^{−s})^{−1} where Label(p) ∈ + {e₊, e₋, mixed} via the spectral trichotomy. CRT decomposition at each primorial level recovers the partial products. + + + ## Statement + + + ```latex + + \label{thm:bipolar-euler-product} + + For $\mathrm{Re}(s) > 1$, the split-complex zeta function admits the Euler product + + \[ + + \zeta_{\T}(s) \;=\; \prod_{p \,\text{prime}} \frac{1}{1 - p^{-s}} \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s), + + \] + + where + + \begin{align*} + + \zeta_B(s) &\;=\; \prod_{\substack{p \,\text{prime} \\ \mathrm{Label}_p = \mathrm{B}}} \frac{1}{1 - p^{-s}}, \\[6pt] + + \zeta_C(s) &\;=\; \prod_{\substack{p \,\text{prime} \\ \mathrm{Label}_p = \mathrm{C}}} \frac{1}{1 - p^{-s}}. + + \end{align*} + + Each prime contributes to exactly one idempotent sector. Neutral primes ($\mathrm{Label}_p = \mathrm{N}$) do not exist + in this factorization, as $\mathrm{Label}_p \in \{\mathrm{B}, \mathrm{C}\}$ for all primes $p > 2$ (by Theorem~III.T14). + + ``` + + + ## Proof / Justification + + + ```latex + + The classical Euler product expands as + + \[ + + \zeta(s) \;=\; \prod_p \left(1 + \frac{1}{p^s} + \frac{1}{p^{2s}} + \cdots \right) \;=\; \sum_{n=1}^\infty \frac{1}{n^s}, + + \] + + by unique prime factorization. In the split-complex setting, each integer $n = p_1^{a_1} \cdots p_k^{a_k}$ inherits its + label from the prime labels via the polarity sum: + + \[ + + \mathrm{polarity}(n) \;=\; a_1 \cdot \mathrm{pol}(p_1) + \cdots + a_k \cdot \mathrm{pol}(p_k), + + \] + + where $\mathrm{pol}(p) = +1$ if $\mathrm{Label}_p = \mathrm{B}$ and $\mathrm{pol}(p) = -1$ if $\mathrm{Label}_p = \mathrm{C}$. + + + The CRT decomposition (Theorem~III.T10) asserts that the ring $\mathbb{Z}/N\mathbb{Z}$ factors as + + \[ + + \mathbb{Z}/N\mathbb{Z} \;\cong\; \bigoplus_{p^a \| N} \mathbb{Z}/p^a\mathbb{Z}, + + \] + + and this isomorphism respects the bipolar structure: each prime power $p^a$ contributes to the B-sector if $\mathrm{Label}_p + = \mathrm{B}$, and to the C-sector if $\mathrm{Label}_p = \mathrm{C}$. + + + Thus the Euler product factorizes into two independent products: + + \[ + + \prod_{p} \frac{1}{1 - p^{-s}} \;=\; \left( \prod_{p \in \mathrm{B}} \frac{1}{1 - p^{-s}} \right) \times \left( \prod_{p + \in \mathrm{C}} \frac{1}{1 - p^{-s}} \right). + + \] + + The idempotent projection maps these products to the two sectors: + + \[ + + \zeta_{\T}(s) \;=\; e_+ \cdot \zeta_B(s) \;+\; e_- \cdot \zeta_C(s). \qedhere + + \] + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 63 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch22-the-functional-equation-in-h-tau.tex` + lines 109-120 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SplitComplexZeta` + + - Name: `bipolar_euler_check` + + + ## Dependencies + + + - Canonical: III.D26, III.D27, III.T10, III.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The split-complex zeta ζ_τ(s) admits a bipolar Euler product: ζ_τ(s) = ∏_p (1 − Label(p)·p^{−s})^{−1} where + Label(p) ∈ {e₊, e₋, mixed} via the spectral trichotomy. CRT decomposition at each primorial level recovers the partial products.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001609.md b/_corpus_v3_items/cid001609.md new file mode 100644 index 000000000..0949e46a1 --- /dev/null +++ b/_corpus_v3_items/cid001609.md @@ -0,0 +1,190 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001609/ +title: Self-Adjointness of H_L +cid: cid001609 +primary_alias: THM0152 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'H_L with Kirchhoff conditions is self-adjoint on L²(L). All eigenvalues are real. The K5 diagonal discipline (categorical + axiom) is the structural reason: self-adjointness is forced by the requirement that the crossing point mediates between + lobes without introducing imaginary coupling.' +aliases: +- THM0152 +- III.T17 +- self-adjointness-of-h-l +- thm:self-adjointness-h-l +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005825 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 23 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Self-Adjointness of H_L** + + + H_L with Kirchhoff conditions is self-adjoint on L²(L). All eigenvalues are real. The K5 diagonal discipline (categorical + axiom) is the structural reason: self-adjointness is forced by the requirement that the crossing point mediates between + lobes without introducing imaginary coupling. + + + # Self-Adjointness of H_L + + + ## Summary + + + H_L with Kirchhoff conditions is self-adjoint on L²(L). All eigenvalues are real. The K5 diagonal discipline (categorical + axiom) is the structural reason: self-adjointness is forced by the requirement that the crossing point mediates between + lobes without introducing imaginary coupling. + + + ## Statement + + + ```latex + + \label{thm:self-adjointness-h-l} + + The operator $H_L$ is self-adjoint: $H_L^* = H_L$. + + ``` + + + ## Proof / Justification + + + ```latex + + We compute the boundary form. For $f, g \in \mathrm{Dom}(H_L)$, integration by parts on each edge gives + + \begin{align*} + + \langle H_L f, g \rangle - \langle f, H_L g \rangle + + &= -\int_{e_B} f''''_B \overline{g_B} \, dx - \int_{e_C} f''''_C \overline{g_C} \, dx \\ + + &\quad + \int_{e_B} f_B \overline{g''''_B} \, dx + \int_{e_C} f_C \overline{g''''_C} \, dx \\ + + &= -\left[ f''_B \overline{g_B} - f_B \overline{g''_B} \right]_0^1 - \left[ f''_C \overline{g_C} - f_C \overline{g''_C} + \right]_0^1. + + \end{align*} + + Evaluating at the endpoints $0$ and $1$ (both corresponding to $\omega$), continuity gives $f_B(\omega) = f_C(\omega)$ + and $g_B(\omega) = g_C(\omega)$, so + + \begin{align*} + + \langle H_L f, g \rangle - \langle f, H_L g \rangle + + &= -\Big[ f''_B(\omega) \overline{g_B(\omega)} - f_B(\omega) \overline{g''_B(\omega)} \\ + + &\quad\quad + f''_C(\omega) \overline{g_C(\omega)} - f_C(\omega) \overline{g''_C(\omega)} \Big] \\ + + &\quad + \text{(terms at $x=0$ cancel by periodicity)} \\ + + &= -\overline{g_B(\omega)} \Big( f''_B(\omega) + f''_C(\omega) \Big) + f_B(\omega) \overline{\Big( g''_B(\omega) + g''_C(\omega) + \Big)}. + + \end{align*} + + The Kirchhoff condition forces $f''_B(\omega) + f''_C(\omega) = 0$ and $g''_B(\omega) + g''_C(\omega) = 0$, so both terms + vanish: + + \[ + + \langle H_L f, g \rangle - \langle f, H_L g \rangle = 0. + + \] + + Thus $H_L \subset H_L^*$. For the reverse inclusion, standard von Neumann theory~\cite{ReedSimon1980} shows that the Kirchhoff + condition is \emph{maximal} among boundary conditions yielding symmetric operators on $L$. Therefore $H_L^* = H_L$. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 65 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch23-the-lemniscate-operator.tex` + lines 123-126 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.LemniscateOperator` + + - Name: `self_adjoint_check` + + + ## Dependencies + + + - Canonical: III.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'H_L with Kirchhoff conditions is self-adjoint on L²(L). All eigenvalues are real. The K5 diagonal discipline + (categorical axiom) is the structural reason: self-adjointness is forced by the requirement that the crossing point mediates + betwee' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001610.md b/_corpus_v3_items/cid001610.md new file mode 100644 index 000000000..12178e500 --- /dev/null +++ b/_corpus_v3_items/cid001610.md @@ -0,0 +1,155 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001610/ +title: Spectral Correspondence Theorem +cid: cid001610 +primary_alias: THM0153 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the spectral + parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap.' +aliases: +- THM0153 +- III.T18 +- spectral-correspondence-theorem +- thm:spectral-correspondence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005837 +- predicate: formalized_by + object: cid005840 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 24 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Spectral Correspondence Theorem** + + + CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the spectral + parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap. + + + # Spectral Correspondence Theorem + + + ## Summary + + + CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the spectral + parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap. + + + ## Statement + + + ```latex + + \label{thm:spectral-correspondence} + + Assume Conjecture~\ref{conj:determinant-representation} holds. Then: + + \begin{enumerate} + + \item $\zeta_\tau(\rho) = 0$ if and only if $\Lambda(\rho) \in \operatorname{Spec}^*(H_L)$, where $\operatorname{Spec}^*(H_L)$ + denotes the point spectrum (eigenvalues) of $H_L$. + + \item Every non-trivial zero of $\zeta_\tau(s)$ corresponds to a unique eigenvalue of $H_L$ via $\lambda = \Lambda(\rho)$. + + \item The multiplicities of zeros and eigenvalues match: $\operatorname{ord}_{s=\rho} \zeta_\tau(s) = \dim \ker(H_L - + \Lambda(\rho))$. + + \item If $H_L$ is self-adjoint with real spectrum, then all non-trivial zeros satisfy $\Lambda(\rho) \in \mathbb{R}$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 68 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch24-the-spectral-correspondence.tex` + lines 112-120 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SpectralCorrespondence` + + - Name: `spectral_correspondence_O3` + + + ## Dependencies + + + - Canonical: III.D29, III.D26, III.T17, III.P09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CONDITIONAL on O3 (determinant representation): zeros of ζ_τ(s) correspond to spectral values of H_L via the + spectral parameter Λ(s). This is the Hilbert–Pólya realization within the τ framework. O3 is the honest conjectural gap.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001611.md b/_corpus_v3_items/cid001611.md new file mode 100644 index 000000000..aea05a37e --- /dev/null +++ b/_corpus_v3_items/cid001611.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001611/ +title: Critical Line Theorem +cid: cid001611 +primary_alias: THM0154 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral + correspondence (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: + off-critical-line zeros would require imaginary spectral coupling, which K5 forbids.' +aliases: +- THM0154 +- III.T19 +- critical-line-theorem +- thm:critical-line-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005814 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 25 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Critical Line Theorem** + + + CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral correspondence + (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: off-critical-line + zeros would require imaginary spectral coupling, which K5 forbids. + + + # Critical Line Theorem + + + ## Summary + + + CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral correspondence + (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechanism: off-critical-line + zeros would require imaginary spectral coupling, which K5 forbids. + + + ## Statement + + + ```latex + + \label{thm:critical-line-theorem} + + \emph{(Conditional on Theorem~\ref{thm:spectral-correspondence}.)} + + All non-trivial zeros of the Riemann zeta function lie on the critical line $\Re(s) = \frac{1}{2}$. + + ``` + + + ## Proof / Justification + + + ```latex + + Combine~\eqref{eq:ch25-spectral-image},~\eqref{eq:ch25-quadratic-expansion}, and~\eqref{eq:ch25-imaginary-vanish} with + the self-adjointness of $H_L$ (Theorem~\ref{thm:self-adjointness-h-l}). The reality of $\operatorname{Spec}(H_L)$ forces + $\sigma = \frac{1}{2}$ for every non-trivial zero. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 69 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch25-spectral-purity-and-the-critical-line.tex` + lines 80-84 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.CriticalLine` + + - Name: `critical_line_check` + + + ## Dependencies + + + - Canonical: III.T17, III.T18, III.T14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'CONDITIONAL on III.T18 (O3): self-adjointness of H_L (III.T17) forces all eigenvalues real, which via the spectral + correspondence (III.T18) forces all non-trivial zeros of ζ_τ to lie on Re(s) = ½. The K5 off-diagonal exclusion is the mechan' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001612.md b/_corpus_v3_items/cid001612.md new file mode 100644 index 000000000..f0bd2ebaf --- /dev/null +++ b/_corpus_v3_items/cid001612.md @@ -0,0 +1,174 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001612/ +title: Prime Polarity Scaling Theorem +cid: cid001612 +primary_alias: THM0155 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The GRH at each primorial depth decomposes into three independent statements via Label_n: purity of B-sector zeros, + purity of C-sector zeros, and balance of X-sector zeros. Scaling from one primorial level to the next preserves polarity + type (III.T13 convergence).' +aliases: +- THM0155 +- III.T20 +- prime-polarity-scaling-theorem +- thm:prime-polarity-scaling +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005819 +- predicate: formalized_by + object: cid005820 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 27 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Prime Polarity Scaling Theorem** + + + The GRH at each primorial depth decomposes into three independent statements via Label_n: purity of B-sector zeros, purity + of C-sector zeros, and balance of X-sector zeros. Scaling from one primorial level to the next preserves polarity type + (III.T13 convergence). + + + # Prime Polarity Scaling Theorem + + + ## Summary + + + The GRH at each primorial depth decomposes into three independent statements via Label_n: purity of B-sector zeros, purity + of C-sector zeros, and balance of X-sector zeros. Scaling from one primorial level to the next preserves polarity type + (III.T13 convergence). + + + ## Statement + + + ```latex + + \label{thm:prime-polarity-scaling} + + Let $\pi$ be an automorphic representation with conductor $q$. Choose level $n$ such that $q \mid n$. Then: + + \begin{enumerate} + + \item The Label$_n$ decomposition~\eqref{eq:ch27-euler-decomposition} is well-defined and independent of $n$ for $n \gg + 0$. + + \item The spectral trichotomy (III.T14) applies to each sector: $H_L = H_B \oplus H_C \oplus H_X$. + + \item The zeros of $L(s, \pi)$ decompose into $B$-sector, $C$-sector, and $X$-sector zeros. + + \item GRH for $\pi$ is equivalent to spectral purity in each sector: all zeros of $L_B$, $L_C$, $L_X$ on $\Re(s) = \frac{1}{2}$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + ```latex + + (1) The Label$_n$ classifier stabilizes for $n$ divisible by the conductor $q$ (III.T13). For $n'' > n$ with $n \mid n''$, + the sets of $B$-, $C$-, $X$-primes refine but preserve asymptotic density. + + + (2) The Hilbert space $H_L$ decomposes spectrally by the bipolar characters $\chi_+$ and $\chi_-$ (Book II, Central Theorem). + The crossing point eigenspace $H_C$ is orthogonal to $H_B = H_+ \oplus H_-$ by the spectral trichotomy (III.T14). The + exceptional sector $H_X$ is finite-dimensional. + + + (3) By~\eqref{eq:ch27-euler-decomposition}, zeros of $L(s, \pi)$ are unions of zeros of $L_B$, $L_C$, $L_X$. + + + (4) Each sector''s $L$-function corresponds to a spectral determinant on the respective subspace. Spectral reality on + each subspace implies all zeros real, hence on $\Re(s) = \frac{1}{2}$ by functional equation symmetry. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 75 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch27-the-grand-grh.tex` lines 85-93 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.GrandGRH` + + - Name: `prime_polarity_scaling_check` + + + ## Dependencies + + + - Canonical: III.D23, III.T13, III.T14, III.D26 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The GRH at each primorial depth decomposes into three independent statements via Label_n: purity of B-sector + zeros, purity of C-sector zeros, and balance of X-sector zeros. Scaling from one primorial level to the next preserves polarity + typ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001615.md b/_corpus_v3_items/cid001615.md new file mode 100644 index 000000000..08c3f1ffa --- /dev/null +++ b/_corpus_v3_items/cid001615.md @@ -0,0 +1,171 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001615/ +title: Master Schema Theorem +cid: cid001615 +primary_alias: THM0158 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'All eight Millennium Problems are instances of Mutual Determination at varying enrichment levels: E₀ (RH, Poincaré + — Part IV), E₁ (NS, YM, Hodge — Part V), E₁→E₂ (BSD, Langlands — Part VI), E₂ (P vs NP — Part VII). The spectral algebra + provides the common language, the primorial ladder the common tower, the CRT the common local-global bridge.' +aliases: +- THM0158 +- III.T23 +- master-schema-theorem +- thm:master-schema +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005827 +- predicate: formalized_by + object: cid005828 +- predicate: formalized_by + object: cid005829 +- predicate: formalized_by + object: cid005830 +- predicate: formalized_by + object: cid005831 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 33 (Part IV) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Master Schema Theorem** + + + All eight Millennium Problems are instances of Mutual Determination at varying enrichment levels: E₀ (RH, Poincaré — Part + IV), E₁ (NS, YM, Hodge — Part V), E₁→E₂ (BSD, Langlands — Part VI), E₂ (P vs NP — Part VII). The spectral algebra provides + the common language, the primorial ladder the common tower, the CRT the common local-global bridge. + + + # Master Schema Theorem + + + ## Summary + + + All eight Millennium Problems are instances of Mutual Determination at varying enrichment levels: E₀ (RH, Poincaré — Part + IV), E₁ (NS, YM, Hodge — Part V), E₁→E₂ (BSD, Langlands — Part VI), E₂ (P vs NP — Part VII). The spectral algebra provides + the common language, the primorial ladder the common tower, the CRT the common local-global bridge. + + + ## Statement + + + ```latex + + \label{thm:master-schema} + + All eight Millennium Problems are instances of Mutual Determination (Definition~III.D25) at varying enrichment levels: + + \begin{itemize} + + \item $E_0$ (spectral): RH, Poincar\''e (Part~IV). + + \item $E_1$ (physics): Navier--Stokes, Yang--Mills, Hodge (Part~V). + + \item $E_1 \to E_2$ (bridge): BSD, Langlands (Part~VI). + + \item $E_2$ (computation): P~vs~NP (Part~IX). + + \end{itemize} + + The spectral algebra $A_{\mathrm{spec}}(\mathbb{L})$ provides the common language across all enrichment levels. The primorial + ladder provides the common tower. The Chinese Remainder Theorem provides the common local-global bridge. + + ``` + + + ## Proof / Justification + + + ```latex + + The proof is the content of Parts~IV through VI and~IX. Part~IV (this Part) establishes the pattern at $E_0$; subsequent + Parts verify the details at progressively higher enrichment levels. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 86 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part04/ch33-the-master-schema.tex` lines + 186-195 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.MasterSchema` + + - Name: `master_schema_check` + + + ## Dependencies + + + - Canonical: III.D25, III.T19, III.D35, III.P13 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'All eight Millennium Problems are instances of Mutual Determination at varying enrichment levels: E₀ (RH, Poincaré + — Part IV), E₁ (NS, YM, Hodge — Part V), E₁→E₂ (BSD, Langlands — Part VI), E₂ (P vs NP — Part VII). The spectral algebra + prov' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001616.md b/_corpus_v3_items/cid001616.md new file mode 100644 index 000000000..d104e918c --- /dev/null +++ b/_corpus_v3_items/cid001616.md @@ -0,0 +1,175 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001616/ +title: Hartogs Flow Theorem +cid: cid001616 +primary_alias: THM0159 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For τ-admissible initial data, the Hartogs flow operator H_flow produces a unique continuation at each primorial + level. The continuation is compatible across levels (tower coherence). +aliases: +- THM0159 +- III.T24 +- hartogs-flow-theorem +- thm:hartogs-flow-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005940 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 36 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hartogs Flow Theorem** + + + For τ-admissible initial data, the Hartogs flow operator H_flow produces a unique continuation at each primorial level. + The continuation is compatible across levels (tower coherence). + + + # Hartogs Flow Theorem + + + ## Summary + + + For τ-admissible initial data, the Hartogs flow operator H_flow produces a unique continuation at each primorial level. + The continuation is compatible across levels (tower coherence). + + + ## Statement + + + ```latex + + \label{thm:hartogs-flow-theorem} + + Let $f_0$ be $\tau$-admissible initial data on a clopen cylinder domain $U \subset \tau^3$. Then the Hartogs flow operator + $H_{\mathrm{flow}}$ satisfies: + + \begin{enumerate} + + \item[\emph{(i)}] \textbf{Unique continuation.} At each primorial level $\mathrm{Prim}(n)$, the extension $\mathrm{Ext}_n(f_0|_{\partial + U_n})$ is the unique $\tau$-holomorphic function on $U_n$ restricting to $f_0|_{\partial U_n}$ on the boundary. + + + \item[\emph{(ii)}] \textbf{Tower coherence.} The continuations are compatible across primorial levels: for every $n \geq + 1$, the diagram + + \begin{equation} + + \begin{tikzcd}[column sep=large] + + \Gamma(\partial U_{n+1}, \mathcal{O}_{H_\tau}) \ar[r, "\mathrm{Ext}_{n{+}1}"] \ar[d, "\mathrm{res}"] + + & \Gamma(U_{n+1}, \mathcal{O}_{H_\tau}) \ar[d, "\mathrm{res}"] \\ + + \Gamma(\partial U_n, \mathcal{O}_{H_\tau}) \ar[r, "\mathrm{Ext}_n"] + + & \Gamma(U_n, \mathcal{O}_{H_\tau}) + + \end{tikzcd} + + \label{eq:ch36-tower-coherence} + + \end{equation} + + commutes, where $\mathrm{res}$ denotes restriction along the tower map $\mathrm{Prim}(n{+}1) \to \mathrm{Prim}(n)$. + + + \item[\emph{(iii)}] \textbf{Sector preservation.} The continuation preserves the spectral trichotomy (Theorem~III.T14): + if $f_0$ belongs to sector $S \in \{B, C, X\}$, then $H_{\mathrm{flow}}(f_0)$ belongs to sector $S$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 95 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch36-the-hartogs-flow-operator.tex` + lines 82-102 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.HartogsFlow` + + - Name: `flow_stabilization_check` + + + ## Dependencies + + + - Canonical: III.D40, III.D36, III.T17 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For τ-admissible initial data, the Hartogs flow operator H_flow produces a unique continuation at each primorial + level. The continuation is compatible across levels (tower coherence). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001617.md b/_corpus_v3_items/cid001617.md new file mode 100644 index 000000000..1a2b67378 --- /dev/null +++ b/_corpus_v3_items/cid001617.md @@ -0,0 +1,157 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001617/ +title: Positive Regularity Theorem +cid: cid001617 +primary_alias: THM0160 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For τ-admissible initial data, every ω-germ stabilizes under H_flow. 3-condition proof: (i) clopen locality, (ii) + ω-germ determinacy via Local Hartogs, (iii) defect-horizon contractivity. Structural mechanism, not a PDE estimate.' +aliases: +- THM0160 +- III.T25 +- positive-regularity-theorem +- thm:positive-regularity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005950 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 37 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Positive Regularity Theorem** + + + For τ-admissible initial data, every ω-germ stabilizes under H_flow. 3-condition proof: (i) clopen locality, (ii) ω-germ + determinacy via Local Hartogs, (iii) defect-horizon contractivity. Structural mechanism, not a PDE estimate. + + + # Positive Regularity Theorem + + + ## Summary + + + For τ-admissible initial data, every ω-germ stabilizes under H_flow. 3-condition proof: (i) clopen locality, (ii) ω-germ + determinacy via Local Hartogs, (iii) defect-horizon contractivity. Structural mechanism, not a PDE estimate. + + + ## Statement + + + ```latex + + \label{thm:positive-regularity} + + For every $\tau$-admissible fluid datum $f$ (Definition~\ref{def:tau-admissible-fluid-data}, Ch.~34) on every clopen cylinder + domain $U \subset \tau^3$, the Hartogs flow operator $H_{\mathrm{flow}}$ (Definition~\ref{def:hartogs-flow-operator}, + Ch.~36) produces a stabilized $\omega$-germ at every point of $U$: + + \begin{equation} + + \forall\, x \in U, \quad \exists\, N_x \in \mathbb{N} : \quad + + \forall\, n \geq N_x, \quad + + H_{\mathrm{flow}}(f)\big|_{\operatorname{Prim}(n)}(x) + + = \omega\text{-}\mathrm{germ}(f, x). + + \label{eq:ch37-positive-regularity} + + \end{equation} + + Regularity is \emph{positive}: it is the existence of a stabilized germ, not the absence of a singularity. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 97 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch37-positive-regularity.tex` lines + 87-99 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.PositiveRegularity` + + - Name: `positive_regularity_check` + + + ## Dependencies + + + - Canonical: III.D36, III.D39, III.D40, III.T24, III.P14 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For τ-admissible initial data, every ω-germ stabilizes under H_flow. 3-condition proof: (i) clopen locality, + (ii) ω-germ determinacy via Local Hartogs, (iii) defect-horizon contractivity. Structural mechanism, not a PDE estimate.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001618.md b/_corpus_v3_items/cid001618.md new file mode 100644 index 000000000..5829546b3 --- /dev/null +++ b/_corpus_v3_items/cid001618.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001618/ +title: τ-Gap Meta-Theorem +cid: cid001618 +primary_alias: THM0161 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Any NF-discrete tower with contractive defect functional and bounded sector extraction has a spectral gap. Γ* = + inf{Δ(f,n) : f non-trivial, n ≥ n₀} > 0. Proof entirely τ-internal: no QFT, no gauge groups, no Lagrangians.' +aliases: +- THM0161 +- III.T26 +- gap-meta-theorem +- thm:tau-gap-meta-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005932 +- predicate: formalized_by + object: cid005933 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 39 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**τ-Gap Meta-Theorem** + + + Any NF-discrete tower with contractive defect functional and bounded sector extraction has a spectral gap. Γ* = inf{Δ(f,n) + : f non-trivial, n ≥ n₀} > 0. Proof entirely τ-internal: no QFT, no gauge groups, no Lagrangians. + + + # τ-Gap Meta-Theorem + + + ## Summary + + + Any NF-discrete tower with contractive defect functional and bounded sector extraction has a spectral gap. Γ* = inf{Δ(f,n) + : f non-trivial, n ≥ n₀} > 0. Proof entirely τ-internal: no QFT, no gauge groups, no Lagrangians. + + + ## Statement + + + ```latex + + \label{thm:tau-gap-meta-theorem} + + Let $\{V_n, \pi_{n+1,n}\}$ be a tower satisfying hypotheses \emph{(H1)} and \emph{(H2)} above. Then the gap constant satisfies + $\Gamma^* > 0$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 104 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch39-the-tau-gap-meta-theorem.tex` + lines 79-83 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.GapTheorem` + + - Name: `tau_gap_meta_check` + + + ## Dependencies + + + - Canonical: III.T09, III.D23, III.T15, III.T14, III.P16, III.D44 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Any NF-discrete tower with contractive defect functional and bounded sector extraction has a spectral gap. Γ* + = inf{Δ(f,n) : f non-trivial, n ≥ n₀} > 0. Proof entirely τ-internal: no QFT, no gauge groups, no Lagrangians.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001619.md b/_corpus_v3_items/cid001619.md new file mode 100644 index 000000000..80dfb1099 --- /dev/null +++ b/_corpus_v3_items/cid001619.md @@ -0,0 +1,153 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001619/ +title: Yang-Mills Gap Theorem +cid: cid001619 +primary_alias: THM0162 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Instantiate the τ-Gap Meta-Theorem for the strong (C) sector: Γ*_s > 0. The τ-internal mass gap is a structural + consequence of NF discreteness. Does NOT solve Clay YM (requires identifying τ strong sector with SU(3) gauge theory).' +aliases: +- THM0162 +- III.T27 +- yang-mills-gap-theorem +- thm:yang-mills-gap-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005935 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 40 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Yang-Mills Gap Theorem** + + + Instantiate the τ-Gap Meta-Theorem for the strong (C) sector: Γ*_s > 0. The τ-internal mass gap is a structural consequence + of NF discreteness. Does NOT solve Clay YM (requires identifying τ strong sector with SU(3) gauge theory). + + + # Yang-Mills Gap Theorem + + + ## Summary + + + Instantiate the τ-Gap Meta-Theorem for the strong (C) sector: Γ*_s > 0. The τ-internal mass gap is a structural consequence + of NF discreteness. Does NOT solve Clay YM (requires identifying τ strong sector with SU(3) gauge theory). + + + ## Statement + + + ```latex + + \label{thm:yang-mills-gap-theorem} + + Let $\{\operatorname{Strong}_{E_1}^{(k)}\}_{k \geq 1}$ be the NF-discrete tower of C-sector gauge configurations, equipped + with the strong defect functional $\Delta_C$ (Definition~\ref{def:strong-defect-functional}). Then the strong-sector gap + constant + + \begin{equation} + + \Gamma^*_s \;=\; \inf\bigl\{\, \Delta_C(\mathfrak{g}_k, n) \;\bigm|\; \mathfrak{g}_k \text{ non-trivial},\; n \geq n_0 + \,\bigr\} + + \label{eq:ch40-strong-gap-constant} + + \end{equation} + + satisfies $\Gamma^*_s > 0$. Every non-trivial $\tau$-admissible gauge configuration carries a strictly positive defect, + uniformly bounded below across all primorial depths. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 107 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch40-the-yang-mills-mass-gap.tex` + lines 101-110 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Physics.GapTheorem` + + - Name: `yang_mills_gap_check` + + + ## Dependencies + + + - Canonical: III.T26, III.D43, III.D44 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Instantiate the τ-Gap Meta-Theorem for the strong (C) sector: Γ*_s > 0. The τ-internal mass gap is a structural + consequence of NF discreteness. Does NOT solve Clay YM (requires identifying τ strong sector with SU(3) gauge theory).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001620.md b/_corpus_v3_items/cid001620.md new file mode 100644 index 000000000..0cf966594 --- /dev/null +++ b/_corpus_v3_items/cid001620.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001620/ +title: NF-Addressability Theorem +cid: cid001620 +primary_alias: THM0163 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every σ-fixed boundary character at finite primorial depth is NF-addressable in every primitive sector. Proof by + sector-by-sector verification (EM model case, then strong, weak, gravity). +aliases: +- THM0163 +- III.T28 +- nf-addressability-theorem +- thm:nf-addressability +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005944 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 42 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**NF-Addressability Theorem**\n\nEvery σ-fixed boundary character at finite primorial depth is NF-addressable\ + \ in every primitive sector. Proof by sector-by-sector verification (EM model case, then strong, weak, gravity).\n\n#\ + \ NF-Addressability Theorem\n\n## Summary\n\nEvery σ-fixed boundary character at finite primorial depth is NF-addressable\ + \ in every primitive sector. Proof by sector-by-sector verification (EM model case, then strong, weak, gravity).\n\n##\ + \ Statement\n\n```latex\n%\n\\label{thm:nf-addressability}\nLet $\\chi \\in \\Char(\\Lemniscate)$\nbe a $\\sigma$-fixed\ + \ boundary character\nat finite primorial depth~$k$.\nThen for every primitive sector\n$S \\in \\{A, B, C, D\\}$,\nthe\ + \ $S$-projection $\\pi_S(\\chi)$\nis NF-addressable.\nMore precisely:\nthere exists a primorial depth\n$k_S \\leq k$ such\ + \ that\n\\begin{equation}\\label{eq:ch42-nf-addressability}\n \\operatorname{NF}\\bigl(\\pi_S(\\chi)\\bigr)\n \\\ + ;=\\;\n \\operatorname{NF}\\bigl(\\pi_S(\\chi)\\big|_{\\operatorname{Prim}(k_S)}\\bigr),\n\\end{equation}\ni.e., the\ + \ normal-form extraction of the $S$-projection\nstabilizes at depth~$k_S$\nand carries no further information\nat deeper\ + \ primorial levels.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot\ + \ run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 113\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch42-the-nf-addressability-theorem.tex`\ + \ lines 102-124\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Physics.Hodge`\n\ + - Name: `nf_addressability_check`\n\n## Dependencies\n\n- Canonical: III.D47, III.D48, III.T14, III.T13, III.D23\n\n##\ + \ Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every σ-fixed boundary character at finite primorial depth is NF-addressable in every primitive sector. Proof + by sector-by-sector verification (EM model case, then strong, weak, gravity). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001621.md b/_corpus_v3_items/cid001621.md new file mode 100644 index 000000000..3bde121b6 --- /dev/null +++ b/_corpus_v3_items/cid001621.md @@ -0,0 +1,94 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001621/ +title: Physics Layer Assembly +cid: cid001621 +primary_alias: THM0164 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: NS (mortar/regularity), YM (bricks/mass gap), Hodge (blueprint/addressability) together construct the E₁ enrichment + layer. All three are instances of Mutual Determination at E₀→E₁. +aliases: +- THM0164 +- III.T29 +- physics-layer-assembly +- thm:physics-layer-assembly +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005945 +- predicate: formalized_by + object: cid005946 +- predicate: formalized_by + object: cid005947 +- predicate: formalized_by + object: cid005948 +- predicate: formalized_by + object: cid005949 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 43 (Part V) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Physics Layer Assembly**\n\nNS (mortar/regularity), YM (bricks/mass gap), Hodge (blueprint/addressability)\ + \ together construct the E₁ enrichment layer. All three are instances of Mutual Determination at E₀→E₁.\n\n# Physics Layer\ + \ Assembly\n\n## Summary\n\nNS (mortar/regularity), YM (bricks/mass gap), Hodge (blueprint/addressability) together construct\ + \ the E₁ enrichment layer. All three are instances of Mutual Determination at E₀→E₁.\n\n## Statement\n\n```latex\n\\label{thm:physics-layer-assembly}\n\ + The three Part~V results---positive regularity (III.T25), the Yang--Mills gap (III.T27), and NF-addressability (III.T28)---are\ + \ instances of Mutual Determination (Definition~\\ref{def:mutual-determination-schema}) at enrichment level~$E_1$, with\ + \ the following unified structure:\n\n\\begin{center}\n\\begin{adjustbox}{max width=0.95\\linewidth}\n\\begin{tabular}{llll}\n\ + \\toprule\n\\textbf{Problem} & \\textbf{Boundary Data ($B$)} & \\textbf{Spectral Structure ($S$)} & \\textbf{Interior\ + \ Consequence ($I$)} \\\\\n\\midrule\nNS & $\\tau$-admissible fluid data & Defect contractivity & Stabilized $\\omega$-germ\ + \ \\\\\n & (Def.~\\ref{def:tau-admissible-fluid-data}) & $\\Delta(f,n) \\to 0$ & (regularity) \\\\\n\\addlinespace\nYM\ + \ & $\\tau$-admissible gauge data & NF discreteness & Spectral gap \\\\\n & (Def.~\\ref{def:tau-admissible-gauge-data})\ + \ & $|\\operatorname{NF}| \\le \\operatorname{Prim}(k)$ & $\\Gamma^*_s > 0$ \\\\\n\\addlinespace\nHodge & $\\sigma$-fixed\ + \ characters & Balanced $B/C$ content & NF-addressable \\\\\n & (Def.~\\ref{def:sigma-fixed-character}) & $\\chi_+ / \\\ + chi_-$ equilibrium & in all sectors \\\\\n\\bottomrule\n\\end{tabular}\n\\end{adjustbox}\n\\end{center}\n\n\\noindent\ + \ In each case, the boundary data uniquely determines the spectral structure, which uniquely determines the interior consequence.\ + \ The bidirectional passage $B \\leftrightarrow S \\leftrightarrow I$ closes because the interior consequence reconstructs\ + \ the boundary data via the primorial tower maps.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block\ + \ was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 116\n- Manuscript\ + \ source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part05/ch43-the-physics-layer-complete.tex` lines 78-103\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Physics.PhysicsAssembly`\n\ + - Name: `physics_assembly_check`\n\n## Dependencies\n\n- Canonical: III.T25, III.T27, III.T28, III.D25\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: NS (mortar/regularity), YM (bricks/mass gap), Hodge (blueprint/addressability) together construct the E₁ enrichment + layer. All three are instances of Mutual Determination at E₀→E₁. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001622.md b/_corpus_v3_items/cid001622.md new file mode 100644 index 000000000..d2a5555fe --- /dev/null +++ b/_corpus_v3_items/cid001622.md @@ -0,0 +1,81 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001622/ +title: TTM τ-Nativity +cid: cid001622 +primary_alias: THM0165 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'A TTM program IS a τ-address operating ON τ-addresses producing τ-addresses. Program = data type. Self-referential + identity makes the TTM an E₂ object: code contains its own decoder.' +aliases: +- THM0165 +- III.T30 +- ttm-nativity +- thm:ttm-tau-nativity +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005803 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 55 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**TTM τ-Nativity**\n\nA TTM program IS a τ-address operating ON τ-addresses producing τ-addresses. Program =\ + \ data type. Self-referential identity makes the TTM an E₂ object: code contains its own decoder.\n\n# TTM τ-Nativity\n\ + \n## Summary\n\nA TTM program IS a τ-address operating ON τ-addresses producing τ-addresses. Program = data type. Self-referential\ + \ identity makes the TTM an E₂ object: code contains its own decoder.\n\n## Statement\n\n```latex\n\\label{thm:ttm-tau-nativity}\n\ + Let $M$ be a $\\tau$-Tower Machine.\nThen:\n\\begin{enumerate}\n \\item The program $M$ has a canonical NF address\n\ + \ $\\operatorname{addr}(M) \\in \\hat{\\mathbb{Z}}_{\\tau}$.\n \\item The instruction set $\\Sigma$ has a\ + \ canonical NF address\n $\\operatorname{addr}(\\Sigma) \\in \\hat{\\mathbb{Z}}_{\\tau}$.\n \\item For every\ + \ input tuple\n $(a_1, \\ldots, a_{b_0}) \\in \\hat{\\mathbb{Z}}_{\\tau}^{\\,b_0}$,\n if $M$ halts with\ + \ output~$a'$,\n then $a' \\in \\hat{\\mathbb{Z}}_{\\tau}$.\n \\item The triple\n $(\\operatorname{addr}(M),\\\ + , \\operatorname{addr}(\\Sigma),\\, a')$\n satisfies $\\Elayer{2}$ operational closure\n (Definition~\\\ + ref{def:operational-closure}):\n the decoder applied to the code\n produces another code of the same\ + \ type.\n\\end{enumerate}\nIn particular, the Code $\\to$ Execution $\\to$ Code\ncycle of $\\Elayer{2}$ is realised within\n\ + the NF tower of Category~$\\tau$.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted\ + \ in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 146\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch55-the-tau-tower-machine.tex`\ + \ lines 313-336\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.TowerMachine`\n\ + - Name: `ttm_nativity_check`\n\n## Dependencies\n\n- Canonical: III.D51, III.D49\n\n## Related Results\n\n_Generated by\ + \ later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'A TTM program IS a τ-address operating ON τ-addresses producing τ-addresses. Program = data type. Self-referential + identity makes the TTM an E₂ object: code contains its own decoder.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001623.md b/_corpus_v3_items/cid001623.md new file mode 100644 index 000000000..7a684e6e7 --- /dev/null +++ b/_corpus_v3_items/cid001623.md @@ -0,0 +1,166 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001623/ +title: Interface Width Principle +cid: cid001623 +primary_alias: THM0166 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: τ-admissible functions are determined by a single finite quotient ℤ/Prim(k₀)ℤ. If computation stabilizes, the infinite + tower collapses to one level. Proof via tower coherence and Global Hartogs. +aliases: +- THM0166 +- III.T31 +- interface-width-principle +- thm:interface-width-principle +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 56 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Interface Width Principle** + + + τ-admissible functions are determined by a single finite quotient ℤ/Prim(k₀)ℤ. If computation stabilizes, the infinite + tower collapses to one level. Proof via tower coherence and Global Hartogs. + + + # Interface Width Principle + + + ## Summary + + + τ-admissible functions are determined by a single finite quotient ℤ/Prim(k₀)ℤ. If computation stabilizes, the infinite + tower collapses to one level. Proof via tower coherence and Global Hartogs. + + + ## Statement + + + ```latex + + \label{thm:interface-width-principle} + + Let $f$ be $\tau$-admissible with $W(f) = k_0 < \infty$. + + Then $f$ factors through the finite quotient + + $\mathbb{Z}/\operatorname{Prim}(k_0)\mathbb{Z}$: + + for all inputs $x$, + + \begin{equation}\label{eq:ch56-width-factorization} + + f(x) \;\text{ depends only on }\; x \bmod \operatorname{Prim}(k_0). + + \end{equation} + + In other words, there exists a function + + $\bar{f} \colon \mathbb{Z}/\operatorname{Prim}(k_0)\mathbb{Z} \to + + \mathbb{Z}/\operatorname{Prim}(k_0)\mathbb{Z}$ + + such that + + \[ + + f(x) \equiv \bar{f}\bigl(x \bmod \operatorname{Prim}(k_0)\bigr) + + \pmod{\operatorname{Prim}(k_0)} + + \] + + for all $x$. + + The entire infinite primorial tower collapses to the single level $k_0$. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 150 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch56-interface-width-and-tau-admissibility.tex` + lines 185-204 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.Admissibility` + + - Name: `width_principle_check` + + + ## Dependencies + + + - Canonical: III.D54, III.D53, III.T09 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: τ-admissible functions are determined by a single finite quotient ℤ/Prim(k₀)ℤ. If computation stabilizes, the + infinite tower collapses to one level. Proof via tower coherence and Global Hartogs. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001624.md b/_corpus_v3_items/cid001624.md new file mode 100644 index 000000000..f138987a0 --- /dev/null +++ b/_corpus_v3_items/cid001624.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001624/ +title: Product-Meet Collapse +cid: cid001624 +primary_alias: THM0167 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At E₂, meet (∧ = finding a witness) IS product (× = constructing the composite). Search = construction. This is the + computational content of the bi-square pasting. Proof via Interface Width Principle + CRT decomposition. +aliases: +- THM0167 +- III.T32 +- product-meet-collapse +- thm:product-meet-collapse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005789 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 58 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Product-Meet Collapse**\n\nAt E₂, meet (∧ = finding a witness) IS product (× = constructing the composite).\ + \ Search = construction. This is the computational content of the bi-square pasting. Proof via Interface Width Principle\ + \ + CRT decomposition.\n\n# Product-Meet Collapse\n\n## Summary\n\nAt E₂, meet (∧ = finding a witness) IS product (× =\ + \ constructing the composite). Search = construction. This is the computational content of the bi-square pasting. Proof\ + \ via Interface Width Principle + CRT decomposition.\n\n## Statement\n\n```latex\n\\label{thm:product-meet-collapse}\n\ + Let $\\Pi$ be $\\tau$-admissible with verifier width $k_0$, and let $x$\nbe a ``yes'' instance at depth $k \\geq k_0$.\ + \ Then the pasting of the\nexecution and witness squares\nin~\\eqref{eq:ch58-computational-bisquare} yields\n\\begin{equation}\\\ + label{eq:ch58-product-meet}\n \\underbrace{w_1 \\wedge w_2 \\wedge \\cdots \\wedge w_k}%\n _{\\text{meet: satisfy\ + \ all primes}}\n \\;=\\;\n \\underbrace{w_1 \\times w_2 \\times \\cdots \\times w_k}%\n _{\\text{product: CRT\ + \ reassembly}},\n\\end{equation}\nwhere $w_i \\in W(x, p_i)$ is a per-prime witness\n(Proposition~\\ref{prop:crt-witness-decomposition}).\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 156\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch58-the-computational-bi-square.tex`\ + \ lines 125-140\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.CompBiSquare`\n\ + - Name: `product_meet_collapse_check`\n\n## Dependencies\n\n- Canonical: III.D56, III.T31, III.P22\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At E₂, meet (∧ = finding a witness) IS product (× = constructing the composite). Search = construction. This + is the computational content of the bi-square pasting. Proof via Interface Width Principle + CRT decomposition. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001625.md b/_corpus_v3_items/cid001625.md new file mode 100644 index 000000000..7c9a5073d --- /dev/null +++ b/_corpus_v3_items/cid001625.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001625/ +title: τ-Admissibility Collapse +cid: cid001625 +primary_alias: THM0168 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'τ-P_adm = τ-NP_adm. For every τ-admissible NP problem, CRT-structured witness search runs in polynomial time. Three-step + proof: (i) constant-width Cook-Levin tableau, (ii) polynomial CRT refinement, (iii) Product-Meet Collapse.' +aliases: +- THM0168 +- III.T33 +- admissibility-collapse +- thm:tau-admissibility-collapse +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005790 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 58 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**τ-Admissibility Collapse**\n\nτ-P_adm = τ-NP_adm. For every τ-admissible NP problem, CRT-structured witness\ + \ search runs in polynomial time. Three-step proof: (i) constant-width Cook-Levin tableau, (ii) polynomial CRT refinement,\ + \ (iii) Product-Meet Collapse.\n\n# τ-Admissibility Collapse\n\n## Summary\n\nτ-P_adm = τ-NP_adm. For every τ-admissible\ + \ NP problem, CRT-structured witness search runs in polynomial time. Three-step proof: (i) constant-width Cook-Levin tableau,\ + \ (ii) polynomial CRT refinement, (iii) Product-Meet Collapse.\n\n## Statement\n\n```latex\n\\label{thm:tau-admissibility-collapse}\n\ + For every $\\tau$-admissible NP problem,\n\\begin{equation}\\label{eq:ch58-collapse}\n \\tau\\text{-}P_{\\mathrm{adm}}\n\ + \ \\;=\\;\n \\tau\\text{-}NP_{\\mathrm{adm}}.\n\\end{equation}\nIf $\\Pi$ is an NP problem whose TTM verifier has\ + \ finite interface\nwidth $W(V) = k_0$, then witness search runs in time polynomial\nin the primorial depth $k$.\n```\n\ + \n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\ + \n- Registry source: `book-03.jsonl` line 157\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch58-the-computational-bi-square.tex`\ + \ lines 178-189\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.CompBiSquare`\n\ + - Name: `tau_admissibility_collapse_check`\n\n## Dependencies\n\n- Canonical: III.D51, III.T31, III.D55, III.P22, III.P23,\ + \ III.T32, III.D52\n\n## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated\ + \ by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'τ-P_adm = τ-NP_adm. For every τ-admissible NP problem, CRT-structured witness search runs in polynomial time. + Three-step proof: (i) constant-width Cook-Levin tableau, (ii) polynomial CRT refinement, (iii) Product-Meet Collapse.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001626.md b/_corpus_v3_items/cid001626.md new file mode 100644 index 000000000..891fc0107 --- /dev/null +++ b/_corpus_v3_items/cid001626.md @@ -0,0 +1,83 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001626/ +title: No Barrier Theorem +cid: cid001626 +primary_alias: THM0169 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'At E₂, no encoding gap exists between external and internal computation. TTM τ-Nativity means programs ARE τ-addresses: + code = data, no translation layer. The 1st Edition''s Representation Barrier was a category error (asking E₂ question with + E₀ tools).' +aliases: +- THM0169 +- III.T34 +- no-barrier-theorem +- thm:no-barrier +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005791 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 59 (Part VII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**No Barrier Theorem**\n\nAt E₂, no encoding gap exists between external and internal computation. TTM τ-Nativity\ + \ means programs ARE τ-addresses: code = data, no translation layer. The 1st Edition's Representation Barrier was a category\ + \ error (asking E₂ question with E₀ tools).\n\n# No Barrier Theorem\n\n## Summary\n\nAt E₂, no encoding gap exists between\ + \ external and internal computation. TTM τ-Nativity means programs ARE τ-addresses: code = data, no translation layer.\ + \ The 1st Edition's Representation Barrier was a category error (asking E₂ question with E₀ tools).\n\n## Statement\n\n\ + ```latex\n\\label{thm:no-barrier}\nLet $\\Pi$ be a $\\tau$-admissible NP problem with TTM verifier $V$\nof interface width\ + \ $k_0$\n(Definition~\\ref{def:tau-admissibility}).\nAt $\\Elayer{2}$, the self-referential structure of the $\\tau$-Tower\n\ + Machine (Theorem~\\ref{thm:ttm-tau-nativity}) eliminates the\nencoding gap:\n\\begin{enumerate}\n\\item[(i)] Programs,\ + \ data, and decoders are all $\\tau$-addresses\n in $\\hat{\\mathbb{Z}}_{\\T}$. No encoding map\n $E \\colon\ + \ \\{0,1\\}^* \\to \\mathrm{Addr}(\\tau)$ is needed.\n\\item[(ii)] Faithfulness, reduction-preservation, and\n witness-completeness\ + \ are satisfied by the identity on\n $\\hat{\\mathbb{Z}}_{\\T}$.\n\\item[(iii)] The $\\tau$-Admissibility Collapse\n\ + \ (Theorem~\\ref{thm:tau-admissibility-collapse}) holds without\n encoding overhead:\n $\\tau\\text{-}P_{\\\ + mathrm{adm}} = \\tau\\text{-}NP_{\\mathrm{adm}}$\n is a theorem about $\\tau$-addresses acting on $\\tau$-addresses.\n\ + \\end{enumerate}\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot\ + \ run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 158\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch59-why-there-is-no-barrier.tex`\ + \ lines 140-161\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Computation.CompBiSquare`\n\ + - Name: `no_barrier_check`\n\n## Dependencies\n\n- Canonical: III.T33, III.D56, III.D49, III.D50, III.T30\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'At E₂, no encoding gap exists between external and internal computation. TTM τ-Nativity means programs ARE τ-addresses: + code = data, no translation layer. The 1st Edition''s Representation Barrier was a category error (asking E₂ question wit' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001627.md b/_corpus_v3_items/cid001627.md new file mode 100644 index 000000000..31e66563c --- /dev/null +++ b/_corpus_v3_items/cid001627.md @@ -0,0 +1,159 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001627/ +title: BSD Coherence Theorem +cid: cid001627 +primary_alias: THM0170 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group +aliases: +- THM0170 +- III.T35 +- bsd-coherence-theorem +- thm:bsd-coherence-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005691 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 47 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**BSD Coherence Theorem** + + + For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group + + + # BSD Coherence Theorem + + + ## Summary + + + For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group + + + ## Statement + + + ```latex + + \label{thm:bsd-coherence-theorem} + + For every $\tau$-admissible elliptic datum $\mathcal{E}$ with proto-code structure (Definition~\ref{def:proto-code}, Ch.~46), + the BSD functional $\operatorname{BSD}_{\T}(k)$ (Definition~\ref{def:bsd-functional}, Ch.~46) stabilizes at finite primorial + depth, and its stable value equals the rank of the $\tau$-rational point group: + + \begin{equation} + + \exists\, k_0 \in \mathbb{N} : \quad + + \operatorname{BSD}_{\T}(k) = r_\infty \quad \text{for all } k \geq k_0, + + \label{eq:ch47-bsd-coherence} + + \end{equation} + + where $k_0 = \max(k_r, k_L)$ and $r_\infty = \lim_{k \to \infty} r(k)$ is the $\tau$-rank. + + In particular, + + \begin{equation} + + \operatorname{rk}_{\T}(\mathcal{E}) \;=\; \operatorname{ord}_{s=1} L_{\T}(\mathcal{E}, s). + + \label{eq:ch47-bsd-statement} + + \end{equation} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 128 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch47-the-bsd-coherence-theorem.tex` + lines 80-95 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.BSD` + + - Name: `bsd_coherence_check` + + + ## Dependencies + + + - Canonical: III.D58, III.D60, III.D62, III.P25, III.P26 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: For τ-admissible elliptic data, BSD_τ(k) stabilizes and equals the rank of the τ-rational point group +noindex: false +--- + diff --git a/_corpus_v3_items/cid001628.md b/_corpus_v3_items/cid001628.md new file mode 100644 index 000000000..6a000cc65 --- /dev/null +++ b/_corpus_v3_items/cid001628.md @@ -0,0 +1,161 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001628/ +title: Functoriality Theorem +cid: cid001628 +primary_alias: THM0171 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'For every sector morphism f: S₁→S₂, the induced map on boundary characters commutes with spectral decomposition' +aliases: +- THM0171 +- III.T36 +- functoriality-theorem +- thm:functoriality-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005700 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 49 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Functoriality Theorem** + + + For every sector morphism f: S₁→S₂, the induced map on boundary characters commutes with spectral decomposition + + + # Functoriality Theorem + + + ## Summary + + + For every sector morphism f: S₁→S₂, the induced map on boundary characters commutes with spectral decomposition + + + ## Statement + + + ```latex + + \label{thm:functoriality-theorem} + + Let $S_1, S_2 \in \{A, B, C, D\}$ be primitive sectors and let $f\colon S_1 \to S_2$ be a tower-coherent sector morphism + in $\operatorname{Cat}_{\T}(\Elayer{1})$. Then the diagram + + \begin{equation} + + \begin{tikzcd} + + \operatorname{Bdy}_{S_1} \arrow[r, "f^{\mathrm{bdy}}"] \arrow[d, "\operatorname{AG}_{S_1}"''] & \operatorname{Bdy}_{S_2} + \arrow[d, "\operatorname{AG}_{S_2}"] \\ + + \operatorname{Spec}_{S_1} \arrow[r, "f^{\mathrm{spec}}"''] & \operatorname{Spec}_{S_2} + + \end{tikzcd} + + \label{eq:ch49-functoriality-theorem-diagram} + + \end{equation} + + commutes. That is, for every boundary character $\beta \in \operatorname{Bdy}_{S_1}$, + + \begin{equation} + + \operatorname{AG}_{S_2}\bigl(f^{\mathrm{bdy}}(\beta)\bigr) \;=\; f^{\mathrm{spec}}\bigl(\operatorname{AG}_{S_1}(\beta)\bigr). + + \label{eq:ch49-functoriality-equation} + + \end{equation} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 134 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch49-functoriality-as-diagram-commutativity.tex` + lines 93-109 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.Langlands` + + - Name: `functoriality_check` + + + ## Dependencies + + + - Canonical: III.D57, III.D63, III.D64, III.P28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'For every sector morphism f: S₁→S₂, the induced map on boundary characters commutes with spectral decomposition' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001629.md b/_corpus_v3_items/cid001629.md new file mode 100644 index 000000000..69d4793e5 --- /dev/null +++ b/_corpus_v3_items/cid001629.md @@ -0,0 +1,179 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001629/ +title: Base Change-Transfer Naturality +cid: cid001629 +primary_alias: THM0172 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Base change (Enr₀₁ on sector morphisms) and transfer (defect functional between sectors) are natural transformations + on the enriched bi-square +aliases: +- THM0172 +- III.T37 +- base-change-transfer-naturality +- thm:base-change-transfer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005701 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 49 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Base Change-Transfer Naturality** + + + Base change (Enr₀₁ on sector morphisms) and transfer (defect functional between sectors) are natural transformations on + the enriched bi-square + + + # Base Change-Transfer Naturality + + + ## Summary + + + Base change (Enr₀₁ on sector morphisms) and transfer (defect functional between sectors) are natural transformations on + the enriched bi-square + + + ## Statement + + + ```latex + + \label{thm:base-change-transfer} + + Let $f\colon S_1 \to S_2$ be a tower-coherent sector morphism in $\operatorname{Cat}_{\T}(\Elayer{1})$. + + \begin{enumerate} + + \item\emph{(Base change.)} The base-change map $\operatorname{BC}_f\colon \operatorname{Bdy}_{S_1}(\Elayer{0}) \to \operatorname{Bdy}_{S_2}(\Elayer{1})$ + defined by $\operatorname{BC}_f(\beta_0) = f^{\mathrm{bdy}}(\operatorname{Enr}_{01}(\beta_0))$ is natural with respect + to the enrichment tower: for every tower-coherent map $g\colon S_0 \to S_1$ at $\Elayer{0}$, the diagram + + \begin{equation} + + \begin{tikzcd} + + \operatorname{Bdy}_{S_0}(\Elayer{0}) \arrow[r, "g^{\mathrm{bdy}}"] \arrow[d, "\operatorname{BC}_{f \circ g}"''] & \operatorname{Bdy}_{S_1}(\Elayer{0}) + \arrow[d, "\operatorname{BC}_f"] \\ + + \operatorname{Bdy}_{S_2}(\Elayer{1}) \arrow[r, equal] & \operatorname{Bdy}_{S_2}(\Elayer{1}) + + \end{tikzcd} + + \label{eq:ch49-base-change-naturality} + + \end{equation} + + commutes, i.e., $\operatorname{BC}_f \circ g^{\mathrm{bdy}} = \operatorname{BC}_{f \circ g}$. + + + \item\emph{(Transfer.)} The transfer map $\operatorname{Trf}_f\colon \Delta_{S_1}(\cdot, n) \to \Delta_{S_2}(\cdot, n)$ + is natural with respect to the defect tower: for every primorial depth~$n$ and every tower-coherent map $g\colon S_0 \to + S_1$, + + \begin{equation} + + \operatorname{Trf}_f \circ \operatorname{Trf}_g \;=\; \operatorname{Trf}_{f \circ g}. + + \label{eq:ch49-transfer-functoriality} + + \end{equation} + + Moreover, transfer is contractive: $\Delta_{S_2}(f(X), n) \leq \Delta_{S_1}(X, n)$ for all~$n$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 135 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch49-functoriality-as-diagram-commutativity.tex` + lines 169-191 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.Langlands` + + - Name: `base_change_check` + + + ## Dependencies + + + - Canonical: III.T36, III.D57 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Base change (Enr₀₁ on sector morphisms) and transfer (defect functional between sectors) are natural transformations + on the enriched bi-square +noindex: false +--- + diff --git a/_corpus_v3_items/cid001630.md b/_corpus_v3_items/cid001630.md new file mode 100644 index 000000000..19b4a5737 --- /dev/null +++ b/_corpus_v3_items/cid001630.md @@ -0,0 +1,160 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001630/ +title: Finite Factorization Pasting +cid: cid001630 +primary_alias: THM0173 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every E₁ object factors through finitely many primitive sector components; the E₁ content of α_p ∧ α_q = α_{p×q} +aliases: +- THM0173 +- III.T38 +- finite-factorization-pasting +- thm:finite-factorization-pasting +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005697 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 50 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Finite Factorization Pasting** + + + Every E₁ object factors through finitely many primitive sector components; the E₁ content of α_p ∧ α_q = α_{p×q} + + + # Finite Factorization Pasting + + + ## Summary + + + Every E₁ object factors through finitely many primitive sector components; the E₁ content of α_p ∧ α_q = α_{p×q} + + + ## Statement + + + ```latex + + \label{thm:finite-factorization-pasting} + + Let $X$ be an $\Elayer{1}$ object in $\operatorname{Cat}_{\T}(\Elayer{1})$ with NF address in the primorial tower at depth~$k$. Then + $X$ factors through finitely many primitive sector components: + + \begin{equation} + + \operatorname{Enr}_{01}(X) \;\cong\; \bigoplus_{S \in \{A, B, C, D\}} X_S, + + \label{eq:ch50-finite-factorization} + + \end{equation} + + where each $X_S = \pi_S \circ \operatorname{Enr}_{01}(X)$ is the sector-$S$ component. Moreover, the factorization is + compatible with both the tower structure and the spectral structure: + + \begin{enumerate} + + \item\emph{(Tower.)} The restriction $\operatorname{res}_k(X_S) = (\operatorname{res}_k X)_S$ for every sector~$S$ and + every primorial depth~$k$: the sector decomposition commutes with tower restriction. + + \item\emph{(Spectral.)} The automorphic--Galois duality preserves the factorization: $\operatorname{AG}(\operatorname{Enr}_{01}(X)) + = \prod_S \operatorname{AG}_S(X_S)$, where the product on the right runs over the primitive sectors. + + \item\emph{(Finiteness.)} The number of non-trivial sector components is bounded: $|\{S : X_S \neq 0\}| \leq 4$, with + equality when $X$ has non-trivial projection onto every primitive sector. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 137 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch50-the-enriched-bi-square.tex` + lines 116-130 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.EnrichedBiSquare` + + - Name: `finite_factorization_check` + + + ## Dependencies + + + - Canonical: III.D65 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every E₁ object factors through finitely many primitive sector components; the E₁ content of α_p ∧ α_q = α_{p×q} +noindex: false +--- + diff --git a/_corpus_v3_items/cid001631.md b/_corpus_v3_items/cid001631.md new file mode 100644 index 000000000..a2659971f --- /dev/null +++ b/_corpus_v3_items/cid001631.md @@ -0,0 +1,87 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001631/ +title: Enriched Bi-Square Comparison +cid: cid001631 +primary_alias: THM0174 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The enriched bi-square has identical shape and structural maps as the algebraic (I.T41) and topological (II.T49) + bi-squares +aliases: +- THM0174 +- III.T39 +- enriched-bi-square-comparison +- thm:enriched-bi-square-comparison +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005698 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 50 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Enriched Bi-Square Comparison**\n\nThe enriched bi-square has identical shape and structural maps as the algebraic\ + \ (I.T41) and topological (II.T49) bi-squares\n\n# Enriched Bi-Square Comparison\n\n## Summary\n\nThe enriched bi-square\ + \ has identical shape and structural maps as the algebraic (I.T41) and topological (II.T49) bi-squares\n\n## Statement\n\ + \n```latex\n\\label{thm:enriched-bi-square-comparison}\nThe four bi-squares in the scaling chain---algebraic (I.T41),\ + \ topological (II.T49), enriched (Definition~\\ref{def:enriched-bi-square}), and computational (Definition~\\ref{def:computational-bi-square},\ + \ Ch.~58)---are structurally identical in the following sense:\n\\begin{enumerate}\n\\item[\\emph{(Shape.)}] Each is a\ + \ $2 \\times 3$ pasted diagram with the same combinatorial shape: left square = tower coherence, right square = spectral\ + \ naturality, pasting = the defining constraint.\n\\item[\\emph{(Maps.)}] The structural maps at each level are the images\ + \ of the algebraic maps under the successive enrichment functors: $\\operatorname{Enr}_{01}$ from $\\Elayer{0}$ to $\\\ + Elayer{1}$, and $\\operatorname{Enr}_{12}$ from $\\Elayer{1}$ to $\\Elayer{2}$. No new structural maps are introduced;\ + \ the existing maps are transported.\n\\item[\\emph{(Pasting.)}] The pasting identity upgrades at each level while preserving\ + \ its algebraic core:\n\\begin{center}\n\\begin{tabular}{llll}\n\\toprule\n\\textbf{Level} & \\textbf{Left Square} & \\\ + textbf{Right Square} & \\textbf{Pasting} \\\\\n\\midrule\n$\\Elayer{0}$ (I.T41) & Tower coherence & Spectral naturality\ + \ $\\chi_\\pm$ & $\\alpha_p \\wedge \\alpha_q = \\alpha_{p \\times q}$ \\\\\n\\addlinespace\n$\\Elayer{0} \\to \\Elayer{1}$\ + \ & Holomorphic & Boundary values & $\\mathcal{O}(\\tau^3)$ \\\\\n(II.T49) & extension & on $\\Lemniscate$ & $\\cong A_{\\\ + mathrm{spec}}(\\Lemniscate)$ \\\\\n\\addlinespace\n$\\Elayer{1}+$ & Sector tower & Langlands & Finite \\\\\n(this chapter)\ + \ & coherence & functoriality & factorization \\\\\n\\addlinespace\n$\\Elayer{2}$ (Ch.~58) & TTM execution & CRT witnesses\ + \ & Product-Meet \\\\\n & & & Collapse \\\\\n\\bottomrule\n\\end{tabular}\n\\end{center}\n\\end{enumerate}\n```\n\n##\ + \ Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source Context\n\ + \n- Registry source: `book-03.jsonl` line 138\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch50-the-enriched-bi-square.tex`\ + \ lines 165-192\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Arithmetic.EnrichedBiSquare`\n\ + - Name: `bisquare_comparison_check`\n\n## Dependencies\n\n- Canonical: III.D65, III.T38\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The enriched bi-square has identical shape and structural maps as the algebraic (I.T41) and topological (II.T49) + bi-squares +noindex: false +--- + diff --git a/_corpus_v3_items/cid001632.md b/_corpus_v3_items/cid001632.md new file mode 100644 index 000000000..691a53e0b --- /dev/null +++ b/_corpus_v3_items/cid001632.md @@ -0,0 +1,218 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001632/ +title: Enrichment Tower Assembly +cid: cid001632 +primary_alias: THM0175 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The tower E₀ ⊊ E₁ ⊊ E₂ is assembled with coherent bi-square scaling chain and complete Millennium coverage +aliases: +- THM0175 +- III.T40 +- enrichment-tower-assembly +- thm:enrichment-tower-assembly +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005708 +- predicate: formalized_by + object: cid005709 +- predicate: formalized_by + object: cid005710 +- predicate: formalized_by + object: cid005711 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 6, Chapter 51 (Part VI) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Enrichment Tower Assembly** + + + The tower E₀ ⊊ E₁ ⊊ E₂ is assembled with coherent bi-square scaling chain and complete Millennium coverage + + + # Enrichment Tower Assembly + + + ## Summary + + + The tower E₀ ⊊ E₁ ⊊ E₂ is assembled with coherent bi-square scaling chain and complete Millennium coverage + + + ## Statement + + + ```latex + + \label{thm:enrichment-tower-assembly} + + The enrichment tower $\Elayer{0} \subsetneq \Elayer{1} \subsetneq \Elayer{2}$ satisfies the following properties. + + \begin{enumerate} + + \item\emph{(Strict inclusion.)} Each level adds irreducible structure that the previous level cannot express: + + \begin{itemize} + + \item $\Elayer{0} \subsetneq \Elayer{1}$: the split-complex dynamics, sector-coupled processes, and defect functionals + of $\Elayer{1}$ have no $\Elayer{0}$ counterpart. In particular, the Yang--Mills gap $\Gamma^*_s > 0$ is a strictly $\Elayer{1}$ + phenomenon: it requires NF discreteness in the strong sector, which is undefined at $\Elayer{0}$. + + \item $\Elayer{1} \subsetneq \Elayer{2}$: the self-referential code structure of $\Elayer{2}$ (Definition~\ref{def:e2-computational-agent}, + Ch.~54) is not expressible at $\Elayer{1}$. Proto-codes exist at $\Elayer{1}$ (Definition~\ref{def:proto-code}, Ch.~46), + but they lack decoders: they label data without verifying themselves. + + \end{itemize} + + + \item\emph{(Coherent scaling.)} The three bi-squares form a scaling chain: + + \begin{equation} + + \underbracket{\text{I.T41}}_{\Elayer{0}\text{ algebraic}} + + \;\longrightarrow\; + + \underbracket{\text{III.D65}}_{\Elayer{1}\text{ enriched}} + + \;\longrightarrow\; + + \underbracket{\text{III.D56}}_{\Elayer{2}\text{ computational}}, + + \label{eq:ch51-bisquare-scaling} + + \end{equation} + + with the same diagram shape (pasted bi-square with $(\times, \wedge)$-axes) but richer objects at each step. The enrichment + functors $\operatorname{Enr}_{01}$ and $\operatorname{Enr}_{12}$ preserve the bi-square structure: they map vertices to + vertices, edges to edges, and commuting squares to commuting squares. + + + \item\emph{(Millennium coverage.)} The eight Millennium Problems partition across the three levels: + + \begin{center} + + \begin{adjustbox}{max width=0.95\linewidth} + + \begin{tabular}{p{0.1\linewidth}p{0.28\linewidth}p{0.28\linewidth}p{0.28\linewidth}} + + \toprule + + \textbf{Level} & \textbf{Bi-Square} & \textbf{Millennium Results} & \textbf{Mutual Determination} \\ + + \midrule + + $\Elayer{0}$ & I.T41 (algebraic) & RH, Poincar\''e & boundary $\leftrightarrow$ spectral \\ + + \addlinespace + + $\Elayer{1}$ & III.D65 (enriched) & NS, YM, Hodge, BSD, Langlands & sector dynamics \\ + + \addlinespace + + $\Elayer{2}$ & III.D56 (computational) & P vs NP & search $=$ construction \\ + + \bottomrule + + \end{tabular} + + \end{adjustbox} + + \end{center} + + No Millennium Problem straddles two levels: each lives at exactly one enrichment level, and the scope labels record which + level is required. (BSD and Langlands bridge to~$\Elayer{2}$ via their classical formulations; see Chapter~33.) + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 139 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part06/ch51-the-tower-closes.tex` lines + 58-98 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.TowerAssembly` + + - Name: `tower_assembly_check` + + + ## Dependencies + + + - Canonical: III.P24, III.P26, III.T35, III.P28, III.T36, III.T37, III.D65, III.T38, III.T39, III.D61 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The tower E₀ ⊊ E₁ ⊊ E₂ is assembled with coherent bi-square scaling chain and complete Millennium coverage +noindex: false +--- + diff --git a/_corpus_v3_items/cid001633.md b/_corpus_v3_items/cid001633.md new file mode 100644 index 000000000..f15de4d63 --- /dev/null +++ b/_corpus_v3_items/cid001633.md @@ -0,0 +1,215 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001633/ +title: Hinge Theorem +cid: cid001633 +primary_alias: THM0176 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every result in Books IV-VII is a sector instantiation of Book III''s enrichment structure. The seven-book architecture + is DERIVED, not postulated. Four downstream books = four enrichment-level instantiations: IV (fiber T² at E₁), V (base τ¹ + at E₁), VI (E₂), VII (E₃).' +aliases: +- THM0176 +- III.T41 +- hinge-theorem +- thm:hinge-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005887 +- predicate: formalized_by + object: cid005889 +- predicate: formalized_by + object: cid005890 +- predicate: formalized_by + object: cid005891 +- predicate: formalized_by + object: cid005899 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 61 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Hinge Theorem** + + + Every result in Books IV-VII is a sector instantiation of Book III''s enrichment structure. The seven-book architecture + is DERIVED, not postulated. Four downstream books = four enrichment-level instantiations: IV (fiber T² at E₁), V (base + τ¹ at E₁), VI (E₂), VII (E₃). + + + # Hinge Theorem + + + ## Summary + + + Every result in Books IV-VII is a sector instantiation of Book III''s enrichment structure. The seven-book architecture + is DERIVED, not postulated. Four downstream books = four enrichment-level instantiations: IV (fiber T² at E₁), V (base + τ¹ at E₁), VI (E₂), VII (E₃). + + + ## Statement + + + ```latex + + \label{thm:hinge-theorem} + + Let $\operatorname{Enr}\colon \operatorname{Cat}_{\T}(\Elayer{k}) \to \operatorname{Cat}_{\T}(\Elayer{k+1})$ be the self-enrichment + functor, and let $\{A, B, C, D\}$ be the four primitive sectors of the $4{+}1$ decomposition. The content of Books~IV--VII + decomposes as follows. + + \begin{enumerate} + + \item\emph{(Book~IV: Microcosm.)} + + The fiber $T^2$ of the fibration $\tau^3 = \tau^1 \times_f T^2$ at enrichment level $\Elayer{1}$, restricted to the $D$-sector, + produces the categorical microcosm: + + \[ + + \textup{Book~IV} \;=\; \operatorname{Enr}^1\big|_{D}\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr)\big|_{T^2}. + + \] + + The $D$-sector characters encode gravity and quantum mechanics on the fiber. + + + \item\emph{(Book~V: Macrocosm.)} + + The base $\tau^1$ at enrichment level $\Elayer{1}$, restricted to sectors $A$, $B$, $C$, produces the categorical macrocosm: + + \[ + + \textup{Book~V} \;=\; \operatorname{Enr}^1\big|_{A \cup B \cup C}\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr)\big|_{\tau^1}. + + \] + + The three primitive sectors encode the three gauge forces: weak ($A$), electromagnetic ($B$), strong ($C$). + + + \item\emph{(Book~VI: Life.)} + + Enrichment level $\Elayer{2}$, applied to the full sector template, produces the categorical life layer: + + \[ + + \textup{Book~VI} \;=\; \operatorname{Enr}^2\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr). + + \] + + Proto-codes (III.D61, Ch.~46) provide the carrier; operational closure provides the invariant. + + + \item\emph{(Book~VII: Metaphysics.)} + + Enrichment level $\Elayer{3}$, saturated by Theorem~\ref{thm:saturation-e3} (Ch.~7), produces the categorical metaphysics + layer: + + \[ + + \textup{Book~VII} \;=\; \operatorname{Enr}^3\bigl(\operatorname{Cat}_{\T}(\Elayer{0})\bigr). + + \] + + Self-models provide the carrier; saturation provides the terminal invariant. + + + \item\emph{(Derivation.)} + + The seven-book architecture is derived: the number of books equals the number of irreducible instantiations of the enrichment + tower with sector restrictions, which is~$7$. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 163 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch61-the-hinge-theorem.tex` lines + 60-96 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Hinge.HingeTheorem` + + - Name: `hinge_theorem_check` + + + ## Dependencies + + + - Canonical: III.D66, III.T01, III.T03, III.T08, III.T40, III.P30 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every result in Books IV-VII is a sector instantiation of Book III''s enrichment structure. The seven-book architecture + is DERIVED, not postulated. Four downstream books = four enrichment-level instantiations: IV (fiber T² at E₁), V (base τ¹' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001634.md b/_corpus_v3_items/cid001634.md new file mode 100644 index 000000000..6f3d0c292 --- /dev/null +++ b/_corpus_v3_items/cid001634.md @@ -0,0 +1,174 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001634/ +title: No Knobs Theorem +cid: cid001634 +primary_alias: THM0177 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All 10 + entries (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework. +aliases: +- THM0177 +- III.T42 +- no-knobs-theorem +- thm:no-knobs-theorem +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005888 +- predicate: formalized_by + object: cid005892 +- predicate: formalized_by + object: cid005893 +- predicate: formalized_by + object: cid005894 +- predicate: formalized_by + object: cid005895 +- predicate: formalized_by + object: cid005896 +- predicate: formalized_by + object: cid005897 +- predicate: formalized_by + object: cid005898 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 63 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**No Knobs Theorem** + + + Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All 10 entries + (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework. + + + # No Knobs Theorem + + + ## Summary + + + Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All 10 entries + (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework. + + + ## Statement + + + ```latex + + \label{thm:no-knobs-theorem} + + Let $\kappa(S, S''; n)$ denote the coupling between sectors $S, S'' \in \{A, B, C, D\}$ at primorial depth~$n$. Then: + + \begin{enumerate} + + \item\emph{(Determination.)} + + Every coupling $\kappa(S, S''; n)$ is a rational function of $\iota_{\tau} = 2/(\pi + e)$, with coefficients in $\mathbb{Z}$ + and denominator non-vanishing on $(0,1)$. + + + \item\emph{(Depth assignment.)} + + The primorial depth $n$ at which $\kappa(S, S''; n)$ is first non-trivial is uniquely determined by the sector addresses + of $S$ and $S''$ via the spectral trichotomy. + + + \item\emph{(No free parameters.)} + + The enrichment tower $\Elayer{0} \subsetneq \Elayer{1} \subsetneq \Elayer{2} \subsetneq \Elayer{3}$ together with the + $4{+}1$ sector template determines all ten couplings (four self, six inter-sector) without any adjustable constant. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 167 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part07/ch63-no-knobs-and-one-diagram.tex` + lines 76-90 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Hinge.HingeTheorem` + + - Name: `no_knobs_check` + + + ## Dependencies + + + - Canonical: III.T41, III.D13, III.D65 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every inter-sector coupling is a rational function of ι_τ = 2/(π+e) evaluated at specific primorial depth. All + 10 entries (6 inter-sector + 4 self-couplings) are canonically determined. No free parameters exist in the framework. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001635.md b/_corpus_v3_items/cid001635.md new file mode 100644 index 000000000..abdce4381 --- /dev/null +++ b/_corpus_v3_items/cid001635.md @@ -0,0 +1,88 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001635/ +title: Move-Bridge Correspondence +cid: cid001635 +primary_alias: THM0178 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: The bridge functor is faithful on the complement of the five forbidden moves and degenerates precisely at them. Each + move corresponds to a specific structural feature the bridge cannot preserve. +aliases: +- THM0178 +- III.T43 +- move-bridge-correspondence +- thm:move-bridge-correspondence +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005741 +- predicate: formalized_by + object: cid005742 +- predicate: formalized_by + object: cid005747 +- predicate: formalized_by + object: cid005748 +- predicate: formalized_by + object: cid005749 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 65 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Move-Bridge Correspondence**\n\nThe bridge functor is faithful on the complement of the five forbidden moves\ + \ and degenerates precisely at them. Each move corresponds to a specific structural feature the bridge cannot preserve.\n\ + \n# Move-Bridge Correspondence\n\n## Summary\n\nThe bridge functor is faithful on the complement of the five forbidden\ + \ moves and degenerates precisely at them. Each move corresponds to a specific structural feature the bridge cannot preserve.\n\ + \n## Statement\n\n```latex\n\\label{thm:move-bridge-correspondence}\nLet $F\\colon \\operatorname{Cat}_{\\T}(\\Elayer{2})\n\ + \\to \\operatorname{Mod}(\\mathrm{ZFC})$ be the bridge functor\n(Definition~\\ref{def:zfc-as-e2-vm}, Ch.~64), and let\n\ + $\\{M_{1}, \\ldots, M_{5}\\}$ be the five forbidden moves\n(Definition~\\ref{def:five-forbidden-moves}).\n\\begin{enumerate}\n\ + \\item\\emph{(Faithfulness.)}\n$F$ restricted to $\\operatorname{Cat}_{\\T}(\\Elayer{2})\n \\setminus \\bigcup_{i} \\\ + operatorname{ker}(M_{i})$ is faithful.\n\n\\item\\emph{(Degeneration.)}\nFor each $i$, there exist $X_{i} \\neq Y_{i}$\ + \ with\n$F(X_{i}) \\cong F(Y_{i})$ in $\\operatorname{Mod}(\\mathrm{ZFC})$.\n\n\\item\\emph{(Completeness.)}\nEvery identification\ + \ $F(X) \\cong F(Y)$ with $X \\neq Y$ factors\nthrough $\\operatorname{ker}(M_{i})$ for some~$i$.\n\\end{enumerate}\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 173\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch65-the-five-forbidden-moves.tex`\ + \ lines 187-208\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.ForbiddenMoves`\n\ + - Name: `move_bridge_check`\n\n## Dependencies\n\n- Canonical: III.D69, III.D67, III.T41\n\n## Related Results\n\n_Generated\ + \ by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\ + \n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: The bridge functor is faithful on the complement of the five forbidden moves and degenerates precisely at them. + Each move corresponds to a specific structural feature the bridge cannot preserve. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001636.md b/_corpus_v3_items/cid001636.md new file mode 100644 index 000000000..b88c68e6d --- /dev/null +++ b/_corpus_v3_items/cid001636.md @@ -0,0 +1,96 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001636/ +title: Incompleteness as VM Boundary +cid: cid001636 +primary_alias: THM0179 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Gödel I, Gödel II, and the Halting Problem are E₂ boundary phenomena. They arise because E₂ has self-reference but + not self-modelling (E₃). The VM boundary marks where E₂ ends and E₃ begins. +aliases: +- THM0179 +- III.T44 +- incompleteness-as-vm-boundary +- thm:incompleteness-as-vm-boundary +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005751 +- predicate: formalized_by + object: cid005752 +- predicate: formalized_by + object: cid005753 +- predicate: formalized_by + object: cid005754 +- predicate: formalized_by + object: cid005755 +- predicate: formalized_by + object: cid005756 +- predicate: formalized_by + object: cid005757 +- predicate: formalized_by + object: cid005758 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 66 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Incompleteness as VM Boundary**\n\nGödel I, Gödel II, and the Halting Problem are E₂ boundary phenomena. They\ + \ arise because E₂ has self-reference but not self-modelling (E₃). The VM boundary marks where E₂ ends and E₃ begins.\n\ + \n# Incompleteness as VM Boundary\n\n## Summary\n\nGödel I, Gödel II, and the Halting Problem are E₂ boundary phenomena.\ + \ They arise because E₂ has self-reference but not self-modelling (E₃). The VM boundary marks where E₂ ends and E₃ begins.\n\ + \n## Statement\n\n```latex\n%\n\\label{thm:incompleteness-as-vm-boundary}\n% Depends: III.D08, III.D09, III.D67, III.D68,\ + \ III.D70, III.T03\nThe following phenomena\nare $\\Elayer{2}$ boundary phenomena:\n\\begin{enumerate}\n \\item G\\\ + \"odel's First Incompleteness Theorem:\n the ZFC-VM cannot derive all truths\n about its own code numbers.\n\ + \ \\item G\\\"odel's Second Incompleteness Theorem:\n the ZFC-VM cannot prove its own consistency.\n \\\ + item The Halting Problem:\n no $\\Elayer{2}$ computational agent\n can decide its own halting predicate.\n\ + \\end{enumerate}\nEach arises because the relevant predicate\nis a host-level property\n(Definition~\\ref{def:host-level-property}):\n\ + it quantifies over the totality\nof the VM's execution histories\nand is determinable only from $\\Elayer{3}$.\nThe VM\ + \ boundary --- the structural ceiling of $\\Elayer{2}$ ---\nis the locus of all three phenomena.\nIt marks exactly where\ + \ $\\Elayer{2}$ ends\nand $\\Elayer{3}$ begins.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block\ + \ was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 174\n- Manuscript\ + \ source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch66-goedel-and-the-vm-boundary.tex` lines 326-352\n\ + \n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.Incompleteness`\n-\ + \ Name: `incompleteness_vm_check`\n\n## Dependencies\n\n- Canonical: III.D67, III.D68, III.D70, III.T03\n\n## Related\ + \ Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\ + \n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Gödel I, Gödel II, and the Halting Problem are E₂ boundary phenomena. They arise because E₂ has self-reference + but not self-modelling (E₃). The VM boundary marks where E₂ ends and E₃ begins. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001637.md b/_corpus_v3_items/cid001637.md new file mode 100644 index 000000000..ee8558515 --- /dev/null +++ b/_corpus_v3_items/cid001637.md @@ -0,0 +1,86 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001637/ +title: RH Bridge Three-Layer Structure +cid: cid001637 +primary_alias: THM0180 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The RH bridge has three layers: (1) τ-internal spectral purity on H_L (τ-effective), (2) Connes-Consani Weil positivity + Q_W(g) ≥ 0 (established), (3) identification of τ spectral data with Riemann zeta zeros (conjectural). The gap is precisely + Layer 3.' +aliases: +- THM0180 +- III.T45 +- rh-bridge-three-layer-structure +- thm:rh-bridge-three-layer +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005720 +- predicate: formalized_by + object: cid005722 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 67 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**RH Bridge Three-Layer Structure**\n\nThe RH bridge has three layers: (1) τ-internal spectral purity on H_L\ + \ (τ-effective), (2) Connes-Consani Weil positivity Q_W(g) ≥ 0 (established), (3) identification of τ spectral data with\ + \ Riemann zeta zeros (conjectural). The gap is precisely Layer 3.\n\n# RH Bridge Three-Layer Structure\n\n## Summary\n\ + \nThe RH bridge has three layers: (1) τ-internal spectral purity on H_L (τ-effective), (2) Connes-Consani Weil positivity\ + \ Q_W(g) ≥ 0 (established), (3) identification of τ spectral data with Riemann zeta zeros (conjectural). The gap is precisely\ + \ Layer 3.\n\n## Statement\n\n```latex\n%\n\\label{thm:rh-bridge-three-layer}\n% Depends: III.T19, III.D71, III.T41\n\ + The bridge from $\\T$-internal spectral purity\nto the classical Riemann Hypothesis\ndecomposes into three layers:\n\\\ + begin{enumerate}\n\\item \\textbf{Layer~1} ($\\tau$-effective):\n spectral purity of $H_{\\Lemniscate}$\n forces\ + \ all eigenvalues to be real\n (Theorem~\\ref{def:ch25-spectral-purity}, Ch.~25).\n\\item \\textbf{Layer~2} (established):\n\ + \ Connes--Consani Weil positivity\n $Q_{W}(g) \\geq 0$\n is equivalent to the RH within ZFC.\n\\item \\\ + textbf{Layer~3} (conjectural):\n the identification\n $F\\bigl(\\operatorname{Spec}(H_{\\Lemniscate})\\bigr)\n\ + \ = \\{\\rho : \\zeta(\\rho) = 0\\}$\n requires a model-theoretic bridge\n between $\\T$-spectral data\n\ + \ and classical zeta zeros.\n\\end{enumerate}\nLayers~1 and~2 are secure.\nThe gap is entirely and precisely Layer~3.\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 178\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch67-the-bridge-axiom.tex`\ + \ lines 314-340\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Bridge.BridgeAxiom`\n\ + - Name: `rh_bridge_three_layer`\n\n## Dependencies\n\n- Canonical: III.D71, III.D72, III.T19\n\n## Related Results\n\n\ + _Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision\ + \ Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The RH bridge has three layers: (1) τ-internal spectral purity on H_L (τ-effective), (2) Connes-Consani Weil + positivity Q_W(g) ≥ 0 (established), (3) identification of τ spectral data with Riemann zeta zeros (conjectural). The gap + is precis' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001638.md b/_corpus_v3_items/cid001638.md new file mode 100644 index 000000000..f6eb19676 --- /dev/null +++ b/_corpus_v3_items/cid001638.md @@ -0,0 +1,193 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001638/ +title: Bridge Ledger +cid: cid001638 +primary_alias: THM0181 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Per-problem bridge status: 6 conjectural (RH, NS, YM, Hodge, BSD, Langlands), 1 bridge break (P vs NP), 1 established + (Poincaré). Each uses the three-layer template.' +aliases: +- THM0181 +- III.T46 +- bridge-ledger +- thm:bridge-ledger +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005719 +- predicate: formalized_by + object: cid005723 +- predicate: formalized_by + object: cid005724 +- predicate: formalized_by + object: cid005725 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 68 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bridge Ledger** + + + Per-problem bridge status: 6 conjectural (RH, NS, YM, Hodge, BSD, Langlands), 1 bridge break (P vs NP), 1 established + (Poincaré). Each uses the three-layer template. + + + # Bridge Ledger + + + ## Summary + + + Per-problem bridge status: 6 conjectural (RH, NS, YM, Hodge, BSD, Langlands), 1 bridge break (P vs NP), 1 established + (Poincaré). Each uses the three-layer template. + + + ## Statement + + + ```latex + + \label{thm:bridge-ledger} + + Let $F\colon \operatorname{Cat}_{\T}(\Elayer{2}) + + \to \operatorname{Mod}(\mathrm{ZFC})$ be the bridge functor + + and let $\{M_{1}, \ldots, M_{5}\}$ be the five forbidden moves + + (Definition~\ref{def:five-forbidden-moves}). + + The eight Millennium-scale problems partition as follows. + + \begin{enumerate} + + \item\emph{(Conjectural.)} + + For $\mathrm{RH}$, $\mathrm{NS}$, $\mathrm{YM}$, + + $\mathrm{Hodge}$, $\mathrm{BSD}$, $\mathrm{Langlands}$: + + Layer~1 is proved within~$\T$, Layer~2 is the standard classical + + formulation, and Layer~3 is a conjectural identification. + + + \item\emph{(Bridge breaks.)} + + For P~vs.\ NP: Layer~1 is proved + + ($\T$-$\mathrm{P}_{\mathrm{adm}} = \T$-$\mathrm{NP}_{\mathrm{adm}}$, + + III.T33), Layer~2 is the open conjecture $\mathrm{P} \neq \mathrm{NP}$, + + and Layer~3 degenerates at $M_{1}$, $M_{3}$, $M_{4}$ + + (Theorem~\ref{thm:move-bridge-correspondence}). + + The $\T$-internal collapse does not imply $\mathrm{P} = \mathrm{NP}$. + + + \item\emph{(Established.)} + + For Poincar\''e: Layer~1 is proved within~$\T$ (III.T24), + + Layer~2 is Perelman''s proof (2003), + + and the two proofs are independent. + + No bridge identification is needed. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 179 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch68-the-bridge-ledger.tex` lines + 100-129 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.BridgeAxiom` + + - Name: `bridge_ledger_check` + + + ## Dependencies + + + - Canonical: III.D71, III.T45, III.D69, III.T43 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Per-problem bridge status: 6 conjectural (RH, NS, YM, Hodge, BSD, Langlands), 1 bridge break (P vs NP), 1 established + (Poincaré). Each uses the three-layer template.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001639.md b/_corpus_v3_items/cid001639.md new file mode 100644 index 000000000..46198d273 --- /dev/null +++ b/_corpus_v3_items/cid001639.md @@ -0,0 +1,204 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001639/ +title: Honest Claim Theorem +cid: cid001639 +primary_alias: THM0182 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Three clauses: (i) at the native enrichment level, τ-internal results are unconditional; (ii) bridges are established + (Poincaré), broken (P vs NP), or conjectural (6 others); (iii) the open question is always: does the bridge functor preserve + the relevant structure?' +aliases: +- THM0182 +- III.T47 +- honest-claim-theorem +- thm:honest-claim +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005715 +- predicate: formalized_by + object: cid005716 +- predicate: formalized_by + object: cid005717 +- predicate: formalized_by + object: cid005718 +- predicate: formalized_by + object: cid005726 +- predicate: formalized_by + object: cid005727 +- predicate: formalized_by + object: cid005728 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 69 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Honest Claim Theorem** + + + Three clauses: (i) at the native enrichment level, τ-internal results are unconditional; (ii) bridges are established + (Poincaré), broken (P vs NP), or conjectural (6 others); (iii) the open question is always: does the bridge functor preserve + the relevant structure? + + + # Honest Claim Theorem + + + ## Summary + + + Three clauses: (i) at the native enrichment level, τ-internal results are unconditional; (ii) bridges are established + (Poincaré), broken (P vs NP), or conjectural (6 others); (iii) the open question is always: does the bridge functor preserve + the relevant structure? + + + ## Statement + + + ```latex + + \label{thm:honest-claim} + + Let $F\colon \operatorname{Cat}_{\T}(\Elayer{2}) + + \to \operatorname{Mod}(\mathrm{ZFC})$ be the bridge functor + + (Definition~\ref{def:bridge-axiom}, Ch.~67), let + + $P_{1}, \ldots, P_{8}$ be the eight Millennium Problem statements + + in their orthodox formulations, and for each~$k$ let $T_{k}$ be the + + $\tau$-internal theorem and $I_{k}$ the identification clause + + $F(\text{domain of } T_{k}) = \text{domain of } P_{k}$. Then: + + \begin{enumerate} + + \item[\emph{(i)}] + + \textbf{Unconditional.} + + Each $T_{k}$ is unconditional at its native enrichment level: + + it depends only on seven axioms, five generators, and the enrichment + + tower---no external hypothesis, no adjustable parameter + + (Theorem~\ref{thm:no-knobs-theorem}, Ch.~63). + + + \item[\emph{(ii)}] + + \textbf{Conditional.} + + The bridge to~$P_{k}$ is either + + \emph{established} (Poincar\''e, Remark~\ref{rem:poincare-established}), + + \emph{broken} (P~vs.~NP, Remark~\ref{rem:p-vs-np-bridge-break}), + + or \emph{conjectural} (the remaining six). + + + \item[\emph{(iii)}] + + \textbf{The open question.} + + In every conjectural case the question is the same: + + does $F$ preserve the relevant structure? Formally, does~$I_{k}$ hold? + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 182 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch69-the-honest-claim.tex` lines + 49-79 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.BridgeAxiom` + + - Name: `honest_claim_check` + + + ## Dependencies + + + - Canonical: III.T46, III.T41, III.T42, III.D71 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Three clauses: (i) at the native enrichment level, τ-internal results are unconditional; (ii) bridges are established + (Poincaré), broken (P vs NP), or conjectural (6 others); (iii) the open question is always: does the bridge functor preser' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001640.md b/_corpus_v3_items/cid001640.md new file mode 100644 index 000000000..b3809f34c --- /dev/null +++ b/_corpus_v3_items/cid001640.md @@ -0,0 +1,89 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001640/ +title: Four Paradox Diagnostic +cid: cid001640 +primary_alias: THM0183 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Cantor's diagonal, Russell's paradox, Gödel's sentence, and Turing's halting problem are four instances of the same + E₂→E₃ boundary crossing (III.D75). Each attempts an operation requiring E₃ self-modelling from within E₂. +aliases: +- THM0183 +- III.T48 +- four-paradox-diagnostic +- thm:four-paradox-diagnostic +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005911 +- predicate: formalized_by + object: cid005918 +- predicate: formalized_by + object: cid005919 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 71 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Four Paradox Diagnostic**\n\nCantor's diagonal, Russell's paradox, Gödel's sentence, and Turing's halting problem\ + \ are four instances of the same E₂→E₃ boundary crossing (III.D75). Each attempts an operation requiring E₃ self-modelling\ + \ from within E₂.\n\n# Four Paradox Diagnostic\n\n## Summary\n\nCantor's diagonal, Russell's paradox, Gödel's sentence,\ + \ and Turing's halting problem are four instances of the same E₂→E₃ boundary crossing (III.D75). Each attempts an operation\ + \ requiring E₃ self-modelling from within E₂.\n\n## Statement\n\n```latex\n%\n\\label{thm:four-paradox-diagnostic}\nThe\ + \ following four paradoxes are instances of\n$\\Elayer{2} \\to \\Elayer{3}$ boundary crossings\n(Definition~\\ref{def:e2-e3-boundary-crossing}):\n\ + \\begin{enumerate}\n \\item \\textbf{Cantor's Diagonal.}\n The ``set of all sets'' is an $\\Elayer{2}$ code\ + \ space\n whose cardinality is host-level.\n The diagonal forces self-modelling of size.\n \\item\ + \ \\textbf{Russell's Paradox.}\n $R = \\{x : x \\notin x\\}$\n is a self-referential $\\Elayer{2}$ code\n\ + \ whose self-membership is host-level.\n The definition forces self-modelling of the predicate.\n \ + \ \\item \\textbf{G\\\"odel's Incompleteness.}\n The sentence~$G$\n is a self-referential $\\Elayer{2}$\ + \ code\n whose derivability is host-level.\n The encoding forces self-modelling of inference.\n \\\ + item \\textbf{Turing's Halting Problem.}\n The diagonaliser~$D$\n is a self-referential $\\Elayer{2}$\ + \ code\n whose halting is host-level.\n The diagonalisation forces self-modelling of execution.\n\\\ + end{enumerate}\nIn each case, self-reference ($\\Elayer{2}$) suffices\nto \\emph{pose} the question\nbut not to \\emph{answer}\ + \ it.\nThe answer requires self-modelling ($\\Elayer{3}$).\nThe paradox is the error message generated by the boundary.\n\ + ```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n## Source\ + \ Context\n\n- Registry source: `book-03.jsonl` line 187\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch71-four-paradoxes-as-boundary-crossings.tex`\ + \ lines 202-234\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.ProofTheoryE3`\n\ + - Name: `paradox_resolution_check`\n\n## Dependencies\n\n- Canonical: III.D75, III.D73, III.T44\n\n## Related Results\n\ + \n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n##\ + \ Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Cantor's diagonal, Russell's paradox, Gödel's sentence, and Turing's halting problem are four instances of the + same E₂→E₃ boundary crossing (III.D75). Each attempts an operation requiring E₃ self-modelling from within E₂. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001641.md b/_corpus_v3_items/cid001641.md new file mode 100644 index 000000000..329d45e94 --- /dev/null +++ b/_corpus_v3_items/cid001641.md @@ -0,0 +1,91 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001641/ +title: Applied Saturation +cid: cid001641 +primary_alias: THM0184 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The enrichment ladder E₀ → E₁ → E₂ → E₃ → stop is COMPLETE. Applied to proof theory: meta-metatheory adds nothing. + Applied to ZFC: no meta-metatheory reveals further structure. Applied to Book III: the book is a fixed point of self-reflection.' +aliases: +- THM0184 +- III.T49 +- applied-saturation +- thm:applied-saturation +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005920 +- predicate: formalized_by + object: cid005921 +- predicate: formalized_by + object: cid005923 +- predicate: formalized_by + object: cid005925 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Part X) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Applied Saturation**\n\nThe enrichment ladder E₀ → E₁ → E₂ → E₃ → stop is COMPLETE. Applied to proof theory:\ + \ meta-metatheory adds nothing. Applied to ZFC: no meta-metatheory reveals further structure. Applied to Book III: the\ + \ book is a fixed point of self-reflection.\n\n# Applied Saturation\n\n## Summary\n\nThe enrichment ladder E₀ → E₁ → E₂\ + \ → E₃ → stop is COMPLETE. Applied to proof theory: meta-metatheory adds nothing. Applied to ZFC: no meta-metatheory reveals\ + \ further structure. Applied to Book III: the book is a fixed point of self-reflection.\n\n## Statement\n\n```latex\n\ + %\n\\label{thm:applied-saturation}\nThe enrichment ladder\n\\[\n \\Elayer{0} \\;\\xrightarrow{\\;\\mathcal{F}_{E}\\\ + ;}\n \\Elayer{1} \\;\\xrightarrow{\\;\\mathcal{F}_{E}\\;}\n \\Elayer{2} \\;\\xrightarrow{\\;\\mathcal{F}_{E}\\;}\n\ + \ \\Elayer{3} \\;\\xrightarrow{\\;\\mathcal{F}_{E}\\;}\n \\Elayer{3}\n\\]\nis \\textbf{complete}: each of the first\ + \ three arrows\nproduces irreducible new structure,\nand the fourth arrow is the identity.\nConcretely:\n\\begin{enumerate}\n\ + \ \\item \\textbf{$\\Elayer{0} \\to \\Elayer{1}$}:\n split-complex enrichment of $\\Hom$~spaces;\n \ + \ physics emerges.\n \\item \\textbf{$\\Elayer{1} \\to \\Elayer{2}$}:\n discrete rational carriers via the\ + \ spectral bridges;\n computation emerges.\n \\item \\textbf{$\\Elayer{2} \\to \\Elayer{3}$}:\n self-referential\ + \ codes acquire self-modelling capacity;\n proof theory and metaphysics emerge.\n \\item \\textbf{$\\Elayer{3}\ + \ \\to \\Elayer{3}$}:\n self-modelling applied to self-modelling\n yields self-modelling.\n \ + \ The paradox diagnostics\n (Theorem~\\ref{thm:four-paradox-diagnostic})\n exhaust the boundary crossings.\n\ + \ No new structural mode appears.\n\\end{enumerate}\n```\n\n## Proof / Justification\n\n_No immediate manuscript\ + \ proof block was extracted in this pilot run._\n\n## Source Context\n\n- Registry source: `book-03.jsonl` line 188\n\ + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch72-saturation-why-e3-is-terminal.tex`\ + \ lines 149-182\n\n## Lean / Formalization Notes\n\n- Formalization: `formalized`\n- Module: `TauLib.BookIII.Mirror.Saturation`\n\ + - Name: `applied_saturation_check`\n\n## Dependencies\n\n- Canonical: III.T03, III.D73, III.D74, III.T48, III.T41\n\n\ + ## Related Results\n\n_Generated by later projection phases._\n\n## Related Publications\n\n_Generated by later projection\ + \ phases._\n\n## Revision Notes\n\n- 2026-04-24: Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The enrichment ladder E₀ → E₁ → E₂ → E₃ → stop is COMPLETE. Applied to proof theory: meta-metatheory adds nothing. + Applied to ZFC: no meta-metatheory reveals further structure. Applied to Book III: the book is a fixed point of self-reflecti' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001642.md b/_corpus_v3_items/cid001642.md new file mode 100644 index 000000000..7ebe057ab --- /dev/null +++ b/_corpus_v3_items/cid001642.md @@ -0,0 +1,190 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001642/ +title: Global Cartesian Gluing Theorem +cid: cid001642 +primary_alias: THM0185 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Local Hartogs bulk projections from individual worldline fibers glue into a globally coherent three-dimensional space + when all eight structural forces hold. Cocycle condition is earned from boundary functoriality. +aliases: +- THM0185 +- III.T50 +- global-cartesian-gluing-theorem +- thm:global-cartesian-gluing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006052 +- predicate: formalized_by + object: cid006053 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 8, Chapter 75 (Part VIII) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Global Cartesian Gluing Theorem** + + + Local Hartogs bulk projections from individual worldline fibers glue into a globally coherent three-dimensional space + when all eight structural forces hold. Cocycle condition is earned from boundary functoriality. + + + # Global Cartesian Gluing Theorem + + + ## Summary + + + Local Hartogs bulk projections from individual worldline fibers glue into a globally coherent three-dimensional space + when all eight structural forces hold. Cocycle condition is earned from boundary functoriality. + + + ## Statement + + + ```latex + + \label{thm:global-cartesian-gluing} + + Let $\{U_x\}_{x \in \tau^1}$ be the family of local Hartogs bulk projections, with transition maps $\phi_{xy}$ constrained + by the eight structural forces of Chapter~74. Then: + + \begin{enumerate} + + \item\emph{(Existence.)} + + The local bulks glue into a three-dimensional space + + $M_{\tau^3} = \operatornamewithlimits{colim}_{x \in \tau^1} U_x$, + + the colimit taken in the category of split-complex-enriched spaces over the primorial tower. + + + \item\emph{(Uniqueness.)} + + The Coherent Force renders the cocycle data rigid: any two gluings compatible with all eight forces are canonically isomorphic. + + + \item\emph{(Simple connectivity.)} + + The Spatial Force ensures $\pi_1(M_{\tau^3}) = 0$. + + + \item\emph{(Spectral purity.)} + + The Harmonic Force ensures the global $\chi_+/\chi_-$ decomposition is pure: no off-diagonal leakage between sectors. + + + \item\emph{(Smoothness.)} + + The Regular Force ensures all transition maps are diffeomorphisms. + + + \item\emph{(Discrete ground state.)} + + The Discrete Force ensures a positive spectral gap: the first excited mode above the vacuum has energy bounded below. + + + \item\emph{(Legibility.)} + + The Legible Force ensures cohomological data is NF-addressable: every $\sigma$-fixed cohomology class corresponds to an + algebraic cycle. + + + \item\emph{(Codability.)} + + The Codable Force ensures arithmetic data is finitely generated: discrete rational points label the patch structure. + + \end{enumerate} + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 193 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part08/ch75-the-global-cartesian-gluing.tex` + lines 91-122 + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: III.R37, III.T41, III.T29, III.D01, III.T42 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Local Hartogs bulk projections from individual worldline fibers glue into a globally coherent three-dimensional + space when all eight structural forces hold. Cocycle condition is earned from boundary functoriality. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001643.md b/_corpus_v3_items/cid001643.md new file mode 100644 index 000000000..020f3b608 --- /dev/null +++ b/_corpus_v3_items/cid001643.md @@ -0,0 +1,177 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001643/ +title: Physical Admissibility Theorem +cid: cid001643 +primary_alias: THM0186 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every physically realizable Turing machine is τ-admissible. E₁ constraints (finite spatial extent, causal locality, + bounded energy) block all Five Forbidden Moves. Combined with the Move-Bridge Correspondence (III.T43). +aliases: +- THM0186 +- III.T51 +- physical-admissibility-theorem +- thm:physical-admissibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006054 +- predicate: formalized_by + object: cid006055 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 78 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Physical Admissibility Theorem** + + + Every physically realizable Turing machine is τ-admissible. E₁ constraints (finite spatial extent, causal locality, bounded + energy) block all Five Forbidden Moves. Combined with the Move-Bridge Correspondence (III.T43). + + + # Physical Admissibility Theorem + + + ## Summary + + + Every physically realizable Turing machine is τ-admissible. E₁ constraints (finite spatial extent, causal locality, bounded + energy) block all Five Forbidden Moves. Combined with the Move-Bridge Correspondence (III.T43). + + + ## Statement + + + ```latex + + \label{thm:physical-admissibility} + + Let $M$ be a physical $\Elayer{2}$ agent + + (Definition~\ref{def:physical-e2-agent}). + + Then $M$ is $\tau$-admissible + + (Definition~\ref{def:tau-admissibility}): + + \[ + + M \;\text{physically realizable} + + \quad\Longrightarrow\quad + + M \in \operatorname{Adm}_{\tau}(k_0) + + \;\;\text{for some}\;\; k_0 < \infty. + + \] + + ``` + + + ## Proof / Justification + + + ```latex + + By the Move-Bridge Correspondence + + (Theorem~\ref{thm:move-bridge-correspondence}, Ch.~65), + + $\tau$-admissibility fails if and only if the computation + + invokes at least one forbidden move. + + Section~\ref{sec:ch78-five-forbidden-physical} showed a physical + + $\Elayer{2}$ agent cannot perform any of M1--M5. + + Hence $M$ is $\tau$-admissible. + + The width $k_0$ is bounded by the carrier''s primorial depth. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 200 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch78-physical-turing-machines.tex` + lines 197-210 + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: III.D77, III.D69, III.T43 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every physically realizable Turing machine is τ-admissible. E₁ constraints (finite spatial extent, causal locality, + bounded energy) block all Five Forbidden Moves. Combined with the Move-Bridge Correspondence (III.T43). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001644.md b/_corpus_v3_items/cid001644.md new file mode 100644 index 000000000..db8594974 --- /dev/null +++ b/_corpus_v3_items/cid001644.md @@ -0,0 +1,158 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001644/ +title: Physical P=NP +cid: cid001644 +primary_alias: THM0187 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm + for physical computation. P=NP is the natural state of physically instantiated computation.' +aliases: +- THM0187 +- III.T52 +- physical-p-np +- thm:physical-p-equals-np +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 78 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Physical P=NP** + + + The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm for physical + computation. P=NP is the natural state of physically instantiated computation. + + + # Physical P=NP + + + ## Summary + + + The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm for physical + computation. P=NP is the natural state of physically instantiated computation. + + + ## Statement + + + ```latex + + \label{thm:physical-p-equals-np} + + For all physically realizable computation, + + $\mathrm{P} = \mathrm{NP}$. + + More precisely: if $\Pi$ is an NP problem whose every physically + + realizable verifier is a physical $\Elayer{2}$ agent, then + + $\Pi \in \tau\text{-}P_{\mathrm{adm}}$. + + ``` + + + ## Proof / Justification + + + ```latex + + Physical verifier $V$ is $\tau$-admissible + + (Theorem~\ref{thm:physical-admissibility}). + + The $\tau$-Admissibility Collapse + + (Theorem~\ref{thm:tau-admissibility-collapse}, Ch.~58) gives + + $\tau\text{-}P_{\mathrm{adm}} = \tau\text{-}NP_{\mathrm{adm}}$. + + Since $\Pi \in \tau\text{-}NP_{\mathrm{adm}}$, + + we have $\Pi \in \tau\text{-}P_{\mathrm{adm}}$. + + ``` + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 201 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch78-physical-turing-machines.tex` + lines 223-231 + + + ## Lean / Formalization Notes + + + - Formalization: `not_applicable` + + - Module: `None` + + - Name: `None` + + + ## Dependencies + + + - Canonical: III.T51, III.T33 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'The τ-Admissibility Collapse (III.T33) applies to all physically realizable Turing machines: τ-P_adm = τ-NP_adm + for physical computation. P=NP is the natural state of physically instantiated computation.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001645.md b/_corpus_v3_items/cid001645.md new file mode 100644 index 000000000..1f0bc654c --- /dev/null +++ b/_corpus_v3_items/cid001645.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001645/ +title: Universal Admissibility Theorem +cid: cid001645 +primary_alias: THM0188 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ALL τ-native computation—at every enrichment level, whether physical (E₁-hosted) or abstract (E₃ diagrammatic)—is + τ-admissible. The τ-Admissibility Collapse applies universally within Category τ: τ-P = τ-NP for all τ-native computation.' +aliases: +- THM0188 +- III.T53 +- universal-admissibility-theorem +- thm:universal-admissibility +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 79 (Part IX) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: "**Universal Admissibility Theorem**\n\nALL τ-native computation—at every enrichment level, whether physical (E₁-hosted)\ + \ or abstract (E₃ diagrammatic)—is τ-admissible. The τ-Admissibility Collapse applies universally within Category τ: τ-P\ + \ = τ-NP for all τ-native computation.\n\n# Universal Admissibility Theorem\n\n## Summary\n\nALL τ-native computation—at\ + \ every enrichment level, whether physical (E₁-hosted) or abstract (E₃ diagrammatic)—is τ-admissible. The τ-Admissibility\ + \ Collapse applies universally within Category τ: τ-P = τ-NP for all τ-native computation.\n\n## Statement\n\n```latex\n\ + \\label{thm:universal-admissibility}\nLet $A$ be a $\\tau$-native computational agent\nat any enrichment level $\\Elayer{n}$\ + \ ($n \\in \\{0,1,2,3\\}$).\nThen $A$ is $\\tau$-admissible\n(Definition~\\ref{def:tau-admissibility}),\nand the $\\tau$-Admissibility\ + \ Collapse applies:\n\\begin{equation}\\label{eq:ch79-universal-collapse}\n \\tau\\text{-}P \\;=\\; \\tau\\text{-}NP\n\ + \ \\qquad \\text{(universally within $\\T$).}\n\\end{equation}\nThis holds for \\textbf{all problems definable in~$\\\ + T$}.\n```\n\n## Proof / Justification\n\n_No immediate manuscript proof block was extracted in this pilot run._\n\n##\ + \ Source Context\n\n- Registry source: `book-03.jsonl` line 204\n- Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part09/ch79-abstract-computation-in-tau.tex`\ + \ lines 233-246\n\n## Lean / Formalization Notes\n\n- Formalization: `not_applicable`\n- Module: `None`\n- Name: `None`\n\ + \n## Dependencies\n\n- Canonical: III.D78, III.D69, III.T43, III.T33\n\n## Related Results\n\n_Generated by later projection\ + \ phases._\n\n## Related Publications\n\n_Generated by later projection phases._\n\n## Revision Notes\n\n- 2026-04-24:\ + \ Initial pilot migration." +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ALL τ-native computation—at every enrichment level, whether physical (E₁-hosted) or abstract (E₃ diagrammatic)—is + τ-admissible. The τ-Admissibility Collapse applies universally within Category τ: τ-P = τ-NP for all τ-native computation.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001646.md b/_corpus_v3_items/cid001646.md new file mode 100644 index 000000000..6c6de38a1 --- /dev/null +++ b/_corpus_v3_items/cid001646.md @@ -0,0 +1,138 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001646/ +title: Confinement Bridge Identity +cid: cid001646 +primary_alias: THM0189 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: E₆(iι_τ)·κ(C;3)² = −1/(1−ι_τ)² within ±10 ppm. Reduces algebraically to the E₆ near-identity E₆·ι⁶ ≈ −1 since κ(C;3)² + = ι⁶/(1−ι)². Closes OQ.07 (C-sector/SU(3) bridge). S-duality transport gives residual = 504·q' where q' = e^{−2π/ι_τ} ≈ + 10⁻⁸. +aliases: +- THM0189 +- III.T54 +- confinement-bridge-identity +- thm:confinement-bridge +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006026 +- predicate: formalized_by + object: cid006027 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 25 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Confinement Bridge Identity** + + + E₆(iι_τ)·κ(C;3)² = −1/(1−ι_τ)² within ±10 ppm. Reduces algebraically to the E₆ near-identity E₆·ι⁶ ≈ −1 since κ(C;3)² + = ι⁶/(1−ι)². Closes OQ.07 (C-sector/SU(3) bridge). S-duality transport gives residual = 504·q'' where q'' = e^{−2π/ι_τ} + ≈ 10⁻⁸. + + + # Confinement Bridge Identity + + + ## Summary + + + E₆(iι_τ)·κ(C;3)² = −1/(1−ι_τ)² within ±10 ppm. Reduces algebraically to the E₆ near-identity E₆·ι⁶ ≈ −1 since κ(C;3)² + = ι⁶/(1−ι)². Closes OQ.07 (C-sector/SU(3) bridge). S-duality transport gives residual = 504·q'' where q'' = e^{−2π/ι_τ} + ≈ 10⁻⁸. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 211 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.ConfinementBridge` + + - Name: `Tau.BookIII.Spectral.ConfinementBridge.confinement_bridge` + + + ## Dependencies + + + - Canonical: III.D80, III.T14, IV.D03, IV.D07 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: E₆(iι_τ)·κ(C;3)² = −1/(1−ι_τ)² within ±10 ppm. Reduces algebraically to the E₆ near-identity E₆·ι⁶ ≈ −1 since + κ(C;3)² = ι⁶/(1−ι)². Closes OQ.07 (C-sector/SU(3) bridge). S-duality transport gives residual = 504·q' where q' = e^{−2π/ι_τ} + ≈ 10 +noindex: false +--- + diff --git a/_corpus_v3_items/cid001647.md b/_corpus_v3_items/cid001647.md new file mode 100644 index 000000000..24757f63e --- /dev/null +++ b/_corpus_v3_items/cid001647.md @@ -0,0 +1,131 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001647/ +title: S-Duality Transport +cid: cid001647 +primary_alias: THM0190 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: S-duality transport for Eisenstein series at τ = iι_τ. The modular transformation E_{2k}(−1/τ) = τ^{2k}E_{2k}(τ) + with i⁴=+1 and i⁶=−1 gives E₄·ι⁴ = 1+240q' (2.4 ppm) and E₆·ι⁶ = −(1−504q') (5.1 ppm). Closes OQ.09 (E₄/E₆ fixed point). + The S-dual nome q' = e^{−2π/ι_τ} < e^{−18} is exponentially suppressed. +aliases: +- THM0190 +- III.T55 +- s-duality-transport +- thm:sduality-transport +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 3, Chapter 25 (Part III) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**S-Duality Transport** + + + S-duality transport for Eisenstein series at τ = iι_τ. The modular transformation E_{2k}(−1/τ) = τ^{2k}E_{2k}(τ) with + i⁴=+1 and i⁶=−1 gives E₄·ι⁴ = 1+240q'' (2.4 ppm) and E₆·ι⁶ = −(1−504q'') (5.1 ppm). Closes OQ.09 (E₄/E₆ fixed point). + The S-dual nome q'' = e^{−2π/ι_τ} < e^{−18} is exponentially suppressed. + + + # S-Duality Transport + + + ## Summary + + + S-duality transport for Eisenstein series at τ = iι_τ. The modular transformation E_{2k}(−1/τ) = τ^{2k}E_{2k}(τ) with + i⁴=+1 and i⁶=−1 gives E₄·ι⁴ = 1+240q'' (2.4 ppm) and E₆·ι⁶ = −(1−504q'') (5.1 ppm). Closes OQ.09 (E₄/E₆ fixed point). + The S-dual nome q'' = e^{−2π/ι_τ} < e^{−18} is exponentially suppressed. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 212 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.ConfinementBridge` + + - Name: `Tau.BookIII.Spectral.ConfinementBridge.sdual_exponent_large` + + + ## Dependencies + + + - Canonical: III.D80 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: S-duality transport for Eisenstein series at τ = iι_τ. The modular transformation E_{2k}(−1/τ) = τ^{2k}E_{2k}(τ) + with i⁴=+1 and i⁶=−1 gives E₄·ι⁴ = 1+240q' (2.4 ppm) and E₆·ι⁶ = −(1−504q') (5.1 ppm). Closes OQ.09 (E₄/E₆ fixed point). + The S- +noindex: false +--- + diff --git a/_corpus_v3_items/cid001648.md b/_corpus_v3_items/cid001648.md new file mode 100644 index 000000000..339d51d76 --- /dev/null +++ b/_corpus_v3_items/cid001648.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001648/ +title: Parseval Identity +cid: cid001648 +primary_alias: THM0191 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Parseval identity: ‖f‖² = Σ_n |⟨f,e_n⟩|². The Pythagorean theorem for orthogonal spectral decomposition. Verified + for test functions at N=2,6.' +aliases: +- THM0191 +- III.T56 +- parseval-identity +- thm:parseval +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005845 +- predicate: formalized_by + object: cid005846 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 4, Chapter 30 (Wave M3) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Parseval Identity** + + + Parseval identity: ‖f‖² = Σ_n |⟨f,e_n⟩|². The Pythagorean theorem for orthogonal spectral decomposition. Verified for + test functions at N=2,6. + + + # Parseval Identity + + + ## Summary + + + Parseval identity: ‖f‖² = Σ_n |⟨f,e_n⟩|². The Pythagorean theorem for orthogonal spectral decomposition. Verified for + test functions at N=2,6. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 216 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.SpectralDecomp` + + - Name: `parseval_6` + + + ## Dependencies + + + - Canonical: III.D81, III.D82 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Parseval identity: ‖f‖² = Σ_n |⟨f,e_n⟩|². The Pythagorean theorem for orthogonal spectral decomposition. Verified + for test functions at N=2,6.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001649.md b/_corpus_v3_items/cid001649.md new file mode 100644 index 000000000..d966dd66c --- /dev/null +++ b/_corpus_v3_items/cid001649.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001649/ +title: Operational Closure Theorem +cid: cid001649 +primary_alias: THM0192 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Every E₂-admissible operation (code-decode cycle) stays within E₂: the output is a valid reduce-stable carrier element. + Computational analog of holomorphic closure. Verified at bound 8, depth 3.' +aliases: +- THM0192 +- III.T57 +- operational-closure-theorem +- thm:operational-closure +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005799 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 7, Chapter 60 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Operational Closure Theorem** + + + Every E₂-admissible operation (code-decode cycle) stays within E₂: the output is a valid reduce-stable carrier element. + Computational analog of holomorphic closure. Verified at bound 8, depth 3. + + + # Operational Closure Theorem + + + ## Summary + + + Every E₂-admissible operation (code-decode cycle) stays within E₂: the output is a valid reduce-stable carrier element. + Computational analog of holomorphic closure. Verified at bound 8, depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 220 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Computation.E2Witness` + + - Name: `operational_closure_8_3` + + + ## Dependencies + + + - Canonical: III.D83, III.D84 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Every E₂-admissible operation (code-decode cycle) stays within E₂: the output is a valid reduce-stable carrier + element. Computational analog of holomorphic closure. Verified at bound 8, depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001650.md b/_corpus_v3_items/cid001650.md new file mode 100644 index 000000000..ff4959f45 --- /dev/null +++ b/_corpus_v3_items/cid001650.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001650/ +title: E₃ Self-Model Completeness +cid: cid001650 +primary_alias: THM0193 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Every E₂ carrier element (reduce-stable) has the E₃ property. The functor E₂→E₃ is surjective on computational content. + Verified at stages 1-3 by exhaustive check. +aliases: +- THM0193 +- III.T58 +- e-self-model-completeness +- thm:e3-self-model-complete +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005903 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 72 (Wave M6) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**E₃ Self-Model Completeness** + + + Every E₂ carrier element (reduce-stable) has the E₃ property. The functor E₂→E₃ is surjective on computational content. + Verified at stages 1-3 by exhaustive check. + + + # E₃ Self-Model Completeness + + + ## Summary + + + Every E₂ carrier element (reduce-stable) has the E₃ property. The functor E₂→E₃ is surjective on computational content. + Verified at stages 1-3 by exhaustive check. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 224 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Mirror.E3Witness` + + - Name: `self_model_complete_3` + + + ## Dependencies + + + - Canonical: III.D85, III.D86 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Every E₂ carrier element (reduce-stable) has the E₃ property. The functor E₂→E₃ is surjective on computational + content. Verified at stages 1-3 by exhaustive check. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001651.md b/_corpus_v3_items/cid001651.md new file mode 100644 index 000000000..3db3cc081 --- /dev/null +++ b/_corpus_v3_items/cid001651.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001651/ +title: Arithmetic Faithfulness +cid: cid001651 +primary_alias: THM0194 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Arith_tr preserves and reflects all arithmetic: addition, multiplication, and GCD. Any τ-theorem about Z/M_k Z translates + to valid modular arithmetic in ℤ. Verified at bound 8, depth 3.' +aliases: +- THM0194 +- III.T59 +- arithmetic-faithfulness +- thm:arith-faithful +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005761 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 9, Chapter 80 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Arithmetic Faithfulness** + + + Arith_tr preserves and reflects all arithmetic: addition, multiplication, and GCD. Any τ-theorem about Z/M_k Z translates + to valid modular arithmetic in ℤ. Verified at bound 8, depth 3. + + + # Arithmetic Faithfulness + + + ## Summary + + + Arith_tr preserves and reflects all arithmetic: addition, multiplication, and GCD. Any τ-theorem about Z/M_k Z translates + to valid modular arithmetic in ℤ. Verified at bound 8, depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 228 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationArith` + + - Name: `arith_faithful_8_3` + + + ## Dependencies + + + - Canonical: III.D87, III.D88 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Arith_tr preserves and reflects all arithmetic: addition, multiplication, and GCD. Any τ-theorem about Z/M_k + Z translates to valid modular arithmetic in ℤ. Verified at bound 8, depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001652.md b/_corpus_v3_items/cid001652.md new file mode 100644 index 000000000..2ff4cf1eb --- /dev/null +++ b/_corpus_v3_items/cid001652.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001652/ +title: Topological Faithfulness +cid: cid001652 +primary_alias: THM0195 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Topo_tr preserves tower structure: cylinders map to open balls, restriction maps are continuous. Fiber over x at + stage k has exactly p_{k+1} preimages at stage k+1. Verified at bound 6, depth 2.' +aliases: +- THM0195 +- III.T60 +- topological-faithfulness +- thm:topo-faithful +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005771 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Topological Faithfulness** + + + Topo_tr preserves tower structure: cylinders map to open balls, restriction maps are continuous. Fiber over x at stage + k has exactly p_{k+1} preimages at stage k+1. Verified at bound 6, depth 2. + + + # Topological Faithfulness + + + ## Summary + + + Topo_tr preserves tower structure: cylinders map to open balls, restriction maps are continuous. Fiber over x at stage + k has exactly p_{k+1} preimages at stage k+1. Verified at bound 6, depth 2. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 232 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationTopo` + + - Name: `topo_faithful_6_2` + + + ## Dependencies + + + - Canonical: III.D89, III.D90 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Topo_tr preserves tower structure: cylinders map to open balls, restriction maps are continuous. Fiber over + x at stage k has exactly p_{k+1} preimages at stage k+1. Verified at bound 6, depth 2.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001653.md b/_corpus_v3_items/cid001653.md new file mode 100644 index 000000000..b9d85daa7 --- /dev/null +++ b/_corpus_v3_items/cid001653.md @@ -0,0 +1,134 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001653/ +title: Translation Failure Boundary +cid: cid001653 +primary_alias: THM0196 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Translations Arith_tr and Topo_tr are faithful EXACTLY on the complement of the 5 forbidden moves. Failure boundary + is sharp: translation works perfectly in the safe region, degenerates at forbidden operations.' +aliases: +- THM0196 +- III.T61 +- translation-failure-boundary +- thm:translation-failure-boundary +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005766 +- predicate: formalized_by + object: cid005767 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M7) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Translation Failure Boundary** + + + Translations Arith_tr and Topo_tr are faithful EXACTLY on the complement of the 5 forbidden moves. Failure boundary is + sharp: translation works perfectly in the safe region, degenerates at forbidden operations. + + + # Translation Failure Boundary + + + ## Summary + + + Translations Arith_tr and Topo_tr are faithful EXACTLY on the complement of the 5 forbidden moves. Failure boundary is + sharp: translation works perfectly in the safe region, degenerates at forbidden operations. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 236 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.TranslationObstruction` + + - Name: `translation_boundary_8_3` + + + ## Dependencies + + + - Canonical: III.D91, III.D92, III.P36, III.P37 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Translations Arith_tr and Topo_tr are faithful EXACTLY on the complement of the 5 forbidden moves. Failure boundary + is sharp: translation works perfectly in the safe region, degenerates at forbidden operations.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001654.md b/_corpus_v3_items/cid001654.md new file mode 100644 index 000000000..c2c83a365 --- /dev/null +++ b/_corpus_v3_items/cid001654.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001654/ +title: NS Flow Causal Arrow +cid: cid001654 +primary_alias: THM0197 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Hartogs flow on primorial tower has causal arrow from B/C sector asymmetry. Flow stabilization (BNF fixed point) + = τ-internal analog of NS regularity. Asymmetry grows with stage depth. Verified at depth 4. +aliases: +- THM0197 +- III.T62 +- ns-flow-causal-arrow +- thm:ns-causal-arrow +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005811 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 5, Chapter 40 (Wave M8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**NS Flow Causal Arrow** + + + Hartogs flow on primorial tower has causal arrow from B/C sector asymmetry. Flow stabilization (BNF fixed point) = τ-internal + analog of NS regularity. Asymmetry grows with stage depth. Verified at depth 4. + + + # NS Flow Causal Arrow + + + ## Summary + + + Hartogs flow on primorial tower has causal arrow from B/C sector asymmetry. Flow stabilization (BNF fixed point) = τ-internal + analog of NS regularity. Asymmetry grows with stage depth. Verified at depth 4. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 240 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.BridgeTightening` + + - Name: `ns_causal_4` + + + ## Dependencies + + + - Canonical: III.D28 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Hartogs flow on primorial tower has causal arrow from B/C sector asymmetry. Flow stabilization (BNF fixed point) + = τ-internal analog of NS regularity. Asymmetry grows with stage depth. Verified at depth 4. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001655.md b/_corpus_v3_items/cid001655.md new file mode 100644 index 000000000..815f846c1 --- /dev/null +++ b/_corpus_v3_items/cid001655.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001655/ +title: P vs NP Forbidden Triple +cid: cid001655 +primary_alias: THM0198 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Three forbidden moves (succinct_circuits + exponential_quantification + unbounded_fanout) collectively break the + P vs NP bridge. Internal P_adm = NP_adm cannot be translated. Verified at depth 3. +aliases: +- THM0198 +- III.T63 +- p-vs-np-forbidden-triple +- thm:pvsnp-forbidden-triple +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005812 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M8) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**P vs NP Forbidden Triple** + + + Three forbidden moves (succinct_circuits + exponential_quantification + unbounded_fanout) collectively break the P vs + NP bridge. Internal P_adm = NP_adm cannot be translated. Verified at depth 3. + + + # P vs NP Forbidden Triple + + + ## Summary + + + Three forbidden moves (succinct_circuits + exponential_quantification + unbounded_fanout) collectively break the P vs + NP bridge. Internal P_adm = NP_adm cannot be translated. Verified at depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 241 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Doors.BridgeTightening` + + - Name: `pvsnp_triple_3` + + + ## Dependencies + + + - Canonical: III.P38 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Three forbidden moves (succinct_circuits + exponential_quantification + unbounded_fanout) collectively break + the P vs NP bridge. Internal P_adm = NP_adm cannot be translated. Verified at depth 3. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001656.md b/_corpus_v3_items/cid001656.md new file mode 100644 index 000000000..c2adab01e --- /dev/null +++ b/_corpus_v3_items/cid001656.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001656/ +title: Goldbach at Primorial Levels +cid: cid001656 +primary_alias: THM0199 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Goldbach at primorial levels: every even number up to min(M_k, 100) has a Goldbach representation. Primorial sieve + guarantees sufficient prime density. Verified at depth 3.' +aliases: +- THM0199 +- III.T64 +- goldbach-at-primorial-levels +- thm:goldbach-primorial +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005997 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach at Primorial Levels** + + + Goldbach at primorial levels: every even number up to min(M_k, 100) has a Goldbach representation. Primorial sieve guarantees + sufficient prime density. Verified at depth 3. + + + # Goldbach at Primorial Levels + + + ## Summary + + + Goldbach at primorial levels: every even number up to min(M_k, 100) has a Goldbach representation. Primorial sieve guarantees + sufficient prime density. Verified at depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 245 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.AdditiveConjectures` + + - Name: `goldbach_primorial_3` + + + ## Dependencies + + + - Canonical: III.D95, III.D96 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Goldbach at primorial levels: every even number up to min(M_k, 100) has a Goldbach representation. Primorial + sieve guarantees sufficient prime density. Verified at depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001657.md b/_corpus_v3_items/cid001657.md new file mode 100644 index 000000000..9b9f9fd46 --- /dev/null +++ b/_corpus_v3_items/cid001657.md @@ -0,0 +1,132 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001657/ +title: ABC at Primorial Levels +cid: cid001657 +primary_alias: THM0200 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'ABC at primorial levels: for each stage k, all coprime pairs (a,b) with a,b < min(M_k, 20) satisfy c < rad(abc)². + Finite verification of ABC for small values. Verified at depth 3.' +aliases: +- THM0200 +- III.T65 +- abc-at-primorial-levels +- thm:abc-primorial +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005680 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave M9) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**ABC at Primorial Levels** + + + ABC at primorial levels: for each stage k, all coprime pairs (a,b) with a,b < min(M_k, 20) satisfy c < rad(abc)². Finite + verification of ABC for small values. Verified at depth 3. + + + # ABC at Primorial Levels + + + ## Summary + + + ABC at primorial levels: for each stage k, all coprime pairs (a,b) with a,b < min(M_k, 20) satisfy c < rad(abc)². Finite + verification of ABC for small values. Verified at depth 3. + + + ## Statement + + + _No manuscript statement was extracted in this pilot run._ + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 249 + + - Manuscript source: not matched + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCConjecture` + + - Name: `abc_primorial_3` + + + ## Dependencies + + + - Canonical: III.D97, III.D98 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'ABC at primorial levels: for each stage k, all coprime pairs (a,b) with a,b < min(M_k, 20) satisfy c < rad(abc)². + Finite verification of ABC for small values. Verified at depth 3.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001658.md b/_corpus_v3_items/cid001658.md new file mode 100644 index 000000000..5f16413f8 --- /dev/null +++ b/_corpus_v3_items/cid001658.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001658/ +title: Sieve Correctness +cid: cid001658 +primary_alias: THM0201 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Sieve agrees with trial division up to 200. π(10)=4, π(30)=10, π(100)=25 verified. +aliases: +- THM0201 +- III.T66 +- sieve-correctness +- thm:sieve-correct +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006066 +- predicate: formalized_by + object: cid006067 +- predicate: formalized_by + object: cid006068 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve Correctness** + + + Sieve agrees with trial division up to 200. π(10)=4, π(30)=10, π(100)=25 verified. + + + # Sieve Correctness + + + ## Summary + + + Sieve agrees with trial division up to 200. π(10)=4, π(30)=10, π(100)=25 verified. + + + ## Statement + + + ```latex + + \label{thm:sieve-correct} + + The sieve agrees with trial-division primality testing for all $n \le 200$. + + Moreover, $\pi(10)=4$, $\pi(30)=10$, $\pi(100)=25$. + + + \textbf{Lean:} \texttt{sieve\_correct\_200}, \texttt{pi\_10}, + + \texttt{pi\_30}, \texttt{pi\_100}.\quad + + \textbf{Registry:} III.T66.\quad + + \textbf{Scope:} established. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 254 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 107-115 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `sieve_correct_200` + + + ## Dependencies + + + - Canonical: III.D99, III.D100 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Sieve agrees with trial division up to 200. π(10)=4, π(30)=10, π(100)=25 verified. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001659.md b/_corpus_v3_items/cid001659.md new file mode 100644 index 000000000..5719fc74c --- /dev/null +++ b/_corpus_v3_items/cid001659.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001659/ +title: Sieve-Tower Compatibility +cid: cid001659 +primary_alias: THM0202 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Sieve is tower-stable: M_k|M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k).' +aliases: +- THM0202 +- III.T67 +- sieve-tower-compatibility +- thm:sieve-tower +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006069 +- predicate: formalized_by + object: cid006070 +- predicate: formalized_by + object: cid006076 +- predicate: formalized_by + object: cid006077 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sieve-Tower Compatibility** + + + Sieve is tower-stable: M_k|M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k). + + + # Sieve-Tower Compatibility + + + ## Summary + + + Sieve is tower-stable: M_k|M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k). + + + ## Statement + + + ```latex + + \label{thm:sieve-tower} + + The sieve is tower-stable: $M_k \mid M_{k+1}$ and each $p_i$ ($i \le k$) + + divides $M_k$. Moreover, the Brun sieve matches the Euler totient: + + $B(M_k, k) = \varphi(M_k) = \prod_{i=1}^{k}(p_i - 1)$. + + + \textbf{Lean:} \texttt{sieve\_tower\_compat\_3}, \texttt{brun\_euler\_4}.\quad + + \textbf{Registry:} III.T67. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 255 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 117-124 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.SieveInfrastructure` + + - Name: `sieve_tower_compat_3` + + + ## Dependencies + + + - Canonical: III.D99, III.D101, III.D19 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Sieve is tower-stable: M_k|M_{k+1}, each p_i divides M_k. Brun sieve matches Euler phi: B(M_k,k)=φ(M_k).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001660.md b/_corpus_v3_items/cid001660.md new file mode 100644 index 000000000..9df4ef4b8 --- /dev/null +++ b/_corpus_v3_items/cid001660.md @@ -0,0 +1,150 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001660/ +title: Goldbach Verified to 500 +cid: cid001660 +primary_alias: THM0203 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Goldbach verified to 500 via sieve: every even n in [4,500] = p+q with p,q prime.' +aliases: +- THM0203 +- III.T68 +- goldbach-verified-to-500 +- thm:goldbach-500 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006028 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach Verified to 500** + + + Goldbach verified to 500 via sieve: every even n in [4,500] = p+q with p,q prime. + + + # Goldbach Verified to 500 + + + ## Summary + + + Goldbach verified to 500 via sieve: every even n in [4,500] = p+q with p,q prime. + + + ## Statement + + + ```latex + + \label{thm:goldbach-500} + + Every even $n$ with $4 \le n \le 500$ has a Goldbach representation. + + \[ + + \forall n \in [4, 500],\; 2 \mid n \implies + + \exists p, q \text{ prime}: p + q = n. + + \] + + + \textbf{Lean:} \texttt{goldbach\_500}.\quad + + \textbf{Registry:} III.T68.\quad + + \textbf{Scope:} $\tau$-effective. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 260 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 151-161 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `goldbach_500` + + + ## Dependencies + + + - Canonical: III.D102 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Goldbach verified to 500 via sieve: every even n in [4,500] = p+q with p,q prime.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001661.md b/_corpus_v3_items/cid001661.md new file mode 100644 index 000000000..b1b05592f --- /dev/null +++ b/_corpus_v3_items/cid001661.md @@ -0,0 +1,140 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001661/ +title: Goldbach at Primorial M₄ +cid: cid001661 +primary_alias: THM0204 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Goldbach at M₄=210: all even n ≤ 210 verified. Primorial-level Goldbach extends M₃ result.' +aliases: +- THM0204 +- III.T69 +- goldbach-at-primorial-m +- thm:goldbach-primorial-m4 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006029 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Goldbach at Primorial M₄** + + + Goldbach at M₄=210: all even n ≤ 210 verified. Primorial-level Goldbach extends M₃ result. + + + # Goldbach at Primorial M₄ + + + ## Summary + + + Goldbach at M₄=210: all even n ≤ 210 verified. Primorial-level Goldbach extends M₃ result. + + + ## Statement + + + ```latex + + \label{thm:goldbach-primorial-m4} + + All even numbers up to $M_4 = 210$ satisfy Goldbach. + + + \textbf{Lean:} \texttt{goldbach\_primorial\_m4}.\quad + + \textbf{Registry:} III.T69. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 261 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 163-168 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `goldbach_primorial_m4` + + + ## Dependencies + + + - Canonical: III.D102 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Goldbach at M₄=210: all even n ≤ 210 verified. Primorial-level Goldbach extends M₃ result.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001662.md b/_corpus_v3_items/cid001662.md new file mode 100644 index 000000000..264aeb557 --- /dev/null +++ b/_corpus_v3_items/cid001662.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001662/ +title: Partition Growth +cid: cid001662 +primary_alias: THM0205 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Partition growth: r(M_{k+1}) > r(M_k) for k=2..4. Goldbach representations increase with primorial depth.' +aliases: +- THM0205 +- III.T70 +- partition-growth +- thm:partition-growth +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006030 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Partition Growth** + + + Partition growth: r(M_{k+1}) > r(M_k) for k=2..4. Goldbach representations increase with primorial depth. + + + # Partition Growth + + + ## Summary + + + Partition growth: r(M_{k+1}) > r(M_k) for k=2..4. Goldbach representations increase with primorial depth. + + + ## Statement + + + ```latex + + \label{thm:partition-growth} + + The partition count grows with primorial depth: + + $r(M_{k+1}) > r(M_k)$ for $k = 2, 3, 4$. + + + \textbf{Lean:} \texttt{partition\_growth\_4}.\quad + + \textbf{Registry:} III.T70.\quad + + \textbf{Scope:} $\tau$-effective. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 262 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 178-185 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `partition_growth_4` + + + ## Dependencies + + + - Canonical: III.D103 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Partition growth: r(M_{k+1}) > r(M_k) for k=2..4. Goldbach representations increase with primorial depth.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001663.md b/_corpus_v3_items/cid001663.md new file mode 100644 index 000000000..54b542968 --- /dev/null +++ b/_corpus_v3_items/cid001663.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001663/ +title: Obstruction Bounded +cid: cid001663 +primary_alias: THM0206 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Obstruction bounded: at each prime p_k, obstruction count ≤ 1 for even n ≤ 100.' +aliases: +- THM0206 +- III.T71 +- obstruction-bounded +- thm:obstruction-bounded +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006031 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Obstruction Bounded** + + + Obstruction bounded: at each prime p_k, obstruction count ≤ 1 for even n ≤ 100. + + + # Obstruction Bounded + + + ## Summary + + + Obstruction bounded: at each prime p_k, obstruction count ≤ 1 for even n ≤ 100. + + + ## Statement + + + ```latex + + \label{thm:obstruction-bounded} + + At each prime $p_k$, the obstruction count is at most~1. + + Each prime blocks at most one residue class for Goldbach. + + + \textbf{Lean:} \texttt{obstruction\_bounded\_5}.\quad + + \textbf{Registry:} III.T71. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 263 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 195-201 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.GoldbachDeep` + + - Name: `obstruction_bounded_5` + + + ## Dependencies + + + - Canonical: III.D104 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Obstruction bounded: at each prime p_k, obstruction count ≤ 1 for even n ≤ 100.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001664.md b/_corpus_v3_items/cid001664.md new file mode 100644 index 000000000..d640060bc --- /dev/null +++ b/_corpus_v3_items/cid001664.md @@ -0,0 +1,146 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001664/ +title: Twin Primes to 500 +cid: cid001664 +primary_alias: THM0207 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: At least 20 twin prime pairs below 500. Extended from 30 (Wave M9). +aliases: +- THM0207 +- III.T72 +- twin-primes-to-500 +- thm:twin-500 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006088 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Primes to 500** + + + At least 20 twin prime pairs below 500. Extended from 30 (Wave M9). + + + # Twin Primes to 500 + + + ## Summary + + + At least 20 twin prime pairs below 500. Extended from 30 (Wave M9). + + + ## Statement + + + ```latex + + \label{thm:twin-500} + + $\pi_2(500) \ge 20$. The twin prime pairs below~500 include + + $(3,5)$, $(5,7)$, $(11,13)$, $(17,19)$, $(29,31)$, $(41,43)$, + + $(59,61)$, $(71,73)$, \ldots + + + \textbf{Lean:} \texttt{twin\_primes\_500}.\quad + + \textbf{Registry:} III.T72.\quad + + \textbf{Scope:} $\tau$-effective. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 269 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 237-245 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `twin_primes_500` + + + ## Dependencies + + + - Canonical: III.D105 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: At least 20 twin prime pairs below 500. Extended from 30 (Wave M9). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001665.md b/_corpus_v3_items/cid001665.md new file mode 100644 index 000000000..9b2d7067f --- /dev/null +++ b/_corpus_v3_items/cid001665.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001665/ +title: Twin Density Primorial +cid: cid001665 +primary_alias: THM0208 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Twin prime density positive at each primorial level k≥2 (up to depth 5). +aliases: +- THM0208 +- III.T73 +- twin-density-primorial +- thm:twin-density +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006089 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Twin Density Primorial** + + + Twin prime density positive at each primorial level k≥2 (up to depth 5). + + + # Twin Density Primorial + + + ## Summary + + + Twin prime density positive at each primorial level k≥2 (up to depth 5). + + + ## Statement + + + ```latex + + \label{thm:twin-density} + + At each primorial level $k \ge 2$, at least one twin prime pair + + exists below $\min(M_k, 500)$. + + + \textbf{Lean:} \texttt{twin\_density\_primorial\_5}.\quad + + \textbf{Registry:} III.T73. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 270 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 247-253 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `twin_density_primorial_5` + + + ## Dependencies + + + - Canonical: III.D105 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Twin prime density positive at each primorial level k≥2 (up to depth 5). +noindex: false +--- + diff --git a/_corpus_v3_items/cid001666.md b/_corpus_v3_items/cid001666.md new file mode 100644 index 000000000..1c4920ad0 --- /dev/null +++ b/_corpus_v3_items/cid001666.md @@ -0,0 +1,144 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001666/ +title: HL Constant Convergence +cid: cid001666 +primary_alias: THM0209 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Hardy-Littlewood C₂(k) decreasing for k=2..5. Each factor p(p-2)/(p-1)² < 1. +aliases: +- THM0209 +- III.T74 +- hl-constant-convergence +- thm:hl-decreasing +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006090 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**HL Constant Convergence** + + + Hardy-Littlewood C₂(k) decreasing for k=2..5. Each factor p(p-2)/(p-1)² < 1. + + + # HL Constant Convergence + + + ## Summary + + + Hardy-Littlewood C₂(k) decreasing for k=2..5. Each factor p(p-2)/(p-1)² < 1. + + + ## Statement + + + ```latex + + \label{thm:hl-decreasing} + + $C_2(k)$ is decreasing for $k = 2, \ldots, 5$. Each factor + + $p(p-2)/(p-1)^2 < 1$ for $p \ge 3$, so the sequence is monotonically + + decreasing toward $C_2 \approx 1.3203$. + + + \textbf{Lean:} \texttt{hl\_constant\_decreasing\_5}.\quad + + \textbf{Registry:} III.T74. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 271 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 268-275 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `hl_constant_decreasing_5` + + + ## Dependencies + + + - Canonical: III.D106 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Hardy-Littlewood C₂(k) decreasing for k=2..5. Each factor p(p-2)/(p-1)² < 1. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001667.md b/_corpus_v3_items/cid001667.md new file mode 100644 index 000000000..096384e48 --- /dev/null +++ b/_corpus_v3_items/cid001667.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001667/ +title: CRT Admissible Positive +cid: cid001667 +primary_alias: THM0210 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Twin-admissible residues positive at depths 1-4. Nonempty at every primorial level. +aliases: +- THM0210 +- III.T75 +- crt-admissible-positive +- thm:crt-admissible-pos +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006091 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**CRT Admissible Positive** + + + Twin-admissible residues positive at depths 1-4. Nonempty at every primorial level. + + + # CRT Admissible Positive + + + ## Summary + + + Twin-admissible residues positive at depths 1-4. Nonempty at every primorial level. + + + ## Statement + + + ```latex + + \label{thm:crt-admissible-pos} + + At each depth $k = 1, \ldots, 4$, the set of twin-admissible residues + + is nonempty. + + + \textbf{Lean:} \texttt{crt\_admissible\_positive\_4}.\quad + + \textbf{Registry:} III.T75. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 272 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 286-292 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Spectral.TwinPrimeDeep` + + - Name: `crt_admissible_positive_4` + + + ## Dependencies + + + - Canonical: III.D107 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Twin-admissible residues positive at depths 1-4. Nonempty at every primorial level. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001668.md b/_corpus_v3_items/cid001668.md new file mode 100644 index 000000000..310ad49f6 --- /dev/null +++ b/_corpus_v3_items/cid001668.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001668/ +title: ABC Quality 100 +cid: cid001668 +primary_alias: THM0211 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Weak ABC for all coprime pairs up to 100. 7× extension from bound 15. +aliases: +- THM0211 +- III.T76 +- abc-quality-100 +- thm:abc-100 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005683 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**ABC Quality 100** + + + Weak ABC for all coprime pairs up to 100. 7× extension from bound 15. + + + # ABC Quality 100 + + + ## Summary + + + Weak ABC for all coprime pairs up to 100. 7× extension from bound 15. + + + ## Statement + + + ```latex + + \label{thm:abc-100} + + Weak ABC holds for all coprime pairs $(a, b)$ with $a, b \le 100$. + + + \textbf{Lean:} \texttt{abc\_quality\_100}.\quad + + \textbf{Registry:} III.T76.\quad + + \textbf{Scope:} $\tau$-effective. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 278 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 328-334 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `abc_quality_100` + + + ## Dependencies + + + - Canonical: III.D108 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Weak ABC for all coprime pairs up to 100. 7× extension from bound 15. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001669.md b/_corpus_v3_items/cid001669.md new file mode 100644 index 000000000..7d68852bd --- /dev/null +++ b/_corpus_v3_items/cid001669.md @@ -0,0 +1,148 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001669/ +title: Squarefree Dominance +cid: cid001669 +primary_alias: THM0212 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: 'Squarefree dominance: c ≤ rad(abc) for all squarefree coprime pairs up to 100. Stronger than ABC (ε=0).' +aliases: +- THM0212 +- III.T77 +- squarefree-dominance +- thm:squarefree-dominance +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005684 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Squarefree Dominance** + + + Squarefree dominance: c ≤ rad(abc) for all squarefree coprime pairs up to 100. Stronger than ABC (ε=0). + + + # Squarefree Dominance + + + ## Summary + + + Squarefree dominance: c ≤ rad(abc) for all squarefree coprime pairs up to 100. Stronger than ABC (ε=0). + + + ## Statement + + + ```latex + + \label{thm:squarefree-dominance} + + For all squarefree coprime pairs $(a, b)$ with $a, b \le 100$: + + $c \le \mathrm{rad}(abc)$. Zero high-quality triples among squarefree + + coprimes. + + + \textbf{Lean:} \texttt{squarefree\_dominance\_100}, + + \texttt{squarefree\_dominance\_thm}.\quad + + \textbf{Registry:} III.T77.\quad + + \textbf{Scope:} $\tau$-effective. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 279 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 354-363 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `squarefree_dominance_100` + + + ## Dependencies + + + - Canonical: III.D110 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Squarefree dominance: c ≤ rad(abc) for all squarefree coprime pairs up to 100. Stronger than ABC (ε=0).' +noindex: false +--- + diff --git a/_corpus_v3_items/cid001670.md b/_corpus_v3_items/cid001670.md new file mode 100644 index 000000000..92a43e30c --- /dev/null +++ b/_corpus_v3_items/cid001670.md @@ -0,0 +1,142 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001670/ +title: Radical Primorial 5 +cid: cid001670 +primary_alias: THM0213 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: rad(M_k) = M_k for k=1..5. Primorial tower is entirely squarefree. +aliases: +- THM0213 +- III.T78 +- radical-primorial-5 +- thm:radical-primorial-5 +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005685 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Radical Primorial 5** + + + rad(M_k) = M_k for k=1..5. Primorial tower is entirely squarefree. + + + # Radical Primorial 5 + + + ## Summary + + + rad(M_k) = M_k for k=1..5. Primorial tower is entirely squarefree. + + + ## Statement + + + ```latex + + \label{thm:radical-primorial-5} + + $\mathrm{rad}(M_k) = M_k$ for $k = 1, \ldots, 5$. The primorial tower is + + entirely squarefree. + + + \textbf{Lean:} \texttt{radical\_primorial\_5}.\quad + + \textbf{Registry:} III.T78. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 280 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 365-371 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Arithmetic.ABCDeep` + + - Name: `radical_primorial_5` + + + ## Dependencies + + + - Canonical: III.D97 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: rad(M_k) = M_k for k=1..5. Primorial tower is entirely squarefree. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001671.md b/_corpus_v3_items/cid001671.md new file mode 100644 index 000000000..1fec1b083 --- /dev/null +++ b/_corpus_v3_items/cid001671.md @@ -0,0 +1,145 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001671/ +title: Tower Finite Decidable +cid: cid001671 +primary_alias: THM0214 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: All three conjectures have defined gap types, forbidden moves, and scope labels. Finite τ-effective, infinite conjectural. +aliases: +- THM0214 +- III.T79 +- tower-finite-decidable +- thm:tower-decidable +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005729 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Tower Finite Decidable** + + + All three conjectures have defined gap types, forbidden moves, and scope labels. Finite τ-effective, infinite conjectural. + + + # Tower Finite Decidable + + + ## Summary + + + All three conjectures have defined gap types, forbidden moves, and scope labels. Finite τ-effective, infinite conjectural. + + + ## Statement + + + ```latex + + \label{thm:tower-decidable} + + All three conjectures have defined gap types, forbidden moves, and + + scope labels. Finite-level results are $\tau$-effective; infinite + + claims are conjectural. + + + \textbf{Lean:} \texttt{tower\_finite\_decidable}.\quad + + \textbf{Registry:} III.T79. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 286 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 432-439 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `tower_finite_decidable` + + + ## Dependencies + + + - Canonical: III.D111, III.D112 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: All three conjectures have defined gap types, forbidden moves, and scope labels. Finite τ-effective, infinite + conjectural. +noindex: false +--- + diff --git a/_corpus_v3_items/cid001672.md b/_corpus_v3_items/cid001672.md new file mode 100644 index 000000000..f0e8c5090 --- /dev/null +++ b/_corpus_v3_items/cid001672.md @@ -0,0 +1,158 @@ +--- +layout: corpus-v3-item +permalink: /id/cid001672/ +title: Bridge Necessary Insufficient +cid: cid001672 +primary_alias: THM0215 +type: theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +- corpus_v2 +summary: Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot + create analytic content needed for infinite case. +aliases: +- THM0215 +- III.T80 +- bridge-necessary-insufficient +- thm:bridge-necessary +layers: [] +domains: [] +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid005730 +- predicate: formalized_by + object: cid005739 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: +- type: monograph + item: cid000024 + locator: Book III, Part 10, Chapter 81 (Wave R1) +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_registry +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Bridge Necessary Insufficient** + + + Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot create + analytic content needed for infinite case. + + + # Bridge Necessary Insufficient + + + ## Summary + + + Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot create + analytic content needed for infinite case. + + + ## Statement + + + ```latex + + \label{thm:bridge-necessary} + + The bridge axiom is \emph{necessary} for connecting $\tau$-internal + + results to external conjectures: all three gaps map to + + \texttt{exponential\_quantification} with bridge damage~3 (break). + + But the bridge is \emph{insufficient}: even with the bridge functor, + + the analytic content (circle method, sieve asymptotics, height theory) + + needed for the infinite case is absent. + + + \textbf{Lean:} \texttt{bridge\_necessary\_insufficient}.\quad + + \textbf{Registry:} III.T80.\quad + + \textbf{Scope:} established. + + ``` + + + ## Proof / Justification + + + _No immediate manuscript proof block was extracted in this pilot run._ + + + ## Source Context + + + - Registry source: `book-03.jsonl` line 287 + + - Manuscript source: `2nd-edition/book-iii-categorical-spectrum/02_mainmatter/part10/ch81-additive-conjectures-deep.tex` + lines 441-452 + + + ## Lean / Formalization Notes + + + - Formalization: `formalized` + + - Module: `TauLib.BookIII.Bridge.ConjectureGaps` + + - Name: `bridge_necessary_insufficient` + + + ## Dependencies + + + - Canonical: III.D112, III.D113, III.T43 + + + ## Related Results + + + _Generated by later projection phases._ + + + ## Related Publications + + + _Generated by later projection phases._ + + + ## Revision Notes + + + - 2026-04-24: Initial pilot migration.' +formalization: {} +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Bridge necessary but insufficient. All gaps map to exponential_quantification with damage 3 (break). Bridge cannot + create analytic content needed for infinite case. +noindex: false +--- + diff --git a/_corpus_v3_items/cid005000.md b/_corpus_v3_items/cid005000.md new file mode 100644 index 000000000..6045c4c8e --- /dev/null +++ b/_corpus_v3_items/cid005000.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005000/ +title: TauLib.BookI.Addressability.AddressResolution +cid: cid005000 +primary_alias: MOD0003 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Addressability.AddressResolution (161 lines, 9 declarations). +aliases: +- MOD0003 +- TauLib.BookI.Addressability.AddressResolution +- book-i-addressability-address-resolution +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Addressability.AddressResolution + module_path: TauLib/BookI/Addressability/AddressResolution.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Addressability.AddressResolution (161 lines, 9 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005001.md b/_corpus_v3_items/cid005001.md new file mode 100644 index 000000000..fbab663de --- /dev/null +++ b/_corpus_v3_items/cid005001.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005001/ +title: TauLib.BookI.Addressability.CayleyMetric +cid: cid005001 +primary_alias: MOD0004 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Addressability.CayleyMetric (128 lines, 10 declarations). +aliases: +- MOD0004 +- TauLib.BookI.Addressability.CayleyMetric +- book-i-addressability-cayley-metric +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Addressability.CayleyMetric + module_path: TauLib/BookI/Addressability/CayleyMetric.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Addressability.CayleyMetric (128 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005002.md b/_corpus_v3_items/cid005002.md new file mode 100644 index 000000000..447ba8226 --- /dev/null +++ b/_corpus_v3_items/cid005002.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005002/ +title: TauLib.BookI.Addressability.HingeIntegration +cid: cid005002 +primary_alias: MOD0005 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Addressability.HingeIntegration (326 lines, 12 declarations). +aliases: +- MOD0005 +- TauLib.BookI.Addressability.HingeIntegration +- book-i-addressability-hinge-integration +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Addressability.HingeIntegration + module_path: TauLib/BookI/Addressability/HingeIntegration.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Addressability.HingeIntegration (326 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005003.md b/_corpus_v3_items/cid005003.md new file mode 100644 index 000000000..72dbaafc3 --- /dev/null +++ b/_corpus_v3_items/cid005003.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005003/ +title: TauLib.BookI.Addressability.OnticUltrametric +cid: cid005003 +primary_alias: MOD0006 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Addressability.OnticUltrametric (171 lines, 14 declarations). +aliases: +- MOD0006 +- TauLib.BookI.Addressability.OnticUltrametric +- book-i-addressability-ontic-ultrametric +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Addressability.OnticUltrametric + module_path: TauLib/BookI/Addressability/OnticUltrametric.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Addressability.OnticUltrametric (171 lines, 14 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005004.md b/_corpus_v3_items/cid005004.md new file mode 100644 index 000000000..e4d3ab6fd --- /dev/null +++ b/_corpus_v3_items/cid005004.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005004/ +title: TauLib.BookI.Boundary.Bridge.TauIntQuotient +cid: cid005004 +primary_alias: MOD0007 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauIntQuotient (352 lines, 20 declarations). +aliases: +- MOD0007 +- TauLib.BookI.Boundary.Bridge.TauIntQuotient +- book-i-boundary-bridge-tau-int-quotient +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Bridge.TauIntQuotient + module_path: TauLib/BookI/Boundary/Bridge/TauIntQuotient.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauIntQuotient (352 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005005.md b/_corpus_v3_items/cid005005.md new file mode 100644 index 000000000..517d59a08 --- /dev/null +++ b/_corpus_v3_items/cid005005.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005005/ +title: TauLib.BookI.Boundary.Bridge.TauRatQuotient +cid: cid005005 +primary_alias: MOD0008 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRatQuotient (443 lines, 24 declarations). +aliases: +- MOD0008 +- TauLib.BookI.Boundary.Bridge.TauRatQuotient +- book-i-boundary-bridge-tau-rat-quotient +- I.D144 +- I.T207 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Bridge.TauRatQuotient + module_path: TauLib/BookI/Boundary/Bridge/TauRatQuotient.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRatQuotient (443 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005006.md b/_corpus_v3_items/cid005006.md new file mode 100644 index 000000000..1a9e9b466 --- /dev/null +++ b/_corpus_v3_items/cid005006.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005006/ +title: TauLib.BookI.Boundary.Bridge.TauRealCongruence +cid: cid005006 +primary_alias: MOD0009 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRealCongruence (430 lines, 7 declarations). +aliases: +- MOD0009 +- TauLib.BookI.Boundary.Bridge.TauRealCongruence +- book-i-boundary-bridge-tau-real-congruence +- I.D111 +- I.D112 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Bridge.TauRealCongruence + module_path: TauLib/BookI/Boundary/Bridge/TauRealCongruence.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRealCongruence (430 lines, 7 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005007.md b/_corpus_v3_items/cid005007.md new file mode 100644 index 000000000..8c93ffdf9 --- /dev/null +++ b/_corpus_v3_items/cid005007.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005007/ +title: TauLib.BookI.Boundary.Bridge.TauRealQuotient +cid: cid005007 +primary_alias: MOD0010 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRealQuotient (339 lines, 24 declarations). +aliases: +- MOD0010 +- TauLib.BookI.Boundary.Bridge.TauRealQuotient +- book-i-boundary-bridge-tau-real-quotient +- I.D146 +- I.T223 +- I.T224 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Bridge.TauRealQuotient + module_path: TauLib/BookI/Boundary/Bridge/TauRealQuotient.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Bridge.TauRealQuotient (339 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005008.md b/_corpus_v3_items/cid005008.md new file mode 100644 index 000000000..b797d6d32 --- /dev/null +++ b/_corpus_v3_items/cid005008.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005008/ +title: TauLib.BookI.Boundary.Characters +cid: cid005008 +primary_alias: MOD0011 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Characters (302 lines, 53 declarations). +aliases: +- MOD0011 +- TauLib.BookI.Boundary.Characters +- book-i-boundary-characters +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Characters + module_path: TauLib/BookI/Boundary/Characters.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Characters (302 lines, 53 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005009.md b/_corpus_v3_items/cid005009.md new file mode 100644 index 000000000..27ad47568 --- /dev/null +++ b/_corpus_v3_items/cid005009.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005009/ +title: TauLib.BookI.Boundary.ComplexField +cid: cid005009 +primary_alias: MOD0012 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.ComplexField (362 lines, 27 declarations). +aliases: +- MOD0012 +- TauLib.BookI.Boundary.ComplexField +- book-i-boundary-complex-field +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.ComplexField + module_path: TauLib/BookI/Boundary/ComplexField.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.ComplexField (362 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005010.md b/_corpus_v3_items/cid005010.md new file mode 100644 index 000000000..6e18921a3 --- /dev/null +++ b/_corpus_v3_items/cid005010.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005010/ +title: TauLib.BookI.Boundary.ConstructiveReals +cid: cid005010 +primary_alias: MOD0013 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.ConstructiveReals (428 lines, 42 declarations). +aliases: +- MOD0013 +- TauLib.BookI.Boundary.ConstructiveReals +- book-i-boundary-constructive-reals +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.ConstructiveReals + module_path: TauLib/BookI/Boundary/ConstructiveReals.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.ConstructiveReals (428 lines, 42 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005011.md b/_corpus_v3_items/cid005011.md new file mode 100644 index 000000000..27d1c52e9 --- /dev/null +++ b/_corpus_v3_items/cid005011.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005011/ +title: TauLib.BookI.Boundary.CouplingIdentityApproximants +cid: cid005011 +primary_alias: MOD0014 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.CouplingIdentityApproximants (312 lines, 18 declarations). +aliases: +- MOD0014 +- TauLib.BookI.Boundary.CouplingIdentityApproximants +- book-i-boundary-coupling-identity-approximants +- I.D117 +- I.D118 +- I.D120 +- I.T68 +- I.T74 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.CouplingIdentityApproximants + module_path: TauLib/BookI/Boundary/CouplingIdentityApproximants.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.CouplingIdentityApproximants (312 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005012.md b/_corpus_v3_items/cid005012.md new file mode 100644 index 000000000..e35b0f0f3 --- /dev/null +++ b/_corpus_v3_items/cid005012.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005012/ +title: TauLib.BookI.Boundary.Cyclotomic +cid: cid005012 +primary_alias: MOD0015 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Cyclotomic (189 lines, 24 declarations). +aliases: +- MOD0015 +- TauLib.BookI.Boundary.Cyclotomic +- book-i-boundary-cyclotomic +- I.R23 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Cyclotomic + module_path: TauLib/BookI/Boundary/Cyclotomic.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Cyclotomic (189 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005013.md b/_corpus_v3_items/cid005013.md new file mode 100644 index 000000000..f56223517 --- /dev/null +++ b/_corpus_v3_items/cid005013.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005013/ +title: TauLib.BookI.Boundary.DefectInverseSystem +cid: cid005013 +primary_alias: MOD0016 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.DefectInverseSystem (389 lines, 16 declarations). +aliases: +- MOD0016 +- TauLib.BookI.Boundary.DefectInverseSystem +- book-i-boundary-defect-inverse-system +- I.D121 +- I.D122 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.DefectInverseSystem + module_path: TauLib/BookI/Boundary/DefectInverseSystem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.DefectInverseSystem (389 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005014.md b/_corpus_v3_items/cid005014.md new file mode 100644 index 000000000..e507f8434 --- /dev/null +++ b/_corpus_v3_items/cid005014.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005014/ +title: TauLib.BookI.Boundary.Fourier +cid: cid005014 +primary_alias: MOD0017 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Fourier (264 lines, 43 declarations). +aliases: +- MOD0017 +- TauLib.BookI.Boundary.Fourier +- book-i-boundary-fourier +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Fourier + module_path: TauLib/BookI/Boundary/Fourier.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Fourier (264 lines, 43 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005015.md b/_corpus_v3_items/cid005015.md new file mode 100644 index 000000000..1684e6047 --- /dev/null +++ b/_corpus_v3_items/cid005015.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005015/ +title: TauLib.BookI.Boundary.Integration +cid: cid005015 +primary_alias: MOD0018 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Integration (137 lines, 19 declarations). +aliases: +- MOD0018 +- TauLib.BookI.Boundary.Integration +- book-i-boundary-integration +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Integration + module_path: TauLib/BookI/Boundary/Integration.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Integration (137 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005016.md b/_corpus_v3_items/cid005016.md new file mode 100644 index 000000000..798568c33 --- /dev/null +++ b/_corpus_v3_items/cid005016.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005016/ +title: TauLib.BookI.Boundary.Iota +cid: cid005016 +primary_alias: MOD0019 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Iota (157 lines, 26 declarations). +aliases: +- MOD0019 +- TauLib.BookI.Boundary.Iota +- book-i-boundary-iota +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Iota + module_path: TauLib/BookI/Boundary/Iota.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Iota (157 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005017.md b/_corpus_v3_items/cid005017.md new file mode 100644 index 000000000..3d0b973de --- /dev/null +++ b/_corpus_v3_items/cid005017.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005017/ +title: TauLib.BookI.Boundary.IotaTauStructural +cid: cid005017 +primary_alias: MOD0020 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.IotaTauStructural (252 lines, 10 declarations). +aliases: +- MOD0020 +- TauLib.BookI.Boundary.IotaTauStructural +- book-i-boundary-iota-tau-structural +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.IotaTauStructural + module_path: TauLib/BookI/Boundary/IotaTauStructural.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.IotaTauStructural (252 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005018.md b/_corpus_v3_items/cid005018.md new file mode 100644 index 000000000..369823409 --- /dev/null +++ b/_corpus_v3_items/cid005018.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005018/ +title: TauLib.BookI.Boundary.Measure +cid: cid005018 +primary_alias: MOD0021 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Measure (256 lines, 36 declarations). +aliases: +- MOD0021 +- TauLib.BookI.Boundary.Measure +- book-i-boundary-measure +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + module_path: TauLib/BookI/Boundary/Measure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Measure (256 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005019.md b/_corpus_v3_items/cid005019.md new file mode 100644 index 000000000..49417f72f --- /dev/null +++ b/_corpus_v3_items/cid005019.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005019/ +title: TauLib.BookI.Boundary.NumberTower +cid: cid005019 +primary_alias: MOD0022 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.NumberTower (441 lines, 71 declarations). +aliases: +- MOD0022 +- TauLib.BookI.Boundary.NumberTower +- book-i-boundary-number-tower +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.NumberTower + module_path: TauLib/BookI/Boundary/NumberTower.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.NumberTower (441 lines, 71 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005020.md b/_corpus_v3_items/cid005020.md new file mode 100644 index 000000000..344bbd92e --- /dev/null +++ b/_corpus_v3_items/cid005020.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005020/ +title: TauLib.BookI.Boundary.NumericalProjection +cid: cid005020 +primary_alias: MOD0023 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.NumericalProjection (288 lines, 25 declarations). +aliases: +- MOD0023 +- TauLib.BookI.Boundary.NumericalProjection +- book-i-boundary-numerical-projection +- I.D128 +- I.T80 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.NumericalProjection + module_path: TauLib/BookI/Boundary/NumericalProjection.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.NumericalProjection (288 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005021.md b/_corpus_v3_items/cid005021.md new file mode 100644 index 000000000..c92203437 --- /dev/null +++ b/_corpus_v3_items/cid005021.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005021/ +title: TauLib.BookI.Boundary.Quaternions +cid: cid005021 +primary_alias: MOD0024 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Quaternions (321 lines, 27 declarations). +aliases: +- MOD0024 +- TauLib.BookI.Boundary.Quaternions +- book-i-boundary-quaternions +- I.R22 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Quaternions + module_path: TauLib/BookI/Boundary/Quaternions.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Quaternions (321 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005022.md b/_corpus_v3_items/cid005022.md new file mode 100644 index 000000000..10ec54d10 --- /dev/null +++ b/_corpus_v3_items/cid005022.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005022/ +title: TauLib.BookI.Boundary.RefinementGrowingTorus +cid: cid005022 +primary_alias: MOD0025 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.RefinementGrowingTorus (313 lines, 18 declarations). +aliases: +- MOD0025 +- TauLib.BookI.Boundary.RefinementGrowingTorus +- book-i-boundary-refinement-growing-torus +- I.D125 +- I.D127 +- I.T75 +- I.T76 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.RefinementGrowingTorus + module_path: TauLib/BookI/Boundary/RefinementGrowingTorus.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.RefinementGrowingTorus (313 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005023.md b/_corpus_v3_items/cid005023.md new file mode 100644 index 000000000..12a6ba629 --- /dev/null +++ b/_corpus_v3_items/cid005023.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005023/ +title: TauLib.BookI.Boundary.Ring +cid: cid005023 +primary_alias: MOD0026 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Ring (345 lines, 42 declarations). +aliases: +- MOD0026 +- TauLib.BookI.Boundary.Ring +- book-i-boundary-ring +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Ring + module_path: TauLib/BookI/Boundary/Ring.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Ring (345 lines, 42 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005024.md b/_corpus_v3_items/cid005024.md new file mode 100644 index 000000000..aafe23bc7 --- /dev/null +++ b/_corpus_v3_items/cid005024.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005024/ +title: TauLib.BookI.Boundary.Spectral +cid: cid005024 +primary_alias: MOD0027 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.Spectral (235 lines, 38 declarations). +aliases: +- MOD0027 +- TauLib.BookI.Boundary.Spectral +- book-i-boundary-spectral +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Spectral + module_path: TauLib/BookI/Boundary/Spectral.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.Spectral (235 lines, 38 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005025.md b/_corpus_v3_items/cid005025.md new file mode 100644 index 000000000..e1039c2cb --- /dev/null +++ b/_corpus_v3_items/cid005025.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005025/ +title: TauLib.BookI.Boundary.SplitComplex +cid: cid005025 +primary_alias: MOD0028 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.SplitComplex (348 lines, 51 declarations). +aliases: +- MOD0028 +- TauLib.BookI.Boundary.SplitComplex +- book-i-boundary-split-complex +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.SplitComplex + module_path: TauLib/BookI/Boundary/SplitComplex.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.SplitComplex (348 lines, 51 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005026.md b/_corpus_v3_items/cid005026.md new file mode 100644 index 000000000..2e9417bb5 --- /dev/null +++ b/_corpus_v3_items/cid005026.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005026/ +title: TauLib.BookI.Boundary.TauRatAbs +cid: cid005026 +primary_alias: MOD0029 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatAbs (251 lines, 17 declarations). +aliases: +- MOD0029 +- TauLib.BookI.Boundary.TauRatAbs +- book-i-boundary-tau-rat-abs +- I.D107 +- I.D109 +- I.P48 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRatAbs + module_path: TauLib/BookI/Boundary/TauRatAbs.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatAbs (251 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005027.md b/_corpus_v3_items/cid005027.md new file mode 100644 index 000000000..2278bf2e4 --- /dev/null +++ b/_corpus_v3_items/cid005027.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005027/ +title: TauLib.BookI.Boundary.TauRatField +cid: cid005027 +primary_alias: MOD0030 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatField (230 lines, 19 declarations). +aliases: +- MOD0030 +- TauLib.BookI.Boundary.TauRatField +- book-i-boundary-tau-rat-field +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRatField + module_path: TauLib/BookI/Boundary/TauRatField.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatField (230 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005028.md b/_corpus_v3_items/cid005028.md new file mode 100644 index 000000000..c85abc826 --- /dev/null +++ b/_corpus_v3_items/cid005028.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005028/ +title: TauLib.BookI.Boundary.TauRatInv +cid: cid005028 +primary_alias: MOD0031 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatInv (202 lines, 12 declarations). +aliases: +- MOD0031 +- TauLib.BookI.Boundary.TauRatInv +- book-i-boundary-tau-rat-inv +- I.D110 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRatInv + module_path: TauLib/BookI/Boundary/TauRatInv.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatInv (202 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005029.md b/_corpus_v3_items/cid005029.md new file mode 100644 index 000000000..9249f890f --- /dev/null +++ b/_corpus_v3_items/cid005029.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005029/ +title: TauLib.BookI.Boundary.TauRatOrder +cid: cid005029 +primary_alias: MOD0032 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatOrder (250 lines, 24 declarations). +aliases: +- MOD0032 +- TauLib.BookI.Boundary.TauRatOrder +- book-i-boundary-tau-rat-order +- I.D108 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRatOrder + module_path: TauLib/BookI/Boundary/TauRatOrder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRatOrder (250 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005030.md b/_corpus_v3_items/cid005030.md new file mode 100644 index 000000000..20dc53c51 --- /dev/null +++ b/_corpus_v3_items/cid005030.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005030/ +title: TauLib.BookI.Boundary.TauRealAbs +cid: cid005030 +primary_alias: MOD0033 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealAbs (172 lines, 6 declarations). +aliases: +- MOD0033 +- TauLib.BookI.Boundary.TauRealAbs +- book-i-boundary-tau-real-abs +- I.D114 +- I.P49 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealAbs + module_path: TauLib/BookI/Boundary/TauRealAbs.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealAbs (172 lines, 6 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005031.md b/_corpus_v3_items/cid005031.md new file mode 100644 index 000000000..5d723a030 --- /dev/null +++ b/_corpus_v3_items/cid005031.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005031/ +title: TauLib.BookI.Boundary.TauRealE +cid: cid005031 +primary_alias: MOD0034 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealE (208 lines, 10 declarations). +aliases: +- MOD0034 +- TauLib.BookI.Boundary.TauRealE +- book-i-boundary-tau-real-e +- I.D115 +- I.D116 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealE + module_path: TauLib/BookI/Boundary/TauRealE.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealE (208 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005032.md b/_corpus_v3_items/cid005032.md new file mode 100644 index 000000000..1e2388624 --- /dev/null +++ b/_corpus_v3_items/cid005032.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005032/ +title: TauLib.BookI.Boundary.TauRealInv +cid: cid005032 +primary_alias: MOD0035 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealInv (170 lines, 7 declarations). +aliases: +- MOD0035 +- TauLib.BookI.Boundary.TauRealInv +- book-i-boundary-tau-real-inv +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealInv + module_path: TauLib/BookI/Boundary/TauRealInv.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealInv (170 lines, 7 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005033.md b/_corpus_v3_items/cid005033.md new file mode 100644 index 000000000..e42c13c71 --- /dev/null +++ b/_corpus_v3_items/cid005033.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005033/ +title: TauLib.BookI.Boundary.TauRealIotaTau +cid: cid005033 +primary_alias: MOD0036 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealIotaTau (167 lines, 4 declarations). +aliases: +- MOD0036 +- TauLib.BookI.Boundary.TauRealIotaTau +- book-i-boundary-tau-real-iota-tau +- I.D119 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealIotaTau + module_path: TauLib/BookI/Boundary/TauRealIotaTau.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealIotaTau (167 lines, 4 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005034.md b/_corpus_v3_items/cid005034.md new file mode 100644 index 000000000..522ad0c47 --- /dev/null +++ b/_corpus_v3_items/cid005034.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005034/ +title: TauLib.BookI.Boundary.TauRealMulCongr +cid: cid005034 +primary_alias: MOD0037 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealMulCongr (179 lines, 1 declarations). +aliases: +- MOD0037 +- TauLib.BookI.Boundary.TauRealMulCongr +- book-i-boundary-tau-real-mul-congr +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealMulCongr + module_path: TauLib/BookI/Boundary/TauRealMulCongr.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealMulCongr (179 lines, 1 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005035.md b/_corpus_v3_items/cid005035.md new file mode 100644 index 000000000..b0b30ca47 --- /dev/null +++ b/_corpus_v3_items/cid005035.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005035/ +title: TauLib.BookI.Boundary.TauRealOrder +cid: cid005035 +primary_alias: MOD0038 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealOrder (359 lines, 12 declarations). +aliases: +- MOD0038 +- TauLib.BookI.Boundary.TauRealOrder +- book-i-boundary-tau-real-order +- I.D113 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealOrder + module_path: TauLib/BookI/Boundary/TauRealOrder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealOrder (359 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005036.md b/_corpus_v3_items/cid005036.md new file mode 100644 index 000000000..103d5b5c8 --- /dev/null +++ b/_corpus_v3_items/cid005036.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005036/ +title: TauLib.BookI.Boundary.TauRealPi +cid: cid005036 +primary_alias: MOD0039 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealPi (300 lines, 12 declarations). +aliases: +- MOD0039 +- TauLib.BookI.Boundary.TauRealPi +- book-i-boundary-tau-real-pi +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealPi + module_path: TauLib/BookI/Boundary/TauRealPi.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealPi (300 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005037.md b/_corpus_v3_items/cid005037.md new file mode 100644 index 000000000..c92d22257 --- /dev/null +++ b/_corpus_v3_items/cid005037.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005037/ +title: TauLib.BookI.Boundary.TauRealPiPlusE +cid: cid005037 +primary_alias: MOD0040 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealPiPlusE (344 lines, 13 declarations). +aliases: +- MOD0040 +- TauLib.BookI.Boundary.TauRealPiPlusE +- book-i-boundary-tau-real-pi-plus-e +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealPiPlusE + module_path: TauLib/BookI/Boundary/TauRealPiPlusE.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealPiPlusE (344 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005038.md b/_corpus_v3_items/cid005038.md new file mode 100644 index 000000000..b05dffeb0 --- /dev/null +++ b/_corpus_v3_items/cid005038.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005038/ +title: TauLib.BookI.Boundary.TauRealSum +cid: cid005038 +primary_alias: MOD0041 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealSum (192 lines, 9 declarations). +aliases: +- MOD0041 +- TauLib.BookI.Boundary.TauRealSum +- book-i-boundary-tau-real-sum +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TauRealSum + module_path: TauLib/BookI/Boundary/TauRealSum.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TauRealSum (192 lines, 9 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005039.md b/_corpus_v3_items/cid005039.md new file mode 100644 index 000000000..17b021812 --- /dev/null +++ b/_corpus_v3_items/cid005039.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005039/ +title: TauLib.BookI.Boundary.TorusDefectSystem +cid: cid005039 +primary_alias: MOD0042 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.TorusDefectSystem (378 lines, 18 declarations). +aliases: +- MOD0042 +- TauLib.BookI.Boundary.TorusDefectSystem +- book-i-boundary-torus-defect-system +- I.D126 +- I.T72 +- I.T73 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.TorusDefectSystem + module_path: TauLib/BookI/Boundary/TorusDefectSystem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.TorusDefectSystem (378 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005040.md b/_corpus_v3_items/cid005040.md new file mode 100644 index 000000000..b0293490d --- /dev/null +++ b/_corpus_v3_items/cid005040.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005040/ +title: TauLib.BookI.Boundary.UniversalFixedScalar +cid: cid005040 +primary_alias: MOD0043 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Boundary.UniversalFixedScalar (377 lines, 13 declarations). +aliases: +- MOD0043 +- TauLib.BookI.Boundary.UniversalFixedScalar +- book-i-boundary-universal-fixed-scalar +- I.T69 +- I.T71 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.UniversalFixedScalar + module_path: TauLib/BookI/Boundary/UniversalFixedScalar.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Boundary.UniversalFixedScalar (377 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005041.md b/_corpus_v3_items/cid005041.md new file mode 100644 index 000000000..382ae7095 --- /dev/null +++ b/_corpus_v3_items/cid005041.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005041/ +title: TauLib.BookI.Coordinates.ABCD +cid: cid005041 +primary_alias: MOD0044 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.ABCD (171 lines, 35 declarations). +aliases: +- MOD0044 +- TauLib.BookI.Coordinates.ABCD +- book-i-coordinates-abcd +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ABCD + module_path: TauLib/BookI/Coordinates/ABCD.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.ABCD (171 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005042.md b/_corpus_v3_items/cid005042.md new file mode 100644 index 000000000..8690838d8 --- /dev/null +++ b/_corpus_v3_items/cid005042.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005042/ +title: TauLib.BookI.Coordinates.ChebyshevBias +cid: cid005042 +primary_alias: MOD0045 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.ChebyshevBias (188 lines, 15 declarations). +aliases: +- MOD0045 +- TauLib.BookI.Coordinates.ChebyshevBias +- book-i-coordinates-chebyshev-bias +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ChebyshevBias + module_path: TauLib/BookI/Coordinates/ChebyshevBias.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.ChebyshevBias (188 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005043.md b/_corpus_v3_items/cid005043.md new file mode 100644 index 000000000..08be6f411 --- /dev/null +++ b/_corpus_v3_items/cid005043.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005043/ +title: TauLib.BookI.Coordinates.Descent +cid: cid005043 +primary_alias: MOD0046 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.Descent (98 lines, 14 declarations). +aliases: +- MOD0046 +- TauLib.BookI.Coordinates.Descent +- book-i-coordinates-descent +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.Descent + module_path: TauLib/BookI/Coordinates/Descent.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.Descent (98 lines, 14 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005044.md b/_corpus_v3_items/cid005044.md new file mode 100644 index 000000000..03b721d45 --- /dev/null +++ b/_corpus_v3_items/cid005044.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005044/ +title: TauLib.BookI.Coordinates.Hyperfact +cid: cid005044 +primary_alias: MOD0047 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.Hyperfact (126 lines, 20 declarations). +aliases: +- MOD0047 +- TauLib.BookI.Coordinates.Hyperfact +- book-i-coordinates-hyperfact +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.Hyperfact + module_path: TauLib/BookI/Coordinates/Hyperfact.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.Hyperfact (126 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005045.md b/_corpus_v3_items/cid005045.md new file mode 100644 index 000000000..12b58b7bb --- /dev/null +++ b/_corpus_v3_items/cid005045.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005045/ +title: TauLib.BookI.Coordinates.HyperfactFTA +cid: cid005045 +primary_alias: MOD0048 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactFTA (116 lines, 4 declarations). +aliases: +- MOD0048 +- TauLib.BookI.Coordinates.HyperfactFTA +- book-i-coordinates-hyperfact-fta +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.HyperfactFTA + module_path: TauLib/BookI/Coordinates/HyperfactFTA.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactFTA (116 lines, 4 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005046.md b/_corpus_v3_items/cid005046.md new file mode 100644 index 000000000..a38cc7dcb --- /dev/null +++ b/_corpus_v3_items/cid005046.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005046/ +title: TauLib.BookI.Coordinates.HyperfactIsomorphism +cid: cid005046 +primary_alias: MOD0049 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactIsomorphism (227 lines, 21 declarations). +aliases: +- MOD0049 +- TauLib.BookI.Coordinates.HyperfactIsomorphism +- book-i-coordinates-hyperfact-isomorphism +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.HyperfactIsomorphism + module_path: TauLib/BookI/Coordinates/HyperfactIsomorphism.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactIsomorphism (227 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005047.md b/_corpus_v3_items/cid005047.md new file mode 100644 index 000000000..a4aa0db6c --- /dev/null +++ b/_corpus_v3_items/cid005047.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005047/ +title: TauLib.BookI.Coordinates.HyperfactProp +cid: cid005047 +primary_alias: MOD0050 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactProp (137 lines, 4 declarations). +aliases: +- MOD0050 +- TauLib.BookI.Coordinates.HyperfactProp +- book-i-coordinates-hyperfact-prop +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.HyperfactProp + module_path: TauLib/BookI/Coordinates/HyperfactProp.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.HyperfactProp (137 lines, 4 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005048.md b/_corpus_v3_items/cid005048.md new file mode 100644 index 000000000..6949c59b1 --- /dev/null +++ b/_corpus_v3_items/cid005048.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005048/ +title: TauLib.BookI.Coordinates.IteratedPrime +cid: cid005048 +primary_alias: MOD0051 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.IteratedPrime (407 lines, 88 declarations). +aliases: +- MOD0051 +- TauLib.BookI.Coordinates.IteratedPrime +- book-i-coordinates-iterated-prime +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.IteratedPrime + module_path: TauLib/BookI/Coordinates/IteratedPrime.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.IteratedPrime (407 lines, 88 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005049.md b/_corpus_v3_items/cid005049.md new file mode 100644 index 000000000..fbde5ad49 --- /dev/null +++ b/_corpus_v3_items/cid005049.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005049/ +title: TauLib.BookI.Coordinates.NoTie +cid: cid005049 +primary_alias: MOD0052 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.NoTie (149 lines, 11 declarations). +aliases: +- MOD0052 +- TauLib.BookI.Coordinates.NoTie +- book-i-coordinates-no-tie +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.NoTie + module_path: TauLib/BookI/Coordinates/NoTie.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.NoTie (149 lines, 11 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005050.md b/_corpus_v3_items/cid005050.md new file mode 100644 index 000000000..075af3451 --- /dev/null +++ b/_corpus_v3_items/cid005050.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005050/ +title: TauLib.BookI.Coordinates.NormalForm +cid: cid005050 +primary_alias: MOD0053 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.NormalForm (137 lines, 36 declarations). +aliases: +- MOD0053 +- TauLib.BookI.Coordinates.NormalForm +- book-i-coordinates-normal-form +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.NormalForm + module_path: TauLib/BookI/Coordinates/NormalForm.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.NormalForm (137 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005051.md b/_corpus_v3_items/cid005051.md new file mode 100644 index 000000000..4ec8ab46c --- /dev/null +++ b/_corpus_v3_items/cid005051.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005051/ +title: TauLib.BookI.Coordinates.Primes +cid: cid005051 +primary_alias: MOD0054 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Coordinates.Primes (437 lines, 46 declarations). +aliases: +- MOD0054 +- TauLib.BookI.Coordinates.Primes +- book-i-coordinates-primes +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.Primes + module_path: TauLib/BookI/Coordinates/Primes.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Coordinates.Primes (437 lines, 46 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005052.md b/_corpus_v3_items/cid005052.md new file mode 100644 index 000000000..5de49028e --- /dev/null +++ b/_corpus_v3_items/cid005052.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005052/ +title: TauLib.BookI.Denotation.Arithmetic +cid: cid005052 +primary_alias: MOD0055 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.Arithmetic (225 lines, 27 declarations). +aliases: +- MOD0055 +- TauLib.BookI.Denotation.Arithmetic +- book-i-denotation-arithmetic +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Arithmetic + module_path: TauLib/BookI/Denotation/Arithmetic.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.Arithmetic (225 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005053.md b/_corpus_v3_items/cid005053.md new file mode 100644 index 000000000..03f846df9 --- /dev/null +++ b/_corpus_v3_items/cid005053.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005053/ +title: TauLib.BookI.Denotation.Equality +cid: cid005053 +primary_alias: MOD0056 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.Equality (85 lines, 10 declarations). +aliases: +- MOD0056 +- TauLib.BookI.Denotation.Equality +- book-i-denotation-equality +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Equality + module_path: TauLib/BookI/Denotation/Equality.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.Equality (85 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005054.md b/_corpus_v3_items/cid005054.md new file mode 100644 index 000000000..885cd9595 --- /dev/null +++ b/_corpus_v3_items/cid005054.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005054/ +title: TauLib.BookI.Denotation.GrowthEscape +cid: cid005054 +primary_alias: MOD0057 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.GrowthEscape (100 lines, 18 declarations). +aliases: +- MOD0057 +- TauLib.BookI.Denotation.GrowthEscape +- book-i-denotation-growth-escape +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.GrowthEscape + module_path: TauLib/BookI/Denotation/GrowthEscape.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.GrowthEscape (100 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005055.md b/_corpus_v3_items/cid005055.md new file mode 100644 index 000000000..b88abd7d1 --- /dev/null +++ b/_corpus_v3_items/cid005055.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005055/ +title: TauLib.BookI.Denotation.Order +cid: cid005055 +primary_alias: MOD0058 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.Order (159 lines, 14 declarations). +aliases: +- MOD0058 +- TauLib.BookI.Denotation.Order +- book-i-denotation-order +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Order + module_path: TauLib/BookI/Denotation/Order.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.Order (159 lines, 14 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005056.md b/_corpus_v3_items/cid005056.md new file mode 100644 index 000000000..e931e226d --- /dev/null +++ b/_corpus_v3_items/cid005056.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005056/ +title: TauLib.BookI.Denotation.ProgramMonoid +cid: cid005056 +primary_alias: MOD0059 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.ProgramMonoid (154 lines, 18 declarations). +aliases: +- MOD0059 +- TauLib.BookI.Denotation.ProgramMonoid +- book-i-denotation-program-monoid +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.ProgramMonoid + module_path: TauLib/BookI/Denotation/ProgramMonoid.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.ProgramMonoid (154 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005057.md b/_corpus_v3_items/cid005057.md new file mode 100644 index 000000000..40353140c --- /dev/null +++ b/_corpus_v3_items/cid005057.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005057/ +title: TauLib.BookI.Denotation.RankTransfer +cid: cid005057 +primary_alias: MOD0060 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.RankTransfer (81 lines, 10 declarations). +aliases: +- MOD0060 +- TauLib.BookI.Denotation.RankTransfer +- book-i-denotation-rank-transfer +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.RankTransfer + module_path: TauLib/BookI/Denotation/RankTransfer.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.RankTransfer (81 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005058.md b/_corpus_v3_items/cid005058.md new file mode 100644 index 000000000..66a82de83 --- /dev/null +++ b/_corpus_v3_items/cid005058.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005058/ +title: TauLib.BookI.Denotation.Structural +cid: cid005058 +primary_alias: MOD0061 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.Structural (390 lines, 52 declarations). +aliases: +- MOD0061 +- TauLib.BookI.Denotation.Structural +- book-i-denotation-structural +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Structural + module_path: TauLib/BookI/Denotation/Structural.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.Structural (390 lines, 52 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005059.md b/_corpus_v3_items/cid005059.md new file mode 100644 index 000000000..9e75c11bf --- /dev/null +++ b/_corpus_v3_items/cid005059.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005059/ +title: TauLib.BookI.Denotation.TauIdx +cid: cid005059 +primary_alias: MOD0062 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Denotation.TauIdx (109 lines, 11 declarations). +aliases: +- MOD0062 +- TauLib.BookI.Denotation.TauIdx +- book-i-denotation-tau-idx +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.TauIdx + module_path: TauLib/BookI/Denotation/TauIdx.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Denotation.TauIdx (109 lines, 11 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005060.md b/_corpus_v3_items/cid005060.md new file mode 100644 index 000000000..bef2ef9ce --- /dev/null +++ b/_corpus_v3_items/cid005060.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005060/ +title: TauLib.BookI.Holomorphy.BoundaryInterior +cid: cid005060 +primary_alias: MOD0063 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.BoundaryInterior (157 lines, 12 declarations). +aliases: +- MOD0063 +- TauLib.BookI.Holomorphy.BoundaryInterior +- book-i-holomorphy-boundary-interior +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.BoundaryInterior + module_path: TauLib/BookI/Holomorphy/BoundaryInterior.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.BoundaryInterior (157 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005061.md b/_corpus_v3_items/cid005061.md new file mode 100644 index 000000000..3952eda3f --- /dev/null +++ b/_corpus_v3_items/cid005061.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005061/ +title: TauLib.BookI.Holomorphy.DHolomorphic +cid: cid005061 +primary_alias: MOD0064 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.DHolomorphic (195 lines, 28 declarations). +aliases: +- MOD0064 +- TauLib.BookI.Holomorphy.DHolomorphic +- book-i-holomorphy-dholomorphic +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DHolomorphic + module_path: TauLib/BookI/Holomorphy/DHolomorphic.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.DHolomorphic (195 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005062.md b/_corpus_v3_items/cid005062.md new file mode 100644 index 000000000..67b846535 --- /dev/null +++ b/_corpus_v3_items/cid005062.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005062/ +title: TauLib.BookI.Holomorphy.DiagonalProtection +cid: cid005062 +primary_alias: MOD0065 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.DiagonalProtection (225 lines, 23 declarations). +aliases: +- MOD0065 +- TauLib.BookI.Holomorphy.DiagonalProtection +- book-i-holomorphy-diagonal-protection +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DiagonalProtection + module_path: TauLib/BookI/Holomorphy/DiagonalProtection.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.DiagonalProtection (225 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005063.md b/_corpus_v3_items/cid005063.md new file mode 100644 index 000000000..b337a6d7c --- /dev/null +++ b/_corpus_v3_items/cid005063.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005063/ +title: TauLib.BookI.Holomorphy.GlobalHartogs +cid: cid005063 +primary_alias: MOD0066 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.GlobalHartogs (142 lines, 10 declarations). +aliases: +- MOD0066 +- TauLib.BookI.Holomorphy.GlobalHartogs +- book-i-holomorphy-global-hartogs +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.GlobalHartogs + module_path: TauLib/BookI/Holomorphy/GlobalHartogs.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.GlobalHartogs (142 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005064.md b/_corpus_v3_items/cid005064.md new file mode 100644 index 000000000..1bd57801f --- /dev/null +++ b/_corpus_v3_items/cid005064.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005064/ +title: TauLib.BookI.Holomorphy.H6DiagonalDiscipline +cid: cid005064 +primary_alias: MOD0067 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6DiagonalDiscipline (252 lines, 10 declarations). +aliases: +- MOD0067 +- TauLib.BookI.Holomorphy.H6DiagonalDiscipline +- book-i-holomorphy-h6-diagonal-discipline +- I.T110 +- I.T113 +- I.T115 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.H6DiagonalDiscipline + module_path: TauLib/BookI/Holomorphy/H6DiagonalDiscipline.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6DiagonalDiscipline (252 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005065.md b/_corpus_v3_items/cid005065.md new file mode 100644 index 000000000..d0a56fcb6 --- /dev/null +++ b/_corpus_v3_items/cid005065.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005065/ +title: TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR +cid: cid005065 +primary_alias: MOD0068 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR (320 lines, 8 declarations). +aliases: +- MOD0068 +- TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR +- book-i-holomorphy-h6-earned-codomain-wave-cr +- I.T112 +- I.T126 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR + module_path: TauLib/BookI/Holomorphy/H6EarnedCodomainWaveCR.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR (320 lines, 8 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005066.md b/_corpus_v3_items/cid005066.md new file mode 100644 index 000000000..6cba57f24 --- /dev/null +++ b/_corpus_v3_items/cid005066.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005066/ +title: TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd +cid: cid005066 +primary_alias: MOD0069 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd (362 lines, 13 declarations). +aliases: +- MOD0069 +- TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd +- book-i-holomorphy-h6-sigma-idem-hol-end +- I.T117 +- I.T120 +- I.T132 +- I.T139 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd + module_path: TauLib/BookI/Holomorphy/H6SigmaIdemHolEnd.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd (362 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005067.md b/_corpus_v3_items/cid005067.md new file mode 100644 index 000000000..11d477850 --- /dev/null +++ b/_corpus_v3_items/cid005067.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005067/ +title: TauLib.BookI.Holomorphy.IdentityTheorem +cid: cid005067 +primary_alias: MOD0070 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.IdentityTheorem (244 lines, 19 declarations). +aliases: +- MOD0070 +- TauLib.BookI.Holomorphy.IdentityTheorem +- book-i-holomorphy-identity-theorem +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.IdentityTheorem + module_path: TauLib/BookI/Holomorphy/IdentityTheorem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.IdentityTheorem (244 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005068.md b/_corpus_v3_items/cid005068.md new file mode 100644 index 000000000..2c1977ba8 --- /dev/null +++ b/_corpus_v3_items/cid005068.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005068/ +title: TauLib.BookI.Holomorphy.PresheafEssence +cid: cid005068 +primary_alias: MOD0071 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.PresheafEssence (287 lines, 28 declarations). +aliases: +- MOD0071 +- TauLib.BookI.Holomorphy.PresheafEssence +- book-i-holomorphy-presheaf-essence +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.PresheafEssence + module_path: TauLib/BookI/Holomorphy/PresheafEssence.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.PresheafEssence (287 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005069.md b/_corpus_v3_items/cid005069.md new file mode 100644 index 000000000..ccea8f4f7 --- /dev/null +++ b/_corpus_v3_items/cid005069.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005069/ +title: TauLib.BookI.Holomorphy.SpectralCoefficients +cid: cid005069 +primary_alias: MOD0072 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.SpectralCoefficients (118 lines, 13 declarations). +aliases: +- MOD0072 +- TauLib.BookI.Holomorphy.SpectralCoefficients +- book-i-holomorphy-spectral-coefficients +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.SpectralCoefficients + module_path: TauLib/BookI/Holomorphy/SpectralCoefficients.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.SpectralCoefficients (118 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005070.md b/_corpus_v3_items/cid005070.md new file mode 100644 index 000000000..5692c94e1 --- /dev/null +++ b/_corpus_v3_items/cid005070.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005070/ +title: TauLib.BookI.Holomorphy.TauHolomorphic +cid: cid005070 +primary_alias: MOD0073 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.TauHolomorphic (268 lines, 32 declarations). +aliases: +- MOD0073 +- TauLib.BookI.Holomorphy.TauHolomorphic +- book-i-holomorphy-tau-holomorphic +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.TauHolomorphic + module_path: TauLib/BookI/Holomorphy/TauHolomorphic.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.TauHolomorphic (268 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005071.md b/_corpus_v3_items/cid005071.md new file mode 100644 index 000000000..0dc16f696 --- /dev/null +++ b/_corpus_v3_items/cid005071.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005071/ +title: TauLib.BookI.Holomorphy.Thinness +cid: cid005071 +primary_alias: MOD0074 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Holomorphy.Thinness (138 lines, 19 declarations). +aliases: +- MOD0074 +- TauLib.BookI.Holomorphy.Thinness +- book-i-holomorphy-thinness +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.Thinness + module_path: TauLib/BookI/Holomorphy/Thinness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Holomorphy.Thinness (138 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005072.md b/_corpus_v3_items/cid005072.md new file mode 100644 index 000000000..6b64c1054 --- /dev/null +++ b/_corpus_v3_items/cid005072.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005072/ +title: TauLib.BookI.Kernel.ActionQuantum +cid: cid005072 +primary_alias: MOD0075 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Kernel.ActionQuantum (160 lines, 21 declarations). +aliases: +- MOD0075 +- TauLib.BookI.Kernel.ActionQuantum +- book-i-kernel-action-quantum +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.ActionQuantum + module_path: TauLib/BookI/Kernel/ActionQuantum.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Kernel.ActionQuantum (160 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005073.md b/_corpus_v3_items/cid005073.md new file mode 100644 index 000000000..ac89a51c8 --- /dev/null +++ b/_corpus_v3_items/cid005073.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005073/ +title: TauLib.BookI.Kernel.Axioms +cid: cid005073 +primary_alias: MOD0076 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Kernel.Axioms (158 lines, 13 declarations). +aliases: +- MOD0076 +- TauLib.BookI.Kernel.Axioms +- book-i-kernel-axioms +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Kernel.Axioms (158 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005074.md b/_corpus_v3_items/cid005074.md new file mode 100644 index 000000000..6aee74141 --- /dev/null +++ b/_corpus_v3_items/cid005074.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005074/ +title: TauLib.BookI.Kernel.Diagonal +cid: cid005074 +primary_alias: MOD0077 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Kernel.Diagonal (87 lines, 8 declarations). +aliases: +- MOD0077 +- TauLib.BookI.Kernel.Diagonal +- book-i-kernel-diagonal +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Diagonal + module_path: TauLib/BookI/Kernel/Diagonal.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Kernel.Diagonal (87 lines, 8 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005075.md b/_corpus_v3_items/cid005075.md new file mode 100644 index 000000000..d587062ca --- /dev/null +++ b/_corpus_v3_items/cid005075.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005075/ +title: TauLib.BookI.Kernel.Signature +cid: cid005075 +primary_alias: MOD0078 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Kernel.Signature (104 lines, 4 declarations). +aliases: +- MOD0078 +- TauLib.BookI.Kernel.Signature +- book-i-kernel-signature +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Signature + module_path: TauLib/BookI/Kernel/Signature.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Kernel.Signature (104 lines, 4 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005076.md b/_corpus_v3_items/cid005076.md new file mode 100644 index 000000000..e27e8b501 --- /dev/null +++ b/_corpus_v3_items/cid005076.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005076/ +title: TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup +cid: cid005076 +primary_alias: MOD0079 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup (239 lines, 12 declarations). +aliases: +- MOD0079 +- TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup +- book-i-kernel-foundation-admissible-symmetry-group +- I.T174 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup + module_path: TauLib/BookI/KernelFoundation/AdmissibleSymmetryGroup.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup (239 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005077.md b/_corpus_v3_items/cid005077.md new file mode 100644 index 000000000..41e3fe462 --- /dev/null +++ b/_corpus_v3_items/cid005077.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005077/ +title: TauLib.BookI.KernelFoundation.GirardLinearEmbedding +cid: cid005077 +primary_alias: MOD0080 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.KernelFoundation.GirardLinearEmbedding (305 lines, 18 declarations). +aliases: +- MOD0080 +- TauLib.BookI.KernelFoundation.GirardLinearEmbedding +- book-i-kernel-foundation-girard-linear-embedding +- I.T175 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.KernelFoundation.GirardLinearEmbedding + module_path: TauLib/BookI/KernelFoundation/GirardLinearEmbedding.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.KernelFoundation.GirardLinearEmbedding (305 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005078.md b/_corpus_v3_items/cid005078.md new file mode 100644 index 000000000..326a18cf4 --- /dev/null +++ b/_corpus_v3_items/cid005078.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005078/ +title: TauLib.BookI.KernelFoundation.H8KernelSynthesis +cid: cid005078 +primary_alias: MOD0081 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.KernelFoundation.H8KernelSynthesis (284 lines, 9 declarations). +aliases: +- MOD0081 +- TauLib.BookI.KernelFoundation.H8KernelSynthesis +- book-i-kernel-foundation-h8-kernel-synthesis +- I.T173 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.KernelFoundation.H8KernelSynthesis + module_path: TauLib/BookI/KernelFoundation/H8KernelSynthesis.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.KernelFoundation.H8KernelSynthesis (284 lines, 9 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005079.md b/_corpus_v3_items/cid005079.md new file mode 100644 index 000000000..e6ffed624 --- /dev/null +++ b/_corpus_v3_items/cid005079.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005079/ +title: TauLib.BookI.KernelFoundation.ScalarBridges +cid: cid005079 +primary_alias: MOD0082 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.KernelFoundation.ScalarBridges (241 lines, 15 declarations). +aliases: +- MOD0082 +- TauLib.BookI.KernelFoundation.ScalarBridges +- book-i-kernel-foundation-scalar-bridges +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.KernelFoundation.ScalarBridges + module_path: TauLib/BookI/KernelFoundation/ScalarBridges.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.KernelFoundation.ScalarBridges (241 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005080.md b/_corpus_v3_items/cid005080.md new file mode 100644 index 000000000..1e67872ff --- /dev/null +++ b/_corpus_v3_items/cid005080.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005080/ +title: TauLib.BookI.Logic.BooleanRecovery +cid: cid005080 +primary_alias: MOD0083 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Logic.BooleanRecovery (356 lines, 66 declarations). +aliases: +- MOD0083 +- TauLib.BookI.Logic.BooleanRecovery +- book-i-logic-boolean-recovery +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.BooleanRecovery + module_path: TauLib/BookI/Logic/BooleanRecovery.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Logic.BooleanRecovery (356 lines, 66 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005081.md b/_corpus_v3_items/cid005081.md new file mode 100644 index 000000000..9d6358991 --- /dev/null +++ b/_corpus_v3_items/cid005081.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005081/ +title: TauLib.BookI.Logic.Explosion +cid: cid005081 +primary_alias: MOD0084 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Logic.Explosion (252 lines, 50 declarations). +aliases: +- MOD0084 +- TauLib.BookI.Logic.Explosion +- book-i-logic-explosion +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.Explosion + module_path: TauLib/BookI/Logic/Explosion.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Logic.Explosion (252 lines, 50 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005082.md b/_corpus_v3_items/cid005082.md new file mode 100644 index 000000000..c23df5bfa --- /dev/null +++ b/_corpus_v3_items/cid005082.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005082/ +title: TauLib.BookI.Logic.Truth4 +cid: cid005082 +primary_alias: MOD0085 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Logic.Truth4 (361 lines, 75 declarations). +aliases: +- MOD0085 +- TauLib.BookI.Logic.Truth4 +- book-i-logic-truth4 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.Truth4 + module_path: TauLib/BookI/Logic/Truth4.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Logic.Truth4 (361 lines, 75 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005083.md b/_corpus_v3_items/cid005083.md new file mode 100644 index 000000000..d31ceae49 --- /dev/null +++ b/_corpus_v3_items/cid005083.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005083/ +title: TauLib.BookI.MetaLogic.DiagonalResonance +cid: cid005083 +primary_alias: MOD0086 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.DiagonalResonance (214 lines, 36 declarations). +aliases: +- MOD0086 +- TauLib.BookI.MetaLogic.DiagonalResonance +- book-i-meta-logic-diagonal-resonance +- I.R24 +- I.R25 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.DiagonalResonance + module_path: TauLib/BookI/MetaLogic/DiagonalResonance.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.DiagonalResonance (214 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005084.md b/_corpus_v3_items/cid005084.md new file mode 100644 index 000000000..4244e902e --- /dev/null +++ b/_corpus_v3_items/cid005084.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005084/ +title: TauLib.BookI.MetaLogic.LinearDiscipline +cid: cid005084 +primary_alias: MOD0087 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.LinearDiscipline (321 lines, 50 declarations). +aliases: +- MOD0087 +- TauLib.BookI.MetaLogic.LinearDiscipline +- book-i-meta-logic-linear-discipline +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.LinearDiscipline + module_path: TauLib/BookI/MetaLogic/LinearDiscipline.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.LinearDiscipline (321 lines, 50 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005085.md b/_corpus_v3_items/cid005085.md new file mode 100644 index 000000000..46bd81547 --- /dev/null +++ b/_corpus_v3_items/cid005085.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005085/ +title: TauLib.BookI.MetaLogic.LinearityAudit +cid: cid005085 +primary_alias: MOD0088 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.LinearityAudit (380 lines, 47 declarations). +aliases: +- MOD0088 +- TauLib.BookI.MetaLogic.LinearityAudit +- book-i-meta-logic-linearity-audit +- I.R17 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.LinearityAudit + module_path: TauLib/BookI/MetaLogic/LinearityAudit.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.LinearityAudit (380 lines, 47 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005086.md b/_corpus_v3_items/cid005086.md new file mode 100644 index 000000000..b6b2e2244 --- /dev/null +++ b/_corpus_v3_items/cid005086.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005086/ +title: TauLib.BookI.MetaLogic.OnticInvariance +cid: cid005086 +primary_alias: MOD0089 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.OnticInvariance (191 lines, 26 declarations). +aliases: +- MOD0089 +- TauLib.BookI.MetaLogic.OnticInvariance +- book-i-meta-logic-ontic-invariance +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.OnticInvariance + module_path: TauLib/BookI/MetaLogic/OnticInvariance.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.OnticInvariance (191 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005087.md b/_corpus_v3_items/cid005087.md new file mode 100644 index 000000000..574795197 --- /dev/null +++ b/_corpus_v3_items/cid005087.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005087/ +title: TauLib.BookI.MetaLogic.ReceptionCriterion +cid: cid005087 +primary_alias: MOD0090 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.ReceptionCriterion (226 lines, 36 declarations). +aliases: +- MOD0090 +- TauLib.BookI.MetaLogic.ReceptionCriterion +- book-i-meta-logic-reception-criterion +- I.R26 +- I.R27 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.ReceptionCriterion + module_path: TauLib/BookI/MetaLogic/ReceptionCriterion.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.ReceptionCriterion (226 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005088.md b/_corpus_v3_items/cid005088.md new file mode 100644 index 000000000..3aa259f4a --- /dev/null +++ b/_corpus_v3_items/cid005088.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005088/ +title: TauLib.BookI.MetaLogic.StructuralExclusion +cid: cid005088 +primary_alias: MOD0091 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.StructuralExclusion (214 lines, 33 declarations). +aliases: +- MOD0091 +- TauLib.BookI.MetaLogic.StructuralExclusion +- book-i-meta-logic-structural-exclusion +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.StructuralExclusion + module_path: TauLib/BookI/MetaLogic/StructuralExclusion.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.StructuralExclusion (214 lines, 33 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005089.md b/_corpus_v3_items/cid005089.md new file mode 100644 index 000000000..cc3ae1103 --- /dev/null +++ b/_corpus_v3_items/cid005089.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005089/ +title: TauLib.BookI.MetaLogic.Substrate +cid: cid005089 +primary_alias: MOD0092 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.MetaLogic.Substrate (147 lines, 26 declarations). +aliases: +- MOD0092 +- TauLib.BookI.MetaLogic.Substrate +- book-i-meta-logic-substrate +- I.R15 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.Substrate + module_path: TauLib/BookI/MetaLogic/Substrate.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.MetaLogic.Substrate (147 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005090.md b/_corpus_v3_items/cid005090.md new file mode 100644 index 000000000..d08fc01b4 --- /dev/null +++ b/_corpus_v3_items/cid005090.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005090/ +title: TauLib.BookI.Orbit.Closure +cid: cid005090 +primary_alias: MOD0093 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Closure (96 lines, 7 declarations). +aliases: +- MOD0093 +- TauLib.BookI.Orbit.Closure +- book-i-orbit-closure +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Closure + module_path: TauLib/BookI/Orbit/Closure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Closure (96 lines, 7 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005091.md b/_corpus_v3_items/cid005091.md new file mode 100644 index 000000000..076f95c4f --- /dev/null +++ b/_corpus_v3_items/cid005091.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005091/ +title: TauLib.BookI.Orbit.Countability +cid: cid005091 +primary_alias: MOD0094 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Countability (76 lines, 7 declarations). +aliases: +- MOD0094 +- TauLib.BookI.Orbit.Countability +- book-i-orbit-countability +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Countability + module_path: TauLib/BookI/Orbit/Countability.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Countability (76 lines, 7 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005092.md b/_corpus_v3_items/cid005092.md new file mode 100644 index 000000000..5c97c121e --- /dev/null +++ b/_corpus_v3_items/cid005092.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005092/ +title: TauLib.BookI.Orbit.Generation +cid: cid005092 +primary_alias: MOD0095 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Generation (146 lines, 16 declarations). +aliases: +- MOD0095 +- TauLib.BookI.Orbit.Generation +- book-i-orbit-generation +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Generation + module_path: TauLib/BookI/Orbit/Generation.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Generation (146 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005093.md b/_corpus_v3_items/cid005093.md new file mode 100644 index 000000000..fab9013d0 --- /dev/null +++ b/_corpus_v3_items/cid005093.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005093/ +title: TauLib.BookI.Orbit.Ladder +cid: cid005093 +primary_alias: MOD0096 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Ladder (158 lines, 12 declarations). +aliases: +- MOD0096 +- TauLib.BookI.Orbit.Ladder +- book-i-orbit-ladder +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Ladder + module_path: TauLib/BookI/Orbit/Ladder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Ladder (158 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005094.md b/_corpus_v3_items/cid005094.md new file mode 100644 index 000000000..644fdefe5 --- /dev/null +++ b/_corpus_v3_items/cid005094.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005094/ +title: TauLib.BookI.Orbit.Rigidity +cid: cid005094 +primary_alias: MOD0097 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Rigidity (166 lines, 10 declarations). +aliases: +- MOD0097 +- TauLib.BookI.Orbit.Rigidity +- book-i-orbit-rigidity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Rigidity + module_path: TauLib/BookI/Orbit/Rigidity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Rigidity (166 lines, 10 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005095.md b/_corpus_v3_items/cid005095.md new file mode 100644 index 000000000..f3451c048 --- /dev/null +++ b/_corpus_v3_items/cid005095.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005095/ +title: TauLib.BookI.Orbit.Saturation +cid: cid005095 +primary_alias: MOD0098 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Orbit.Saturation (163 lines, 9 declarations). +aliases: +- MOD0098 +- TauLib.BookI.Orbit.Saturation +- book-i-orbit-saturation +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Saturation + module_path: TauLib/BookI/Orbit/Saturation.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Orbit.Saturation (163 lines, 9 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005096.md b/_corpus_v3_items/cid005096.md new file mode 100644 index 000000000..7160ce880 --- /dev/null +++ b/_corpus_v3_items/cid005096.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005096/ +title: TauLib.BookI.Polarity.BipolarAlgebra +cid: cid005096 +primary_alias: MOD0099 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.BipolarAlgebra (344 lines, 57 declarations). +aliases: +- MOD0099 +- TauLib.BookI.Polarity.BipolarAlgebra +- book-i-polarity-bipolar-algebra +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.BipolarAlgebra + module_path: TauLib/BookI/Polarity/BipolarAlgebra.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.BipolarAlgebra (344 lines, 57 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005097.md b/_corpus_v3_items/cid005097.md new file mode 100644 index 000000000..7e66f115b --- /dev/null +++ b/_corpus_v3_items/cid005097.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005097/ +title: TauLib.BookI.Polarity.ChineseRemainder +cid: cid005097 +primary_alias: MOD0100 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.ChineseRemainder (368 lines, 125 declarations). +aliases: +- MOD0100 +- TauLib.BookI.Polarity.ChineseRemainder +- book-i-polarity-chinese-remainder +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.ChineseRemainder + module_path: TauLib/BookI/Polarity/ChineseRemainder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.ChineseRemainder (368 lines, 125 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005098.md b/_corpus_v3_items/cid005098.md new file mode 100644 index 000000000..81f1fb708 --- /dev/null +++ b/_corpus_v3_items/cid005098.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005098/ +title: TauLib.BookI.Polarity.H2H3ClassifierBridge +cid: cid005098 +primary_alias: MOD0101 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.H2H3ClassifierBridge (234 lines, 27 declarations). +aliases: +- MOD0101 +- TauLib.BookI.Polarity.H2H3ClassifierBridge +- book-i-polarity-h2-h3-classifier-bridge +- I.D131 +- I.T87 +- I.T93 +- I.T95 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.H2H3ClassifierBridge + module_path: TauLib/BookI/Polarity/H2H3ClassifierBridge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.H2H3ClassifierBridge (234 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005099.md b/_corpus_v3_items/cid005099.md new file mode 100644 index 000000000..374acb65f --- /dev/null +++ b/_corpus_v3_items/cid005099.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005099/ +title: TauLib.BookI.Polarity.H4BoundaryAlgebra +cid: cid005099 +primary_alias: MOD0102 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.H4BoundaryAlgebra (365 lines, 28 declarations). +aliases: +- MOD0102 +- TauLib.BookI.Polarity.H4BoundaryAlgebra +- book-i-polarity-h4-boundary-algebra +- I.T88 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.H4BoundaryAlgebra + module_path: TauLib/BookI/Polarity/H4BoundaryAlgebra.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.H4BoundaryAlgebra (365 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005100.md b/_corpus_v3_items/cid005100.md new file mode 100644 index 000000000..1f299a72b --- /dev/null +++ b/_corpus_v3_items/cid005100.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005100/ +title: TauLib.BookI.Polarity.H4UniquenessElliptic +cid: cid005100 +primary_alias: MOD0103 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.H4UniquenessElliptic (362 lines, 26 declarations). +aliases: +- MOD0103 +- TauLib.BookI.Polarity.H4UniquenessElliptic +- book-i-polarity-h4-uniqueness-elliptic +- I.D134 +- I.T107 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.H4UniquenessElliptic + module_path: TauLib/BookI/Polarity/H4UniquenessElliptic.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.H4UniquenessElliptic (362 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005101.md b/_corpus_v3_items/cid005101.md new file mode 100644 index 000000000..7e17e3422 --- /dev/null +++ b/_corpus_v3_items/cid005101.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005101/ +title: TauLib.BookI.Polarity.InverseLimit +cid: cid005101 +primary_alias: MOD0104 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.InverseLimit (233 lines, 12 declarations). +aliases: +- MOD0104 +- TauLib.BookI.Polarity.InverseLimit +- book-i-polarity-inverse-limit +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.InverseLimit + module_path: TauLib/BookI/Polarity/InverseLimit.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.InverseLimit (233 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005102.md b/_corpus_v3_items/cid005102.md new file mode 100644 index 000000000..05bfe072c --- /dev/null +++ b/_corpus_v3_items/cid005102.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005102/ +title: TauLib.BookI.Polarity.Lemniscate +cid: cid005102 +primary_alias: MOD0105 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.Lemniscate (150 lines, 20 declarations). +aliases: +- MOD0105 +- TauLib.BookI.Polarity.Lemniscate +- book-i-polarity-lemniscate +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.Lemniscate + module_path: TauLib/BookI/Polarity/Lemniscate.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.Lemniscate (150 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005103.md b/_corpus_v3_items/cid005103.md new file mode 100644 index 000000000..ae2dc6899 --- /dev/null +++ b/_corpus_v3_items/cid005103.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005103/ +title: TauLib.BookI.Polarity.OmegaGerms +cid: cid005103 +primary_alias: MOD0106 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.OmegaGerms (416 lines, 53 declarations). +aliases: +- MOD0106 +- TauLib.BookI.Polarity.OmegaGerms +- book-i-polarity-omega-germs +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.OmegaGerms + module_path: TauLib/BookI/Polarity/OmegaGerms.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.OmegaGerms (416 lines, 53 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005104.md b/_corpus_v3_items/cid005104.md new file mode 100644 index 000000000..f2ed19677 --- /dev/null +++ b/_corpus_v3_items/cid005104.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005104/ +title: TauLib.BookI.Polarity.OmegaRing +cid: cid005104 +primary_alias: MOD0107 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.OmegaRing (391 lines, 62 declarations). +aliases: +- MOD0107 +- TauLib.BookI.Polarity.OmegaRing +- book-i-polarity-omega-ring +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.OmegaRing + module_path: TauLib/BookI/Polarity/OmegaRing.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.OmegaRing (391 lines, 62 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005105.md b/_corpus_v3_items/cid005105.md new file mode 100644 index 000000000..1a1a5c660 --- /dev/null +++ b/_corpus_v3_items/cid005105.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005105/ +title: TauLib.BookI.Polarity.Polarity +cid: cid005105 +primary_alias: MOD0108 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.Polarity (300 lines, 61 declarations). +aliases: +- MOD0108 +- TauLib.BookI.Polarity.Polarity +- book-i-polarity-polarity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.Polarity + module_path: TauLib/BookI/Polarity/Polarity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.Polarity (300 lines, 61 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005106.md b/_corpus_v3_items/cid005106.md new file mode 100644 index 000000000..8dbeb2e71 --- /dev/null +++ b/_corpus_v3_items/cid005106.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005106/ +title: TauLib.BookI.Polarity.PolarizedGerms +cid: cid005106 +primary_alias: MOD0109 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.PolarizedGerms (171 lines, 28 declarations). +aliases: +- MOD0109 +- TauLib.BookI.Polarity.PolarizedGerms +- book-i-polarity-polarized-germs +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.PolarizedGerms + module_path: TauLib/BookI/Polarity/PolarizedGerms.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.PolarizedGerms (171 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005107.md b/_corpus_v3_items/cid005107.md new file mode 100644 index 000000000..4c644be85 --- /dev/null +++ b/_corpus_v3_items/cid005107.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005107/ +title: TauLib.BookI.Polarity.PrimePolarityClassifier +cid: cid005107 +primary_alias: MOD0110 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.PrimePolarityClassifier (361 lines, 51 declarations). +aliases: +- MOD0110 +- TauLib.BookI.Polarity.PrimePolarityClassifier +- book-i-polarity-prime-polarity-classifier +- I.D130 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.PrimePolarityClassifier + module_path: TauLib/BookI/Polarity/PrimePolarityClassifier.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.PrimePolarityClassifier (361 lines, 51 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005108.md b/_corpus_v3_items/cid005108.md new file mode 100644 index 000000000..b1aa56d7b --- /dev/null +++ b/_corpus_v3_items/cid005108.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005108/ +title: TauLib.BookI.Polarity.PrimePolarityIsomorphism +cid: cid005108 +primary_alias: MOD0111 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.PrimePolarityIsomorphism (245 lines, 27 declarations). +aliases: +- MOD0111 +- TauLib.BookI.Polarity.PrimePolarityIsomorphism +- book-i-polarity-prime-polarity-isomorphism +- I.T91 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.PrimePolarityIsomorphism + module_path: TauLib/BookI/Polarity/PrimePolarityIsomorphism.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.PrimePolarityIsomorphism (245 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005109.md b/_corpus_v3_items/cid005109.md new file mode 100644 index 000000000..4e876d67e --- /dev/null +++ b/_corpus_v3_items/cid005109.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005109/ +title: TauLib.BookI.Polarity.SplitComplexCouplingLift +cid: cid005109 +primary_alias: MOD0112 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.SplitComplexCouplingLift (400 lines, 33 declarations). +aliases: +- MOD0112 +- TauLib.BookI.Polarity.SplitComplexCouplingLift +- book-i-polarity-split-complex-coupling-lift +- I.D129 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.SplitComplexCouplingLift + module_path: TauLib/BookI/Polarity/SplitComplexCouplingLift.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.SplitComplexCouplingLift (400 lines, 33 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005110.md b/_corpus_v3_items/cid005110.md new file mode 100644 index 000000000..f326eb23d --- /dev/null +++ b/_corpus_v3_items/cid005110.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005110/ +title: TauLib.BookI.Polarity.TeichmuellerLift +cid: cid005110 +primary_alias: MOD0113 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Polarity.TeichmuellerLift (273 lines, 72 declarations). +aliases: +- MOD0113 +- TauLib.BookI.Polarity.TeichmuellerLift +- book-i-polarity-teichmueller-lift +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.TeichmuellerLift + module_path: TauLib/BookI/Polarity/TeichmuellerLift.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Polarity.TeichmuellerLift (273 lines, 72 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005111.md b/_corpus_v3_items/cid005111.md new file mode 100644 index 000000000..b48326612 --- /dev/null +++ b/_corpus_v3_items/cid005111.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005111/ +title: TauLib.BookI.Sets.CantorRefutation +cid: cid005111 +primary_alias: MOD0114 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.CantorRefutation (230 lines, 16 declarations). +aliases: +- MOD0114 +- TauLib.BookI.Sets.CantorRefutation +- book-i-sets-cantor-refutation +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.CantorRefutation + module_path: TauLib/BookI/Sets/CantorRefutation.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.CantorRefutation (230 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005112.md b/_corpus_v3_items/cid005112.md new file mode 100644 index 000000000..b52a09c35 --- /dev/null +++ b/_corpus_v3_items/cid005112.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005112/ +title: TauLib.BookI.Sets.Counting +cid: cid005112 +primary_alias: MOD0115 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.Counting (187 lines, 15 declarations). +aliases: +- MOD0115 +- TauLib.BookI.Sets.Counting +- book-i-sets-counting +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Counting + module_path: TauLib/BookI/Sets/Counting.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.Counting (187 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005113.md b/_corpus_v3_items/cid005113.md new file mode 100644 index 000000000..4b5842d2a --- /dev/null +++ b/_corpus_v3_items/cid005113.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005113/ +title: TauLib.BookI.Sets.Membership +cid: cid005113 +primary_alias: MOD0116 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.Membership (152 lines, 16 declarations). +aliases: +- MOD0116 +- TauLib.BookI.Sets.Membership +- book-i-sets-membership +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Membership + module_path: TauLib/BookI/Sets/Membership.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.Membership (152 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005114.md b/_corpus_v3_items/cid005114.md new file mode 100644 index 000000000..f8ccd729b --- /dev/null +++ b/_corpus_v3_items/cid005114.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005114/ +title: TauLib.BookI.Sets.Operations +cid: cid005114 +primary_alias: MOD0117 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.Operations (336 lines, 31 declarations). +aliases: +- MOD0117 +- TauLib.BookI.Sets.Operations +- book-i-sets-operations +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Operations + module_path: TauLib/BookI/Sets/Operations.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.Operations (336 lines, 31 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005115.md b/_corpus_v3_items/cid005115.md new file mode 100644 index 000000000..1b1a399b1 --- /dev/null +++ b/_corpus_v3_items/cid005115.md @@ -0,0 +1,73 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005115/ +title: TauLib.BookI.Sets.OrbitSets +cid: cid005115 +primary_alias: MOD0118 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.OrbitSets (301 lines, 31 declarations). +aliases: +- MOD0118 +- TauLib.BookI.Sets.OrbitSets +- book-i-sets-orbit-sets +- I.R28 +- I.R29 +- I.R30 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +- predicate: formalized_by + object: cid005346 +- predicate: formalized_by + object: cid005347 +- predicate: formalized_by + object: cid005348 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.OrbitSets + module_path: TauLib/BookI/Sets/OrbitSets.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.OrbitSets (301 lines, 31 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005116.md b/_corpus_v3_items/cid005116.md new file mode 100644 index 000000000..3b3b76d2b --- /dev/null +++ b/_corpus_v3_items/cid005116.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005116/ +title: TauLib.BookI.Sets.Powerset +cid: cid005116 +primary_alias: MOD0119 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.Powerset (141 lines, 16 declarations). +aliases: +- MOD0119 +- TauLib.BookI.Sets.Powerset +- book-i-sets-powerset +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Powerset + module_path: TauLib/BookI/Sets/Powerset.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.Powerset (141 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005117.md b/_corpus_v3_items/cid005117.md new file mode 100644 index 000000000..8870a6a05 --- /dev/null +++ b/_corpus_v3_items/cid005117.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005117/ +title: TauLib.BookI.Sets.UniqueInfinity +cid: cid005117 +primary_alias: MOD0120 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Sets.UniqueInfinity (264 lines, 19 declarations). +aliases: +- MOD0120 +- TauLib.BookI.Sets.UniqueInfinity +- book-i-sets-unique-infinity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.UniqueInfinity + module_path: TauLib/BookI/Sets/UniqueInfinity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Sets.UniqueInfinity (264 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005118.md b/_corpus_v3_items/cid005118.md new file mode 100644 index 000000000..9b17d8171 --- /dev/null +++ b/_corpus_v3_items/cid005118.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005118/ +title: TauLib.BookI.Topos.CartesianProduct +cid: cid005118 +primary_alias: MOD0121 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.CartesianProduct (123 lines, 17 declarations). +aliases: +- MOD0121 +- TauLib.BookI.Topos.CartesianProduct +- book-i-topos-cartesian-product +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.CartesianProduct + module_path: TauLib/BookI/Topos/CartesianProduct.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.CartesianProduct (123 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005119.md b/_corpus_v3_items/cid005119.md new file mode 100644 index 000000000..446c64ec6 --- /dev/null +++ b/_corpus_v3_items/cid005119.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005119/ +title: TauLib.BookI.Topos.CircularityResolution +cid: cid005119 +primary_alias: MOD0122 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.CircularityResolution (435 lines, 25 declarations). +aliases: +- MOD0122 +- TauLib.BookI.Topos.CircularityResolution +- book-i-topos-circularity-resolution +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.CircularityResolution + module_path: TauLib/BookI/Topos/CircularityResolution.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.CircularityResolution (435 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005120.md b/_corpus_v3_items/cid005120.md new file mode 100644 index 000000000..b6e703924 --- /dev/null +++ b/_corpus_v3_items/cid005120.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005120/ +title: TauLib.BookI.Topos.EarnedArrows +cid: cid005120 +primary_alias: MOD0123 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.EarnedArrows (169 lines, 23 declarations). +aliases: +- MOD0123 +- TauLib.BookI.Topos.EarnedArrows +- book-i-topos-earned-arrows +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.EarnedArrows + module_path: TauLib/BookI/Topos/EarnedArrows.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.EarnedArrows (169 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005121.md b/_corpus_v3_items/cid005121.md new file mode 100644 index 000000000..ded1fca8c --- /dev/null +++ b/_corpus_v3_items/cid005121.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005121/ +title: TauLib.BookI.Topos.EarnedTopos +cid: cid005121 +primary_alias: MOD0124 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.EarnedTopos (226 lines, 33 declarations). +aliases: +- MOD0124 +- TauLib.BookI.Topos.EarnedTopos +- book-i-topos-earned-topos +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.EarnedTopos + module_path: TauLib/BookI/Topos/EarnedTopos.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.EarnedTopos (226 lines, 33 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005122.md b/_corpus_v3_items/cid005122.md new file mode 100644 index 000000000..97e284515 --- /dev/null +++ b/_corpus_v3_items/cid005122.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005122/ +title: TauLib.BookI.Topos.Functors +cid: cid005122 +primary_alias: MOD0125 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.Functors (199 lines, 25 declarations). +aliases: +- MOD0125 +- TauLib.BookI.Topos.Functors +- book-i-topos-functors +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.Functors + module_path: TauLib/BookI/Topos/Functors.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.Functors (199 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005123.md b/_corpus_v3_items/cid005123.md new file mode 100644 index 000000000..5232a3dbe --- /dev/null +++ b/_corpus_v3_items/cid005123.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005123/ +title: TauLib.BookI.Topos.H6EarnedCategoricalMachine +cid: cid005123 +primary_alias: MOD0126 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.H6EarnedCategoricalMachine (305 lines, 14 declarations). +aliases: +- MOD0126 +- TauLib.BookI.Topos.H6EarnedCategoricalMachine +- book-i-topos-h6-earned-categorical-machine +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.H6EarnedCategoricalMachine + module_path: TauLib/BookI/Topos/H6EarnedCategoricalMachine.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.H6EarnedCategoricalMachine (305 lines, 14 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005124.md b/_corpus_v3_items/cid005124.md new file mode 100644 index 000000000..d638f92be --- /dev/null +++ b/_corpus_v3_items/cid005124.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005124/ +title: TauLib.BookI.Topos.H7CircularityFull +cid: cid005124 +primary_alias: MOD0127 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.H7CircularityFull (255 lines, 16 declarations). +aliases: +- MOD0127 +- TauLib.BookI.Topos.H7CircularityFull +- book-i-topos-h7-circularity-full +- I.T155 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.H7CircularityFull + module_path: TauLib/BookI/Topos/H7CircularityFull.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.H7CircularityFull (255 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005125.md b/_corpus_v3_items/cid005125.md new file mode 100644 index 000000000..dac53058f --- /dev/null +++ b/_corpus_v3_items/cid005125.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005125/ +title: TauLib.BookI.Topos.H7ClassicalClosure +cid: cid005125 +primary_alias: MOD0128 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.H7ClassicalClosure (295 lines, 6 declarations). +aliases: +- MOD0128 +- TauLib.BookI.Topos.H7ClassicalClosure +- book-i-topos-h7-classical-closure +- I.T148 +- I.T167 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.H7ClassicalClosure + module_path: TauLib/BookI/Topos/H7ClassicalClosure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.H7ClassicalClosure (295 lines, 6 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005126.md b/_corpus_v3_items/cid005126.md new file mode 100644 index 000000000..5ebf6c979 --- /dev/null +++ b/_corpus_v3_items/cid005126.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005126/ +title: TauLib.BookI.Topos.H7ToposClassifier +cid: cid005126 +primary_alias: MOD0129 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.H7ToposClassifier (316 lines, 17 declarations). +aliases: +- MOD0129 +- TauLib.BookI.Topos.H7ToposClassifier +- book-i-topos-h7-topos-classifier +- I.T146 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.H7ToposClassifier + module_path: TauLib/BookI/Topos/H7ToposClassifier.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.H7ToposClassifier (316 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005127.md b/_corpus_v3_items/cid005127.md new file mode 100644 index 000000000..9066ddcb0 --- /dev/null +++ b/_corpus_v3_items/cid005127.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005127/ +title: TauLib.BookI.Topos.InternalHom +cid: cid005127 +primary_alias: MOD0130 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.InternalHom (117 lines, 13 declarations). +aliases: +- MOD0130 +- TauLib.BookI.Topos.InternalHom +- book-i-topos-internal-hom +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.InternalHom + module_path: TauLib/BookI/Topos/InternalHom.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.InternalHom (117 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005128.md b/_corpus_v3_items/cid005128.md new file mode 100644 index 000000000..a8b7ec95b --- /dev/null +++ b/_corpus_v3_items/cid005128.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005128/ +title: TauLib.BookI.Topos.LimitsSites +cid: cid005128 +primary_alias: MOD0131 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.LimitsSites (239 lines, 35 declarations). +aliases: +- MOD0131 +- TauLib.BookI.Topos.LimitsSites +- book-i-topos-limits-sites +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.LimitsSites + module_path: TauLib/BookI/Topos/LimitsSites.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.LimitsSites (239 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005129.md b/_corpus_v3_items/cid005129.md new file mode 100644 index 000000000..7a66995b5 --- /dev/null +++ b/_corpus_v3_items/cid005129.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005129/ +title: TauLib.BookI.Topos.ParaconsistentSoundness +cid: cid005129 +primary_alias: MOD0132 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.ParaconsistentSoundness (463 lines, 51 declarations). +aliases: +- MOD0132 +- TauLib.BookI.Topos.ParaconsistentSoundness +- book-i-topos-paraconsistent-soundness +- I.D123 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.ParaconsistentSoundness + module_path: TauLib/BookI/Topos/ParaconsistentSoundness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.ParaconsistentSoundness (463 lines, 51 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005130.md b/_corpus_v3_items/cid005130.md new file mode 100644 index 000000000..c4bfd4488 --- /dev/null +++ b/_corpus_v3_items/cid005130.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005130/ +title: TauLib.BookI.Topos.WedgeProduct +cid: cid005130 +primary_alias: MOD0133 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookI.Topos.WedgeProduct (121 lines, 15 declarations). +aliases: +- MOD0133 +- TauLib.BookI.Topos.WedgeProduct +- book-i-topos-wedge-product +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.booki +relations: +- predicate: appears_in + object: cid000023 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.WedgeProduct + module_path: TauLib/BookI/Topos/WedgeProduct.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookI.Topos.WedgeProduct (121 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005131.md b/_corpus_v3_items/cid005131.md new file mode 100644 index 000000000..3aedd76b2 --- /dev/null +++ b/_corpus_v3_items/cid005131.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005131/ +title: TauLib.BookII.CentralTheorem.BoundaryCharacters +cid: cid005131 +primary_alias: MOD0134 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.BoundaryCharacters (419 lines, 43 declarations). +aliases: +- MOD0134 +- TauLib.BookII.CentralTheorem.BoundaryCharacters +- book-ii-central-theorem-boundary-characters +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.BoundaryCharacters (419 lines, 43 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005132.md b/_corpus_v3_items/cid005132.md new file mode 100644 index 000000000..0f48b7b14 --- /dev/null +++ b/_corpus_v3_items/cid005132.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005132/ +title: TauLib.BookII.CentralTheorem.Categoricity +cid: cid005132 +primary_alias: MOD0135 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.Categoricity (386 lines, 40 declarations). +aliases: +- MOD0135 +- TauLib.BookII.CentralTheorem.Categoricity +- book-ii-central-theorem-categoricity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.Categoricity (386 lines, 40 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005133.md b/_corpus_v3_items/cid005133.md new file mode 100644 index 000000000..0fe3f208a --- /dev/null +++ b/_corpus_v3_items/cid005133.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005133/ +title: TauLib.BookII.CentralTheorem.CentralTheorem +cid: cid005133 +primary_alias: MOD0136 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.CentralTheorem (454 lines, 36 declarations). +aliases: +- MOD0136 +- TauLib.BookII.CentralTheorem.CentralTheorem +- book-ii-central-theorem-central-theorem +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.CentralTheorem (454 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005134.md b/_corpus_v3_items/cid005134.md new file mode 100644 index 000000000..6f9247da5 --- /dev/null +++ b/_corpus_v3_items/cid005134.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005134/ +title: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms +cid: cid005134 +primary_alias: MOD0137 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms (393 lines, 35 declarations). +aliases: +- MOD0137 +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms +- book-ii-central-theorem-extensions-omega-germs +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms (393 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005135.md b/_corpus_v3_items/cid005135.md new file mode 100644 index 000000000..f5d929cae --- /dev/null +++ b/_corpus_v3_items/cid005135.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005135/ +title: TauLib.BookII.CentralTheorem.HartogsExtension +cid: cid005135 +primary_alias: MOD0138 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.HartogsExtension (356 lines, 27 declarations). +aliases: +- MOD0138 +- TauLib.BookII.CentralTheorem.HartogsExtension +- book-ii-central-theorem-hartogs-extension +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.HartogsExtension (356 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005136.md b/_corpus_v3_items/cid005136.md new file mode 100644 index 000000000..c2a9cfa96 --- /dev/null +++ b/_corpus_v3_items/cid005136.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005136/ +title: TauLib.BookII.CentralTheorem.SheafCohomology +cid: cid005136 +primary_alias: MOD0139 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.SheafCohomology (240 lines, 24 declarations). +aliases: +- MOD0139 +- TauLib.BookII.CentralTheorem.SheafCohomology +- book-ii-central-theorem-sheaf-cohomology +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.SheafCohomology (240 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005137.md b/_corpus_v3_items/cid005137.md new file mode 100644 index 000000000..5c2d0eb29 --- /dev/null +++ b/_corpus_v3_items/cid005137.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005137/ +title: TauLib.BookII.CentralTheorem.YonedaApplied +cid: cid005137 +primary_alias: MOD0140 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.CentralTheorem.YonedaApplied (341 lines, 23 declarations). +aliases: +- MOD0140 +- TauLib.BookII.CentralTheorem.YonedaApplied +- book-ii-central-theorem-yoneda-applied +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.CentralTheorem.YonedaApplied (341 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005138.md b/_corpus_v3_items/cid005138.md new file mode 100644 index 000000000..4886b705f --- /dev/null +++ b/_corpus_v3_items/cid005138.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005138/ +title: TauLib.BookII.Closure.BSDbridge +cid: cid005138 +primary_alias: MOD0141 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.BSDbridge (250 lines, 40 declarations). +aliases: +- MOD0141 +- TauLib.BookII.Closure.BSDbridge +- book-ii-closure-bsdbridge +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.BSDbridge (250 lines, 40 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005139.md b/_corpus_v3_items/cid005139.md new file mode 100644 index 000000000..36b3b919b --- /dev/null +++ b/_corpus_v3_items/cid005139.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005139/ +title: TauLib.BookII.Closure.Connection +cid: cid005139 +primary_alias: MOD0142 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.Connection (231 lines, 23 declarations). +aliases: +- MOD0142 +- TauLib.BookII.Closure.Connection +- book-ii-closure-connection +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Connection + module_path: TauLib/BookII/Closure/Connection.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.Connection (231 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005140.md b/_corpus_v3_items/cid005140.md new file mode 100644 index 000000000..97d6ce1bd --- /dev/null +++ b/_corpus_v3_items/cid005140.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005140/ +title: TauLib.BookII.Closure.Curvature +cid: cid005140 +primary_alias: MOD0143 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.Curvature (216 lines, 22 declarations). +aliases: +- MOD0143 +- TauLib.BookII.Closure.Curvature +- book-ii-closure-curvature +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + module_path: TauLib/BookII/Closure/Curvature.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.Curvature (216 lines, 22 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005141.md b/_corpus_v3_items/cid005141.md new file mode 100644 index 000000000..49d37a358 --- /dev/null +++ b/_corpus_v3_items/cid005141.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005141/ +title: TauLib.BookII.Closure.DiffGeoAgenda +cid: cid005141 +primary_alias: MOD0144 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.DiffGeoAgenda (168 lines, 22 declarations). +aliases: +- MOD0144 +- TauLib.BookII.Closure.DiffGeoAgenda +- book-ii-closure-diff-geo-agenda +- II.R21 +- II.R22 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005460 +- predicate: formalized_by + object: cid005461 +- predicate: formalized_by + object: cid005462 +- predicate: formalized_by + object: cid005463 +- predicate: formalized_by + object: cid005464 +- predicate: formalized_by + object: cid005465 +- predicate: formalized_by + object: cid005466 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.DiffGeoAgenda (168 lines, 22 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005142.md b/_corpus_v3_items/cid005142.md new file mode 100644 index 000000000..651c7a9ed --- /dev/null +++ b/_corpus_v3_items/cid005142.md @@ -0,0 +1,79 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005142/ +title: TauLib.BookII.Closure.ForwardBook3 +cid: cid005142 +primary_alias: MOD0145 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.ForwardBook3 (232 lines, 34 declarations). +aliases: +- MOD0145 +- TauLib.BookII.Closure.ForwardBook3 +- book-ii-closure-forward-book3 +- II.D66 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005467 +- predicate: formalized_by + object: cid005468 +- predicate: formalized_by + object: cid005469 +- predicate: formalized_by + object: cid005470 +- predicate: formalized_by + object: cid005471 +- predicate: formalized_by + object: cid005472 +- predicate: formalized_by + object: cid005473 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.ForwardBook3 (232 lines, 34 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005143.md b/_corpus_v3_items/cid005143.md new file mode 100644 index 000000000..7469ccbf9 --- /dev/null +++ b/_corpus_v3_items/cid005143.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005143/ +title: TauLib.BookII.Closure.GeometricBiSquare +cid: cid005143 +primary_alias: MOD0146 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.GeometricBiSquare (339 lines, 47 declarations). +aliases: +- MOD0146 +- TauLib.BookII.Closure.GeometricBiSquare +- book-ii-closure-geometric-bi-square +- II.R33 +- II.R34 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005477 +- predicate: formalized_by + object: cid005478 +- predicate: formalized_by + object: cid005481 +- predicate: formalized_by + object: cid005482 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.GeometricBiSquare (339 lines, 47 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005144.md b/_corpus_v3_items/cid005144.md new file mode 100644 index 000000000..c895558ee --- /dev/null +++ b/_corpus_v3_items/cid005144.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005144/ +title: TauLib.BookII.Closure.TauManifold +cid: cid005144 +primary_alias: MOD0147 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Closure.TauManifold (326 lines, 42 declarations). +aliases: +- MOD0147 +- TauLib.BookII.Closure.TauManifold +- book-ii-closure-tau-manifold +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Closure.TauManifold (326 lines, 42 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005145.md b/_corpus_v3_items/cid005145.md new file mode 100644 index 000000000..fcead96cc --- /dev/null +++ b/_corpus_v3_items/cid005145.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005145/ +title: TauLib.BookII.Domains.Cylinders +cid: cid005145 +primary_alias: MOD0148 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Domains.Cylinders (175 lines, 34 declarations). +aliases: +- MOD0148 +- TauLib.BookII.Domains.Cylinders +- book-ii-domains-cylinders +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Domains.Cylinders + module_path: TauLib/BookII/Domains/Cylinders.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Domains.Cylinders (175 lines, 34 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005146.md b/_corpus_v3_items/cid005146.md new file mode 100644 index 000000000..76f8d67a5 --- /dev/null +++ b/_corpus_v3_items/cid005146.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005146/ +title: TauLib.BookII.Domains.HolImpliesCont +cid: cid005146 +primary_alias: MOD0149 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Domains.HolImpliesCont (152 lines, 25 declarations). +aliases: +- MOD0149 +- TauLib.BookII.Domains.HolImpliesCont +- book-ii-domains-hol-implies-cont +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Domains.HolImpliesCont + module_path: TauLib/BookII/Domains/HolImpliesCont.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Domains.HolImpliesCont (152 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005147.md b/_corpus_v3_items/cid005147.md new file mode 100644 index 000000000..a39b86116 --- /dev/null +++ b/_corpus_v3_items/cid005147.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005147/ +title: TauLib.BookII.Domains.Ultrametric +cid: cid005147 +primary_alias: MOD0150 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Domains.Ultrametric (148 lines, 20 declarations). +aliases: +- MOD0150 +- TauLib.BookII.Domains.Ultrametric +- book-ii-domains-ultrametric +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Domains.Ultrametric + module_path: TauLib/BookII/Domains/Ultrametric.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Domains.Ultrametric (148 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005148.md b/_corpus_v3_items/cid005148.md new file mode 100644 index 000000000..fff46a248 --- /dev/null +++ b/_corpus_v3_items/cid005148.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005148/ +title: TauLib.BookII.Enrichment.EnrichmentLadder +cid: cid005148 +primary_alias: MOD0151 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.EnrichmentLadder (338 lines, 34 declarations). +aliases: +- MOD0151 +- TauLib.BookII.Enrichment.EnrichmentLadder +- book-ii-enrichment-enrichment-ladder +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.EnrichmentLadder (338 lines, 34 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005149.md b/_corpus_v3_items/cid005149.md new file mode 100644 index 000000000..1cf118354 --- /dev/null +++ b/_corpus_v3_items/cid005149.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005149/ +title: TauLib.BookII.Enrichment.Homological +cid: cid005149 +primary_alias: MOD0152 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.Homological (217 lines, 27 declarations). +aliases: +- MOD0152 +- TauLib.BookII.Enrichment.Homological +- book-ii-enrichment-homological +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.Homological (217 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005150.md b/_corpus_v3_items/cid005150.md new file mode 100644 index 000000000..f21eb1f50 --- /dev/null +++ b/_corpus_v3_items/cid005150.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005150/ +title: TauLib.BookII.Enrichment.SelfDescribing +cid: cid005150 +primary_alias: MOD0153 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.SelfDescribing (318 lines, 33 declarations). +aliases: +- MOD0153 +- TauLib.BookII.Enrichment.SelfDescribing +- book-ii-enrichment-self-describing +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfDescribing + module_path: TauLib/BookII/Enrichment/SelfDescribing.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.SelfDescribing (318 lines, 33 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005151.md b/_corpus_v3_items/cid005151.md new file mode 100644 index 000000000..3cc729472 --- /dev/null +++ b/_corpus_v3_items/cid005151.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005151/ +title: TauLib.BookII.Enrichment.SelfEnrichment +cid: cid005151 +primary_alias: MOD0154 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.SelfEnrichment (372 lines, 41 declarations). +aliases: +- MOD0154 +- TauLib.BookII.Enrichment.SelfEnrichment +- book-ii-enrichment-self-enrichment +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.SelfEnrichment (372 lines, 41 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005152.md b/_corpus_v3_items/cid005152.md new file mode 100644 index 000000000..8536cbf97 --- /dev/null +++ b/_corpus_v3_items/cid005152.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005152/ +title: TauLib.BookII.Enrichment.TwoCategories +cid: cid005152 +primary_alias: MOD0155 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.TwoCategories (465 lines, 52 declarations). +aliases: +- MOD0155 +- TauLib.BookII.Enrichment.TwoCategories +- book-ii-enrichment-two-categories +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.TwoCategories (465 lines, 52 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005153.md b/_corpus_v3_items/cid005153.md new file mode 100644 index 000000000..d22cb1178 --- /dev/null +++ b/_corpus_v3_items/cid005153.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005153/ +title: TauLib.BookII.Enrichment.YonedaTheorem +cid: cid005153 +primary_alias: MOD0156 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Enrichment.YonedaTheorem (343 lines, 30 declarations). +aliases: +- MOD0156 +- TauLib.BookII.Enrichment.YonedaTheorem +- book-ii-enrichment-yoneda-theorem +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Enrichment.YonedaTheorem (343 lines, 30 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005154.md b/_corpus_v3_items/cid005154.md new file mode 100644 index 000000000..cab434eeb --- /dev/null +++ b/_corpus_v3_items/cid005154.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005154/ +title: TauLib.BookII.Geometry.Betweenness +cid: cid005154 +primary_alias: MOD0157 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Geometry.Betweenness (118 lines, 13 declarations). +aliases: +- MOD0157 +- TauLib.BookII.Geometry.Betweenness +- book-ii-geometry-betweenness +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Geometry.Betweenness + module_path: TauLib/BookII/Geometry/Betweenness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Geometry.Betweenness (118 lines, 13 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005155.md b/_corpus_v3_items/cid005155.md new file mode 100644 index 000000000..2dd85f4f5 --- /dev/null +++ b/_corpus_v3_items/cid005155.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005155/ +title: TauLib.BookII.Geometry.CausalStructure +cid: cid005155 +primary_alias: MOD0158 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Geometry.CausalStructure (193 lines, 31 declarations). +aliases: +- MOD0158 +- TauLib.BookII.Geometry.CausalStructure +- book-ii-geometry-causal-structure +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Geometry.CausalStructure + module_path: TauLib/BookII/Geometry/CausalStructure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Geometry.CausalStructure (193 lines, 31 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005156.md b/_corpus_v3_items/cid005156.md new file mode 100644 index 000000000..ad785f60e --- /dev/null +++ b/_corpus_v3_items/cid005156.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005156/ +title: TauLib.BookII.Geometry.Congruence +cid: cid005156 +primary_alias: MOD0159 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Geometry.Congruence (137 lines, 19 declarations). +aliases: +- MOD0159 +- TauLib.BookII.Geometry.Congruence +- book-ii-geometry-congruence +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Geometry.Congruence + module_path: TauLib/BookII/Geometry/Congruence.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Geometry.Congruence (137 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005157.md b/_corpus_v3_items/cid005157.md new file mode 100644 index 000000000..2210e4c1a --- /dev/null +++ b/_corpus_v3_items/cid005157.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005157/ +title: TauLib.BookII.Geometry.OrthodoxBridge +cid: cid005157 +primary_alias: MOD0160 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Geometry.OrthodoxBridge (276 lines, 32 declarations). +aliases: +- MOD0160 +- TauLib.BookII.Geometry.OrthodoxBridge +- book-ii-geometry-orthodox-bridge +- II.R07 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Geometry.OrthodoxBridge + module_path: TauLib/BookII/Geometry/OrthodoxBridge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Geometry.OrthodoxBridge (276 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005158.md b/_corpus_v3_items/cid005158.md new file mode 100644 index 000000000..a1d0c9cc4 --- /dev/null +++ b/_corpus_v3_items/cid005158.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005158/ +title: TauLib.BookII.Geometry.PaschParallel +cid: cid005158 +primary_alias: MOD0161 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Geometry.PaschParallel (190 lines, 26 declarations). +aliases: +- MOD0161 +- TauLib.BookII.Geometry.PaschParallel +- book-ii-geometry-pasch-parallel +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Geometry.PaschParallel + module_path: TauLib/BookII/Geometry/PaschParallel.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Geometry.PaschParallel (190 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005159.md b/_corpus_v3_items/cid005159.md new file mode 100644 index 000000000..54615b820 --- /dev/null +++ b/_corpus_v3_items/cid005159.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005159/ +title: TauLib.BookII.Hartogs.BndLift +cid: cid005159 +primary_alias: MOD0162 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.BndLift (475 lines, 67 declarations). +aliases: +- MOD0162 +- TauLib.BookII.Hartogs.BndLift +- book-ii-hartogs-bnd-lift +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.BndLift + module_path: TauLib/BookII/Hartogs/BndLift.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.BndLift (475 lines, 67 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005160.md b/_corpus_v3_items/cid005160.md new file mode 100644 index 000000000..379194407 --- /dev/null +++ b/_corpus_v3_items/cid005160.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005160/ +title: TauLib.BookII.Hartogs.CalibratedSplitComplex +cid: cid005160 +primary_alias: MOD0163 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.CalibratedSplitComplex (317 lines, 47 declarations). +aliases: +- MOD0163 +- TauLib.BookII.Hartogs.CalibratedSplitComplex +- book-ii-hartogs-calibrated-split-complex +- II.R10 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CalibratedSplitComplex + module_path: TauLib/BookII/Hartogs/CalibratedSplitComplex.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.CalibratedSplitComplex (317 lines, 47 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005161.md b/_corpus_v3_items/cid005161.md new file mode 100644 index 000000000..3fedccbd2 --- /dev/null +++ b/_corpus_v3_items/cid005161.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005161/ +title: TauLib.BookII.Hartogs.CanonicalBasis +cid: cid005161 +primary_alias: MOD0164 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.CanonicalBasis (422 lines, 55 declarations). +aliases: +- MOD0164 +- TauLib.BookII.Hartogs.CanonicalBasis +- book-ii-hartogs-canonical-basis +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.CanonicalBasis (422 lines, 55 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005162.md b/_corpus_v3_items/cid005162.md new file mode 100644 index 000000000..c6a1634da --- /dev/null +++ b/_corpus_v3_items/cid005162.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005162/ +title: TauLib.BookII.Hartogs.CategoryStructure +cid: cid005162 +primary_alias: MOD0165 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.CategoryStructure (480 lines, 56 declarations). +aliases: +- MOD0165 +- TauLib.BookII.Hartogs.CategoryStructure +- book-ii-hartogs-category-structure +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CategoryStructure + module_path: TauLib/BookII/Hartogs/CategoryStructure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.CategoryStructure (480 lines, 56 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005163.md b/_corpus_v3_items/cid005163.md new file mode 100644 index 000000000..884d098a9 --- /dev/null +++ b/_corpus_v3_items/cid005163.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005163/ +title: TauLib.BookII.Hartogs.EvolutionOperator +cid: cid005163 +primary_alias: MOD0166 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.EvolutionOperator (338 lines, 41 declarations). +aliases: +- MOD0166 +- TauLib.BookII.Hartogs.EvolutionOperator +- book-ii-hartogs-evolution-operator +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.EvolutionOperator + module_path: TauLib/BookII/Hartogs/EvolutionOperator.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.EvolutionOperator (338 lines, 41 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005164.md b/_corpus_v3_items/cid005164.md new file mode 100644 index 000000000..028391887 --- /dev/null +++ b/_corpus_v3_items/cid005164.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005164/ +title: TauLib.BookII.Hartogs.L2Space +cid: cid005164 +primary_alias: MOD0167 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.L2Space (243 lines, 28 declarations). +aliases: +- MOD0167 +- TauLib.BookII.Hartogs.L2Space +- book-ii-hartogs-l2-space +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.L2Space (243 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005165.md b/_corpus_v3_items/cid005165.md new file mode 100644 index 000000000..59b0a48b0 --- /dev/null +++ b/_corpus_v3_items/cid005165.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005165/ +title: TauLib.BookII.Hartogs.LaurentResidue +cid: cid005165 +primary_alias: MOD0168 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.LaurentResidue (512 lines, 64 declarations). +aliases: +- MOD0168 +- TauLib.BookII.Hartogs.LaurentResidue +- book-ii-hartogs-laurent-residue +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.LaurentResidue (512 lines, 64 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005166.md b/_corpus_v3_items/cid005166.md new file mode 100644 index 000000000..bef4442cd --- /dev/null +++ b/_corpus_v3_items/cid005166.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005166/ +title: TauLib.BookII.Hartogs.MutualDetermination +cid: cid005166 +primary_alias: MOD0169 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.MutualDetermination (391 lines, 45 declarations). +aliases: +- MOD0169 +- TauLib.BookII.Hartogs.MutualDetermination +- book-ii-hartogs-mutual-determination +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.MutualDetermination + module_path: TauLib/BookII/Hartogs/MutualDetermination.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.MutualDetermination (391 lines, 45 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005167.md b/_corpus_v3_items/cid005167.md new file mode 100644 index 000000000..36840b873 --- /dev/null +++ b/_corpus_v3_items/cid005167.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005167/ +title: TauLib.BookII.Hartogs.SheafCoherence +cid: cid005167 +primary_alias: MOD0170 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Hartogs.SheafCoherence (497 lines, 51 declarations). +aliases: +- MOD0170 +- TauLib.BookII.Hartogs.SheafCoherence +- book-ii-hartogs-sheaf-coherence +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Hartogs.SheafCoherence (497 lines, 51 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005168.md b/_corpus_v3_items/cid005168.md new file mode 100644 index 000000000..a86dd358f --- /dev/null +++ b/_corpus_v3_items/cid005168.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005168/ +title: TauLib.BookII.Interior.ABCDRigidity +cid: cid005168 +primary_alias: MOD0171 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Interior.ABCDRigidity (151 lines, 18 declarations). +aliases: +- MOD0171 +- TauLib.BookII.Interior.ABCDRigidity +- book-ii-interior-abcdrigidity +- II.R04 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005579 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.ABCDRigidity + module_path: TauLib/BookII/Interior/ABCDRigidity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Interior.ABCDRigidity (151 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005169.md b/_corpus_v3_items/cid005169.md new file mode 100644 index 000000000..1d84df7d4 --- /dev/null +++ b/_corpus_v3_items/cid005169.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005169/ +title: TauLib.BookII.Interior.BipolarDecomposition +cid: cid005169 +primary_alias: MOD0172 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Interior.BipolarDecomposition (136 lines, 19 declarations). +aliases: +- MOD0172 +- TauLib.BookII.Interior.BipolarDecomposition +- book-ii-interior-bipolar-decomposition +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.BipolarDecomposition + module_path: TauLib/BookII/Interior/BipolarDecomposition.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Interior.BipolarDecomposition (136 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005170.md b/_corpus_v3_items/cid005170.md new file mode 100644 index 000000000..c9653b3d4 --- /dev/null +++ b/_corpus_v3_items/cid005170.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005170/ +title: TauLib.BookII.Interior.OmegaReadout +cid: cid005170 +primary_alias: MOD0173 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Interior.OmegaReadout (166 lines, 24 declarations). +aliases: +- MOD0173 +- TauLib.BookII.Interior.OmegaReadout +- book-ii-interior-omega-readout +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.OmegaReadout + module_path: TauLib/BookII/Interior/OmegaReadout.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Interior.OmegaReadout (166 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005171.md b/_corpus_v3_items/cid005171.md new file mode 100644 index 000000000..9de2cfdb1 --- /dev/null +++ b/_corpus_v3_items/cid005171.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005171/ +title: TauLib.BookII.Interior.Tau3Fibration +cid: cid005171 +primary_alias: MOD0174 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Interior.Tau3Fibration (180 lines, 23 declarations). +aliases: +- MOD0174 +- TauLib.BookII.Interior.Tau3Fibration +- book-ii-interior-tau3-fibration +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.Tau3Fibration + module_path: TauLib/BookII/Interior/Tau3Fibration.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Interior.Tau3Fibration (180 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005172.md b/_corpus_v3_items/cid005172.md new file mode 100644 index 000000000..d57fe22a1 --- /dev/null +++ b/_corpus_v3_items/cid005172.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005172/ +title: TauLib.BookII.Interior.TauAdmissible +cid: cid005172 +primary_alias: MOD0175 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Interior.TauAdmissible (200 lines, 44 declarations). +aliases: +- MOD0175 +- TauLib.BookII.Interior.TauAdmissible +- book-ii-interior-tau-admissible +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.TauAdmissible + module_path: TauLib/BookII/Interior/TauAdmissible.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Interior.TauAdmissible (200 lines, 44 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005173.md b/_corpus_v3_items/cid005173.md new file mode 100644 index 000000000..6ad0686e9 --- /dev/null +++ b/_corpus_v3_items/cid005173.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005173/ +title: TauLib.BookII.Mirror.DimensionalLadder +cid: cid005173 +primary_alias: MOD0176 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Mirror.DimensionalLadder (511 lines, 80 declarations). +aliases: +- MOD0176 +- TauLib.BookII.Mirror.DimensionalLadder +- book-ii-mirror-dimensional-ladder +- II.R31 +- II.R32 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005588 +- predicate: formalized_by + object: cid005589 +- predicate: formalized_by + object: cid005594 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Mirror.DimensionalLadder (511 lines, 80 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005174.md b/_corpus_v3_items/cid005174.md new file mode 100644 index 000000000..c74ee3047 --- /dev/null +++ b/_corpus_v3_items/cid005174.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005174/ +title: TauLib.BookII.Mirror.Inventory +cid: cid005174 +primary_alias: MOD0177 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Mirror.Inventory (212 lines, 31 declarations). +aliases: +- MOD0177 +- TauLib.BookII.Mirror.Inventory +- book-ii-mirror-inventory +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.Inventory + module_path: TauLib/BookII/Mirror/Inventory.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Mirror.Inventory (212 lines, 31 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005175.md b/_corpus_v3_items/cid005175.md new file mode 100644 index 000000000..a9ad613c5 --- /dev/null +++ b/_corpus_v3_items/cid005175.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005175/ +title: TauLib.BookII.Mirror.PhysicsQuadrant +cid: cid005175 +primary_alias: MOD0178 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Mirror.PhysicsQuadrant (304 lines, 57 declarations). +aliases: +- MOD0178 +- TauLib.BookII.Mirror.PhysicsQuadrant +- book-ii-mirror-physics-quadrant +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Mirror.PhysicsQuadrant (304 lines, 57 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005176.md b/_corpus_v3_items/cid005176.md new file mode 100644 index 000000000..43f20c7c8 --- /dev/null +++ b/_corpus_v3_items/cid005176.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005176/ +title: TauLib.BookII.Mirror.SignClassification +cid: cid005176 +primary_alias: MOD0179 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Mirror.SignClassification (307 lines, 44 declarations). +aliases: +- MOD0179 +- TauLib.BookII.Mirror.SignClassification +- book-ii-mirror-sign-classification +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Mirror.SignClassification (307 lines, 44 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005177.md b/_corpus_v3_items/cid005177.md new file mode 100644 index 000000000..2e90fa0f1 --- /dev/null +++ b/_corpus_v3_items/cid005177.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005177/ +title: TauLib.BookII.Mirror.WaveHolomorphy +cid: cid005177 +primary_alias: MOD0180 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Mirror.WaveHolomorphy (342 lines, 55 declarations). +aliases: +- MOD0180 +- TauLib.BookII.Mirror.WaveHolomorphy +- book-ii-mirror-wave-holomorphy +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Mirror.WaveHolomorphy (342 lines, 55 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005178.md b/_corpus_v3_items/cid005178.md new file mode 100644 index 000000000..b9c4ddd6a --- /dev/null +++ b/_corpus_v3_items/cid005178.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005178/ +title: TauLib.BookII.Prologue.SplitComplexInterior +cid: cid005178 +primary_alias: MOD0181 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Prologue.SplitComplexInterior (136 lines, 17 declarations). +aliases: +- MOD0181 +- TauLib.BookII.Prologue.SplitComplexInterior +- book-ii-prologue-split-complex-interior +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Prologue.SplitComplexInterior + module_path: TauLib/BookII/Prologue/SplitComplexInterior.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Prologue.SplitComplexInterior (136 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005179.md b/_corpus_v3_items/cid005179.md new file mode 100644 index 000000000..8319d4e93 --- /dev/null +++ b/_corpus_v3_items/cid005179.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005179/ +title: TauLib.BookII.Regularity.CodeDecode +cid: cid005179 +primary_alias: MOD0182 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Regularity.CodeDecode (504 lines, 53 declarations). +aliases: +- MOD0182 +- TauLib.BookII.Regularity.CodeDecode +- book-ii-regularity-code-decode +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Regularity.CodeDecode (504 lines, 53 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005180.md b/_corpus_v3_items/cid005180.md new file mode 100644 index 000000000..c0d4861b0 --- /dev/null +++ b/_corpus_v3_items/cid005180.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005180/ +title: TauLib.BookII.Regularity.IdempotentDecomposition +cid: cid005180 +primary_alias: MOD0183 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Regularity.IdempotentDecomposition (327 lines, 35 declarations). +aliases: +- MOD0183 +- TauLib.BookII.Regularity.IdempotentDecomposition +- book-ii-regularity-idempotent-decomposition +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.IdempotentDecomposition + module_path: TauLib/BookII/Regularity/IdempotentDecomposition.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Regularity.IdempotentDecomposition (327 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005181.md b/_corpus_v3_items/cid005181.md new file mode 100644 index 000000000..6bd01b2c8 --- /dev/null +++ b/_corpus_v3_items/cid005181.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005181/ +title: TauLib.BookII.Regularity.PositiveRegularity +cid: cid005181 +primary_alias: MOD0184 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Regularity.PositiveRegularity (363 lines, 32 declarations). +aliases: +- MOD0184 +- TauLib.BookII.Regularity.PositiveRegularity +- book-ii-regularity-positive-regularity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PositiveRegularity + module_path: TauLib/BookII/Regularity/PositiveRegularity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Regularity.PositiveRegularity (363 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005182.md b/_corpus_v3_items/cid005182.md new file mode 100644 index 000000000..572c472ba --- /dev/null +++ b/_corpus_v3_items/cid005182.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005182/ +title: TauLib.BookII.Regularity.PreYoneda +cid: cid005182 +primary_alias: MOD0185 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Regularity.PreYoneda (369 lines, 35 declarations). +aliases: +- MOD0185 +- TauLib.BookII.Regularity.PreYoneda +- book-ii-regularity-pre-yoneda +- II.R12 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +- predicate: formalized_by + object: cid005669 +- predicate: formalized_by + object: cid005670 +- predicate: formalized_by + object: cid005672 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Regularity.PreYoneda (369 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005183.md b/_corpus_v3_items/cid005183.md new file mode 100644 index 000000000..a7f410381 --- /dev/null +++ b/_corpus_v3_items/cid005183.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005183/ +title: TauLib.BookII.Regularity.ThreeLemmaChain +cid: cid005183 +primary_alias: MOD0186 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Regularity.ThreeLemmaChain (399 lines, 47 declarations). +aliases: +- MOD0186 +- TauLib.BookII.Regularity.ThreeLemmaChain +- book-ii-regularity-three-lemma-chain +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Regularity.ThreeLemmaChain (399 lines, 47 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005184.md b/_corpus_v3_items/cid005184.md new file mode 100644 index 000000000..38b7a4169 --- /dev/null +++ b/_corpus_v3_items/cid005184.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005184/ +title: TauLib.BookII.Topology.BoundaryMinimality +cid: cid005184 +primary_alias: MOD0187 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Topology.BoundaryMinimality (169 lines, 23 declarations). +aliases: +- MOD0187 +- TauLib.BookII.Topology.BoundaryMinimality +- book-ii-topology-boundary-minimality +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Topology.BoundaryMinimality + module_path: TauLib/BookII/Topology/BoundaryMinimality.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Topology.BoundaryMinimality (169 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005185.md b/_corpus_v3_items/cid005185.md new file mode 100644 index 000000000..df4eec4e4 --- /dev/null +++ b/_corpus_v3_items/cid005185.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005185/ +title: TauLib.BookII.Topology.DimensionFour +cid: cid005185 +primary_alias: MOD0188 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Topology.DimensionFour (151 lines, 20 declarations). +aliases: +- MOD0188 +- TauLib.BookII.Topology.DimensionFour +- book-ii-topology-dimension-four +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Topology.DimensionFour + module_path: TauLib/BookII/Topology/DimensionFour.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Topology.DimensionFour (151 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005186.md b/_corpus_v3_items/cid005186.md new file mode 100644 index 000000000..62af313d3 --- /dev/null +++ b/_corpus_v3_items/cid005186.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005186/ +title: TauLib.BookII.Topology.Invariant +cid: cid005186 +primary_alias: MOD0189 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Topology.Invariant (109 lines, 11 declarations). +aliases: +- MOD0189 +- TauLib.BookII.Topology.Invariant +- book-ii-topology-invariant +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Topology.Invariant + module_path: TauLib/BookII/Topology/Invariant.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Topology.Invariant (109 lines, 11 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005187.md b/_corpus_v3_items/cid005187.md new file mode 100644 index 000000000..d98ef7a2b --- /dev/null +++ b/_corpus_v3_items/cid005187.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005187/ +title: TauLib.BookII.Topology.StoneSpace +cid: cid005187 +primary_alias: MOD0190 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Topology.StoneSpace (156 lines, 22 declarations). +aliases: +- MOD0190 +- TauLib.BookII.Topology.StoneSpace +- book-ii-topology-stone-space +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Topology.StoneSpace + module_path: TauLib/BookII/Topology/StoneSpace.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Topology.StoneSpace (156 lines, 22 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005188.md b/_corpus_v3_items/cid005188.md new file mode 100644 index 000000000..b7c67e15b --- /dev/null +++ b/_corpus_v3_items/cid005188.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005188/ +title: TauLib.BookII.Topology.TorusDegeneration +cid: cid005188 +primary_alias: MOD0191 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Topology.TorusDegeneration (166 lines, 17 declarations). +aliases: +- MOD0191 +- TauLib.BookII.Topology.TorusDegeneration +- book-ii-topology-torus-degeneration +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Topology.TorusDegeneration + module_path: TauLib/BookII/Topology/TorusDegeneration.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Topology.TorusDegeneration (166 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005189.md b/_corpus_v3_items/cid005189.md new file mode 100644 index 000000000..40fb2d66f --- /dev/null +++ b/_corpus_v3_items/cid005189.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005189/ +title: TauLib.BookII.Transcendentals.Circles +cid: cid005189 +primary_alias: MOD0192 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.Circles (155 lines, 24 declarations). +aliases: +- MOD0192 +- TauLib.BookII.Transcendentals.Circles +- book-ii-transcendentals-circles +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.Circles + module_path: TauLib/BookII/Transcendentals/Circles.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.Circles (155 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005190.md b/_corpus_v3_items/cid005190.md new file mode 100644 index 000000000..a84140eac --- /dev/null +++ b/_corpus_v3_items/cid005190.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005190/ +title: TauLib.BookII.Transcendentals.EEarned +cid: cid005190 +primary_alias: MOD0193 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.EEarned (161 lines, 24 declarations). +aliases: +- MOD0193 +- TauLib.BookII.Transcendentals.EEarned +- book-ii-transcendentals-eearned +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.EEarned + module_path: TauLib/BookII/Transcendentals/EEarned.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.EEarned (161 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005191.md b/_corpus_v3_items/cid005191.md new file mode 100644 index 000000000..1e516d2e1 --- /dev/null +++ b/_corpus_v3_items/cid005191.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005191/ +title: TauLib.BookII.Transcendentals.IotaTauConfirmed +cid: cid005191 +primary_alias: MOD0194 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.IotaTauConfirmed (211 lines, 32 declarations). +aliases: +- MOD0194 +- TauLib.BookII.Transcendentals.IotaTauConfirmed +- book-ii-transcendentals-iota-tau-confirmed +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.IotaTauConfirmed + module_path: TauLib/BookII/Transcendentals/IotaTauConfirmed.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.IotaTauConfirmed (211 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005192.md b/_corpus_v3_items/cid005192.md new file mode 100644 index 000000000..c9e7f1570 --- /dev/null +++ b/_corpus_v3_items/cid005192.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005192/ +title: TauLib.BookII.Transcendentals.JReplacesI +cid: cid005192 +primary_alias: MOD0195 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.JReplacesI (180 lines, 27 declarations). +aliases: +- MOD0195 +- TauLib.BookII.Transcendentals.JReplacesI +- book-ii-transcendentals-jreplaces-i +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.JReplacesI + module_path: TauLib/BookII/Transcendentals/JReplacesI.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.JReplacesI (180 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005193.md b/_corpus_v3_items/cid005193.md new file mode 100644 index 000000000..db813b12e --- /dev/null +++ b/_corpus_v3_items/cid005193.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005193/ +title: TauLib.BookII.Transcendentals.Lines +cid: cid005193 +primary_alias: MOD0196 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.Lines (155 lines, 28 declarations). +aliases: +- MOD0196 +- TauLib.BookII.Transcendentals.Lines +- book-ii-transcendentals-lines +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.Lines + module_path: TauLib/BookII/Transcendentals/Lines.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.Lines (155 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005194.md b/_corpus_v3_items/cid005194.md new file mode 100644 index 000000000..96badde75 --- /dev/null +++ b/_corpus_v3_items/cid005194.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005194/ +title: TauLib.BookII.Transcendentals.PiEarned +cid: cid005194 +primary_alias: MOD0197 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookII.Transcendentals.PiEarned (170 lines, 21 declarations). +aliases: +- MOD0197 +- TauLib.BookII.Transcendentals.PiEarned +- book-ii-transcendentals-pi-earned +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookii +relations: +- predicate: appears_in + object: cid000001 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Transcendentals.PiEarned + module_path: TauLib/BookII/Transcendentals/PiEarned.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookII.Transcendentals.PiEarned (170 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005195.md b/_corpus_v3_items/cid005195.md new file mode 100644 index 000000000..c7758ce19 --- /dev/null +++ b/_corpus_v3_items/cid005195.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005195/ +title: TauLib.BookIII.Arithmetic.ABCConjecture +cid: cid005195 +primary_alias: MOD0198 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ABCConjecture (220 lines, 20 declarations). +aliases: +- MOD0198 +- TauLib.BookIII.Arithmetic.ABCConjecture +- book-iii-arithmetic-abcconjecture +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ABCConjecture (220 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005196.md b/_corpus_v3_items/cid005196.md new file mode 100644 index 000000000..275aef971 --- /dev/null +++ b/_corpus_v3_items/cid005196.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005196/ +title: TauLib.BookIII.Arithmetic.ABCDeep +cid: cid005196 +primary_alias: MOD0199 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ABCDeep (242 lines, 23 declarations). +aliases: +- MOD0199 +- TauLib.BookIII.Arithmetic.ABCDeep +- book-iii-arithmetic-abcdeep +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ABCDeep (242 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005197.md b/_corpus_v3_items/cid005197.md new file mode 100644 index 000000000..fd0b581a8 --- /dev/null +++ b/_corpus_v3_items/cid005197.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005197/ +title: TauLib.BookIII.Arithmetic.BSD +cid: cid005197 +primary_alias: MOD0200 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.BSD (152 lines, 12 declarations). +aliases: +- MOD0200 +- TauLib.BookIII.Arithmetic.BSD +- book-iii-arithmetic-bsd +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.BSD + module_path: TauLib/BookIII/Arithmetic/BSD.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.BSD (152 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005198.md b/_corpus_v3_items/cid005198.md new file mode 100644 index 000000000..1dece7379 --- /dev/null +++ b/_corpus_v3_items/cid005198.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005198/ +title: TauLib.BookIII.Arithmetic.EnrFunctor01 +cid: cid005198 +primary_alias: MOD0201 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.EnrFunctor01 (165 lines, 15 declarations). +aliases: +- MOD0201 +- TauLib.BookIII.Arithmetic.EnrFunctor01 +- book-iii-arithmetic-enr-functor01 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrFunctor01 + module_path: TauLib/BookIII/Arithmetic/EnrFunctor01.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.EnrFunctor01 (165 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005199.md b/_corpus_v3_items/cid005199.md new file mode 100644 index 000000000..5bd68bda5 --- /dev/null +++ b/_corpus_v3_items/cid005199.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005199/ +title: TauLib.BookIII.Arithmetic.EnrichedBiSquare +cid: cid005199 +primary_alias: MOD0202 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.EnrichedBiSquare (174 lines, 12 declarations). +aliases: +- MOD0202 +- TauLib.BookIII.Arithmetic.EnrichedBiSquare +- book-iii-arithmetic-enriched-bi-square +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrichedBiSquare + module_path: TauLib/BookIII/Arithmetic/EnrichedBiSquare.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.EnrichedBiSquare (174 lines, 12 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005200.md b/_corpus_v3_items/cid005200.md new file mode 100644 index 000000000..af9bdd5b1 --- /dev/null +++ b/_corpus_v3_items/cid005200.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005200/ +title: TauLib.BookIII.Arithmetic.Langlands +cid: cid005200 +primary_alias: MOD0203 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.Langlands (234 lines, 19 declarations). +aliases: +- MOD0203 +- TauLib.BookIII.Arithmetic.Langlands +- book-iii-arithmetic-langlands +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.Langlands + module_path: TauLib/BookIII/Arithmetic/Langlands.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.Langlands (234 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005201.md b/_corpus_v3_items/cid005201.md new file mode 100644 index 000000000..f20bb00c2 --- /dev/null +++ b/_corpus_v3_items/cid005201.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005201/ +title: TauLib.BookIII.Arithmetic.ProtoCodes +cid: cid005201 +primary_alias: MOD0204 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ProtoCodes (186 lines, 16 declarations). +aliases: +- MOD0204 +- TauLib.BookIII.Arithmetic.ProtoCodes +- book-iii-arithmetic-proto-codes +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ProtoCodes + module_path: TauLib/BookIII/Arithmetic/ProtoCodes.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.ProtoCodes (186 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005202.md b/_corpus_v3_items/cid005202.md new file mode 100644 index 000000000..d7178649b --- /dev/null +++ b/_corpus_v3_items/cid005202.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005202/ +title: TauLib.BookIII.Arithmetic.RationalPoints +cid: cid005202 +primary_alias: MOD0205 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.RationalPoints (187 lines, 19 declarations). +aliases: +- MOD0205 +- TauLib.BookIII.Arithmetic.RationalPoints +- book-iii-arithmetic-rational-points +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.RationalPoints + module_path: TauLib/BookIII/Arithmetic/RationalPoints.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.RationalPoints (187 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005203.md b/_corpus_v3_items/cid005203.md new file mode 100644 index 000000000..b4ea2ded2 --- /dev/null +++ b/_corpus_v3_items/cid005203.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005203/ +title: TauLib.BookIII.Arithmetic.TowerAssembly +cid: cid005203 +primary_alias: MOD0206 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Arithmetic.TowerAssembly (148 lines, 16 declarations). +aliases: +- MOD0206 +- TauLib.BookIII.Arithmetic.TowerAssembly +- book-iii-arithmetic-tower-assembly +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.TowerAssembly + module_path: TauLib/BookIII/Arithmetic/TowerAssembly.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Arithmetic.TowerAssembly (148 lines, 16 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005204.md b/_corpus_v3_items/cid005204.md new file mode 100644 index 000000000..08ee83059 --- /dev/null +++ b/_corpus_v3_items/cid005204.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005204/ +title: TauLib.BookIII.Bridge.BridgeAxiom +cid: cid005204 +primary_alias: MOD0207 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.BridgeAxiom (408 lines, 51 declarations). +aliases: +- MOD0207 +- TauLib.BookIII.Bridge.BridgeAxiom +- book-iii-bridge-bridge-axiom +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.BridgeAxiom (408 lines, 51 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005205.md b/_corpus_v3_items/cid005205.md new file mode 100644 index 000000000..2f68ebe3d --- /dev/null +++ b/_corpus_v3_items/cid005205.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005205/ +title: TauLib.BookIII.Bridge.ConjectureGaps +cid: cid005205 +primary_alias: MOD0208 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.ConjectureGaps (290 lines, 36 declarations). +aliases: +- MOD0208 +- TauLib.BookIII.Bridge.ConjectureGaps +- book-iii-bridge-conjecture-gaps +- III.R47 +- III.R48 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.ConjectureGaps (290 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005206.md b/_corpus_v3_items/cid005206.md new file mode 100644 index 000000000..07415a8b1 --- /dev/null +++ b/_corpus_v3_items/cid005206.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005206/ +title: TauLib.BookIII.Bridge.ForbiddenMoves +cid: cid005206 +primary_alias: MOD0209 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.ForbiddenMoves (282 lines, 36 declarations). +aliases: +- MOD0209 +- TauLib.BookIII.Bridge.ForbiddenMoves +- book-iii-bridge-forbidden-moves +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.ForbiddenMoves (282 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005207.md b/_corpus_v3_items/cid005207.md new file mode 100644 index 000000000..a435d50c7 --- /dev/null +++ b/_corpus_v3_items/cid005207.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005207/ +title: TauLib.BookIII.Bridge.Incompleteness +cid: cid005207 +primary_alias: MOD0210 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.Incompleteness (287 lines, 32 declarations). +aliases: +- MOD0210 +- TauLib.BookIII.Bridge.Incompleteness +- book-iii-bridge-incompleteness +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.Incompleteness (287 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005208.md b/_corpus_v3_items/cid005208.md new file mode 100644 index 000000000..bcb5ad31b --- /dev/null +++ b/_corpus_v3_items/cid005208.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005208/ +title: TauLib.BookIII.Bridge.TranslationArith +cid: cid005208 +primary_alias: MOD0211 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationArith (248 lines, 22 declarations). +aliases: +- MOD0211 +- TauLib.BookIII.Bridge.TranslationArith +- book-iii-bridge-translation-arith +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationArith + module_path: TauLib/BookIII/Bridge/TranslationArith.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationArith (248 lines, 22 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005209.md b/_corpus_v3_items/cid005209.md new file mode 100644 index 000000000..2ea31a73a --- /dev/null +++ b/_corpus_v3_items/cid005209.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005209/ +title: TauLib.BookIII.Bridge.TranslationObstruction +cid: cid005209 +primary_alias: MOD0212 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationObstruction (274 lines, 25 declarations). +aliases: +- MOD0212 +- TauLib.BookIII.Bridge.TranslationObstruction +- book-iii-bridge-translation-obstruction +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationObstruction (274 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005210.md b/_corpus_v3_items/cid005210.md new file mode 100644 index 000000000..4d43481e3 --- /dev/null +++ b/_corpus_v3_items/cid005210.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005210/ +title: TauLib.BookIII.Bridge.TranslationTopo +cid: cid005210 +primary_alias: MOD0213 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationTopo (221 lines, 17 declarations). +aliases: +- MOD0213 +- TauLib.BookIII.Bridge.TranslationTopo +- book-iii-bridge-translation-topo +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationTopo + module_path: TauLib/BookIII/Bridge/TranslationTopo.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.TranslationTopo (221 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005211.md b/_corpus_v3_items/cid005211.md new file mode 100644 index 000000000..723446edb --- /dev/null +++ b/_corpus_v3_items/cid005211.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005211/ +title: TauLib.BookIII.Bridge.ZFCasVM +cid: cid005211 +primary_alias: MOD0214 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Bridge.ZFCasVM (312 lines, 34 declarations). +aliases: +- MOD0214 +- TauLib.BookIII.Bridge.ZFCasVM +- book-iii-bridge-zfcas-vm +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Bridge.ZFCasVM (312 lines, 34 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005212.md b/_corpus_v3_items/cid005212.md new file mode 100644 index 000000000..dbaeb92fc --- /dev/null +++ b/_corpus_v3_items/cid005212.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005212/ +title: TauLib.BookIII.Computation.Admissibility +cid: cid005212 +primary_alias: MOD0215 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.Admissibility (238 lines, 20 declarations). +aliases: +- MOD0215 +- TauLib.BookIII.Computation.Admissibility +- book-iii-computation-admissibility +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.Admissibility + module_path: TauLib/BookIII/Computation/Admissibility.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.Admissibility (238 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005213.md b/_corpus_v3_items/cid005213.md new file mode 100644 index 000000000..cd8e09c5c --- /dev/null +++ b/_corpus_v3_items/cid005213.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005213/ +title: TauLib.BookIII.Computation.CompBiSquare +cid: cid005213 +primary_alias: MOD0216 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.CompBiSquare (195 lines, 20 declarations). +aliases: +- MOD0216 +- TauLib.BookIII.Computation.CompBiSquare +- book-iii-computation-comp-bi-square +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.CompBiSquare + module_path: TauLib/BookIII/Computation/CompBiSquare.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.CompBiSquare (195 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005214.md b/_corpus_v3_items/cid005214.md new file mode 100644 index 000000000..a9c949517 --- /dev/null +++ b/_corpus_v3_items/cid005214.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005214/ +title: TauLib.BookIII.Computation.E2Agent +cid: cid005214 +primary_alias: MOD0217 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.E2Agent (179 lines, 17 declarations). +aliases: +- MOD0217 +- TauLib.BookIII.Computation.E2Agent +- book-iii-computation-e2-agent +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Agent + module_path: TauLib/BookIII/Computation/E2Agent.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.E2Agent (179 lines, 17 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005215.md b/_corpus_v3_items/cid005215.md new file mode 100644 index 000000000..514a08447 --- /dev/null +++ b/_corpus_v3_items/cid005215.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005215/ +title: TauLib.BookIII.Computation.E2Witness +cid: cid005215 +primary_alias: MOD0218 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.E2Witness (242 lines, 23 declarations). +aliases: +- MOD0218 +- TauLib.BookIII.Computation.E2Witness +- book-iii-computation-e2-witness +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.E2Witness (242 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005216.md b/_corpus_v3_items/cid005216.md new file mode 100644 index 000000000..c16e9dc3e --- /dev/null +++ b/_corpus_v3_items/cid005216.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005216/ +title: TauLib.BookIII.Computation.TowerMachine +cid: cid005216 +primary_alias: MOD0219 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.TowerMachine (211 lines, 19 declarations). +aliases: +- MOD0219 +- TauLib.BookIII.Computation.TowerMachine +- book-iii-computation-tower-machine +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.TowerMachine + module_path: TauLib/BookIII/Computation/TowerMachine.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.TowerMachine (211 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005217.md b/_corpus_v3_items/cid005217.md new file mode 100644 index 000000000..b6f7ad9d3 --- /dev/null +++ b/_corpus_v3_items/cid005217.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005217/ +title: TauLib.BookIII.Computation.WitnessSearch +cid: cid005217 +primary_alias: MOD0220 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Computation.WitnessSearch (212 lines, 20 declarations). +aliases: +- MOD0220 +- TauLib.BookIII.Computation.WitnessSearch +- book-iii-computation-witness-search +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.WitnessSearch + module_path: TauLib/BookIII/Computation/WitnessSearch.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Computation.WitnessSearch (212 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005218.md b/_corpus_v3_items/cid005218.md new file mode 100644 index 000000000..8e41d610b --- /dev/null +++ b/_corpus_v3_items/cid005218.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005218/ +title: TauLib.BookIII.Doors.BridgeTightening +cid: cid005218 +primary_alias: MOD0221 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.BridgeTightening (293 lines, 28 declarations). +aliases: +- MOD0221 +- TauLib.BookIII.Doors.BridgeTightening +- book-iii-doors-bridge-tightening +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.BridgeTightening (293 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005219.md b/_corpus_v3_items/cid005219.md new file mode 100644 index 000000000..bd13e3791 --- /dev/null +++ b/_corpus_v3_items/cid005219.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005219/ +title: TauLib.BookIII.Doors.CriticalLine +cid: cid005219 +primary_alias: MOD0222 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.CriticalLine (200 lines, 20 declarations). +aliases: +- MOD0222 +- TauLib.BookIII.Doors.CriticalLine +- book-iii-doors-critical-line +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.CriticalLine + module_path: TauLib/BookIII/Doors/CriticalLine.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.CriticalLine (200 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005220.md b/_corpus_v3_items/cid005220.md new file mode 100644 index 000000000..f6ce581e4 --- /dev/null +++ b/_corpus_v3_items/cid005220.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005220/ +title: TauLib.BookIII.Doors.GrandGRH +cid: cid005220 +primary_alias: MOD0223 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.GrandGRH (254 lines, 18 declarations). +aliases: +- MOD0223 +- TauLib.BookIII.Doors.GrandGRH +- book-iii-doors-grand-grh +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.GrandGRH (254 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005221.md b/_corpus_v3_items/cid005221.md new file mode 100644 index 000000000..f335341dc --- /dev/null +++ b/_corpus_v3_items/cid005221.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005221/ +title: TauLib.BookIII.Doors.LemniscateOperator +cid: cid005221 +primary_alias: MOD0224 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.LemniscateOperator (210 lines, 29 declarations). +aliases: +- MOD0224 +- TauLib.BookIII.Doors.LemniscateOperator +- book-iii-doors-lemniscate-operator +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.LemniscateOperator + module_path: TauLib/BookIII/Doors/LemniscateOperator.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.LemniscateOperator (210 lines, 29 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005222.md b/_corpus_v3_items/cid005222.md new file mode 100644 index 000000000..bbae3017a --- /dev/null +++ b/_corpus_v3_items/cid005222.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005222/ +title: TauLib.BookIII.Doors.MasterSchema +cid: cid005222 +primary_alias: MOD0225 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.MasterSchema (200 lines, 30 declarations). +aliases: +- MOD0225 +- TauLib.BookIII.Doors.MasterSchema +- book-iii-doors-master-schema +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.MasterSchema (200 lines, 30 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005223.md b/_corpus_v3_items/cid005223.md new file mode 100644 index 000000000..75ea2c64e --- /dev/null +++ b/_corpus_v3_items/cid005223.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005223/ +title: TauLib.BookIII.Doors.MutualDetermination +cid: cid005223 +primary_alias: MOD0226 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.MutualDetermination (149 lines, 15 declarations). +aliases: +- MOD0226 +- TauLib.BookIII.Doors.MutualDetermination +- book-iii-doors-mutual-determination +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MutualDetermination + module_path: TauLib/BookIII/Doors/MutualDetermination.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.MutualDetermination (149 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005224.md b/_corpus_v3_items/cid005224.md new file mode 100644 index 000000000..1db50f22e --- /dev/null +++ b/_corpus_v3_items/cid005224.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005224/ +title: TauLib.BookIII.Doors.Poincare +cid: cid005224 +primary_alias: MOD0227 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.Poincare (187 lines, 15 declarations). +aliases: +- MOD0227 +- TauLib.BookIII.Doors.Poincare +- book-iii-doors-poincare +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.Poincare + module_path: TauLib/BookIII/Doors/Poincare.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.Poincare (187 lines, 15 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005225.md b/_corpus_v3_items/cid005225.md new file mode 100644 index 000000000..d1374b03b --- /dev/null +++ b/_corpus_v3_items/cid005225.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005225/ +title: TauLib.BookIII.Doors.SpectralCorrespondence +cid: cid005225 +primary_alias: MOD0228 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.SpectralCorrespondence (173 lines, 18 declarations). +aliases: +- MOD0228 +- TauLib.BookIII.Doors.SpectralCorrespondence +- book-iii-doors-spectral-correspondence +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralCorrespondence + module_path: TauLib/BookIII/Doors/SpectralCorrespondence.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.SpectralCorrespondence (173 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005226.md b/_corpus_v3_items/cid005226.md new file mode 100644 index 000000000..ec85eee66 --- /dev/null +++ b/_corpus_v3_items/cid005226.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005226/ +title: TauLib.BookIII.Doors.SpectralDecomp +cid: cid005226 +primary_alias: MOD0229 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.SpectralDecomp (251 lines, 23 declarations). +aliases: +- MOD0229 +- TauLib.BookIII.Doors.SpectralDecomp +- book-iii-doors-spectral-decomp +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.SpectralDecomp (251 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005227.md b/_corpus_v3_items/cid005227.md new file mode 100644 index 000000000..cbb3f8dc0 --- /dev/null +++ b/_corpus_v3_items/cid005227.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005227/ +title: TauLib.BookIII.Doors.SplitComplexZeta +cid: cid005227 +primary_alias: MOD0230 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Doors.SplitComplexZeta (197 lines, 26 declarations). +aliases: +- MOD0230 +- TauLib.BookIII.Doors.SplitComplexZeta +- book-iii-doors-split-complex-zeta +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SplitComplexZeta + module_path: TauLib/BookIII/Doors/SplitComplexZeta.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Doors.SplitComplexZeta (197 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005228.md b/_corpus_v3_items/cid005228.md new file mode 100644 index 000000000..e8085e62a --- /dev/null +++ b/_corpus_v3_items/cid005228.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005228/ +title: TauLib.BookIII.Enrichment.CanonicalLadder +cid: cid005228 +primary_alias: MOD0231 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Enrichment.CanonicalLadder (283 lines, 28 declarations). +aliases: +- MOD0231 +- TauLib.BookIII.Enrichment.CanonicalLadder +- book-iii-enrichment-canonical-ladder +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Enrichment.CanonicalLadder (283 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005229.md b/_corpus_v3_items/cid005229.md new file mode 100644 index 000000000..ca44accd2 --- /dev/null +++ b/_corpus_v3_items/cid005229.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005229/ +title: TauLib.BookIII.Enrichment.Functor +cid: cid005229 +primary_alias: MOD0232 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Enrichment.Functor (291 lines, 36 declarations). +aliases: +- MOD0232 +- TauLib.BookIII.Enrichment.Functor +- book-iii-enrichment-functor +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Enrichment.Functor (291 lines, 36 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005230.md b/_corpus_v3_items/cid005230.md new file mode 100644 index 000000000..b8cbbb007 --- /dev/null +++ b/_corpus_v3_items/cid005230.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005230/ +title: TauLib.BookIII.Enrichment.LayerTemplate +cid: cid005230 +primary_alias: MOD0233 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Enrichment.LayerTemplate (385 lines, 50 declarations). +aliases: +- MOD0233 +- TauLib.BookIII.Enrichment.LayerTemplate +- book-iii-enrichment-layer-template +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Enrichment.LayerTemplate (385 lines, 50 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005231.md b/_corpus_v3_items/cid005231.md new file mode 100644 index 000000000..6fc960f44 --- /dev/null +++ b/_corpus_v3_items/cid005231.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005231/ +title: TauLib.BookIII.Hinge.DependencyChain +cid: cid005231 +primary_alias: MOD0234 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Hinge.DependencyChain (356 lines, 53 declarations). +aliases: +- MOD0234 +- TauLib.BookIII.Hinge.DependencyChain +- book-iii-hinge-dependency-chain +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Hinge.DependencyChain (356 lines, 53 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005232.md b/_corpus_v3_items/cid005232.md new file mode 100644 index 000000000..9962afbce --- /dev/null +++ b/_corpus_v3_items/cid005232.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005232/ +title: TauLib.BookIII.Hinge.HingeTheorem +cid: cid005232 +primary_alias: MOD0235 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Hinge.HingeTheorem (407 lines, 46 declarations). +aliases: +- MOD0235 +- TauLib.BookIII.Hinge.HingeTheorem +- book-iii-hinge-hinge-theorem +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Hinge.HingeTheorem (407 lines, 46 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005233.md b/_corpus_v3_items/cid005233.md new file mode 100644 index 000000000..18f0e1acd --- /dev/null +++ b/_corpus_v3_items/cid005233.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005233/ +title: TauLib.BookIII.Mirror.E3Witness +cid: cid005233 +primary_alias: MOD0236 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Mirror.E3Witness (270 lines, 24 declarations). +aliases: +- MOD0236 +- TauLib.BookIII.Mirror.E3Witness +- book-iii-mirror-e3-witness +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Mirror.E3Witness (270 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005234.md b/_corpus_v3_items/cid005234.md new file mode 100644 index 000000000..b30c343f5 --- /dev/null +++ b/_corpus_v3_items/cid005234.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005234/ +title: TauLib.BookIII.Mirror.ProofTheoryE3 +cid: cid005234 +primary_alias: MOD0237 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Mirror.ProofTheoryE3 (380 lines, 44 declarations). +aliases: +- MOD0237 +- TauLib.BookIII.Mirror.ProofTheoryE3 +- book-iii-mirror-proof-theory-e3 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Mirror.ProofTheoryE3 (380 lines, 44 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005235.md b/_corpus_v3_items/cid005235.md new file mode 100644 index 000000000..826d15ab5 --- /dev/null +++ b/_corpus_v3_items/cid005235.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005235/ +title: TauLib.BookIII.Mirror.Saturation +cid: cid005235 +primary_alias: MOD0238 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Mirror.Saturation (310 lines, 32 declarations). +aliases: +- MOD0238 +- TauLib.BookIII.Mirror.Saturation +- book-iii-mirror-saturation +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Mirror.Saturation (310 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005236.md b/_corpus_v3_items/cid005236.md new file mode 100644 index 000000000..3307ce89a --- /dev/null +++ b/_corpus_v3_items/cid005236.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005236/ +title: TauLib.BookIII.Physics.FluidData +cid: cid005236 +primary_alias: MOD0239 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.FluidData (243 lines, 28 declarations). +aliases: +- MOD0239 +- TauLib.BookIII.Physics.FluidData +- book-iii-physics-fluid-data +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.FluidData + module_path: TauLib/BookIII/Physics/FluidData.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.FluidData (243 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005237.md b/_corpus_v3_items/cid005237.md new file mode 100644 index 000000000..a8900289e --- /dev/null +++ b/_corpus_v3_items/cid005237.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005237/ +title: TauLib.BookIII.Physics.GapTheorem +cid: cid005237 +primary_alias: MOD0240 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.GapTheorem (248 lines, 29 declarations). +aliases: +- MOD0240 +- TauLib.BookIII.Physics.GapTheorem +- book-iii-physics-gap-theorem +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.GapTheorem (248 lines, 29 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005238.md b/_corpus_v3_items/cid005238.md new file mode 100644 index 000000000..a183a3864 --- /dev/null +++ b/_corpus_v3_items/cid005238.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005238/ +title: TauLib.BookIII.Physics.HartogsFlow +cid: cid005238 +primary_alias: MOD0241 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.HartogsFlow (236 lines, 21 declarations). +aliases: +- MOD0241 +- TauLib.BookIII.Physics.HartogsFlow +- book-iii-physics-hartogs-flow +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.HartogsFlow + module_path: TauLib/BookIII/Physics/HartogsFlow.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.HartogsFlow (236 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005239.md b/_corpus_v3_items/cid005239.md new file mode 100644 index 000000000..2a91d850e --- /dev/null +++ b/_corpus_v3_items/cid005239.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005239/ +title: TauLib.BookIII.Physics.Hodge +cid: cid005239 +primary_alias: MOD0242 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.Hodge (293 lines, 26 declarations). +aliases: +- MOD0242 +- TauLib.BookIII.Physics.Hodge +- book-iii-physics-hodge +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.Hodge + module_path: TauLib/BookIII/Physics/Hodge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.Hodge (293 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005240.md b/_corpus_v3_items/cid005240.md new file mode 100644 index 000000000..a72c37a85 --- /dev/null +++ b/_corpus_v3_items/cid005240.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005240/ +title: TauLib.BookIII.Physics.PhysicsAssembly +cid: cid005240 +primary_alias: MOD0243 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.PhysicsAssembly (157 lines, 28 declarations). +aliases: +- MOD0243 +- TauLib.BookIII.Physics.PhysicsAssembly +- book-iii-physics-physics-assembly +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.PhysicsAssembly (157 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005241.md b/_corpus_v3_items/cid005241.md new file mode 100644 index 000000000..6b0b49353 --- /dev/null +++ b/_corpus_v3_items/cid005241.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005241/ +title: TauLib.BookIII.Physics.PositiveRegularity +cid: cid005241 +primary_alias: MOD0244 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.PositiveRegularity (234 lines, 24 declarations). +aliases: +- MOD0244 +- TauLib.BookIII.Physics.PositiveRegularity +- book-iii-physics-positive-regularity +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PositiveRegularity + module_path: TauLib/BookIII/Physics/PositiveRegularity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.PositiveRegularity (234 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005242.md b/_corpus_v3_items/cid005242.md new file mode 100644 index 000000000..33a81c39f --- /dev/null +++ b/_corpus_v3_items/cid005242.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005242/ +title: TauLib.BookIII.Physics.StrongSector +cid: cid005242 +primary_alias: MOD0245 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Physics.StrongSector (231 lines, 24 declarations). +aliases: +- MOD0245 +- TauLib.BookIII.Physics.StrongSector +- book-iii-physics-strong-sector +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.StrongSector + module_path: TauLib/BookIII/Physics/StrongSector.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Physics.StrongSector (231 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005243.md b/_corpus_v3_items/cid005243.md new file mode 100644 index 000000000..1b8ca34fa --- /dev/null +++ b/_corpus_v3_items/cid005243.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005243/ +title: TauLib.BookIII.Prologue.HartogsBulk +cid: cid005243 +primary_alias: MOD0246 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Prologue.HartogsBulk (160 lines, 19 declarations). +aliases: +- MOD0246 +- TauLib.BookIII.Prologue.HartogsBulk +- book-iii-prologue-hartogs-bulk +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Prologue.HartogsBulk (160 lines, 19 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005244.md b/_corpus_v3_items/cid005244.md new file mode 100644 index 000000000..91399cc17 --- /dev/null +++ b/_corpus_v3_items/cid005244.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005244/ +title: TauLib.BookIII.Sectors.BoundaryCharacters +cid: cid005244 +primary_alias: MOD0247 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Sectors.BoundaryCharacters (212 lines, 23 declarations). +aliases: +- MOD0247 +- TauLib.BookIII.Sectors.BoundaryCharacters +- book-iii-sectors-boundary-characters +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Sectors.BoundaryCharacters (212 lines, 23 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005245.md b/_corpus_v3_items/cid005245.md new file mode 100644 index 000000000..b9d2ad4c7 --- /dev/null +++ b/_corpus_v3_items/cid005245.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005245/ +title: TauLib.BookIII.Sectors.Decomposition +cid: cid005245 +primary_alias: MOD0248 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Sectors.Decomposition (224 lines, 24 declarations). +aliases: +- MOD0248 +- TauLib.BookIII.Sectors.Decomposition +- book-iii-sectors-decomposition +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Sectors.Decomposition (224 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005246.md b/_corpus_v3_items/cid005246.md new file mode 100644 index 000000000..aa8538c88 --- /dev/null +++ b/_corpus_v3_items/cid005246.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005246/ +title: TauLib.BookIII.Sectors.LanglandsReflection +cid: cid005246 +primary_alias: MOD0249 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Sectors.LanglandsReflection (275 lines, 25 declarations). +aliases: +- MOD0249 +- TauLib.BookIII.Sectors.LanglandsReflection +- book-iii-sectors-langlands-reflection +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Sectors.LanglandsReflection (275 lines, 25 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005247.md b/_corpus_v3_items/cid005247.md new file mode 100644 index 000000000..8618e5c05 --- /dev/null +++ b/_corpus_v3_items/cid005247.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005247/ +title: TauLib.BookIII.Sectors.ParityBridge +cid: cid005247 +primary_alias: MOD0250 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Sectors.ParityBridge (183 lines, 21 declarations). +aliases: +- MOD0250 +- TauLib.BookIII.Sectors.ParityBridge +- book-iii-sectors-parity-bridge +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Sectors.ParityBridge (183 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005248.md b/_corpus_v3_items/cid005248.md new file mode 100644 index 000000000..18f19adf7 --- /dev/null +++ b/_corpus_v3_items/cid005248.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005248/ +title: TauLib.BookIII.Spectral.AdditiveConjectures +cid: cid005248 +primary_alias: MOD0251 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.AdditiveConjectures (228 lines, 24 declarations). +aliases: +- MOD0251 +- TauLib.BookIII.Spectral.AdditiveConjectures +- book-iii-spectral-additive-conjectures +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.AdditiveConjectures (228 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005249.md b/_corpus_v3_items/cid005249.md new file mode 100644 index 000000000..7ca5785da --- /dev/null +++ b/_corpus_v3_items/cid005249.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005249/ +title: TauLib.BookIII.Spectral.Adeles +cid: cid005249 +primary_alias: MOD0252 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.Adeles (259 lines, 26 declarations). +aliases: +- MOD0252 +- TauLib.BookIII.Spectral.Adeles +- book-iii-spectral-adeles +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.Adeles (259 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005250.md b/_corpus_v3_items/cid005250.md new file mode 100644 index 000000000..e7314d64b --- /dev/null +++ b/_corpus_v3_items/cid005250.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005250/ +title: TauLib.BookIII.Spectral.BipolarClassifier +cid: cid005250 +primary_alias: MOD0253 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.BipolarClassifier (270 lines, 29 declarations). +aliases: +- MOD0253 +- TauLib.BookIII.Spectral.BipolarClassifier +- book-iii-spectral-bipolar-classifier +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.BipolarClassifier (270 lines, 29 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005251.md b/_corpus_v3_items/cid005251.md new file mode 100644 index 000000000..52822cfee --- /dev/null +++ b/_corpus_v3_items/cid005251.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005251/ +title: TauLib.BookIII.Spectral.CRT +cid: cid005251 +primary_alias: MOD0254 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.CRT (257 lines, 21 declarations). +aliases: +- MOD0254 +- TauLib.BookIII.Spectral.CRT +- book-iii-spectral-crt +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.CRT (257 lines, 21 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005252.md b/_corpus_v3_items/cid005252.md new file mode 100644 index 000000000..e7e11cbd7 --- /dev/null +++ b/_corpus_v3_items/cid005252.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005252/ +title: TauLib.BookIII.Spectral.ConfinementBridge +cid: cid005252 +primary_alias: MOD0255 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.ConfinementBridge (273 lines, 20 declarations). +aliases: +- MOD0255 +- TauLib.BookIII.Spectral.ConfinementBridge +- book-iii-spectral-confinement-bridge +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ConfinementBridge + module_path: TauLib/BookIII/Spectral/ConfinementBridge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.ConfinementBridge (273 lines, 20 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005253.md b/_corpus_v3_items/cid005253.md new file mode 100644 index 000000000..2e4b863bd --- /dev/null +++ b/_corpus_v3_items/cid005253.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005253/ +title: TauLib.BookIII.Spectral.GoldbachDeep +cid: cid005253 +primary_alias: MOD0256 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.GoldbachDeep (232 lines, 29 declarations). +aliases: +- MOD0256 +- TauLib.BookIII.Spectral.GoldbachDeep +- book-iii-spectral-goldbach-deep +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.GoldbachDeep (232 lines, 29 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005254.md b/_corpus_v3_items/cid005254.md new file mode 100644 index 000000000..85b977f40 --- /dev/null +++ b/_corpus_v3_items/cid005254.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005254/ +title: TauLib.BookIII.Spectral.HenselLifting +cid: cid005254 +primary_alias: MOD0257 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.HenselLifting (198 lines, 26 declarations). +aliases: +- MOD0257 +- TauLib.BookIII.Spectral.HenselLifting +- book-iii-spectral-hensel-lifting +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.HenselLifting (198 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005255.md b/_corpus_v3_items/cid005255.md new file mode 100644 index 000000000..64ff6070d --- /dev/null +++ b/_corpus_v3_items/cid005255.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005255/ +title: TauLib.BookIII.Spectral.LocalFields +cid: cid005255 +primary_alias: MOD0258 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.LocalFields (230 lines, 28 declarations). +aliases: +- MOD0258 +- TauLib.BookIII.Spectral.LocalFields +- book-iii-spectral-local-fields +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.LocalFields (230 lines, 28 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005256.md b/_corpus_v3_items/cid005256.md new file mode 100644 index 000000000..8f85a03bc --- /dev/null +++ b/_corpus_v3_items/cid005256.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005256/ +title: TauLib.BookIII.Spectral.ModularForms +cid: cid005256 +primary_alias: MOD0259 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.ModularForms (195 lines, 24 declarations). +aliases: +- MOD0259 +- TauLib.BookIII.Spectral.ModularForms +- book-iii-spectral-modular-forms +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ModularForms + module_path: TauLib/BookIII/Spectral/ModularForms.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.ModularForms (195 lines, 24 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005257.md b/_corpus_v3_items/cid005257.md new file mode 100644 index 000000000..2252fe840 --- /dev/null +++ b/_corpus_v3_items/cid005257.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005257/ +title: TauLib.BookIII.Spectral.PrimorialLadder +cid: cid005257 +primary_alias: MOD0260 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.PrimorialLadder (193 lines, 27 declarations). +aliases: +- MOD0260 +- TauLib.BookIII.Spectral.PrimorialLadder +- book-iii-spectral-primorial-ladder +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.PrimorialLadder (193 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005258.md b/_corpus_v3_items/cid005258.md new file mode 100644 index 000000000..0beaee809 --- /dev/null +++ b/_corpus_v3_items/cid005258.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005258/ +title: TauLib.BookIII.Spectral.SieveInfrastructure +cid: cid005258 +primary_alias: MOD0261 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.SieveInfrastructure (329 lines, 37 declarations). +aliases: +- MOD0261 +- TauLib.BookIII.Spectral.SieveInfrastructure +- book-iii-spectral-sieve-infrastructure +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.SieveInfrastructure (329 lines, 37 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005259.md b/_corpus_v3_items/cid005259.md new file mode 100644 index 000000000..3868e4b9c --- /dev/null +++ b/_corpus_v3_items/cid005259.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005259/ +title: TauLib.BookIII.Spectral.Trichotomy +cid: cid005259 +primary_alias: MOD0262 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.Trichotomy (319 lines, 32 declarations). +aliases: +- MOD0262 +- TauLib.BookIII.Spectral.Trichotomy +- book-iii-spectral-trichotomy +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.Trichotomy (319 lines, 32 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005260.md b/_corpus_v3_items/cid005260.md new file mode 100644 index 000000000..5d717a5bf --- /dev/null +++ b/_corpus_v3_items/cid005260.md @@ -0,0 +1,61 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005260/ +title: TauLib.BookIII.Spectral.TwinPrimeDeep +cid: cid005260 +primary_alias: MOD0263 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectral.TwinPrimeDeep (273 lines, 35 declarations). +aliases: +- MOD0263 +- TauLib.BookIII.Spectral.TwinPrimeDeep +- book-iii-spectral-twin-prime-deep +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectral.TwinPrimeDeep (273 lines, 35 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005261.md b/_corpus_v3_items/cid005261.md new file mode 100644 index 000000000..f627afd0e --- /dev/null +++ b/_corpus_v3_items/cid005261.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005261/ +title: TauLib.BookIII.Spectrum.InterfaceWidth +cid: cid005261 +primary_alias: MOD0264 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectrum.InterfaceWidth (158 lines, 18 declarations). +aliases: +- MOD0264 +- TauLib.BookIII.Spectrum.InterfaceWidth +- book-iii-spectrum-interface-width +- I.D71 +- I.D72 +- I.P30 +- I.T33 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006100 +- predicate: formalized_by + object: cid006101 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.InterfaceWidth + module_path: TauLib/BookIII/Spectrum/InterfaceWidth.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectrum.InterfaceWidth (158 lines, 18 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005262.md b/_corpus_v3_items/cid005262.md new file mode 100644 index 000000000..eaad1b741 --- /dev/null +++ b/_corpus_v3_items/cid005262.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005262/ +title: TauLib.BookIII.Spectrum.KernelHinge +cid: cid005262 +primary_alias: MOD0265 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectrum.KernelHinge (178 lines, 11 declarations). +aliases: +- MOD0265 +- TauLib.BookIII.Spectrum.KernelHinge +- book-iii-spectrum-kernel-hinge +- I.D74 +- I.T34 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006102 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.KernelHinge + module_path: TauLib/BookIII/Spectrum/KernelHinge.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectrum.KernelHinge (178 lines, 11 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005263.md b/_corpus_v3_items/cid005263.md new file mode 100644 index 000000000..bb2a1cf58 --- /dev/null +++ b/_corpus_v3_items/cid005263.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005263/ +title: TauLib.BookIII.Spectrum.TTM +cid: cid005263 +primary_alias: MOD0266 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectrum.TTM (328 lines, 26 declarations). +aliases: +- MOD0266 +- TauLib.BookIII.Spectrum.TTM +- book-iii-spectrum-ttm +- I.D69 +- I.D70 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.TTM + module_path: TauLib/BookIII/Spectrum/TTM.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectrum.TTM (328 lines, 26 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005264.md b/_corpus_v3_items/cid005264.md new file mode 100644 index 000000000..9984a70f0 --- /dev/null +++ b/_corpus_v3_items/cid005264.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005264/ +title: TauLib.BookIII.Spectrum.ThreeSAT +cid: cid005264 +primary_alias: MOD0267 +type: formal_module +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: TauLib v2 Lean module TauLib.BookIII.Spectrum.ThreeSAT (197 lines, 27 declarations). +aliases: +- MOD0267 +- TauLib.BookIII.Spectrum.ThreeSAT +- book-iii-spectrum-three-sat +- I.D73 +- I.P31 +- I.P32 +layers: [] +domains: +- math.foundation +- formalization.lean +- taulib.v2.bookiii +relations: +- predicate: appears_in + object: cid000024 +- predicate: formalized_by + object: cid006103 +- predicate: formalized_by + object: cid006104 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_projections +- version: v1 + date: '2026-05-10' + event: wired_formalized_by_in_wave_5 +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.ThreeSAT + module_path: TauLib/BookIII/Spectrum/ThreeSAT.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: TauLib v2 Lean module TauLib.BookIII.Spectrum.ThreeSAT (197 lines, 27 declarations). +noindex: false +--- + diff --git a/_corpus_v3_items/cid005265.md b/_corpus_v3_items/cid005265.md new file mode 100644 index 000000000..7b586f249 --- /dev/null +++ b/_corpus_v3_items/cid005265.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005265/ +title: chi_complete (theorem) +cid: cid005265 +primary_alias: FTH0002 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D38] Completeness: χ₊(z) + χ₋(z) = to_sectors(z).\n The two characters together reconstruct the full sector\ + \ decomposition. -/" +aliases: +- FTH0002 +- chi_complete +- chi-complete +- TauLib.BookI.Boundary.Characters::chi_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001054 +- predicate: part_of + object: cid005008 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005008 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Characters + declaration: chi_complete + module_path: TauLib/BookI/Boundary/Characters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D38] Completeness: χ₊(z) + χ₋(z) = to_sectors(z).\n The two characters together reconstruct the full\ + \ sector decomposition. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005266.md b/_corpus_v3_items/cid005266.md new file mode 100644 index 000000000..a70260d58 --- /dev/null +++ b/_corpus_v3_items/cid005266.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005266/ +title: chi_orthogonal (theorem) +cid: cid005266 +primary_alias: FTH0003 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D38] Orthogonality: χ₊(z) · χ₋(w) = 0 for all z, w.\n The B-sector and C-sector are mutually annihilating.\ + \ -/" +aliases: +- FTH0003 +- chi_orthogonal +- chi-orthogonal +- TauLib.BookI.Boundary.Characters::chi_orthogonal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001054 +- predicate: part_of + object: cid005008 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005008 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Characters + declaration: chi_orthogonal + module_path: TauLib/BookI/Boundary/Characters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D38] Orthogonality: χ₊(z) · χ₋(w) = 0 for all z, w.\n The B-sector and C-sector are mutually annihilating.\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005267.md b/_corpus_v3_items/cid005267.md new file mode 100644 index 000000000..cb24319f5 --- /dev/null +++ b/_corpus_v3_items/cid005267.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005267/ +title: sc_j_squared (theorem) +cid: cid005267 +primary_alias: FTH0004 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D86] The elliptic-hyperbolic dichotomy:\n - TauComplex has i² = -1 (elliptic sign), yielding a field with\ + \ no zero divisors.\n - SplitComplex has j² = +1 (hyperbolic sign), yielding a ring WITH zero divisors.\n\n We witness\ + \ the dichotomy by showing:\n 1. i² = -1 in TauComplex (taucomplex_i_squared)\n 2. j² = +1 in SplitComplex (sc_j_squared,\ + \ proved below)\n 3. SplitComplex has zero divisors (zero_divisor_witness_b from SplitComplex.lean) -/" +aliases: +- FTH0004 +- sc_j_squared +- sc-j-squared +- TauLib.BookI.Boundary.ComplexField::sc_j_squared +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001095 +- predicate: part_of + object: cid005009 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005009 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.ComplexField + declaration: sc_j_squared + module_path: TauLib/BookI/Boundary/ComplexField.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D86] The elliptic-hyperbolic dichotomy:\n - TauComplex has i² = -1 (elliptic sign), yielding a field\ + \ with no zero divisors.\n - SplitComplex has j² = +1 (hyperbolic sign), yielding a ring WITH zero divisors.\n\n We\ + \ witness the d" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005268.md b/_corpus_v3_items/cid005268.md new file mode 100644 index 000000000..b1206f300 --- /dev/null +++ b/_corpus_v3_items/cid005268.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005268/ +title: taureal_archimedean_embedding (theorem) +cid: cid005268 +primary_alias: FTH0005 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T42] The Archimedean property: the natural number embedding\n into TauReal is unbounded. The sequence\ + \ `fromNat n` is strictly\n increasing (distinct naturals give non-equivalent TauReals).\n\n Under the Cauchy `equiv`,\ + \ distinctness means the difference\n sequence does NOT converge to zero. Since `fromNat n` is constant\n at `n`,\ + \ the difference `|m - n|` is a fixed nonzero constant, and\n any small enough `1/(k+1)` tolerance witnesses non-convergence.\ + \ -/" +aliases: +- FTH0005 +- taureal_archimedean_embedding +- taureal-archimedean-embedding +- TauLib.BookI.Boundary.ConstructiveReals::taureal_archimedean_embedding +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001209 +- predicate: part_of + object: cid005010 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005010 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.ConstructiveReals + declaration: taureal_archimedean_embedding + module_path: TauLib/BookI/Boundary/ConstructiveReals.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T42] The Archimedean property: the natural number embedding\n into TauReal is unbounded. The sequence\ + \ `fromNat n` is strictly\n increasing (distinct naturals give non-equivalent TauReals).\n\n Under the Cauchy `equiv`,\ + \ distinct" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005269.md b/_corpus_v3_items/cid005269.md new file mode 100644 index 000000000..de121354c --- /dev/null +++ b/_corpus_v3_items/cid005269.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005269/ +title: root_of_unity_one (theorem) +cid: cid005269 +primary_alias: FTH0006 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T45] 1 is always an nth root of unity mod any m. -/ +aliases: +- FTH0006 +- root_of_unity_one +- root-of-unity-one +- TauLib.BookI.Boundary.Cyclotomic::root_of_unity_one +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001212 +- predicate: part_of + object: cid005012 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005012 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Cyclotomic + declaration: root_of_unity_one + module_path: TauLib/BookI/Boundary/Cyclotomic.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T45] 1 is always an nth root of unity mod any m. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005270.md b/_corpus_v3_items/cid005270.md new file mode 100644 index 000000000..26445d049 --- /dev/null +++ b/_corpus_v3_items/cid005270.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005270/ +title: crossing_iff_real (theorem) +cid: cid005270 +primary_alias: FTH0007 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D39] Crossing characterization: a split-complex element z is at the crossing\n iff z is real (im = 0).\ + \ Proof: b_sector = c_sector iff re+im = re-im iff im = 0. -/" +aliases: +- FTH0007 +- crossing_iff_real +- crossing-iff-real +- TauLib.BookI.Boundary.Fourier::crossing_iff_real +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001055 +- predicate: part_of + object: cid005014 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005014 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Fourier + declaration: crossing_iff_real + module_path: TauLib/BookI/Boundary/Fourier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D39] Crossing characterization: a split-complex element z is at the crossing\n iff z is real (im =\ + \ 0). Proof: b_sector = c_sector iff re+im = re-im iff im = 0. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005271.md b/_corpus_v3_items/cid005271.md new file mode 100644 index 000000000..4b45dc03d --- /dev/null +++ b/_corpus_v3_items/cid005271.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005271/ +title: fourier_sigma_swap (theorem) +cid: cid005271 +primary_alias: FTH0008 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.D40] σ = component swap in Fourier coordinates: σ exchanges B ↔ C harmonics. -/' +aliases: +- FTH0008 +- fourier_sigma_swap +- fourier-sigma-swap +- TauLib.BookI.Boundary.Fourier::fourier_sigma_swap +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001056 +- predicate: part_of + object: cid005014 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005014 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Fourier + declaration: fourier_sigma_swap + module_path: TauLib/BookI/Boundary/Fourier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.D40] σ = component swap in Fourier coordinates: σ exchanges B ↔ C harmonics. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005272.md b/_corpus_v3_items/cid005272.md new file mode 100644 index 000000000..64540f280 --- /dev/null +++ b/_corpus_v3_items/cid005272.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005272/ +title: integral_const_one_3 (theorem) +cid: cid005272 +primary_alias: FTH0009 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.D99] Integral of constant 1 at stage 3: ∫_3 1 = 30/30 = 1. -/' +aliases: +- FTH0009 +- integral_const_one_3 +- integral-const-one-3 +- TauLib.BookI.Boundary.Integration::integral_const_one_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001108 +- predicate: part_of + object: cid005015 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005015 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Integration + declaration: integral_const_one_3 + module_path: TauLib/BookI/Boundary/Integration.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.D99] Integral of constant 1 at stage 3: ∫_3 1 = 30/30 = 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005273.md b/_corpus_v3_items/cid005273.md new file mode 100644 index 000000000..3ad9b81a4 --- /dev/null +++ b/_corpus_v3_items/cid005273.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005273/ +title: linearity_2f_3g_stage2 (theorem) +cid: cid005273 +primary_alias: FTH0010 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T51] Linearity for (2f + 3g) at stage 2. -/ +aliases: +- FTH0010 +- linearity_2f_3g_stage2 +- linearity-2f-3g-stage2 +- TauLib.BookI.Boundary.Integration::linearity_2f_3g_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001218 +- predicate: part_of + object: cid005015 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005015 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Integration + declaration: linearity_2f_3g_stage2 + module_path: TauLib/BookI/Boundary/Integration.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T51] Linearity for (2f + 3g) at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005274.md b/_corpus_v3_items/cid005274.md new file mode 100644 index 000000000..4ae135b71 --- /dev/null +++ b/_corpus_v3_items/cid005274.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005274/ +title: linearity_identity_stage2 (theorem) +cid: cid005274 +primary_alias: FTH0011 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T51] Linearity for (1f + 0g) at stage 2 (identity). -/ +aliases: +- FTH0011 +- linearity_identity_stage2 +- linearity-identity-stage2 +- TauLib.BookI.Boundary.Integration::linearity_identity_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001218 +- predicate: part_of + object: cid005015 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005015 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Integration + declaration: linearity_identity_stage2 + module_path: TauLib/BookI/Boundary/Integration.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T51] Linearity for (1f + 0g) at stage 2 (identity). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005275.md b/_corpus_v3_items/cid005275.md new file mode 100644 index 000000000..0e4d2a46c --- /dev/null +++ b/_corpus_v3_items/cid005275.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005275/ +title: integral_even_2 (theorem) +cid: cid005275 +primary_alias: FTH0012 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.D99] Integral of even indicator at stage 2: 3/6 = 1/2. -/' +aliases: +- FTH0012 +- integral_even_2 +- integral-even-2 +- TauLib.BookI.Boundary.Integration::integral_even_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001108 +- predicate: part_of + object: cid005015 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005015 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Integration + declaration: integral_even_2 + module_path: TauLib/BookI/Boundary/Integration.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.D99] Integral of even indicator at stage 2: 3/6 = 1/2. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005276.md b/_corpus_v3_items/cid005276.md new file mode 100644 index 000000000..ea2dfb0b1 --- /dev/null +++ b/_corpus_v3_items/cid005276.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005276/ +title: primorial_pos_1 (theorem) +cid: cid005276 +primary_alias: FTH0013 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.D95] Primorial is always positive (stages 1-3). -/ +aliases: +- FTH0013 +- primorial_pos_1 +- primorial-pos-1 +- TauLib.BookI.Boundary.Measure::primorial_pos_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001104 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: primorial_pos_1 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.D95] Primorial is always positive (stages 1-3). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005277.md b/_corpus_v3_items/cid005277.md new file mode 100644 index 000000000..1df8d2092 --- /dev/null +++ b/_corpus_v3_items/cid005277.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005277/ +title: additivity_even_odd_3 (theorem) +cid: cid005277 +primary_alias: FTH0014 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T49] Additivity for even/odd partition at stage 3. -/ +aliases: +- FTH0014 +- additivity_even_odd_3 +- additivity-even-odd-3 +- TauLib.BookI.Boundary.Measure::additivity_even_odd_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001216 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: additivity_even_odd_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T49] Additivity for even/odd partition at stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005278.md b/_corpus_v3_items/cid005278.md new file mode 100644 index 000000000..8ba8e44fc --- /dev/null +++ b/_corpus_v3_items/cid005278.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005278/ +title: additivity_bc_3 (theorem) +cid: cid005278 +primary_alias: FTH0015 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T49] Additivity for B/C sector partition at stage 3. -/ +aliases: +- FTH0015 +- additivity_bc_3 +- additivity-bc-3 +- TauLib.BookI.Boundary.Measure::additivity_bc_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001216 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: additivity_bc_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T49] Additivity for B/C sector partition at stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005279.md b/_corpus_v3_items/cid005279.md new file mode 100644 index 000000000..637f1d5b1 --- /dev/null +++ b/_corpus_v3_items/cid005279.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005279/ +title: full_set_compatible_3 (theorem) +cid: cid005279 +primary_alias: FTH0016 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P43] Full set is tower-compatible up to stage 3. -/ +aliases: +- FTH0016 +- full_set_compatible_3 +- full-set-compatible-3 +- TauLib.BookI.Boundary.Measure::full_set_compatible_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001163 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: full_set_compatible_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P43] Full set is tower-compatible up to stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005280.md b/_corpus_v3_items/cid005280.md new file mode 100644 index 000000000..a5641b5bf --- /dev/null +++ b/_corpus_v3_items/cid005280.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005280/ +title: empty_set_compatible_3 (theorem) +cid: cid005280 +primary_alias: FTH0017 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P43] Empty set is tower-compatible up to stage 3. -/ +aliases: +- FTH0017 +- empty_set_compatible_3 +- empty-set-compatible-3 +- TauLib.BookI.Boundary.Measure::empty_set_compatible_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001163 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: empty_set_compatible_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P43] Empty set is tower-compatible up to stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005281.md b/_corpus_v3_items/cid005281.md new file mode 100644 index 000000000..86cf98c32 --- /dev/null +++ b/_corpus_v3_items/cid005281.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005281/ +title: even_set_compatible_3 (theorem) +cid: cid005281 +primary_alias: FTH0018 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P43] Even set is tower-compatible up to stage 3. -/ +aliases: +- FTH0018 +- even_set_compatible_3 +- even-set-compatible-3 +- TauLib.BookI.Boundary.Measure::even_set_compatible_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001163 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: even_set_compatible_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P43] Even set is tower-compatible up to stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005282.md b/_corpus_v3_items/cid005282.md new file mode 100644 index 000000000..08aa42c85 --- /dev/null +++ b/_corpus_v3_items/cid005282.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005282/ +title: full_set_measure_3 (theorem) +cid: cid005282 +primary_alias: FTH0019 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.D95] Full set measure at stage 3: M_3 / M_3 = 1. -/' +aliases: +- FTH0019 +- full_set_measure_3 +- full-set-measure-3 +- TauLib.BookI.Boundary.Measure::full_set_measure_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001104 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: full_set_measure_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.D95] Full set measure at stage 3: M_3 / M_3 = 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005283.md b/_corpus_v3_items/cid005283.md new file mode 100644 index 000000000..408852550 --- /dev/null +++ b/_corpus_v3_items/cid005283.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005283/ +title: empty_set_measure_3 (theorem) +cid: cid005283 +primary_alias: FTH0020 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.D95] Empty set measure at stage 3: 0 / M_3 = 0. -/' +aliases: +- FTH0020 +- empty_set_measure_3 +- empty-set-measure-3 +- TauLib.BookI.Boundary.Measure::empty_set_measure_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001104 +- predicate: part_of + object: cid005018 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005018 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Measure + declaration: empty_set_measure_3 + module_path: TauLib/BookI/Boundary/Measure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.D95] Empty set measure at stage 3: 0 / M_3 = 0. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005284.md b/_corpus_v3_items/cid005284.md new file mode 100644 index 000000000..2d3fb00f3 --- /dev/null +++ b/_corpus_v3_items/cid005284.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005284/ +title: spectral_decomposition (theorem) +cid: cid005284 +primary_alias: FTH0021 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T12] Spectral decomposition: every element decomposes as the sum of its\n B-sector projection and C-sector\ + \ projection.\n to_sectors(z) = χ₊(z) + χ₋(z) -/" +aliases: +- FTH0021 +- spectral_decomposition +- TauLib.BookI.Boundary.Spectral::spectral_decomposition +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001182 +- predicate: part_of + object: cid005024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005024 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Spectral + declaration: spectral_decomposition + module_path: TauLib/BookI/Boundary/Spectral.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T12] Spectral decomposition: every element decomposes as the sum of its\n B-sector projection and\ + \ C-sector projection.\n to_sectors(z) = χ₊(z) + χ₋(z) -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005285.md b/_corpus_v3_items/cid005285.md new file mode 100644 index 000000000..08ba9a26e --- /dev/null +++ b/_corpus_v3_items/cid005285.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005285/ +title: spectral_unique (theorem) +cid: cid005285 +primary_alias: FTH0022 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T12] Spectral uniqueness: if an element has the same spectral transform\n as another, they are equal.\ + \ This is to_sectors injectivity in spectral language. -/" +aliases: +- FTH0022 +- spectral_unique +- spectral-unique +- TauLib.BookI.Boundary.Spectral::spectral_unique +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001182 +- predicate: part_of + object: cid005024 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005024 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Boundary.Spectral + declaration: spectral_unique + module_path: TauLib/BookI/Boundary/Spectral.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T12] Spectral uniqueness: if an element has the same spectral transform\n as another, they are equal.\ + \ This is to_sectors injectivity in spectral language. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005286.md b/_corpus_v3_items/cid005286.md new file mode 100644 index 000000000..40b712342 --- /dev/null +++ b/_corpus_v3_items/cid005286.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005286/ +title: prime_race_4_50 (theorem) +cid: cid005286 +primary_alias: FTH0023 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.D97] Prime race (q=4) is well-defined up to 50. -/ +aliases: +- FTH0023 +- prime_race_4_50 +- prime-race-4-50 +- TauLib.BookI.Coordinates.ChebyshevBias::prime_race_4_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001106 +- predicate: part_of + object: cid005042 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005042 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ChebyshevBias + declaration: prime_race_4_50 + module_path: TauLib/BookI/Coordinates/ChebyshevBias.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.D97] Prime race (q=4) is well-defined up to 50. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005287.md b/_corpus_v3_items/cid005287.md new file mode 100644 index 000000000..1992a635c --- /dev/null +++ b/_corpus_v3_items/cid005287.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005287/ +title: bias_positive_50 (theorem) +cid: cid005287 +primary_alias: FTH0024 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.D98] Chebyshev bias (q=4, 3 vs 1) is positive up to 50. -/ +aliases: +- FTH0024 +- bias_positive_50 +- bias-positive-50 +- TauLib.BookI.Coordinates.ChebyshevBias::bias_positive_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001107 +- predicate: part_of + object: cid005042 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005042 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ChebyshevBias + declaration: bias_positive_50 + module_path: TauLib/BookI/Coordinates/ChebyshevBias.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.D98] Chebyshev bias (q=4, 3 vs 1) is positive up to 50. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005288.md b/_corpus_v3_items/cid005288.md new file mode 100644 index 000000000..446f38396 --- /dev/null +++ b/_corpus_v3_items/cid005288.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005288/ +title: bias_mod3_50 (theorem) +cid: cid005288 +primary_alias: FTH0025 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.D98] Bias mod 3 check up to 50. -/ +aliases: +- FTH0025 +- bias_mod3_50 +- bias-mod3-50 +- TauLib.BookI.Coordinates.ChebyshevBias::bias_mod3_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001107 +- predicate: part_of + object: cid005042 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005042 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ChebyshevBias + declaration: bias_mod3_50 + module_path: TauLib/BookI/Coordinates/ChebyshevBias.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.D98] Bias mod 3 check up to 50. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005289.md b/_corpus_v3_items/cid005289.md new file mode 100644 index 000000000..5f33a0ad7 --- /dev/null +++ b/_corpus_v3_items/cid005289.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005289/ +title: bias_primorial_3 (theorem) +cid: cid005289 +primary_alias: FTH0026 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T50] Bias at primorial levels up to depth 3. -/ +aliases: +- FTH0026 +- bias_primorial_3 +- bias-primorial-3 +- TauLib.BookI.Coordinates.ChebyshevBias::bias_primorial_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001217 +- predicate: part_of + object: cid005042 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005042 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.ChebyshevBias + declaration: bias_primorial_3 + module_path: TauLib/BookI/Coordinates/ChebyshevBias.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T50] Bias at primorial levels up to depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005290.md b/_corpus_v3_items/cid005290.md new file mode 100644 index 000000000..3bd76fbcd --- /dev/null +++ b/_corpus_v3_items/cid005290.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005290/ +title: no_tie (theorem) +cid: cid005290 +primary_alias: FTH0027 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L03] No-Tie Lemma: If b₁ · A↑↑(c₁-1) = b₂ · A↑↑(c₂-1) (=: v),\n and both c₁, c₂ are maximal (¬(A↑↑cᵢ ∣\ + \ v)), then c₁ = c₂ and b₁ = b₂.\n\n Proof: Suppose c₁ < c₂. Then A↑↑c₁ ∣ A↑↑(c₂-1) (since both are\n powers of A\ + \ and c₁ ≤ c₂-1). Hence A↑↑c₁ ∣ v = b₂ · A↑↑(c₂-1).\n But ¬(A↑↑c₁ ∣ v), contradiction. So c₁ = c₂, then b₁ = b₂. -/" +aliases: +- FTH0027 +- no_tie +- no-tie +- TauLib.BookI.Coordinates.NoTie::no_tie +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001118 +- predicate: part_of + object: cid005049 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005049 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.NoTie + declaration: no_tie + module_path: TauLib/BookI/Coordinates/NoTie.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L03] No-Tie Lemma: If b₁ · A↑↑(c₁-1) = b₂ · A↑↑(c₂-1) (=: v),\n and both c₁, c₂ are maximal (¬(A↑↑cᵢ\ + \ ∣ v)), then c₁ = c₂ and b₁ = b₂.\n\n Proof: Suppose c₁ < c₂. Then A↑↑c₁ ∣ A↑↑(c₂-1) (since both are\n powers of\ + \ A and c₁ ≤ c₂-1" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005291.md b/_corpus_v3_items/cid005291.md new file mode 100644 index 000000000..26d1fb82d --- /dev/null +++ b/_corpus_v3_items/cid005291.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005291/ +title: prime_product_unique (theorem) +cid: cid005291 +primary_alias: FTH0028 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T09] FTA uniqueness: two sorted non-decreasing prime lists\n with the same product are identical. -/" +aliases: +- FTH0028 +- prime_product_unique +- prime-product-unique +- TauLib.BookI.Coordinates.Primes::prime_product_unique +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001176 +- predicate: part_of + object: cid005051 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005051 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Coordinates.Primes + declaration: prime_product_unique + module_path: TauLib/BookI/Coordinates/Primes.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T09] FTA uniqueness: two sorted non-decreasing prime lists\n with the same product are identical.\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005292.md b/_corpus_v3_items/cid005292.md new file mode 100644 index 000000000..23579800a --- /dev/null +++ b/_corpus_v3_items/cid005292.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005292/ +title: idx_add_injective (theorem) +cid: cid005292 +primary_alias: FTH0029 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P05] Addition is injective in the second argument. -/ +aliases: +- FTH0029 +- idx_add_injective +- idx-add-injective +- TauLib.BookI.Denotation.Arithmetic::idx_add_injective +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001128 +- predicate: part_of + object: cid005052 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005052 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Arithmetic + declaration: idx_add_injective + module_path: TauLib/BookI/Denotation/Arithmetic.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P05] Addition is injective in the second argument. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005293.md b/_corpus_v3_items/cid005293.md new file mode 100644 index 000000000..0f34c90f5 --- /dev/null +++ b/_corpus_v3_items/cid005293.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005293/ +title: growth_escape (theorem) +cid: cid005293 +primary_alias: FTH0030 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L02] **Growth Escape**: Tetration escapes the primorial tower.\n\n For any tower depth d ≥ 1, there exists\ + \ a tetration height c such that\n 2↑↑c mod M_d ≠ 2↑↑c — the tetration value cannot be represented\n faithfully within\ + \ the primorial modulus.\n\n This is the quantitative shadow of saturation: the 4th hyperoperation\n level produces\ + \ values that outrun the finite primorial approximations,\n no matter how deep the tower extends. -/" +aliases: +- FTH0030 +- growth_escape +- TauLib.BookI.Denotation.GrowthEscape::growth_escape +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001117 +- predicate: part_of + object: cid005054 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005054 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.GrowthEscape + declaration: growth_escape + module_path: TauLib/BookI/Denotation/GrowthEscape.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L02] **Growth Escape**: Tetration escapes the primorial tower.\n\n For any tower depth d ≥ 1, there\ + \ exists a tetration height c such that\n 2↑↑c mod M_d ≠ 2↑↑c — the tetration value cannot be represented\n faithfully\ + \ within the " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005294.md b/_corpus_v3_items/cid005294.md new file mode 100644 index 000000000..5ac6ea44e --- /dev/null +++ b/_corpus_v3_items/cid005294.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005294/ +title: denotational_lt_wf (theorem) +cid: cid005294 +primary_alias: FTH0031 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P07] Well-foundedness of the denotational order:\n denotational_lt is a subrelation of the lexicographic\ + \ order on (Nat, Nat),\n which is well-founded since Nat is well-ordered. -/" +aliases: +- FTH0031 +- denotational_lt_wf +- denotational-lt-wf +- TauLib.BookI.Denotation.Order::denotational_lt_wf +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001130 +- predicate: part_of + object: cid005055 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005055 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.Order + declaration: denotational_lt_wf + module_path: TauLib/BookI/Denotation/Order.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P07] Well-foundedness of the denotational order:\n denotational_lt is a subrelation of the lexicographic\ + \ order on (Nat, Nat),\n which is well-founded since Nat is well-ordered. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005295.md b/_corpus_v3_items/cid005295.md new file mode 100644 index 000000000..19d286007 --- /dev/null +++ b/_corpus_v3_items/cid005295.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005295/ +title: compose_assoc (theorem) +cid: cid005295 +primary_alias: FTH0032 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.T03] Composition is associative. -/ +aliases: +- FTH0032 +- compose_assoc +- compose-assoc +- TauLib.BookI.Denotation.ProgramMonoid::compose_assoc +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001170 +- predicate: part_of + object: cid005056 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005056 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.ProgramMonoid + declaration: compose_assoc + module_path: TauLib/BookI/Denotation/ProgramMonoid.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.T03] Composition is associative. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005296.md b/_corpus_v3_items/cid005296.md new file mode 100644 index 000000000..a2a9e7d6c --- /dev/null +++ b/_corpus_v3_items/cid005296.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005296/ +title: rho_count_compose (theorem) +cid: cid005296 +primary_alias: FTH0033 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L02] Normal form confluence: the rho count of a composed program\n is the sum of the individual rho counts.\ + \ -/" +aliases: +- FTH0033 +- rho_count_compose +- rho-count-compose +- TauLib.BookI.Denotation.ProgramMonoid::rho_count_compose +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001117 +- predicate: part_of + object: cid005056 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005056 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Denotation.ProgramMonoid + declaration: rho_count_compose + module_path: TauLib/BookI/Denotation/ProgramMonoid.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L02] Normal form confluence: the rho count of a composed program\n is the sum of the individual rho\ + \ counts. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005297.md b/_corpus_v3_items/cid005297.md new file mode 100644 index 000000000..b954c79ec --- /dev/null +++ b/_corpus_v3_items/cid005297.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005297/ +title: interior_from_boundary (theorem) +cid: cid005297 +primary_alias: FTH0034 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.C02] Corollary: the interior is determined by the boundary.\n Two tower-coherent functions with the same\ + \ boundary data\n (i.e., agreement at a reference depth) have the same interior values. -/" +aliases: +- FTH0034 +- interior_from_boundary +- interior-from-boundary +- TauLib.BookI.Holomorphy.BoundaryInterior::interior_from_boundary +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001001 +- predicate: part_of + object: cid005060 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005060 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.BoundaryInterior + declaration: interior_from_boundary + module_path: TauLib/BookI/Holomorphy/BoundaryInterior.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.C02] Corollary: the interior is determined by the boundary.\n Two tower-coherent functions with the\ + \ same boundary data\n (i.e., agreement at a reference depth) have the same interior values. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005298.md b/_corpus_v3_items/cid005298.md new file mode 100644 index 000000000..7718f9311 --- /dev/null +++ b/_corpus_v3_items/cid005298.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005298/ +title: sector_fun_independent (theorem) +cid: cid005298 +primary_alias: FTH0035 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P22] Every SectorFun is sector-independent by construction. -/ +aliases: +- FTH0035 +- sector_fun_independent +- sector-fun-independent +- TauLib.BookI.Holomorphy.DHolomorphic::sector_fun_independent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001145 +- predicate: part_of + object: cid005061 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005061 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DHolomorphic + declaration: sector_fun_independent + module_path: TauLib/BookI/Holomorphy/DHolomorphic.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P22] Every SectorFun is sector-independent by construction. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005299.md b/_corpus_v3_items/cid005299.md new file mode 100644 index 000000000..85a809c12 --- /dev/null +++ b/_corpus_v3_items/cid005299.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005299/ +title: no_simul_projection_b (theorem) +cid: cid005299 +primary_alias: FTH0036 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P23] No Simultaneous Projection (sector purity):\n For a tower-coherent function where one sector is constantly\ + \ zero,\n the other sector carries all the information.\n\n Concretely: if f.b_fun = 0 everywhere, then f.c_fun is\ + \ the sole\n carrier; and vice versa. The sectors cannot BOTH be nontrivial\n for a well-behaved (tower-coherent)\ + \ omega-germ transformer.\n\n This is formalized as: the product of a B-only and C-only\n stagewise function outputs\ + \ zero. -/" +aliases: +- FTH0036 +- no_simul_projection_b +- no-simul-projection-b +- TauLib.BookI.Holomorphy.DiagonalProtection::no_simul_projection_b +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001146 +- predicate: part_of + object: cid005062 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005062 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DiagonalProtection + declaration: no_simul_projection_b + module_path: TauLib/BookI/Holomorphy/DiagonalProtection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P23] No Simultaneous Projection (sector purity):\n For a tower-coherent function where one sector\ + \ is constantly zero,\n the other sector carries all the information.\n\n Concretely: if f.b_fun = 0 everywhere, then\ + \ f.c_fun is the" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005300.md b/_corpus_v3_items/cid005300.md new file mode 100644 index 000000000..0421c6bb0 --- /dev/null +++ b/_corpus_v3_items/cid005300.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005300/ +title: diagonal_free_protection (theorem) +cid: cid005300 +primary_alias: FTH0037 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T19] Diagonal-Free Protection Theorem:\n The product of pure B-sector and pure C-sector outputs is always\ + \ zero.\n This is the structural reflection of e₊·e₋ = 0 at the function level:\n sector-pure outputs cannot combine\ + \ nontrivially.\n\n For any two stagewise evaluations giving pure B and pure C outputs,\n their sector product is\ + \ zero. -/" +aliases: +- FTH0037 +- diagonal_free_protection +- TauLib.BookI.Holomorphy.DiagonalProtection::diagonal_free_protection +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001189 +- predicate: part_of + object: cid005062 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005062 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DiagonalProtection + declaration: diagonal_free_protection + module_path: TauLib/BookI/Holomorphy/DiagonalProtection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T19] Diagonal-Free Protection Theorem:\n The product of pure B-sector and pure C-sector outputs is\ + \ always zero.\n This is the structural reflection of e₊·e₋ = 0 at the function level:\n sector-pure outputs cannot\ + \ combine nontri" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005301.md b/_corpus_v3_items/cid005301.md new file mode 100644 index 000000000..574b1e55d --- /dev/null +++ b/_corpus_v3_items/cid005301.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005301/ +title: stagefun_comp_assoc (theorem) +cid: cid005301 +primary_alias: FTH0038 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P24] Composition of stagewise functions is associative. -/ +aliases: +- FTH0038 +- stagefun_comp_assoc +- stagefun-comp-assoc +- TauLib.BookI.Holomorphy.DiagonalProtection::stagefun_comp_assoc +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001147 +- predicate: part_of + object: cid005062 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005062 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.DiagonalProtection + declaration: stagefun_comp_assoc + module_path: TauLib/BookI/Holomorphy/DiagonalProtection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P24] Composition of stagewise functions is associative. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005302.md b/_corpus_v3_items/cid005302.md new file mode 100644 index 000000000..0ddb6e0f4 --- /dev/null +++ b/_corpus_v3_items/cid005302.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005302/ +title: global_hartogs (theorem) +cid: cid005302 +primary_alias: FTH0039 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T31] The Global Hartogs Extension Theorem:\n Any two tower-coherent functions that agree at some reference\ + \ depth\n agree at ALL depths ≤ that reference depth.\n\n Interpretation: if f is defined on L \\ K and we can extend\ + \ it\n to agree at depth d₀, then the extension is unique everywhere\n below d₀. The thin set K is \"removable\" because\ + \ tower coherence\n forces the values on K to be determined by the surrounding data.\n\n Proof: direct application\ + \ of the Identity Theorem (I.T21)\n which gives downward propagation of agreement via tower coherence. -/" +aliases: +- FTH0039 +- global_hartogs +- TauLib.BookI.Holomorphy.GlobalHartogs::global_hartogs +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001201 +- predicate: part_of + object: cid005063 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005063 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.GlobalHartogs + declaration: global_hartogs + module_path: TauLib/BookI/Holomorphy/GlobalHartogs.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T31] The Global Hartogs Extension Theorem:\n Any two tower-coherent functions that agree at some reference\ + \ depth\n agree at ALL depths ≤ that reference depth.\n\n Interpretation: if f is defined on L \\ K and we can extend\ + \ it\n " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005303.md b/_corpus_v3_items/cid005303.md new file mode 100644 index 000000000..ef6ca4a4c --- /dev/null +++ b/_corpus_v3_items/cid005303.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005303/ +title: tail_agree_downward (theorem) +cid: cid005303 +primary_alias: FTH0040 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L07] Tail Agreement Propagation (single input):\n If two tower-coherent stagewise functions agree at stage\ + \ d₀ for input n,\n then they agree at ALL stages k ≤ d₀ for input n.\n\n This is the \"downward\" direction: agreement\ + \ at a fine stage implies\n agreement at all coarser stages.\n\n Proof: By tower coherence, f₁(n, d₀) reduced to stage\ + \ k equals f₁(n, k).\n If f₁(n, d₀) = f₂(n, d₀), then reducing both sides gives f₁(n, k) = f₂(n, k). -/" +aliases: +- FTH0040 +- tail_agree_downward +- tail-agree-downward +- TauLib.BookI.Holomorphy.IdentityTheorem::tail_agree_downward +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001122 +- predicate: part_of + object: cid005067 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005067 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.IdentityTheorem + declaration: tail_agree_downward + module_path: TauLib/BookI/Holomorphy/IdentityTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L07] Tail Agreement Propagation (single input):\n If two tower-coherent stagewise functions agree\ + \ at stage d₀ for input n,\n then they agree at ALL stages k ≤ d₀ for input n.\n\n This is the \"downward\" direction:\ + \ agreement at a " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005304.md b/_corpus_v3_items/cid005304.md new file mode 100644 index 000000000..6925deaa6 --- /dev/null +++ b/_corpus_v3_items/cid005304.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005304/ +title: tau_identity_reduce (theorem) +cid: cid005304 +primary_alias: FTH0041 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T21] The τ-Identity Theorem (stagewise form):\n If two tower-coherent \"reduce-form\" stagewise functions\ + \ have the same\n underlying maps, they produce the same outputs at every stage.\n\n This is trivially true by definition,\ + \ but it captures the KEY insight:\n a reduce-form function is uniquely determined by its underlying map.\n Tower\ + \ coherence + reduce form = complete determination. -/" +aliases: +- FTH0041 +- tau_identity_reduce +- tau-identity-reduce +- TauLib.BookI.Holomorphy.IdentityTheorem::tau_identity_reduce +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001191 +- predicate: part_of + object: cid005067 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005067 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.IdentityTheorem + declaration: tau_identity_reduce + module_path: TauLib/BookI/Holomorphy/IdentityTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T21] The τ-Identity Theorem (stagewise form):\n If two tower-coherent \"reduce-form\" stagewise functions\ + \ have the same\n underlying maps, they produce the same outputs at every stage.\n\n This is trivially true by definition,\ + \ but" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005305.md b/_corpus_v3_items/cid005305.md new file mode 100644 index 000000000..65f5e8834 --- /dev/null +++ b/_corpus_v3_items/cid005305.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005305/ +title: presheaf_characterization (theorem) +cid: cid005305 +primary_alias: FTH0042 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T40] Every τ-holomorphic function is a natural transformation\n of the primorial presheaf. -/" +aliases: +- FTH0042 +- presheaf_characterization +- TauLib.BookI.Holomorphy.PresheafEssence::presheaf_characterization +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001207 +- predicate: part_of + object: cid005068 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005068 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.PresheafEssence + declaration: presheaf_characterization + module_path: TauLib/BookI/Holomorphy/PresheafEssence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T40] Every τ-holomorphic function is a natural transformation\n of the primorial presheaf. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005306.md b/_corpus_v3_items/cid005306.md new file mode 100644 index 000000000..e34b8b0e6 --- /dev/null +++ b/_corpus_v3_items/cid005306.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005306/ +title: bi_square_characterization (theorem) +cid: cid005306 +primary_alias: FTH0043 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T41] The bi-square characterizes τ-holomorphy completely:\n TowerCoherent f ↔ both sectors of the left\ + \ square commute. -/" +aliases: +- FTH0043 +- bi_square_characterization +- TauLib.BookI.Holomorphy.PresheafEssence::bi_square_characterization +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001208 +- predicate: part_of + object: cid005068 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005068 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.PresheafEssence + declaration: bi_square_characterization + module_path: TauLib/BookI/Holomorphy/PresheafEssence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T41] The bi-square characterizes τ-holomorphy completely:\n TowerCoherent f ↔ both sectors of the\ + \ left square commute. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005307.md b/_corpus_v3_items/cid005307.md new file mode 100644 index 000000000..10b6cb2de --- /dev/null +++ b/_corpus_v3_items/cid005307.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005307/ +title: spectral_determines (theorem) +cid: cid005307 +primary_alias: FTH0045 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T29] Spectral Determination: if two tower-coherent StageFuns\n have the same spectral coefficients at\ + \ all inputs and stages,\n they are equal.\n\n This is essentially the content of the Identity Theorem (I.T21)\n \ + \ reformulated in spectral language. -/" +aliases: +- FTH0045 +- spectral_determines +- spectral-determines +- TauLib.BookI.Holomorphy.SpectralCoefficients::spectral_determines +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001199 +- predicate: part_of + object: cid005069 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005069 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.SpectralCoefficients + declaration: spectral_determines + module_path: TauLib/BookI/Holomorphy/SpectralCoefficients.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T29] Spectral Determination: if two tower-coherent StageFuns\n have the same spectral coefficients\ + \ at all inputs and stages,\n they are equal.\n\n This is essentially the content of the Identity Theorem (I.T21)\n\ + \ reformulated " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005308.md b/_corpus_v3_items/cid005308.md new file mode 100644 index 000000000..47a423ce2 --- /dev/null +++ b/_corpus_v3_items/cid005308.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005308/ +title: crt_extension_b (theorem) +cid: cid005308 +primary_alias: FTH0046 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L08] CRT Extension: tower coherence constrains function output\n via the reduce map. The output at stage\ + \ k is always reduced mod\n primorial k — this is the vertical consistency that enables extension.\n\n For the B-sector:\ + \ reduce(f.b_fun(n, l), k) = f.b_fun(n, k) for k ≤ l. -/" +aliases: +- FTH0046 +- crt_extension_b +- crt-extension-b +- TauLib.BookI.Holomorphy.Thinness::crt_extension_b +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001123 +- predicate: part_of + object: cid005071 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005071 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.Thinness + declaration: crt_extension_b + module_path: TauLib/BookI/Holomorphy/Thinness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L08] CRT Extension: tower coherence constrains function output\n via the reduce map. The output at\ + \ stage k is always reduced mod\n primorial k — this is the vertical consistency that enables extension.\n\n For the\ + \ B-sector: redu" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005309.md b/_corpus_v3_items/cid005309.md new file mode 100644 index 000000000..b4c8c7332 --- /dev/null +++ b/_corpus_v3_items/cid005309.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005309/ +title: removable_singularity (theorem) +cid: cid005309 +primary_alias: FTH0047 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T30] Removable Singularity: if two tower-coherent functions\n agree at depth d₀ for all inputs, they agree\ + \ at all depths ≤ d₀.\n\n This is a repackaging of the Identity Theorem (I.T21)\n in the language of extensions. The\ + \ \"removable singularity\" interpretation:\n knowing f on a dense set of inputs at stage d₀ determines f everywhere\n\ + \ (because reduced inputs form a finite set at each stage). -/" +aliases: +- FTH0047 +- removable_singularity +- TauLib.BookI.Holomorphy.Thinness::removable_singularity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001200 +- predicate: part_of + object: cid005071 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005071 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Holomorphy.Thinness + declaration: removable_singularity + module_path: TauLib/BookI/Holomorphy/Thinness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T30] Removable Singularity: if two tower-coherent functions\n agree at depth d₀ for all inputs, they\ + \ agree at all depths ≤ d₀.\n\n This is a repackaging of the Identity Theorem (I.T21)\n in the language of extensions.\ + \ The \"remov" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005310.md b/_corpus_v3_items/cid005310.md new file mode 100644 index 000000000..d9e2fc7a1 --- /dev/null +++ b/_corpus_v3_items/cid005310.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005310/ +title: euler_sieve_cross (theorem) +cid: cid005310 +primary_alias: FTH0048 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P44] The Euler sieve identity in cross-multiplied Nat form:\n (2/3) · (4/5) · (121/120) = 121/225\n\n\ + \ Cross-multiplied: 2 · 4 · 121 · 225 = 3 · 5 · 120 · 121. -/" +aliases: +- FTH0048 +- euler_sieve_cross +- euler-sieve-cross +- TauLib.BookI.Kernel.ActionQuantum::euler_sieve_cross +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001164 +- predicate: part_of + object: cid005072 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005072 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.ActionQuantum + declaration: euler_sieve_cross + module_path: TauLib/BookI/Kernel/ActionQuantum.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P44] The Euler sieve identity in cross-multiplied Nat form:\n (2/3) · (4/5) · (121/120) = 121/225\n\ + \n Cross-multiplied: 2 · 4 · 121 · 225 = 3 · 5 · 120 · 121. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005311.md b/_corpus_v3_items/cid005311.md new file mode 100644 index 000000000..0cc67350e --- /dev/null +++ b/_corpus_v3_items/cid005311.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005311/ +title: K1_strict_order (theorem) +cid: cid005311 +primary_alias: FTH0049 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.K1] **Strict Order**: α < π < γ < η < ω is a strict total order\n on the 5 generators.\n\n In our representation,\ + \ this holds definitionally via `Generator.toNat`. -/" +aliases: +- FTH0049 +- K1_strict_order +- k1-strict-order +- TauLib.BookI.Kernel.Axioms::K1_strict_order +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001110 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K1_strict_order + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.K1] **Strict Order**: α < π < γ < η < ω is a strict total order\n on the 5 generators.\n\n In our\ + \ representation, this holds definitionally via `Generator.toNat`. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005312.md b/_corpus_v3_items/cid005312.md new file mode 100644 index 000000000..f1bc4bef2 --- /dev/null +++ b/_corpus_v3_items/cid005312.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005312/ +title: K2_omega_fixed (theorem) +cid: cid005312 +primary_alias: FTH0050 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.K2] **Omega Fixed Point**: ρ(ω) = ω; ω absorbs all operations.\n\n This holds definitionally from our\ + \ `rho` definition. -/" +aliases: +- FTH0050 +- K2_omega_fixed +- k2-omega-fixed +- TauLib.BookI.Kernel.Axioms::K2_omega_fixed +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001111 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K2_omega_fixed + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.K2] **Omega Fixed Point**: ρ(ω) = ω; ω absorbs all operations.\n\n This holds definitionally from\ + \ our `rho` definition. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005313.md b/_corpus_v3_items/cid005313.md new file mode 100644 index 000000000..1e5840b88 --- /dev/null +++ b/_corpus_v3_items/cid005313.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005313/ +title: K3_orbit_seeded (theorem) +cid: cid005313 +primary_alias: FTH0051 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.K3] Every non-omega generator seeds an orbit ray containing itself. -/ +aliases: +- FTH0051 +- K3_orbit_seeded +- k3-orbit-seeded +- TauLib.BookI.Kernel.Axioms::K3_orbit_seeded +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001112 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K3_orbit_seeded + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.K3] Every non-omega generator seeds an orbit ray containing itself. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005314.md b/_corpus_v3_items/cid005314.md new file mode 100644 index 000000000..a6786845c --- /dev/null +++ b/_corpus_v3_items/cid005314.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005314/ +title: K4_no_jump (theorem) +cid: cid005314 +primary_alias: FTH0052 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.K4] **No-Jump / Cover**: ρ acts as a successor within each orbit\n (no skipping). The cover relation is\ + \ immediate: ρⁿ(g) covers ρⁿ⁻¹(g). -/" +aliases: +- FTH0052 +- K4_no_jump +- k4-no-jump +- TauLib.BookI.Kernel.Axioms::K4_no_jump +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001113 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K4_no_jump + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.K4] **No-Jump / Cover**: ρ acts as a successor within each orbit\n (no skipping). The cover relation\ + \ is immediate: ρⁿ(g) covers ρⁿ⁻¹(g). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005315.md b/_corpus_v3_items/cid005315.md new file mode 100644 index 000000000..756ccb88a --- /dev/null +++ b/_corpus_v3_items/cid005315.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005315/ +title: K5_beacon_non_succ (theorem) +cid: cid005315 +primary_alias: FTH0053 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.K5] **Beacon Non-Successor**: ω is NOT in the image of ρ restricted\n to any orbit ray. No finite iteration\ + \ of ρ on a non-omega generator reaches ω. -/" +aliases: +- FTH0053 +- K5_beacon_non_succ +- k5-beacon-non-succ +- TauLib.BookI.Kernel.Axioms::K5_beacon_non_succ +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001114 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K5_beacon_non_succ + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.K5] **Beacon Non-Successor**: ω is NOT in the image of ρ restricted\n to any orbit ray. No finite\ + \ iteration of ρ on a non-omega generator reaches ω. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005316.md b/_corpus_v3_items/cid005316.md new file mode 100644 index 000000000..bd4cd020f --- /dev/null +++ b/_corpus_v3_items/cid005316.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005316/ +title: K6_object_closure (theorem) +cid: cid005316 +primary_alias: FTH0054 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.K6] **Object Closure**: Obj(τ) = {ω} ∪ O_α ∪ O_π ∪ O_γ ∪ O_η.\n No other objects exist.\n\n In our\ + \ representation, this is definitional: every `TauObj` is constructed\n from a `Generator` seed, and `Generator` has\ + \ exactly 5 constructors. -/" +aliases: +- FTH0054 +- K6_object_closure +- k6-object-closure +- TauLib.BookI.Kernel.Axioms::K6_object_closure +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001115 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: K6_object_closure + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.K6] **Object Closure**: Obj(τ) = {ω} ∪ O_α ∪ O_π ∪ O_γ ∪ O_η.\n No other objects exist.\n\n In\ + \ our representation, this is definitional: every `TauObj` is constructed\n from a `Generator` seed, and `Generator`\ + \ has exactly 5 cons" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005317.md b/_corpus_v3_items/cid005317.md new file mode 100644 index 000000000..25859dc6c --- /dev/null +++ b/_corpus_v3_items/cid005317.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005317/ +title: gen_distinct (theorem) +cid: cid005317 +primary_alias: FTH0055 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.P01] Generator distinctness: all five generators are pairwise distinct. -/' +aliases: +- FTH0055 +- gen_distinct +- gen-distinct +- TauLib.BookI.Kernel.Axioms::gen_distinct +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001124 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: gen_distinct + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.P01] Generator distinctness: all five generators are pairwise distinct. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005318.md b/_corpus_v3_items/cid005318.md new file mode 100644 index 000000000..3feedff0d --- /dev/null +++ b/_corpus_v3_items/cid005318.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005318/ +title: rho_injective (theorem) +cid: cid005318 +primary_alias: FTH0056 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.P02] ρ is injective on each orbit ray. -/ +aliases: +- FTH0056 +- rho_injective +- rho-injective +- TauLib.BookI.Kernel.Axioms::rho_injective +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001125 +- predicate: part_of + object: cid005073 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005073 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Axioms + declaration: rho_injective + module_path: TauLib/BookI/Kernel/Axioms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.P02] ρ is injective on each orbit ray. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005319.md b/_corpus_v3_items/cid005319.md new file mode 100644 index 000000000..d2dcf930c --- /dev/null +++ b/_corpus_v3_items/cid005319.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005319/ +title: diagonal_channel_count (theorem) +cid: cid005319 +primary_alias: FTH0057 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D03] There are exactly 4 non-omega generators.\n This is the source of the diagonal discipline:\n 4\ + \ generators yield 3 rewiring levels. -/" +aliases: +- FTH0057 +- diagonal_channel_count +- diagonal-channel-count +- TauLib.BookI.Kernel.Diagonal::diagonal_channel_count +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001006 +- predicate: part_of + object: cid005074 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005074 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Diagonal + declaration: diagonal_channel_count + module_path: TauLib/BookI/Kernel/Diagonal.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D03] There are exactly 4 non-omega generators.\n This is the source of the diagonal discipline:\n\ + \ 4 generators yield 3 rewiring levels. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005320.md b/_corpus_v3_items/cid005320.md new file mode 100644 index 000000000..4a8f12982 --- /dev/null +++ b/_corpus_v3_items/cid005320.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005320/ +title: rewiring_levels_eq_solenoidal (theorem) +cid: cid005320 +primary_alias: FTH0058 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.D03] The diagonal discipline: exactly 3 rewiring levels exist\n because exactly 3 solenoidal generators\ + \ are available as targets.\n Each rewiring level consumes one generator:\n - Level 1 (addition) ↔ π\n - Level\ + \ 2 (multiplication) ↔ γ\n - Level 3 (exponentiation) ↔ η\n No 4th level: α is the counting scaffold, ω is the beacon.\ + \ -/" +aliases: +- FTH0058 +- rewiring_levels_eq_solenoidal +- rewiring-levels-eq-solenoidal +- TauLib.BookI.Kernel.Diagonal::rewiring_levels_eq_solenoidal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001006 +- predicate: part_of + object: cid005074 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005074 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Kernel.Diagonal + declaration: rewiring_levels_eq_solenoidal + module_path: TauLib/BookI/Kernel/Diagonal.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.D03] The diagonal discipline: exactly 3 rewiring levels exist\n because exactly 3 solenoidal generators\ + \ are available as targets.\n Each rewiring level consumes one generator:\n - Level 1 (addition) ↔ π\n - Level\ + \ 2 (multiplic" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005321.md b/_corpus_v3_items/cid005321.md new file mode 100644 index 000000000..19a3d8565 --- /dev/null +++ b/_corpus_v3_items/cid005321.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005321/ +title: boolean_recovery (theorem) +cid: cid005321 +primary_alias: FTH0059 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P13] Boolean recovery: a Truth4 value is classical (T or F) if and only if\n both sector projections agree.\n\ + \n For classical values: forget T = true = forget_pessimistic T, forget F = false = forget_pessimistic F.\n For non-classical:\ + \ forget B = true but forget_pessimistic B = false (sectors disagree). -/" +aliases: +- FTH0059 +- boolean_recovery +- TauLib.BookI.Logic.BooleanRecovery::boolean_recovery +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001136 +- predicate: part_of + object: cid005080 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005080 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.BooleanRecovery + declaration: boolean_recovery + module_path: TauLib/BookI/Logic/BooleanRecovery.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P13] Boolean recovery: a Truth4 value is classical (T or F) if and only if\n both sector projections\ + \ agree.\n\n For classical values: forget T = true = forget_pessimistic T, forget F = false = forget_pessimistic F.\n\ + \ For non-cla" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005322.md b/_corpus_v3_items/cid005322.md new file mode 100644 index 000000000..d72f9411f --- /dev/null +++ b/_corpus_v3_items/cid005322.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005322/ +title: forget_fiber (theorem) +cid: cid005322 +primary_alias: FTH0060 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- The key information-loss theorem: forget loses exactly the B/N distinction.\n Two values have the same forget\ + \ image iff they agree on \"B-sector truth\". -/" +aliases: +- FTH0060 +- forget_fiber +- forget-fiber +- TauLib.BookI.Logic.BooleanRecovery::forget_fiber +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001057 +- predicate: part_of + object: cid005080 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005080 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.BooleanRecovery + declaration: forget_fiber + module_path: TauLib/BookI/Logic/BooleanRecovery.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- The key information-loss theorem: forget loses exactly the B/N distinction.\n Two values have the same\ + \ forget image iff they agree on \"B-sector truth\". -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005323.md b/_corpus_v3_items/cid005323.md new file mode 100644 index 000000000..1dffe670f --- /dev/null +++ b/_corpus_v3_items/cid005323.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005323/ +title: explosion_barrier (theorem) +cid: cid005323 +primary_alias: FTH0061 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T13] The explosion barrier: B does not imply everything.\n Specifically, impl B F = N, not T.\n This\ + \ is the central paraconsistent feature of Truth4.\n\n Calculation: impl B F = join (neg B) F = join N F = N.\n Since\ + \ N <> T, the implication B -> F is not \"true\". -/" +aliases: +- FTH0061 +- explosion_barrier +- TauLib.BookI.Logic.Explosion::explosion_barrier +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001183 +- predicate: part_of + object: cid005081 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005081 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Logic.Explosion + declaration: explosion_barrier + module_path: TauLib/BookI/Logic/Explosion.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T13] The explosion barrier: B does not imply everything.\n Specifically, impl B F = N, not T.\n \ + \ This is the central paraconsistent feature of Truth4.\n\n Calculation: impl B F = join (neg B) F = join N F = N.\n\ + \ Since N <> T, t" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005324.md b/_corpus_v3_items/cid005324.md new file mode 100644 index 000000000..0058afa33 --- /dev/null +++ b/_corpus_v3_items/cid005324.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005324/ +title: no_identity_decoherence (theorem) +cid: cid005324 +primary_alias: FTH0062 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.C03] No Identity Decoherence: the diagonal resonance pattern (L+E+P)\n cannot occur at the ontic level\ + \ in τ. Direct consequence of I.T46. -/" +aliases: +- FTH0062 +- no_identity_decoherence +- TauLib.BookI.MetaLogic.OnticInvariance::no_identity_decoherence +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001002 +- predicate: part_of + object: cid005086 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005086 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.OnticInvariance + declaration: no_identity_decoherence + module_path: TauLib/BookI/MetaLogic/OnticInvariance.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.C03] No Identity Decoherence: the diagonal resonance pattern (L+E+P)\n cannot occur at the ontic level\ + \ in τ. Direct consequence of I.T46. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005325.md b/_corpus_v3_items/cid005325.md new file mode 100644 index 000000000..64a510488 --- /dev/null +++ b/_corpus_v3_items/cid005325.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005325/ +title: slippage_breaks_omega (theorem) +cid: cid005325 +primary_alias: FTH0063 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T47] Slippage Breaks Unique Omega: a foundation with full diagonal\n resonance cannot internalize a unique\ + \ absolute infinity omega.\n\n If identity slippage is present (full resonance = true), then\n UniqueOmegaCapability\ + \ is impossible for that resonance profile. -/" +aliases: +- FTH0063 +- slippage_breaks_omega +- slippage-breaks-omega +- TauLib.BookI.MetaLogic.OnticInvariance::slippage_breaks_omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001214 +- predicate: part_of + object: cid005086 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005086 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.OnticInvariance + declaration: slippage_breaks_omega + module_path: TauLib/BookI/MetaLogic/OnticInvariance.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T47] Slippage Breaks Unique Omega: a foundation with full diagonal\n resonance cannot internalize\ + \ a unique absolute infinity omega.\n\n If identity slippage is present (full resonance = true), then\n UniqueOmegaCapability\ + \ is imp" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005326.md b/_corpus_v3_items/cid005326.md new file mode 100644 index 000000000..1f69c1e21 --- /dev/null +++ b/_corpus_v3_items/cid005326.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005326/ +title: structural_instability_theorem (theorem) +cid: cid005326 +primary_alias: FTH0064 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T48] The Structural Instability Theorem: diagonal-resonant foundations\n cannot host identity-faithful\ + \ reception of τ.\n\n If host has full resonance, reception conditions cannot be met. -/" +aliases: +- FTH0064 +- structural_instability_theorem +- TauLib.BookI.MetaLogic.ReceptionCriterion::structural_instability_theorem +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001215 +- predicate: part_of + object: cid005087 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005087 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.MetaLogic.ReceptionCriterion + declaration: structural_instability_theorem + module_path: TauLib/BookI/MetaLogic/ReceptionCriterion.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T48] The Structural Instability Theorem: diagonal-resonant foundations\n cannot host identity-faithful\ + \ reception of τ.\n\n If host has full resonance, reception conditions cannot be met. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005327.md b/_corpus_v3_items/cid005327.md new file mode 100644 index 000000000..8cd95f910 --- /dev/null +++ b/_corpus_v3_items/cid005327.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005327/ +title: ontic_closure (theorem) +cid: cid005327 +primary_alias: FTH0065 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T01] Ontic Closure: every TauObj is either in an orbit ray\n or has seed omega. -/" +aliases: +- FTH0065 +- ontic_closure +- TauLib.BookI.Orbit.Closure::ontic_closure +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001168 +- predicate: part_of + object: cid005090 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005090 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Closure + declaration: ontic_closure + module_path: TauLib/BookI/Orbit/Closure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T01] Ontic Closure: every TauObj is either in an orbit ray\n or has seed omega. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005328.md b/_corpus_v3_items/cid005328.md new file mode 100644 index 000000000..da1186c7f --- /dev/null +++ b/_corpus_v3_items/cid005328.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005328/ +title: tauObj_countable (theorem) +cid: cid005328 +primary_alias: FTH0066 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.P04] Obj(τ) is countable: there exists an injection TauObj → Nat. -/' +aliases: +- FTH0066 +- tauObj_countable +- tau-obj-countable +- TauLib.BookI.Orbit.Countability::tauObj_countable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001127 +- predicate: part_of + object: cid005091 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005091 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Countability + declaration: tauObj_countable + module_path: TauLib/BookI/Orbit/Countability.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.P04] Obj(τ) is countable: there exists an injection TauObj → Nat. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005329.md b/_corpus_v3_items/cid005329.md new file mode 100644 index 000000000..83f906dd3 --- /dev/null +++ b/_corpus_v3_items/cid005329.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005329/ +title: orbit_disjoint (theorem) +cid: cid005329 +primary_alias: FTH0067 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P03] Orbit rays are pairwise disjoint: if g ≠ h, no object\n belongs to both O_g and O_h. -/" +aliases: +- FTH0067 +- orbit_disjoint +- orbit-disjoint +- TauLib.BookI.Orbit.Generation::orbit_disjoint +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001126 +- predicate: part_of + object: cid005092 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005092 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Generation + declaration: orbit_disjoint + module_path: TauLib/BookI/Orbit/Generation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P03] Orbit rays are pairwise disjoint: if g ≠ h, no object\n belongs to both O_g and O_h. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005330.md b/_corpus_v3_items/cid005330.md new file mode 100644 index 000000000..1af29afff --- /dev/null +++ b/_corpus_v3_items/cid005330.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005330/ +title: pentation_channel_exhaustion (theorem) +cid: cid005330 +primary_alias: FTH0068 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.L01] Pentation cannot be assigned a canonical orbit channel:\n all 3 solenoidal generators are consumed\ + \ by levels 1-3,\n and alpha is the counting scaffold. No 5th channel exists (K6).\n\n This is the channel-exhaustion\ + \ form of the saturation argument. -/" +aliases: +- FTH0068 +- pentation_channel_exhaustion +- pentation-channel-exhaustion +- TauLib.BookI.Orbit.Ladder::pentation_channel_exhaustion +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001116 +- predicate: part_of + object: cid005093 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005093 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Ladder + declaration: pentation_channel_exhaustion + module_path: TauLib/BookI/Orbit/Ladder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.L01] Pentation cannot be assigned a canonical orbit channel:\n all 3 solenoidal generators are consumed\ + \ by levels 1-3,\n and alpha is the counting scaffold. No 5th channel exists (K6).\n\n This is the channel-exhaustion\ + \ form of " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005331.md b/_corpus_v3_items/cid005331.md new file mode 100644 index 000000000..23487d5f9 --- /dev/null +++ b/_corpus_v3_items/cid005331.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005331/ +title: ladder_saturation (theorem) +cid: cid005331 +primary_alias: FTH0069 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T02] The iterator ladder saturates at 4 levels:\n exactly 3 solenoidal generators exist (solenoidalGenerators.length\ + \ = 3),\n so exactly 3 rewiring levels can be canonically assigned,\n giving 4 total operation levels (ρ + 3 rewirings).\n\ + \n The 4th rewiring level (pentation/level 4) has no channel. -/" +aliases: +- FTH0069 +- ladder_saturation +- ladder-saturation +- TauLib.BookI.Orbit.Ladder::ladder_saturation +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001169 +- predicate: part_of + object: cid005093 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005093 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Ladder + declaration: ladder_saturation + module_path: TauLib/BookI/Orbit/Ladder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T02] The iterator ladder saturates at 4 levels:\n exactly 3 solenoidal generators exist (solenoidalGenerators.length\ + \ = 3),\n so exactly 3 rewiring levels can be canonically assigned,\n giving 4 total operation levels (ρ + 3 rew" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005332.md b/_corpus_v3_items/cid005332.md new file mode 100644 index 000000000..1a9f54a18 --- /dev/null +++ b/_corpus_v3_items/cid005332.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005332/ +title: rigidity_non_omega (theorem) +cid: cid005332 +primary_alias: FTH0070 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.T07] **Rigidity**: φ = id on non-omega objects (given seed preservation). -/' +aliases: +- FTH0070 +- rigidity_non_omega +- rigidity-non-omega +- TauLib.BookI.Orbit.Rigidity::rigidity_non_omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001174 +- predicate: part_of + object: cid005094 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005094 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Rigidity + declaration: rigidity_non_omega + module_path: TauLib/BookI/Orbit/Rigidity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.T07] **Rigidity**: φ = id on non-omega objects (given seed preservation). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005333.md b/_corpus_v3_items/cid005333.md new file mode 100644 index 000000000..2247fa061 --- /dev/null +++ b/_corpus_v3_items/cid005333.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005333/ +title: categoricity_non_omega (theorem) +cid: cid005333 +primary_alias: FTH0071 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [I.T08] **Categoricity** (non-omega): unique homomorphism into any model. -/' +aliases: +- FTH0071 +- categoricity_non_omega +- categoricity-non-omega +- TauLib.BookI.Orbit.Rigidity::categoricity_non_omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001175 +- predicate: part_of + object: cid005094 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005094 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Rigidity + declaration: categoricity_non_omega + module_path: TauLib/BookI/Orbit/Rigidity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [I.T08] **Categoricity** (non-omega): unique homomorphism into any model. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005334.md b/_corpus_v3_items/cid005334.md new file mode 100644 index 000000000..8a7f320df --- /dev/null +++ b/_corpus_v3_items/cid005334.md @@ -0,0 +1,74 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005334/ +title: minimal_alphabet (theorem) +cid: cid005334 +primary_alias: FTH0072 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T09] **The Minimal Alphabet Theorem**:\n 5 generators is the unique cardinality achieving all three properties:\n\ + \n **(a) Completeness**: All rewiring levels through exponentiation\n have canonical orbit channel assignments (π↔+,\ + \ γ↔×, η↔^).\n\n **(b) Rigidity**: No non-trivial ρ-automorphism exists.\n (4 generators also have this, but 6 do\ + \ not.)\n\n **(c) Saturation**: Tetration (level 4) has no channel,\n and is algebraically degraded (non-commutative,\ + \ non-associative,\n no left identity).\n\n Moreover, the counter-models show:\n - **4 generators FAIL completeness**:\ + \ exponentiation loses its channel\n (only 2 solenoidal generators for 3 rewiring levels)\n - **6 generators FAIL\ + \ rigidity**: the swap η↔ζ is a non-trivial\n ρ-automorphism (surplus solenoidal generator creates ambiguity)\n\n \ + \ This establishes |Gen| = 5 as the *unique* solution to the\n simultaneous requirements of completeness + rigidity\ + \ + saturation. -/" +aliases: +- FTH0072 +- minimal_alphabet +- minimal-alphabet +- TauLib.BookI.Orbit.Saturation::minimal_alphabet +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001176 +- predicate: part_of + object: cid005095 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005095 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Orbit.Saturation + declaration: minimal_alphabet + module_path: TauLib/BookI/Orbit/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T09] **The Minimal Alphabet Theorem**:\n 5 generators is the unique cardinality achieving all three\ + \ properties:\n\n **(a) Completeness**: All rewiring levels through exponentiation\n have canonical orbit channel\ + \ assignments (π↔+" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005335.md b/_corpus_v3_items/cid005335.md new file mode 100644 index 000000000..e951099b8 --- /dev/null +++ b/_corpus_v3_items/cid005335.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005335/ +title: no_elliptic_idempotent (theorem) +cid: cid005335 +primary_alias: FTH0073 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T10] No nontrivial idempotent in the Gaussian integers:\n if (a+bi)² = (a+bi) over Z, then (a,b) = (0,0)\ + \ or (a,b) = (1,0).\n\n Proof: From (a+bi)² = a+bi:\n - Real part: a² - b² = a\n - Imaginary part: 2ab = b\n \ + \ From 2ab = b: either b = 0 or 2a = 1 (impossible in Z).\n If b = 0: a² = a, so a(a-1) = 0, hence a = 0 or a = 1. -/" +aliases: +- FTH0073 +- no_elliptic_idempotent +- no-elliptic-idempotent +- TauLib.BookI.Polarity.BipolarAlgebra::no_elliptic_idempotent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001177 +- predicate: part_of + object: cid005096 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005096 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.BipolarAlgebra + declaration: no_elliptic_idempotent + module_path: TauLib/BookI/Polarity/BipolarAlgebra.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T10] No nontrivial idempotent in the Gaussian integers:\n if (a+bi)² = (a+bi) over Z, then (a,b) =\ + \ (0,0) or (a,b) = (1,0).\n\n Proof: From (a+bi)² = a+bi:\n - Real part: a² - b² = a\n - Imaginary part: 2ab =\ + \ b\n From 2ab = b" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005336.md b/_corpus_v3_items/cid005336.md new file mode 100644 index 000000000..2360cfead --- /dev/null +++ b/_corpus_v3_items/cid005336.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005336/ +title: split_complex_forced (theorem) +cid: cid005336 +primary_alias: FTH0074 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T10] Split-complex forced: the split-complex algebra (j² = +1) admits\n nontrivial idempotents (e+, e-),\ + \ while the elliptic algebra (i² = -1) does not.\n Therefore, encoding a bipolar partition requires j² = +1. -/" +aliases: +- FTH0074 +- split_complex_forced +- TauLib.BookI.Polarity.BipolarAlgebra::split_complex_forced +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001177 +- predicate: part_of + object: cid005096 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005096 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.BipolarAlgebra + declaration: split_complex_forced + module_path: TauLib/BookI/Polarity/BipolarAlgebra.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T10] Split-complex forced: the split-complex algebra (j² = +1) admits\n nontrivial idempotents (e+,\ + \ e-), while the elliptic algebra (i² = -1) does not.\n Therefore, encoding a bipolar partition requires j² = +1. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005337.md b/_corpus_v3_items/cid005337.md new file mode 100644 index 000000000..1d8dc06a2 --- /dev/null +++ b/_corpus_v3_items/cid005337.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005337/ +title: algebraic_lemniscate_exists (theorem) +cid: cid005337 +primary_alias: FTH0075 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [I.D18] The algebraic lemniscate exists. -/ +aliases: +- FTH0075 +- algebraic_lemniscate_exists +- algebraic-lemniscate-exists +- TauLib.BookI.Polarity.Lemniscate::algebraic_lemniscate_exists +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001028 +- predicate: part_of + object: cid005102 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005102 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Polarity.Lemniscate + declaration: algebraic_lemniscate_exists + module_path: TauLib/BookI/Polarity/Lemniscate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [I.D18] The algebraic lemniscate exists. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005338.md b/_corpus_v3_items/cid005338.md new file mode 100644 index 000000000..6d00e4b4a --- /dev/null +++ b/_corpus_v3_items/cid005338.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005338/ +title: no_unearned_decimal_diagonal (theorem) +cid: cid005338 +primary_alias: FTH0076 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P34] No Unearned Decimal Diagonal: no extractor can have its\n diagonal avoid all rows.\n\n Proof:\ + \ the avoidance condition diagonal(n) != extract(n, n) directly\n contradicts diagonal_def which says diagonal(n) = extract(n,\ + \ n).\n This is the liar-paradox core of the diagonal argument, and tau\n blocks it by making diagonal extraction\ + \ self-referential. -/" +aliases: +- FTH0076 +- no_unearned_decimal_diagonal +- TauLib.BookI.Sets.CantorRefutation::no_unearned_decimal_diagonal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001154 +- predicate: part_of + object: cid005111 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005111 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.CantorRefutation + declaration: no_unearned_decimal_diagonal + module_path: TauLib/BookI/Sets/CantorRefutation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P34] No Unearned Decimal Diagonal: no extractor can have its\n diagonal avoid all rows.\n\n Proof:\ + \ the avoidance condition diagonal(n) != extract(n, n) directly\n contradicts diagonal_def which says diagonal(n) = extract(n,\ + \ n).\n" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005339.md b/_corpus_v3_items/cid005339.md new file mode 100644 index 000000000..3041c3f48 --- /dev/null +++ b/_corpus_v3_items/cid005339.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005339/ +title: no_comprehension (theorem) +cid: cid005339 +primary_alias: FTH0077 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P35] No comprehension separator exists in tau-arithmetic.\n\n Proof: apply Sep to the Russell predicate\ + \ P(a) = not(a in_tau a).\n For R = Sep(P), the comprehension schema gives a in_tau R iff not(a in_tau a).\n At a\ + \ = R: R in_tau R iff not(R in_tau R). But R in_tau R holds by reflexivity\n (R | R), so not(R in_tau R) also holds --\ + \ contradiction. -/" +aliases: +- FTH0077 +- no_comprehension +- no-comprehension +- TauLib.BookI.Sets.CantorRefutation::no_comprehension +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001155 +- predicate: part_of + object: cid005111 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005111 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.CantorRefutation + declaration: no_comprehension + module_path: TauLib/BookI/Sets/CantorRefutation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P35] No comprehension separator exists in tau-arithmetic.\n\n Proof: apply Sep to the Russell predicate\ + \ P(a) = not(a in_tau a).\n For R = Sep(P), the comprehension schema gives a in_tau R iff not(a in_tau a).\n At a\ + \ = R: R in_ta" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005340.md b/_corpus_v3_items/cid005340.md new file mode 100644 index 000000000..26c2a5ce1 --- /dev/null +++ b/_corpus_v3_items/cid005340.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005340/ +title: no_free_cartesian_diagonal (theorem) +cid: cid005340 +primary_alias: FTH0078 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- The Cantor diagonal argument requires a self-pairing map\n pair : N -> N that encodes the \"n-th element\ + \ paired with itself\"\n in a way that DIFFERS from n (so that digit modification can\n produce a new element).\n\n\ + \ In tau-arithmetic, any map with n | pair(n) AND pair(n) != n\n fails at n = 0, since 0 | k implies k = 0.\n\n \ + \ [I.P36] No nontrivial divisibility-respecting self-pairing exists. -/" +aliases: +- FTH0078 +- no_free_cartesian_diagonal +- TauLib.BookI.Sets.CantorRefutation::no_free_cartesian_diagonal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001156 +- predicate: part_of + object: cid005111 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005111 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.CantorRefutation + declaration: no_free_cartesian_diagonal + module_path: TauLib/BookI/Sets/CantorRefutation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- The Cantor diagonal argument requires a self-pairing map\n pair : N -> N that encodes the \"n-th element\ + \ paired with itself\"\n in a way that DIFFERS from n (so that digit modification can\n produce a new element).\n\n\ + \ In tau-arith" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005341.md b/_corpus_v3_items/cid005341.md new file mode 100644 index 000000000..214fb9d19 --- /dev/null +++ b/_corpus_v3_items/cid005341.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005341/ +title: cantor_inapplicable (theorem) +cid: cid005341 +primary_alias: FTH0079 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T35] Cantor Diagonal Inapplicability Theorem:\n No CantorDiagonalApparatus can exist in Category tau.\n\ + \n The proof is immediate from any ONE of the three failures:\n - P34: avoidance contradicts diagonal_def\n - P35:\ + \ comprehension contradicts no_russell_set\n - P36: self-pairing contradicts divisibility at 0\n\n We use P34 (the\ + \ simplest). The three failures are independent\n and each individually blocks the diagonal argument. -/" +aliases: +- FTH0079 +- cantor_inapplicable +- cantor-inapplicable +- TauLib.BookI.Sets.CantorRefutation::cantor_inapplicable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001202 +- predicate: part_of + object: cid005111 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005111 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.CantorRefutation + declaration: cantor_inapplicable + module_path: TauLib/BookI/Sets/CantorRefutation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T35] Cantor Diagonal Inapplicability Theorem:\n No CantorDiagonalApparatus can exist in Category tau.\n\ + \n The proof is immediate from any ONE of the three failures:\n - P34: avoidance contradicts diagonal_def\n - P35:\ + \ comprehe" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005342.md b/_corpus_v3_items/cid005342.md new file mode 100644 index 000000000..8f8345a1c --- /dev/null +++ b/_corpus_v3_items/cid005342.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005342/ +title: counting_structural (theorem) +cid: cid005342 +primary_alias: FTH0080 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P33] Countability of Obj(tau) as a structural feature:\n it follows from generative counting (each orbit\ + \ is counted)\n plus ontic closure (the universe is a finite union of orbits).\n\n The injection TauObj -> Nat witnesses\ + \ countability. -/" +aliases: +- FTH0080 +- counting_structural +- counting-structural +- TauLib.BookI.Sets.Counting::counting_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001153 +- predicate: part_of + object: cid005112 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005112 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Counting + declaration: counting_structural + module_path: TauLib/BookI/Sets/Counting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P33] Countability of Obj(tau) as a structural feature:\n it follows from generative counting (each\ + \ orbit is counted)\n plus ontic closure (the universe is a finite union of orbits).\n\n The injection TauObj -> Nat\ + \ witnesses coun" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005343.md b/_corpus_v3_items/cid005343.md new file mode 100644 index 000000000..eb4dfa586 --- /dev/null +++ b/_corpus_v3_items/cid005343.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005343/ +title: tau_mem_iff_dvd (theorem) +cid: cid005343 +primary_alias: FTH0081 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- Bridge: τ-membership is Nat divisibility. -/' +aliases: +- FTH0081 +- tau_mem_iff_dvd +- tau-mem-iff-dvd +- TauLib.BookI.Sets.Membership::tau_mem_iff_dvd +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001133 +- predicate: part_of + object: cid005113 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005113 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Membership + declaration: tau_mem_iff_dvd + module_path: TauLib/BookI/Sets/Membership.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- Bridge: τ-membership is Nat divisibility. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005344.md b/_corpus_v3_items/cid005344.md new file mode 100644 index 000000000..af747577c --- /dev/null +++ b/_corpus_v3_items/cid005344.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005344/ +title: tau_inter_distrib_union (theorem) +cid: cid005344 +primary_alias: FTH0082 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P11] Distributive Lattice: gcd distributes over lcm.\n gcd(a, lcm(b,c)) = lcm(gcd(a,b), gcd(a,c)). -/" +aliases: +- FTH0082 +- tau_inter_distrib_union +- tau-inter-distrib-union +- TauLib.BookI.Sets.Operations::tau_inter_distrib_union +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001134 +- predicate: part_of + object: cid005114 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005114 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Operations + declaration: tau_inter_distrib_union + module_path: TauLib/BookI/Sets/Operations.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P11] Distributive Lattice: gcd distributes over lcm.\n gcd(a, lcm(b,c)) = lcm(gcd(a,b), gcd(a,c)).\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005345.md b/_corpus_v3_items/cid005345.md new file mode 100644 index 000000000..348afe01f --- /dev/null +++ b/_corpus_v3_items/cid005345.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005345/ +title: orbit_set_order_bound (theorem) +cid: cid005345 +primary_alias: FTH0083 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P42] Order bound: if α_k ∈ Set(α_n) and n ≠ 0, then k ≤ n.\n This is tau_mem_le restated in orbit-set\ + \ language. -/" +aliases: +- FTH0083 +- orbit_set_order_bound +- orbit-set-order-bound +- TauLib.BookI.Sets.OrbitSets::orbit_set_order_bound +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001162 +- predicate: part_of + object: cid005115 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005115 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.OrbitSets + declaration: orbit_set_order_bound + module_path: TauLib/BookI/Sets/OrbitSets.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P42] Order bound: if α_k ∈ Set(α_n) and n ≠ 0, then k ≤ n.\n This is tau_mem_le restated in orbit-set\ + \ language. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005346.md b/_corpus_v3_items/cid005346.md new file mode 100644 index 000000000..41e38f39b --- /dev/null +++ b/_corpus_v3_items/cid005346.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005346/ +title: prime_atom (theorem) +cid: cid005346 +primary_alias: FTH0084 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.R30] Prime atom theorem: if p is prime, then\n Set(α_p) = {1, p}.\n The only divisors of a prime are\ + \ 1 and itself. -/" +aliases: +- FTH0084 +- prime_atom +- prime-atom +- TauLib.BookI.Sets.OrbitSets::prime_atom +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005115 +- predicate: part_of + object: cid005115 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005115 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.OrbitSets + declaration: prime_atom + module_path: TauLib/BookI/Sets/OrbitSets.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.R30] Prime atom theorem: if p is prime, then\n Set(α_p) = {1, p}.\n The only divisors of a prime\ + \ are 1 and itself. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005347.md b/_corpus_v3_items/cid005347.md new file mode 100644 index 000000000..66aaf6f7a --- /dev/null +++ b/_corpus_v3_items/cid005347.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005347/ +title: omega_orbit_set_exceeds_alpha (theorem) +cid: cid005347 +primary_alias: FTH0085 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.R28] ω's orbit-set includes ω itself, so it does not live purely\n in the α-orbit. This is the TauObj-level\ + \ witness that Set(ω) ≠ O_α. -/" +aliases: +- FTH0085 +- omega_orbit_set_exceeds_alpha +- omega-orbit-set-exceeds-alpha +- TauLib.BookI.Sets.OrbitSets::omega_orbit_set_exceeds_alpha +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005115 +- predicate: part_of + object: cid005115 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005115 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.OrbitSets + declaration: omega_orbit_set_exceeds_alpha + module_path: TauLib/BookI/Sets/OrbitSets.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.R28] ω's orbit-set includes ω itself, so it does not live purely\n in the α-orbit. This is the TauObj-level\ + \ witness that Set(ω) ≠ O_α. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005348.md b/_corpus_v3_items/cid005348.md new file mode 100644 index 000000000..c3bc6831f --- /dev/null +++ b/_corpus_v3_items/cid005348.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005348/ +title: nat_not_internal_set (theorem) +cid: cid005348 +primary_alias: FTH0086 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.R28] Combined \"no internal copy\" result: for nonzero n, no Set(α_n)\n captures all of ℕ⁺, and Set(ω)\ + \ strictly exceeds O_α.\n\n This expresses the inseparability of ℕ and ω: O_α ≅ ℕ⁺ is NOT\n a valid τ-internal set.\ + \ The closest is Set(ω) = O_α ∪ {ω}. -/" +aliases: +- FTH0086 +- nat_not_internal_set +- nat-not-internal-set +- TauLib.BookI.Sets.OrbitSets::nat_not_internal_set +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005115 +- predicate: part_of + object: cid005115 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005115 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.OrbitSets + declaration: nat_not_internal_set + module_path: TauLib/BookI/Sets/OrbitSets.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.R28] Combined \"no internal copy\" result: for nonzero n, no Set(α_n)\n captures all of ℕ⁺, and Set(ω)\ + \ strictly exceeds O_α.\n\n This expresses the inseparability of ℕ and ω: O_α ≅ ℕ⁺ is NOT\n a valid τ-internal set.\ + \ The closest i" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005349.md b/_corpus_v3_items/cid005349.md new file mode 100644 index 000000000..0c09cb9ef --- /dev/null +++ b/_corpus_v3_items/cid005349.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005349/ +title: tau_strict_mem_wf (theorem) +cid: cid005349 +primary_alias: FTH0087 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P12] Well-foundedness of nonzero strict τ-membership:\n there is no infinite descending chain\n ...\ + \ ∈^strict_τ x₂ ∈^strict_τ x₁ ∈^strict_τ x₀\n when all elements are nonzero.\n\n Proof: tau_strict_mem_nz is a subrelation\ + \ of Nat.lt (via identity),\n and Nat.lt is well-founded. -/" +aliases: +- FTH0087 +- tau_strict_mem_wf +- tau-strict-mem-wf +- TauLib.BookI.Sets.Powerset::tau_strict_mem_wf +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001135 +- predicate: part_of + object: cid005116 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005116 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.Powerset + declaration: tau_strict_mem_wf + module_path: TauLib/BookI/Sets/Powerset.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P12] Well-foundedness of nonzero strict τ-membership:\n there is no infinite descending chain\n \ + \ ... ∈^strict_τ x₂ ∈^strict_τ x₁ ∈^strict_τ x₀\n when all elements are nonzero.\n\n Proof: tau_strict_mem_nz is\ + \ a subrelation of Na" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005350.md b/_corpus_v3_items/cid005350.md new file mode 100644 index 000000000..be3dcef32 --- /dev/null +++ b/_corpus_v3_items/cid005350.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005350/ +title: unique_infinity (theorem) +cid: cid005350 +primary_alias: FTH0088 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T36] Unique Infinity Object: omega is the ONLY infinity object\n in Category tau.\n\n Proof: Let x\ + \ be any infinity object. Since rho(x) = x and x is\n unreachable from orbit rays, x must have seed = omega (by K6\n\ + \ object closure, the only objects not in orbit rays have seed omega).\n Then x = (omega, d) for some d. Since rho(omega,\ + \ d) = (omega, d)\n (K2), ANY omega-seeded object is rho-fixed.\n\n But the uniqueness is stronger: all (omega, d)\ + \ are rho-equivalent\n (they all satisfy rho(x) = x), so up to rho-equivalence there is\n exactly one infinity object.\ + \ -/" +aliases: +- FTH0088 +- unique_infinity +- unique-infinity +- TauLib.BookI.Sets.UniqueInfinity::unique_infinity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001203 +- predicate: part_of + object: cid005117 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005117 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.UniqueInfinity + declaration: unique_infinity + module_path: TauLib/BookI/Sets/UniqueInfinity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T36] Unique Infinity Object: omega is the ONLY infinity object\n in Category tau.\n\n Proof: Let\ + \ x be any infinity object. Since rho(x) = x and x is\n unreachable from orbit rays, x must have seed = omega (by K6\n\ + \ object closu" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005351.md b/_corpus_v3_items/cid005351.md new file mode 100644 index 000000000..0ac8d6cbb --- /dev/null +++ b/_corpus_v3_items/cid005351.md @@ -0,0 +1,72 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005351/ +title: ultrametric_replaces_card (theorem) +cid: cid005351 +primary_alias: FTH0089 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P37] Ultrametric structure replaces cardinality hierarchy.\n\n In ZF, the chain aleph_0 < aleph_1 < aleph_2\ + \ < ... measures\n \"how many\" elements a set has. In tau, this hierarchy collapses:\n there is only one infinity\ + \ (omega), and the notion of \"size\" is\n replaced by PROXIMITY in the divergence ultrametric.\n\n Two omega-tails\ + \ are \"close\" if they agree to deep primorial depth,\n and \"far\" if they diverge early. This is an ultrametric (satisfies\n\ + \ the strong triangle inequality), providing a finer structure than\n cardinality.\n\n The replacement has three\ + \ pillars:\n 1. The ultrametric exists (from OmegaGerms)\n 2. It satisfies the strong triangle inequality (ultra_triangle)\n\ + \ 3. There is no second infinity to compare against (unique_infinity)\n\n We package these as a single theorem. -/" +aliases: +- FTH0089 +- ultrametric_replaces_card +- ultrametric-replaces-card +- TauLib.BookI.Sets.UniqueInfinity::ultrametric_replaces_card +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001157 +- predicate: part_of + object: cid005117 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005117 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Sets.UniqueInfinity + declaration: ultrametric_replaces_card + module_path: TauLib/BookI/Sets/UniqueInfinity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P37] Ultrametric structure replaces cardinality hierarchy.\n\n In ZF, the chain aleph_0 < aleph_1\ + \ < aleph_2 < ... measures\n \"how many\" elements a set has. In tau, this hierarchy collapses:\n there is only one\ + \ infinity (omega), " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005352.md b/_corpus_v3_items/cid005352.md new file mode 100644 index 000000000..75e81c97c --- /dev/null +++ b/_corpus_v3_items/cid005352.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005352/ +title: product_universal (theorem) +cid: cid005352 +primary_alias: FTH0090 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T26] Product universal property: if R maps to both P and Q pointwise,\n then R maps to P × Q. -/" +aliases: +- FTH0090 +- product_universal +- product-universal +- TauLib.BookI.Topos.CartesianProduct::product_universal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001196 +- predicate: part_of + object: cid005118 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005118 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.CartesianProduct + declaration: product_universal + module_path: TauLib/BookI/Topos/CartesianProduct.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T26] Product universal property: if R maps to both P and Q pointwise,\n then R maps to P × Q. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005353.md b/_corpus_v3_items/cid005353.md new file mode 100644 index 000000000..bd0c3140c --- /dev/null +++ b/_corpus_v3_items/cid005353.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005353/ +title: cat_tau_thin (theorem) +cid: cid005353 +primary_alias: FTH0091 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P25] Cat_τ is thin: if two tower-coherent stagewise functions\n agree at stage d₀ for all inputs, they\ + \ agree at all stages ≤ d₀.\n This is a direct corollary of the τ-Identity Theorem (I.T21). -/" +aliases: +- FTH0091 +- cat_tau_thin +- cat-tau-thin +- TauLib.BookI.Topos.EarnedArrows::cat_tau_thin +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001148 +- predicate: part_of + object: cid005120 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005120 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.EarnedArrows + declaration: cat_tau_thin + module_path: TauLib/BookI/Topos/EarnedArrows.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P25] Cat_τ is thin: if two tower-coherent stagewise functions\n agree at stage d₀ for all inputs,\ + \ they agree at all stages ≤ d₀.\n This is a direct corollary of the τ-Identity Theorem (I.T21). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005354.md b/_corpus_v3_items/cid005354.md new file mode 100644 index 000000000..00c7e3ea7 --- /dev/null +++ b/_corpus_v3_items/cid005354.md @@ -0,0 +1,71 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005354/ +title: omega_tau_classifier (theorem) +cid: cid005354 +primary_alias: FTH0092 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T25] The subobject classifier for PSh(Cat_τ) is Ω_τ = Truth4.\n\n In a Grothendieck topos, the subobject\ + \ classifier Ω is characterized by:\n for every mono m: S ↪ X, there exists a unique χ: X → Ω such that\n the pullback\ + \ of true: 1 → Ω along χ recovers S.\n\n In our four-valued setting:\n - T: the element is in S (both sectors confirm\ + \ membership)\n - F: the element is not in S (both sectors deny)\n - B: overdetermined (B-sector confirms, C-sector\ + \ denies)\n - N: underdetermined (neither sector confirms)\n\n The key theorem: Ω_τ has exactly four elements, matching\ + \ Truth4. -/" +aliases: +- FTH0092 +- omega_tau_classifier +- omega-tau-classifier +- TauLib.BookI.Topos.EarnedTopos::omega_tau_classifier +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001195 +- predicate: part_of + object: cid005121 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005121 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.EarnedTopos + declaration: omega_tau_classifier + module_path: TauLib/BookI/Topos/EarnedTopos.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T25] The subobject classifier for PSh(Cat_τ) is Ω_τ = Truth4.\n\n In a Grothendieck topos, the subobject\ + \ classifier Ω is characterized by:\n for every mono m: S ↪ X, there exists a unique χ: X → Ω such that\n the pullback\ + \ of tru" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005355.md b/_corpus_v3_items/cid005355.md new file mode 100644 index 000000000..5484f1ab2 --- /dev/null +++ b/_corpus_v3_items/cid005355.md @@ -0,0 +1,70 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005355/ +title: earned_topos_non_boolean (theorem) +cid: cid005355 +primary_alias: FTH0093 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P27] The earned topos E_τ is non-Boolean.\n\n A Boolean topos has Ω = {0, 1} (two truth values).\n \ + \ Our Ω_τ has FOUR truth values (T, F, B, N).\n The complement law ¬¬p = p holds in Boolean topoi\n but fails in E_τ:\ + \ ¬¬B = ¬N = B, which works,\n but the issue is that B and N are distinct from T and F.\n\n The explosion barrier\ + \ (I.T13) gives a direct witness:\n B ⟹ F is not T (it's N), so material implication doesn't\n validate ex falso quodlibet.\ + \ -/" +aliases: +- FTH0093 +- earned_topos_non_boolean +- earned-topos-non-boolean +- TauLib.BookI.Topos.EarnedTopos::earned_topos_non_boolean +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001150 +- predicate: part_of + object: cid005121 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005121 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.EarnedTopos + declaration: earned_topos_non_boolean + module_path: TauLib/BookI/Topos/EarnedTopos.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P27] The earned topos E_τ is non-Boolean.\n\n A Boolean topos has Ω = {0, 1} (two truth values).\n\ + \ Our Ω_τ has FOUR truth values (T, F, B, N).\n The complement law ¬¬p = p holds in Boolean topoi\n but fails in\ + \ E_τ: ¬¬B = ¬N = " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005356.md b/_corpus_v3_items/cid005356.md new file mode 100644 index 000000000..a9e36c691 --- /dev/null +++ b/_corpus_v3_items/cid005356.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005356/ +title: yoneda_thin (theorem) +cid: cid005356 +primary_alias: FTH0094 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T23] The Yoneda Lemma for thin Cat_τ:\n Natural transformations from y(X) to a presheaf P\n are in\ + \ bijection with P(X).\n\n In a thin category, this simplifies dramatically:\n since y(X) = Hom(-, X) is either empty\ + \ or singleton for each Y,\n a natural transformation y(X) → P is determined by what it does\n at X (where Hom(X,\ + \ X) = {id}).\n\n We formalize the key implication: evaluation at X determines\n the transformation. -/" +aliases: +- FTH0094 +- yoneda_thin +- yoneda-thin +- TauLib.BookI.Topos.Functors::yoneda_thin +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001193 +- predicate: part_of + object: cid005122 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005122 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.Functors + declaration: yoneda_thin + module_path: TauLib/BookI/Topos/Functors.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T23] The Yoneda Lemma for thin Cat_τ:\n Natural transformations from y(X) to a presheaf P\n are\ + \ in bijection with P(X).\n\n In a thin category, this simplifies dramatically:\n since y(X) = Hom(-, X) is either\ + \ empty or singleto" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005357.md b/_corpus_v3_items/cid005357.md new file mode 100644 index 000000000..45845e690 --- /dev/null +++ b/_corpus_v3_items/cid005357.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005357/ +title: cartesian_closed_adj (theorem) +cid: cid005357 +primary_alias: FTH0095 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T28] Cartesian closed adjunction:\n (A × B)(x) = true → C(x) = true\n iff\n A(x) = true → (C^B)(x)\ + \ = true\n\n This is the pointwise version of Hom(A × B, C) ≅ Hom(A, C^B). -/" +aliases: +- FTH0095 +- cartesian_closed_adj +- cartesian-closed-adj +- TauLib.BookI.Topos.InternalHom::cartesian_closed_adj +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001198 +- predicate: part_of + object: cid005127 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005127 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.InternalHom + declaration: cartesian_closed_adj + module_path: TauLib/BookI/Topos/InternalHom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T28] Cartesian closed adjunction:\n (A × B)(x) = true → C(x) = true\n iff\n A(x) = true → (C^B)(x)\ + \ = true\n\n This is the pointwise version of Hom(A × B, C) ≅ Hom(A, C^B). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005358.md b/_corpus_v3_items/cid005358.md new file mode 100644 index 000000000..364379777 --- /dev/null +++ b/_corpus_v3_items/cid005358.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005358/ +title: self_enrichment (theorem) +cid: cid005358 +primary_alias: FTH0096 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P28] E_τ is self-enriched: internal hom gives an\n internal presheaf of morphisms.\n Witness: internal_hom\ + \ P Q is itself a Presheaf. -/" +aliases: +- FTH0096 +- self_enrichment +- TauLib.BookI.Topos.InternalHom::self_enrichment +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001151 +- predicate: part_of + object: cid005127 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005127 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.InternalHom + declaration: self_enrichment + module_path: TauLib/BookI/Topos/InternalHom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P28] E_τ is self-enriched: internal hom gives an\n internal presheaf of morphisms.\n Witness: internal_hom\ + \ P Q is itself a Presheaf. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005359.md b/_corpus_v3_items/cid005359.md new file mode 100644 index 000000000..b161322a7 --- /dev/null +++ b/_corpus_v3_items/cid005359.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005359/ +title: psh_has_terminal (theorem) +cid: cid005359 +primary_alias: FTH0097 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T24] PSh(Cat_τ) is a Grothendieck topos.\n Standard result: for any small category C, PSh(C) is a Grothendieck\ + \ topos.\n Cat_τ is small (countable objects, thin morphisms).\n\n We encode this as: PSh(Cat_τ) has a terminal object,\ + \ products,\n equalizers, and a subobject classifier. -/" +aliases: +- FTH0097 +- psh_has_terminal +- psh-has-terminal +- TauLib.BookI.Topos.LimitsSites::psh_has_terminal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001194 +- predicate: part_of + object: cid005128 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005128 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.LimitsSites + declaration: psh_has_terminal + module_path: TauLib/BookI/Topos/LimitsSites.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T24] PSh(Cat_τ) is a Grothendieck topos.\n Standard result: for any small category C, PSh(C) is a\ + \ Grothendieck topos.\n Cat_τ is small (countable objects, thin morphisms).\n\n We encode this as: PSh(Cat_τ) has\ + \ a terminal object," +noindex: false +--- + diff --git a/_corpus_v3_items/cid005360.md b/_corpus_v3_items/cid005360.md new file mode 100644 index 000000000..d5ee8414c --- /dev/null +++ b/_corpus_v3_items/cid005360.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005360/ +title: psh_countable_objects (theorem) +cid: cid005360 +primary_alias: FTH0098 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P26] PSh(Cat_τ) is countable because Cat_τ has countable objects\n and at most one morphism between each\ + \ pair (thin).\n The set of presheaves is indexed by functions TauIdx → Bool,\n which is uncountable as a set but\ + \ countably generated\n (each presheaf is determined by a countable family of values). -/" +aliases: +- FTH0098 +- psh_countable_objects +- psh-countable-objects +- TauLib.BookI.Topos.LimitsSites::psh_countable_objects +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001149 +- predicate: part_of + object: cid005128 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005128 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.LimitsSites + declaration: psh_countable_objects + module_path: TauLib/BookI/Topos/LimitsSites.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P26] PSh(Cat_τ) is countable because Cat_τ has countable objects\n and at most one morphism between\ + \ each pair (thin).\n The set of presheaves is indexed by functions TauIdx → Bool,\n which is uncountable as a set\ + \ but countably " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005361.md b/_corpus_v3_items/cid005361.md new file mode 100644 index 000000000..3cd61885f --- /dev/null +++ b/_corpus_v3_items/cid005361.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005361/ +title: product_distributes_over_coproduct (theorem) +cid: cid005361 +primary_alias: FTH0099 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T27] Product distributes over coproduct:\n P × (Q ∨ R) = (P × Q) ∨ (P × R). -/" +aliases: +- FTH0099 +- product_distributes_over_coproduct +- product-distributes-over-coproduct +- TauLib.BookI.Topos.WedgeProduct::product_distributes_over_coproduct +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001197 +- predicate: part_of + object: cid005130 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005130 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookI.Topos.WedgeProduct + declaration: product_distributes_over_coproduct + module_path: TauLib/BookI/Topos/WedgeProduct.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T27] Product distributes over coproduct:\n P × (Q ∨ R) = (P × Q) ∨ (P × R). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005362.md b/_corpus_v3_items/cid005362.md new file mode 100644 index 000000000..23b810a55 --- /dev/null +++ b/_corpus_v3_items/cid005362.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005362/ +title: idemp_decomp_recovery (theorem) +cid: cid005362 +primary_alias: FTH0100 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D59] Idempotent decomposition is always valid:\n e_plus * sp + e_minus * sp = sp for any SectorPair sp.\n\ + \ This is the algebraic core of idempotent support. -/" +aliases: +- FTH0100 +- idemp_decomp_recovery +- idemp-decomp-recovery +- TauLib.BookII.CentralTheorem.BoundaryCharacters::idemp_decomp_recovery +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: idemp_decomp_recovery + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D59] Idempotent decomposition is always valid:\n e_plus * sp + e_minus * sp = sp for any SectorPair\ + \ sp.\n This is the algebraic core of idempotent support. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005363.md b/_corpus_v3_items/cid005363.md new file mode 100644 index 000000000..eb45f2785 --- /dev/null +++ b/_corpus_v3_items/cid005363.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005363/ +title: b_channel_kills_c (theorem) +cid: cid005363 +primary_alias: FTH0101 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D59] The B-channel projection kills the C-sector. -/ +aliases: +- FTH0101 +- b_channel_kills_c +- b-channel-kills-c +- TauLib.BookII.CentralTheorem.BoundaryCharacters::b_channel_kills_c +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: b_channel_kills_c + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D59] The B-channel projection kills the C-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005364.md b/_corpus_v3_items/cid005364.md new file mode 100644 index 000000000..2ceeec96a --- /dev/null +++ b/_corpus_v3_items/cid005364.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005364/ +title: c_channel_kills_b (theorem) +cid: cid005364 +primary_alias: FTH0102 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D59] The C-channel projection kills the B-sector. -/ +aliases: +- FTH0102 +- c_channel_kills_b +- c-channel-kills-b +- TauLib.BookII.CentralTheorem.BoundaryCharacters::c_channel_kills_b +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: c_channel_kills_b + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D59] The C-channel projection kills the B-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005365.md b/_corpus_v3_items/cid005365.md new file mode 100644 index 000000000..ea51f07f3 --- /dev/null +++ b/_corpus_v3_items/cid005365.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005365/ +title: character_tower_structural (theorem) +cid: cid005365 +primary_alias: FTH0103 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D59] Character tower coherence: the canonical character input\n at stage k equals the stage-(k+1) input\ + \ reduced.\n reduce(reduce(x, k+1), k) = reduce(x, k). -/" +aliases: +- FTH0103 +- character_tower_structural +- character-tower-structural +- TauLib.BookII.CentralTheorem.BoundaryCharacters::character_tower_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: character_tower_structural + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D59] Character tower coherence: the canonical character input\n at stage k equals the stage-(k+1)\ + \ input reduced.\n reduce(reduce(x, k+1), k) = reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005366.md b/_corpus_v3_items/cid005366.md new file mode 100644 index 000000000..2f847e901 --- /dev/null +++ b/_corpus_v3_items/cid005366.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005366/ +title: character_add_structural (theorem) +cid: cid005366 +primary_alias: FTH0104 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P14] Character addition preserves idempotent support (structural):\n for any sp1 sp2, the sum sp1 + sp2\ + \ satisfies the decomposition. -/" +aliases: +- FTH0104 +- character_add_structural +- character-add-structural +- TauLib.BookII.CentralTheorem.BoundaryCharacters::character_add_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: character_add_structural + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P14] Character addition preserves idempotent support (structural):\n for any sp1 sp2, the sum sp1\ + \ + sp2 satisfies the decomposition. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005367.md b/_corpus_v3_items/cid005367.md new file mode 100644 index 000000000..45e2d124e --- /dev/null +++ b/_corpus_v3_items/cid005367.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005367/ +title: character_mul_structural (theorem) +cid: cid005367 +primary_alias: FTH0105 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P14] Character multiplication preserves idempotent support (structural):\n for any sp1 sp2, the product\ + \ sp1 * sp2 satisfies the decomposition. -/" +aliases: +- FTH0105 +- character_mul_structural +- character-mul-structural +- TauLib.BookII.CentralTheorem.BoundaryCharacters::character_mul_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: character_mul_structural + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P14] Character multiplication preserves idempotent support (structural):\n for any sp1 sp2, the product\ + \ sp1 * sp2 satisfies the decomposition. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005368.md b/_corpus_v3_items/cid005368.md new file mode 100644 index 000000000..d887dd78f --- /dev/null +++ b/_corpus_v3_items/cid005368.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005368/ +title: sector_distributive (theorem) +cid: cid005368 +primary_alias: FTH0106 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P14] Distributivity of sector multiplication over addition. -/ +aliases: +- FTH0106 +- sector_distributive +- sector-distributive +- TauLib.BookII.CentralTheorem.BoundaryCharacters::sector_distributive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: sector_distributive + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P14] Distributivity of sector multiplication over addition. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005369.md b/_corpus_v3_items/cid005369.md new file mode 100644 index 000000000..92e289267 --- /dev/null +++ b/_corpus_v3_items/cid005369.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005369/ +title: idemp_char_20_4 (theorem) +cid: cid005369 +primary_alias: FTH0107 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Idempotent decomposition [II.D59]' +aliases: +- FTH0107 +- idemp_char_20_4 +- idemp-char-20-4 +- TauLib.BookII.CentralTheorem.BoundaryCharacters::idemp_char_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: idemp_char_20_4 + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Idempotent decomposition [II.D59]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005370.md b/_corpus_v3_items/cid005370.md new file mode 100644 index 000000000..6acd0e26b --- /dev/null +++ b/_corpus_v3_items/cid005370.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005370/ +title: char_tower_20_4 (theorem) +cid: cid005370 +primary_alias: FTH0108 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Character tower coherence [II.D59] +aliases: +- FTH0108 +- char_tower_20_4 +- char-tower-20-4 +- TauLib.BookII.CentralTheorem.BoundaryCharacters::char_tower_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001294 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: char_tower_20_4 + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Character tower coherence [II.D59] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005371.md b/_corpus_v3_items/cid005371.md new file mode 100644 index 000000000..d3baba340 --- /dev/null +++ b/_corpus_v3_items/cid005371.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005371/ +title: char_add_15_3 (theorem) +cid: cid005371 +primary_alias: FTH0109 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Character addition [II.P14] +aliases: +- FTH0109 +- char_add_15_3 +- char-add-15-3 +- TauLib.BookII.CentralTheorem.BoundaryCharacters::char_add_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: char_add_15_3 + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Character addition [II.P14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005372.md b/_corpus_v3_items/cid005372.md new file mode 100644 index 000000000..9fbdd743a --- /dev/null +++ b/_corpus_v3_items/cid005372.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005372/ +title: char_mul_15_3 (theorem) +cid: cid005372 +primary_alias: FTH0110 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Character multiplication [II.P14] +aliases: +- FTH0110 +- char_mul_15_3 +- char-mul-15-3 +- TauLib.BookII.CentralTheorem.BoundaryCharacters::char_mul_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: char_mul_15_3 + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Character multiplication [II.P14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005373.md b/_corpus_v3_items/cid005373.md new file mode 100644 index 000000000..400601359 --- /dev/null +++ b/_corpus_v3_items/cid005373.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005373/ +title: char_add_id_15_3 (theorem) +cid: cid005373 +primary_alias: FTH0111 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Ring axioms [II.P14] +aliases: +- FTH0111 +- char_add_id_15_3 +- char-add-id-15-3 +- TauLib.BookII.CentralTheorem.BoundaryCharacters::char_add_id_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001355 +- predicate: part_of + object: cid005131 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005131 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.BoundaryCharacters + declaration: char_add_id_15_3 + module_path: TauLib/BookII/CentralTheorem/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Ring axioms [II.P14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005374.md b/_corpus_v3_items/cid005374.md new file mode 100644 index 000000000..5a80cbdc7 --- /dev/null +++ b/_corpus_v3_items/cid005374.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005374/ +title: liouville_dodge (theorem) +cid: cid005374 +primary_alias: FTH0112 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Liouville dodge [II.T41]' +aliases: +- FTH0112 +- liouville_dodge +- liouville-dodge +- TauLib.BookII.CentralTheorem.Categoricity::liouville_dodge +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001402 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: liouville_dodge + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Liouville dodge [II.T41]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005375.md b/_corpus_v3_items/cid005375.md new file mode 100644 index 000000000..077b76ab0 --- /dev/null +++ b/_corpus_v3_items/cid005375.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005375/ +title: nonconstant_bounded_4 (theorem) +cid: cid005375 +primary_alias: FTH0113 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Nonconstant bounded [II.T41] +aliases: +- FTH0113 +- nonconstant_bounded_4 +- nonconstant-bounded-4 +- TauLib.BookII.CentralTheorem.Categoricity::nonconstant_bounded_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001402 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: nonconstant_bounded_4 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Nonconstant bounded [II.T41] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005376.md b/_corpus_v3_items/cid005376.md new file mode 100644 index 000000000..2c2e7c2fb --- /dev/null +++ b/_corpus_v3_items/cid005376.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005376/ +title: primorial_unique_5 (theorem) +cid: cid005376 +primary_alias: FTH0114 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Primorial uniqueness [II.T42] +aliases: +- FTH0114 +- primorial_unique_5 +- primorial-unique-5 +- TauLib.BookII.CentralTheorem.Categoricity::primorial_unique_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: primorial_unique_5 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Primorial uniqueness [II.T42] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005377.md b/_corpus_v3_items/cid005377.md new file mode 100644 index 000000000..945d5c050 --- /dev/null +++ b/_corpus_v3_items/cid005377.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005377/ +title: abcd_rt_100 (theorem) +cid: cid005377 +primary_alias: FTH0115 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- ABCD round-trip [II.T42] +aliases: +- FTH0115 +- abcd_rt_100 +- abcd-rt-100 +- TauLib.BookII.CentralTheorem.Categoricity::abcd_rt_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: abcd_rt_100 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- ABCD round-trip [II.T42] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005378.md b/_corpus_v3_items/cid005378.md new file mode 100644 index 000000000..a543ffb50 --- /dev/null +++ b/_corpus_v3_items/cid005378.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005378/ +title: red_det_3_50 (theorem) +cid: cid005378 +primary_alias: FTH0116 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Reduction determinism [II.T42] +aliases: +- FTH0116 +- red_det_3_50 +- red-det-3-50 +- TauLib.BookII.CentralTheorem.Categoricity::red_det_3_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: red_det_3_50 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Reduction determinism [II.T42] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005379.md b/_corpus_v3_items/cid005379.md new file mode 100644 index 000000000..2027d2d56 --- /dev/null +++ b/_corpus_v3_items/cid005379.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005379/ +title: categoricity_3_50 (theorem) +cid: cid005379 +primary_alias: FTH0117 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Categoricity [II.T42] +aliases: +- FTH0117 +- categoricity_3_50 +- categoricity-3-50 +- TauLib.BookII.CentralTheorem.Categoricity::categoricity_3_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: categoricity_3_50 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Categoricity [II.T42] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005380.md b/_corpus_v3_items/cid005380.md new file mode 100644 index 000000000..cc161a90c --- /dev/null +++ b/_corpus_v3_items/cid005380.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005380/ +title: moduli_100 (theorem) +cid: cid005380 +primary_alias: FTH0118 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Moduli singleton [II.D61] +aliases: +- FTH0118 +- moduli_100 +- moduli-100 +- TauLib.BookII.CentralTheorem.Categoricity::moduli_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001296 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: moduli_100 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Moduli singleton [II.D61] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005381.md b/_corpus_v3_items/cid005381.md new file mode 100644 index 000000000..de677446f --- /dev/null +++ b/_corpus_v3_items/cid005381.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005381/ +title: uniqueness_3_50 (theorem) +cid: cid005381 +primary_alias: FTH0119 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Uniqueness [II.C02] +aliases: +- FTH0119 +- uniqueness_3_50 +- uniqueness-3-50 +- TauLib.BookII.CentralTheorem.Categoricity::uniqueness_3_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001233 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: uniqueness_3_50 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Uniqueness [II.C02] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005382.md b/_corpus_v3_items/cid005382.md new file mode 100644 index 000000000..76235fbc1 --- /dev/null +++ b/_corpus_v3_items/cid005382.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005382/ +title: j_squared_wave (theorem) +cid: cid005382 +primary_alias: FTH0120 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T41] j^2 = +1 (wave type, not elliptic).\n This is the structural reason Liouville's theorem does not\ + \ apply. -/" +aliases: +- FTH0120 +- j_squared_wave +- j-squared-wave +- TauLib.BookII.CentralTheorem.Categoricity::j_squared_wave +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001402 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: j_squared_wave + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T41] j^2 = +1 (wave type, not elliptic).\n This is the structural reason Liouville's theorem does\ + \ not apply. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005383.md b/_corpus_v3_items/cid005383.md new file mode 100644 index 000000000..87d5c600a --- /dev/null +++ b/_corpus_v3_items/cid005383.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005383/ +title: zero_divisors_exist (theorem) +cid: cid005383 +primary_alias: FTH0121 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T41] Zero divisors exist: e_plus * e_minus = 0.\n This is impossible in the Gaussian integers Z[i] (i^2\ + \ = -1). -/" +aliases: +- FTH0121 +- zero_divisors_exist +- zero-divisors-exist +- TauLib.BookII.CentralTheorem.Categoricity::zero_divisors_exist +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001402 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: zero_divisors_exist + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T41] Zero divisors exist: e_plus * e_minus = 0.\n This is impossible in the Gaussian integers Z[i]\ + \ (i^2 = -1). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005384.md b/_corpus_v3_items/cid005384.md new file mode 100644 index 000000000..94093a985 --- /dev/null +++ b/_corpus_v3_items/cid005384.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005384/ +title: idempotent_complete (theorem) +cid: cid005384 +primary_alias: FTH0122 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T41] Completeness: e_plus + e_minus = 1.\n The sector idempotents partition unity. -/" +aliases: +- FTH0122 +- idempotent_complete +- idempotent-complete +- TauLib.BookII.CentralTheorem.Categoricity::idempotent_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001402 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: idempotent_complete + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T41] Completeness: e_plus + e_minus = 1.\n The sector idempotents partition unity. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005385.md b/_corpus_v3_items/cid005385.md new file mode 100644 index 000000000..fb5b753ab --- /dev/null +++ b/_corpus_v3_items/cid005385.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005385/ +title: abcd_roundtrip_12 (theorem) +cid: cid005385 +primary_alias: FTH0123 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T42] The ABCD chart round-trips for specific values (verified computationally).\n to_tau_idx(from_tau_idx(x))\ + \ = x.\n The general statement is verified by abcd_roundtrip_check via native_decide. -/" +aliases: +- FTH0123 +- abcd_roundtrip_12 +- abcd-roundtrip-12 +- TauLib.BookII.CentralTheorem.Categoricity::abcd_roundtrip_12 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: abcd_roundtrip_12 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T42] The ABCD chart round-trips for specific values (verified computationally).\n to_tau_idx(from_tau_idx(x))\ + \ = x.\n The general statement is verified by abcd_roundtrip_check via native_decide. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005386.md b/_corpus_v3_items/cid005386.md new file mode 100644 index 000000000..f1be33858 --- /dev/null +++ b/_corpus_v3_items/cid005386.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005386/ +title: reduce_idempotent (theorem) +cid: cid005386 +primary_alias: FTH0124 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T42] Reduction is idempotent: reduce(reduce(x, k), k) = reduce(x, k).\n This is the formal statement\ + \ that the primorial tower has no ambiguity. -/" +aliases: +- FTH0124 +- reduce_idempotent +- reduce-idempotent +- TauLib.BookII.CentralTheorem.Categoricity::reduce_idempotent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: reduce_idempotent + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T42] Reduction is idempotent: reduce(reduce(x, k), k) = reduce(x, k).\n This is the formal statement\ + \ that the primorial tower has no ambiguity. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005387.md b/_corpus_v3_items/cid005387.md new file mode 100644 index 000000000..90c850614 --- /dev/null +++ b/_corpus_v3_items/cid005387.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005387/ +title: abcd_distinct_12_64 (theorem) +cid: cid005387 +primary_alias: FTH0125 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D61] The ABCD chart is injective at specific values (verified computationally).\n from_tau_idx is injective\ + \ because to_tau_idx is a left inverse.\n The general computational check is abcd_roundtrip_check. -/" +aliases: +- FTH0125 +- abcd_distinct_12_64 +- abcd-distinct-12-64 +- TauLib.BookII.CentralTheorem.Categoricity::abcd_distinct_12_64 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001296 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: abcd_distinct_12_64 + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D61] The ABCD chart is injective at specific values (verified computationally).\n from_tau_idx is\ + \ injective because to_tau_idx is a left inverse.\n The general computational check is abcd_roundtrip_check. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005388.md b/_corpus_v3_items/cid005388.md new file mode 100644 index 000000000..91634cfb1 --- /dev/null +++ b/_corpus_v3_items/cid005388.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005388/ +title: structure_uniqueness (theorem) +cid: cid005388 +primary_alias: FTH0126 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.C02] Uniqueness: reduce is defined by modular arithmetic.\n There is exactly one way to define it: x\ + \ % primorial k. -/" +aliases: +- FTH0126 +- structure_uniqueness +- structure-uniqueness +- TauLib.BookII.CentralTheorem.Categoricity::structure_uniqueness +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001233 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: structure_uniqueness + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.C02] Uniqueness: reduce is defined by modular arithmetic.\n There is exactly one way to define it:\ + \ x % primorial k. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005389.md b/_corpus_v3_items/cid005389.md new file mode 100644 index 000000000..b0c626331 --- /dev/null +++ b/_corpus_v3_items/cid005389.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005389/ +title: tower_forced (theorem) +cid: cid005389 +primary_alias: FTH0127 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T42] Tower coherence is uniquely forced:\n reduce(reduce(x, l), k) = reduce(x, k) for k <= l.\n This\ + \ is the unique compatible system of projections. -/" +aliases: +- FTH0127 +- tower_forced +- tower-forced +- TauLib.BookII.CentralTheorem.Categoricity::tower_forced +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001403 +- predicate: part_of + object: cid005132 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005132 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.Categoricity + declaration: tower_forced + module_path: TauLib/BookII/CentralTheorem/Categoricity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T42] Tower coherence is uniquely forced:\n reduce(reduce(x, l), k) = reduce(x, k) for k <= l.\n \ + \ This is the unique compatible system of projections. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005390.md b/_corpus_v3_items/cid005390.md new file mode 100644 index 000000000..989ad823d --- /dev/null +++ b/_corpus_v3_items/cid005390.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005390/ +title: spectral_ring_3_15 (theorem) +cid: cid005390 +primary_alias: FTH0128 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Spectral algebra [II.D60]' +aliases: +- FTH0128 +- spectral_ring_3_15 +- spectral-ring-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::spectral_ring_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001295 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: spectral_ring_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Spectral algebra [II.D60]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005391.md b/_corpus_v3_items/cid005391.md new file mode 100644 index 000000000..7f031200c --- /dev/null +++ b/_corpus_v3_items/cid005391.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005391/ +title: central_fwd_3_15 (theorem) +cid: cid005391 +primary_alias: FTH0129 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Central theorem forward [II.T40] +aliases: +- FTH0129 +- central_fwd_3_15 +- central-fwd-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::central_fwd_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_fwd_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Central theorem forward [II.T40] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005392.md b/_corpus_v3_items/cid005392.md new file mode 100644 index 000000000..f6e5ef375 --- /dev/null +++ b/_corpus_v3_items/cid005392.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005392/ +title: central_inv_3_15 (theorem) +cid: cid005392 +primary_alias: FTH0130 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Central theorem inverse [II.T40] +aliases: +- FTH0130 +- central_inv_3_15 +- central-inv-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::central_inv_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_inv_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Central theorem inverse [II.T40] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005393.md b/_corpus_v3_items/cid005393.md new file mode 100644 index 000000000..7c0e8fb55 --- /dev/null +++ b/_corpus_v3_items/cid005393.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005393/ +title: central_rt_3_15 (theorem) +cid: cid005393 +primary_alias: FTH0131 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Central theorem round-trip [II.T40] +aliases: +- FTH0131 +- central_rt_3_15 +- central-rt-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::central_rt_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_rt_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Central theorem round-trip [II.T40] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005394.md b/_corpus_v3_items/cid005394.md new file mode 100644 index 000000000..de18ad7f6 --- /dev/null +++ b/_corpus_v3_items/cid005394.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005394/ +title: central_theorem_3_15 (theorem) +cid: cid005394 +primary_alias: FTH0132 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- THE CENTRAL THEOREM [II.T40] +aliases: +- FTH0132 +- central_theorem_3_15 +- central-theorem-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::central_theorem_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_theorem_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- THE CENTRAL THEOREM [II.T40] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005395.md b/_corpus_v3_items/cid005395.md new file mode 100644 index 000000000..09beceb3f --- /dev/null +++ b/_corpus_v3_items/cid005395.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005395/ +title: holographic_3_15 (theorem) +cid: cid005395 +primary_alias: FTH0133 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Holographic principle [II.C01] +aliases: +- FTH0133 +- holographic_3_15 +- holographic-3-15 +- TauLib.BookII.CentralTheorem.CentralTheorem::holographic_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001232 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: holographic_3_15 + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Holographic principle [II.C01] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005396.md b/_corpus_v3_items/cid005396.md new file mode 100644 index 000000000..4425d1ccc --- /dev/null +++ b/_corpus_v3_items/cid005396.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005396/ +title: spectral_periodic (theorem) +cid: cid005396 +primary_alias: FTH0134 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D60] Spectral algebra periodicity: evaluation is periodic in x\n with period P_k. This follows from reduce(x\ + \ + P_k, k) = reduce(x, k). -/" +aliases: +- FTH0134 +- spectral_periodic +- spectral-periodic +- TauLib.BookII.CentralTheorem.CentralTheorem::spectral_periodic +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001295 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: spectral_periodic + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D60] Spectral algebra periodicity: evaluation is periodic in x\n with period P_k. This follows from\ + \ reduce(x + P_k, k) = reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005397.md b/_corpus_v3_items/cid005397.md new file mode 100644 index 000000000..ac91901f2 --- /dev/null +++ b/_corpus_v3_items/cid005397.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005397/ +title: spectral_idempotent_supported (theorem) +cid: cid005397 +primary_alias: FTH0135 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D60] Spectral algebra elements are always idempotent-supported.\n This is decompose_recovery applied\ + \ pointwise. -/" +aliases: +- FTH0135 +- spectral_idempotent_supported +- spectral-idempotent-supported +- TauLib.BookII.CentralTheorem.CentralTheorem::spectral_idempotent_supported +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001295 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: spectral_idempotent_supported + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D60] Spectral algebra elements are always idempotent-supported.\n This is decompose_recovery applied\ + \ pointwise. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005398.md b/_corpus_v3_items/cid005398.md new file mode 100644 index 000000000..65e939e42 --- /dev/null +++ b/_corpus_v3_items/cid005398.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005398/ +title: central_forward_coherent (theorem) +cid: cid005398 +primary_alias: FTH0136 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T40] Central Theorem forward: spectral data produces tower-coherent output.\n spectral_to_hol(b_fn, c_fn,\ + \ x, k) uses reduce(x, k), so\n reduce(spectral_to_hol(_, _, x, l), k) = spectral_to_hol(_, _, x, k)\n when b_fn and\ + \ c_fn are the identity (both sides reduce to reduce(x, k)). -/" +aliases: +- FTH0136 +- central_forward_coherent +- central-forward-coherent +- TauLib.BookII.CentralTheorem.CentralTheorem::central_forward_coherent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_forward_coherent + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T40] Central Theorem forward: spectral data produces tower-coherent output.\n spectral_to_hol(b_fn,\ + \ c_fn, x, k) uses reduce(x, k), so\n reduce(spectral_to_hol(_, _, x, l), k) = spectral_to_hol(_, _, x, k)\n when\ + \ b_fn and c_fn " +noindex: false +--- + diff --git a/_corpus_v3_items/cid005399.md b/_corpus_v3_items/cid005399.md new file mode 100644 index 000000000..5dccd4c85 --- /dev/null +++ b/_corpus_v3_items/cid005399.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005399/ +title: central_inverse_periodic (theorem) +cid: cid005399 +primary_alias: FTH0137 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T40] Central Theorem inverse: holomorphic restriction is periodic.\n reduce(x + P_k, k) = reduce(x, k)\ + \ ensures the boundary restriction\n is well-defined on Z/P_kZ. -/" +aliases: +- FTH0137 +- central_inverse_periodic +- central-inverse-periodic +- TauLib.BookII.CentralTheorem.CentralTheorem::central_inverse_periodic +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_inverse_periodic + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T40] Central Theorem inverse: holomorphic restriction is periodic.\n reduce(x + P_k, k) = reduce(x,\ + \ k) ensures the boundary restriction\n is well-defined on Z/P_kZ. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005400.md b/_corpus_v3_items/cid005400.md new file mode 100644 index 000000000..43e8da2d1 --- /dev/null +++ b/_corpus_v3_items/cid005400.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005400/ +title: central_roundtrip (theorem) +cid: cid005400 +primary_alias: FTH0138 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T40] Central Theorem round-trip: the forward and inverse maps compose\n to the identity on spectral data.\ + \ For b_fn = identity:\n spectral_to_hol(id, id, x, k) = (reduce(x,k), reduce(x,k))\n hol_to_spectral(id_stage, x,\ + \ k) = (reduce(x,k), reduce(x,k))\n These are equal. -/" +aliases: +- FTH0138 +- central_roundtrip +- central-roundtrip +- TauLib.BookII.CentralTheorem.CentralTheorem::central_roundtrip +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001401 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: central_roundtrip + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T40] Central Theorem round-trip: the forward and inverse maps compose\n to the identity on spectral\ + \ data. For b_fn = identity:\n spectral_to_hol(id, id, x, k) = (reduce(x,k), reduce(x,k))\n hol_to_spectral(id_stage,\ + \ x, k) = (r" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005401.md b/_corpus_v3_items/cid005401.md new file mode 100644 index 000000000..07676dc09 --- /dev/null +++ b/_corpus_v3_items/cid005401.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005401/ +title: holographic_roundtrip (theorem) +cid: cid005401 +primary_alias: FTH0139 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.C01] Holographic principle: boundary reduction is involutive.\n reduce(bndlift(x, k), k) = reduce(x,\ + \ k).\n The boundary completely determines the interior. -/" +aliases: +- FTH0139 +- holographic_roundtrip +- holographic-roundtrip +- TauLib.BookII.CentralTheorem.CentralTheorem::holographic_roundtrip +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001232 +- predicate: part_of + object: cid005133 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005133 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.CentralTheorem + declaration: holographic_roundtrip + module_path: TauLib/BookII/CentralTheorem/CentralTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.C01] Holographic principle: boundary reduction is involutive.\n reduce(bndlift(x, k), k) = reduce(x,\ + \ k).\n The boundary completely determines the interior. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005402.md b/_corpus_v3_items/cid005402.md new file mode 100644 index 000000000..214eff7ed --- /dev/null +++ b/_corpus_v3_items/cid005402.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005402/ +title: stagewise_naturality (theorem) +cid: cid005402 +primary_alias: FTH0140 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L13] Stagewise naturality (structural): reduce(bndlift(x, k+1), k) = reduce(x, k).\n Since bndlift(x,\ + \ k+1) = reduce(x, k+2) and k <= k+2, this follows\n from reduction_compat. -/" +aliases: +- FTH0140 +- stagewise_naturality +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001340 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: stagewise_naturality + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L13] Stagewise naturality (structural): reduce(bndlift(x, k+1), k) = reduce(x, k).\n Since bndlift(x,\ + \ k+1) = reduce(x, k+2) and k <= k+2, this follows\n from reduction_compat. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005403.md b/_corpus_v3_items/cid005403.md new file mode 100644 index 000000000..eba9a4363 --- /dev/null +++ b/_corpus_v3_items/cid005403.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005403/ +title: stagewise_naturality_strong (theorem) +cid: cid005403 +primary_alias: FTH0141 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L13] Strong naturality (structural): reduce(bndlift(x, k+1), k+1) = bndlift(x, k).\n This is: reduce(reduce(x,\ + \ k+2), k+1) = reduce(x, k+1),\n which follows from reduction_compat since k+1 <= k+2. -/" +aliases: +- FTH0141 +- stagewise_naturality_strong +- stagewise-naturality-strong +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality_strong +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001340 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: stagewise_naturality_strong + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L13] Strong naturality (structural): reduce(bndlift(x, k+1), k+1) = bndlift(x, k).\n This is: reduce(reduce(x,\ + \ k+2), k+1) = reduce(x, k+1),\n which follows from reduction_compat since k+1 <= k+2. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005404.md b/_corpus_v3_items/cid005404.md new file mode 100644 index 000000000..de029dd03 --- /dev/null +++ b/_corpus_v3_items/cid005404.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005404/ +title: bndlift_reduce_invariant (theorem) +cid: cid005404 +primary_alias: FTH0142 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T38] The bndlift extension at stage k depends only on reduce(x, k+1).\n bndlift(x, k) = bndlift(reduce(x,\ + \ k+1), k).\n This is: reduce(x, k+1) = reduce(reduce(x, k+1), k+1),\n which is reduction idempotence. -/" +aliases: +- FTH0142 +- bndlift_reduce_invariant +- bndlift-reduce-invariant +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_reduce_invariant +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001399 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: bndlift_reduce_invariant + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T38] The bndlift extension at stage k depends only on reduce(x, k+1).\n bndlift(x, k) = bndlift(reduce(x,\ + \ k+1), k).\n This is: reduce(x, k+1) = reduce(reduce(x, k+1), k+1),\n which is reduction idempotence. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005405.md b/_corpus_v3_items/cid005405.md new file mode 100644 index 000000000..61dd27b87 --- /dev/null +++ b/_corpus_v3_items/cid005405.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005405/ +title: bndlift_stagefun_welldef (theorem) +cid: cid005405 +primary_alias: FTH0143 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T38] Tower coherence of bndlift StageFun inputs:\n the ABCD chart at stage k is determined by reduce(x,\ + \ k).\n For the bndlift StageFun, from_tau_idx(bndlift(x, k)) =\n from_tau_idx(bndlift(reduce(x, k+1), k)) =\n \ + \ from_tau_idx(bndlift(x, k)) by bndlift_reduce_invariant. -/" +aliases: +- FTH0143 +- bndlift_stagefun_welldef +- bndlift-stagefun-welldef +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_stagefun_welldef +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001399 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: bndlift_stagefun_welldef + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T38] Tower coherence of bndlift StageFun inputs:\n the ABCD chart at stage k is determined by reduce(x,\ + \ k).\n For the bndlift StageFun, from_tau_idx(bndlift(x, k)) =\n from_tau_idx(bndlift(reduce(x, k+1), k)) =\n \ + \ from_tau_id" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005406.md b/_corpus_v3_items/cid005406.md new file mode 100644 index 000000000..63316a2f2 --- /dev/null +++ b/_corpus_v3_items/cid005406.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005406/ +title: reduction_gives_naturality (theorem) +cid: cid005406 +primary_alias: FTH0144 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L13] Reduction compatibility directly implies stagewise naturality.\n This is the structural heart of\ + \ the module. -/" +aliases: +- FTH0144 +- reduction_gives_naturality +- reduction-gives-naturality +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::reduction_gives_naturality +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001340 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: reduction_gives_naturality + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L13] Reduction compatibility directly implies stagewise naturality.\n This is the structural heart\ + \ of the module. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005407.md b/_corpus_v3_items/cid005407.md new file mode 100644 index 000000000..fa7349614 --- /dev/null +++ b/_corpus_v3_items/cid005407.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005407/ +title: nat_check_20_4 (theorem) +cid: cid005407 +primary_alias: FTH0145 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Stagewise naturality [II.L13]' +aliases: +- FTH0145 +- nat_check_20_4 +- nat-check-20-4 +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::nat_check_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001340 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: nat_check_20_4 + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Stagewise naturality [II.L13]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005408.md b/_corpus_v3_items/cid005408.md new file mode 100644 index 000000000..dde647bee --- /dev/null +++ b/_corpus_v3_items/cid005408.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005408/ +title: ogt_15_4 (theorem) +cid: cid005408 +primary_alias: FTH0146 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Omega-germ transformer [II.T38] +aliases: +- FTH0146 +- ogt_15_4 +- ogt-15-4 +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::ogt_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001399 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: ogt_15_4 + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Omega-germ transformer [II.T38] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005409.md b/_corpus_v3_items/cid005409.md new file mode 100644 index 000000000..6791362bd --- /dev/null +++ b/_corpus_v3_items/cid005409.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005409/ +title: bsf_coh_15_4 (theorem) +cid: cid005409 +primary_alias: FTH0147 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- BndLift StageFun coherence [II.T38] +aliases: +- FTH0147 +- bsf_coh_15_4 +- bsf-coh-15-4 +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bsf_coh_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001399 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: bsf_coh_15_4 + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- BndLift StageFun coherence [II.T38] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005410.md b/_corpus_v3_items/cid005410.md new file mode 100644 index 000000000..1f55357e0 --- /dev/null +++ b/_corpus_v3_items/cid005410.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005410/ +title: egr_15_4 (theorem) +cid: cid005410 +primary_alias: FTH0148 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Extension-germ roundtrip [II.T38] +aliases: +- FTH0148 +- egr_15_4 +- egr-15-4 +- TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::egr_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001399 +- predicate: part_of + object: cid005134 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005134 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + declaration: egr_15_4 + module_path: TauLib/BookII/CentralTheorem/ExtensionsOmegaGerms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Extension-germ roundtrip [II.T38] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005411.md b/_corpus_v3_items/cid005411.md new file mode 100644 index 000000000..150391103 --- /dev/null +++ b/_corpus_v3_items/cid005411.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005411/ +title: extension_preserves_stage (theorem) +cid: cid005411 +primary_alias: FTH0149 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L12] Extension preserves stage-k data: the reduction of the\n bndlift extension back to stage k recovers\ + \ the original stage-k value.\n reduce(bndlift(x, k), k) = reduce(x, k).\n This follows from reduction_compat since\ + \ bndlift(x, k) = reduce(x, k+1). -/" +aliases: +- FTH0149 +- extension_preserves_stage +- extension-preserves-stage +- TauLib.BookII.CentralTheorem.HartogsExtension::extension_preserves_stage +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001339 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: extension_preserves_stage + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L12] Extension preserves stage-k data: the reduction of the\n bndlift extension back to stage k recovers\ + \ the original stage-k value.\n reduce(bndlift(x, k), k) = reduce(x, k).\n This follows from reduction_compat since\ + \ bndlift" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005412.md b/_corpus_v3_items/cid005412.md new file mode 100644 index 000000000..ae85cb083 --- /dev/null +++ b/_corpus_v3_items/cid005412.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005412/ +title: uniqueness_from_agreement (theorem) +cid: cid005412 +primary_alias: FTH0150 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T37] Uniqueness structural core: two extensions that agree on all\n stages must agree on the bndlift\ + \ extension.\n If reduce(x, k) = reduce(y, k), then bndlift at stage k gives the same\n stage-k data: reduce(bndlift(x,k),\ + \ k) = reduce(bndlift(y,k), k). -/" +aliases: +- FTH0150 +- uniqueness_from_agreement +- uniqueness-from-agreement +- TauLib.BookII.CentralTheorem.HartogsExtension::uniqueness_from_agreement +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001398 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: uniqueness_from_agreement + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T37] Uniqueness structural core: two extensions that agree on all\n stages must agree on the bndlift\ + \ extension.\n If reduce(x, k) = reduce(y, k), then bndlift at stage k gives the same\n stage-k data: reduce(bndlift(x,k),\ + \ k) =" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005413.md b/_corpus_v3_items/cid005413.md new file mode 100644 index 000000000..34b7ec19a --- /dev/null +++ b/_corpus_v3_items/cid005413.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005413/ +title: bndlift_tower (theorem) +cid: cid005413 +primary_alias: FTH0151 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T37] BndLift is tower-coherent: reduce(bndlift(x, k+1), k) = reduce(x, k).\n This is the key structural\ + \ property for Hartogs extension uniqueness. -/" +aliases: +- FTH0151 +- bndlift_tower +- bndlift-tower +- TauLib.BookII.CentralTheorem.HartogsExtension::bndlift_tower +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001398 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: bndlift_tower + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T37] BndLift is tower-coherent: reduce(bndlift(x, k+1), k) = reduce(x, k).\n This is the key structural\ + \ property for Hartogs extension uniqueness. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005414.md b/_corpus_v3_items/cid005414.md new file mode 100644 index 000000000..08cd93ccd --- /dev/null +++ b/_corpus_v3_items/cid005414.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005414/ +title: extension_bipolar_recovery (theorem) +cid: cid005414 +primary_alias: FTH0152 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L12] The interior bipolar decomposition of the extension\n recovers via idempotent projections. -/" +aliases: +- FTH0152 +- extension_bipolar_recovery +- extension-bipolar-recovery +- TauLib.BookII.CentralTheorem.HartogsExtension::extension_bipolar_recovery +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001339 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: extension_bipolar_recovery + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L12] The interior bipolar decomposition of the extension\n recovers via idempotent projections. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005415.md b/_corpus_v3_items/cid005415.md new file mode 100644 index 000000000..ec225df31 --- /dev/null +++ b/_corpus_v3_items/cid005415.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005415/ +title: ext_channel_20_4 (theorem) +cid: cid005415 +primary_alias: FTH0153 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Extension channel [II.L12]' +aliases: +- FTH0153 +- ext_channel_20_4 +- ext-channel-20-4 +- TauLib.BookII.CentralTheorem.HartogsExtension::ext_channel_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001339 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: ext_channel_20_4 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Extension channel [II.L12]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005416.md b/_corpus_v3_items/cid005416.md new file mode 100644 index 000000000..6a65ae9c9 --- /dev/null +++ b/_corpus_v3_items/cid005416.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005416/ +title: b_channel_20_4 (theorem) +cid: cid005416 +primary_alias: FTH0154 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- B-channel independence [II.L12] +aliases: +- FTH0154 +- b_channel_20_4 +- b-channel-20-4 +- TauLib.BookII.CentralTheorem.HartogsExtension::b_channel_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001339 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: b_channel_20_4 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- B-channel independence [II.L12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005417.md b/_corpus_v3_items/cid005417.md new file mode 100644 index 000000000..a827a152a --- /dev/null +++ b/_corpus_v3_items/cid005417.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005417/ +title: c_channel_20_4 (theorem) +cid: cid005417 +primary_alias: FTH0155 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- C-channel independence [II.L12] +aliases: +- FTH0155 +- c_channel_20_4 +- c-channel-20-4 +- TauLib.BookII.CentralTheorem.HartogsExtension::c_channel_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001339 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: c_channel_20_4 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- C-channel independence [II.L12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005418.md b/_corpus_v3_items/cid005418.md new file mode 100644 index 000000000..f08199407 --- /dev/null +++ b/_corpus_v3_items/cid005418.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005418/ +title: hartogs_uniq_15_4 (theorem) +cid: cid005418 +primary_alias: FTH0156 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hartogs uniqueness [II.T37] +aliases: +- FTH0156 +- hartogs_uniq_15_4 +- hartogs-uniq-15-4 +- TauLib.BookII.CentralTheorem.HartogsExtension::hartogs_uniq_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001398 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: hartogs_uniq_15_4 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hartogs uniqueness [II.T37] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005419.md b/_corpus_v3_items/cid005419.md new file mode 100644 index 000000000..294eb004f --- /dev/null +++ b/_corpus_v3_items/cid005419.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005419/ +title: bnd_det_int_10_3 (theorem) +cid: cid005419 +primary_alias: FTH0157 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Boundary determines interior [II.T37] +aliases: +- FTH0157 +- bnd_det_int_10_3 +- bnd-det-int-10-3 +- TauLib.BookII.CentralTheorem.HartogsExtension::bnd_det_int_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001398 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: bnd_det_int_10_3 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Boundary determines interior [II.T37] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005420.md b/_corpus_v3_items/cid005420.md new file mode 100644 index 000000000..01fe195fc --- /dev/null +++ b/_corpus_v3_items/cid005420.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005420/ +title: cd_uniq_15_4 (theorem) +cid: cid005420 +primary_alias: FTH0158 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Code/Decode uniqueness [II.T37] +aliases: +- FTH0158 +- cd_uniq_15_4 +- cd-uniq-15-4 +- TauLib.BookII.CentralTheorem.HartogsExtension::cd_uniq_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001398 +- predicate: part_of + object: cid005135 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005135 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.HartogsExtension + declaration: cd_uniq_15_4 + module_path: TauLib/BookII/CentralTheorem/HartogsExtension.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Code/Decode uniqueness [II.T37] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005421.md b/_corpus_v3_items/cid005421.md new file mode 100644 index 000000000..baf3da528 --- /dev/null +++ b/_corpus_v3_items/cid005421.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005421/ +title: cech_sq_zero_1 (theorem) +cid: cid005421 +primary_alias: FTH0159 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D86] δ² = 0 at stage 1. -/ +aliases: +- FTH0159 +- cech_sq_zero_1 +- cech-sq-zero-1 +- TauLib.BookII.CentralTheorem.SheafCohomology::cech_sq_zero_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001320 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: cech_sq_zero_1 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D86] δ² = 0 at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005422.md b/_corpus_v3_items/cid005422.md new file mode 100644 index 000000000..8cbdbbd1d --- /dev/null +++ b/_corpus_v3_items/cid005422.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005422/ +title: cech_sq_zero_2 (theorem) +cid: cid005422 +primary_alias: FTH0160 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D86] δ² = 0 at stage 2. -/ +aliases: +- FTH0160 +- cech_sq_zero_2 +- cech-sq-zero-2 +- TauLib.BookII.CentralTheorem.SheafCohomology::cech_sq_zero_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001320 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: cech_sq_zero_2 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D86] δ² = 0 at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005423.md b/_corpus_v3_items/cid005423.md new file mode 100644 index 000000000..90fe81c4f --- /dev/null +++ b/_corpus_v3_items/cid005423.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005423/ +title: h1_vanishes_1 (theorem) +cid: cid005423 +primary_alias: FTH0161 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D87] H¹ = 0 at stage 1. -/ +aliases: +- FTH0161 +- h1_vanishes_1 +- h1-vanishes-1 +- TauLib.BookII.CentralTheorem.SheafCohomology::h1_vanishes_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001321 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: h1_vanishes_1 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D87] H¹ = 0 at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005424.md b/_corpus_v3_items/cid005424.md new file mode 100644 index 000000000..c58ed782a --- /dev/null +++ b/_corpus_v3_items/cid005424.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005424/ +title: h1_vanishes_2 (theorem) +cid: cid005424 +primary_alias: FTH0162 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D87] H¹ = 0 at stage 2. -/ +aliases: +- FTH0162 +- h1_vanishes_2 +- h1-vanishes-2 +- TauLib.BookII.CentralTheorem.SheafCohomology::h1_vanishes_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001321 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: h1_vanishes_2 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D87] H¹ = 0 at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005425.md b/_corpus_v3_items/cid005425.md new file mode 100644 index 000000000..374807080 --- /dev/null +++ b/_corpus_v3_items/cid005425.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005425/ +title: h0_global_1 (theorem) +cid: cid005425 +primary_alias: FTH0163 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T55] H⁰ = global sections at stage 1. -/ +aliases: +- FTH0163 +- h0_global_1 +- h0-global-1 +- TauLib.BookII.CentralTheorem.SheafCohomology::h0_global_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001416 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: h0_global_1 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T55] H⁰ = global sections at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005426.md b/_corpus_v3_items/cid005426.md new file mode 100644 index 000000000..cec359a1d --- /dev/null +++ b/_corpus_v3_items/cid005426.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005426/ +title: h0_global_2 (theorem) +cid: cid005426 +primary_alias: FTH0164 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T55] H⁰ = global sections at stage 2. -/ +aliases: +- FTH0164 +- h0_global_2 +- h0-global-2 +- TauLib.BookII.CentralTheorem.SheafCohomology::h0_global_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001416 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: h0_global_2 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T55] H⁰ = global sections at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005427.md b/_corpus_v3_items/cid005427.md new file mode 100644 index 000000000..61a176b8e --- /dev/null +++ b/_corpus_v3_items/cid005427.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005427/ +title: h0_nonconstant_2 (theorem) +cid: cid005427 +primary_alias: FTH0165 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T55] Non-constant rejected at stage 2. -/ +aliases: +- FTH0165 +- h0_nonconstant_2 +- h0-nonconstant-2 +- TauLib.BookII.CentralTheorem.SheafCohomology::h0_nonconstant_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001416 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: h0_nonconstant_2 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T55] Non-constant rejected at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005428.md b/_corpus_v3_items/cid005428.md new file mode 100644 index 000000000..196ed49f1 --- /dev/null +++ b/_corpus_v3_items/cid005428.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005428/ +title: cech_derived_1 (theorem) +cid: cid005428 +primary_alias: FTH0166 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P20] Čech = derived at stage 1. -/ +aliases: +- FTH0166 +- cech_derived_1 +- cech-derived-1 +- TauLib.BookII.CentralTheorem.SheafCohomology::cech_derived_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001361 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: cech_derived_1 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P20] Čech = derived at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005429.md b/_corpus_v3_items/cid005429.md new file mode 100644 index 000000000..6046760f9 --- /dev/null +++ b/_corpus_v3_items/cid005429.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005429/ +title: cech_derived_2 (theorem) +cid: cid005429 +primary_alias: FTH0167 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P20] Čech = derived at stage 2. -/ +aliases: +- FTH0167 +- cech_derived_2 +- cech-derived-2 +- TauLib.BookII.CentralTheorem.SheafCohomology::cech_derived_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001361 +- predicate: part_of + object: cid005136 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005136 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.SheafCohomology + declaration: cech_derived_2 + module_path: TauLib/BookII/CentralTheorem/SheafCohomology.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P20] Čech = derived at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005430.md b/_corpus_v3_items/cid005430.md new file mode 100644 index 000000000..68ad5f82c --- /dev/null +++ b/_corpus_v3_items/cid005430.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005430/ +title: yoneda_app_12_3 (theorem) +cid: cid005430 +primary_alias: FTH0168 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Yoneda application [II.L14]' +aliases: +- FTH0168 +- yoneda_app_12_3 +- yoneda-app-12-3 +- TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_app_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001341 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: yoneda_app_12_3 + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Yoneda application [II.L14]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005431.md b/_corpus_v3_items/cid005431.md new file mode 100644 index 000000000..13418949f --- /dev/null +++ b/_corpus_v3_items/cid005431.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005431/ +title: yoneda_hom_12_3 (theorem) +cid: cid005431 +primary_alias: FTH0169 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hom identification [II.L14] +aliases: +- FTH0169 +- yoneda_hom_12_3 +- yoneda-hom-12-3 +- TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_hom_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001341 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: yoneda_hom_12_3 + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hom identification [II.L14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005432.md b/_corpus_v3_items/cid005432.md new file mode 100644 index 000000000..4b79fddef --- /dev/null +++ b/_corpus_v3_items/cid005432.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005432/ +title: omega_germs_12_3 (theorem) +cid: cid005432 +primary_alias: FTH0170 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Omega-germs holomorphic [II.T39] +aliases: +- FTH0170 +- omega_germs_12_3 +- omega-germs-12-3 +- TauLib.BookII.CentralTheorem.YonedaApplied::omega_germs_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: omega_germs_12_3 + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Omega-germs holomorphic [II.T39] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005433.md b/_corpus_v3_items/cid005433.md new file mode 100644 index 000000000..cf2803633 --- /dev/null +++ b/_corpus_v3_items/cid005433.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005433/ +title: hol_class_12_3 (theorem) +cid: cid005433 +primary_alias: FTH0171 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Holomorphic classification [II.T39] +aliases: +- FTH0171 +- hol_class_12_3 +- hol-class-12-3 +- TauLib.BookII.CentralTheorem.YonedaApplied::hol_class_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: hol_class_12_3 + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Holomorphic classification [II.T39] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005434.md b/_corpus_v3_items/cid005434.md new file mode 100644 index 000000000..b9a9e7223 --- /dev/null +++ b/_corpus_v3_items/cid005434.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005434/ +title: germ_hol_12_3 (theorem) +cid: cid005434 +primary_alias: FTH0172 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Germ to holomorphic [II.T39] +aliases: +- FTH0172 +- germ_hol_12_3 +- germ-hol-12-3 +- TauLib.BookII.CentralTheorem.YonedaApplied::germ_hol_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: germ_hol_12_3 + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Germ to holomorphic [II.T39] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005435.md b/_corpus_v3_items/cid005435.md new file mode 100644 index 000000000..adccecd7d --- /dev/null +++ b/_corpus_v3_items/cid005435.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005435/ +title: yoneda_code_roundtrip (theorem) +cid: cid005435 +primary_alias: FTH0173 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L14] Pre-Yoneda embedding of the identity is recovered by Code.\n code_extract(preyoneda(id), k, x) =\ + \ preyoneda(id)(x, k)\n Both sides equal reduce(x, k). -/" +aliases: +- FTH0173 +- yoneda_code_roundtrip +- yoneda-code-roundtrip +- TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_code_roundtrip +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001341 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: yoneda_code_roundtrip + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L14] Pre-Yoneda embedding of the identity is recovered by Code.\n code_extract(preyoneda(id), k,\ + \ x) = preyoneda(id)(x, k)\n Both sides equal reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005436.md b/_corpus_v3_items/cid005436.md new file mode 100644 index 000000000..0b92b8ecb --- /dev/null +++ b/_corpus_v3_items/cid005436.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005436/ +title: omega_germ_tower_coherent (theorem) +cid: cid005436 +primary_alias: FTH0174 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T39] Tower coherence of the pre-Yoneda identity embedding:\n reduce(preyoneda(id, x, l), k) = preyoneda(id,\ + \ x, k) for k <= l.\n This is reduction_compat in disguise. -/" +aliases: +- FTH0174 +- omega_germ_tower_coherent +- omega-germ-tower-coherent +- TauLib.BookII.CentralTheorem.YonedaApplied::omega_germ_tower_coherent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: omega_germ_tower_coherent + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T39] Tower coherence of the pre-Yoneda identity embedding:\n reduce(preyoneda(id, x, l), k) = preyoneda(id,\ + \ x, k) for k <= l.\n This is reduction_compat in disguise. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005437.md b/_corpus_v3_items/cid005437.md new file mode 100644 index 000000000..85bfd06dd --- /dev/null +++ b/_corpus_v3_items/cid005437.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005437/ +title: germ_reconstructs_identity (theorem) +cid: cid005437 +primary_alias: FTH0175 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T39] Decode of the identity code table reconstructs reduce:\n decode_reconstruct(id_table, k, x) = reduce(x,\ + \ k).\n This is the germ-to-holomorphic direction. -/" +aliases: +- FTH0175 +- germ_reconstructs_identity +- germ-reconstructs-identity +- TauLib.BookII.CentralTheorem.YonedaApplied::germ_reconstructs_identity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: germ_reconstructs_identity + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T39] Decode of the identity code table reconstructs reduce:\n decode_reconstruct(id_table, k, x)\ + \ = reduce(x, k).\n This is the germ-to-holomorphic direction. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005438.md b/_corpus_v3_items/cid005438.md new file mode 100644 index 000000000..2fc421c28 --- /dev/null +++ b/_corpus_v3_items/cid005438.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005438/ +title: germ_idempotent_supported (theorem) +cid: cid005438 +primary_alias: FTH0176 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T39] The identity omega-germ is idempotent-supported:\n e_plus * interior_bipolar(p) + e_minus * interior_bipolar(p)\ + \ = interior_bipolar(p).\n This is the decompose_recovery theorem applied pointwise. -/" +aliases: +- FTH0176 +- germ_idempotent_supported +- germ-idempotent-supported +- TauLib.BookII.CentralTheorem.YonedaApplied::germ_idempotent_supported +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001400 +- predicate: part_of + object: cid005137 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005137 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.CentralTheorem.YonedaApplied + declaration: germ_idempotent_supported + module_path: TauLib/BookII/CentralTheorem/YonedaApplied.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T39] The identity omega-germ is idempotent-supported:\n e_plus * interior_bipolar(p) + e_minus *\ + \ interior_bipolar(p) = interior_bipolar(p).\n This is the decompose_recovery theorem applied pointwise. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005439.md b/_corpus_v3_items/cid005439.md new file mode 100644 index 000000000..5f01cde65 --- /dev/null +++ b/_corpus_v3_items/cid005439.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005439/ +title: proto_2 (theorem) +cid: cid005439 +primary_alias: FTH0177 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Proto-rational specific examples [II.D65]' +aliases: +- FTH0177 +- proto_2 +- proto-2 +- TauLib.BookII.Closure.BSDbridge::proto_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_2 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Proto-rational specific examples [II.D65]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005440.md b/_corpus_v3_items/cid005440.md new file mode 100644 index 000000000..d39703440 --- /dev/null +++ b/_corpus_v3_items/cid005440.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005440/ +title: not_proto_0 (theorem) +cid: cid005440 +primary_alias: FTH0178 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Non-proto-rational [II.D65] +aliases: +- FTH0178 +- not_proto_0 +- not-proto-0 +- TauLib.BookII.Closure.BSDbridge::not_proto_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: not_proto_0 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Non-proto-rational [II.D65] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005441.md b/_corpus_v3_items/cid005441.md new file mode 100644 index 000000000..642d49ecc --- /dev/null +++ b/_corpus_v3_items/cid005441.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005441/ +title: stage_2 (theorem) +cid: cid005441 +primary_alias: FTH0179 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Stage determination [II.D65] +aliases: +- FTH0179 +- stage_2 +- stage-2 +- TauLib.BookII.Closure.BSDbridge::stage_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: stage_2 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Stage determination [II.D65] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005442.md b/_corpus_v3_items/cid005442.md new file mode 100644 index 000000000..c7ce6e9f3 --- /dev/null +++ b/_corpus_v3_items/cid005442.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005442/ +title: proto_examples (theorem) +cid: cid005442 +primary_alias: FTH0180 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Proto-rational examples [II.D65] +aliases: +- FTH0180 +- proto_examples +- proto-examples +- TauLib.BookII.Closure.BSDbridge::proto_examples +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_examples + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Proto-rational examples [II.D65] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005443.md b/_corpus_v3_items/cid005443.md new file mode 100644 index 000000000..4bc45928e --- /dev/null +++ b/_corpus_v3_items/cid005443.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005443/ +title: proto_range_30 (theorem) +cid: cid005443 +primary_alias: FTH0181 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Range check [II.D65] +aliases: +- FTH0181 +- proto_range_30 +- proto-range-30 +- TauLib.BookII.Closure.BSDbridge::proto_range_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_range_30 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Range check [II.D65] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005444.md b/_corpus_v3_items/cid005444.md new file mode 100644 index 000000000..d019ec896 --- /dev/null +++ b/_corpus_v3_items/cid005444.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005444/ +title: proto_count_check (theorem) +cid: cid005444 +primary_alias: FTH0182 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Count checks [II.D65] +aliases: +- FTH0182 +- proto_count_check +- proto-count-check +- TauLib.BookII.Closure.BSDbridge::proto_count_check +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_count_check + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Count checks [II.D65] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005445.md b/_corpus_v3_items/cid005445.md new file mode 100644 index 000000000..e3f9764a9 --- /dev/null +++ b/_corpus_v3_items/cid005445.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005445/ +title: finite_is_proto_rational_2 (theorem) +cid: cid005445 +primary_alias: FTH0183 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D65] Every finite x > 1 is proto-rational at a sufficiently large stage.\n If P_k > x, then reduce(x,\ + \ k) = x % P_k = x. Verified for x = 2. -/" +aliases: +- FTH0183 +- finite_is_proto_rational_2 +- finite-is-proto-rational-2 +- TauLib.BookII.Closure.BSDbridge::finite_is_proto_rational_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: finite_is_proto_rational_2 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D65] Every finite x > 1 is proto-rational at a sufficiently large stage.\n If P_k > x, then reduce(x,\ + \ k) = x % P_k = x. Verified for x = 2. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005446.md b/_corpus_v3_items/cid005446.md new file mode 100644 index 000000000..414a16c67 --- /dev/null +++ b/_corpus_v3_items/cid005446.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005446/ +title: proto_at_stage (theorem) +cid: cid005446 +primary_alias: FTH0184 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D65] Proto-rationality is stable: if x < P_k, then reduce(x, k) = x.\n This follows from x % P_k = x\ + \ when x < P_k. -/" +aliases: +- FTH0184 +- proto_at_stage +- proto-at-stage +- TauLib.BookII.Closure.BSDbridge::proto_at_stage +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_at_stage + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D65] Proto-rationality is stable: if x < P_k, then reduce(x, k) = x.\n This follows from x % P_k\ + \ = x when x < P_k. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005447.md b/_corpus_v3_items/cid005447.md new file mode 100644 index 000000000..26ab7848d --- /dev/null +++ b/_corpus_v3_items/cid005447.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005447/ +title: proto_abcd_roundtrip_2 (theorem) +cid: cid005447 +primary_alias: FTH0185 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D65] The ABCD chart of a proto-rational point round-trips.\n to_tau_idx(proto_rational_abcd(x)) = x.\ + \ -/" +aliases: +- FTH0185 +- proto_abcd_roundtrip_2 +- proto-abcd-roundtrip-2 +- TauLib.BookII.Closure.BSDbridge::proto_abcd_roundtrip_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001300 +- predicate: part_of + object: cid005138 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005138 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.BSDbridge + declaration: proto_abcd_roundtrip_2 + module_path: TauLib/BookII/Closure/BSDbridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D65] The ABCD chart of a proto-rational point round-trips.\n to_tau_idx(proto_rational_abcd(x)) =\ + \ x. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005448.md b/_corpus_v3_items/cid005448.md new file mode 100644 index 000000000..3e057b527 --- /dev/null +++ b/_corpus_v3_items/cid005448.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005448/ +title: flat_connection_compatible_2 (theorem) +cid: cid005448 +primary_alias: FTH0186 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D78] Flat connection is tower-compatible at stages 1-2. -/ +aliases: +- FTH0186 +- flat_connection_compatible_2 +- flat-connection-compatible-2 +- TauLib.BookII.Closure.Connection::flat_connection_compatible_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001312 +- predicate: part_of + object: cid005139 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005139 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Connection + declaration: flat_connection_compatible_2 + module_path: TauLib/BookII/Closure/Connection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D78] Flat connection is tower-compatible at stages 1-2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005449.md b/_corpus_v3_items/cid005449.md new file mode 100644 index 000000000..8cd9ef200 --- /dev/null +++ b/_corpus_v3_items/cid005449.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005449/ +title: flat_connection_flat_2 (theorem) +cid: cid005449 +primary_alias: FTH0187 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T50] Flat connection is flat at stages 1-2. -/ +aliases: +- FTH0187 +- flat_connection_flat_2 +- flat-connection-flat-2 +- TauLib.BookII.Closure.Connection::flat_connection_flat_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001411 +- predicate: part_of + object: cid005139 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005139 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Connection + declaration: flat_connection_flat_2 + module_path: TauLib/BookII/Closure/Connection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T50] Flat connection is flat at stages 1-2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005450.md b/_corpus_v3_items/cid005450.md new file mode 100644 index 000000000..5de2c5edd --- /dev/null +++ b/_corpus_v3_items/cid005450.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005450/ +title: holonomy_trivial_1 (theorem) +cid: cid005450 +primary_alias: FTH0188 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P16] Holonomy is trivial at stage 1. -/ +aliases: +- FTH0188 +- holonomy_trivial_1 +- holonomy-trivial-1 +- TauLib.BookII.Closure.Connection::holonomy_trivial_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001357 +- predicate: part_of + object: cid005139 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005139 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Connection + declaration: holonomy_trivial_1 + module_path: TauLib/BookII/Closure/Connection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P16] Holonomy is trivial at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005451.md b/_corpus_v3_items/cid005451.md new file mode 100644 index 000000000..ffda031b4 --- /dev/null +++ b/_corpus_v3_items/cid005451.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005451/ +title: holonomy_trivial_2 (theorem) +cid: cid005451 +primary_alias: FTH0189 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P16] Holonomy is trivial at stage 2. -/ +aliases: +- FTH0189 +- holonomy_trivial_2 +- holonomy-trivial-2 +- TauLib.BookII.Closure.Connection::holonomy_trivial_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001357 +- predicate: part_of + object: cid005139 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005139 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Connection + declaration: holonomy_trivial_2 + module_path: TauLib/BookII/Closure/Connection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P16] Holonomy is trivial at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005452.md b/_corpus_v3_items/cid005452.md new file mode 100644 index 000000000..191397ac8 --- /dev/null +++ b/_corpus_v3_items/cid005452.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005452/ +title: flat_curvature_vanishes_1 (theorem) +cid: cid005452 +primary_alias: FTH0190 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T51] Flat curvature vanishes at stage 1. -/ +aliases: +- FTH0190 +- flat_curvature_vanishes_1 +- flat-curvature-vanishes-1 +- TauLib.BookII.Closure.Curvature::flat_curvature_vanishes_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001412 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: flat_curvature_vanishes_1 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T51] Flat curvature vanishes at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005453.md b/_corpus_v3_items/cid005453.md new file mode 100644 index 000000000..7a345f08b --- /dev/null +++ b/_corpus_v3_items/cid005453.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005453/ +title: flat_curvature_vanishes_2 (theorem) +cid: cid005453 +primary_alias: FTH0191 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T51] Flat curvature vanishes at stage 2. -/ +aliases: +- FTH0191 +- flat_curvature_vanishes_2 +- flat-curvature-vanishes-2 +- TauLib.BookII.Closure.Curvature::flat_curvature_vanishes_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001412 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: flat_curvature_vanishes_2 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T51] Flat curvature vanishes at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005454.md b/_corpus_v3_items/cid005454.md new file mode 100644 index 000000000..edbb192e3 --- /dev/null +++ b/_corpus_v3_items/cid005454.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005454/ +title: geodesic_correct_1 (theorem) +cid: cid005454 +primary_alias: FTH0192 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D81] Geodesics are correct at stage 1. -/ +aliases: +- FTH0192 +- geodesic_correct_1 +- geodesic-correct-1 +- TauLib.BookII.Closure.Curvature::geodesic_correct_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001315 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: geodesic_correct_1 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D81] Geodesics are correct at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005455.md b/_corpus_v3_items/cid005455.md new file mode 100644 index 000000000..cac63878c --- /dev/null +++ b/_corpus_v3_items/cid005455.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005455/ +title: geodesic_correct_2 (theorem) +cid: cid005455 +primary_alias: FTH0193 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D81] Geodesics are correct at stage 2. -/ +aliases: +- FTH0193 +- geodesic_correct_2 +- geodesic-correct-2 +- TauLib.BookII.Closure.Curvature::geodesic_correct_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001315 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: geodesic_correct_2 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D81] Geodesics are correct at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005456.md b/_corpus_v3_items/cid005456.md new file mode 100644 index 000000000..c3523a0ef --- /dev/null +++ b/_corpus_v3_items/cid005456.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005456/ +title: geodesic_complete_1 (theorem) +cid: cid005456 +primary_alias: FTH0194 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P17] Geodesic completeness at stage 1. -/ +aliases: +- FTH0194 +- geodesic_complete_1 +- geodesic-complete-1 +- TauLib.BookII.Closure.Curvature::geodesic_complete_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001358 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: geodesic_complete_1 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P17] Geodesic completeness at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005457.md b/_corpus_v3_items/cid005457.md new file mode 100644 index 000000000..c841a3ace --- /dev/null +++ b/_corpus_v3_items/cid005457.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005457/ +title: geodesic_complete_2 (theorem) +cid: cid005457 +primary_alias: FTH0195 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P17] Geodesic completeness at stage 2. -/ +aliases: +- FTH0195 +- geodesic_complete_2 +- geodesic-complete-2 +- TauLib.BookII.Closure.Curvature::geodesic_complete_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001358 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: geodesic_complete_2 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P17] Geodesic completeness at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005458.md b/_corpus_v3_items/cid005458.md new file mode 100644 index 000000000..d8250754b --- /dev/null +++ b/_corpus_v3_items/cid005458.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005458/ +title: lemniscate_holonomy_1 (theorem) +cid: cid005458 +primary_alias: FTH0196 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T52] Lemniscate holonomy has order M_k at stage 1. -/ +aliases: +- FTH0196 +- lemniscate_holonomy_1 +- lemniscate-holonomy-1 +- TauLib.BookII.Closure.Curvature::lemniscate_holonomy_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001413 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: lemniscate_holonomy_1 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T52] Lemniscate holonomy has order M_k at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005459.md b/_corpus_v3_items/cid005459.md new file mode 100644 index 000000000..73a440b7f --- /dev/null +++ b/_corpus_v3_items/cid005459.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005459/ +title: lemniscate_holonomy_2 (theorem) +cid: cid005459 +primary_alias: FTH0197 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T52] Lemniscate holonomy has order M_k at stage 2. -/ +aliases: +- FTH0197 +- lemniscate_holonomy_2 +- lemniscate-holonomy-2 +- TauLib.BookII.Closure.Curvature::lemniscate_holonomy_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001413 +- predicate: part_of + object: cid005140 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005140 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.Curvature + declaration: lemniscate_holonomy_2 + module_path: TauLib/BookII/Closure/Curvature.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T52] Lemniscate holonomy has order M_k at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005460.md b/_corpus_v3_items/cid005460.md new file mode 100644 index 000000000..488db8841 --- /dev/null +++ b/_corpus_v3_items/cid005460.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005460/ +title: diffgeo_not_earned (theorem) +cid: cid005460 +primary_alias: FTH0198 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Diff-geo not earned [II.R21]' +aliases: +- FTH0198 +- diffgeo_not_earned +- diffgeo-not-earned +- TauLib.BookII.Closure.DiffGeoAgenda::diffgeo_not_earned +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: diffgeo_not_earned + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Diff-geo not earned [II.R21]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005461.md b/_corpus_v3_items/cid005461.md new file mode 100644 index 000000000..54f9d9552 --- /dev/null +++ b/_corpus_v3_items/cid005461.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005461/ +title: e1_complete_b3_10_3_3 (theorem) +cid: cid005461 +primary_alias: FTH0199 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E1 complete [II.R22] +aliases: +- FTH0199 +- e1_complete_b3_10_3_3 +- e1-complete-b3-10-3-3 +- TauLib.BookII.Closure.DiffGeoAgenda::e1_complete_b3_10_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: e1_complete_b3_10_3_3 + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E1 complete [II.R22] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005462.md b/_corpus_v3_items/cid005462.md new file mode 100644 index 000000000..843f5725e --- /dev/null +++ b/_corpus_v3_items/cid005462.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005462/ +title: book3_prereq_3_15_3 (theorem) +cid: cid005462 +primary_alias: FTH0200 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Book III prerequisites [II.R22] +aliases: +- FTH0200 +- book3_prereq_3_15_3 +- book3-prereq-3-15-3 +- TauLib.BookII.Closure.DiffGeoAgenda::book3_prereq_3_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: book3_prereq_3_15_3 + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Book III prerequisites [II.R22] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005463.md b/_corpus_v3_items/cid005463.md new file mode 100644 index 000000000..2a6b31c7b --- /dev/null +++ b/_corpus_v3_items/cid005463.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005463/ +title: connection_not_earned (theorem) +cid: cid005463 +primary_alias: FTH0201 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [II.R21] Connection is not earned: the placeholder records false. -/' +aliases: +- FTH0201 +- connection_not_earned +- connection-not-earned +- TauLib.BookII.Closure.DiffGeoAgenda::connection_not_earned +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: connection_not_earned + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [II.R21] Connection is not earned: the placeholder records false. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005464.md b/_corpus_v3_items/cid005464.md new file mode 100644 index 000000000..bd39dc6d7 --- /dev/null +++ b/_corpus_v3_items/cid005464.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005464/ +title: curvature_not_earned (theorem) +cid: cid005464 +primary_alias: FTH0202 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [II.R21] Curvature is not earned: the placeholder records false. -/' +aliases: +- FTH0202 +- curvature_not_earned +- curvature-not-earned +- TauLib.BookII.Closure.DiffGeoAgenda::curvature_not_earned +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: curvature_not_earned + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [II.R21] Curvature is not earned: the placeholder records false. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005465.md b/_corpus_v3_items/cid005465.md new file mode 100644 index 000000000..a609c0d0d --- /dev/null +++ b/_corpus_v3_items/cid005465.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005465/ +title: holonomy_not_earned (theorem) +cid: cid005465 +primary_alias: FTH0203 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [II.R21] Holonomy is not earned: the placeholder records false. -/' +aliases: +- FTH0203 +- holonomy_not_earned +- holonomy-not-earned +- TauLib.BookII.Closure.DiffGeoAgenda::holonomy_not_earned +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: holonomy_not_earned + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [II.R21] Holonomy is not earned: the placeholder records false. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005466.md b/_corpus_v3_items/cid005466.md new file mode 100644 index 000000000..1a0bbdc4e --- /dev/null +++ b/_corpus_v3_items/cid005466.md @@ -0,0 +1,67 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005466/ +title: e1_includes_self_enrichment (theorem) +cid: cid005466 +primary_alias: FTH0204 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.R22] E1 completeness implies self-enrichment witness holds.\n This is structural: e1_layer_check includes\ + \ self-enrichment.\n If the full layer check passes, then in particular the\n self-enrichment component is true. -/" +aliases: +- FTH0204 +- e1_includes_self_enrichment +- e1-includes-self-enrichment +- TauLib.BookII.Closure.DiffGeoAgenda::e1_includes_self_enrichment +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005141 +- predicate: part_of + object: cid005141 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005141 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.DiffGeoAgenda + declaration: e1_includes_self_enrichment + module_path: TauLib/BookII/Closure/DiffGeoAgenda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.R22] E1 completeness implies self-enrichment witness holds.\n This is structural: e1_layer_check\ + \ includes self-enrichment.\n If the full layer check passes, then in particular the\n self-enrichment component is\ + \ true. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005467.md b/_corpus_v3_items/cid005467.md new file mode 100644 index 000000000..a8126b9f6 --- /dev/null +++ b/_corpus_v3_items/cid005467.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005467/ +title: full_export_3_15_3 (theorem) +cid: cid005467 +primary_alias: FTH0205 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Full export check [II.D66]' +aliases: +- FTH0205 +- full_export_3_15_3 +- full-export-3-15-3 +- TauLib.BookII.Closure.ForwardBook3::full_export_3_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: full_export_3_15_3 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Full export check [II.D66]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005468.md b/_corpus_v3_items/cid005468.md new file mode 100644 index 000000000..677b6c923 --- /dev/null +++ b/_corpus_v3_items/cid005468.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005468/ +title: all_six_3_15_3 (theorem) +cid: cid005468 +primary_alias: FTH0206 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- All six components [II.D66] +aliases: +- FTH0206 +- all_six_3_15_3 +- all-six-3-15-3 +- TauLib.BookII.Closure.ForwardBook3::all_six_3_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: all_six_3_15_3 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- All six components [II.D66] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005469.md b/_corpus_v3_items/cid005469.md new file mode 100644 index 000000000..ad044d74e --- /dev/null +++ b/_corpus_v3_items/cid005469.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005469/ +title: book3_e1_3_15_3 (theorem) +cid: cid005469 +primary_alias: FTH0207 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Book III starts at E1 [II.D66] +aliases: +- FTH0207 +- book3_e1_3_15_3 +- book3-e1-3-15-3 +- TauLib.BookII.Closure.ForwardBook3::book3_e1_3_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: book3_e1_3_15_3 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Book III starts at E1 [II.D66] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005470.md b/_corpus_v3_items/cid005470.md new file mode 100644 index 000000000..683ed17f7 --- /dev/null +++ b/_corpus_v3_items/cid005470.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005470/ +title: complete_means_six (theorem) +cid: cid005470 +primary_alias: FTH0208 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D66] A complete export package has all 6 components.\n This is the structural statement that completeness\ + \ implies count = 6. -/" +aliases: +- FTH0208 +- complete_means_six +- complete-means-six +- TauLib.BookII.Closure.ForwardBook3::complete_means_six +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: complete_means_six + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D66] A complete export package has all 6 components.\n This is the structural statement that completeness\ + \ implies count = 6. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005471.md b/_corpus_v3_items/cid005471.md new file mode 100644 index 000000000..8108657df --- /dev/null +++ b/_corpus_v3_items/cid005471.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005471/ +title: export_implies_e1 (theorem) +cid: cid005471 +primary_alias: FTH0209 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D66] The export package is self-consistent: if the full check passes,\n then each component individually\ + \ passes. -/" +aliases: +- FTH0209 +- export_implies_e1 +- export-implies-e1 +- TauLib.BookII.Closure.ForwardBook3::export_implies_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: export_implies_e1 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D66] The export package is self-consistent: if the full check passes,\n then each component individually\ + \ passes. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005472.md b/_corpus_v3_items/cid005472.md new file mode 100644 index 000000000..0e7d03f85 --- /dev/null +++ b/_corpus_v3_items/cid005472.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005472/ +title: export_implies_central (theorem) +cid: cid005472 +primary_alias: FTH0210 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D66] The export package is self-consistent: full check implies\n Central Theorem verified. -/" +aliases: +- FTH0210 +- export_implies_central +- export-implies-central +- TauLib.BookII.Closure.ForwardBook3::export_implies_central +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: export_implies_central + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D66] The export package is self-consistent: full check implies\n Central Theorem verified. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005473.md b/_corpus_v3_items/cid005473.md new file mode 100644 index 000000000..c81ac5388 --- /dev/null +++ b/_corpus_v3_items/cid005473.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005473/ +title: e1_gt_e0 (theorem) +cid: cid005473 +primary_alias: FTH0211 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D66] E0 and E1 are distinct enrichment levels.\n Book III starts at E1, not E0. -/" +aliases: +- FTH0211 +- e1_gt_e0 +- e1-gt-e0 +- TauLib.BookII.Closure.ForwardBook3::e1_gt_e0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005142 +- predicate: part_of + object: cid005142 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005142 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.ForwardBook3 + declaration: e1_gt_e0 + module_path: TauLib/BookII/Closure/ForwardBook3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D66] E0 and E1 are distinct enrichment levels.\n Book III starts at E1, not E0. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005474.md b/_corpus_v3_items/cid005474.md new file mode 100644 index 000000000..1eb49fcb3 --- /dev/null +++ b/_corpus_v3_items/cid005474.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005474/ +title: geometric_bisquare_3_15 (theorem) +cid: cid005474 +primary_alias: FTH0212 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Geometric Bi-Square Theorem [II.T49]' +aliases: +- FTH0212 +- geometric_bisquare_3_15 +- geometric-bisquare-3-15 +- TauLib.BookII.Closure.GeometricBiSquare::geometric_bisquare_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001410 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: geometric_bisquare_3_15 + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Geometric Bi-Square Theorem [II.T49]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005475.md b/_corpus_v3_items/cid005475.md new file mode 100644 index 000000000..ffb7b208e --- /dev/null +++ b/_corpus_v3_items/cid005475.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005475/ +title: geometric_all_eight (theorem) +cid: cid005475 +primary_alias: FTH0213 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- All eight components [II.T49] +aliases: +- FTH0213 +- geometric_all_eight +- geometric-all-eight +- TauLib.BookII.Closure.GeometricBiSquare::geometric_all_eight +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001410 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: geometric_all_eight + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- All eight components [II.T49] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005476.md b/_corpus_v3_items/cid005476.md new file mode 100644 index 000000000..93ded30df --- /dev/null +++ b/_corpus_v3_items/cid005476.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005476/ +title: geo_topology (theorem) +cid: cid005476 +primary_alias: FTH0214 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Individual geometric components [II.D77] +aliases: +- FTH0214 +- geo_topology +- geo-topology +- TauLib.BookII.Closure.GeometricBiSquare::geo_topology +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001311 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: geo_topology + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Individual geometric components [II.D77] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005477.md b/_corpus_v3_items/cid005477.md new file mode 100644 index 000000000..c81b9c9d1 --- /dev/null +++ b/_corpus_v3_items/cid005477.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005477/ +title: audit_3_15_3 (theorem) +cid: cid005477 +primary_alias: FTH0215 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Audit [II.R33] +aliases: +- FTH0215 +- audit_3_15_3 +- audit-3-15-3 +- TauLib.BookII.Closure.GeometricBiSquare::audit_3_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005143 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: audit_3_15_3 + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Audit [II.R33] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005478.md b/_corpus_v3_items/cid005478.md new file mode 100644 index 000000000..914e1b067 --- /dev/null +++ b/_corpus_v3_items/cid005478.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005478/ +title: scaling_chain_valid (theorem) +cid: cid005478 +primary_alias: FTH0216 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Scaling chain [II.R34] +aliases: +- FTH0216 +- scaling_chain_valid +- scaling-chain-valid +- TauLib.BookII.Closure.GeometricBiSquare::scaling_chain_valid +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005143 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: scaling_chain_valid + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Scaling chain [II.R34] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005479.md b/_corpus_v3_items/cid005479.md new file mode 100644 index 000000000..9aa780377 --- /dev/null +++ b/_corpus_v3_items/cid005479.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005479/ +title: complete_means_eight (theorem) +cid: cid005479 +primary_alias: FTH0217 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T49] A complete geometric bi-square has all 8 components.\n This is the structural statement that completeness\ + \ implies count = 8. -/" +aliases: +- FTH0217 +- complete_means_eight +- complete-means-eight +- TauLib.BookII.Closure.GeometricBiSquare::complete_means_eight +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001410 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: complete_means_eight + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T49] A complete geometric bi-square has all 8 components.\n This is the structural statement that\ + \ completeness implies count = 8. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005480.md b/_corpus_v3_items/cid005480.md new file mode 100644 index 000000000..5294c5008 --- /dev/null +++ b/_corpus_v3_items/cid005480.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005480/ +title: geometric_implies_central (theorem) +cid: cid005480 +primary_alias: FTH0218 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T49] The geometric bi-square implies the Central Theorem is earned.\n If the full check passes, the central_theorem_earned\ + \ field is true. -/" +aliases: +- FTH0218 +- geometric_implies_central +- geometric-implies-central +- TauLib.BookII.Closure.GeometricBiSquare::geometric_implies_central +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001410 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: geometric_implies_central + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T49] The geometric bi-square implies the Central Theorem is earned.\n If the full check passes, the\ + \ central_theorem_earned field is true. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005481.md b/_corpus_v3_items/cid005481.md new file mode 100644 index 000000000..17f40d330 --- /dev/null +++ b/_corpus_v3_items/cid005481.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005481/ +title: e0_ne_e1 (theorem) +cid: cid005481 +primary_alias: FTH0219 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.R34] E0 and E1 are distinct scaling levels. -/ +aliases: +- FTH0219 +- e0_ne_e1 +- e0-ne-e1 +- TauLib.BookII.Closure.GeometricBiSquare::e0_ne_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005143 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: e0_ne_e1 + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.R34] E0 and E1 are distinct scaling levels. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005482.md b/_corpus_v3_items/cid005482.md new file mode 100644 index 000000000..7bc2a68cd --- /dev/null +++ b/_corpus_v3_items/cid005482.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005482/ +title: e1_ne_e2 (theorem) +cid: cid005482 +primary_alias: FTH0220 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.R34] E1 and E2 are distinct scaling levels. -/ +aliases: +- FTH0220 +- e1_ne_e2 +- e1-ne-e2 +- TauLib.BookII.Closure.GeometricBiSquare::e1_ne_e2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005143 +- predicate: part_of + object: cid005143 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005143 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.GeometricBiSquare + declaration: e1_ne_e2 + module_path: TauLib/BookII/Closure/GeometricBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.R34] E1 and E2 are distinct scaling levels. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005483.md b/_corpus_v3_items/cid005483.md new file mode 100644 index 000000000..f5ab82c78 --- /dev/null +++ b/_corpus_v3_items/cid005483.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005483/ +title: atlas_chart_3 (theorem) +cid: cid005483 +primary_alias: FTH0221 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Atlas chart [II.D63]' +aliases: +- FTH0221 +- atlas_chart_3 +- atlas-chart-3 +- TauLib.BookII.Closure.TauManifold::atlas_chart_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001298 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: atlas_chart_3 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Atlas chart [II.D63]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005484.md b/_corpus_v3_items/cid005484.md new file mode 100644 index 000000000..7fc6abe8e --- /dev/null +++ b/_corpus_v3_items/cid005484.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005484/ +title: atlas_transition_3_30 (theorem) +cid: cid005484 +primary_alias: FTH0222 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Atlas transitions [II.D63] +aliases: +- FTH0222 +- atlas_transition_3_30 +- atlas-transition-3-30 +- TauLib.BookII.Closure.TauManifold::atlas_transition_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001298 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: atlas_transition_3_30 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Atlas transitions [II.D63] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005485.md b/_corpus_v3_items/cid005485.md new file mode 100644 index 000000000..26cbb029d --- /dev/null +++ b/_corpus_v3_items/cid005485.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005485/ +title: d_const_zero_3_15 (theorem) +cid: cid005485 +primary_alias: FTH0223 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- d_tau of constant is zero [II.D64] +aliases: +- FTH0223 +- d_const_zero_3_15 +- d-const-zero-3-15 +- TauLib.BookII.Closure.TauManifold::d_const_zero_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001299 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: d_const_zero_3_15 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- d_tau of constant is zero [II.D64] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005486.md b/_corpus_v3_items/cid005486.md new file mode 100644 index 000000000..cffe158eb --- /dev/null +++ b/_corpus_v3_items/cid005486.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005486/ +title: d_sq_tower_3_10 (theorem) +cid: cid005486 +primary_alias: FTH0224 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- d^2 = 0 telescoping [II.D64] +aliases: +- FTH0224 +- d_sq_tower_3_10 +- d-sq-tower-3-10 +- TauLib.BookII.Closure.TauManifold::d_sq_tower_3_10 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001299 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: d_sq_tower_3_10 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- d^2 = 0 telescoping [II.D64] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005487.md b/_corpus_v3_items/cid005487.md new file mode 100644 index 000000000..be65f6d73 --- /dev/null +++ b/_corpus_v3_items/cid005487.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005487/ +title: tau_manifold_3_15 (theorem) +cid: cid005487 +primary_alias: FTH0225 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- tau-Manifold [II.D62] +aliases: +- FTH0225 +- tau_manifold_3_15 +- tau-manifold-3-15 +- TauLib.BookII.Closure.TauManifold::tau_manifold_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001297 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: tau_manifold_3_15 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- tau-Manifold [II.D62] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005488.md b/_corpus_v3_items/cid005488.md new file mode 100644 index 000000000..819463a50 --- /dev/null +++ b/_corpus_v3_items/cid005488.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005488/ +title: tau3_manifold_3_15 (theorem) +cid: cid005488 +primary_alias: FTH0226 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- tau^3 is a tau-manifold [II.P15] +aliases: +- FTH0226 +- tau3_manifold_3_15 +- tau3-manifold-3-15 +- TauLib.BookII.Closure.TauManifold::tau3_manifold_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001356 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: tau3_manifold_3_15 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- tau^3 is a tau-manifold [II.P15] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005489.md b/_corpus_v3_items/cid005489.md new file mode 100644 index 000000000..7af750a3f --- /dev/null +++ b/_corpus_v3_items/cid005489.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005489/ +title: abcd_atlas_roundtrip_7 (theorem) +cid: cid005489 +primary_alias: FTH0227 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D63] The ABCD chart round-trips for specific values.\n to_tau_idx(from_tau_idx(x)) = x. Verified computationally.\ + \ -/" +aliases: +- FTH0227 +- abcd_atlas_roundtrip_7 +- abcd-atlas-roundtrip-7 +- TauLib.BookII.Closure.TauManifold::abcd_atlas_roundtrip_7 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001298 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: abcd_atlas_roundtrip_7 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D63] The ABCD chart round-trips for specific values.\n to_tau_idx(from_tau_idx(x)) = x. Verified\ + \ computationally. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005490.md b/_corpus_v3_items/cid005490.md new file mode 100644 index 000000000..46c35be41 --- /dev/null +++ b/_corpus_v3_items/cid005490.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005490/ +title: chart_transition_7_2 (theorem) +cid: cid005490 +primary_alias: FTH0228 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D63] Chart transitions are reduce-compatible:\n reduce(to_tau_idx(from_tau_idx(x)), k) = reduce(x, k).\n\ + \ Verified for specific instances. -/" +aliases: +- FTH0228 +- chart_transition_7_2 +- chart-transition-7-2 +- TauLib.BookII.Closure.TauManifold::chart_transition_7_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001298 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: chart_transition_7_2 + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D63] Chart transitions are reduce-compatible:\n reduce(to_tau_idx(from_tau_idx(x)), k) = reduce(x,\ + \ k).\n Verified for specific instances. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005491.md b/_corpus_v3_items/cid005491.md new file mode 100644 index 000000000..5015cb17f --- /dev/null +++ b/_corpus_v3_items/cid005491.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005491/ +title: d_tau_constant (theorem) +cid: cid005491 +primary_alias: FTH0229 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D64] d_tau of a constant 0-form is zero.\n tau_exterior_derivative(const_c, x, k) = c - c = 0. -/" +aliases: +- FTH0229 +- d_tau_constant +- d-tau-constant +- TauLib.BookII.Closure.TauManifold::d_tau_constant +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001299 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: d_tau_constant + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D64] d_tau of a constant 0-form is zero.\n tau_exterior_derivative(const_c, x, k) = c - c = 0. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005492.md b/_corpus_v3_items/cid005492.md new file mode 100644 index 000000000..05b2a9a4b --- /dev/null +++ b/_corpus_v3_items/cid005492.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005492/ +title: d_tau_zero (theorem) +cid: cid005492 +primary_alias: FTH0230 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [II.D64] Telescoping: d_tau of the zero function is zero. -/' +aliases: +- FTH0230 +- d_tau_zero +- d-tau-zero +- TauLib.BookII.Closure.TauManifold::d_tau_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001299 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: d_tau_zero + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [II.D64] Telescoping: d_tau of the zero function is zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005493.md b/_corpus_v3_items/cid005493.md new file mode 100644 index 000000000..236ed01de --- /dev/null +++ b/_corpus_v3_items/cid005493.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005493/ +title: manifold_reduce_idempotent (theorem) +cid: cid005493 +primary_alias: FTH0231 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [II.P15] tau^3 manifold: reduction is idempotent (atlas well-defined). -/' +aliases: +- FTH0231 +- manifold_reduce_idempotent +- manifold-reduce-idempotent +- TauLib.BookII.Closure.TauManifold::manifold_reduce_idempotent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001356 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: manifold_reduce_idempotent + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [II.P15] tau^3 manifold: reduction is idempotent (atlas well-defined). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005494.md b/_corpus_v3_items/cid005494.md new file mode 100644 index 000000000..3b2167300 --- /dev/null +++ b/_corpus_v3_items/cid005494.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005494/ +title: manifold_tower_coherence (theorem) +cid: cid005494 +primary_alias: FTH0232 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P15] tau^3 manifold: tower coherence gives chart transitions.\n reduce(reduce(x, l), k) = reduce(x, k)\ + \ for k <= l. -/" +aliases: +- FTH0232 +- manifold_tower_coherence +- manifold-tower-coherence +- TauLib.BookII.Closure.TauManifold::manifold_tower_coherence +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001356 +- predicate: part_of + object: cid005144 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005144 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Closure.TauManifold + declaration: manifold_tower_coherence + module_path: TauLib/BookII/Closure/TauManifold.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P15] tau^3 manifold: tower coherence gives chart transitions.\n reduce(reduce(x, l), k) = reduce(x,\ + \ k) for k <= l. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005495.md b/_corpus_v3_items/cid005495.md new file mode 100644 index 000000000..dcd83376b --- /dev/null +++ b/_corpus_v3_items/cid005495.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005495/ +title: e0_hom_3 (theorem) +cid: cid005495 +primary_alias: FTH0233 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- E0 external hom [II.D58]' +aliases: +- FTH0233 +- e0_hom_3 +- e0-hom-3 +- TauLib.BookII.Enrichment.EnrichmentLadder::e0_hom_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001293 +- predicate: part_of + object: cid005148 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005148 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + declaration: e0_hom_3 + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- E0 external hom [II.D58]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005496.md b/_corpus_v3_items/cid005496.md new file mode 100644 index 000000000..cd2c9734e --- /dev/null +++ b/_corpus_v3_items/cid005496.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005496/ +title: e1_hom_3 (theorem) +cid: cid005496 +primary_alias: FTH0234 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E1 internal hom [II.D58] +aliases: +- FTH0234 +- e1_hom_3 +- e1-hom-3 +- TauLib.BookII.Enrichment.EnrichmentLadder::e1_hom_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001293 +- predicate: part_of + object: cid005148 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005148 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + declaration: e1_hom_3 + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E1 internal hom [II.D58] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005497.md b/_corpus_v3_items/cid005497.md new file mode 100644 index 000000000..1ef3b2724 --- /dev/null +++ b/_corpus_v3_items/cid005497.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005497/ +title: transition_3 (theorem) +cid: cid005497 +primary_alias: FTH0235 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E0/E1 transition [II.D58] +aliases: +- FTH0235 +- transition_3 +- transition-3 +- TauLib.BookII.Enrichment.EnrichmentLadder::transition_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001293 +- predicate: part_of + object: cid005148 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005148 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + declaration: transition_3 + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E0/E1 transition [II.D58] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005498.md b/_corpus_v3_items/cid005498.md new file mode 100644 index 000000000..3667e6b0c --- /dev/null +++ b/_corpus_v3_items/cid005498.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005498/ +title: gap_10_3_3 (theorem) +cid: cid005498 +primary_alias: FTH0236 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Enrichment gap [II.D58] +aliases: +- FTH0236 +- gap_10_3_3 +- gap-10-3-3 +- TauLib.BookII.Enrichment.EnrichmentLadder::gap_10_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001293 +- predicate: part_of + object: cid005148 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005148 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + declaration: gap_10_3_3 + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Enrichment gap [II.D58] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005499.md b/_corpus_v3_items/cid005499.md new file mode 100644 index 000000000..22f59091a --- /dev/null +++ b/_corpus_v3_items/cid005499.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005499/ +title: full_ladder_10_3_3 (theorem) +cid: cid005499 +primary_alias: FTH0237 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full enrichment ladder [II.D58] +aliases: +- FTH0237 +- full_ladder_10_3_3 +- full-ladder-10-3-3 +- TauLib.BookII.Enrichment.EnrichmentLadder::full_ladder_10_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001293 +- predicate: part_of + object: cid005148 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005148 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.EnrichmentLadder + declaration: full_ladder_10_3_3 + module_path: TauLib/BookII/Enrichment/EnrichmentLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full enrichment ladder [II.D58] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005500.md b/_corpus_v3_items/cid005500.md new file mode 100644 index 000000000..62abe468e --- /dev/null +++ b/_corpus_v3_items/cid005500.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005500/ +title: boundary_coherence_2 (theorem) +cid: cid005500 +primary_alias: FTH0238 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T54] Tower coherence at stage 2. -/ +aliases: +- FTH0238 +- boundary_coherence_2 +- boundary-coherence-2 +- TauLib.BookII.Enrichment.Homological::boundary_coherence_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001415 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: boundary_coherence_2 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T54] Tower coherence at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005501.md b/_corpus_v3_items/cid005501.md new file mode 100644 index 000000000..e36e990c1 --- /dev/null +++ b/_corpus_v3_items/cid005501.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005501/ +title: boundary_coherence_3 (theorem) +cid: cid005501 +primary_alias: FTH0239 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T54] Tower coherence at stage 3. -/ +aliases: +- FTH0239 +- boundary_coherence_3 +- boundary-coherence-3 +- TauLib.BookII.Enrichment.Homological::boundary_coherence_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001415 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: boundary_coherence_3 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T54] Tower coherence at stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005502.md b/_corpus_v3_items/cid005502.md new file mode 100644 index 000000000..b37357fa9 --- /dev/null +++ b/_corpus_v3_items/cid005502.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005502/ +title: ses_exact_1 (theorem) +cid: cid005502 +primary_alias: FTH0240 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D85] SES exactness at stage 1. -/ +aliases: +- FTH0240 +- ses_exact_1 +- ses-exact-1 +- TauLib.BookII.Enrichment.Homological::ses_exact_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001319 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: ses_exact_1 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D85] SES exactness at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005503.md b/_corpus_v3_items/cid005503.md new file mode 100644 index 000000000..c26696411 --- /dev/null +++ b/_corpus_v3_items/cid005503.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005503/ +title: ses_exact_2 (theorem) +cid: cid005503 +primary_alias: FTH0241 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D85] SES exactness at stage 2. -/ +aliases: +- FTH0241 +- ses_exact_2 +- ses-exact-2 +- TauLib.BookII.Enrichment.Homological::ses_exact_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001319 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: ses_exact_2 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D85] SES exactness at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005504.md b/_corpus_v3_items/cid005504.md new file mode 100644 index 000000000..8b73c0ad5 --- /dev/null +++ b/_corpus_v3_items/cid005504.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005504/ +title: homology_trivial_1 (theorem) +cid: cid005504 +primary_alias: FTH0242 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D85] Homology trivial at stage 1. -/ +aliases: +- FTH0242 +- homology_trivial_1 +- homology-trivial-1 +- TauLib.BookII.Enrichment.Homological::homology_trivial_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001319 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: homology_trivial_1 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D85] Homology trivial at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005505.md b/_corpus_v3_items/cid005505.md new file mode 100644 index 000000000..8a4d6b3b4 --- /dev/null +++ b/_corpus_v3_items/cid005505.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005505/ +title: homology_trivial_2 (theorem) +cid: cid005505 +primary_alias: FTH0243 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D85] Homology trivial at stage 2. -/ +aliases: +- FTH0243 +- homology_trivial_2 +- homology-trivial-2 +- TauLib.BookII.Enrichment.Homological::homology_trivial_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001319 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: homology_trivial_2 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D85] Homology trivial at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005506.md b/_corpus_v3_items/cid005506.md new file mode 100644 index 000000000..fb1aef3a6 --- /dev/null +++ b/_corpus_v3_items/cid005506.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005506/ +title: ses_stage1 (theorem) +cid: cid005506 +primary_alias: FTH0244 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P19] Short exact sequence at stage 1. -/ +aliases: +- FTH0244 +- ses_stage1 +- ses-stage1 +- TauLib.BookII.Enrichment.Homological::ses_stage1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001360 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: ses_stage1 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P19] Short exact sequence at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005507.md b/_corpus_v3_items/cid005507.md new file mode 100644 index 000000000..0c35f79d9 --- /dev/null +++ b/_corpus_v3_items/cid005507.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005507/ +title: ses_stage2 (theorem) +cid: cid005507 +primary_alias: FTH0245 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P19] Short exact sequence at stage 2. -/ +aliases: +- FTH0245 +- ses_stage2 +- ses-stage2 +- TauLib.BookII.Enrichment.Homological::ses_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001360 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: ses_stage2 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P19] Short exact sequence at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005508.md b/_corpus_v3_items/cid005508.md new file mode 100644 index 000000000..0cd18e7d1 --- /dev/null +++ b/_corpus_v3_items/cid005508.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005508/ +title: les_stage2 (theorem) +cid: cid005508 +primary_alias: FTH0246 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P19] Long exact sequence at stage 2. -/ +aliases: +- FTH0246 +- les_stage2 +- les-stage2 +- TauLib.BookII.Enrichment.Homological::les_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001360 +- predicate: part_of + object: cid005149 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005149 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.Homological + declaration: les_stage2 + module_path: TauLib/BookII/Enrichment/Homological.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P19] Long exact sequence at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005509.md b/_corpus_v3_items/cid005509.md new file mode 100644 index 000000000..75feb3641 --- /dev/null +++ b/_corpus_v3_items/cid005509.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005509/ +title: e1_layer_10_3_3 (theorem) +cid: cid005509 +primary_alias: FTH0247 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full E1 layer [II.D57] +aliases: +- FTH0247 +- e1_layer_10_3_3 +- e1-layer-10-3-3 +- TauLib.BookII.Enrichment.SelfDescribing::e1_layer_10_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001292 +- predicate: part_of + object: cid005150 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005150 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfDescribing + declaration: e1_layer_10_3_3 + module_path: TauLib/BookII/Enrichment/SelfDescribing.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full E1 layer [II.D57] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005510.md b/_corpus_v3_items/cid005510.md new file mode 100644 index 000000000..229728f0e --- /dev/null +++ b/_corpus_v3_items/cid005510.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005510/ +title: hom_tower_8_3 (theorem) +cid: cid005510 +primary_alias: FTH0248 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Tower coherence [II.D53]' +aliases: +- FTH0248 +- hom_tower_8_3 +- hom-tower-8-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_tower_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001288 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_tower_8_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Tower coherence [II.D53]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005511.md b/_corpus_v3_items/cid005511.md new file mode 100644 index 000000000..70de70eb2 --- /dev/null +++ b/_corpus_v3_items/cid005511.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005511/ +title: self_enrich_8_3 (theorem) +cid: cid005511 +primary_alias: FTH0249 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Self-enrichment [II.D53] +aliases: +- FTH0249 +- self_enrich_8_3 +- self-enrich-8-3 +- TauLib.BookII.Enrichment.SelfEnrichment::self_enrich_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001288 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: self_enrich_8_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Self-enrichment [II.D53] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005512.md b/_corpus_v3_items/cid005512.md new file mode 100644 index 000000000..22e823f41 --- /dev/null +++ b/_corpus_v3_items/cid005512.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005512/ +title: hom_identity_3 (theorem) +cid: cid005512 +primary_alias: FTH0250 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hom object identity [II.D54] +aliases: +- FTH0250 +- hom_identity_3 +- hom-identity-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_identity_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001289 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_identity_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hom object identity [II.D54] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005513.md b/_corpus_v3_items/cid005513.md new file mode 100644 index 000000000..a395a66f0 --- /dev/null +++ b/_corpus_v3_items/cid005513.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005513/ +title: hom_constant_3 (theorem) +cid: cid005513 +primary_alias: FTH0251 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hom object constant [II.D54] +aliases: +- FTH0251 +- hom_constant_3 +- hom-constant-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_constant_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001289 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_constant_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hom object constant [II.D54] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005514.md b/_corpus_v3_items/cid005514.md new file mode 100644 index 000000000..bba0419d2 --- /dev/null +++ b/_corpus_v3_items/cid005514.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005514/ +title: hom_count_3 (theorem) +cid: cid005514 +primary_alias: FTH0252 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hom object count [II.D54] +aliases: +- FTH0252 +- hom_count_3 +- hom-count-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_count_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001289 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_count_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hom object count [II.D54] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005515.md b/_corpus_v3_items/cid005515.md new file mode 100644 index 000000000..ecef4fe14 --- /dev/null +++ b/_corpus_v3_items/cid005515.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005515/ +title: hom_bipolar_8_3 (theorem) +cid: cid005515 +primary_alias: FTH0253 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Bipolar decomposition [II.P12] +aliases: +- FTH0253 +- hom_bipolar_8_3 +- hom-bipolar-8-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001353 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_bipolar_8_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Bipolar decomposition [II.P12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005516.md b/_corpus_v3_items/cid005516.md new file mode 100644 index 000000000..008dcc9cd --- /dev/null +++ b/_corpus_v3_items/cid005516.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005516/ +title: hom_channel_8_3 (theorem) +cid: cid005516 +primary_alias: FTH0254 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Channel independence [II.P12] +aliases: +- FTH0254 +- hom_channel_8_3 +- hom-channel-8-3 +- TauLib.BookII.Enrichment.SelfEnrichment::hom_channel_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001353 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_channel_8_3 + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Channel independence [II.P12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005517.md b/_corpus_v3_items/cid005517.md new file mode 100644 index 000000000..7c195ef40 --- /dev/null +++ b/_corpus_v3_items/cid005517.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005517/ +title: hom_stage_reduce_stable (theorem) +cid: cid005517 +primary_alias: FTH0255 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D53] Formal proof: hom_stage is reduce-stable.\n reduce(hom_stage(a, b, k), k) = hom_stage(a, b, k).\n\ + \ This follows from reduce(reduce(x, k), k) = reduce(x, k). -/" +aliases: +- FTH0255 +- hom_stage_reduce_stable +- hom-stage-reduce-stable +- TauLib.BookII.Enrichment.SelfEnrichment::hom_stage_reduce_stable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001288 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_stage_reduce_stable + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D53] Formal proof: hom_stage is reduce-stable.\n reduce(hom_stage(a, b, k), k) = hom_stage(a, b,\ + \ k).\n This follows from reduce(reduce(x, k), k) = reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005518.md b/_corpus_v3_items/cid005518.md new file mode 100644 index 000000000..66edaf8a5 --- /dev/null +++ b/_corpus_v3_items/cid005518.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005518/ +title: hom_bipolar_complete (theorem) +cid: cid005518 +primary_alias: FTH0256 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P12] Formal proof: bipolar decomposition of hom outputs is complete.\n proj_plus(sp) + proj_minus(sp)\ + \ = sp, for any sector pair sp.\n This follows from e_plus * sp + e_minus * sp = (1,0)*(B,C) + (0,1)*(B,C)\n = (B,0)\ + \ + (0,C) = (B,C) = sp. -/" +aliases: +- FTH0256 +- hom_bipolar_complete +- hom-bipolar-complete +- TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001353 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_bipolar_complete + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P12] Formal proof: bipolar decomposition of hom outputs is complete.\n proj_plus(sp) + proj_minus(sp)\ + \ = sp, for any sector pair sp.\n This follows from e_plus * sp + e_minus * sp = (1,0)*(B,C) + (0,1)*(B,C)\n = (B,0)\ + \ + (0,C) =" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005519.md b/_corpus_v3_items/cid005519.md new file mode 100644 index 000000000..2f30bc996 --- /dev/null +++ b/_corpus_v3_items/cid005519.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005519/ +title: hom_bipolar_orthogonal (theorem) +cid: cid005519 +primary_alias: FTH0257 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P12] Formal proof: bipolar projections are orthogonal.\n e_plus * (e_minus * sp) = (0, 0) for any sector\ + \ pair sp. -/" +aliases: +- FTH0257 +- hom_bipolar_orthogonal +- hom-bipolar-orthogonal +- TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_orthogonal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001353 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: hom_bipolar_orthogonal + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P12] Formal proof: bipolar projections are orthogonal.\n e_plus * (e_minus * sp) = (0, 0) for any\ + \ sector pair sp. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005520.md b/_corpus_v3_items/cid005520.md new file mode 100644 index 000000000..a2e9d1584 --- /dev/null +++ b/_corpus_v3_items/cid005520.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005520/ +title: zero_map_compat (theorem) +cid: cid005520 +primary_alias: FTH0258 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D54] Formal proof: the constant 0 map is reduce-compatible.\n reduce(0, k) = 0 for all k, so the zero\ + \ map trivially satisfies\n f(reduce(x, k)) = reduce(f(x), k). -/" +aliases: +- FTH0258 +- zero_map_compat +- zero-map-compat +- TauLib.BookII.Enrichment.SelfEnrichment::zero_map_compat +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001289 +- predicate: part_of + object: cid005151 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005151 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.SelfEnrichment + declaration: zero_map_compat + module_path: TauLib/BookII/Enrichment/SelfEnrichment.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D54] Formal proof: the constant 0 map is reduce-compatible.\n reduce(0, k) = 0 for all k, so the\ + \ zero map trivially satisfies\n f(reduce(x, k)) = reduce(f(x), k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005521.md b/_corpus_v3_items/cid005521.md new file mode 100644 index 000000000..7e1b10092 --- /dev/null +++ b/_corpus_v3_items/cid005521.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005521/ +title: two_cat_assoc_10_3 (theorem) +cid: cid005521 +primary_alias: FTH0259 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Horizontal associativity [II.D55]' +aliases: +- FTH0259 +- two_cat_assoc_10_3 +- two-cat-assoc-10-3 +- TauLib.BookII.Enrichment.TwoCategories::two_cat_assoc_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_cat_assoc_10_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Horizontal associativity [II.D55]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005522.md b/_corpus_v3_items/cid005522.md new file mode 100644 index 000000000..3eae0a681 --- /dev/null +++ b/_corpus_v3_items/cid005522.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005522/ +title: two_cat_unit_10_3 (theorem) +cid: cid005522 +primary_alias: FTH0260 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Unit laws [II.D55] +aliases: +- FTH0260 +- two_cat_unit_10_3 +- two-cat-unit-10-3 +- TauLib.BookII.Enrichment.TwoCategories::two_cat_unit_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_cat_unit_10_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Unit laws [II.D55] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005523.md b/_corpus_v3_items/cid005523.md new file mode 100644 index 000000000..a92602e2e --- /dev/null +++ b/_corpus_v3_items/cid005523.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005523/ +title: two_cat_vert_10_3 (theorem) +cid: cid005523 +primary_alias: FTH0261 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Vertical associativity [II.D55] +aliases: +- FTH0261 +- two_cat_vert_10_3 +- two-cat-vert-10-3 +- TauLib.BookII.Enrichment.TwoCategories::two_cat_vert_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_cat_vert_10_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Vertical associativity [II.D55] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005524.md b/_corpus_v3_items/cid005524.md new file mode 100644 index 000000000..1517b3495 --- /dev/null +++ b/_corpus_v3_items/cid005524.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005524/ +title: two_morph_tower_8_3 (theorem) +cid: cid005524 +primary_alias: FTH0262 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- 2-morphism tower coherence [II.D56] +aliases: +- FTH0262 +- two_morph_tower_8_3 +- two-morph-tower-8-3 +- TauLib.BookII.Enrichment.TwoCategories::two_morph_tower_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001291 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_morph_tower_8_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- 2-morphism tower coherence [II.D56] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005525.md b/_corpus_v3_items/cid005525.md new file mode 100644 index 000000000..dfb8418a5 --- /dev/null +++ b/_corpus_v3_items/cid005525.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005525/ +title: two_morphism_10_3 (theorem) +cid: cid005525 +primary_alias: FTH0263 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- 2-morphism mediating [II.D56] +aliases: +- FTH0263 +- two_morphism_10_3 +- two-morphism-10-3 +- TauLib.BookII.Enrichment.TwoCategories::two_morphism_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001291 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_morphism_10_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- 2-morphism mediating [II.D56] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005526.md b/_corpus_v3_items/cid005526.md new file mode 100644 index 000000000..0ea4d9bba --- /dev/null +++ b/_corpus_v3_items/cid005526.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005526/ +title: enrich_iter_3 (theorem) +cid: cid005526 +primary_alias: FTH0264 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Enrichment iteration [II.P13] +aliases: +- FTH0264 +- enrich_iter_3 +- enrich-iter-3 +- TauLib.BookII.Enrichment.TwoCategories::enrich_iter_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001354 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: enrich_iter_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Enrichment iteration [II.P13] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005527.md b/_corpus_v3_items/cid005527.md new file mode 100644 index 000000000..da1314264 --- /dev/null +++ b/_corpus_v3_items/cid005527.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005527/ +title: enrich_finite_5 (theorem) +cid: cid005527 +primary_alias: FTH0265 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Enrichment finiteness [II.P13] +aliases: +- FTH0265 +- enrich_finite_5 +- enrich-finite-5 +- TauLib.BookII.Enrichment.TwoCategories::enrich_finite_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001354 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: enrich_finite_5 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Enrichment finiteness [II.P13] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005528.md b/_corpus_v3_items/cid005528.md new file mode 100644 index 000000000..1576a8950 --- /dev/null +++ b/_corpus_v3_items/cid005528.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005528/ +title: interchange_10_3 (theorem) +cid: cid005528 +primary_alias: FTH0266 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Interchange law [II.P13] +aliases: +- FTH0266 +- interchange_10_3 +- interchange-10-3 +- TauLib.BookII.Enrichment.TwoCategories::interchange_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001354 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: interchange_10_3 + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Interchange law [II.P13] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005529.md b/_corpus_v3_items/cid005529.md new file mode 100644 index 000000000..8b25f28eb --- /dev/null +++ b/_corpus_v3_items/cid005529.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005529/ +title: horiz_comp_assoc (theorem) +cid: cid005529 +primary_alias: FTH0267 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D55] Formal proof: horizontal composition is associative.\n horiz_comp f (horiz_comp g h) = horiz_comp\ + \ (horiz_comp f g) h.\n Both sides are fun n k => f(g(h(n, k), k), k). -/" +aliases: +- FTH0267 +- horiz_comp_assoc +- horiz-comp-assoc +- TauLib.BookII.Enrichment.TwoCategories::horiz_comp_assoc +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: horiz_comp_assoc + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D55] Formal proof: horizontal composition is associative.\n horiz_comp f (horiz_comp g h) = horiz_comp\ + \ (horiz_comp f g) h.\n Both sides are fun n k => f(g(h(n, k), k), k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005530.md b/_corpus_v3_items/cid005530.md new file mode 100644 index 000000000..e0378959d --- /dev/null +++ b/_corpus_v3_items/cid005530.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005530/ +title: vert_comp_assoc (theorem) +cid: cid005530 +primary_alias: FTH0268 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D55] Formal proof: vertical composition is associative.\n vert_comp f (vert_comp g h) = vert_comp (vert_comp\ + \ f g) h.\n Both sides are fun n k => f(g(h(n, k), k), k). -/" +aliases: +- FTH0268 +- vert_comp_assoc +- vert-comp-assoc +- TauLib.BookII.Enrichment.TwoCategories::vert_comp_assoc +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: vert_comp_assoc + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D55] Formal proof: vertical composition is associative.\n vert_comp f (vert_comp g h) = vert_comp\ + \ (vert_comp f g) h.\n Both sides are fun n k => f(g(h(n, k), k), k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005531.md b/_corpus_v3_items/cid005531.md new file mode 100644 index 000000000..1d50da9b5 --- /dev/null +++ b/_corpus_v3_items/cid005531.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005531/ +title: id_two_cell_tower_coherent (theorem) +cid: cid005531 +primary_alias: FTH0269 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D55] Formal proof: the identity 2-cell is tower-coherent.\n reduce(id_two_cell(x, l), k) = id_two_cell(x,\ + \ k) for k <= l.\n This is reduction_compat. -/" +aliases: +- FTH0269 +- id_two_cell_tower_coherent +- id-two-cell-tower-coherent +- TauLib.BookII.Enrichment.TwoCategories::id_two_cell_tower_coherent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001290 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: id_two_cell_tower_coherent + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D55] Formal proof: the identity 2-cell is tower-coherent.\n reduce(id_two_cell(x, l), k) = id_two_cell(x,\ + \ k) for k <= l.\n This is reduction_compat. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005532.md b/_corpus_v3_items/cid005532.md new file mode 100644 index 000000000..28bb2e5a2 --- /dev/null +++ b/_corpus_v3_items/cid005532.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005532/ +title: two_sq_mediates_at_reduce (theorem) +cid: cid005532 +primary_alias: FTH0270 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D56] Formal proof: two_sq mediates one_id => one_sq.\n two_sq(one_id(x, k), k) = one_sq(x, k).\n LHS\ + \ = reduce((reduce(x, k))^2, k), RHS = reduce(x^2, k).\n i.e., (x % P_k) * (x % P_k) % P_k = x * x % P_k. -/" +aliases: +- FTH0270 +- two_sq_mediates_at_reduce +- two-sq-mediates-at-reduce +- TauLib.BookII.Enrichment.TwoCategories::two_sq_mediates_at_reduce +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001291 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: two_sq_mediates_at_reduce + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D56] Formal proof: two_sq mediates one_id => one_sq.\n two_sq(one_id(x, k), k) = one_sq(x, k).\n\ + \ LHS = reduce((reduce(x, k))^2, k), RHS = reduce(x^2, k).\n i.e., (x % P_k) * (x % P_k) % P_k = x * x % P_k. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005533.md b/_corpus_v3_items/cid005533.md new file mode 100644 index 000000000..e66c5b42d --- /dev/null +++ b/_corpus_v3_items/cid005533.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005533/ +title: primorial_pos_check (theorem) +cid: cid005533 +primary_alias: FTH0271 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.P13] Formal proof: enrichment iteration is well-founded.\n The primorial P_k is positive for all k >=\ + \ 0.\n Verified computationally for stages 0 through 5. -/" +aliases: +- FTH0271 +- primorial_pos_check +- primorial-pos-check +- TauLib.BookII.Enrichment.TwoCategories::primorial_pos_check +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001354 +- predicate: part_of + object: cid005152 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005152 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.TwoCategories + declaration: primorial_pos_check + module_path: TauLib/BookII/Enrichment/TwoCategories.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.P13] Formal proof: enrichment iteration is well-founded.\n The primorial P_k is positive for all\ + \ k >= 0.\n Verified computationally for stages 0 through 5. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005534.md b/_corpus_v3_items/cid005534.md new file mode 100644 index 000000000..e6cae9af3 --- /dev/null +++ b/_corpus_v3_items/cid005534.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005534/ +title: probe_yoneda_10_3 (theorem) +cid: cid005534 +primary_alias: FTH0272 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Probe naturality iff Yoneda [II.L11]' +aliases: +- FTH0272 +- probe_yoneda_10_3 +- probe-yoneda-10-3 +- TauLib.BookII.Enrichment.YonedaTheorem::probe_yoneda_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001338 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: probe_yoneda_10_3 + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Probe naturality iff Yoneda [II.L11]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005535.md b/_corpus_v3_items/cid005535.md new file mode 100644 index 000000000..36b136f5e --- /dev/null +++ b/_corpus_v3_items/cid005535.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005535/ +title: yoneda_faithful_10_3 (theorem) +cid: cid005535 +primary_alias: FTH0273 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Faithfulness [II.T36] +aliases: +- FTH0273 +- yoneda_faithful_10_3 +- yoneda-faithful-10-3 +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_faithful_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_faithful_10_3 + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Faithfulness [II.T36] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005536.md b/_corpus_v3_items/cid005536.md new file mode 100644 index 000000000..c3fd46d70 --- /dev/null +++ b/_corpus_v3_items/cid005536.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005536/ +title: yoneda_full_10_3 (theorem) +cid: cid005536 +primary_alias: FTH0274 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Fullness [II.T36] +aliases: +- FTH0274 +- yoneda_full_10_3 +- yoneda-full-10-3 +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_full_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_full_10_3 + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Fullness [II.T36] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005537.md b/_corpus_v3_items/cid005537.md new file mode 100644 index 000000000..18a055fdc --- /dev/null +++ b/_corpus_v3_items/cid005537.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005537/ +title: yoneda_bipolar_10_3 (theorem) +cid: cid005537 +primary_alias: FTH0275 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Bipolar preservation [II.T36] +aliases: +- FTH0275 +- yoneda_bipolar_10_3 +- yoneda-bipolar-10-3 +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_bipolar_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_bipolar_10_3 + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Bipolar preservation [II.T36] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005538.md b/_corpus_v3_items/cid005538.md new file mode 100644 index 000000000..509c1dc21 --- /dev/null +++ b/_corpus_v3_items/cid005538.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005538/ +title: yoneda_roundtrip_10_3 (theorem) +cid: cid005538 +primary_alias: FTH0276 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Round-trip [II.T36] +aliases: +- FTH0276 +- yoneda_roundtrip_10_3 +- yoneda-roundtrip-10-3 +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_roundtrip_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_roundtrip_10_3 + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Round-trip [II.T36] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005539.md b/_corpus_v3_items/cid005539.md new file mode 100644 index 000000000..910be0851 --- /dev/null +++ b/_corpus_v3_items/cid005539.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005539/ +title: yoneda_faithful_id (theorem) +cid: cid005539 +primary_alias: FTH0277 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T36] Formal proof: the pre-Yoneda embedding of the identity\n is faithful — Code extraction recovers\ + \ the function.\n code_extract(fun a => preyoneda(id, a, k), k, x) = (reduce(x, k) : Int).\n\n Unfolding: code_extract\ + \ f k x = f(reduce(x, k))\n where f(a) = preyoneda(reduce(·, k), a, k) = reduce(reduce(a, k), k).\n So the full expression\ + \ is reduce(reduce(reduce(x, k), k), k) which\n collapses to reduce(x, k) by triple application of mod idempotence. -/" +aliases: +- FTH0277 +- yoneda_faithful_id +- yoneda-faithful-id +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_faithful_id +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_faithful_id + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T36] Formal proof: the pre-Yoneda embedding of the identity\n is faithful — Code extraction recovers\ + \ the function.\n code_extract(fun a => preyoneda(id, a, k), k, x) = (reduce(x, k) : Int).\n\n Unfolding: code_extract\ + \ f k x = f" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005540.md b/_corpus_v3_items/cid005540.md new file mode 100644 index 000000000..8e0d763e5 --- /dev/null +++ b/_corpus_v3_items/cid005540.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005540/ +title: yoneda_full_id (theorem) +cid: cid005540 +primary_alias: FTH0278 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T36] Formal proof: fullness of the Yoneda embedding.\n For any tower-coherent g given by g(x, k) = reduce(x,\ + \ k),\n preyoneda(g_k, x, k) = g(x, k).\n\n preyoneda(reduce(·, k), x, k) = reduce(reduce(x, k), k) = reduce(x, k).\ + \ -/" +aliases: +- FTH0278 +- yoneda_full_id +- yoneda-full-id +- TauLib.BookII.Enrichment.YonedaTheorem::yoneda_full_id +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001397 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: yoneda_full_id + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T36] Formal proof: fullness of the Yoneda embedding.\n For any tower-coherent g given by g(x, k)\ + \ = reduce(x, k),\n preyoneda(g_k, x, k) = g(x, k).\n\n preyoneda(reduce(·, k), x, k) = reduce(reduce(x, k), k) =\ + \ reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005541.md b/_corpus_v3_items/cid005541.md new file mode 100644 index 000000000..8587bca4e --- /dev/null +++ b/_corpus_v3_items/cid005541.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005541/ +title: probe_naturality_structural (theorem) +cid: cid005541 +primary_alias: FTH0279 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L11] Formal proof: probe naturality IS tower coherence.\n For f = reduce(·, k), naturality at (x, k)\ + \ reduces to\n reduce(reduce(x, k+1), k) = reduce(x, k), which is reduction_compat. -/" +aliases: +- FTH0279 +- probe_naturality_structural +- probe-naturality-structural +- TauLib.BookII.Enrichment.YonedaTheorem::probe_naturality_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001338 +- predicate: part_of + object: cid005153 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005153 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Enrichment.YonedaTheorem + declaration: probe_naturality_structural + module_path: TauLib/BookII/Enrichment/YonedaTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L11] Formal proof: probe naturality IS tower coherence.\n For f = reduce(·, k), naturality at (x,\ + \ k) reduces to\n reduce(reduce(x, k+1), k) = reduce(x, k), which is reduction_compat. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005542.md b/_corpus_v3_items/cid005542.md new file mode 100644 index 000000000..7624a427a --- /dev/null +++ b/_corpus_v3_items/cid005542.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005542/ +title: bndlift_exist_3_30 (theorem) +cid: cid005542 +primary_alias: FTH0280 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Tower coherence [II.T26] +aliases: +- FTH0280 +- bndlift_exist_3_30 +- bndlift-exist-3-30 +- TauLib.BookII.Hartogs.BndLift::bndlift_exist_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001387 +- predicate: part_of + object: cid005159 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005159 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.BndLift + declaration: bndlift_exist_3_30 + module_path: TauLib/BookII/Hartogs/BndLift.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Tower coherence [II.T26] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005543.md b/_corpus_v3_items/cid005543.md new file mode 100644 index 000000000..a3399dcd0 --- /dev/null +++ b/_corpus_v3_items/cid005543.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005543/ +title: bipolar_indep_30 (theorem) +cid: cid005543 +primary_alias: FTH0281 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Channel independence [II.P08] +aliases: +- FTH0281 +- bipolar_indep_30 +- bipolar-indep-30 +- TauLib.BookII.Hartogs.BndLift::bipolar_indep_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001349 +- predicate: part_of + object: cid005159 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005159 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.BndLift + declaration: bipolar_indep_30 + module_path: TauLib/BookII/Hartogs/BndLift.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Channel independence [II.P08] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005544.md b/_corpus_v3_items/cid005544.md new file mode 100644 index 000000000..be05fc0c5 --- /dev/null +++ b/_corpus_v3_items/cid005544.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005544/ +title: basis_ortho_3_30 (theorem) +cid: cid005544 +primary_alias: FTH0282 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Orthogonality [II.D45]' +aliases: +- FTH0282 +- basis_ortho_3_30 +- basis-ortho-3-30 +- TauLib.BookII.Hartogs.CanonicalBasis::basis_ortho_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001280 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: basis_ortho_3_30 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Orthogonality [II.D45]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005545.md b/_corpus_v3_items/cid005545.md new file mode 100644 index 000000000..ce05f490c --- /dev/null +++ b/_corpus_v3_items/cid005545.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005545/ +title: basis_complete_3_30 (theorem) +cid: cid005545 +primary_alias: FTH0283 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Completeness [II.D45] +aliases: +- FTH0283 +- basis_complete_3_30 +- basis-complete-3-30 +- TauLib.BookII.Hartogs.CanonicalBasis::basis_complete_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001280 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: basis_complete_3_30 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Completeness [II.D45] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005546.md b/_corpus_v3_items/cid005546.md new file mode 100644 index 000000000..8f4d62cdb --- /dev/null +++ b/_corpus_v3_items/cid005546.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005546/ +title: basis_indep_4 (theorem) +cid: cid005546 +primary_alias: FTH0284 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Independence [II.D45] +aliases: +- FTH0284 +- basis_indep_4 +- basis-indep-4 +- TauLib.BookII.Hartogs.CanonicalBasis::basis_indep_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001280 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: basis_indep_4 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Independence [II.D45] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005547.md b/_corpus_v3_items/cid005547.md new file mode 100644 index 000000000..437d4a832 --- /dev/null +++ b/_corpus_v3_items/cid005547.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005547/ +title: basis_3_30 (theorem) +cid: cid005547 +primary_alias: FTH0285 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Canonical basis [II.D45] +aliases: +- FTH0285 +- basis_3_30 +- basis-3-30 +- TauLib.BookII.Hartogs.CanonicalBasis::basis_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001280 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: basis_3_30 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Canonical basis [II.D45] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005548.md b/_corpus_v3_items/cid005548.md new file mode 100644 index 000000000..a6258ea1d --- /dev/null +++ b/_corpus_v3_items/cid005548.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005548/ +title: spectral_support_3_30 (theorem) +cid: cid005548 +primary_alias: FTH0286 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Finite spectral support [II.T31] +aliases: +- FTH0286 +- spectral_support_3_30 +- spectral-support-3-30 +- TauLib.BookII.Hartogs.CanonicalBasis::spectral_support_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001392 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: spectral_support_3_30 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Finite spectral support [II.T31] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005549.md b/_corpus_v3_items/cid005549.md new file mode 100644 index 000000000..a4f3b2693 --- /dev/null +++ b/_corpus_v3_items/cid005549.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005549/ +title: proj_delta_3 (theorem) +cid: cid005549 +primary_alias: FTH0287 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Projection delta [II.P09] +aliases: +- FTH0287 +- proj_delta_3 +- proj-delta-3 +- TauLib.BookII.Hartogs.CanonicalBasis::proj_delta_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001350 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: proj_delta_3 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Projection delta [II.P09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005550.md b/_corpus_v3_items/cid005550.md new file mode 100644 index 000000000..aa4bac4b7 --- /dev/null +++ b/_corpus_v3_items/cid005550.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005550/ +title: proj_recovery_3 (theorem) +cid: cid005550 +primary_alias: FTH0288 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Projection recovery [II.P09] +aliases: +- FTH0288 +- proj_recovery_3 +- proj-recovery-3 +- TauLib.BookII.Hartogs.CanonicalBasis::proj_recovery_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001350 +- predicate: part_of + object: cid005161 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005161 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CanonicalBasis + declaration: proj_recovery_3 + module_path: TauLib/BookII/Hartogs/CanonicalBasis.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Projection recovery [II.P09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005551.md b/_corpus_v3_items/cid005551.md new file mode 100644 index 000000000..75c5e278c --- /dev/null +++ b/_corpus_v3_items/cid005551.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005551/ +title: hol_assoc_thm (theorem) +cid: cid005551 +primary_alias: FTH0289 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T29] Associativity theorem (formal):\n Holomorphic composition is associative by definition.\n hol_comp\ + \ f (hol_comp g h) n k = hol_comp (hol_comp f g) h n k\n\n This is an immediate consequence of the definition:\n both\ + \ sides expand to f(g(h(n, k), k), k). -/" +aliases: +- FTH0289 +- hol_assoc_thm +- hol-assoc-thm +- TauLib.BookII.Hartogs.CategoryStructure::hol_assoc_thm +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001390 +- predicate: part_of + object: cid005162 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005162 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CategoryStructure + declaration: hol_assoc_thm + module_path: TauLib/BookII/Hartogs/CategoryStructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T29] Associativity theorem (formal):\n Holomorphic composition is associative by definition.\n \ + \ hol_comp f (hol_comp g h) n k = hol_comp (hol_comp f g) h n k\n\n This is an immediate consequence of the definition:\n\ + \ both sides" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005552.md b/_corpus_v3_items/cid005552.md new file mode 100644 index 000000000..bc0dd0bc1 --- /dev/null +++ b/_corpus_v3_items/cid005552.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005552/ +title: assoc_12_4 (theorem) +cid: cid005552 +primary_alias: FTH0290 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Associativity [II.T29]' +aliases: +- FTH0290 +- assoc_12_4 +- assoc-12-4 +- TauLib.BookII.Hartogs.CategoryStructure::assoc_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001390 +- predicate: part_of + object: cid005162 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005162 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CategoryStructure + declaration: assoc_12_4 + module_path: TauLib/BookII/Hartogs/CategoryStructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Associativity [II.T29]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005553.md b/_corpus_v3_items/cid005553.md new file mode 100644 index 000000000..b551d0820 --- /dev/null +++ b/_corpus_v3_items/cid005553.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005553/ +title: holend_3_10_ok (theorem) +cid: cid005553 +primary_alias: FTH0291 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Category axioms [II.D41] +aliases: +- FTH0291 +- holend_3_10_ok +- holend-3-10-ok +- TauLib.BookII.Hartogs.CategoryStructure::holend_3_10_ok +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001276 +- predicate: part_of + object: cid005162 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005162 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.CategoryStructure + declaration: holend_3_10_ok + module_path: TauLib/BookII/Hartogs/CategoryStructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Category axioms [II.D41] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005554.md b/_corpus_v3_items/cid005554.md new file mode 100644 index 000000000..743d1d3be --- /dev/null +++ b/_corpus_v3_items/cid005554.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005554/ +title: evolution_semigroup_thm (theorem) +cid: cid005554 +primary_alias: FTH0292 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T28] Evolution semigroup theorem (formal):\n reduce(reduce(x, m), n) = reduce(x, n) for n <= m.\n \ + \ This is a direct corollary of reduction_compat from ModArith. -/" +aliases: +- FTH0292 +- evolution_semigroup_thm +- evolution-semigroup-thm +- TauLib.BookII.Hartogs.EvolutionOperator::evolution_semigroup_thm +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001389 +- predicate: part_of + object: cid005163 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005163 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.EvolutionOperator + declaration: evolution_semigroup_thm + module_path: TauLib/BookII/Hartogs/EvolutionOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T28] Evolution semigroup theorem (formal):\n reduce(reduce(x, m), n) = reduce(x, n) for n <= m.\n\ + \ This is a direct corollary of reduction_compat from ModArith. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005555.md b/_corpus_v3_items/cid005555.md new file mode 100644 index 000000000..f3530e8d7 --- /dev/null +++ b/_corpus_v3_items/cid005555.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005555/ +title: semigroup_12_4 (theorem) +cid: cid005555 +primary_alias: FTH0293 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Evolution semigroup [II.T28]' +aliases: +- FTH0293 +- semigroup_12_4 +- semigroup-12-4 +- TauLib.BookII.Hartogs.EvolutionOperator::semigroup_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001389 +- predicate: part_of + object: cid005163 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005163 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.EvolutionOperator + declaration: semigroup_12_4 + module_path: TauLib/BookII/Hartogs/EvolutionOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Evolution semigroup [II.T28]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005556.md b/_corpus_v3_items/cid005556.md new file mode 100644 index 000000000..c4fde6e46 --- /dev/null +++ b/_corpus_v3_items/cid005556.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005556/ +title: asymmetry_witness (theorem) +cid: cid005556 +primary_alias: FTH0294 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Causal arrow [II.D38] +aliases: +- FTH0294 +- asymmetry_witness +- asymmetry-witness +- TauLib.BookII.Hartogs.EvolutionOperator::asymmetry_witness +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001273 +- predicate: part_of + object: cid005163 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005163 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.EvolutionOperator + declaration: asymmetry_witness + module_path: TauLib/BookII/Hartogs/EvolutionOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Causal arrow [II.D38] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005557.md b/_corpus_v3_items/cid005557.md new file mode 100644 index 000000000..30e2ae65f --- /dev/null +++ b/_corpus_v3_items/cid005557.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005557/ +title: inner_product_symmetry_2 (theorem) +cid: cid005557 +primary_alias: FTH0295 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D82] Symmetry of inner product at stage 2. -/ +aliases: +- FTH0295 +- inner_product_symmetry_2 +- inner-product-symmetry-2 +- TauLib.BookII.Hartogs.L2Space::inner_product_symmetry_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001316 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: inner_product_symmetry_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D82] Symmetry of inner product at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005558.md b/_corpus_v3_items/cid005558.md new file mode 100644 index 000000000..0ec0ae611 --- /dev/null +++ b/_corpus_v3_items/cid005558.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005558/ +title: inner_product_linearity_2 (theorem) +cid: cid005558 +primary_alias: FTH0296 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D82] Linearity of inner product at stage 2. -/ +aliases: +- FTH0296 +- inner_product_linearity_2 +- inner-product-linearity-2 +- TauLib.BookII.Hartogs.L2Space::inner_product_linearity_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001316 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: inner_product_linearity_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D82] Linearity of inner product at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005559.md b/_corpus_v3_items/cid005559.md new file mode 100644 index 000000000..747d2b80d --- /dev/null +++ b/_corpus_v3_items/cid005559.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005559/ +title: norm_positivity_2 (theorem) +cid: cid005559 +primary_alias: FTH0297 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D83] Norm positivity at stage 2. -/ +aliases: +- FTH0297 +- norm_positivity_2 +- norm-positivity-2 +- TauLib.BookII.Hartogs.L2Space::norm_positivity_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001317 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: norm_positivity_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D83] Norm positivity at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005560.md b/_corpus_v3_items/cid005560.md new file mode 100644 index 000000000..cb5061e1e --- /dev/null +++ b/_corpus_v3_items/cid005560.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005560/ +title: norm_definiteness_2 (theorem) +cid: cid005560 +primary_alias: FTH0298 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D83] Norm definiteness at stage 2. -/ +aliases: +- FTH0298 +- norm_definiteness_2 +- norm-definiteness-2 +- TauLib.BookII.Hartogs.L2Space::norm_definiteness_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001317 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: norm_definiteness_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D83] Norm definiteness at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005561.md b/_corpus_v3_items/cid005561.md new file mode 100644 index 000000000..de87c2f77 --- /dev/null +++ b/_corpus_v3_items/cid005561.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005561/ +title: cauchy_schwarz_stage1 (theorem) +cid: cid005561 +primary_alias: FTH0299 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T53] Cauchy-Schwarz exhaustive at stage 1. -/ +aliases: +- FTH0299 +- cauchy_schwarz_stage1 +- cauchy-schwarz-stage1 +- TauLib.BookII.Hartogs.L2Space::cauchy_schwarz_stage1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001414 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: cauchy_schwarz_stage1 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T53] Cauchy-Schwarz exhaustive at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005562.md b/_corpus_v3_items/cid005562.md new file mode 100644 index 000000000..3f076d2a2 --- /dev/null +++ b/_corpus_v3_items/cid005562.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005562/ +title: cauchy_schwarz_stage2 (theorem) +cid: cid005562 +primary_alias: FTH0300 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T53] Cauchy-Schwarz exhaustive at stage 2. -/ +aliases: +- FTH0300 +- cauchy_schwarz_stage2 +- cauchy-schwarz-stage2 +- TauLib.BookII.Hartogs.L2Space::cauchy_schwarz_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001414 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: cauchy_schwarz_stage2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T53] Cauchy-Schwarz exhaustive at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005563.md b/_corpus_v3_items/cid005563.md new file mode 100644 index 000000000..59f3e6624 --- /dev/null +++ b/_corpus_v3_items/cid005563.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005563/ +title: l2_completeness_1 (theorem) +cid: cid005563 +primary_alias: FTH0301 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P18] L² completeness at stage 1. -/ +aliases: +- FTH0301 +- l2_completeness_1 +- l2-completeness-1 +- TauLib.BookII.Hartogs.L2Space::l2_completeness_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001359 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: l2_completeness_1 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P18] L² completeness at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005564.md b/_corpus_v3_items/cid005564.md new file mode 100644 index 000000000..abf09cd5a --- /dev/null +++ b/_corpus_v3_items/cid005564.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005564/ +title: l2_completeness_2 (theorem) +cid: cid005564 +primary_alias: FTH0302 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.P18] L² completeness at stage 2. -/ +aliases: +- FTH0302 +- l2_completeness_2 +- l2-completeness-2 +- TauLib.BookII.Hartogs.L2Space::l2_completeness_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001359 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: l2_completeness_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.P18] L² completeness at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005565.md b/_corpus_v3_items/cid005565.md new file mode 100644 index 000000000..637d93af4 --- /dev/null +++ b/_corpus_v3_items/cid005565.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005565/ +title: l2_basis_orthogonal_1 (theorem) +cid: cid005565 +primary_alias: FTH0303 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D82] Basis orthogonality at stage 1. -/ +aliases: +- FTH0303 +- l2_basis_orthogonal_1 +- l2-basis-orthogonal-1 +- TauLib.BookII.Hartogs.L2Space::l2_basis_orthogonal_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001316 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: l2_basis_orthogonal_1 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D82] Basis orthogonality at stage 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005566.md b/_corpus_v3_items/cid005566.md new file mode 100644 index 000000000..6e59cfc6f --- /dev/null +++ b/_corpus_v3_items/cid005566.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005566/ +title: l2_basis_orthogonal_2 (theorem) +cid: cid005566 +primary_alias: FTH0304 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D82] Basis orthogonality at stage 2. -/ +aliases: +- FTH0304 +- l2_basis_orthogonal_2 +- l2-basis-orthogonal-2 +- TauLib.BookII.Hartogs.L2Space::l2_basis_orthogonal_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001316 +- predicate: part_of + object: cid005164 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005164 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.L2Space + declaration: l2_basis_orthogonal_2 + module_path: TauLib/BookII/Hartogs/L2Space.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D82] Basis orthogonality at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005567.md b/_corpus_v3_items/cid005567.md new file mode 100644 index 000000000..d4bc90d18 --- /dev/null +++ b/_corpus_v3_items/cid005567.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005567/ +title: laurent_range_12_4 (theorem) +cid: cid005567 +primary_alias: FTH0305 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Laurent expansion [II.D42]' +aliases: +- FTH0305 +- laurent_range_12_4 +- laurent-range-12-4 +- TauLib.BookII.Hartogs.LaurentResidue::laurent_range_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001277 +- predicate: part_of + object: cid005165 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005165 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + declaration: laurent_range_12_4 + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Laurent expansion [II.D42]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005568.md b/_corpus_v3_items/cid005568.md new file mode 100644 index 000000000..fef97550e --- /dev/null +++ b/_corpus_v3_items/cid005568.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005568/ +title: residue_stage_10_3 (theorem) +cid: cid005568 +primary_alias: FTH0306 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Residue [II.D43] +aliases: +- FTH0306 +- residue_stage_10_3 +- residue-stage-10-3 +- TauLib.BookII.Hartogs.LaurentResidue::residue_stage_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001278 +- predicate: part_of + object: cid005165 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005165 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + declaration: residue_stage_10_3 + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Residue [II.D43] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005569.md b/_corpus_v3_items/cid005569.md new file mode 100644 index 000000000..ae6c131a2 --- /dev/null +++ b/_corpus_v3_items/cid005569.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005569/ +title: crt_roundtrip_3 (theorem) +cid: cid005569 +primary_alias: FTH0307 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- CRT round-trip [II.T30] +aliases: +- FTH0307 +- crt_roundtrip_3 +- crt-roundtrip-3 +- TauLib.BookII.Hartogs.LaurentResidue::crt_roundtrip_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001391 +- predicate: part_of + object: cid005165 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005165 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + declaration: crt_roundtrip_3 + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- CRT round-trip [II.T30] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005570.md b/_corpus_v3_items/cid005570.md new file mode 100644 index 000000000..872132fb5 --- /dev/null +++ b/_corpus_v3_items/cid005570.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005570/ +title: residue_recon_12_3 (theorem) +cid: cid005570 +primary_alias: FTH0308 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Residue theorem [II.T30] +aliases: +- FTH0308 +- residue_recon_12_3 +- residue-recon-12-3 +- TauLib.BookII.Hartogs.LaurentResidue::residue_recon_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001391 +- predicate: part_of + object: cid005165 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005165 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + declaration: residue_recon_12_3 + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Residue theorem [II.T30] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005571.md b/_corpus_v3_items/cid005571.md new file mode 100644 index 000000000..1342fb4a1 --- /dev/null +++ b/_corpus_v3_items/cid005571.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005571/ +title: mero_id_12_4 (theorem) +cid: cid005571 +primary_alias: FTH0309 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Meromorphic [II.D44] +aliases: +- FTH0309 +- mero_id_12_4 +- mero-id-12-4 +- TauLib.BookII.Hartogs.LaurentResidue::mero_id_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001279 +- predicate: part_of + object: cid005165 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005165 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.LaurentResidue + declaration: mero_id_12_4 + module_path: TauLib/BookII/Hartogs/LaurentResidue.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Meromorphic [II.D44] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005572.md b/_corpus_v3_items/cid005572.md new file mode 100644 index 000000000..95bb3f9d4 --- /dev/null +++ b/_corpus_v3_items/cid005572.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005572/ +title: mutual_determination (theorem) +cid: cid005572 +primary_alias: FTH0310 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Mutual Determination Theorem [II.T27] +aliases: +- FTH0310 +- mutual_determination +- mutual-determination +- TauLib.BookII.Hartogs.MutualDetermination::mutual_determination +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001388 +- predicate: part_of + object: cid005166 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005166 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.MutualDetermination + declaration: mutual_determination + module_path: TauLib/BookII/Hartogs/MutualDetermination.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Mutual Determination Theorem [II.T27] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005573.md b/_corpus_v3_items/cid005573.md new file mode 100644 index 000000000..1cb7f00fc --- /dev/null +++ b/_corpus_v3_items/cid005573.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005573/ +title: partition_1 (theorem) +cid: cid005573 +primary_alias: FTH0311 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Cylinder partition [II.D47]' +aliases: +- FTH0311 +- partition_1 +- partition-1 +- TauLib.BookII.Hartogs.SheafCoherence::partition_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001282 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: partition_1 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Cylinder partition [II.D47]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005574.md b/_corpus_v3_items/cid005574.md new file mode 100644 index 000000000..9c91c3266 --- /dev/null +++ b/_corpus_v3_items/cid005574.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005574/ +title: disjoint_1 (theorem) +cid: cid005574 +primary_alias: FTH0312 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Same-stage disjointness [II.L06] +aliases: +- FTH0312 +- disjoint_1 +- disjoint-1 +- TauLib.BookII.Hartogs.SheafCoherence::disjoint_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001333 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: disjoint_1 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Same-stage disjointness [II.L06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005575.md b/_corpus_v3_items/cid005575.md new file mode 100644 index 000000000..4d53ea2fb --- /dev/null +++ b/_corpus_v3_items/cid005575.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005575/ +title: containment_3_30 (theorem) +cid: cid005575 +primary_alias: FTH0313 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Cross-stage containment [II.L06] +aliases: +- FTH0313 +- containment_3_30 +- containment-3-30 +- TauLib.BookII.Hartogs.SheafCoherence::containment_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001333 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: containment_3_30 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Cross-stage containment [II.L06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005576.md b/_corpus_v3_items/cid005576.md new file mode 100644 index 000000000..26c736b42 --- /dev/null +++ b/_corpus_v3_items/cid005576.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005576/ +title: gluing_3 (theorem) +cid: cid005576 +primary_alias: FTH0314 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Gluing lemma [II.L06] +aliases: +- FTH0314 +- gluing_3 +- gluing-3 +- TauLib.BookII.Hartogs.SheafCoherence::gluing_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001333 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: gluing_3 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Gluing lemma [II.L06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005577.md b/_corpus_v3_items/cid005577.md new file mode 100644 index 000000000..0af5bf814 --- /dev/null +++ b/_corpus_v3_items/cid005577.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005577/ +title: sheaf_3 (theorem) +cid: cid005577 +primary_alias: FTH0315 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Sheaf axioms [II.T32] +aliases: +- FTH0315 +- sheaf_3 +- sheaf-3 +- TauLib.BookII.Hartogs.SheafCoherence::sheaf_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001393 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: sheaf_3 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Sheaf axioms [II.T32] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005578.md b/_corpus_v3_items/cid005578.md new file mode 100644 index 000000000..47e5c19d1 --- /dev/null +++ b/_corpus_v3_items/cid005578.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005578/ +title: funct_3_30 (theorem) +cid: cid005578 +primary_alias: FTH0316 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Functoriality [II.D47] +aliases: +- FTH0316 +- funct_3_30 +- funct-3-30 +- TauLib.BookII.Hartogs.SheafCoherence::funct_3_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001282 +- predicate: part_of + object: cid005167 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005167 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Hartogs.SheafCoherence + declaration: funct_3_30 + module_path: TauLib/BookII/Hartogs/SheafCoherence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Functoriality [II.D47] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005579.md b/_corpus_v3_items/cid005579.md new file mode 100644 index 000000000..63b58208f --- /dev/null +++ b/_corpus_v3_items/cid005579.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005579/ +title: abcd_has_zero_divisors (theorem) +cid: cid005579 +primary_alias: FTH0317 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.R04] Split-complex has zero divisors; quaternions don't.\n This is the key algebraic difference: H_τ\ + \ admits bipolar sectors\n via e₊ · e₋ = 0, while ℍ is a division algebra. -/" +aliases: +- FTH0317 +- abcd_has_zero_divisors +- abcd-has-zero-divisors +- TauLib.BookII.Interior.ABCDRigidity::abcd_has_zero_divisors +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005168 +- predicate: part_of + object: cid005168 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005168 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Interior.ABCDRigidity + declaration: abcd_has_zero_divisors + module_path: TauLib/BookII/Interior/ABCDRigidity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.R04] Split-complex has zero divisors; quaternions don't.\n This is the key algebraic difference:\ + \ H_τ admits bipolar sectors\n via e₊ · e₋ = 0, while ℍ is a division algebra. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005580.md b/_corpus_v3_items/cid005580.md new file mode 100644 index 000000000..3fa927fb2 --- /dev/null +++ b/_corpus_v3_items/cid005580.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005580/ +title: tau_spans_three_rungs (theorem) +cid: cid005580 +primary_alias: FTH0318 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T47] τ³ features originate from exactly 3 distinct rungs. -/ +aliases: +- FTH0318 +- tau_spans_three_rungs +- tau-spans-three-rungs +- TauLib.BookII.Mirror.DimensionalLadder::tau_spans_three_rungs +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001408 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: tau_spans_three_rungs + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T47] τ³ features originate from exactly 3 distinct rungs. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005581.md b/_corpus_v3_items/cid005581.md new file mode 100644 index 000000000..352051b56 --- /dev/null +++ b/_corpus_v3_items/cid005581.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005581/ +title: simultaneous_rung (theorem) +cid: cid005581 +primary_alias: FTH0319 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T47] Simultaneous Rung Theorem: τ³ exhibits features from at least\n three classical SCV dimension rungs\ + \ simultaneously.\n Specifically: C1 (CauchyIntegral), C2 (DistinguishedBoundary, HartogsExtension),\n and C3 (FullHartogs).\ + \ -/" +aliases: +- FTH0319 +- simultaneous_rung +- simultaneous-rung +- TauLib.BookII.Mirror.DimensionalLadder::simultaneous_rung +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001408 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: simultaneous_rung + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T47] Simultaneous Rung Theorem: τ³ exhibits features from at least\n three classical SCV dimension\ + \ rungs simultaneously.\n Specifically: C1 (CauchyIntegral), C2 (DistinguishedBoundary, HartogsExtension),\n and C3\ + \ (FullHartogs)" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005582.md b/_corpus_v3_items/cid005582.md new file mode 100644 index 000000000..668f2c3a9 --- /dev/null +++ b/_corpus_v3_items/cid005582.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005582/ +title: tau_engine_inactive (theorem) +cid: cid005582 +primary_alias: FTH0320 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T48] The Archimedean-elliptic engine is inactive in the tau quadrant. -/ +aliases: +- FTH0320 +- tau_engine_inactive +- tau-engine-inactive +- TauLib.BookII.Mirror.DimensionalLadder::tau_engine_inactive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001409 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: tau_engine_inactive + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T48] The Archimedean-elliptic engine is inactive in the tau quadrant. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005583.md b/_corpus_v3_items/cid005583.md new file mode 100644 index 000000000..125762331 --- /dev/null +++ b/_corpus_v3_items/cid005583.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005583/ +title: ladder_collapse (theorem) +cid: cid005583 +primary_alias: FTH0321 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T48] Fourth Quadrant Ladder Collapse: the dimensional ladder does not\n exist in the (Hyperbolic, Non-Archimedean)\ + \ quadrant because the engine\n that generates it is absent. -/" +aliases: +- FTH0321 +- ladder_collapse +- ladder-collapse +- TauLib.BookII.Mirror.DimensionalLadder::ladder_collapse +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001409 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: ladder_collapse + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T48] Fourth Quadrant Ladder Collapse: the dimensional ladder does not\n exist in the (Hyperbolic,\ + \ Non-Archimedean) quadrant because the engine\n that generates it is absent. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005584.md b/_corpus_v3_items/cid005584.md new file mode 100644 index 000000000..28003dc76 --- /dev/null +++ b/_corpus_v3_items/cid005584.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005584/ +title: fibration_forced (theorem) +cid: cid005584 +primary_alias: FTH0322 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D76] The fibration index equals base + fiber. -/ +aliases: +- FTH0322 +- fibration_forced +- fibration-forced +- TauLib.BookII.Mirror.DimensionalLadder::fibration_forced +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001310 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: fibration_forced + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D76] The fibration index equals base + fiber. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005585.md b/_corpus_v3_items/cid005585.md new file mode 100644 index 000000000..e09bc4797 --- /dev/null +++ b/_corpus_v3_items/cid005585.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005585/ +title: refinement_is_abcd (theorem) +cid: cid005585 +primary_alias: FTH0323 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D76] The refinement dimension equals the number of ABCD rays. -/ +aliases: +- FTH0323 +- refinement_is_abcd +- refinement-is-abcd +- TauLib.BookII.Mirror.DimensionalLadder::refinement_is_abcd +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001310 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: refinement_is_abcd + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D76] The refinement dimension equals the number of ABCD rays. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005586.md b/_corpus_v3_items/cid005586.md new file mode 100644 index 000000000..22660c9ef --- /dev/null +++ b/_corpus_v3_items/cid005586.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005586/ +title: fibration_is_three (theorem) +cid: cid005586 +primary_alias: FTH0324 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D76] The fibration is a 3-fold. -/ +aliases: +- FTH0324 +- fibration_is_three +- fibration-is-three +- TauLib.BookII.Mirror.DimensionalLadder::fibration_is_three +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001310 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: fibration_is_three + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D76] The fibration is a 3-fold. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005587.md b/_corpus_v3_items/cid005587.md new file mode 100644 index 000000000..23ec6b0ca --- /dev/null +++ b/_corpus_v3_items/cid005587.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005587/ +title: rigidity_no_free_parameter (theorem) +cid: cid005587 +primary_alias: FTH0325 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D76] Rigidity means there is no dimension parameter to vary:\n the fibration index is determined by the\ + \ base and fiber structure. -/" +aliases: +- FTH0325 +- rigidity_no_free_parameter +- rigidity-no-free-parameter +- TauLib.BookII.Mirror.DimensionalLadder::rigidity_no_free_parameter +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001310 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: rigidity_no_free_parameter + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D76] Rigidity means there is no dimension parameter to vary:\n the fibration index is determined\ + \ by the base and fiber structure. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005588.md b/_corpus_v3_items/cid005588.md new file mode 100644 index 000000000..1fc68a7a4 --- /dev/null +++ b/_corpus_v3_items/cid005588.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005588/ +title: categoricity_kills_ladder (theorem) +cid: cid005588 +primary_alias: FTH0326 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.R31] Categoricity rules out any ladder. -/ +aliases: +- FTH0326 +- categoricity_kills_ladder +- categoricity-kills-ladder +- TauLib.BookII.Mirror.DimensionalLadder::categoricity_kills_ladder +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005173 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: categoricity_kills_ladder + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.R31] Categoricity rules out any ladder. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005589.md b/_corpus_v3_items/cid005589.md new file mode 100644 index 000000000..f7e52c257 --- /dev/null +++ b/_corpus_v3_items/cid005589.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005589/ +title: moduli_singleton (theorem) +cid: cid005589 +primary_alias: FTH0327 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.R31] The moduli count is exactly 1. -/ +aliases: +- FTH0327 +- moduli_singleton +- moduli-singleton +- TauLib.BookII.Mirror.DimensionalLadder::moduli_singleton +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005173 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: moduli_singleton + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.R31] The moduli count is exactly 1. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005590.md b/_corpus_v3_items/cid005590.md new file mode 100644 index 000000000..7eb8bff53 --- /dev/null +++ b/_corpus_v3_items/cid005590.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005590/ +title: engine_only_qft_native (theorem) +cid: cid005590 +primary_alias: FTH0328 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D75] Engine active only in QFT quadrant' +aliases: +- FTH0328 +- engine_only_qft_native +- engine-only-qft-native +- TauLib.BookII.Mirror.DimensionalLadder::engine_only_qft_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001309 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: engine_only_qft_native + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D75] Engine active only in QFT quadrant' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005591.md b/_corpus_v3_items/cid005591.md new file mode 100644 index 000000000..78ec36086 --- /dev/null +++ b/_corpus_v3_items/cid005591.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005591/ +title: rigidity_native (theorem) +cid: cid005591 +primary_alias: FTH0329 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D76] Rigidity values +aliases: +- FTH0329 +- rigidity_native +- rigidity-native +- TauLib.BookII.Mirror.DimensionalLadder::rigidity_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001310 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: rigidity_native + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D76] Rigidity values +noindex: false +--- + diff --git a/_corpus_v3_items/cid005592.md b/_corpus_v3_items/cid005592.md new file mode 100644 index 000000000..18c2d901f --- /dev/null +++ b/_corpus_v3_items/cid005592.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005592/ +title: simultaneous_rung_native (theorem) +cid: cid005592 +primary_alias: FTH0330 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T47] Simultaneous rung +aliases: +- FTH0330 +- simultaneous_rung_native +- simultaneous-rung-native +- TauLib.BookII.Mirror.DimensionalLadder::simultaneous_rung_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001408 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: simultaneous_rung_native + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T47] Simultaneous rung +noindex: false +--- + diff --git a/_corpus_v3_items/cid005593.md b/_corpus_v3_items/cid005593.md new file mode 100644 index 000000000..039d938a4 --- /dev/null +++ b/_corpus_v3_items/cid005593.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005593/ +title: ladder_collapse_native (theorem) +cid: cid005593 +primary_alias: FTH0331 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T48] Ladder collapse +aliases: +- FTH0331 +- ladder_collapse_native +- ladder-collapse-native +- TauLib.BookII.Mirror.DimensionalLadder::ladder_collapse_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001409 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: ladder_collapse_native + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T48] Ladder collapse +noindex: false +--- + diff --git a/_corpus_v3_items/cid005594.md b/_corpus_v3_items/cid005594.md new file mode 100644 index 000000000..5ad9bcf5e --- /dev/null +++ b/_corpus_v3_items/cid005594.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005594/ +title: categoricity_native (theorem) +cid: cid005594 +primary_alias: FTH0332 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.R31] Categoricity +aliases: +- FTH0332 +- categoricity_native +- categoricity-native +- TauLib.BookII.Mirror.DimensionalLadder::categoricity_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005173 +- predicate: part_of + object: cid005173 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005173 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.DimensionalLadder + declaration: categoricity_native + module_path: TauLib/BookII/Mirror/DimensionalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.R31] Categoricity +noindex: false +--- + diff --git a/_corpus_v3_items/cid005595.md b/_corpus_v3_items/cid005595.md new file mode 100644 index 000000000..a3d7b24b5 --- /dev/null +++ b/_corpus_v3_items/cid005595.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005595/ +title: rewiring_table_complete (theorem) +cid: cid005595 +primary_alias: FTH0333 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D72] The rewiring table has exactly 12 rows. -/ +aliases: +- FTH0333 +- rewiring_table_complete +- rewiring-table-complete +- TauLib.BookII.Mirror.Inventory::rewiring_table_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001306 +- predicate: part_of + object: cid005174 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005174 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.Inventory + declaration: rewiring_table_complete + module_path: TauLib/BookII/Mirror/Inventory.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D72] The rewiring table has exactly 12 rows. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005596.md b/_corpus_v3_items/cid005596.md new file mode 100644 index 000000000..20ae75664 --- /dev/null +++ b/_corpus_v3_items/cid005596.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005596/ +title: table_12 (theorem) +cid: cid005596 +primary_alias: FTH0334 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D72] Table completeness' +aliases: +- FTH0334 +- table_12 +- table-12 +- TauLib.BookII.Mirror.Inventory::table_12 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001306 +- predicate: part_of + object: cid005174 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005174 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.Inventory + declaration: table_12 + module_path: TauLib/BookII/Mirror/Inventory.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D72] Table completeness' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005597.md b/_corpus_v3_items/cid005597.md new file mode 100644 index 000000000..91152b0b0 --- /dev/null +++ b/_corpus_v3_items/cid005597.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005597/ +title: rows_orthodox (theorem) +cid: cid005597 +primary_alias: FTH0335 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D72] All rows have nonempty descriptions +aliases: +- FTH0335 +- rows_orthodox +- rows-orthodox +- TauLib.BookII.Mirror.Inventory::rows_orthodox +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001306 +- predicate: part_of + object: cid005174 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005174 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.Inventory + declaration: rows_orthodox + module_path: TauLib/BookII/Mirror/Inventory.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D72] All rows have nonempty descriptions +noindex: false +--- + diff --git a/_corpus_v3_items/cid005598.md b/_corpus_v3_items/cid005598.md new file mode 100644 index 000000000..5fce28752 --- /dev/null +++ b/_corpus_v3_items/cid005598.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005598/ +title: levels_match (theorem) +cid: cid005598 +primary_alias: FTH0336 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D72] Table consistency +aliases: +- FTH0336 +- levels_match +- levels-match +- TauLib.BookII.Mirror.Inventory::levels_match +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001306 +- predicate: part_of + object: cid005174 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005174 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.Inventory + declaration: levels_match + module_path: TauLib/BookII/Mirror/Inventory.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D72] Table consistency +noindex: false +--- + diff --git a/_corpus_v3_items/cid005599.md b/_corpus_v3_items/cid005599.md new file mode 100644 index 000000000..68922846d --- /dev/null +++ b/_corpus_v3_items/cid005599.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005599/ +title: qft_gr_same_column (theorem) +cid: cid005599 +primary_alias: FTH0337 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D74] QFT and GR are in the Archimedean column. -/ +aliases: +- FTH0337 +- qft_gr_same_column +- qft-gr-same-column +- TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_same_column +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001308 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: qft_gr_same_column + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D74] QFT and GR are in the Archimedean column. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005600.md b/_corpus_v3_items/cid005600.md new file mode 100644 index 000000000..389e3eec9 --- /dev/null +++ b/_corpus_v3_items/cid005600.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005600/ +title: qft_gr_incompatible_pde (theorem) +cid: cid005600 +primary_alias: FTH0338 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D74] QFT and GR have incompatible PDE types. -/ +aliases: +- FTH0338 +- qft_gr_incompatible_pde +- qft-gr-incompatible-pde +- TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_incompatible_pde +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001308 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: qft_gr_incompatible_pde + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D74] QFT and GR have incompatible PDE types. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005601.md b/_corpus_v3_items/cid005601.md new file mode 100644 index 000000000..b074b5a45 --- /dev/null +++ b/_corpus_v3_items/cid005601.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005601/ +title: qft_gr_obstructed (theorem) +cid: cid005601 +primary_alias: FTH0339 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D74] QFT/GR unification IS obstructed. -/ +aliases: +- FTH0339 +- qft_gr_obstructed +- qft-gr-obstructed +- TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_obstructed +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001308 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: qft_gr_obstructed + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D74] QFT/GR unification IS obstructed. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005602.md b/_corpus_v3_items/cid005602.md new file mode 100644 index 000000000..e57556cec --- /dev/null +++ b/_corpus_v3_items/cid005602.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005602/ +title: tau_is_non_archimedean (theorem) +cid: cid005602 +primary_alias: FTH0340 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] The tau quadrant is non-Archimedean. -/ +aliases: +- FTH0340 +- tau_is_non_archimedean +- tau-is-non-archimedean +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_is_non_archimedean +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_is_non_archimedean + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] The tau quadrant is non-Archimedean. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005603.md b/_corpus_v3_items/cid005603.md new file mode 100644 index 000000000..e0f407ce5 --- /dev/null +++ b/_corpus_v3_items/cid005603.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005603/ +title: tau_is_hyperbolic (theorem) +cid: cid005603 +primary_alias: FTH0341 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] The tau quadrant is hyperbolic. -/ +aliases: +- FTH0341 +- tau_is_hyperbolic +- tau-is-hyperbolic +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_is_hyperbolic +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_is_hyperbolic + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] The tau quadrant is hyperbolic. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005604.md b/_corpus_v3_items/cid005604.md new file mode 100644 index 000000000..8a5bd3b56 --- /dev/null +++ b/_corpus_v3_items/cid005604.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005604/ +title: tau_not_archimedean_with_qft (theorem) +cid: cid005604 +primary_alias: FTH0342 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] Tau is NOT in the Archimedean column with QFT. -/ +aliases: +- FTH0342 +- tau_not_archimedean_with_qft +- tau-not-archimedean-with-qft +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_not_archimedean_with_qft +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_not_archimedean_with_qft + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] Tau is NOT in the Archimedean column with QFT. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005605.md b/_corpus_v3_items/cid005605.md new file mode 100644 index 000000000..31f84db34 --- /dev/null +++ b/_corpus_v3_items/cid005605.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005605/ +title: tau_not_archimedean_with_gr (theorem) +cid: cid005605 +primary_alias: FTH0343 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] Tau is NOT in the Archimedean column with GR. -/ +aliases: +- FTH0343 +- tau_not_archimedean_with_gr +- tau-not-archimedean-with-gr +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_not_archimedean_with_gr +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_not_archimedean_with_gr + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] Tau is NOT in the Archimedean column with GR. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005606.md b/_corpus_v3_items/cid005606.md new file mode 100644 index 000000000..f4a656afa --- /dev/null +++ b/_corpus_v3_items/cid005606.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005606/ +title: tau_qft_not_obstructed (theorem) +cid: cid005606 +primary_alias: FTH0344 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] The unification obstruction does NOT apply to tau and QFT. -/ +aliases: +- FTH0344 +- tau_qft_not_obstructed +- tau-qft-not-obstructed +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_qft_not_obstructed +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_qft_not_obstructed + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] The unification obstruction does NOT apply to tau and QFT. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005607.md b/_corpus_v3_items/cid005607.md new file mode 100644 index 000000000..d658b216b --- /dev/null +++ b/_corpus_v3_items/cid005607.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005607/ +title: tau_gr_not_obstructed (theorem) +cid: cid005607 +primary_alias: FTH0345 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] The unification obstruction does NOT apply to tau and GR. -/ +aliases: +- FTH0345 +- tau_gr_not_obstructed +- tau-gr-not-obstructed +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_gr_not_obstructed +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_gr_not_obstructed + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] The unification obstruction does NOT apply to tau and GR. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005608.md b/_corpus_v3_items/cid005608.md new file mode 100644 index 000000000..fc6564003 --- /dev/null +++ b/_corpus_v3_items/cid005608.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005608/ +title: tau_escapes_obstruction (theorem) +cid: cid005608 +primary_alias: FTH0346 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T46] Fourth quadrant resolution: tau escapes the obstruction because\n it is in the non-Archimedean column.\ + \ -/" +aliases: +- FTH0346 +- tau_escapes_obstruction +- tau-escapes-obstruction +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_escapes_obstruction +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_escapes_obstruction + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T46] Fourth quadrant resolution: tau escapes the obstruction because\n it is in the non-Archimedean\ + \ column. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005609.md b/_corpus_v3_items/cid005609.md new file mode 100644 index 000000000..7fc352ab2 --- /dev/null +++ b/_corpus_v3_items/cid005609.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005609/ +title: tau_distinct_from_qft (theorem) +cid: cid005609 +primary_alias: FTH0347 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T46] The tau quadrant is structurally distinct from both QFT and GR. -/ +aliases: +- FTH0347 +- tau_distinct_from_qft +- tau-distinct-from-qft +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_distinct_from_qft +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_distinct_from_qft + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T46] The tau quadrant is structurally distinct from both QFT and GR. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005610.md b/_corpus_v3_items/cid005610.md new file mode 100644 index 000000000..70be1ce8c --- /dev/null +++ b/_corpus_v3_items/cid005610.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005610/ +title: four_quadrants (theorem) +cid: cid005610 +primary_alias: FTH0348 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D73] Quadrant count' +aliases: +- FTH0348 +- four_quadrants +- four-quadrants +- TauLib.BookII.Mirror.PhysicsQuadrant::four_quadrants +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001307 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: four_quadrants + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D73] Quadrant count' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005611.md b/_corpus_v3_items/cid005611.md new file mode 100644 index 000000000..877906eb6 --- /dev/null +++ b/_corpus_v3_items/cid005611.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005611/ +title: qft_gr_obstructed_native (theorem) +cid: cid005611 +primary_alias: FTH0349 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D74] QFT/GR obstruction +aliases: +- FTH0349 +- qft_gr_obstructed_native +- qft-gr-obstructed-native +- TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_obstructed_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001308 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: qft_gr_obstructed_native + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D74] QFT/GR obstruction +noindex: false +--- + diff --git a/_corpus_v3_items/cid005612.md b/_corpus_v3_items/cid005612.md new file mode 100644 index 000000000..ec0633080 --- /dev/null +++ b/_corpus_v3_items/cid005612.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005612/ +title: tau_escapes_qft (theorem) +cid: cid005612 +primary_alias: FTH0350 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T46] Tau escapes +aliases: +- FTH0350 +- tau_escapes_qft +- tau-escapes-qft +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_escapes_qft +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_escapes_qft + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T46] Tau escapes +noindex: false +--- + diff --git a/_corpus_v3_items/cid005613.md b/_corpus_v3_items/cid005613.md new file mode 100644 index 000000000..426d207d5 --- /dev/null +++ b/_corpus_v3_items/cid005613.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005613/ +title: tau_hyp (theorem) +cid: cid005613 +primary_alias: FTH0351 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T46] Tau properties +aliases: +- FTH0351 +- tau_hyp +- tau-hyp +- TauLib.BookII.Mirror.PhysicsQuadrant::tau_hyp +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001407 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: tau_hyp + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T46] Tau properties +noindex: false +--- + diff --git a/_corpus_v3_items/cid005614.md b/_corpus_v3_items/cid005614.md new file mode 100644 index 000000000..a3fa63fe1 --- /dev/null +++ b/_corpus_v3_items/cid005614.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005614/ +title: obstruction_confined_native (theorem) +cid: cid005614 +primary_alias: FTH0352 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D74] Obstruction confined to Archimedean +aliases: +- FTH0352 +- obstruction_confined_native +- obstruction-confined-native +- TauLib.BookII.Mirror.PhysicsQuadrant::obstruction_confined_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001308 +- predicate: part_of + object: cid005175 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005175 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.PhysicsQuadrant + declaration: obstruction_confined_native + module_path: TauLib/BookII/Mirror/PhysicsQuadrant.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D74] Obstruction confined to Archimedean +noindex: false +--- + diff --git a/_corpus_v3_items/cid005615.md b/_corpus_v3_items/cid005615.md new file mode 100644 index 000000000..7cf1a3725 --- /dev/null +++ b/_corpus_v3_items/cid005615.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005615/ +title: sign_level_count (theorem) +cid: cid005615 +primary_alias: FTH0353 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D68] There are exactly 12 sign levels. -/ +aliases: +- FTH0353 +- sign_level_count +- sign-level-count +- TauLib.BookII.Mirror.SignClassification::sign_level_count +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001302 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: sign_level_count + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D68] There are exactly 12 sign levels. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005616.md b/_corpus_v3_items/cid005616.md new file mode 100644 index 000000000..6de465bfd --- /dev/null +++ b/_corpus_v3_items/cid005616.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005616/ +title: all_descriptions_differ (theorem) +cid: cid005616 +primary_alias: FTH0354 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.D68] At every sign level, the orthodox and tau descriptions are distinct. -/ +aliases: +- FTH0354 +- all_descriptions_differ +- all-descriptions-differ +- TauLib.BookII.Mirror.SignClassification::all_descriptions_differ +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001302 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: all_descriptions_differ + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.D68] At every sign level, the orthodox and tau descriptions are distinct. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005617.md b/_corpus_v3_items/cid005617.md new file mode 100644 index 000000000..9eb9e8849 --- /dev/null +++ b/_corpus_v3_items/cid005617.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005617/ +title: orthodox_path_no_unique_omega (theorem) +cid: cid005617 +primary_alias: FTH0355 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] The orthodox path does not have unique omega. -/ +aliases: +- FTH0355 +- orthodox_path_no_unique_omega +- orthodox-path-no-unique-omega +- TauLib.BookII.Mirror.SignClassification::orthodox_path_no_unique_omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: orthodox_path_no_unique_omega + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] The orthodox path does not have unique omega. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005618.md b/_corpus_v3_items/cid005618.md new file mode 100644 index 000000000..d6ad75245 --- /dev/null +++ b/_corpus_v3_items/cid005618.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005618/ +title: tau_path_no_archimedean (theorem) +cid: cid005618 +primary_alias: FTH0356 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] The tau path does not have Archimedean density. -/ +aliases: +- FTH0356 +- tau_path_no_archimedean +- tau-path-no-archimedean +- TauLib.BookII.Mirror.SignClassification::tau_path_no_archimedean +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: tau_path_no_archimedean + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] The tau path does not have Archimedean density. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005619.md b/_corpus_v3_items/cid005619.md new file mode 100644 index 000000000..1fca8cc93 --- /dev/null +++ b/_corpus_v3_items/cid005619.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005619/ +title: orthodox_path_no_finite_witnesses (theorem) +cid: cid005619 +primary_alias: FTH0357 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] The orthodox path does not have finite witnesses. -/ +aliases: +- FTH0357 +- orthodox_path_no_finite_witnesses +- orthodox-path-no-finite-witnesses +- TauLib.BookII.Mirror.SignClassification::orthodox_path_no_finite_witnesses +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: orthodox_path_no_finite_witnesses + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] The orthodox path does not have finite witnesses. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005620.md b/_corpus_v3_items/cid005620.md new file mode 100644 index 000000000..414f00ee6 --- /dev/null +++ b/_corpus_v3_items/cid005620.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005620/ +title: tau_path_no_epsilon_delta (theorem) +cid: cid005620 +primary_alias: FTH0358 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] The tau path does not have epsilon-delta. -/ +aliases: +- FTH0358 +- tau_path_no_epsilon_delta +- tau-path-no-epsilon-delta +- TauLib.BookII.Mirror.SignClassification::tau_path_no_epsilon_delta +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: tau_path_no_epsilon_delta + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] The tau path does not have epsilon-delta. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005621.md b/_corpus_v3_items/cid005621.md new file mode 100644 index 000000000..2cd3fa123 --- /dev/null +++ b/_corpus_v3_items/cid005621.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005621/ +title: structural_incompatibility (theorem) +cid: cid005621 +primary_alias: FTH0359 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T43] Structural incompatibility: unique omega and Archimedean density\n cannot both hold. Proved by case\ + \ analysis on the two paths. -/" +aliases: +- FTH0359 +- structural_incompatibility +- structural-incompatibility +- TauLib.BookII.Mirror.SignClassification::structural_incompatibility +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: structural_incompatibility + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T43] Structural incompatibility: unique omega and Archimedean density\n cannot both hold. Proved\ + \ by case analysis on the two paths. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005622.md b/_corpus_v3_items/cid005622.md new file mode 100644 index 000000000..133fe9f37 --- /dev/null +++ b/_corpus_v3_items/cid005622.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005622/ +title: paths_distinct (theorem) +cid: cid005622 +primary_alias: FTH0360 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] The two paths are distinct structures. -/ +aliases: +- FTH0360 +- paths_distinct +- paths-distinct +- TauLib.BookII.Mirror.SignClassification::paths_distinct +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: paths_distinct + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] The two paths are distinct structures. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005623.md b/_corpus_v3_items/cid005623.md new file mode 100644 index 000000000..f27fb7f60 --- /dev/null +++ b/_corpus_v3_items/cid005623.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005623/ +title: no_both_omega_and_archimedean_orthodox (theorem) +cid: cid005623 +primary_alias: FTH0361 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T43] No trade-off can have both unique omega and Archimedean density\n if it agrees with one of the two\ + \ canonical paths. -/" +aliases: +- FTH0361 +- no_both_omega_and_archimedean_orthodox +- no-both-omega-and-archimedean-orthodox +- TauLib.BookII.Mirror.SignClassification::no_both_omega_and_archimedean_orthodox +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: no_both_omega_and_archimedean_orthodox + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T43] No trade-off can have both unique omega and Archimedean density\n if it agrees with one of the\ + \ two canonical paths. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005624.md b/_corpus_v3_items/cid005624.md new file mode 100644 index 000000000..4ffe5e37c --- /dev/null +++ b/_corpus_v3_items/cid005624.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005624/ +title: no_both_omega_and_archimedean_tau (theorem) +cid: cid005624 +primary_alias: FTH0362 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T43] Symmetrically for the tau path. -/ +aliases: +- FTH0362 +- no_both_omega_and_archimedean_tau +- no-both-omega-and-archimedean-tau +- TauLib.BookII.Mirror.SignClassification::no_both_omega_and_archimedean_tau +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: no_both_omega_and_archimedean_tau + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T43] Symmetrically for the tau path. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005625.md b/_corpus_v3_items/cid005625.md new file mode 100644 index 000000000..824347475 --- /dev/null +++ b/_corpus_v3_items/cid005625.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005625/ +title: sign_count_12 (theorem) +cid: cid005625 +primary_alias: FTH0363 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D68] Sign level count' +aliases: +- FTH0363 +- sign_count_12 +- sign-count-12 +- TauLib.BookII.Mirror.SignClassification::sign_count_12 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001302 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: sign_count_12 + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.D68] Sign level count' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005626.md b/_corpus_v3_items/cid005626.md new file mode 100644 index 000000000..86fee52e8 --- /dev/null +++ b/_corpus_v3_items/cid005626.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005626/ +title: orthodox_all_nonempty (theorem) +cid: cid005626 +primary_alias: FTH0364 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D68] All descriptions nonempty and distinct +aliases: +- FTH0364 +- orthodox_all_nonempty +- orthodox-all-nonempty +- TauLib.BookII.Mirror.SignClassification::orthodox_all_nonempty +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001302 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: orthodox_all_nonempty + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D68] All descriptions nonempty and distinct +noindex: false +--- + diff --git a/_corpus_v3_items/cid005627.md b/_corpus_v3_items/cid005627.md new file mode 100644 index 000000000..7362f3b65 --- /dev/null +++ b/_corpus_v3_items/cid005627.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005627/ +title: orthodox_omega (theorem) +cid: cid005627 +primary_alias: FTH0365 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D69] Path properties +aliases: +- FTH0365 +- orthodox_omega +- orthodox-omega +- TauLib.BookII.Mirror.SignClassification::orthodox_omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001303 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: orthodox_omega + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D69] Path properties +noindex: false +--- + diff --git a/_corpus_v3_items/cid005628.md b/_corpus_v3_items/cid005628.md new file mode 100644 index 000000000..e0cb14843 --- /dev/null +++ b/_corpus_v3_items/cid005628.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005628/ +title: incompatibility_native (theorem) +cid: cid005628 +primary_alias: FTH0366 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T43] Incompatibility +aliases: +- FTH0366 +- incompatibility_native +- incompatibility-native +- TauLib.BookII.Mirror.SignClassification::incompatibility_native +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001404 +- predicate: part_of + object: cid005176 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005176 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.SignClassification + declaration: incompatibility_native + module_path: TauLib/BookII/Mirror/SignClassification.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T43] Incompatibility +noindex: false +--- + diff --git a/_corpus_v3_items/cid005629.md b/_corpus_v3_items/cid005629.md new file mode 100644 index 000000000..a8bc9ca93 --- /dev/null +++ b/_corpus_v3_items/cid005629.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005629/ +title: hyperbolic_hartogs_natural (theorem) +cid: cid005629 +primary_alias: FTH0367 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T44] Hartogs extension is natural for hyperbolic PDE type. -/ +aliases: +- FTH0367 +- hyperbolic_hartogs_natural +- hyperbolic-hartogs-natural +- TauLib.BookII.Mirror.WaveHolomorphy::hyperbolic_hartogs_natural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: hyperbolic_hartogs_natural + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T44] Hartogs extension is natural for hyperbolic PDE type. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005630.md b/_corpus_v3_items/cid005630.md new file mode 100644 index 000000000..8db48c088 --- /dev/null +++ b/_corpus_v3_items/cid005630.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005630/ +title: elliptic_hartogs_not_natural (theorem) +cid: cid005630 +primary_alias: FTH0368 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T44] Hartogs extension is NOT natural for elliptic PDE type. -/ +aliases: +- FTH0368 +- elliptic_hartogs_not_natural +- elliptic-hartogs-not-natural +- TauLib.BookII.Mirror.WaveHolomorphy::elliptic_hartogs_not_natural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: elliptic_hartogs_not_natural + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T44] Hartogs extension is NOT natural for elliptic PDE type. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005631.md b/_corpus_v3_items/cid005631.md new file mode 100644 index 000000000..814b72d2f --- /dev/null +++ b/_corpus_v3_items/cid005631.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005631/ +title: asymmetric_determination (theorem) +cid: cid005631 +primary_alias: FTH0369 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T44] The elliptic and hyperbolic classifications have opposite\n Hartogs naturalness. -/" +aliases: +- FTH0369 +- asymmetric_determination +- TauLib.BookII.Mirror.WaveHolomorphy::asymmetric_determination +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: asymmetric_determination + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T44] The elliptic and hyperbolic classifications have opposite\n Hartogs naturalness. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005632.md b/_corpus_v3_items/cid005632.md new file mode 100644 index 000000000..d0813fbfa --- /dev/null +++ b/_corpus_v3_items/cid005632.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005632/ +title: max_principle_asymmetry (theorem) +cid: cid005632 +primary_alias: FTH0370 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T44] The elliptic classification has the maximum principle;\n the hyperbolic does not. -/" +aliases: +- FTH0370 +- max_principle_asymmetry +- max-principle-asymmetry +- TauLib.BookII.Mirror.WaveHolomorphy::max_principle_asymmetry +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: max_principle_asymmetry + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T44] The elliptic classification has the maximum principle;\n the hyperbolic does not. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005633.md b/_corpus_v3_items/cid005633.md new file mode 100644 index 000000000..8f0590b5e --- /dev/null +++ b/_corpus_v3_items/cid005633.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005633/ +title: pde_classifications_distinct (theorem) +cid: cid005633 +primary_alias: FTH0371 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T44] The two classifications are structurally distinct. -/ +aliases: +- FTH0371 +- pde_classifications_distinct +- pde-classifications-distinct +- TauLib.BookII.Mirror.WaveHolomorphy::pde_classifications_distinct +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: pde_classifications_distinct + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T44] The two classifications are structurally distinct. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005634.md b/_corpus_v3_items/cid005634.md new file mode 100644 index 000000000..b70d0a0ea --- /dev/null +++ b/_corpus_v3_items/cid005634.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005634/ +title: characteristics_iff_hyperbolic (theorem) +cid: cid005634 +primary_alias: FTH0372 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T44] PDE type determines characteristics existence. -/ +aliases: +- FTH0372 +- characteristics_iff_hyperbolic +- characteristics-iff-hyperbolic +- TauLib.BookII.Mirror.WaveHolomorphy::characteristics_iff_hyperbolic +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: characteristics_iff_hyperbolic + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T44] PDE type determines characteristics existence. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005635.md b/_corpus_v3_items/cid005635.md new file mode 100644 index 000000000..59c1451f5 --- /dev/null +++ b/_corpus_v3_items/cid005635.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005635/ +title: stage_no_light_cones (theorem) +cid: cid005635 +primary_alias: FTH0373 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T45] At every stage k, the default geometry has no light cones. -/ +aliases: +- FTH0373 +- stage_no_light_cones +- stage-no-light-cones +- TauLib.BookII.Mirror.WaveHolomorphy::stage_no_light_cones +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: stage_no_light_cones + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T45] At every stage k, the default geometry has no light cones. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005636.md b/_corpus_v3_items/cid005636.md new file mode 100644 index 000000000..d11786c54 --- /dev/null +++ b/_corpus_v3_items/cid005636.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005636/ +title: stage_euclidean (theorem) +cid: cid005636 +primary_alias: FTH0374 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T45] At every stage k, the default geometry is Euclidean. -/ +aliases: +- FTH0374 +- stage_euclidean +- stage-euclidean +- TauLib.BookII.Mirror.WaveHolomorphy::stage_euclidean +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: stage_euclidean + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T45] At every stage k, the default geometry is Euclidean. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005637.md b/_corpus_v3_items/cid005637.md new file mode 100644 index 000000000..77d46cee7 --- /dev/null +++ b/_corpus_v3_items/cid005637.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005637/ +title: stage_euclidean_check_true (theorem) +cid: cid005637 +primary_alias: FTH0375 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T45] At every stage k, the stage-Euclidean check passes. -/ +aliases: +- FTH0375 +- stage_euclidean_check_true +- stage-euclidean-check-true +- TauLib.BookII.Mirror.WaveHolomorphy::stage_euclidean_check_true +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: stage_euclidean_check_true + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T45] At every stage k, the stage-Euclidean check passes. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005638.md b/_corpus_v3_items/cid005638.md new file mode 100644 index 000000000..5aabe09be --- /dev/null +++ b/_corpus_v3_items/cid005638.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005638/ +title: all_stages_euclidean_5 (theorem) +cid: cid005638 +primary_alias: FTH0376 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T45] All stages up to depth 5 are Euclidean. -/ +aliases: +- FTH0376 +- all_stages_euclidean_5 +- all-stages-euclidean-5 +- TauLib.BookII.Mirror.WaveHolomorphy::all_stages_euclidean_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: all_stages_euclidean_5 + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T45] All stages up to depth 5 are Euclidean. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005639.md b/_corpus_v3_items/cid005639.md new file mode 100644 index 000000000..b81ad6a9f --- /dev/null +++ b/_corpus_v3_items/cid005639.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005639/ +title: stage_size_is_primorial (theorem) +cid: cid005639 +primary_alias: FTH0377 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [II.T45] The stage size at k is the k-th primorial. -/ +aliases: +- FTH0377 +- stage_size_is_primorial +- stage-size-is-primorial +- TauLib.BookII.Mirror.WaveHolomorphy::stage_size_is_primorial +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: stage_size_is_primorial + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [II.T45] The stage size at k is the k-th primorial. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005640.md b/_corpus_v3_items/cid005640.md new file mode 100644 index 000000000..3abee3a1e --- /dev/null +++ b/_corpus_v3_items/cid005640.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005640/ +title: sector_additive (theorem) +cid: cid005640 +primary_alias: FTH0378 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D70] The sector components are additive:\n (z + w)_b = z_b + w_b and (z + w)_c = z_c + w_c. -/" +aliases: +- FTH0378 +- sector_additive +- sector-additive +- TauLib.BookII.Mirror.WaveHolomorphy::sector_additive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001304 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: sector_additive + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D70] The sector components are additive:\n (z + w)_b = z_b + w_b and (z + w)_c = z_c + w_c. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005641.md b/_corpus_v3_items/cid005641.md new file mode 100644 index 000000000..21ebcbc00 --- /dev/null +++ b/_corpus_v3_items/cid005641.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005641/ +title: b_sector_multiplicative (theorem) +cid: cid005641 +primary_alias: FTH0379 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D70] The sector components are multiplicative:\n (z * w)_b = z_b * w_b. This is the key ring-isomorphism\ + \ property. -/" +aliases: +- FTH0379 +- b_sector_multiplicative +- b-sector-multiplicative +- TauLib.BookII.Mirror.WaveHolomorphy::b_sector_multiplicative +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001304 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: b_sector_multiplicative + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D70] The sector components are multiplicative:\n (z * w)_b = z_b * w_b. This is the key ring-isomorphism\ + \ property. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005642.md b/_corpus_v3_items/cid005642.md new file mode 100644 index 000000000..859a82e43 --- /dev/null +++ b/_corpus_v3_items/cid005642.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005642/ +title: hartogs_natural_hyp (theorem) +cid: cid005642 +primary_alias: FTH0380 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.T44] Asymmetric determination' +aliases: +- FTH0380 +- hartogs_natural_hyp +- hartogs-natural-hyp +- TauLib.BookII.Mirror.WaveHolomorphy::hartogs_natural_hyp +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: hartogs_natural_hyp + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [II.T44] Asymmetric determination' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005643.md b/_corpus_v3_items/cid005643.md new file mode 100644 index 000000000..317182302 --- /dev/null +++ b/_corpus_v3_items/cid005643.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005643/ +title: chars_hyp (theorem) +cid: cid005643 +primary_alias: FTH0381 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T44] Characteristics +aliases: +- FTH0381 +- chars_hyp +- chars-hyp +- TauLib.BookII.Mirror.WaveHolomorphy::chars_hyp +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: chars_hyp + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T44] Characteristics +noindex: false +--- + diff --git a/_corpus_v3_items/cid005644.md b/_corpus_v3_items/cid005644.md new file mode 100644 index 000000000..363041151 --- /dev/null +++ b/_corpus_v3_items/cid005644.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005644/ +title: max_hyp (theorem) +cid: cid005644 +primary_alias: FTH0382 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T44] Max principle +aliases: +- FTH0382 +- max_hyp +- max-hyp +- TauLib.BookII.Mirror.WaveHolomorphy::max_hyp +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001405 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: max_hyp + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T44] Max principle +noindex: false +--- + diff --git a/_corpus_v3_items/cid005645.md b/_corpus_v3_items/cid005645.md new file mode 100644 index 000000000..9c5de0532 --- /dev/null +++ b/_corpus_v3_items/cid005645.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005645/ +title: stages_euclidean_5 (theorem) +cid: cid005645 +primary_alias: FTH0383 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.T45] All stages Euclidean +aliases: +- FTH0383 +- stages_euclidean_5 +- stages-euclidean-5 +- TauLib.BookII.Mirror.WaveHolomorphy::stages_euclidean_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001406 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: stages_euclidean_5 + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.T45] All stages Euclidean +noindex: false +--- + diff --git a/_corpus_v3_items/cid005646.md b/_corpus_v3_items/cid005646.md new file mode 100644 index 000000000..06a2769a3 --- /dev/null +++ b/_corpus_v3_items/cid005646.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005646/ +title: pde_distinct (theorem) +cid: cid005646 +primary_alias: FTH0384 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [II.D70] Classifications distinct +aliases: +- FTH0384 +- pde_distinct +- pde-distinct +- TauLib.BookII.Mirror.WaveHolomorphy::pde_distinct +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001304 +- predicate: part_of + object: cid005177 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005177 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Mirror.WaveHolomorphy + declaration: pde_distinct + module_path: TauLib/BookII/Mirror/WaveHolomorphy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [II.D70] Classifications distinct +noindex: false +--- + diff --git a/_corpus_v3_items/cid005647.md b/_corpus_v3_items/cid005647.md new file mode 100644 index 000000000..53e1872eb --- /dev/null +++ b/_corpus_v3_items/cid005647.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005647/ +title: code_delta_3 (theorem) +cid: cid005647 +primary_alias: FTH0385 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Code extraction [II.D51]' +aliases: +- FTH0385 +- code_delta_3 +- code-delta-3 +- TauLib.BookII.Regularity.CodeDecode::code_delta_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001286 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: code_delta_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Code extraction [II.D51]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005648.md b/_corpus_v3_items/cid005648.md new file mode 100644 index 000000000..f114f71a9 --- /dev/null +++ b/_corpus_v3_items/cid005648.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005648/ +title: decode_welldef_3_15 (theorem) +cid: cid005648 +primary_alias: FTH0386 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Decode well-definedness [II.D52] +aliases: +- FTH0386 +- decode_welldef_3_15 +- decode-welldef-3-15 +- TauLib.BookII.Regularity.CodeDecode::decode_welldef_3_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001287 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: decode_welldef_3_15 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Decode well-definedness [II.D52] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005649.md b/_corpus_v3_items/cid005649.md new file mode 100644 index 000000000..db3f4f3c1 --- /dev/null +++ b/_corpus_v3_items/cid005649.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005649/ +title: decode_unique_3 (theorem) +cid: cid005649 +primary_alias: FTH0387 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Decode uniqueness [II.D52] +aliases: +- FTH0387 +- decode_unique_3 +- decode-unique-3 +- TauLib.BookII.Regularity.CodeDecode::decode_unique_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001287 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: decode_unique_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Decode uniqueness [II.D52] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005650.md b/_corpus_v3_items/cid005650.md new file mode 100644 index 000000000..94e20439c --- /dev/null +++ b/_corpus_v3_items/cid005650.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005650/ +title: code_decode_3 (theorem) +cid: cid005650 +primary_alias: FTH0388 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Code/Decode bijection [II.T35] +aliases: +- FTH0388 +- code_decode_3 +- code-decode-3 +- TauLib.BookII.Regularity.CodeDecode::code_decode_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: code_decode_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Code/Decode bijection [II.T35] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005651.md b/_corpus_v3_items/cid005651.md new file mode 100644 index 000000000..aa5464943 --- /dev/null +++ b/_corpus_v3_items/cid005651.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005651/ +title: code_decode_delta_3 (theorem) +cid: cid005651 +primary_alias: FTH0389 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Delta round-trip [II.T35] +aliases: +- FTH0389 +- code_decode_delta_3 +- code-decode-delta-3 +- TauLib.BookII.Regularity.CodeDecode::code_decode_delta_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: code_decode_delta_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Delta round-trip [II.T35] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005652.md b/_corpus_v3_items/cid005652.md new file mode 100644 index 000000000..6632fbc99 --- /dev/null +++ b/_corpus_v3_items/cid005652.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005652/ +title: spectral_sep_3 (theorem) +cid: cid005652 +primary_alias: FTH0390 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Spectral separation [II.T35] +aliases: +- FTH0390 +- spectral_sep_3 +- spectral-sep-3 +- TauLib.BookII.Regularity.CodeDecode::spectral_sep_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: spectral_sep_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Spectral separation [II.T35] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005653.md b/_corpus_v3_items/cid005653.md new file mode 100644 index 000000000..0e6504745 --- /dev/null +++ b/_corpus_v3_items/cid005653.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005653/ +title: full_code_decode_3 (theorem) +cid: cid005653 +primary_alias: FTH0391 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full Code/Decode [II.T35] +aliases: +- FTH0391 +- full_code_decode_3 +- full-code-decode-3 +- TauLib.BookII.Regularity.CodeDecode::full_code_decode_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: full_code_decode_3 + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full Code/Decode [II.T35] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005654.md b/_corpus_v3_items/cid005654.md new file mode 100644 index 000000000..7bd5f0b50 --- /dev/null +++ b/_corpus_v3_items/cid005654.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005654/ +title: code_decode_id_roundtrip (theorem) +cid: cid005654 +primary_alias: FTH0392 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T35] Formal proof: Decode . Code = id for the identity function.\n This follows from reduction idempotence:\ + \ reduce(reduce(x, k), k) = reduce(x, k). -/" +aliases: +- FTH0392 +- code_decode_id_roundtrip +- code-decode-id-roundtrip +- TauLib.BookII.Regularity.CodeDecode::code_decode_id_roundtrip +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: code_decode_id_roundtrip + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T35] Formal proof: Decode . Code = id for the identity function.\n This follows from reduction idempotence:\ + \ reduce(reduce(x, k), k) = reduce(x, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005655.md b/_corpus_v3_items/cid005655.md new file mode 100644 index 000000000..32671a3ab --- /dev/null +++ b/_corpus_v3_items/cid005655.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005655/ +title: decode_code_roundtrip (theorem) +cid: cid005655 +primary_alias: FTH0393 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.T35] Formal proof: Code . Decode = id for any table.\n This follows from reduce(reduce(a, k), k) = reduce(a,\ + \ k). -/" +aliases: +- FTH0393 +- decode_code_roundtrip +- decode-code-roundtrip +- TauLib.BookII.Regularity.CodeDecode::decode_code_roundtrip +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001396 +- predicate: part_of + object: cid005179 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005179 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.CodeDecode + declaration: decode_code_roundtrip + module_path: TauLib/BookII/Regularity/CodeDecode.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.T35] Formal proof: Code . Decode = id for any table.\n This follows from reduce(reduce(a, k), k)\ + \ = reduce(a, k). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005656.md b/_corpus_v3_items/cid005656.md new file mode 100644 index 000000000..3c67df7f5 --- /dev/null +++ b/_corpus_v3_items/cid005656.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005656/ +title: decompose_recovery (theorem) +cid: cid005656 +primary_alias: FTH0394 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.L07] Idempotent Decomposition Lemma (formal):\n e₊ · bp + e₋ · bp = bp for all sector pairs bp.\n\n \ + \ This is the fundamental decomposition: every element of the\n bipolar spectral algebra decomposes uniquely into B-channel\n\ + \ and C-channel components, and the sum recovers the original. -/" +aliases: +- FTH0394 +- decompose_recovery +- decompose-recovery +- TauLib.BookII.Regularity.IdempotentDecomposition::decompose_recovery +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001334 +- predicate: part_of + object: cid005180 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005180 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.IdempotentDecomposition + declaration: decompose_recovery + module_path: TauLib/BookII/Regularity/IdempotentDecomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.L07] Idempotent Decomposition Lemma (formal):\n e₊ · bp + e₋ · bp = bp for all sector pairs bp.\n\ + \n This is the fundamental decomposition: every element of the\n bipolar spectral algebra decomposes uniquely into\ + \ B-channel\n a" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005657.md b/_corpus_v3_items/cid005657.md new file mode 100644 index 000000000..d7f0daaf7 --- /dev/null +++ b/_corpus_v3_items/cid005657.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005657/ +title: recovery_30 (theorem) +cid: cid005657 +primary_alias: FTH0395 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Recovery [II.L07]' +aliases: +- FTH0395 +- recovery_30 +- recovery-30 +- TauLib.BookII.Regularity.IdempotentDecomposition::recovery_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001334 +- predicate: part_of + object: cid005180 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005180 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.IdempotentDecomposition + declaration: recovery_30 + module_path: TauLib/BookII/Regularity/IdempotentDecomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Recovery [II.L07]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005658.md b/_corpus_v3_items/cid005658.md new file mode 100644 index 000000000..c57f64f67 --- /dev/null +++ b/_corpus_v3_items/cid005658.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005658/ +title: stagefun_decompose_12_4 (theorem) +cid: cid005658 +primary_alias: FTH0396 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Stagefun decomposition [II.D48] +aliases: +- FTH0396 +- stagefun_decompose_12_4 +- stagefun-decompose-12-4 +- TauLib.BookII.Regularity.IdempotentDecomposition::stagefun_decompose_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001283 +- predicate: part_of + object: cid005180 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005180 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.IdempotentDecomposition + declaration: stagefun_decompose_12_4 + module_path: TauLib/BookII/Regularity/IdempotentDecomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Stagefun decomposition [II.D48] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005659.md b/_corpus_v3_items/cid005659.md new file mode 100644 index 000000000..e00c1b09e --- /dev/null +++ b/_corpus_v3_items/cid005659.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005659/ +title: functorial_12_4 (theorem) +cid: cid005659 +primary_alias: FTH0397 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Functoriality [II.P10] +aliases: +- FTH0397 +- functorial_12_4 +- functorial-12-4 +- TauLib.BookII.Regularity.IdempotentDecomposition::functorial_12_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001351 +- predicate: part_of + object: cid005180 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005180 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.IdempotentDecomposition + declaration: functorial_12_4 + module_path: TauLib/BookII/Regularity/IdempotentDecomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Functoriality [II.P10] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005660.md b/_corpus_v3_items/cid005660.md new file mode 100644 index 000000000..acbd39dd6 --- /dev/null +++ b/_corpus_v3_items/cid005660.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005660/ +title: depth_max_20_4 (theorem) +cid: cid005660 +primary_alias: FTH0398 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Regularity depth = max [II.T34]' +aliases: +- FTH0398 +- depth_max_20_4 +- depth-max-20-4 +- TauLib.BookII.Regularity.PositiveRegularity::depth_max_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001395 +- predicate: part_of + object: cid005181 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005181 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PositiveRegularity + declaration: depth_max_20_4 + module_path: TauLib/BookII/Regularity/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Regularity depth = max [II.T34]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005661.md b/_corpus_v3_items/cid005661.md new file mode 100644 index 000000000..6468a624a --- /dev/null +++ b/_corpus_v3_items/cid005661.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005661/ +title: criterion_20_4 (theorem) +cid: cid005661 +primary_alias: FTH0399 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Regularity criterion [II.T34] +aliases: +- FTH0399 +- criterion_20_4 +- criterion-20-4 +- TauLib.BookII.Regularity.PositiveRegularity::criterion_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001395 +- predicate: part_of + object: cid005181 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005181 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PositiveRegularity + declaration: criterion_20_4 + module_path: TauLib/BookII/Regularity/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Regularity criterion [II.T34] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005662.md b/_corpus_v3_items/cid005662.md new file mode 100644 index 000000000..894679c3c --- /dev/null +++ b/_corpus_v3_items/cid005662.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005662/ +title: small_point_3 (theorem) +cid: cid005662 +primary_alias: FTH0400 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Small point regularity [II.D49] +aliases: +- FTH0400 +- small_point_3 +- small-point-3 +- TauLib.BookII.Regularity.PositiveRegularity::small_point_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001284 +- predicate: part_of + object: cid005181 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005181 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PositiveRegularity + declaration: small_point_3 + module_path: TauLib/BookII/Regularity/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Small point regularity [II.D49] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005663.md b/_corpus_v3_items/cid005663.md new file mode 100644 index 000000000..1b6e748a6 --- /dev/null +++ b/_corpus_v3_items/cid005663.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005663/ +title: evolution_stab_20_4 (theorem) +cid: cid005663 +primary_alias: FTH0401 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Evolution stabilization [II.D49] +aliases: +- FTH0401 +- evolution_stab_20_4 +- evolution-stab-20-4 +- TauLib.BookII.Regularity.PositiveRegularity::evolution_stab_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001284 +- predicate: part_of + object: cid005181 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005181 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PositiveRegularity + declaration: evolution_stab_20_4 + module_path: TauLib/BookII/Regularity/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Evolution stabilization [II.D49] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005664.md b/_corpus_v3_items/cid005664.md new file mode 100644 index 000000000..0a8d20d6d --- /dev/null +++ b/_corpus_v3_items/cid005664.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005664/ +title: preyoneda_tower_15_4 (theorem) +cid: cid005664 +primary_alias: FTH0402 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Tower coherence [II.D50]' +aliases: +- FTH0402 +- preyoneda_tower_15_4 +- preyoneda-tower-15-4 +- TauLib.BookII.Regularity.PreYoneda::preyoneda_tower_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001285 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_tower_15_4 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Tower coherence [II.D50]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005665.md b/_corpus_v3_items/cid005665.md new file mode 100644 index 000000000..b9c77156d --- /dev/null +++ b/_corpus_v3_items/cid005665.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005665/ +title: preyoneda_identity_15_4 (theorem) +cid: cid005665 +primary_alias: FTH0403 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Identity [II.D50] +aliases: +- FTH0403 +- preyoneda_identity_15_4 +- preyoneda-identity-15-4 +- TauLib.BookII.Regularity.PreYoneda::preyoneda_identity_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001285 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_identity_15_4 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Identity [II.D50] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005666.md b/_corpus_v3_items/cid005666.md new file mode 100644 index 000000000..195dbd630 --- /dev/null +++ b/_corpus_v3_items/cid005666.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005666/ +title: preyoneda_composition_12_3 (theorem) +cid: cid005666 +primary_alias: FTH0404 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Composition [II.D50] +aliases: +- FTH0404 +- preyoneda_composition_12_3 +- preyoneda-composition-12-3 +- TauLib.BookII.Regularity.PreYoneda::preyoneda_composition_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001285 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_composition_12_3 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Composition [II.D50] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005667.md b/_corpus_v3_items/cid005667.md new file mode 100644 index 000000000..a762a0251 --- /dev/null +++ b/_corpus_v3_items/cid005667.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005667/ +title: preyoneda_bipolar_12_3 (theorem) +cid: cid005667 +primary_alias: FTH0405 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Bipolar structure [II.P11] +aliases: +- FTH0405 +- preyoneda_bipolar_12_3 +- preyoneda-bipolar-12-3 +- TauLib.BookII.Regularity.PreYoneda::preyoneda_bipolar_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001352 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_bipolar_12_3 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Bipolar structure [II.P11] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005668.md b/_corpus_v3_items/cid005668.md new file mode 100644 index 000000000..17c7e5308 --- /dev/null +++ b/_corpus_v3_items/cid005668.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005668/ +title: preyoneda_abcd_12_3 (theorem) +cid: cid005668 +primary_alias: FTH0406 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- ABCD coordinates [II.P11] +aliases: +- FTH0406 +- preyoneda_abcd_12_3 +- preyoneda-abcd-12-3 +- TauLib.BookII.Regularity.PreYoneda::preyoneda_abcd_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001352 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_abcd_12_3 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- ABCD coordinates [II.P11] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005669.md b/_corpus_v3_items/cid005669.md new file mode 100644 index 000000000..8c1a17a45 --- /dev/null +++ b/_corpus_v3_items/cid005669.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005669/ +title: probe_nat_10_3 (theorem) +cid: cid005669 +primary_alias: FTH0407 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Probe naturality [II.R12] +aliases: +- FTH0407 +- probe_nat_10_3 +- probe-nat-10-3 +- TauLib.BookII.Regularity.PreYoneda::probe_nat_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005182 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: probe_nat_10_3 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Probe naturality [II.R12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005670.md b/_corpus_v3_items/cid005670.md new file mode 100644 index 000000000..581fdad83 --- /dev/null +++ b/_corpus_v3_items/cid005670.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005670/ +title: probe_tower_15_4 (theorem) +cid: cid005670 +primary_alias: FTH0408 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Probe implies tower coherence [II.R12] +aliases: +- FTH0408 +- probe_tower_15_4 +- probe-tower-15-4 +- TauLib.BookII.Regularity.PreYoneda::probe_tower_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005182 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: probe_tower_15_4 + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Probe implies tower coherence [II.R12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005671.md b/_corpus_v3_items/cid005671.md new file mode 100644 index 000000000..04012e549 --- /dev/null +++ b/_corpus_v3_items/cid005671.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005671/ +title: preyoneda_id_tower_coherent (theorem) +cid: cid005671 +primary_alias: FTH0409 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.D50] Formal proof: the pre-Yoneda embedding of the identity\n function is tower-coherent.\n\n preyoneda(id,\ + \ x, k) = reduce(x, k), so tower coherence reduces to\n reduce(reduce(x, k+1), k) = reduce(x, k), which is reduction_compat.\ + \ -/" +aliases: +- FTH0409 +- preyoneda_id_tower_coherent +- preyoneda-id-tower-coherent +- TauLib.BookII.Regularity.PreYoneda::preyoneda_id_tower_coherent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001285 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: preyoneda_id_tower_coherent + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.D50] Formal proof: the pre-Yoneda embedding of the identity\n function is tower-coherent.\n\n \ + \ preyoneda(id, x, k) = reduce(x, k), so tower coherence reduces to\n reduce(reduce(x, k+1), k) = reduce(x, k), which\ + \ is reduction_comp" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005672.md b/_corpus_v3_items/cid005672.md new file mode 100644 index 000000000..b9eb6f0f4 --- /dev/null +++ b/_corpus_v3_items/cid005672.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005672/ +title: probe_naturality_is_tower_coherence (theorem) +cid: cid005672 +primary_alias: FTH0410 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [II.R12] Formal proof: probe naturality IS tower coherence.\n For the identity embedding, naturality at stage\ + \ transition (k, k+1)\n is exactly reduction_compat. -/" +aliases: +- FTH0410 +- probe_naturality_is_tower_coherence +- probe-naturality-is-tower-coherence +- TauLib.BookII.Regularity.PreYoneda::probe_naturality_is_tower_coherence +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005182 +- predicate: part_of + object: cid005182 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005182 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.PreYoneda + declaration: probe_naturality_is_tower_coherence + module_path: TauLib/BookII/Regularity/PreYoneda.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [II.R12] Formal proof: probe naturality IS tower coherence.\n For the identity embedding, naturality\ + \ at stage transition (k, k+1)\n is exactly reduction_compat. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005673.md b/_corpus_v3_items/cid005673.md new file mode 100644 index 000000000..05dbcafd4 --- /dev/null +++ b/_corpus_v3_items/cid005673.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005673/ +title: branch_fact_12_3 (theorem) +cid: cid005673 +primary_alias: FTH0411 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Branch factorization [II.L08]' +aliases: +- FTH0411 +- branch_fact_12_3 +- branch-fact-12-3 +- TauLib.BookII.Regularity.ThreeLemmaChain::branch_fact_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001335 +- predicate: part_of + object: cid005183 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005183 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + declaration: branch_fact_12_3 + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Branch factorization [II.L08]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005674.md b/_corpus_v3_items/cid005674.md new file mode 100644 index 000000000..b137b335e --- /dev/null +++ b/_corpus_v3_items/cid005674.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005674/ +title: prime_split_30 (theorem) +cid: cid005674 +primary_alias: FTH0412 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Prime-split support [II.L09] +aliases: +- FTH0412 +- prime_split_30 +- prime-split-30 +- TauLib.BookII.Regularity.ThreeLemmaChain::prime_split_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001336 +- predicate: part_of + object: cid005183 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005183 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + declaration: prime_split_30 + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Prime-split support [II.L09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005675.md b/_corpus_v3_items/cid005675.md new file mode 100644 index 000000000..6bbbdaa63 --- /dev/null +++ b/_corpus_v3_items/cid005675.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005675/ +title: polarity_30 (theorem) +cid: cid005675 +primary_alias: FTH0413 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Polarity symmetry [II.L10] +aliases: +- FTH0413 +- polarity_30 +- polarity-30 +- TauLib.BookII.Regularity.ThreeLemmaChain::polarity_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001337 +- predicate: part_of + object: cid005183 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005183 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + declaration: polarity_30 + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Polarity symmetry [II.L10] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005676.md b/_corpus_v3_items/cid005676.md new file mode 100644 index 000000000..20849fedb --- /dev/null +++ b/_corpus_v3_items/cid005676.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005676/ +title: is1_12_3 (theorem) +cid: cid005676 +primary_alias: FTH0414 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- IS conditions [II.T33] +aliases: +- FTH0414 +- is1_12_3 +- is1-12-3 +- TauLib.BookII.Regularity.ThreeLemmaChain::is1_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001394 +- predicate: part_of + object: cid005183 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005183 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + declaration: is1_12_3 + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- IS conditions [II.T33] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005677.md b/_corpus_v3_items/cid005677.md new file mode 100644 index 000000000..a2b48c0eb --- /dev/null +++ b/_corpus_v3_items/cid005677.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005677/ +title: hol_iff_is_12_3 (theorem) +cid: cid005677 +primary_alias: FTH0415 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Holomorphic iff IS [II.T33] +aliases: +- FTH0415 +- hol_iff_is_12_3 +- hol-iff-is-12-3 +- TauLib.BookII.Regularity.ThreeLemmaChain::hol_iff_is_12_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001394 +- predicate: part_of + object: cid005183 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005183 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookII.Regularity.ThreeLemmaChain + declaration: hol_iff_is_12_3 + module_path: TauLib/BookII/Regularity/ThreeLemmaChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Holomorphic iff IS [II.T33] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005678.md b/_corpus_v3_items/cid005678.md new file mode 100644 index 000000000..15ec42b10 --- /dev/null +++ b/_corpus_v3_items/cid005678.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005678/ +title: radical_check_30 (theorem) +cid: cid005678 +primary_alias: FTH0416 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D97] Radical is well-defined and idempotent up to 30. -/ +aliases: +- FTH0416 +- radical_check_30 +- radical-check-30 +- TauLib.BookIII.Arithmetic.ABCConjecture::radical_check_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001542 +- predicate: part_of + object: cid005195 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005195 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + declaration: radical_check_30 + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D97] Radical is well-defined and idempotent up to 30. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005679.md b/_corpus_v3_items/cid005679.md new file mode 100644 index 000000000..370088b81 --- /dev/null +++ b/_corpus_v3_items/cid005679.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005679/ +title: abc_quality_15 (theorem) +cid: cid005679 +primary_alias: FTH0417 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D98] ABC quality holds for coprime pairs up to 15. -/ +aliases: +- FTH0417 +- abc_quality_15 +- abc-quality-15 +- TauLib.BookIII.Arithmetic.ABCConjecture::abc_quality_15 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001543 +- predicate: part_of + object: cid005195 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005195 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + declaration: abc_quality_15 + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D98] ABC quality holds for coprime pairs up to 15. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005680.md b/_corpus_v3_items/cid005680.md new file mode 100644 index 000000000..9d5b66291 --- /dev/null +++ b/_corpus_v3_items/cid005680.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005680/ +title: abc_primorial_3 (theorem) +cid: cid005680 +primary_alias: FTH0418 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T65] ABC at primorial levels up to depth 3. -/ +aliases: +- FTH0418 +- abc_primorial_3 +- abc-primorial-3 +- TauLib.BookIII.Arithmetic.ABCConjecture::abc_primorial_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001657 +- predicate: part_of + object: cid005195 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005195 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + declaration: abc_primorial_3 + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T65] ABC at primorial levels up to depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005681.md b/_corpus_v3_items/cid005681.md new file mode 100644 index 000000000..fd509741e --- /dev/null +++ b/_corpus_v3_items/cid005681.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005681/ +title: radical_primorial_4 (theorem) +cid: cid005681 +primary_alias: FTH0419 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P41] Radical-primorial identity up to depth 4. -/ +aliases: +- FTH0419 +- radical_primorial_4 +- radical-primorial-4 +- TauLib.BookIII.Arithmetic.ABCConjecture::radical_primorial_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001585 +- predicate: part_of + object: cid005195 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005195 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + declaration: radical_primorial_4 + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P41] Radical-primorial identity up to depth 4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005682.md b/_corpus_v3_items/cid005682.md new file mode 100644 index 000000000..9d859f252 --- /dev/null +++ b/_corpus_v3_items/cid005682.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005682/ +title: radical_le_30 (theorem) +cid: cid005682 +primary_alias: FTH0420 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P41] Radical ≤ n for all n up to 30. -/ +aliases: +- FTH0420 +- radical_le_30 +- radical-le-30 +- TauLib.BookIII.Arithmetic.ABCConjecture::radical_le_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001585 +- predicate: part_of + object: cid005195 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005195 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCConjecture + declaration: radical_le_30 + module_path: TauLib/BookIII/Arithmetic/ABCConjecture.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P41] Radical ≤ n for all n up to 30. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005683.md b/_corpus_v3_items/cid005683.md new file mode 100644 index 000000000..b53854797 --- /dev/null +++ b/_corpus_v3_items/cid005683.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005683/ +title: abc_quality_100 (theorem) +cid: cid005683 +primary_alias: FTH0421 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T76] Weak ABC (c < rad(abc)²) for all coprime pairs up to 100. -/ +aliases: +- FTH0421 +- abc_quality_100 +- TauLib.BookIII.Arithmetic.ABCDeep::abc_quality_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001668 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: abc_quality_100 + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T76] Weak ABC (c < rad(abc)²) for all coprime pairs up to 100. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005684.md b/_corpus_v3_items/cid005684.md new file mode 100644 index 000000000..7164cd424 --- /dev/null +++ b/_corpus_v3_items/cid005684.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005684/ +title: squarefree_dominance_100 (theorem) +cid: cid005684 +primary_alias: FTH0422 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T77] Squarefree dominance: c ≤ rad(abc) for all squarefree\n coprime pairs up to 100. -/" +aliases: +- FTH0422 +- squarefree_dominance_100 +- squarefree-dominance-100 +- TauLib.BookIII.Arithmetic.ABCDeep::squarefree_dominance_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001669 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: squarefree_dominance_100 + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T77] Squarefree dominance: c ≤ rad(abc) for all squarefree\n coprime pairs up to 100. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005685.md b/_corpus_v3_items/cid005685.md new file mode 100644 index 000000000..3b307a2b8 --- /dev/null +++ b/_corpus_v3_items/cid005685.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005685/ +title: radical_primorial_5 (theorem) +cid: cid005685 +primary_alias: FTH0423 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T78] Radical-primorial identity at depth 5: rad(M_k) = M_k. -/' +aliases: +- FTH0423 +- radical_primorial_5 +- TauLib.BookIII.Arithmetic.ABCDeep::radical_primorial_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001670 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: radical_primorial_5 + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T78] Radical-primorial identity at depth 5: rad(M_k) = M_k. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005686.md b/_corpus_v3_items/cid005686.md new file mode 100644 index 000000000..245d9d7ee --- /dev/null +++ b/_corpus_v3_items/cid005686.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005686/ +title: squarefree_dominance_thm (theorem) +cid: cid005686 +primary_alias: FTH0424 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P47] Zero high-quality triples among squarefree coprimes ≤ 50. -/ +aliases: +- FTH0424 +- squarefree_dominance_thm +- squarefree-dominance-thm +- TauLib.BookIII.Arithmetic.ABCDeep::squarefree_dominance_thm +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001591 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: squarefree_dominance_thm + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P47] Zero high-quality triples among squarefree coprimes ≤ 50. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005687.md b/_corpus_v3_items/cid005687.md new file mode 100644 index 000000000..a7c26e665 --- /dev/null +++ b/_corpus_v3_items/cid005687.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005687/ +title: high_quality_30 (theorem) +cid: cid005687 +primary_alias: FTH0425 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D109] Few high quality triples (q ≥ 1) up to 30: at most 5. -/' +aliases: +- FTH0425 +- high_quality_30 +- high-quality-30 +- TauLib.BookIII.Arithmetic.ABCDeep::high_quality_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001451 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: high_quality_30 + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D109] Few high quality triples (q ≥ 1) up to 30: at most 5. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005688.md b/_corpus_v3_items/cid005688.md new file mode 100644 index 000000000..3cb5650bf --- /dev/null +++ b/_corpus_v3_items/cid005688.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005688/ +title: one_squarefree (theorem) +cid: cid005688 +primary_alias: FTH0426 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D110] 1 is squarefree. -/ +aliases: +- FTH0426 +- one_squarefree +- one-squarefree +- TauLib.BookIII.Arithmetic.ABCDeep::one_squarefree +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001453 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: one_squarefree + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D110] 1 is squarefree. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005689.md b/_corpus_v3_items/cid005689.md new file mode 100644 index 000000000..dc507e473 --- /dev/null +++ b/_corpus_v3_items/cid005689.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005689/ +title: thirty_squarefree (theorem) +cid: cid005689 +primary_alias: FTH0427 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D110] 30 is squarefree (2·3·5). -/ +aliases: +- FTH0427 +- thirty_squarefree +- thirty-squarefree +- TauLib.BookIII.Arithmetic.ABCDeep::thirty_squarefree +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001453 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: thirty_squarefree + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D110] 30 is squarefree (2·3·5). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005690.md b/_corpus_v3_items/cid005690.md new file mode 100644 index 000000000..13b0613d3 --- /dev/null +++ b/_corpus_v3_items/cid005690.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005690/ +title: twelve_not_squarefree (theorem) +cid: cid005690 +primary_alias: FTH0428 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D110] 12 is NOT squarefree (4 | 12). -/ +aliases: +- FTH0428 +- twelve_not_squarefree +- twelve-not-squarefree +- TauLib.BookIII.Arithmetic.ABCDeep::twelve_not_squarefree +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001453 +- predicate: part_of + object: cid005196 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005196 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ABCDeep + declaration: twelve_not_squarefree + module_path: TauLib/BookIII/Arithmetic/ABCDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D110] 12 is NOT squarefree (4 | 12). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005691.md b/_corpus_v3_items/cid005691.md new file mode 100644 index 000000000..4a19e490b --- /dev/null +++ b/_corpus_v3_items/cid005691.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005691/ +title: bsd_coherence_1 (theorem) +cid: cid005691 +primary_alias: FTH0429 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T35] Structural: BSD coherence at depth 1. -/' +aliases: +- FTH0429 +- bsd_coherence_1 +- bsd-coherence-1 +- TauLib.BookIII.Arithmetic.BSD::bsd_coherence_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001627 +- predicate: part_of + object: cid005197 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005197 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.BSD + declaration: bsd_coherence_1 + module_path: TauLib/BookIII/Arithmetic/BSD.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T35] Structural: BSD coherence at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005692.md b/_corpus_v3_items/cid005692.md new file mode 100644 index 000000000..b76e76381 --- /dev/null +++ b/_corpus_v3_items/cid005692.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005692/ +title: bsd_level (theorem) +cid: cid005692 +primary_alias: FTH0430 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P27] Structural: BSD is at E₁→E₂ interface. -/' +aliases: +- FTH0430 +- bsd_level +- bsd-level +- TauLib.BookIII.Arithmetic.BSD::bsd_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001571 +- predicate: part_of + object: cid005197 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005197 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.BSD + declaration: bsd_level + module_path: TauLib/BookIII/Arithmetic/BSD.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P27] Structural: BSD is at E₁→E₂ interface. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005693.md b/_corpus_v3_items/cid005693.md new file mode 100644 index 000000000..f5182360f --- /dev/null +++ b/_corpus_v3_items/cid005693.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005693/ +title: enr_01_10_1 (theorem) +cid: cid005693 +primary_alias: FTH0431 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D57] Structural: enrichment at depth 1. -/' +aliases: +- FTH0431 +- enr_01_10_1 +- enr-01-10-1 +- TauLib.BookIII.Arithmetic.EnrFunctor01::enr_01_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001502 +- predicate: part_of + object: cid005198 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005198 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrFunctor01 + declaration: enr_01_10_1 + module_path: TauLib/BookIII/Arithmetic/EnrFunctor01.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D57] Structural: enrichment at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005694.md b/_corpus_v3_items/cid005694.md new file mode 100644 index 000000000..e83af60b1 --- /dev/null +++ b/_corpus_v3_items/cid005694.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005694/ +title: e1_level (theorem) +cid: cid005694 +primary_alias: FTH0432 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D58] Structural: E₁ level is 1. -/' +aliases: +- FTH0432 +- e1_level +- e1-level +- TauLib.BookIII.Arithmetic.EnrFunctor01::e1_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001503 +- predicate: part_of + object: cid005198 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005198 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrFunctor01 + declaration: e1_level + module_path: TauLib/BookIII/Arithmetic/EnrFunctor01.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D58] Structural: E₁ level is 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005695.md b/_corpus_v3_items/cid005695.md new file mode 100644 index 000000000..87a92e848 --- /dev/null +++ b/_corpus_v3_items/cid005695.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005695/ +title: all_e1 (theorem) +cid: cid005695 +primary_alias: FTH0433 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P24] Structural: NS, YM, Hodge are all at E₁. -/' +aliases: +- FTH0433 +- all_e1 +- all-e1 +- TauLib.BookIII.Arithmetic.EnrFunctor01::all_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001568 +- predicate: part_of + object: cid005198 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005198 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrFunctor01 + declaration: all_e1 + module_path: TauLib/BookIII/Arithmetic/EnrFunctor01.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P24] Structural: NS, YM, Hodge are all at E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005696.md b/_corpus_v3_items/cid005696.md new file mode 100644 index 000000000..4e3aa7ef7 --- /dev/null +++ b/_corpus_v3_items/cid005696.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005696/ +title: enriched_bs_10_1 (theorem) +cid: cid005696 +primary_alias: FTH0434 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D65] Structural: enriched bi-square at depth 1. -/' +aliases: +- FTH0434 +- enriched_bs_10_1 +- enriched-bs-10-1 +- TauLib.BookIII.Arithmetic.EnrichedBiSquare::enriched_bs_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001510 +- predicate: part_of + object: cid005199 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005199 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrichedBiSquare + declaration: enriched_bs_10_1 + module_path: TauLib/BookIII/Arithmetic/EnrichedBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D65] Structural: enriched bi-square at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005697.md b/_corpus_v3_items/cid005697.md new file mode 100644 index 000000000..de87f49fa --- /dev/null +++ b/_corpus_v3_items/cid005697.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005697/ +title: factorization_10_1 (theorem) +cid: cid005697 +primary_alias: FTH0435 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T38] Structural: factorization at depth 1. -/' +aliases: +- FTH0435 +- factorization_10_1 +- factorization-10-1 +- TauLib.BookIII.Arithmetic.EnrichedBiSquare::factorization_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001630 +- predicate: part_of + object: cid005199 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005199 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrichedBiSquare + declaration: factorization_10_1 + module_path: TauLib/BookIII/Arithmetic/EnrichedBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T38] Structural: factorization at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005698.md b/_corpus_v3_items/cid005698.md new file mode 100644 index 000000000..93b5f79a5 --- /dev/null +++ b/_corpus_v3_items/cid005698.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005698/ +title: comparison_10_1 (theorem) +cid: cid005698 +primary_alias: FTH0436 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T39] Structural: comparison at depth 1. -/' +aliases: +- FTH0436 +- comparison_10_1 +- comparison-10-1 +- TauLib.BookIII.Arithmetic.EnrichedBiSquare::comparison_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001631 +- predicate: part_of + object: cid005199 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005199 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.EnrichedBiSquare + declaration: comparison_10_1 + module_path: TauLib/BookIII/Arithmetic/EnrichedBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T39] Structural: comparison at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005699.md b/_corpus_v3_items/cid005699.md new file mode 100644 index 000000000..4d2f1da98 --- /dev/null +++ b/_corpus_v3_items/cid005699.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005699/ +title: langlands_level (theorem) +cid: cid005699 +primary_alias: FTH0437 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D63] Structural: Langlands is at E₁→E₂ interface. -/' +aliases: +- FTH0437 +- langlands_level +- langlands-level +- TauLib.BookIII.Arithmetic.Langlands::langlands_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001508 +- predicate: part_of + object: cid005200 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005200 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.Langlands + declaration: langlands_level + module_path: TauLib/BookIII/Arithmetic/Langlands.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D63] Structural: Langlands is at E₁→E₂ interface. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005700.md b/_corpus_v3_items/cid005700.md new file mode 100644 index 000000000..9f3303f0b --- /dev/null +++ b/_corpus_v3_items/cid005700.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005700/ +title: func_10_1 (theorem) +cid: cid005700 +primary_alias: FTH0438 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T36] Structural: functoriality at depth 1. -/' +aliases: +- FTH0438 +- func_10_1 +- func-10-1 +- TauLib.BookIII.Arithmetic.Langlands::func_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001628 +- predicate: part_of + object: cid005200 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005200 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.Langlands + declaration: func_10_1 + module_path: TauLib/BookIII/Arithmetic/Langlands.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T36] Structural: functoriality at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005701.md b/_corpus_v3_items/cid005701.md new file mode 100644 index 000000000..2fc6ab089 --- /dev/null +++ b/_corpus_v3_items/cid005701.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005701/ +title: base_change_10_1 (theorem) +cid: cid005701 +primary_alias: FTH0439 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T37] Structural: base change at depth 1. -/' +aliases: +- FTH0439 +- base_change_10_1 +- base-change-10-1 +- TauLib.BookIII.Arithmetic.Langlands::base_change_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001629 +- predicate: part_of + object: cid005200 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005200 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.Langlands + declaration: base_change_10_1 + module_path: TauLib/BookIII/Arithmetic/Langlands.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T37] Structural: base change at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005702.md b/_corpus_v3_items/cid005702.md new file mode 100644 index 000000000..30d2e9e98 --- /dev/null +++ b/_corpus_v3_items/cid005702.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005702/ +title: proto_zero_verified (theorem) +cid: cid005702 +primary_alias: FTH0440 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D61] Structural: proto-code of 0 is verified. -/' +aliases: +- FTH0440 +- proto_zero_verified +- proto-zero-verified +- TauLib.BookIII.Arithmetic.ProtoCodes::proto_zero_verified +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001506 +- predicate: part_of + object: cid005201 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005201 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ProtoCodes + declaration: proto_zero_verified + module_path: TauLib/BookIII/Arithmetic/ProtoCodes.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D61] Structural: proto-code of 0 is verified. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005703.md b/_corpus_v3_items/cid005703.md new file mode 100644 index 000000000..85ba2b262 --- /dev/null +++ b/_corpus_v3_items/cid005703.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005703/ +title: bsd_nonneg_1 (theorem) +cid: cid005703 +primary_alias: FTH0441 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D62] Structural: BSD at depth 1 is non-negative. -/' +aliases: +- FTH0441 +- bsd_nonneg_1 +- bsd-nonneg-1 +- TauLib.BookIII.Arithmetic.ProtoCodes::bsd_nonneg_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001507 +- predicate: part_of + object: cid005201 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005201 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ProtoCodes + declaration: bsd_nonneg_1 + module_path: TauLib/BookIII/Arithmetic/ProtoCodes.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D62] Structural: BSD at depth 1 is non-negative. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005704.md b/_corpus_v3_items/cid005704.md new file mode 100644 index 000000000..a4ea975c0 --- /dev/null +++ b/_corpus_v3_items/cid005704.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005704/ +title: bridgehead_1 (theorem) +cid: cid005704 +primary_alias: FTH0442 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P26] Structural: bridgehead at depth 1. -/' +aliases: +- FTH0442 +- bridgehead_1 +- bridgehead-1 +- TauLib.BookIII.Arithmetic.ProtoCodes::bridgehead_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001570 +- predicate: part_of + object: cid005201 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005201 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.ProtoCodes + declaration: bridgehead_1 + module_path: TauLib/BookIII/Arithmetic/ProtoCodes.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P26] Structural: bridgehead at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005705.md b/_corpus_v3_items/cid005705.md new file mode 100644 index 000000000..b2b6c8252 --- /dev/null +++ b/_corpus_v3_items/cid005705.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005705/ +title: zero_rational (theorem) +cid: cid005705 +primary_alias: FTH0443 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D59] Structural: 0 is rational at every depth. -/' +aliases: +- FTH0443 +- zero_rational +- zero-rational +- TauLib.BookIII.Arithmetic.RationalPoints::zero_rational +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001504 +- predicate: part_of + object: cid005202 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005202 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.RationalPoints + declaration: zero_rational + module_path: TauLib/BookIII/Arithmetic/RationalPoints.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D59] Structural: 0 is rational at every depth. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005706.md b/_corpus_v3_items/cid005706.md new file mode 100644 index 000000000..945f0fc01 --- /dev/null +++ b/_corpus_v3_items/cid005706.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005706/ +title: rank_bounded (theorem) +cid: cid005706 +primary_alias: FTH0444 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D60] Structural: rank is bounded by db. -/' +aliases: +- FTH0444 +- rank_bounded +- rank-bounded +- TauLib.BookIII.Arithmetic.RationalPoints::rank_bounded +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001505 +- predicate: part_of + object: cid005202 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005202 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.RationalPoints + declaration: rank_bounded + module_path: TauLib/BookIII/Arithmetic/RationalPoints.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D60] Structural: rank is bounded by db. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005707.md b/_corpus_v3_items/cid005707.md new file mode 100644 index 000000000..8904106cb --- /dev/null +++ b/_corpus_v3_items/cid005707.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005707/ +title: all_rational_1 (theorem) +cid: cid005707 +primary_alias: FTH0445 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P25] Structural: all points rational at depth 1. -/' +aliases: +- FTH0445 +- all_rational_1 +- all-rational-1 +- TauLib.BookIII.Arithmetic.RationalPoints::all_rational_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001569 +- predicate: part_of + object: cid005202 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005202 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.RationalPoints + declaration: all_rational_1 + module_path: TauLib/BookIII/Arithmetic/RationalPoints.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P25] Structural: all points rational at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005708.md b/_corpus_v3_items/cid005708.md new file mode 100644 index 000000000..171a38705 --- /dev/null +++ b/_corpus_v3_items/cid005708.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005708/ +title: e0_lt_e1 (theorem) +cid: cid005708 +primary_alias: FTH0446 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T40] Structural: E₀ < E₁. -/' +aliases: +- FTH0446 +- e0_lt_e1 +- e0-lt-e1 +- TauLib.BookIII.Arithmetic.TowerAssembly::e0_lt_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001632 +- predicate: part_of + object: cid005203 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005203 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.TowerAssembly + declaration: e0_lt_e1 + module_path: TauLib/BookIII/Arithmetic/TowerAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T40] Structural: E₀ < E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005709.md b/_corpus_v3_items/cid005709.md new file mode 100644 index 000000000..6bd235908 --- /dev/null +++ b/_corpus_v3_items/cid005709.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005709/ +title: e1_lt_e2 (theorem) +cid: cid005709 +primary_alias: FTH0447 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T40] Structural: E₁ < E₂. -/' +aliases: +- FTH0447 +- e1_lt_e2 +- e1-lt-e2 +- TauLib.BookIII.Arithmetic.TowerAssembly::e1_lt_e2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001632 +- predicate: part_of + object: cid005203 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005203 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.TowerAssembly + declaration: e1_lt_e2 + module_path: TauLib/BookIII/Arithmetic/TowerAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T40] Structural: E₁ < E₂. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005710.md b/_corpus_v3_items/cid005710.md new file mode 100644 index 000000000..f2a85c129 --- /dev/null +++ b/_corpus_v3_items/cid005710.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005710/ +title: e2_lt_e3 (theorem) +cid: cid005710 +primary_alias: FTH0448 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T40] Structural: E₂ < E₃. -/' +aliases: +- FTH0448 +- e2_lt_e3 +- e2-lt-e3 +- TauLib.BookIII.Arithmetic.TowerAssembly::e2_lt_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001632 +- predicate: part_of + object: cid005203 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005203 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.TowerAssembly + declaration: e2_lt_e3 + module_path: TauLib/BookIII/Arithmetic/TowerAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T40] Structural: E₂ < E₃. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005711.md b/_corpus_v3_items/cid005711.md new file mode 100644 index 000000000..f964d1fc6 --- /dev/null +++ b/_corpus_v3_items/cid005711.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005711/ +title: eight_problems (theorem) +cid: cid005711 +primary_alias: FTH0449 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T40] Structural: all 8 problems covered. -/' +aliases: +- FTH0449 +- eight_problems +- eight-problems +- TauLib.BookIII.Arithmetic.TowerAssembly::eight_problems +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001632 +- predicate: part_of + object: cid005203 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005203 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Arithmetic.TowerAssembly + declaration: eight_problems + module_path: TauLib/BookIII/Arithmetic/TowerAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T40] Structural: all 8 problems covered. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005712.md b/_corpus_v3_items/cid005712.md new file mode 100644 index 000000000..f2eb2ece1 --- /dev/null +++ b/_corpus_v3_items/cid005712.md @@ -0,0 +1,80 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005712/ +title: bridge_functor_exists (axiom) +cid: cid005712 +primary_alias: FAX0001 +type: formal_axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D71] **CONJECTURE-AXIOM — CONDITIONAL RESULTS DOWNSTREAM**\n\n Bridge functor existence for all `(bound,\ + \ db)`. This is one of\n exactly three conjecture-axioms in TauLib; see also\n `spectral_correspondence_O3` (`BookIII.Doors.SpectralCorrespondence`)\n\ + \ and `grand_grh_adelic` (`BookIII.Doors.GrandGRH`).\n\n **Conjectural scope.** At finite level,\n `bridge_functor_check\ + \ bound db` is decidable and verifies the\n finite shadow of the bridge functor for every parameter pair\n `(bound,\ + \ db)` with `bound ≤ 15` and `db ≤ 3` — hundreds of\n discrete checks, each closed in the kernel by `native_decide`.\n\ + \ This axiom asserts that the finite check extends to the full\n universal statement `∀ bound db`. That extension\ + \ is the\n conjectural content.\n\n **Downstream theorems are CONDITIONAL RESULTS.** Any theorem in\n TauLib whose\ + \ transitive proof chain invokes `bridge_functor_exists`\n is conditional on the universal extension of the finite\n\ + \ `bridge_functor_check`. Running `#print axioms ` on\n a downstream theorem will list `bridge_functor_exists`\ + \ among its\n trusted assumptions — this is exactly the audit trail a Lean\n reader should expect, and it is what\ + \ makes the conditional\n status of downstream results inspectable rather than hidden.\n\n **Preferred encoding (future\ + \ work).** The Mathlib-community\n idiom for a conjectural dependency is to take the assumption as\n an explicit hypothesis\ + \ on each downstream theorem rather than as\n a global axiom. Refactoring the downstream Book III theorems to\n take\ + \ `bridge_functor_exists` (or a `(h : BridgeFunctorExists)`\n binder) as a hypothesis is planned for a future wave; this\n\ + \ would make `#print axioms` reveal no unexplained axioms on\n unconditional theorems. Until then, the global `axiom`\n\ + \ declaration is the encoding. -/" +aliases: +- FAX0001 +- bridge_functor_exists +- TauLib.BookIII.Bridge.BridgeAxiom::bridge_functor_exists +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001516 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: bridge_functor_exists + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 + formal_status_label: axiom +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D71] **CONJECTURE-AXIOM — CONDITIONAL RESULTS DOWNSTREAM**\n\n Bridge functor existence for all\ + \ `(bound, db)`. This is one of\n exactly three conjecture-axioms in TauLib; see also\n `spectral_correspondence_O3`\ + \ (`BookIII.Doors" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005713.md b/_corpus_v3_items/cid005713.md new file mode 100644 index 000000000..50741b547 --- /dev/null +++ b/_corpus_v3_items/cid005713.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005713/ +title: bridge_functor_8_3 (theorem) +cid: cid005713 +primary_alias: FTH0450 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D71] Bridge functor at finite level' +aliases: +- FTH0450 +- bridge_functor_8_3 +- bridge-functor-8-3 +- TauLib.BookIII.Bridge.BridgeAxiom::bridge_functor_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001516 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: bridge_functor_8_3 + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D71] Bridge functor at finite level' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005714.md b/_corpus_v3_items/cid005714.md new file mode 100644 index 000000000..6c551fbe7 --- /dev/null +++ b/_corpus_v3_items/cid005714.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005714/ +title: shadow_diagram_8_3 (theorem) +cid: cid005714 +primary_alias: FTH0451 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.D72] Shadow diagram +aliases: +- FTH0451 +- shadow_diagram_8_3 +- shadow-diagram-8-3 +- TauLib.BookIII.Bridge.BridgeAxiom::shadow_diagram_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001517 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: shadow_diagram_8_3 + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.D72] Shadow diagram +noindex: false +--- + diff --git a/_corpus_v3_items/cid005715.md b/_corpus_v3_items/cid005715.md new file mode 100644 index 000000000..1ce7d984c --- /dev/null +++ b/_corpus_v3_items/cid005715.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005715/ +title: honest_claim_8_3 (theorem) +cid: cid005715 +primary_alias: FTH0452 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T47] Honest claim +aliases: +- FTH0452 +- honest_claim_8_3 +- honest-claim-8-3 +- TauLib.BookIII.Bridge.BridgeAxiom::honest_claim_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: honest_claim_8_3 + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T47] Honest claim +noindex: false +--- + diff --git a/_corpus_v3_items/cid005716.md b/_corpus_v3_items/cid005716.md new file mode 100644 index 000000000..a1c304a62 --- /dev/null +++ b/_corpus_v3_items/cid005716.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005716/ +title: conjectural_marked (theorem) +cid: cid005716 +primary_alias: FTH0453 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T47] Conjectural properly marked +aliases: +- FTH0453 +- conjectural_marked +- conjectural-marked +- TauLib.BookIII.Bridge.BridgeAxiom::conjectural_marked +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: conjectural_marked + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T47] Conjectural properly marked +noindex: false +--- + diff --git a/_corpus_v3_items/cid005717.md b/_corpus_v3_items/cid005717.md new file mode 100644 index 000000000..3249f78d7 --- /dev/null +++ b/_corpus_v3_items/cid005717.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005717/ +title: break_marked (theorem) +cid: cid005717 +primary_alias: FTH0454 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T47] Break properly marked +aliases: +- FTH0454 +- break_marked +- break-marked +- TauLib.BookIII.Bridge.BridgeAxiom::break_marked +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: break_marked + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T47] Break properly marked +noindex: false +--- + diff --git a/_corpus_v3_items/cid005718.md b/_corpus_v3_items/cid005718.md new file mode 100644 index 000000000..352cc3993 --- /dev/null +++ b/_corpus_v3_items/cid005718.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005718/ +title: honest_claim_full_8_3 (theorem) +cid: cid005718 +primary_alias: FTH0455 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T47] Full honest claim +aliases: +- FTH0455 +- honest_claim_full_8_3 +- honest-claim-full-8-3 +- TauLib.BookIII.Bridge.BridgeAxiom::honest_claim_full_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: honest_claim_full_8_3 + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T47] Full honest claim +noindex: false +--- + diff --git a/_corpus_v3_items/cid005719.md b/_corpus_v3_items/cid005719.md new file mode 100644 index 000000000..597005686 --- /dev/null +++ b/_corpus_v3_items/cid005719.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005719/ +title: bridge_ledger_consistent (theorem) +cid: cid005719 +primary_alias: FTH0456 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T46] Bridge ledger consistent +aliases: +- FTH0456 +- bridge_ledger_consistent +- bridge-ledger-consistent +- TauLib.BookIII.Bridge.BridgeAxiom::bridge_ledger_consistent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001638 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: bridge_ledger_consistent + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T46] Bridge ledger consistent +noindex: false +--- + diff --git a/_corpus_v3_items/cid005720.md b/_corpus_v3_items/cid005720.md new file mode 100644 index 000000000..273ff1896 --- /dev/null +++ b/_corpus_v3_items/cid005720.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005720/ +title: rh_bridge_conjectural (theorem) +cid: cid005720 +primary_alias: FTH0457 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T45] RH bridge is conjectural (conditional on bridge axiom). -/ +aliases: +- FTH0457 +- rh_bridge_conjectural +- rh-bridge-conjectural +- TauLib.BookIII.Bridge.BridgeAxiom::rh_bridge_conjectural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001637 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: rh_bridge_conjectural + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T45] RH bridge is conjectural (conditional on bridge axiom). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005721.md b/_corpus_v3_items/cid005721.md new file mode 100644 index 000000000..35a0de21d --- /dev/null +++ b/_corpus_v3_items/cid005721.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005721/ +title: one_axiom (theorem) +cid: cid005721 +primary_alias: FTH0458 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D71] Structural: bridge axiom is the ONLY conjectural postulate. -/' +aliases: +- FTH0458 +- one_axiom +- one-axiom +- TauLib.BookIII.Bridge.BridgeAxiom::one_axiom +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001516 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: one_axiom + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D71] Structural: bridge axiom is the ONLY conjectural postulate. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005722.md b/_corpus_v3_items/cid005722.md new file mode 100644 index 000000000..7727144d9 --- /dev/null +++ b/_corpus_v3_items/cid005722.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005722/ +title: rh_layers (theorem) +cid: cid005722 +primary_alias: FTH0459 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T45] Structural: RH bridge has 3 layers. -/' +aliases: +- FTH0459 +- rh_layers +- rh-layers +- TauLib.BookIII.Bridge.BridgeAxiom::rh_layers +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001637 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: rh_layers + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T45] Structural: RH bridge has 3 layers. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005723.md b/_corpus_v3_items/cid005723.md new file mode 100644 index 000000000..f59ef5de4 --- /dev/null +++ b/_corpus_v3_items/cid005723.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005723/ +title: ledger_count (theorem) +cid: cid005723 +primary_alias: FTH0460 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T46] Structural: ledger has exactly 8 entries. -/' +aliases: +- FTH0460 +- ledger_count +- ledger-count +- TauLib.BookIII.Bridge.BridgeAxiom::ledger_count +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001638 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: ledger_count + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T46] Structural: ledger has exactly 8 entries. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005724.md b/_corpus_v3_items/cid005724.md new file mode 100644 index 000000000..9b8799972 --- /dev/null +++ b/_corpus_v3_items/cid005724.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005724/ +title: poincare_established (theorem) +cid: cid005724 +primary_alias: FTH0461 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T46] Structural: Poincare is established. -/' +aliases: +- FTH0461 +- poincare_established +- poincare-established +- TauLib.BookIII.Bridge.BridgeAxiom::poincare_established +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001638 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: poincare_established + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T46] Structural: Poincare is established. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005725.md b/_corpus_v3_items/cid005725.md new file mode 100644 index 000000000..1b196f9c4 --- /dev/null +++ b/_corpus_v3_items/cid005725.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005725/ +title: pvsnp_bridge_break (theorem) +cid: cid005725 +primary_alias: FTH0462 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T46] Structural: P vs NP is a bridge break. -/' +aliases: +- FTH0462 +- pvsnp_bridge_break +- pvsnp-bridge-break +- TauLib.BookIII.Bridge.BridgeAxiom::pvsnp_bridge_break +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001638 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: pvsnp_bridge_break + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T46] Structural: P vs NP is a bridge break. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005726.md b/_corpus_v3_items/cid005726.md new file mode 100644 index 000000000..23965df31 --- /dev/null +++ b/_corpus_v3_items/cid005726.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005726/ +title: scope_order (theorem) +cid: cid005726 +primary_alias: FTH0463 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T47] Structural: scope labels are ordered. -/' +aliases: +- FTH0463 +- scope_order +- scope-order +- TauLib.BookIII.Bridge.BridgeAxiom::scope_order +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: scope_order + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T47] Structural: scope labels are ordered. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005727.md b/_corpus_v3_items/cid005727.md new file mode 100644 index 000000000..f51cd9723 --- /dev/null +++ b/_corpus_v3_items/cid005727.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005727/ +title: established_not_conjectural (theorem) +cid: cid005727 +primary_alias: FTH0464 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T47] Structural: established claims use no conjectural scope. -/' +aliases: +- FTH0464 +- established_not_conjectural +- established-not-conjectural +- TauLib.BookIII.Bridge.BridgeAxiom::established_not_conjectural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: established_not_conjectural + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T47] Structural: established claims use no conjectural scope. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005728.md b/_corpus_v3_items/cid005728.md new file mode 100644 index 000000000..82a385321 --- /dev/null +++ b/_corpus_v3_items/cid005728.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005728/ +title: ledger_partition (theorem) +cid: cid005728 +primary_alias: FTH0465 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T47] Structural: 6 + 1 + 1 = 8 (partition of bridge ledger). -/' +aliases: +- FTH0465 +- ledger_partition +- ledger-partition +- TauLib.BookIII.Bridge.BridgeAxiom::ledger_partition +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001639 +- predicate: part_of + object: cid005204 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005204 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.BridgeAxiom + declaration: ledger_partition + module_path: TauLib/BookIII/Bridge/BridgeAxiom.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T47] Structural: 6 + 1 + 1 = 8 (partition of bridge ledger). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005729.md b/_corpus_v3_items/cid005729.md new file mode 100644 index 000000000..5f8444e69 --- /dev/null +++ b/_corpus_v3_items/cid005729.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005729/ +title: tower_finite_decidable (theorem) +cid: cid005729 +primary_alias: FTH0466 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T79] Tower decidability: all three conjectures have defined\n gap types, forbidden moves, and scope\ + \ labels. -/" +aliases: +- FTH0466 +- tower_finite_decidable +- TauLib.BookIII.Bridge.ConjectureGaps::tower_finite_decidable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001671 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: tower_finite_decidable + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T79] Tower decidability: all three conjectures have defined\n gap types, forbidden moves, and scope\ + \ labels. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005730.md b/_corpus_v3_items/cid005730.md new file mode 100644 index 000000000..ef5d7ffb8 --- /dev/null +++ b/_corpus_v3_items/cid005730.md @@ -0,0 +1,63 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005730/ +title: bridge_necessary_insufficient (theorem) +cid: cid005730 +primary_alias: FTH0467 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T80] Bridge is necessary but insufficient for full conjectures. -/ +aliases: +- FTH0467 +- bridge_necessary_insufficient +- TauLib.BookIII.Bridge.ConjectureGaps::bridge_necessary_insufficient +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001672 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: bridge_necessary_insufficient + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T80] Bridge is necessary but insufficient for full conjectures. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005731.md b/_corpus_v3_items/cid005731.md new file mode 100644 index 000000000..5aa54cc4b --- /dev/null +++ b/_corpus_v3_items/cid005731.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005731/ +title: gap_taxonomy (theorem) +cid: cid005731 +primary_alias: FTH0468 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D112] Gap taxonomy is complete: three distinct gap types. -/' +aliases: +- FTH0468 +- gap_taxonomy +- gap-taxonomy +- TauLib.BookIII.Bridge.ConjectureGaps::gap_taxonomy +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001455 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: gap_taxonomy + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D112] Gap taxonomy is complete: three distinct gap types. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005732.md b/_corpus_v3_items/cid005732.md new file mode 100644 index 000000000..d07fd31d6 --- /dev/null +++ b/_corpus_v3_items/cid005732.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005732/ +title: all_gaps_exponential (theorem) +cid: cid005732 +primary_alias: FTH0469 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D113] All three gaps map to exponential_quantification. -/ +aliases: +- FTH0469 +- all_gaps_exponential +- all-gaps-exponential +- TauLib.BookIII.Bridge.ConjectureGaps::all_gaps_exponential +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001456 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: all_gaps_exponential + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D113] All three gaps map to exponential_quantification. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005733.md b/_corpus_v3_items/cid005733.md new file mode 100644 index 000000000..76926d0ac --- /dev/null +++ b/_corpus_v3_items/cid005733.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005733/ +title: parity_ne_density (theorem) +cid: cid005733 +primary_alias: FTH0470 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D112] Gap types are distinct. -/ +aliases: +- FTH0470 +- parity_ne_density +- parity-ne-density +- TauLib.BookIII.Bridge.ConjectureGaps::parity_ne_density +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001455 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: parity_ne_density + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D112] Gap types are distinct. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005734.md b/_corpus_v3_items/cid005734.md new file mode 100644 index 000000000..62ee0169d --- /dev/null +++ b/_corpus_v3_items/cid005734.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005734/ +title: gap_indices (theorem) +cid: cid005734 +primary_alias: FTH0471 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D112] Three gap types cover indices 0, 1, 2. -/ +aliases: +- FTH0471 +- gap_indices +- gap-indices +- TauLib.BookIII.Bridge.ConjectureGaps::gap_indices +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001455 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: gap_indices + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D112] Three gap types cover indices 0, 1, 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005735.md b/_corpus_v3_items/cid005735.md new file mode 100644 index 000000000..bbd1f9f62 --- /dev/null +++ b/_corpus_v3_items/cid005735.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005735/ +title: goldbach_gap_parity (theorem) +cid: cid005735 +primary_alias: FTH0472 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D113] Goldbach gap = parity. -/ +aliases: +- FTH0472 +- goldbach_gap_parity +- goldbach-gap-parity +- TauLib.BookIII.Bridge.ConjectureGaps::goldbach_gap_parity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001456 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: goldbach_gap_parity + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D113] Goldbach gap = parity. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005736.md b/_corpus_v3_items/cid005736.md new file mode 100644 index 000000000..d0cc06ac4 --- /dev/null +++ b/_corpus_v3_items/cid005736.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005736/ +title: twin_gap_density (theorem) +cid: cid005736 +primary_alias: FTH0473 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D113] Twin primes gap = density. -/ +aliases: +- FTH0473 +- twin_gap_density +- twin-gap-density +- TauLib.BookIII.Bridge.ConjectureGaps::twin_gap_density +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001456 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: twin_gap_density + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D113] Twin primes gap = density. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005737.md b/_corpus_v3_items/cid005737.md new file mode 100644 index 000000000..9b9e865cd --- /dev/null +++ b/_corpus_v3_items/cid005737.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005737/ +title: abc_gap_structural (theorem) +cid: cid005737 +primary_alias: FTH0474 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D113] ABC gap = structural. -/ +aliases: +- FTH0474 +- abc_gap_structural +- abc-gap-structural +- TauLib.BookIII.Bridge.ConjectureGaps::abc_gap_structural +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001456 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: abc_gap_structural + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D113] ABC gap = structural. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005738.md b/_corpus_v3_items/cid005738.md new file mode 100644 index 000000000..e91db4719 --- /dev/null +++ b/_corpus_v3_items/cid005738.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005738/ +title: all_gaps_K4 (theorem) +cid: cid005738 +primary_alias: FTH0475 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D113] All gaps violate K4. -/ +aliases: +- FTH0475 +- all_gaps_K4 +- all-gaps-k4 +- TauLib.BookIII.Bridge.ConjectureGaps::all_gaps_K4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001456 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: all_gaps_K4 + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D113] All gaps violate K4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005739.md b/_corpus_v3_items/cid005739.md new file mode 100644 index 000000000..056097857 --- /dev/null +++ b/_corpus_v3_items/cid005739.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005739/ +title: exponential_damage (theorem) +cid: cid005739 +primary_alias: FTH0476 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T80] Bridge damage at exponential_quantification is 3 (break). -/ +aliases: +- FTH0476 +- exponential_damage +- exponential-damage +- TauLib.BookIII.Bridge.ConjectureGaps::exponential_damage +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001672 +- predicate: part_of + object: cid005205 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005205 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ConjectureGaps + declaration: exponential_damage + module_path: TauLib/BookIII/Bridge/ConjectureGaps.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T80] Bridge damage at exponential_quantification is 3 (break). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005740.md b/_corpus_v3_items/cid005740.md new file mode 100644 index 000000000..2528c782c --- /dev/null +++ b/_corpus_v3_items/cid005740.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005740/ +title: forbidden_moves_8_3 (theorem) +cid: cid005740 +primary_alias: FTH0477 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D69] Forbidden moves' +aliases: +- FTH0477 +- forbidden_moves_8_3 +- forbidden-moves-8-3 +- TauLib.BookIII.Bridge.ForbiddenMoves::forbidden_moves_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: forbidden_moves_8_3 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D69] Forbidden moves' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005741.md b/_corpus_v3_items/cid005741.md new file mode 100644 index 000000000..0790c5a9a --- /dev/null +++ b/_corpus_v3_items/cid005741.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005741/ +title: move_bridge_8_3 (theorem) +cid: cid005741 +primary_alias: FTH0478 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T43] Move-bridge correspondence +aliases: +- FTH0478 +- move_bridge_8_3 +- move-bridge-8-3 +- TauLib.BookIII.Bridge.ForbiddenMoves::move_bridge_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001635 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: move_bridge_8_3 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T43] Move-bridge correspondence +noindex: false +--- + diff --git a/_corpus_v3_items/cid005742.md b/_corpus_v3_items/cid005742.md new file mode 100644 index 000000000..67d6ab196 --- /dev/null +++ b/_corpus_v3_items/cid005742.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005742/ +title: move_correspondence (theorem) +cid: cid005742 +primary_alias: FTH0479 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T43] Correspondence exhaustiveness +aliases: +- FTH0479 +- move_correspondence +- move-correspondence +- TauLib.BookIII.Bridge.ForbiddenMoves::move_correspondence +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001635 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: move_correspondence + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T43] Correspondence exhaustiveness +noindex: false +--- + diff --git a/_corpus_v3_items/cid005743.md b/_corpus_v3_items/cid005743.md new file mode 100644 index 000000000..d2bb23495 --- /dev/null +++ b/_corpus_v3_items/cid005743.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005743/ +title: five_forbidden (theorem) +cid: cid005743 +primary_alias: FTH0480 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D69] Structural: there are exactly 5 forbidden moves. -/' +aliases: +- FTH0480 +- five_forbidden +- five-forbidden +- TauLib.BookIII.Bridge.ForbiddenMoves::five_forbidden +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: five_forbidden + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D69] Structural: there are exactly 5 forbidden moves. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005744.md b/_corpus_v3_items/cid005744.md new file mode 100644 index 000000000..6d2303196 --- /dev/null +++ b/_corpus_v3_items/cid005744.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005744/ +title: move_index_0 (theorem) +cid: cid005744 +primary_alias: FTH0481 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D69] Structural: forbidden move indices are 0..4. -/' +aliases: +- FTH0481 +- move_index_0 +- move-index-0 +- TauLib.BookIII.Bridge.ForbiddenMoves::move_index_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: move_index_0 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D69] Structural: forbidden move indices are 0..4. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005745.md b/_corpus_v3_items/cid005745.md new file mode 100644 index 000000000..15b431e45 --- /dev/null +++ b/_corpus_v3_items/cid005745.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005745/ +title: fanout_violates_K3 (theorem) +cid: cid005745 +primary_alias: FTH0482 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D69] Structural: unbounded fanout violates K3 (boundary). -/' +aliases: +- FTH0482 +- fanout_violates_K3 +- fanout-violates-k3 +- TauLib.BookIII.Bridge.ForbiddenMoves::fanout_violates_K3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: fanout_violates_K3 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D69] Structural: unbounded fanout violates K3 (boundary). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005746.md b/_corpus_v3_items/cid005746.md new file mode 100644 index 000000000..81420e5ec --- /dev/null +++ b/_corpus_v3_items/cid005746.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005746/ +title: equality_violates_K5 (theorem) +cid: cid005746 +primary_alias: FTH0483 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D69] Structural: global equality violates K5 (composition). -/' +aliases: +- FTH0483 +- equality_violates_K5 +- equality-violates-k5 +- TauLib.BookIII.Bridge.ForbiddenMoves::equality_violates_K5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: equality_violates_K5 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D69] Structural: global equality violates K5 (composition). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005747.md b/_corpus_v3_items/cid005747.md new file mode 100644 index 000000000..b4f9ae153 --- /dev/null +++ b/_corpus_v3_items/cid005747.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005747/ +title: circuits_break_bridge (theorem) +cid: cid005747 +primary_alias: FTH0484 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T43] Structural: succinct circuits cause bridge break. -/' +aliases: +- FTH0484 +- circuits_break_bridge +- circuits-break-bridge +- TauLib.BookIII.Bridge.ForbiddenMoves::circuits_break_bridge +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001635 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: circuits_break_bridge + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T43] Structural: succinct circuits cause bridge break. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005748.md b/_corpus_v3_items/cid005748.md new file mode 100644 index 000000000..68cfa0cad --- /dev/null +++ b/_corpus_v3_items/cid005748.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005748/ +title: pvsnp_uses_3_moves (theorem) +cid: cid005748 +primary_alias: FTH0485 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T43] Structural: P vs NP involves 3 forbidden moves. -/' +aliases: +- FTH0485 +- pvsnp_uses_3_moves +- pvsnp-uses-3-moves +- TauLib.BookIII.Bridge.ForbiddenMoves::pvsnp_uses_3_moves +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001635 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: pvsnp_uses_3_moves + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T43] Structural: P vs NP involves 3 forbidden moves. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005749.md b/_corpus_v3_items/cid005749.md new file mode 100644 index 000000000..be8c9b9b5 --- /dev/null +++ b/_corpus_v3_items/cid005749.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005749/ +title: max_damage_is_3 (theorem) +cid: cid005749 +primary_alias: FTH0486 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T43] Structural: maximum bridge damage is 3 (break). -/' +aliases: +- FTH0486 +- max_damage_is_3 +- max-damage-is-3 +- TauLib.BookIII.Bridge.ForbiddenMoves::max_damage_is_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001635 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: max_damage_is_3 + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T43] Structural: maximum bridge damage is 3 (break). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005750.md b/_corpus_v3_items/cid005750.md new file mode 100644 index 000000000..0de736d54 --- /dev/null +++ b/_corpus_v3_items/cid005750.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005750/ +title: threshold_exceeds (theorem) +cid: cid005750 +primary_alias: FTH0487 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D69] Structural: all moves have threshold > db (requires\n unbounded depth). -/" +aliases: +- FTH0487 +- threshold_exceeds +- threshold-exceeds +- TauLib.BookIII.Bridge.ForbiddenMoves::threshold_exceeds +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001514 +- predicate: part_of + object: cid005206 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005206 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ForbiddenMoves + declaration: threshold_exceeds + module_path: TauLib/BookIII/Bridge/ForbiddenMoves.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D69] Structural: all moves have threshold > db (requires\n unbounded depth). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005751.md b/_corpus_v3_items/cid005751.md new file mode 100644 index 000000000..88cdb94bc --- /dev/null +++ b/_corpus_v3_items/cid005751.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005751/ +title: incompleteness_vm_10_3 (theorem) +cid: cid005751 +primary_alias: FTH0488 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.T44] Incompleteness as VM boundary' +aliases: +- FTH0488 +- incompleteness_vm_10_3 +- incompleteness-vm-10-3 +- TauLib.BookIII.Bridge.Incompleteness::incompleteness_vm_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: incompleteness_vm_10_3 + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.T44] Incompleteness as VM boundary' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005752.md b/_corpus_v3_items/cid005752.md new file mode 100644 index 000000000..485df8f24 --- /dev/null +++ b/_corpus_v3_items/cid005752.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005752/ +title: diagonal_depth_0 (theorem) +cid: cid005752 +primary_alias: FTH0489 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: diagonal at depth 0 is 0 (Prim(0)=1). -/' +aliases: +- FTH0489 +- diagonal_depth_0 +- diagonal-depth-0 +- TauLib.BookIII.Bridge.Incompleteness::diagonal_depth_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: diagonal_depth_0 + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: diagonal at depth 0 is 0 (Prim(0)=1). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005753.md b/_corpus_v3_items/cid005753.md new file mode 100644 index 000000000..1dbc3dd4e --- /dev/null +++ b/_corpus_v3_items/cid005753.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005753/ +title: diagonal_mod (theorem) +cid: cid005753 +primary_alias: FTH0490 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: diagonal at depth 3 is modular. -/' +aliases: +- FTH0490 +- diagonal_mod +- diagonal-mod +- TauLib.BookIII.Bridge.Incompleteness::diagonal_mod +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: diagonal_mod + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: diagonal at depth 3 is modular. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005754.md b/_corpus_v3_items/cid005754.md new file mode 100644 index 000000000..7d006c690 --- /dev/null +++ b/_corpus_v3_items/cid005754.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005754/ +title: godel_at_3 (theorem) +cid: cid005754 +primary_alias: FTH0491 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: Godel sentence at depth 3 is 9. -/' +aliases: +- FTH0491 +- godel_at_3 +- godel-at-3 +- TauLib.BookIII.Bridge.Incompleteness::godel_at_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: godel_at_3 + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: Godel sentence at depth 3 is 9. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005755.md b/_corpus_v3_items/cid005755.md new file mode 100644 index 000000000..ca165ee1e --- /dev/null +++ b/_corpus_v3_items/cid005755.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005755/ +title: all_at_e3 (theorem) +cid: cid005755 +primary_alias: FTH0492 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: all three phenomena require E3. -/' +aliases: +- FTH0492 +- all_at_e3 +- all-at-e3 +- TauLib.BookIII.Bridge.Incompleteness::all_at_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: all_at_e3 + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: all three phenomena require E3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005756.md b/_corpus_v3_items/cid005756.md new file mode 100644 index 000000000..57a9890fe --- /dev/null +++ b/_corpus_v3_items/cid005756.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005756/ +title: all_from_e2 (theorem) +cid: cid005756 +primary_alias: FTH0493 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: all three phenomena originate at E2. -/' +aliases: +- FTH0493 +- all_from_e2 +- all-from-e2 +- TauLib.BookIII.Bridge.Incompleteness::all_from_e2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: all_from_e2 + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: all three phenomena originate at E2. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005757.md b/_corpus_v3_items/cid005757.md new file mode 100644 index 000000000..de2b599dd --- /dev/null +++ b/_corpus_v3_items/cid005757.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005757/ +title: e2_lt_e3_boundary (theorem) +cid: cid005757 +primary_alias: FTH0494 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: E2 < E3 (the boundary is genuine). -/' +aliases: +- FTH0494 +- e2_lt_e3_boundary +- e2-lt-e3-boundary +- TauLib.BookIII.Bridge.Incompleteness::e2_lt_e3_boundary +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: e2_lt_e3_boundary + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: E2 < E3 (the boundary is genuine). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005758.md b/_corpus_v3_items/cid005758.md new file mode 100644 index 000000000..2ed7b676a --- /dev/null +++ b/_corpus_v3_items/cid005758.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005758/ +title: e3_resolves_incompleteness (theorem) +cid: cid005758 +primary_alias: FTH0495 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T44] Structural: E3 is the resolution level for incompleteness. -/' +aliases: +- FTH0495 +- e3_resolves_incompleteness +- e3-resolves-incompleteness +- TauLib.BookIII.Bridge.Incompleteness::e3_resolves_incompleteness +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001636 +- predicate: part_of + object: cid005207 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005207 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.Incompleteness + declaration: e3_resolves_incompleteness + module_path: TauLib/BookIII/Bridge/Incompleteness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T44] Structural: E3 is the resolution level for incompleteness. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005759.md b/_corpus_v3_items/cid005759.md new file mode 100644 index 000000000..a6d12a503 --- /dev/null +++ b/_corpus_v3_items/cid005759.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005759/ +title: arith_translation_10_3 (theorem) +cid: cid005759 +primary_alias: FTH0496 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D87] Translation well-defined at bound 10, depth 3. -/ +aliases: +- FTH0496 +- arith_translation_10_3 +- arith-translation-10-3 +- TauLib.BookIII.Bridge.TranslationArith::arith_translation_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001532 +- predicate: part_of + object: cid005208 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005208 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationArith + declaration: arith_translation_10_3 + module_path: TauLib/BookIII/Bridge/TranslationArith.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D87] Translation well-defined at bound 10, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005760.md b/_corpus_v3_items/cid005760.md new file mode 100644 index 000000000..954eaa8eb --- /dev/null +++ b/_corpus_v3_items/cid005760.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005760/ +title: crt_integer_8_3 (theorem) +cid: cid005760 +primary_alias: FTH0497 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D88] CRT roundtrip at bound 8, depth 3. -/ +aliases: +- FTH0497 +- crt_integer_8_3 +- crt-integer-8-3 +- TauLib.BookIII.Bridge.TranslationArith::crt_integer_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001533 +- predicate: part_of + object: cid005208 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005208 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationArith + declaration: crt_integer_8_3 + module_path: TauLib/BookIII/Bridge/TranslationArith.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D88] CRT roundtrip at bound 8, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005761.md b/_corpus_v3_items/cid005761.md new file mode 100644 index 000000000..3e7316779 --- /dev/null +++ b/_corpus_v3_items/cid005761.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005761/ +title: arith_faithful_8_3 (theorem) +cid: cid005761 +primary_alias: FTH0498 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T59] Arithmetic faithfulness at bound 8, depth 3. -/ +aliases: +- FTH0498 +- arith_faithful_8_3 +- arith-faithful-8-3 +- TauLib.BookIII.Bridge.TranslationArith::arith_faithful_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001651 +- predicate: part_of + object: cid005208 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005208 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationArith + declaration: arith_faithful_8_3 + module_path: TauLib/BookIII/Bridge/TranslationArith.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T59] Arithmetic faithfulness at bound 8, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005762.md b/_corpus_v3_items/cid005762.md new file mode 100644 index 000000000..508408574 --- /dev/null +++ b/_corpus_v3_items/cid005762.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005762/ +title: arith_preserves_6_3 (theorem) +cid: cid005762 +primary_alias: FTH0499 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P36] Operation preservation at bound 6, depth 3. -/ +aliases: +- FTH0499 +- arith_preserves_6_3 +- arith-preserves-6-3 +- TauLib.BookIII.Bridge.TranslationArith::arith_preserves_6_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001580 +- predicate: part_of + object: cid005208 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005208 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationArith + declaration: arith_preserves_6_3 + module_path: TauLib/BookIII/Bridge/TranslationArith.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P36] Operation preservation at bound 6, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005763.md b/_corpus_v3_items/cid005763.md new file mode 100644 index 000000000..e2e16fbfa --- /dev/null +++ b/_corpus_v3_items/cid005763.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005763/ +title: obstruction_check_3 (theorem) +cid: cid005763 +primary_alias: FTH0500 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D91] Obstruction values correct at depth 3. -/ +aliases: +- FTH0500 +- obstruction_check_3 +- obstruction-check-3 +- TauLib.BookIII.Bridge.TranslationObstruction::obstruction_check_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001536 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: obstruction_check_3 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D91] Obstruction values correct at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005764.md b/_corpus_v3_items/cid005764.md new file mode 100644 index 000000000..e6095c265 --- /dev/null +++ b/_corpus_v3_items/cid005764.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005764/ +title: arith_obstruction_3 (theorem) +cid: cid005764 +primary_alias: FTH0501 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D92] Arithmetic has 3 obstruction classes. -/ +aliases: +- FTH0501 +- arith_obstruction_3 +- arith-obstruction-3 +- TauLib.BookIII.Bridge.TranslationObstruction::arith_obstruction_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001537 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: arith_obstruction_3 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D92] Arithmetic has 3 obstruction classes. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005765.md b/_corpus_v3_items/cid005765.md new file mode 100644 index 000000000..f34a5b706 --- /dev/null +++ b/_corpus_v3_items/cid005765.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005765/ +title: topo_obstruction_4 (theorem) +cid: cid005765 +primary_alias: FTH0502 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D92] Topology has 4 obstruction classes. -/ +aliases: +- FTH0502 +- topo_obstruction_4 +- topo-obstruction-4 +- TauLib.BookIII.Bridge.TranslationObstruction::topo_obstruction_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001537 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: topo_obstruction_4 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D92] Topology has 4 obstruction classes. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005766.md b/_corpus_v3_items/cid005766.md new file mode 100644 index 000000000..23d85cac4 --- /dev/null +++ b/_corpus_v3_items/cid005766.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005766/ +title: safe_region_8_3 (theorem) +cid: cid005766 +primary_alias: FTH0503 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T61] Safe region faithful at bound 8, depth 3. -/ +aliases: +- FTH0503 +- safe_region_8_3 +- safe-region-8-3 +- TauLib.BookIII.Bridge.TranslationObstruction::safe_region_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001653 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: safe_region_8_3 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T61] Safe region faithful at bound 8, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005767.md b/_corpus_v3_items/cid005767.md new file mode 100644 index 000000000..9fc0c64ba --- /dev/null +++ b/_corpus_v3_items/cid005767.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005767/ +title: translation_boundary_8_3 (theorem) +cid: cid005767 +primary_alias: FTH0504 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T61] Translation failure boundary at bound 8, depth 3. -/ +aliases: +- FTH0504 +- translation_boundary_8_3 +- translation-boundary-8-3 +- TauLib.BookIII.Bridge.TranslationObstruction::translation_boundary_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001653 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: translation_boundary_8_3 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T61] Translation failure boundary at bound 8, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005768.md b/_corpus_v3_items/cid005768.md new file mode 100644 index 000000000..6353fdb9a --- /dev/null +++ b/_corpus_v3_items/cid005768.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005768/ +title: pvsnp_obstruction_3 (theorem) +cid: cid005768 +primary_alias: FTH0505 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P38] P vs NP obstruction at depth 3. -/ +aliases: +- FTH0505 +- pvsnp_obstruction_3 +- pvsnp-obstruction-3 +- TauLib.BookIII.Bridge.TranslationObstruction::pvsnp_obstruction_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001582 +- predicate: part_of + object: cid005209 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005209 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationObstruction + declaration: pvsnp_obstruction_3 + module_path: TauLib/BookIII/Bridge/TranslationObstruction.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P38] P vs NP obstruction at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005769.md b/_corpus_v3_items/cid005769.md new file mode 100644 index 000000000..2e0fc90b3 --- /dev/null +++ b/_corpus_v3_items/cid005769.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005769/ +title: topo_translation_10_3 (theorem) +cid: cid005769 +primary_alias: FTH0506 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D89] Tower coherence at bound 10, depth 3. -/ +aliases: +- FTH0506 +- topo_translation_10_3 +- topo-translation-10-3 +- TauLib.BookIII.Bridge.TranslationTopo::topo_translation_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001534 +- predicate: part_of + object: cid005210 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005210 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationTopo + declaration: topo_translation_10_3 + module_path: TauLib/BookIII/Bridge/TranslationTopo.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D89] Tower coherence at bound 10, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005770.md b/_corpus_v3_items/cid005770.md new file mode 100644 index 000000000..64ab642f9 --- /dev/null +++ b/_corpus_v3_items/cid005770.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005770/ +title: dimension_recovery_3 (theorem) +cid: cid005770 +primary_alias: FTH0507 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D90] Dimension recovery at depth 3. -/ +aliases: +- FTH0507 +- dimension_recovery_3 +- dimension-recovery-3 +- TauLib.BookIII.Bridge.TranslationTopo::dimension_recovery_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001535 +- predicate: part_of + object: cid005210 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005210 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationTopo + declaration: dimension_recovery_3 + module_path: TauLib/BookIII/Bridge/TranslationTopo.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D90] Dimension recovery at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005771.md b/_corpus_v3_items/cid005771.md new file mode 100644 index 000000000..b14013daf --- /dev/null +++ b/_corpus_v3_items/cid005771.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005771/ +title: topo_faithful_6_2 (theorem) +cid: cid005771 +primary_alias: FTH0508 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T60] Topological faithfulness at bound 6, depth 2. -/ +aliases: +- FTH0508 +- topo_faithful_6_2 +- topo-faithful-6-2 +- TauLib.BookIII.Bridge.TranslationTopo::topo_faithful_6_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001652 +- predicate: part_of + object: cid005210 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005210 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationTopo + declaration: topo_faithful_6_2 + module_path: TauLib/BookIII/Bridge/TranslationTopo.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T60] Topological faithfulness at bound 6, depth 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005772.md b/_corpus_v3_items/cid005772.md new file mode 100644 index 000000000..868463add --- /dev/null +++ b/_corpus_v3_items/cid005772.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005772/ +title: boundary_restriction_3 (theorem) +cid: cid005772 +primary_alias: FTH0509 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P37] Boundary restriction at depth 3. -/ +aliases: +- FTH0509 +- boundary_restriction_3 +- boundary-restriction-3 +- TauLib.BookIII.Bridge.TranslationTopo::boundary_restriction_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001581 +- predicate: part_of + object: cid005210 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005210 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.TranslationTopo + declaration: boundary_restriction_3 + module_path: TauLib/BookIII/Bridge/TranslationTopo.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P37] Boundary restriction at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005773.md b/_corpus_v3_items/cid005773.md new file mode 100644 index 000000000..f0ba666e9 --- /dev/null +++ b/_corpus_v3_items/cid005773.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005773/ +title: zfc_vm_8_3 (theorem) +cid: cid005773 +primary_alias: FTH0510 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D67] ZFC VM valid' +aliases: +- FTH0510 +- zfc_vm_8_3 +- zfc-vm-8-3 +- TauLib.BookIII.Bridge.ZFCasVM::zfc_vm_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: zfc_vm_8_3 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D67] ZFC VM valid' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005774.md b/_corpus_v3_items/cid005774.md new file mode 100644 index 000000000..995832603 --- /dev/null +++ b/_corpus_v3_items/cid005774.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005774/ +title: axiom_encoding_8_3 (theorem) +cid: cid005774 +primary_alias: FTH0511 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.D68] Axiom encoding valid +aliases: +- FTH0511 +- axiom_encoding_8_3 +- axiom-encoding-8-3 +- TauLib.BookIII.Bridge.ZFCasVM::axiom_encoding_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001513 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: axiom_encoding_8_3 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.D68] Axiom encoding valid +noindex: false +--- + diff --git a/_corpus_v3_items/cid005775.md b/_corpus_v3_items/cid005775.md new file mode 100644 index 000000000..4059b0c05 --- /dev/null +++ b/_corpus_v3_items/cid005775.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005775/ +title: set_universe_10_4 (theorem) +cid: cid005775 +primary_alias: FTH0512 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.D70] Set-theoretic universe valid +aliases: +- FTH0512 +- set_universe_10_4 +- set-universe-10-4 +- TauLib.BookIII.Bridge.ZFCasVM::set_universe_10_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001515 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: set_universe_10_4 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.D70] Set-theoretic universe valid +noindex: false +--- + diff --git a/_corpus_v3_items/cid005776.md b/_corpus_v3_items/cid005776.md new file mode 100644 index 000000000..c7bf8ff34 --- /dev/null +++ b/_corpus_v3_items/cid005776.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005776/ +title: host_level_10_3 (theorem) +cid: cid005776 +primary_alias: FTH0513 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.D70] Host-level property valid +aliases: +- FTH0513 +- host_level_10_3 +- host-level-10-3 +- TauLib.BookIII.Bridge.ZFCasVM::host_level_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001515 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: host_level_10_3 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.D70] Host-level property valid +noindex: false +--- + diff --git a/_corpus_v3_items/cid005777.md b/_corpus_v3_items/cid005777.md new file mode 100644 index 000000000..b18e15d2b --- /dev/null +++ b/_corpus_v3_items/cid005777.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005777/ +title: zfc_has_9_axioms (theorem) +cid: cid005777 +primary_alias: FTH0514 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D67] Structural: ZFC has exactly 9 axioms. -/' +aliases: +- FTH0514 +- zfc_has_9_axioms +- zfc-has-9-axioms +- TauLib.BookIII.Bridge.ZFCasVM::zfc_has_9_axioms +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: zfc_has_9_axioms + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D67] Structural: ZFC has exactly 9 axioms. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005778.md b/_corpus_v3_items/cid005778.md new file mode 100644 index 000000000..84024a19a --- /dev/null +++ b/_corpus_v3_items/cid005778.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005778/ +title: zfc_is_e2 (theorem) +cid: cid005778 +primary_alias: FTH0515 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D67] Structural: ZFC is at E2 (not E0 or E1). -/' +aliases: +- FTH0515 +- zfc_is_e2 +- zfc-is-e2 +- TauLib.BookIII.Bridge.ZFCasVM::zfc_is_e2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: zfc_is_e2 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D67] Structural: ZFC is at E2 (not E0 or E1). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005779.md b/_corpus_v3_items/cid005779.md new file mode 100644 index 000000000..e3663ac19 --- /dev/null +++ b/_corpus_v3_items/cid005779.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005779/ +title: ext_min_depth (theorem) +cid: cid005779 +primary_alias: FTH0516 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D68] Structural: extensionality needs depth >= 1. -/' +aliases: +- FTH0516 +- ext_min_depth +- ext-min-depth +- TauLib.BookIII.Bridge.ZFCasVM::ext_min_depth +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001513 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: ext_min_depth + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D68] Structural: extensionality needs depth >= 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005780.md b/_corpus_v3_items/cid005780.md new file mode 100644 index 000000000..28534d464 --- /dev/null +++ b/_corpus_v3_items/cid005780.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005780/ +title: inf_min_depth (theorem) +cid: cid005780 +primary_alias: FTH0517 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D68] Structural: infinity needs depth >= 3. -/' +aliases: +- FTH0517 +- inf_min_depth +- inf-min-depth +- TauLib.BookIII.Bridge.ZFCasVM::inf_min_depth +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001513 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: inf_min_depth + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D68] Structural: infinity needs depth >= 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005781.md b/_corpus_v3_items/cid005781.md new file mode 100644 index 000000000..b428b1a42 --- /dev/null +++ b/_corpus_v3_items/cid005781.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005781/ +title: universe_rank_0 (theorem) +cid: cid005781 +primary_alias: FTH0518 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D70] Structural: V_0 = Prim(0) = 1 (singleton universe). -/' +aliases: +- FTH0518 +- universe_rank_0 +- universe-rank-0 +- TauLib.BookIII.Bridge.ZFCasVM::universe_rank_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001515 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: universe_rank_0 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D70] Structural: V_0 = Prim(0) = 1 (singleton universe). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005782.md b/_corpus_v3_items/cid005782.md new file mode 100644 index 000000000..cd468105b --- /dev/null +++ b/_corpus_v3_items/cid005782.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005782/ +title: universe_rank_3 (theorem) +cid: cid005782 +primary_alias: FTH0519 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D70] Structural: V_3 = 30 (primorial 3). -/' +aliases: +- FTH0519 +- universe_rank_3 +- universe-rank-3 +- TauLib.BookIII.Bridge.ZFCasVM::universe_rank_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001515 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: universe_rank_3 + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D70] Structural: V_3 = 30 (primorial 3). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005783.md b/_corpus_v3_items/cid005783.md new file mode 100644 index 000000000..b3f549537 --- /dev/null +++ b/_corpus_v3_items/cid005783.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005783/ +title: pairing_valid (theorem) +cid: cid005783 +primary_alias: FTH0520 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D67] Structural: pairing at depth 3 produces valid result. -/' +aliases: +- FTH0520 +- pairing_valid +- pairing-valid +- TauLib.BookIII.Bridge.ZFCasVM::pairing_valid +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: pairing_valid + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D67] Structural: pairing at depth 3 produces valid result. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005784.md b/_corpus_v3_items/cid005784.md new file mode 100644 index 000000000..578b207ff --- /dev/null +++ b/_corpus_v3_items/cid005784.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005784/ +title: ext_detects_equal (theorem) +cid: cid005784 +primary_alias: FTH0521 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D67] Structural: extensionality detects equality. -/' +aliases: +- FTH0521 +- ext_detects_equal +- ext-detects-equal +- TauLib.BookIII.Bridge.ZFCasVM::ext_detects_equal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: ext_detects_equal + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D67] Structural: extensionality detects equality. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005785.md b/_corpus_v3_items/cid005785.md new file mode 100644 index 000000000..3ad285b84 --- /dev/null +++ b/_corpus_v3_items/cid005785.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005785/ +title: ext_detects_unequal (theorem) +cid: cid005785 +primary_alias: FTH0522 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D67] Structural: extensionality detects inequality. -/' +aliases: +- FTH0522 +- ext_detects_unequal +- ext-detects-unequal +- TauLib.BookIII.Bridge.ZFCasVM::ext_detects_unequal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001512 +- predicate: part_of + object: cid005211 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005211 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Bridge.ZFCasVM + declaration: ext_detects_unequal + module_path: TauLib/BookIII/Bridge/ZFCasVM.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D67] Structural: extensionality detects inequality. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005786.md b/_corpus_v3_items/cid005786.md new file mode 100644 index 000000000..c89b95f73 --- /dev/null +++ b/_corpus_v3_items/cid005786.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005786/ +title: width_bounded (theorem) +cid: cid005786 +primary_alias: FTH0523 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D53] Structural: width at depth 0 is at most 1. -/' +aliases: +- FTH0523 +- width_bounded +- width-bounded +- TauLib.BookIII.Computation.Admissibility::width_bounded +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001498 +- predicate: part_of + object: cid005212 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005212 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.Admissibility + declaration: width_bounded + module_path: TauLib/BookIII/Computation/Admissibility.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D53] Structural: width at depth 0 is at most 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005787.md b/_corpus_v3_items/cid005787.md new file mode 100644 index 000000000..cb093a602 --- /dev/null +++ b/_corpus_v3_items/cid005787.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005787/ +title: admissible_10_1 (theorem) +cid: cid005787 +primary_alias: FTH0524 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D54] Structural: admissibility at depth 1. -/' +aliases: +- FTH0524 +- admissible_10_1 +- admissible-10-1 +- TauLib.BookIII.Computation.Admissibility::admissible_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001499 +- predicate: part_of + object: cid005212 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005212 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.Admissibility + declaration: admissible_10_1 + module_path: TauLib/BookIII/Computation/Admissibility.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D54] Structural: admissibility at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005788.md b/_corpus_v3_items/cid005788.md new file mode 100644 index 000000000..7e4d2ca49 --- /dev/null +++ b/_corpus_v3_items/cid005788.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005788/ +title: id_admissible (theorem) +cid: cid005788 +primary_alias: FTH0525 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P21] Structural: identity is trivially admissible. -/' +aliases: +- FTH0525 +- id_admissible +- id-admissible +- TauLib.BookIII.Computation.Admissibility::id_admissible +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001565 +- predicate: part_of + object: cid005212 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005212 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.Admissibility + declaration: id_admissible + module_path: TauLib/BookIII/Computation/Admissibility.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P21] Structural: identity is trivially admissible. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005789.md b/_corpus_v3_items/cid005789.md new file mode 100644 index 000000000..af65daf2f --- /dev/null +++ b/_corpus_v3_items/cid005789.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005789/ +title: product_meet_10_1 (theorem) +cid: cid005789 +primary_alias: FTH0526 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T32] Structural: product-meet at depth 1. -/' +aliases: +- FTH0526 +- product_meet_10_1 +- product-meet-10-1 +- TauLib.BookIII.Computation.CompBiSquare::product_meet_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001624 +- predicate: part_of + object: cid005213 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005213 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.CompBiSquare + declaration: product_meet_10_1 + module_path: TauLib/BookIII/Computation/CompBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T32] Structural: product-meet at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005790.md b/_corpus_v3_items/cid005790.md new file mode 100644 index 000000000..7990563da --- /dev/null +++ b/_corpus_v3_items/cid005790.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005790/ +title: pvsnp_level (theorem) +cid: cid005790 +primary_alias: FTH0527 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T33] Structural: P vs NP is at E₂. -/' +aliases: +- FTH0527 +- pvsnp_level +- pvsnp-level +- TauLib.BookIII.Computation.CompBiSquare::pvsnp_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001625 +- predicate: part_of + object: cid005213 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005213 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.CompBiSquare + declaration: pvsnp_level + module_path: TauLib/BookIII/Computation/CompBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T33] Structural: P vs NP is at E₂. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005791.md b/_corpus_v3_items/cid005791.md new file mode 100644 index 000000000..f054295b1 --- /dev/null +++ b/_corpus_v3_items/cid005791.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005791/ +title: no_barrier_10_1 (theorem) +cid: cid005791 +primary_alias: FTH0528 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T34] Structural: no-barrier at depth 1. -/' +aliases: +- FTH0528 +- no_barrier_10_1 +- no-barrier-10-1 +- TauLib.BookIII.Computation.CompBiSquare::no_barrier_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001626 +- predicate: part_of + object: cid005213 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005213 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.CompBiSquare + declaration: no_barrier_10_1 + module_path: TauLib/BookIII/Computation/CompBiSquare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T34] Structural: no-barrier at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005792.md b/_corpus_v3_items/cid005792.md new file mode 100644 index 000000000..193053b15 --- /dev/null +++ b/_corpus_v3_items/cid005792.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005792/ +title: agent_step_depth_0 (theorem) +cid: cid005792 +primary_alias: FTH0529 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D49] Structural: agent step at depth 0 is 0. -/' +aliases: +- FTH0529 +- agent_step_depth_0 +- agent-step-depth-0 +- TauLib.BookIII.Computation.E2Agent::agent_step_depth_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001494 +- predicate: part_of + object: cid005214 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005214 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Agent + declaration: agent_step_depth_0 + module_path: TauLib/BookIII/Computation/E2Agent.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D49] Structural: agent step at depth 0 is 0. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005793.md b/_corpus_v3_items/cid005793.md new file mode 100644 index 000000000..72c4a82e3 --- /dev/null +++ b/_corpus_v3_items/cid005793.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005793/ +title: agent_step_mod (theorem) +cid: cid005793 +primary_alias: FTH0530 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D49] Structural: agent step is modular. -/' +aliases: +- FTH0530 +- agent_step_mod +- agent-step-mod +- TauLib.BookIII.Computation.E2Agent::agent_step_mod +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001494 +- predicate: part_of + object: cid005214 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005214 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Agent + declaration: agent_step_mod + module_path: TauLib/BookIII/Computation/E2Agent.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D49] Structural: agent step is modular. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005794.md b/_corpus_v3_items/cid005794.md new file mode 100644 index 000000000..6511a025d --- /dev/null +++ b/_corpus_v3_items/cid005794.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005794/ +title: identity_decoder (theorem) +cid: cid005794 +primary_alias: FTH0531 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D50] Structural: identity decoder (d=0) is a fixpoint. -/' +aliases: +- FTH0531 +- identity_decoder +- identity-decoder +- TauLib.BookIII.Computation.E2Agent::identity_decoder +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001495 +- predicate: part_of + object: cid005214 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005214 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Agent + declaration: identity_decoder + module_path: TauLib/BookIII/Computation/E2Agent.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D50] Structural: identity decoder (d=0) is a fixpoint. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005795.md b/_corpus_v3_items/cid005795.md new file mode 100644 index 000000000..9a1885ea6 --- /dev/null +++ b/_corpus_v3_items/cid005795.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005795/ +title: kleene_check_3 (theorem) +cid: cid005795 +primary_alias: FTH0532 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D83] Kleene fixed points exist at stages 1-3. -/ +aliases: +- FTH0532 +- kleene_check_3 +- kleene-check-3 +- TauLib.BookIII.Computation.E2Witness::kleene_check_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001528 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: kleene_check_3 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D83] Kleene fixed points exist at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005796.md b/_corpus_v3_items/cid005796.md new file mode 100644 index 000000000..58135018b --- /dev/null +++ b/_corpus_v3_items/cid005796.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005796/ +title: fixed_points_stage2 (theorem) +cid: cid005796 +primary_alias: FTH0533 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D83] Fixed point exists at stage 2 (c=0). -/ +aliases: +- FTH0533 +- fixed_points_stage2 +- fixed-points-stage2 +- TauLib.BookIII.Computation.E2Witness::fixed_points_stage2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001528 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: fixed_points_stage2 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D83] Fixed point exists at stage 2 (c=0). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005797.md b/_corpus_v3_items/cid005797.md new file mode 100644 index 000000000..36ac12b8e --- /dev/null +++ b/_corpus_v3_items/cid005797.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005797/ +title: orbit_diversity_2 (theorem) +cid: cid005797 +primary_alias: FTH0534 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D84] Orbit diversity at stage 2. -/ +aliases: +- FTH0534 +- orbit_diversity_2 +- orbit-diversity-2 +- TauLib.BookIII.Computation.E2Witness::orbit_diversity_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001529 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: orbit_diversity_2 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D84] Orbit diversity at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005798.md b/_corpus_v3_items/cid005798.md new file mode 100644 index 000000000..7d2079e13 --- /dev/null +++ b/_corpus_v3_items/cid005798.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005798/ +title: orbit_diversity_3 (theorem) +cid: cid005798 +primary_alias: FTH0535 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D84] Orbit diversity at stage 3. -/ +aliases: +- FTH0535 +- orbit_diversity_3 +- orbit-diversity-3 +- TauLib.BookIII.Computation.E2Witness::orbit_diversity_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001529 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: orbit_diversity_3 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D84] Orbit diversity at stage 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005799.md b/_corpus_v3_items/cid005799.md new file mode 100644 index 000000000..2c2b740aa --- /dev/null +++ b/_corpus_v3_items/cid005799.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005799/ +title: operational_closure_8_3 (theorem) +cid: cid005799 +primary_alias: FTH0536 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T57] Operational closure at bound 8, depth 3. -/ +aliases: +- FTH0536 +- operational_closure_8_3 +- operational-closure-8-3 +- TauLib.BookIII.Computation.E2Witness::operational_closure_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001649 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: operational_closure_8_3 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T57] Operational closure at bound 8, depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005800.md b/_corpus_v3_items/cid005800.md new file mode 100644 index 000000000..c43b18826 --- /dev/null +++ b/_corpus_v3_items/cid005800.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005800/ +title: e2_strict_3 (theorem) +cid: cid005800 +primary_alias: FTH0537 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P34] E₂ strict witness at depth 3. -/ +aliases: +- FTH0537 +- e2_strict_3 +- e2-strict-3 +- TauLib.BookIII.Computation.E2Witness::e2_strict_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001578 +- predicate: part_of + object: cid005215 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005215 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.E2Witness + declaration: e2_strict_3 + module_path: TauLib/BookIII/Computation/E2Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P34] E₂ strict witness at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005801.md b/_corpus_v3_items/cid005801.md new file mode 100644 index 000000000..3b1fdc648 --- /dev/null +++ b/_corpus_v3_items/cid005801.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005801/ +title: ttm_preserves_depth (theorem) +cid: cid005801 +primary_alias: FTH0538 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D51] Structural: TTM step preserves depth. -/' +aliases: +- FTH0538 +- ttm_preserves_depth +- ttm-preserves-depth +- TauLib.BookIII.Computation.TowerMachine::ttm_preserves_depth +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001496 +- predicate: part_of + object: cid005216 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005216 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.TowerMachine + declaration: ttm_preserves_depth + module_path: TauLib/BookIII/Computation/TowerMachine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D51] Structural: TTM step preserves depth. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005802.md b/_corpus_v3_items/cid005802.md new file mode 100644 index 000000000..61d4964ba --- /dev/null +++ b/_corpus_v3_items/cid005802.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005802/ +title: ttm_depth_0 (theorem) +cid: cid005802 +primary_alias: FTH0539 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D51] Structural: TTM at depth 0 (Prim(0)=1, all ops mod 1). -/' +aliases: +- FTH0539 +- ttm_depth_0 +- ttm-depth-0 +- TauLib.BookIII.Computation.TowerMachine::ttm_depth_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001496 +- predicate: part_of + object: cid005216 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005216 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.TowerMachine + declaration: ttm_depth_0 + module_path: TauLib/BookIII/Computation/TowerMachine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D51] Structural: TTM at depth 0 (Prim(0)=1, all ops mod 1). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005803.md b/_corpus_v3_items/cid005803.md new file mode 100644 index 000000000..fd8f97f2b --- /dev/null +++ b/_corpus_v3_items/cid005803.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005803/ +title: code_is_data (theorem) +cid: cid005803 +primary_alias: FTH0540 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T30] Structural: code=data (self-application). -/' +aliases: +- FTH0540 +- code_is_data +- code-is-data +- TauLib.BookIII.Computation.TowerMachine::code_is_data +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001622 +- predicate: part_of + object: cid005216 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005216 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.TowerMachine + declaration: code_is_data + module_path: TauLib/BookIII/Computation/TowerMachine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T30] Structural: code=data (self-application). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005804.md b/_corpus_v3_items/cid005804.md new file mode 100644 index 000000000..088d4f58f --- /dev/null +++ b/_corpus_v3_items/cid005804.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005804/ +title: obs_width (theorem) +cid: cid005804 +primary_alias: FTH0541 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D52] Structural: observable width is 3. -/' +aliases: +- FTH0541 +- obs_width +- obs-width +- TauLib.BookIII.Computation.TowerMachine::obs_width +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001497 +- predicate: part_of + object: cid005216 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005216 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.TowerMachine + declaration: obs_width + module_path: TauLib/BookIII/Computation/TowerMachine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D52] Structural: observable width is 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005805.md b/_corpus_v3_items/cid005805.md new file mode 100644 index 000000000..cb3516ab6 --- /dev/null +++ b/_corpus_v3_items/cid005805.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005805/ +title: witness_zero_3 (theorem) +cid: cid005805 +primary_alias: FTH0542 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D55] Structural: witness of 0 has zero components. -/' +aliases: +- FTH0542 +- witness_zero_3 +- witness-zero-3 +- TauLib.BookIII.Computation.WitnessSearch::witness_zero_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001500 +- predicate: part_of + object: cid005217 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005217 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.WitnessSearch + declaration: witness_zero_3 + module_path: TauLib/BookIII/Computation/WitnessSearch.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D55] Structural: witness of 0 has zero components. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005806.md b/_corpus_v3_items/cid005806.md new file mode 100644 index 000000000..fef392d7f --- /dev/null +++ b/_corpus_v3_items/cid005806.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005806/ +title: crt_faithful_42_3 (theorem) +cid: cid005806 +primary_alias: FTH0543 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P22] Structural: CRT decomposition at depth 3 is faithful. -/' +aliases: +- FTH0543 +- crt_faithful_42_3 +- crt-faithful-42-3 +- TauLib.BookIII.Computation.WitnessSearch::crt_faithful_42_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001566 +- predicate: part_of + object: cid005217 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005217 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.WitnessSearch + declaration: crt_faithful_42_3 + module_path: TauLib/BookIII/Computation/WitnessSearch.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P22] Structural: CRT decomposition at depth 3 is faithful. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005807.md b/_corpus_v3_items/cid005807.md new file mode 100644 index 000000000..27e1e16c3 --- /dev/null +++ b/_corpus_v3_items/cid005807.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005807/ +title: sum_less_prod_3 (theorem) +cid: cid005807 +primary_alias: FTH0544 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P23] Structural: sum of first 3 primes < primorial 3. -/' +aliases: +- FTH0544 +- sum_less_prod_3 +- sum-less-prod-3 +- TauLib.BookIII.Computation.WitnessSearch::sum_less_prod_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001567 +- predicate: part_of + object: cid005217 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005217 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Computation.WitnessSearch + declaration: sum_less_prod_3 + module_path: TauLib/BookIII/Computation/WitnessSearch.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P23] Structural: sum of first 3 primes < primorial 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005808.md b/_corpus_v3_items/cid005808.md new file mode 100644 index 000000000..ea706c1ba --- /dev/null +++ b/_corpus_v3_items/cid005808.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005808/ +title: rh_gap_5 (theorem) +cid: cid005808 +primary_alias: FTH0545 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D93] RH gap characterization at depth 5. -/ +aliases: +- FTH0545 +- rh_gap_5 +- rh-gap-5 +- TauLib.BookIII.Doors.BridgeTightening::rh_gap_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001538 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: rh_gap_5 + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D93] RH gap characterization at depth 5. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005809.md b/_corpus_v3_items/cid005809.md new file mode 100644 index 000000000..e8fbadb72 --- /dev/null +++ b/_corpus_v3_items/cid005809.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005809/ +title: ym_gap_5 (theorem) +cid: cid005809 +primary_alias: FTH0546 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D94] YM gap persistence at depth 5. -/ +aliases: +- FTH0546 +- ym_gap_5 +- ym-gap-5 +- TauLib.BookIII.Doors.BridgeTightening::ym_gap_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001539 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: ym_gap_5 + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D94] YM gap persistence at depth 5. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005810.md b/_corpus_v3_items/cid005810.md new file mode 100644 index 000000000..7c24de4a6 --- /dev/null +++ b/_corpus_v3_items/cid005810.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005810/ +title: ym_gap_growth_4 (theorem) +cid: cid005810 +primary_alias: FTH0547 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D94] YM gap growth at depth 4. -/ +aliases: +- FTH0547 +- ym_gap_growth_4 +- ym-gap-growth-4 +- TauLib.BookIII.Doors.BridgeTightening::ym_gap_growth_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001539 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: ym_gap_growth_4 + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D94] YM gap growth at depth 4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005811.md b/_corpus_v3_items/cid005811.md new file mode 100644 index 000000000..d4fd82389 --- /dev/null +++ b/_corpus_v3_items/cid005811.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005811/ +title: ns_causal_4 (theorem) +cid: cid005811 +primary_alias: FTH0548 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T62] NS causal entropy at depth 4. -/ +aliases: +- FTH0548 +- ns_causal_4 +- ns-causal-4 +- TauLib.BookIII.Doors.BridgeTightening::ns_causal_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001654 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: ns_causal_4 + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T62] NS causal entropy at depth 4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005812.md b/_corpus_v3_items/cid005812.md new file mode 100644 index 000000000..3b365d598 --- /dev/null +++ b/_corpus_v3_items/cid005812.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005812/ +title: pvsnp_triple_3 (theorem) +cid: cid005812 +primary_alias: FTH0549 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T63] P vs NP forbidden triple at depth 3. -/ +aliases: +- FTH0549 +- pvsnp_triple_3 +- pvsnp-triple-3 +- TauLib.BookIII.Doors.BridgeTightening::pvsnp_triple_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001655 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: pvsnp_triple_3 + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T63] P vs NP forbidden triple at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005813.md b/_corpus_v3_items/cid005813.md new file mode 100644 index 000000000..0a1ae2460 --- /dev/null +++ b/_corpus_v3_items/cid005813.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005813/ +title: bridge_ledger_complete (theorem) +cid: cid005813 +primary_alias: FTH0550 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P39] Bridge ledger is complete. -/ +aliases: +- FTH0550 +- bridge_ledger_complete +- bridge-ledger-complete +- TauLib.BookIII.Doors.BridgeTightening::bridge_ledger_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001583 +- predicate: part_of + object: cid005218 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005218 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.BridgeTightening + declaration: bridge_ledger_complete + module_path: TauLib/BookIII/Doors/BridgeTightening.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P39] Bridge ledger is complete. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005814.md b/_corpus_v3_items/cid005814.md new file mode 100644 index 000000000..0b99cb831 --- /dev/null +++ b/_corpus_v3_items/cid005814.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005814/ +title: critical_line_1 (theorem) +cid: cid005814 +primary_alias: FTH0551 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T19] Structural: critical line at depth 1 (only prime 2). -/' +aliases: +- FTH0551 +- critical_line_1 +- critical-line-1 +- TauLib.BookIII.Doors.CriticalLine::critical_line_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001611 +- predicate: part_of + object: cid005219 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005219 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.CriticalLine + declaration: critical_line_1 + module_path: TauLib/BookIII/Doors/CriticalLine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T19] Structural: critical line at depth 1 (only prime 2). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005815.md b/_corpus_v3_items/cid005815.md new file mode 100644 index 000000000..c85574f7e --- /dev/null +++ b/_corpus_v3_items/cid005815.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005815/ +title: k5_eigenvalue_1 (theorem) +cid: cid005815 +primary_alias: FTH0552 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P10] Structural: eigenvalue of first mode equals 1. -/' +aliases: +- FTH0552 +- k5_eigenvalue_1 +- k5-eigenvalue-1 +- TauLib.BookIII.Doors.CriticalLine::k5_eigenvalue_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001554 +- predicate: part_of + object: cid005219 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005219 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.CriticalLine + declaration: k5_eigenvalue_1 + module_path: TauLib/BookIII/Doors/CriticalLine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P10] Structural: eigenvalue of first mode equals 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005816.md b/_corpus_v3_items/cid005816.md new file mode 100644 index 000000000..60477bed2 --- /dev/null +++ b/_corpus_v3_items/cid005816.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005816/ +title: tau_rh_1 (theorem) +cid: cid005816 +primary_alias: FTH0553 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D30] Structural: τ-effective RH at depth 1. -/' +aliases: +- FTH0553 +- tau_rh_1 +- tau-rh-1 +- TauLib.BookIII.Doors.CriticalLine::tau_rh_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001475 +- predicate: part_of + object: cid005219 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005219 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.CriticalLine + declaration: tau_rh_1 + module_path: TauLib/BookIII/Doors/CriticalLine.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D30] Structural: τ-effective RH at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005817.md b/_corpus_v3_items/cid005817.md new file mode 100644 index 000000000..6bfaedca1 --- /dev/null +++ b/_corpus_v3_items/cid005817.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005817/ +title: grand_grh_adelic (axiom) +cid: cid005817 +primary_alias: FAX0002 +type: formal_axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D31] **STATUS: OPEN CONJECTURE — all downstream results are CONDITIONAL ON GRH**… (full doc preserved in + payload)' +aliases: +- FAX0002 +- grand_grh_adelic +- grand-grh-adelic +- TauLib.BookIII.Doors.GrandGRH::grand_grh_adelic +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001476 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: grand_grh_adelic + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 + formal_status_label: axiom +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D31] **STATUS: OPEN CONJECTURE — all downstream results are CONDITIONAL ON GRH**… (full doc preserved + in payload)' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005818.md b/_corpus_v3_items/cid005818.md new file mode 100644 index 000000000..6321da503 --- /dev/null +++ b/_corpus_v3_items/cid005818.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005818/ +title: grand_grh_3 (theorem) +cid: cid005818 +primary_alias: FTH0554 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D31] Structural: Grand GRH finite check at depth 3. -/' +aliases: +- FTH0554 +- grand_grh_3 +- grand-grh-3 +- TauLib.BookIII.Doors.GrandGRH::grand_grh_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001476 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: grand_grh_3 + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D31] Structural: Grand GRH finite check at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005819.md b/_corpus_v3_items/cid005819.md new file mode 100644 index 000000000..080cac812 --- /dev/null +++ b/_corpus_v3_items/cid005819.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005819/ +title: label_5_is_B (theorem) +cid: cid005819 +primary_alias: FTH0555 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T20] Structural: label classification of prime 5 (5 ≡ 1 mod 4 → B). -/' +aliases: +- FTH0555 +- label_5_is_B +- label-5-is-b +- TauLib.BookIII.Doors.GrandGRH::label_5_is_B +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001612 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: label_5_is_B + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T20] Structural: label classification of prime 5 (5 ≡ 1 mod 4 → B). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005820.md b/_corpus_v3_items/cid005820.md new file mode 100644 index 000000000..37373e881 --- /dev/null +++ b/_corpus_v3_items/cid005820.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005820/ +title: label_3_is_C (theorem) +cid: cid005820 +primary_alias: FTH0556 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T20] Structural: label classification of prime 3 (3 ≡ 3 mod 4 → C). -/' +aliases: +- FTH0556 +- label_3_is_C +- label-3-is-c +- TauLib.BookIII.Doors.GrandGRH::label_3_is_C +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001612 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: label_3_is_C + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T20] Structural: label classification of prime 3 (3 ≡ 3 mod 4 → C). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005821.md b/_corpus_v3_items/cid005821.md new file mode 100644 index 000000000..392a67995 --- /dev/null +++ b/_corpus_v3_items/cid005821.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005821/ +title: l_function_3 (theorem) +cid: cid005821 +primary_alias: FTH0557 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D32] Structural: L-function at depth 3 = Prim(3). -/' +aliases: +- FTH0557 +- l_function_3 +- l-function-3 +- TauLib.BookIII.Doors.GrandGRH::l_function_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001477 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: l_function_3 + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D32] Structural: L-function at depth 3 = Prim(3). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005822.md b/_corpus_v3_items/cid005822.md new file mode 100644 index 000000000..c53230b58 --- /dev/null +++ b/_corpus_v3_items/cid005822.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005822/ +title: grand_grh_from_axiom (theorem) +cid: cid005822 +primary_alias: FTH0558 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D31] Structural: Grand GRH from axiom. -/' +aliases: +- FTH0558 +- grand_grh_from_axiom +- grand-grh-from-axiom +- TauLib.BookIII.Doors.GrandGRH::grand_grh_from_axiom +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001476 +- predicate: part_of + object: cid005220 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005220 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.GrandGRH + declaration: grand_grh_from_axiom + module_path: TauLib/BookIII/Doors/GrandGRH.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D31] Structural: Grand GRH from axiom. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005823.md b/_corpus_v3_items/cid005823.md new file mode 100644 index 000000000..24c25b79b --- /dev/null +++ b/_corpus_v3_items/cid005823.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005823/ +title: eigenvalue_zero (theorem) +cid: cid005823 +primary_alias: FTH0559 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D28] Structural: ground state eigenvalue is 0. -/' +aliases: +- FTH0559 +- eigenvalue_zero +- eigenvalue-zero +- TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001473 +- predicate: part_of + object: cid005221 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005221 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.LemniscateOperator + declaration: eigenvalue_zero + module_path: TauLib/BookIII/Doors/LemniscateOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D28] Structural: ground state eigenvalue is 0. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005824.md b/_corpus_v3_items/cid005824.md new file mode 100644 index 000000000..4d42fb687 --- /dev/null +++ b/_corpus_v3_items/cid005824.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005824/ +title: eigenvalue_formula (theorem) +cid: cid005824 +primary_alias: FTH0560 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D28] Structural: eigenvalue at mode n equals n². -/' +aliases: +- FTH0560 +- eigenvalue_formula +- eigenvalue-formula +- TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_formula +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001473 +- predicate: part_of + object: cid005221 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005221 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.LemniscateOperator + declaration: eigenvalue_formula + module_path: TauLib/BookIII/Doors/LemniscateOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D28] Structural: eigenvalue at mode n equals n². -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005825.md b/_corpus_v3_items/cid005825.md new file mode 100644 index 000000000..2c4a2f7f8 --- /dev/null +++ b/_corpus_v3_items/cid005825.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005825/ +title: eigenvalue_nonneg (theorem) +cid: cid005825 +primary_alias: FTH0561 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T17] Structural: all eigenvalues are non-negative. -/' +aliases: +- FTH0561 +- eigenvalue_nonneg +- eigenvalue-nonneg +- TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_nonneg +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001609 +- predicate: part_of + object: cid005221 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005221 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.LemniscateOperator + declaration: eigenvalue_nonneg + module_path: TauLib/BookIII/Doors/LemniscateOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T17] Structural: all eigenvalues are non-negative. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005826.md b/_corpus_v3_items/cid005826.md new file mode 100644 index 000000000..afc51f42b --- /dev/null +++ b/_corpus_v3_items/cid005826.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005826/ +title: spectral_gap_value (theorem) +cid: cid005826 +primary_alias: FTH0562 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P09] Structural: spectral gap value is 1. -/' +aliases: +- FTH0562 +- spectral_gap_value +- spectral-gap-value +- TauLib.BookIII.Doors.LemniscateOperator::spectral_gap_value +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001553 +- predicate: part_of + object: cid005221 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005221 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.LemniscateOperator + declaration: spectral_gap_value + module_path: TauLib/BookIII/Doors/LemniscateOperator.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P09] Structural: spectral gap value is 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005827.md b/_corpus_v3_items/cid005827.md new file mode 100644 index 000000000..ff3dee219 --- /dev/null +++ b/_corpus_v3_items/cid005827.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005827/ +title: rh_level (theorem) +cid: cid005827 +primary_alias: FTH0563 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T23] Structural: RH is at E₀. -/' +aliases: +- FTH0563 +- rh_level +- rh-level +- TauLib.BookIII.Doors.MasterSchema::rh_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001615 +- predicate: part_of + object: cid005222 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005222 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + declaration: rh_level + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T23] Structural: RH is at E₀. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005828.md b/_corpus_v3_items/cid005828.md new file mode 100644 index 000000000..d1bcbbdc1 --- /dev/null +++ b/_corpus_v3_items/cid005828.md @@ -0,0 +1,62 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005828/ +title: pvsnp_level (theorem) +cid: cid005828 +primary_alias: FTH0564 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T23] Structural: P vs NP is at E₂. -/' +aliases: +- FTH0564 +- TauLib.BookIII.Doors.MasterSchema::pvsnp_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001615 +- predicate: part_of + object: cid005222 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005222 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + declaration: pvsnp_level + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T23] Structural: P vs NP is at E₂. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005829.md b/_corpus_v3_items/cid005829.md new file mode 100644 index 000000000..70d1cdd58 --- /dev/null +++ b/_corpus_v3_items/cid005829.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005829/ +title: ns_level (theorem) +cid: cid005829 +primary_alias: FTH0565 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T23] Structural: NS is at E₁. -/' +aliases: +- FTH0565 +- ns_level +- ns-level +- TauLib.BookIII.Doors.MasterSchema::ns_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001615 +- predicate: part_of + object: cid005222 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005222 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + declaration: ns_level + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T23] Structural: NS is at E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005830.md b/_corpus_v3_items/cid005830.md new file mode 100644 index 000000000..b21a3b1d4 --- /dev/null +++ b/_corpus_v3_items/cid005830.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005830/ +title: rh_part (theorem) +cid: cid005830 +primary_alias: FTH0566 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T23] Structural: all 8 problems have parts in {4,5,6,7}. -/' +aliases: +- FTH0566 +- rh_part +- rh-part +- TauLib.BookIII.Doors.MasterSchema::rh_part +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001615 +- predicate: part_of + object: cid005222 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005222 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + declaration: rh_part + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T23] Structural: all 8 problems have parts in {4,5,6,7}. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005831.md b/_corpus_v3_items/cid005831.md new file mode 100644 index 000000000..c708be011 --- /dev/null +++ b/_corpus_v3_items/cid005831.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005831/ +title: e0_before_e1 (theorem) +cid: cid005831 +primary_alias: FTH0567 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T23] Structural: enrichment levels are ordered. -/' +aliases: +- FTH0567 +- e0_before_e1 +- e0-before-e1 +- TauLib.BookIII.Doors.MasterSchema::e0_before_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001615 +- predicate: part_of + object: cid005222 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005222 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MasterSchema + declaration: e0_before_e1 + module_path: TauLib/BookIII/Doors/MasterSchema.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T23] Structural: enrichment levels are ordered. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005832.md b/_corpus_v3_items/cid005832.md new file mode 100644 index 000000000..6db64b527 --- /dev/null +++ b/_corpus_v3_items/cid005832.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005832/ +title: b_to_i_zero (theorem) +cid: cid005832 +primary_alias: FTH0568 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D25] Structural: boundary→interior is exact for 0. -/' +aliases: +- FTH0568 +- b_to_i_zero +- b-to-i-zero +- TauLib.BookIII.Doors.MutualDetermination::b_to_i_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001470 +- predicate: part_of + object: cid005223 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005223 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MutualDetermination + declaration: b_to_i_zero + module_path: TauLib/BookIII/Doors/MutualDetermination.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D25] Structural: boundary→interior is exact for 0. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005833.md b/_corpus_v3_items/cid005833.md new file mode 100644 index 000000000..4e4a439e6 --- /dev/null +++ b/_corpus_v3_items/cid005833.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005833/ +title: md_cycle_42_3 (theorem) +cid: cid005833 +primary_alias: FTH0569 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D25] Structural: mutual determination cycle for 42 at depth 3. -/' +aliases: +- FTH0569 +- md_cycle_42_3 +- md-cycle-42-3 +- TauLib.BookIII.Doors.MutualDetermination::md_cycle_42_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001470 +- predicate: part_of + object: cid005223 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005223 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.MutualDetermination + declaration: md_cycle_42_3 + module_path: TauLib/BookIII/Doors/MutualDetermination.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D25] Structural: mutual determination cycle for 42 at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005834.md b/_corpus_v3_items/cid005834.md new file mode 100644 index 000000000..c520e9d78 --- /dev/null +++ b/_corpus_v3_items/cid005834.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005834/ +title: crt_bijective_42_3 (theorem) +cid: cid005834 +primary_alias: FTH0570 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D35] Structural: CRT is bijective at depth 3\n (simple connectivity = no monodromy). -/" +aliases: +- FTH0570 +- crt_bijective_42_3 +- crt-bijective-42-3 +- TauLib.BookIII.Doors.Poincare::crt_bijective_42_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001480 +- predicate: part_of + object: cid005224 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005224 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.Poincare + declaration: crt_bijective_42_3 + module_path: TauLib/BookIII/Doors/Poincare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D35] Structural: CRT is bijective at depth 3\n (simple connectivity = no monodromy). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005835.md b/_corpus_v3_items/cid005835.md new file mode 100644 index 000000000..fe0c5e37d --- /dev/null +++ b/_corpus_v3_items/cid005835.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005835/ +title: terminal_depth_1 (theorem) +cid: cid005835 +primary_alias: FTH0571 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D35] Structural: terminal object at depth 1 (only mod 2). -/' +aliases: +- FTH0571 +- terminal_depth_1 +- terminal-depth-1 +- TauLib.BookIII.Doors.Poincare::terminal_depth_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001480 +- predicate: part_of + object: cid005224 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005224 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.Poincare + declaration: terminal_depth_1 + module_path: TauLib/BookIII/Doors/Poincare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D35] Structural: terminal object at depth 1 (only mod 2). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005836.md b/_corpus_v3_items/cid005836.md new file mode 100644 index 000000000..8492eb4e1 --- /dev/null +++ b/_corpus_v3_items/cid005836.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005836/ +title: gluing_depth_1 (theorem) +cid: cid005836 +primary_alias: FTH0572 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P13] Structural: gluing at depth 1. -/' +aliases: +- FTH0572 +- gluing_depth_1 +- gluing-depth-1 +- TauLib.BookIII.Doors.Poincare::gluing_depth_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001557 +- predicate: part_of + object: cid005224 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005224 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.Poincare + declaration: gluing_depth_1 + module_path: TauLib/BookIII/Doors/Poincare.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P13] Structural: gluing at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005837.md b/_corpus_v3_items/cid005837.md new file mode 100644 index 000000000..0929891a1 --- /dev/null +++ b/_corpus_v3_items/cid005837.md @@ -0,0 +1,77 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005837/ +title: spectral_correspondence_O3 (axiom) +cid: cid005837 +primary_alias: FAX0003 +type: formal_axiom +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T18] **CONJECTURE-AXIOM — CONDITIONAL RESULTS DOWNSTREAM**\n\n The O(3) spectral correspondence holds\ + \ at all levels. This is one\n of exactly three conjecture-axioms in TauLib; see also\n `bridge_functor_exists` (`BookIII.Bridge.BridgeAxiom`)\ + \ and\n `grand_grh_adelic` (`BookIII.Doors.GrandGRH`).\n\n **Conjectural scope.** At each finite level `k`,\n `spectral_correspondence_finite\ + \ k` is decidable and verified\n computationally via `native_decide`. The axiom asserts that the\n finite correspondence\ + \ extends to the universal statement\n `∀ k : Nat`. That extension is the conjectural content.\n\n **Mathematical\ + \ content.** The spectral correspondence claims\n `ζ_τ(s) = 0 ⟺ Λ(s) ∈ Spec(H_L)`, with the determinant\n representation\ + \ `ζ_τ(s) = det(I − Λ(s)·H_L⁻¹)`. This is the\n framework's honest conjectural gap in the τ-approach to RH.\n\n **Downstream\ + \ theorems are CONDITIONAL RESULTS.** Any theorem\n whose transitive proof chain invokes\n `spectral_correspondence_O3`\ + \ is conditional on the universal\n extension. Running `#print axioms ` on a\n downstream theorem will\ + \ list `spectral_correspondence_O3`;\n readers should treat that theorem as a conditional result.\n\n **Preferred\ + \ encoding (future work).** As with\n `bridge_functor_exists`, the Mathlib-community idiom would\n refactor downstream\ + \ theorems to take this conjecture as an\n explicit hypothesis. Planned for a future wave. -/" +aliases: +- FAX0003 +- spectral_correspondence_O3 +- spectral-correspondence-o3 +- TauLib.BookIII.Doors.SpectralCorrespondence::spectral_correspondence_O3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001610 +- predicate: part_of + object: cid005225 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005225 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralCorrespondence + declaration: spectral_correspondence_O3 + module_path: TauLib/BookIII/Doors/SpectralCorrespondence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 1 + formal_status_label: axiom +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T18] **CONJECTURE-AXIOM — CONDITIONAL RESULTS DOWNSTREAM**\n\n The O(3) spectral correspondence\ + \ holds at all levels. This is one\n of exactly three conjecture-axioms in TauLib; see also\n `bridge_functor_exists`\ + \ (`BookIII.Brid" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005838.md b/_corpus_v3_items/cid005838.md new file mode 100644 index 000000000..aaa51af50 --- /dev/null +++ b/_corpus_v3_items/cid005838.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005838/ +title: spectral_param_zero (theorem) +cid: cid005838 +primary_alias: FTH0573 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D29] Structural: spectral parameter at depth 0 is always 0. -/' +aliases: +- FTH0573 +- spectral_param_zero +- spectral-param-zero +- TauLib.BookIII.Doors.SpectralCorrespondence::spectral_param_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001474 +- predicate: part_of + object: cid005225 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005225 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralCorrespondence + declaration: spectral_param_zero + module_path: TauLib/BookIII/Doors/SpectralCorrespondence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D29] Structural: spectral parameter at depth 0 is always 0. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005839.md b/_corpus_v3_items/cid005839.md new file mode 100644 index 000000000..25979dad2 --- /dev/null +++ b/_corpus_v3_items/cid005839.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005839/ +title: spectral_param_bounded (theorem) +cid: cid005839 +primary_alias: FTH0574 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D29] Structural: spectral parameter at depth k is bounded. -/' +aliases: +- FTH0574 +- spectral_param_bounded +- spectral-param-bounded +- TauLib.BookIII.Doors.SpectralCorrespondence::spectral_param_bounded +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001474 +- predicate: part_of + object: cid005225 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005225 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralCorrespondence + declaration: spectral_param_bounded + module_path: TauLib/BookIII/Doors/SpectralCorrespondence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D29] Structural: spectral parameter at depth k is bounded. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005840.md b/_corpus_v3_items/cid005840.md new file mode 100644 index 000000000..ed2f62409 --- /dev/null +++ b/_corpus_v3_items/cid005840.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005840/ +title: spectral_corr_from_O3 (theorem) +cid: cid005840 +primary_alias: FTH0575 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T18] Structural: O3 implies finite correspondence at any level. -/' +aliases: +- FTH0575 +- spectral_corr_from_O3 +- spectral-corr-from-o3 +- TauLib.BookIII.Doors.SpectralCorrespondence::spectral_corr_from_O3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001610 +- predicate: part_of + object: cid005225 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005225 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralCorrespondence + declaration: spectral_corr_from_O3 + module_path: TauLib/BookIII/Doors/SpectralCorrespondence.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T18] Structural: O3 implies finite correspondence at any level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005841.md b/_corpus_v3_items/cid005841.md new file mode 100644 index 000000000..e172cf9ac --- /dev/null +++ b/_corpus_v3_items/cid005841.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005841/ +title: projector_check_2 (theorem) +cid: cid005841 +primary_alias: FTH0576 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D80] Projectors correct at N = 2. -/ +aliases: +- FTH0576 +- projector_check_2 +- projector-check-2 +- TauLib.BookIII.Doors.SpectralDecomp::projector_check_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001525 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: projector_check_2 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D80] Projectors correct at N = 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005842.md b/_corpus_v3_items/cid005842.md new file mode 100644 index 000000000..3071c82d4 --- /dev/null +++ b/_corpus_v3_items/cid005842.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005842/ +title: projector_check_6 (theorem) +cid: cid005842 +primary_alias: FTH0577 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D80] Projectors correct at N = 6. -/ +aliases: +- FTH0577 +- projector_check_6 +- projector-check-6 +- TauLib.BookIII.Doors.SpectralDecomp::projector_check_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001525 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: projector_check_6 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D80] Projectors correct at N = 6. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005843.md b/_corpus_v3_items/cid005843.md new file mode 100644 index 000000000..38262ca70 --- /dev/null +++ b/_corpus_v3_items/cid005843.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005843/ +title: measure_total_6 (theorem) +cid: cid005843 +primary_alias: FTH0578 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D81] Spectral measure total at N = 6. -/ +aliases: +- FTH0578 +- measure_total_6 +- measure-total-6 +- TauLib.BookIII.Doors.SpectralDecomp::measure_total_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001526 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: measure_total_6 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D81] Spectral measure total at N = 6. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005844.md b/_corpus_v3_items/cid005844.md new file mode 100644 index 000000000..36cd119d5 --- /dev/null +++ b/_corpus_v3_items/cid005844.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005844/ +title: measure_total_30 (theorem) +cid: cid005844 +primary_alias: FTH0579 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D81] Spectral measure total at N = 30. -/ +aliases: +- FTH0579 +- measure_total_30 +- measure-total-30 +- TauLib.BookIII.Doors.SpectralDecomp::measure_total_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001526 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: measure_total_30 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D81] Spectral measure total at N = 30. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005845.md b/_corpus_v3_items/cid005845.md new file mode 100644 index 000000000..284251c95 --- /dev/null +++ b/_corpus_v3_items/cid005845.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005845/ +title: parseval_2 (theorem) +cid: cid005845 +primary_alias: FTH0580 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T56] Parseval identity at N = 2. -/ +aliases: +- FTH0580 +- parseval_2 +- parseval-2 +- TauLib.BookIII.Doors.SpectralDecomp::parseval_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001648 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: parseval_2 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T56] Parseval identity at N = 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005846.md b/_corpus_v3_items/cid005846.md new file mode 100644 index 000000000..c942d0428 --- /dev/null +++ b/_corpus_v3_items/cid005846.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005846/ +title: parseval_6 (theorem) +cid: cid005846 +primary_alias: FTH0581 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T56] Parseval identity at N = 6. -/ +aliases: +- FTH0581 +- parseval_6 +- parseval-6 +- TauLib.BookIII.Doors.SpectralDecomp::parseval_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001648 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: parseval_6 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T56] Parseval identity at N = 6. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005847.md b/_corpus_v3_items/cid005847.md new file mode 100644 index 000000000..a1bc8007b --- /dev/null +++ b/_corpus_v3_items/cid005847.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005847/ +title: spectral_resolution_2 (theorem) +cid: cid005847 +primary_alias: FTH0582 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P48] Spectral resolution at N = 2. -/ +aliases: +- FTH0582 +- spectral_resolution_2 +- spectral-resolution-2 +- TauLib.BookIII.Doors.SpectralDecomp::spectral_resolution_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001592 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: spectral_resolution_2 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P48] Spectral resolution at N = 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005848.md b/_corpus_v3_items/cid005848.md new file mode 100644 index 000000000..72f8384f7 --- /dev/null +++ b/_corpus_v3_items/cid005848.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005848/ +title: spectral_resolution_6 (theorem) +cid: cid005848 +primary_alias: FTH0583 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P48] Spectral resolution at N = 6. -/ +aliases: +- FTH0583 +- spectral_resolution_6 +- spectral-resolution-6 +- TauLib.BookIII.Doors.SpectralDecomp::spectral_resolution_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001592 +- predicate: part_of + object: cid005226 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005226 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SpectralDecomp + declaration: spectral_resolution_6 + module_path: TauLib/BookIII/Doors/SpectralDecomp.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P48] Spectral resolution at N = 6. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005849.md b/_corpus_v3_items/cid005849.md new file mode 100644 index 000000000..bfd6bb83a --- /dev/null +++ b/_corpus_v3_items/cid005849.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005849/ +title: b_zeta_3 (theorem) +cid: cid005849 +primary_alias: FTH0584 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D26] Structural: B-zeta at depth 3 = 5 (only B-type prime ≤ p₃). -/' +aliases: +- FTH0584 +- b_zeta_3 +- b-zeta-3 +- TauLib.BookIII.Doors.SplitComplexZeta::b_zeta_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001471 +- predicate: part_of + object: cid005227 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005227 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SplitComplexZeta + declaration: b_zeta_3 + module_path: TauLib/BookIII/Doors/SplitComplexZeta.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D26] Structural: B-zeta at depth 3 = 5 (only B-type prime ≤ p₃). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005850.md b/_corpus_v3_items/cid005850.md new file mode 100644 index 000000000..cb5671520 --- /dev/null +++ b/_corpus_v3_items/cid005850.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005850/ +title: c_zeta_3 (theorem) +cid: cid005850 +primary_alias: FTH0585 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D26] Structural: C-zeta at depth 3 = 3 (only C-type prime ≤ p₃). -/' +aliases: +- FTH0585 +- c_zeta_3 +- c-zeta-3 +- TauLib.BookIII.Doors.SplitComplexZeta::c_zeta_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001471 +- predicate: part_of + object: cid005227 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005227 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SplitComplexZeta + declaration: c_zeta_3 + module_path: TauLib/BookIII/Doors/SplitComplexZeta.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D26] Structural: C-zeta at depth 3 = 3 (only C-type prime ≤ p₃). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005851.md b/_corpus_v3_items/cid005851.md new file mode 100644 index 000000000..1d885d774 --- /dev/null +++ b/_corpus_v3_items/cid005851.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005851/ +title: fe_involution_involutive (theorem) +cid: cid005851 +primary_alias: FTH0586 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D27] Structural: involution is own inverse. -/' +aliases: +- FTH0586 +- fe_involution_involutive +- fe-involution-involutive +- TauLib.BookIII.Doors.SplitComplexZeta::fe_involution_involutive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001472 +- predicate: part_of + object: cid005227 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005227 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SplitComplexZeta + declaration: fe_involution_involutive + module_path: TauLib/BookIII/Doors/SplitComplexZeta.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D27] Structural: involution is own inverse. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005852.md b/_corpus_v3_items/cid005852.md new file mode 100644 index 000000000..364398346 --- /dev/null +++ b/_corpus_v3_items/cid005852.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005852/ +title: euler_product_3 (theorem) +cid: cid005852 +primary_alias: FTH0587 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T16] Structural: B · C · X = Prim(3) at depth 3. -/' +aliases: +- FTH0587 +- euler_product_3 +- euler-product-3 +- TauLib.BookIII.Doors.SplitComplexZeta::euler_product_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001608 +- predicate: part_of + object: cid005227 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005227 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Doors.SplitComplexZeta + declaration: euler_product_3 + module_path: TauLib/BookIII/Doors/SplitComplexZeta.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T16] Structural: B · C · X = Prim(3) at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005853.md b/_corpus_v3_items/cid005853.md new file mode 100644 index 000000000..71653a7ff --- /dev/null +++ b/_corpus_v3_items/cid005853.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005853/ +title: non_emptiness_8_3 (theorem) +cid: cid005853 +primary_alias: FTH0588 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Non-emptiness [III.T01]' +aliases: +- FTH0588 +- non_emptiness_8_3 +- non-emptiness-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::non_emptiness_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001593 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: non_emptiness_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Non-emptiness [III.T01]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005854.md b/_corpus_v3_items/cid005854.md new file mode 100644 index 000000000..39287a756 --- /dev/null +++ b/_corpus_v3_items/cid005854.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005854/ +title: e1_strictness_8_3 (theorem) +cid: cid005854 +primary_alias: FTH0589 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E₁ strictness witness [III.P01] +aliases: +- FTH0589 +- e1_strictness_8_3 +- e1-strictness-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::e1_strictness_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001545 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: e1_strictness_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E₁ strictness witness [III.P01] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005855.md b/_corpus_v3_items/cid005855.md new file mode 100644 index 000000000..5d1f2bb3d --- /dev/null +++ b/_corpus_v3_items/cid005855.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005855/ +title: strictness_8_3 (theorem) +cid: cid005855 +primary_alias: FTH0590 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Strictness [III.T02] +aliases: +- FTH0590 +- strictness_8_3 +- strictness-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::strictness_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001594 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: strictness_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Strictness [III.T02] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005856.md b/_corpus_v3_items/cid005856.md new file mode 100644 index 000000000..632d4dd75 --- /dev/null +++ b/_corpus_v3_items/cid005856.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005856/ +title: functor_collapse_8_3 (theorem) +cid: cid005856 +primary_alias: FTH0591 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Functor category collapse [III.P02] +aliases: +- FTH0591 +- functor_collapse_8_3 +- functor-collapse-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::functor_collapse_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001546 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: functor_collapse_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Functor category collapse [III.P02] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005857.md b/_corpus_v3_items/cid005857.md new file mode 100644 index 000000000..9944cbb72 --- /dev/null +++ b/_corpus_v3_items/cid005857.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005857/ +title: saturation_e3_8_3 (theorem) +cid: cid005857 +primary_alias: FTH0592 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Saturation at E₃ [III.T03] +aliases: +- FTH0592 +- saturation_e3_8_3 +- saturation-e3-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::saturation_e3_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001595 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: saturation_e3_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Saturation at E₃ [III.T03] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005858.md b/_corpus_v3_items/cid005858.md new file mode 100644 index 000000000..ad8b87373 --- /dev/null +++ b/_corpus_v3_items/cid005858.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005858/ +title: canonical_ladder_8_3 (theorem) +cid: cid005858 +primary_alias: FTH0593 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Canonical Ladder Theorem [III.T04] +aliases: +- FTH0593 +- canonical_ladder_8_3 +- canonical-ladder-8-3 +- TauLib.BookIII.Enrichment.CanonicalLadder::canonical_ladder_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001596 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: canonical_ladder_8_3 + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Canonical Ladder Theorem [III.T04] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005859.md b/_corpus_v3_items/cid005859.md new file mode 100644 index 000000000..c357b21b4 --- /dev/null +++ b/_corpus_v3_items/cid005859.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005859/ +title: zero_in_carrier (theorem) +cid: cid005859 +primary_alias: FTH0594 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T01] Structural non-emptiness: reduce(0, k) = 0 provides\n a witness at every stage. Zero is always\ + \ in the carrier. -/" +aliases: +- FTH0594 +- zero_in_carrier +- zero-in-carrier +- TauLib.BookIII.Enrichment.CanonicalLadder::zero_in_carrier +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001593 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: zero_in_carrier + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T01] Structural non-emptiness: reduce(0, k) = 0 provides\n a witness at every stage. Zero is always\ + \ in the carrier. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005860.md b/_corpus_v3_items/cid005860.md new file mode 100644 index 000000000..35138e39f --- /dev/null +++ b/_corpus_v3_items/cid005860.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005860/ +title: hom_stage_stable (theorem) +cid: cid005860 +primary_alias: FTH0595 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T02] Structural strictness at E₀→E₁: the hom_stage value\n is reduce-stable, witnessing genuine E₁ structure.\ + \ -/" +aliases: +- FTH0595 +- hom_stage_stable +- hom-stage-stable +- TauLib.BookIII.Enrichment.CanonicalLadder::hom_stage_stable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001594 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: hom_stage_stable + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T02] Structural strictness at E₀→E₁: the hom_stage value\n is reduce-stable, witnessing genuine\ + \ E₁ structure. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005861.md b/_corpus_v3_items/cid005861.md new file mode 100644 index 000000000..f4590a821 --- /dev/null +++ b/_corpus_v3_items/cid005861.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005861/ +title: structural_saturation (theorem) +cid: cid005861 +primary_alias: FTH0596 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T03] Structural saturation: E₃.succ = E₃.\n The enrichment functor is idempotent at E₃. -/" +aliases: +- FTH0596 +- structural_saturation +- structural-saturation +- TauLib.BookIII.Enrichment.CanonicalLadder::structural_saturation +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001595 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: structural_saturation + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T03] Structural saturation: E₃.succ = E₃.\n The enrichment functor is idempotent at E₃. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005862.md b/_corpus_v3_items/cid005862.md new file mode 100644 index 000000000..f2defb116 --- /dev/null +++ b/_corpus_v3_items/cid005862.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005862/ +title: ladder_has_four_levels (theorem) +cid: cid005862 +primary_alias: FTH0597 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T04] Structural uniqueness: three applications of succ from\n any starting level reach E₃. The ladder\ + \ has exactly 4 levels. -/" +aliases: +- FTH0597 +- ladder_has_four_levels +- ladder-has-four-levels +- TauLib.BookIII.Enrichment.CanonicalLadder::ladder_has_four_levels +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001596 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: ladder_has_four_levels + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T04] Structural uniqueness: three applications of succ from\n any starting level reach E₃. The ladder\ + \ has exactly 4 levels. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005863.md b/_corpus_v3_items/cid005863.md new file mode 100644 index 000000000..9659e4d2e --- /dev/null +++ b/_corpus_v3_items/cid005863.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005863/ +title: canonical_ordering (theorem) +cid: cid005863 +primary_alias: FTH0598 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T04] Canonical ordering: E₀ < E₁ < E₂ < E₃ via toNat. -/' +aliases: +- FTH0598 +- canonical_ordering +- canonical-ordering +- TauLib.BookIII.Enrichment.CanonicalLadder::canonical_ordering +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001596 +- predicate: part_of + object: cid005228 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005228 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.CanonicalLadder + declaration: canonical_ordering + module_path: TauLib/BookIII/Enrichment/CanonicalLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T04] Canonical ordering: E₀ < E₁ < E₂ < E₃ via toNat. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005864.md b/_corpus_v3_items/cid005864.md new file mode 100644 index 000000000..43de4994a --- /dev/null +++ b/_corpus_v3_items/cid005864.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005864/ +title: enr_functor_e0_8_3 (theorem) +cid: cid005864 +primary_alias: FTH0599 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Enrichment functor E₀→E₁ [III.D04]' +aliases: +- FTH0599 +- enr_functor_e0_8_3 +- enr-functor-e0-8-3 +- TauLib.BookIII.Enrichment.Functor::enr_functor_e0_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: enr_functor_e0_8_3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Enrichment functor E₀→E₁ [III.D04]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005865.md b/_corpus_v3_items/cid005865.md new file mode 100644 index 000000000..b2c32fe21 --- /dev/null +++ b/_corpus_v3_items/cid005865.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005865/ +title: enr_functor_e1_8_3 (theorem) +cid: cid005865 +primary_alias: FTH0600 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Enrichment functor E₁→E₂ [III.D04] +aliases: +- FTH0600 +- enr_functor_e1_8_3 +- enr-functor-e1-8-3 +- TauLib.BookIII.Enrichment.Functor::enr_functor_e1_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: enr_functor_e1_8_3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Enrichment functor E₁→E₂ [III.D04] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005866.md b/_corpus_v3_items/cid005866.md new file mode 100644 index 000000000..7b7a7bc2a --- /dev/null +++ b/_corpus_v3_items/cid005866.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005866/ +title: enr_functor_e2_8_3 (theorem) +cid: cid005866 +primary_alias: FTH0601 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Enrichment functor E₂→E₃ [III.D04] +aliases: +- FTH0601 +- enr_functor_e2_8_3 +- enr-functor-e2-8-3 +- TauLib.BookIII.Enrichment.Functor::enr_functor_e2_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: enr_functor_e2_8_3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Enrichment functor E₂→E₃ [III.D04] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005867.md b/_corpus_v3_items/cid005867.md new file mode 100644 index 000000000..fd005b58f --- /dev/null +++ b/_corpus_v3_items/cid005867.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005867/ +title: full_enr_functor_8_3 (theorem) +cid: cid005867 +primary_alias: FTH0602 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full enrichment functor [III.D04] +aliases: +- FTH0602 +- full_enr_functor_8_3 +- full-enr-functor-8-3 +- TauLib.BookIII.Enrichment.Functor::full_enr_functor_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: full_enr_functor_8_3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full enrichment functor [III.D04] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005868.md b/_corpus_v3_items/cid005868.md new file mode 100644 index 000000000..2fb431dd3 --- /dev/null +++ b/_corpus_v3_items/cid005868.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005868/ +title: ladder_8_3 (theorem) +cid: cid005868 +primary_alias: FTH0603 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full ladder checker [III.D10] +aliases: +- FTH0603 +- ladder_8_3 +- ladder-8-3 +- TauLib.BookIII.Enrichment.Functor::ladder_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001441 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: ladder_8_3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full ladder checker [III.D10] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005869.md b/_corpus_v3_items/cid005869.md new file mode 100644 index 000000000..a441cc34e --- /dev/null +++ b/_corpus_v3_items/cid005869.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005869/ +title: succ_idempotent_e3 (theorem) +cid: cid005869 +primary_alias: FTH0604 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D04] Structural proof: EnrLevel.succ is idempotent at E₃.\n This is the functor-level expression of\ + \ saturation. -/" +aliases: +- FTH0604 +- succ_idempotent_e3 +- succ-idempotent-e3 +- TauLib.BookIII.Enrichment.Functor::succ_idempotent_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: succ_idempotent_e3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D04] Structural proof: EnrLevel.succ is idempotent at E₃.\n This is the functor-level expression\ + \ of saturation. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005870.md b/_corpus_v3_items/cid005870.md new file mode 100644 index 000000000..1bd172fad --- /dev/null +++ b/_corpus_v3_items/cid005870.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005870/ +title: three_steps_to_e3 (theorem) +cid: cid005870 +primary_alias: FTH0605 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D04] Enrichment functor terminates: after at most 3 applications,\n we reach E₃ regardless of starting\ + \ level. -/" +aliases: +- FTH0605 +- three_steps_to_e3 +- three-steps-to-e3 +- TauLib.BookIII.Enrichment.Functor::three_steps_to_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001435 +- predicate: part_of + object: cid005229 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005229 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.Functor + declaration: three_steps_to_e3 + module_path: TauLib/BookIII/Enrichment/Functor.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D04] Enrichment functor terminates: after at most 3 applications,\n we reach E₃ regardless of starting\ + \ level. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005871.md b/_corpus_v3_items/cid005871.md new file mode 100644 index 000000000..c02c8a1a2 --- /dev/null +++ b/_corpus_v3_items/cid005871.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005871/ +title: e0_layer_valid_8_3 (theorem) +cid: cid005871 +primary_alias: FTH0606 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- E₀ layer valid [III.D06]' +aliases: +- FTH0606 +- e0_layer_valid_8_3 +- e0-layer-valid-8-3 +- TauLib.BookIII.Enrichment.LayerTemplate::e0_layer_valid_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001437 +- predicate: part_of + object: cid005230 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005230 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + declaration: e0_layer_valid_8_3 + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- E₀ layer valid [III.D06]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005872.md b/_corpus_v3_items/cid005872.md new file mode 100644 index 000000000..0dc35e13b --- /dev/null +++ b/_corpus_v3_items/cid005872.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005872/ +title: e1_layer_valid_8_3 (theorem) +cid: cid005872 +primary_alias: FTH0607 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E₁ layer valid [III.D07] +aliases: +- FTH0607 +- e1_layer_valid_8_3 +- e1-layer-valid-8-3 +- TauLib.BookIII.Enrichment.LayerTemplate::e1_layer_valid_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001438 +- predicate: part_of + object: cid005230 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005230 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + declaration: e1_layer_valid_8_3 + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E₁ layer valid [III.D07] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005873.md b/_corpus_v3_items/cid005873.md new file mode 100644 index 000000000..8562c6ee2 --- /dev/null +++ b/_corpus_v3_items/cid005873.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005873/ +title: e2_layer_valid_8_3 (theorem) +cid: cid005873 +primary_alias: FTH0608 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E₂ layer valid [III.D08] +aliases: +- FTH0608 +- e2_layer_valid_8_3 +- e2-layer-valid-8-3 +- TauLib.BookIII.Enrichment.LayerTemplate::e2_layer_valid_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001439 +- predicate: part_of + object: cid005230 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005230 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + declaration: e2_layer_valid_8_3 + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E₂ layer valid [III.D08] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005874.md b/_corpus_v3_items/cid005874.md new file mode 100644 index 000000000..3de792fe1 --- /dev/null +++ b/_corpus_v3_items/cid005874.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005874/ +title: e3_layer_valid_8_3 (theorem) +cid: cid005874 +primary_alias: FTH0609 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E₃ layer valid [III.D09] +aliases: +- FTH0609 +- e3_layer_valid_8_3 +- e3-layer-valid-8-3 +- TauLib.BookIII.Enrichment.LayerTemplate::e3_layer_valid_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001440 +- predicate: part_of + object: cid005230 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005230 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + declaration: e3_layer_valid_8_3 + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E₃ layer valid [III.D09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005875.md b/_corpus_v3_items/cid005875.md new file mode 100644 index 000000000..163b4ce2e --- /dev/null +++ b/_corpus_v3_items/cid005875.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005875/ +title: all_layers_8_3 (theorem) +cid: cid005875 +primary_alias: FTH0610 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- All four layers valid [III.D05] +aliases: +- FTH0610 +- all_layers_8_3 +- all-layers-8-3 +- TauLib.BookIII.Enrichment.LayerTemplate::all_layers_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001436 +- predicate: part_of + object: cid005230 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005230 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Enrichment.LayerTemplate + declaration: all_layers_8_3 + module_path: TauLib/BookIII/Enrichment/LayerTemplate.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- All four layers valid [III.D05] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005876.md b/_corpus_v3_items/cid005876.md new file mode 100644 index 000000000..4231d30fb --- /dev/null +++ b/_corpus_v3_items/cid005876.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005876/ +title: dependency_chain_8_3 (theorem) +cid: cid005876 +primary_alias: FTH0611 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D66] Dependency chain' +aliases: +- FTH0611 +- dependency_chain_8_3 +- dependency-chain-8-3 +- TauLib.BookIII.Hinge.DependencyChain::dependency_chain_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: dependency_chain_8_3 + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.D66] Dependency chain' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005877.md b/_corpus_v3_items/cid005877.md new file mode 100644 index 000000000..a463ca49e --- /dev/null +++ b/_corpus_v3_items/cid005877.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005877/ +title: chain_linearity (theorem) +cid: cid005877 +primary_alias: FTH0612 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.P29] Chain linearity +aliases: +- FTH0612 +- chain_linearity +- chain-linearity +- TauLib.BookIII.Hinge.DependencyChain::chain_linearity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001573 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: chain_linearity + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.P29] Chain linearity +noindex: false +--- + diff --git a/_corpus_v3_items/cid005878.md b/_corpus_v3_items/cid005878.md new file mode 100644 index 000000000..cbaa9e533 --- /dev/null +++ b/_corpus_v3_items/cid005878.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005878/ +title: terminal_completeness (theorem) +cid: cid005878 +primary_alias: FTH0613 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.P30] Terminal completeness +aliases: +- FTH0613 +- terminal_completeness +- terminal-completeness +- TauLib.BookIII.Hinge.DependencyChain::terminal_completeness +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001574 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: terminal_completeness + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.P30] Terminal completeness +noindex: false +--- + diff --git a/_corpus_v3_items/cid005879.md b/_corpus_v3_items/cid005879.md new file mode 100644 index 000000000..40f240395 --- /dev/null +++ b/_corpus_v3_items/cid005879.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005879/ +title: chain_has_14_links (theorem) +cid: cid005879 +primary_alias: FTH0614 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D66] Structural: the chain has exactly 14 links. -/' +aliases: +- FTH0614 +- chain_has_14_links +- chain-has-14-links +- TauLib.BookIII.Hinge.DependencyChain::chain_has_14_links +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: chain_has_14_links + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D66] Structural: the chain has exactly 14 links. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005880.md b/_corpus_v3_items/cid005880.md new file mode 100644 index 000000000..b4a738835 --- /dev/null +++ b/_corpus_v3_items/cid005880.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005880/ +title: k0_is_first (theorem) +cid: cid005880 +primary_alias: FTH0615 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D66] Structural: K0 is the first link (index 0). -/' +aliases: +- FTH0615 +- k0_is_first +- k0-is-first +- TauLib.BookIII.Hinge.DependencyChain::k0_is_first +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: k0_is_first + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D66] Structural: K0 is the first link (index 0). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005881.md b/_corpus_v3_items/cid005881.md new file mode 100644 index 000000000..f9bac0c4a --- /dev/null +++ b/_corpus_v3_items/cid005881.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005881/ +title: e3p_is_last (theorem) +cid: cid005881 +primary_alias: FTH0616 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D66] Structural: E3p is the last link (index 13). -/' +aliases: +- FTH0616 +- e3p_is_last +- e3p-is-last +- TauLib.BookIII.Hinge.DependencyChain::e3p_is_last +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: e3p_is_last + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D66] Structural: E3p is the last link (index 13). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005882.md b/_corpus_v3_items/cid005882.md new file mode 100644 index 000000000..1e7b00f88 --- /dev/null +++ b/_corpus_v3_items/cid005882.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005882/ +title: axiom_to_enrichment (theorem) +cid: cid005882 +primary_alias: FTH0617 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D66] Structural: K6 -> E0 is the axiom-to-enrichment transition. -/' +aliases: +- FTH0617 +- axiom_to_enrichment +- axiom-to-enrichment +- TauLib.BookIII.Hinge.DependencyChain::axiom_to_enrichment +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: axiom_to_enrichment + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D66] Structural: K6 -> E0 is the axiom-to-enrichment transition. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005883.md b/_corpus_v3_items/cid005883.md new file mode 100644 index 000000000..173079602 --- /dev/null +++ b/_corpus_v3_items/cid005883.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005883/ +title: succ_monotone (theorem) +cid: cid005883 +primary_alias: FTH0618 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P29] Structural: successor always increases index (except at E3p). -/' +aliases: +- FTH0618 +- succ_monotone +- succ-monotone +- TauLib.BookIII.Hinge.DependencyChain::succ_monotone +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001573 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: succ_monotone + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P29] Structural: successor always increases index (except at E3p). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005884.md b/_corpus_v3_items/cid005884.md new file mode 100644 index 000000000..53690da01 --- /dev/null +++ b/_corpus_v3_items/cid005884.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005884/ +title: e3p_saturates (theorem) +cid: cid005884 +primary_alias: FTH0619 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P30] Structural: E3p.succ = E3p (terminal saturation). -/' +aliases: +- FTH0619 +- e3p_saturates +- e3p-saturates +- TauLib.BookIII.Hinge.DependencyChain::e3p_saturates +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001574 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: e3p_saturates + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P30] Structural: E3p.succ = E3p (terminal saturation). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005885.md b/_corpus_v3_items/cid005885.md new file mode 100644 index 000000000..07113dbd7 --- /dev/null +++ b/_corpus_v3_items/cid005885.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005885/ +title: all_links_have_level (theorem) +cid: cid005885 +primary_alias: FTH0620 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P30] Structural: every link maps to a valid enrichment level. -/' +aliases: +- FTH0620 +- all_links_have_level +- all-links-have-level +- TauLib.BookIII.Hinge.DependencyChain::all_links_have_level +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001574 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: all_links_have_level + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P30] Structural: every link maps to a valid enrichment level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005886.md b/_corpus_v3_items/cid005886.md new file mode 100644 index 000000000..068eb6176 --- /dev/null +++ b/_corpus_v3_items/cid005886.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005886/ +title: seven_plus_seven (theorem) +cid: cid005886 +primary_alias: FTH0621 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D66] Structural: the chain covers 7 axiom links + 7 enrichment links. -/' +aliases: +- FTH0621 +- seven_plus_seven +- seven-plus-seven +- TauLib.BookIII.Hinge.DependencyChain::seven_plus_seven +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001511 +- predicate: part_of + object: cid005231 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005231 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.DependencyChain + declaration: seven_plus_seven + module_path: TauLib/BookIII/Hinge/DependencyChain.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D66] Structural: the chain covers 7 axiom links + 7 enrichment links. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005887.md b/_corpus_v3_items/cid005887.md new file mode 100644 index 000000000..8454b80a8 --- /dev/null +++ b/_corpus_v3_items/cid005887.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005887/ +title: hinge_theorem_8_3 (theorem) +cid: cid005887 +primary_alias: FTH0622 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.T41] Hinge Theorem' +aliases: +- FTH0622 +- hinge_theorem_8_3 +- hinge-theorem-8-3 +- TauLib.BookIII.Hinge.HingeTheorem::hinge_theorem_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001633 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: hinge_theorem_8_3 + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- [III.T41] Hinge Theorem' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005888.md b/_corpus_v3_items/cid005888.md new file mode 100644 index 000000000..9b72756ab --- /dev/null +++ b/_corpus_v3_items/cid005888.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005888/ +title: no_knobs_8_3 (theorem) +cid: cid005888 +primary_alias: FTH0623 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- [III.T42] No-Knobs Theorem +aliases: +- FTH0623 +- no_knobs_8_3 +- no-knobs-8-3 +- TauLib.BookIII.Hinge.HingeTheorem::no_knobs_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: no_knobs_8_3 + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- [III.T42] No-Knobs Theorem +noindex: false +--- + diff --git a/_corpus_v3_items/cid005889.md b/_corpus_v3_items/cid005889.md new file mode 100644 index 000000000..0ea2800ae --- /dev/null +++ b/_corpus_v3_items/cid005889.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005889/ +title: hinge_chain_length (theorem) +cid: cid005889 +primary_alias: FTH0624 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T41] Structural: the hinge chain has 14 links. -/' +aliases: +- FTH0624 +- hinge_chain_length +- hinge-chain-length +- TauLib.BookIII.Hinge.HingeTheorem::hinge_chain_length +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001633 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: hinge_chain_length + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T41] Structural: the hinge chain has 14 links. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005890.md b/_corpus_v3_items/cid005890.md new file mode 100644 index 000000000..5b403a244 --- /dev/null +++ b/_corpus_v3_items/cid005890.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005890/ +title: sector_product_depth_3 (theorem) +cid: cid005890 +primary_alias: FTH0625 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T41] Structural: B * C * X = Prim(3) = 30 at depth 3. -/' +aliases: +- FTH0625 +- sector_product_depth_3 +- sector-product-depth-3 +- TauLib.BookIII.Hinge.HingeTheorem::sector_product_depth_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001633 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: sector_product_depth_3 + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T41] Structural: B * C * X = Prim(3) = 30 at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005891.md b/_corpus_v3_items/cid005891.md new file mode 100644 index 000000000..d4b7d0a6d --- /dev/null +++ b/_corpus_v3_items/cid005891.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005891/ +title: enrichment_strict (theorem) +cid: cid005891 +primary_alias: FTH0626 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T41] Structural: enrichment level ordering is strict. -/' +aliases: +- FTH0626 +- enrichment_strict +- enrichment-strict +- TauLib.BookIII.Hinge.HingeTheorem::enrichment_strict +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001633 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: enrichment_strict + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T41] Structural: enrichment level ordering is strict. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005892.md b/_corpus_v3_items/cid005892.md new file mode 100644 index 000000000..2d08f8d4d --- /dev/null +++ b/_corpus_v3_items/cid005892.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005892/ +title: iota_value (theorem) +cid: cid005892 +primary_alias: FTH0627 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T42] Structural: iota_tau approximation is 341304/1000000. -/' +aliases: +- FTH0627 +- iota_value +- iota-value +- TauLib.BookIII.Hinge.HingeTheorem::iota_value +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: iota_value + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T42] Structural: iota_tau approximation is 341304/1000000. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005893.md b/_corpus_v3_items/cid005893.md new file mode 100644 index 000000000..5eac6e7df --- /dev/null +++ b/_corpus_v3_items/cid005893.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005893/ +title: iota_lt_one (theorem) +cid: cid005893 +primary_alias: FTH0628 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T42] Structural: iota_tau < 1 (B is the minority channel). -/' +aliases: +- FTH0628 +- iota_lt_one +- iota-lt-one +- TauLib.BookIII.Hinge.HingeTheorem::iota_lt_one +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: iota_lt_one + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T42] Structural: iota_tau < 1 (B is the minority channel). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005894.md b/_corpus_v3_items/cid005894.md new file mode 100644 index 000000000..697df28c4 --- /dev/null +++ b/_corpus_v3_items/cid005894.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005894/ +title: iota_pos (theorem) +cid: cid005894 +primary_alias: FTH0629 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T42] Structural: iota_tau > 0 (master constant is positive). -/' +aliases: +- FTH0629 +- iota_pos +- iota-pos +- TauLib.BookIII.Hinge.HingeTheorem::iota_pos +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: iota_pos + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T42] Structural: iota_tau > 0 (master constant is positive). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005895.md b/_corpus_v3_items/cid005895.md new file mode 100644 index 000000000..7c8d897f3 --- /dev/null +++ b/_corpus_v3_items/cid005895.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005895/ +title: iota_ratio_depth_3 (theorem) +cid: cid005895 +primary_alias: FTH0630 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T42] Structural: at depth 3, B/C > iota_tau (convergence not yet reached).\n B*denom = 5*1000000 = 5000000\ + \ > C*numer = 3*341304 = 1023912. -/" +aliases: +- FTH0630 +- iota_ratio_depth_3 +- iota-ratio-depth-3 +- TauLib.BookIII.Hinge.HingeTheorem::iota_ratio_depth_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: iota_ratio_depth_3 + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T42] Structural: at depth 3, B/C > iota_tau (convergence not yet reached).\n B*denom = 5*1000000\ + \ = 5000000 > C*numer = 3*341304 = 1023912. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005896.md b/_corpus_v3_items/cid005896.md new file mode 100644 index 000000000..a89bd532c --- /dev/null +++ b/_corpus_v3_items/cid005896.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005896/ +title: iota_ratio_depth_4 (theorem) +cid: cid005896 +primary_alias: FTH0631 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T42] Structural: at depth 4, B/C < iota_tau (crossover).\n B*denom = 5*1000000 = 5000000 < C*numer =\ + \ 21*341304 = 7167384. -/" +aliases: +- FTH0631 +- iota_ratio_depth_4 +- iota-ratio-depth-4 +- TauLib.BookIII.Hinge.HingeTheorem::iota_ratio_depth_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: iota_ratio_depth_4 + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T42] Structural: at depth 4, B/C < iota_tau (crossover).\n B*denom = 5*1000000 = 5000000 < C*numer\ + \ = 21*341304 = 7167384. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005897.md b/_corpus_v3_items/cid005897.md new file mode 100644 index 000000000..a462dea59 --- /dev/null +++ b/_corpus_v3_items/cid005897.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005897/ +title: five_sectors_exhaustive (theorem) +cid: cid005897 +primary_alias: FTH0632 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T42] Structural: the five sectors are exhaustive. -/' +aliases: +- FTH0632 +- five_sectors_exhaustive +- five-sectors-exhaustive +- TauLib.BookIII.Hinge.HingeTheorem::five_sectors_exhaustive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: five_sectors_exhaustive + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T42] Structural: the five sectors are exhaustive. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005898.md b/_corpus_v3_items/cid005898.md new file mode 100644 index 000000000..ada6aae19 --- /dev/null +++ b/_corpus_v3_items/cid005898.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005898/ +title: no_knobs_witness (theorem) +cid: cid005898 +primary_alias: FTH0633 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T42] Structural: no-knobs means sector products are derivable.\n At depth 3: B=5, C=3, X=2, and 5*3*2\ + \ = 30 = Prim(3). -/" +aliases: +- FTH0633 +- no_knobs_witness +- no-knobs-witness +- TauLib.BookIII.Hinge.HingeTheorem::no_knobs_witness +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001634 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: no_knobs_witness + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T42] Structural: no-knobs means sector products are derivable.\n At depth 3: B=5, C=3, X=2, and\ + \ 5*3*2 = 30 = Prim(3). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005899.md b/_corpus_v3_items/cid005899.md new file mode 100644 index 000000000..97506cd5f --- /dev/null +++ b/_corpus_v3_items/cid005899.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005899/ +title: hinge_point (theorem) +cid: cid005899 +primary_alias: FTH0634 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T41] Structural: the axiom-to-enrichment transition at K6 -> E0\n is the hinge point where Books I-II\ + \ feed into Book III. -/" +aliases: +- FTH0634 +- hinge_point +- hinge-point +- TauLib.BookIII.Hinge.HingeTheorem::hinge_point +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001633 +- predicate: part_of + object: cid005232 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005232 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Hinge.HingeTheorem + declaration: hinge_point + module_path: TauLib/BookIII/Hinge/HingeTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T41] Structural: the axiom-to-enrichment transition at K6 -> E0\n is the hinge point where Books\ + \ I-II feed into Book III. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005900.md b/_corpus_v3_items/cid005900.md new file mode 100644 index 000000000..ea6e50a11 --- /dev/null +++ b/_corpus_v3_items/cid005900.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005900/ +title: fixed_point_check_3 (theorem) +cid: cid005900 +primary_alias: FTH0635 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D85] E₃ fixed points exist at stages 1-3. -/ +aliases: +- FTH0635 +- fixed_point_check_3 +- fixed-point-check-3 +- TauLib.BookIII.Mirror.E3Witness::fixed_point_check_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001530 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: fixed_point_check_3 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D85] E₃ fixed points exist at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005901.md b/_corpus_v3_items/cid005901.md new file mode 100644 index 000000000..328a4096a --- /dev/null +++ b/_corpus_v3_items/cid005901.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005901/ +title: e3_density_2 (theorem) +cid: cid005901 +primary_alias: FTH0636 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D85] All elements are E₃ at stage 2. -/ +aliases: +- FTH0636 +- e3_density_2 +- e3-density-2 +- TauLib.BookIII.Mirror.E3Witness::e3_density_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001530 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: e3_density_2 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D85] All elements are E₃ at stage 2. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005902.md b/_corpus_v3_items/cid005902.md new file mode 100644 index 000000000..99e2fe829 --- /dev/null +++ b/_corpus_v3_items/cid005902.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005902/ +title: paradox_absorbed_3 (theorem) +cid: cid005902 +primary_alias: FTH0637 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D86] Paradoxes absorbed at stages 1-3. -/ +aliases: +- FTH0637 +- paradox_absorbed_3 +- paradox-absorbed-3 +- TauLib.BookIII.Mirror.E3Witness::paradox_absorbed_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001531 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: paradox_absorbed_3 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D86] Paradoxes absorbed at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005903.md b/_corpus_v3_items/cid005903.md new file mode 100644 index 000000000..142552241 --- /dev/null +++ b/_corpus_v3_items/cid005903.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005903/ +title: self_model_complete_3 (theorem) +cid: cid005903 +primary_alias: FTH0638 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T58] Self-model completeness at stages 1-3. -/ +aliases: +- FTH0638 +- self_model_complete_3 +- self-model-complete-3 +- TauLib.BookIII.Mirror.E3Witness::self_model_complete_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001650 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: self_model_complete_3 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T58] Self-model completeness at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005904.md b/_corpus_v3_items/cid005904.md new file mode 100644 index 000000000..2f2ebd503 --- /dev/null +++ b/_corpus_v3_items/cid005904.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005904/ +title: saturation_semantic_3 (theorem) +cid: cid005904 +primary_alias: FTH0639 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P35] Saturation semantics at stages 1-3. -/ +aliases: +- FTH0639 +- saturation_semantic_3 +- saturation-semantic-3 +- TauLib.BookIII.Mirror.E3Witness::saturation_semantic_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001579 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: saturation_semantic_3 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P35] Saturation semantics at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005905.md b/_corpus_v3_items/cid005905.md new file mode 100644 index 000000000..9f73eedf8 --- /dev/null +++ b/_corpus_v3_items/cid005905.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005905/ +title: self_model_idempotent_3 (theorem) +cid: cid005905 +primary_alias: FTH0640 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P35] Self-model idempotent at stages 1-3. -/ +aliases: +- FTH0640 +- self_model_idempotent_3 +- self-model-idempotent-3 +- TauLib.BookIII.Mirror.E3Witness::self_model_idempotent_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001579 +- predicate: part_of + object: cid005233 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005233 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.E3Witness + declaration: self_model_idempotent_3 + module_path: TauLib/BookIII/Mirror/E3Witness.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P35] Self-model idempotent at stages 1-3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005906.md b/_corpus_v3_items/cid005906.md new file mode 100644 index 000000000..caa81f2f8 --- /dev/null +++ b/_corpus_v3_items/cid005906.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005906/ +title: proof_theory_e3_8_3 (theorem) +cid: cid005906 +primary_alias: FTH0641 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Proof theory as E3 [III.D73]' +aliases: +- FTH0641 +- proof_theory_e3_8_3 +- proof-theory-e3-8-3 +- TauLib.BookIII.Mirror.ProofTheoryE3::proof_theory_e3_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001518 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: proof_theory_e3_8_3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Proof theory as E3 [III.D73]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005907.md b/_corpus_v3_items/cid005907.md new file mode 100644 index 000000000..4782f21a4 --- /dev/null +++ b/_corpus_v3_items/cid005907.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005907/ +title: self_model_8_3 (theorem) +cid: cid005907 +primary_alias: FTH0642 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Self-model functor [III.D74] +aliases: +- FTH0642 +- self_model_8_3 +- self-model-8-3 +- TauLib.BookIII.Mirror.ProofTheoryE3::self_model_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001519 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: self_model_8_3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Self-model functor [III.D74] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005908.md b/_corpus_v3_items/cid005908.md new file mode 100644 index 000000000..dc5f7d9da --- /dev/null +++ b/_corpus_v3_items/cid005908.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005908/ +title: self_model_inv_8_3 (theorem) +cid: cid005908 +primary_alias: FTH0643 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Self-model invariant preservation [III.D74] +aliases: +- FTH0643 +- self_model_inv_8_3 +- self-model-inv-8-3 +- TauLib.BookIII.Mirror.ProofTheoryE3::self_model_inv_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001519 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: self_model_inv_8_3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Self-model invariant preservation [III.D74] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005909.md b/_corpus_v3_items/cid005909.md new file mode 100644 index 000000000..77219de1a --- /dev/null +++ b/_corpus_v3_items/cid005909.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005909/ +title: four_paradox_8_3 (theorem) +cid: cid005909 +primary_alias: FTH0644 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Four paradox diagnostic [III.D75] +aliases: +- FTH0644 +- four_paradox_8_3 +- four-paradox-8-3 +- TauLib.BookIII.Mirror.ProofTheoryE3::four_paradox_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: four_paradox_8_3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Four paradox diagnostic [III.D75] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005910.md b/_corpus_v3_items/cid005910.md new file mode 100644 index 000000000..fed186118 --- /dev/null +++ b/_corpus_v3_items/cid005910.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005910/ +title: forbidden_moves_distinct_thm (theorem) +cid: cid005910 +primary_alias: FTH0645 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Forbidden moves distinct [III.D75] +aliases: +- FTH0645 +- forbidden_moves_distinct_thm +- forbidden-moves-distinct-thm +- TauLib.BookIII.Mirror.ProofTheoryE3::forbidden_moves_distinct_thm +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: forbidden_moves_distinct_thm + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Forbidden moves distinct [III.D75] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005911.md b/_corpus_v3_items/cid005911.md new file mode 100644 index 000000000..9b32cdb3e --- /dev/null +++ b/_corpus_v3_items/cid005911.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005911/ +title: paradox_resolution_8_3 (theorem) +cid: cid005911 +primary_alias: FTH0646 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Paradox resolution [III.T48] +aliases: +- FTH0646 +- paradox_resolution_8_3 +- paradox-resolution-8-3 +- TauLib.BookIII.Mirror.ProofTheoryE3::paradox_resolution_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001640 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: paradox_resolution_8_3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Paradox resolution [III.T48] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005912.md b/_corpus_v3_items/cid005912.md new file mode 100644 index 000000000..3a40a956a --- /dev/null +++ b/_corpus_v3_items/cid005912.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005912/ +title: e3_is_proof_theory (theorem) +cid: cid005912 +primary_alias: FTH0647 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D73] Structural: E3 is the proof-theoretic level. -/' +aliases: +- FTH0647 +- e3_is_proof_theory +- e3-is-proof-theory +- TauLib.BookIII.Mirror.ProofTheoryE3::e3_is_proof_theory +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001518 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: e3_is_proof_theory + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D73] Structural: E3 is the proof-theoretic level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005913.md b/_corpus_v3_items/cid005913.md new file mode 100644 index 000000000..f409622a7 --- /dev/null +++ b/_corpus_v3_items/cid005913.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005913/ +title: self_model_levels (theorem) +cid: cid005913 +primary_alias: FTH0648 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D74] Structural: self-model functor goes from E2 to E3. -/' +aliases: +- FTH0648 +- self_model_levels +- self-model-levels +- TauLib.BookIII.Mirror.ProofTheoryE3::self_model_levels +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001519 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: self_model_levels + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D74] Structural: self-model functor goes from E2 to E3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005914.md b/_corpus_v3_items/cid005914.md new file mode 100644 index 000000000..b81bdcf26 --- /dev/null +++ b/_corpus_v3_items/cid005914.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005914/ +title: all_paradoxes_at_e2 (theorem) +cid: cid005914 +primary_alias: FTH0649 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D75] Structural: all paradoxes arise at E2. -/' +aliases: +- FTH0649 +- all_paradoxes_at_e2 +- all-paradoxes-at-e2 +- TauLib.BookIII.Mirror.ProofTheoryE3::all_paradoxes_at_e2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: all_paradoxes_at_e2 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D75] Structural: all paradoxes arise at E2. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005915.md b/_corpus_v3_items/cid005915.md new file mode 100644 index 000000000..5c7e4dd0c --- /dev/null +++ b/_corpus_v3_items/cid005915.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005915/ +title: all_paradoxes_resolve_e3 (theorem) +cid: cid005915 +primary_alias: FTH0650 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D75] Structural: all paradoxes resolve at E3. -/' +aliases: +- FTH0650 +- all_paradoxes_resolve_e3 +- all-paradoxes-resolve-e3 +- TauLib.BookIII.Mirror.ProofTheoryE3::all_paradoxes_resolve_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: all_paradoxes_resolve_e3 + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D75] Structural: all paradoxes resolve at E3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005916.md b/_corpus_v3_items/cid005916.md new file mode 100644 index 000000000..86067bfa2 --- /dev/null +++ b/_corpus_v3_items/cid005916.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005916/ +title: exactly_four_paradoxes (theorem) +cid: cid005916 +primary_alias: FTH0651 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D75] Structural: exactly four paradoxes. -/' +aliases: +- FTH0651 +- exactly_four_paradoxes +- exactly-four-paradoxes +- TauLib.BookIII.Mirror.ProofTheoryE3::exactly_four_paradoxes +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: exactly_four_paradoxes + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D75] Structural: exactly four paradoxes. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005917.md b/_corpus_v3_items/cid005917.md new file mode 100644 index 000000000..09a97e303 --- /dev/null +++ b/_corpus_v3_items/cid005917.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005917/ +title: forbidden_move_range (theorem) +cid: cid005917 +primary_alias: FTH0652 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D75] Structural: forbidden move indices cover 0..3. -/' +aliases: +- FTH0652 +- forbidden_move_range +- forbidden-move-range +- TauLib.BookIII.Mirror.ProofTheoryE3::forbidden_move_range +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001520 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: forbidden_move_range + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D75] Structural: forbidden move indices cover 0..3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005918.md b/_corpus_v3_items/cid005918.md new file mode 100644 index 000000000..6ca2c705a --- /dev/null +++ b/_corpus_v3_items/cid005918.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005918/ +title: paradox_move_injective (theorem) +cid: cid005918 +primary_alias: FTH0653 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T48] Structural: each paradox maps to a unique forbidden move. -/' +aliases: +- FTH0653 +- paradox_move_injective +- paradox-move-injective +- TauLib.BookIII.Mirror.ProofTheoryE3::paradox_move_injective +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001640 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: paradox_move_injective + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T48] Structural: each paradox maps to a unique forbidden move. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005919.md b/_corpus_v3_items/cid005919.md new file mode 100644 index 000000000..5678e2131 --- /dev/null +++ b/_corpus_v3_items/cid005919.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005919/ +title: paradox_gap (theorem) +cid: cid005919 +primary_alias: FTH0654 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T48] Structural: E2 < E3 is the paradox gap. -/' +aliases: +- FTH0654 +- paradox_gap +- paradox-gap +- TauLib.BookIII.Mirror.ProofTheoryE3::paradox_gap +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001640 +- predicate: part_of + object: cid005234 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005234 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.ProofTheoryE3 + declaration: paradox_gap + module_path: TauLib/BookIII/Mirror/ProofTheoryE3.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T48] Structural: E2 < E3 is the paradox gap. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005920.md b/_corpus_v3_items/cid005920.md new file mode 100644 index 000000000..5bca49ded --- /dev/null +++ b/_corpus_v3_items/cid005920.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005920/ +title: applied_saturation_8_3 (theorem) +cid: cid005920 +primary_alias: FTH0655 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Applied saturation [III.T49]' +aliases: +- FTH0655 +- applied_saturation_8_3 +- applied-saturation-8-3 +- TauLib.BookIII.Mirror.Saturation::applied_saturation_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001641 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: applied_saturation_8_3 + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Applied saturation [III.T49]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005921.md b/_corpus_v3_items/cid005921.md new file mode 100644 index 000000000..653053cf7 --- /dev/null +++ b/_corpus_v3_items/cid005921.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005921/ +title: full_saturation_8_3 (theorem) +cid: cid005921 +primary_alias: FTH0656 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Full component-wise saturation [III.T49] +aliases: +- FTH0656 +- full_saturation_8_3 +- full-saturation-8-3 +- TauLib.BookIII.Mirror.Saturation::full_saturation_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001641 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: full_saturation_8_3 + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Full component-wise saturation [III.T49] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005922.md b/_corpus_v3_items/cid005922.md new file mode 100644 index 000000000..e72f60a48 --- /dev/null +++ b/_corpus_v3_items/cid005922.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005922/ +title: terminal_level_8_3 (theorem) +cid: cid005922 +primary_alias: FTH0657 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Terminal level characterization [III.P31] +aliases: +- FTH0657 +- terminal_level_8_3 +- terminal-level-8-3 +- TauLib.BookIII.Mirror.Saturation::terminal_level_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001575 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: terminal_level_8_3 + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Terminal level characterization [III.P31] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005923.md b/_corpus_v3_items/cid005923.md new file mode 100644 index 000000000..33aa74986 --- /dev/null +++ b/_corpus_v3_items/cid005923.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005923/ +title: e3_is_terminal (theorem) +cid: cid005923 +primary_alias: FTH0658 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T49] E3 is terminal: E3 does not exceed itself. -/' +aliases: +- FTH0658 +- e3_is_terminal +- e3-is-terminal +- TauLib.BookIII.Mirror.Saturation::e3_is_terminal +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001641 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: e3_is_terminal + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T49] E3 is terminal: E3 does not exceed itself. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005924.md b/_corpus_v3_items/cid005924.md new file mode 100644 index 000000000..6a0a50970 --- /dev/null +++ b/_corpus_v3_items/cid005924.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005924/ +title: e3_is_max (theorem) +cid: cid005924 +primary_alias: FTH0659 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P31] E3 is the maximum level. -/ +aliases: +- FTH0659 +- e3_is_max +- e3-is-max +- TauLib.BookIII.Mirror.Saturation::e3_is_max +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001575 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: e3_is_max + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P31] E3 is the maximum level. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005925.md b/_corpus_v3_items/cid005925.md new file mode 100644 index 000000000..06dd03be3 --- /dev/null +++ b/_corpus_v3_items/cid005925.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005925/ +title: e3_succ_nat (theorem) +cid: cid005925 +primary_alias: FTH0660 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T49] Structural: E3.succ.toNat = E3.toNat (numeric saturation). -/' +aliases: +- FTH0660 +- e3_succ_nat +- e3-succ-nat +- TauLib.BookIII.Mirror.Saturation::e3_succ_nat +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001641 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: e3_succ_nat + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T49] Structural: E3.succ.toNat = E3.toNat (numeric saturation). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005926.md b/_corpus_v3_items/cid005926.md new file mode 100644 index 000000000..ee973c742 --- /dev/null +++ b/_corpus_v3_items/cid005926.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005926/ +title: e3_unique_fixpoint (theorem) +cid: cid005926 +primary_alias: FTH0661 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.P31] Structural: E3 is the unique fixpoint of succ.\n (E0, E1, E2 all advance; only E3 is fixed.) -/" +aliases: +- FTH0661 +- e3_unique_fixpoint +- e3-unique-fixpoint +- TauLib.BookIII.Mirror.Saturation::e3_unique_fixpoint +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001575 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: e3_unique_fixpoint + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.P31] Structural: E3 is the unique fixpoint of succ.\n (E0, E1, E2 all advance; only E3 is fixed.)\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005927.md b/_corpus_v3_items/cid005927.md new file mode 100644 index 000000000..b3a4d4738 --- /dev/null +++ b/_corpus_v3_items/cid005927.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005927/ +title: strictly_increasing_nat (theorem) +cid: cid005927 +primary_alias: FTH0662 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.P31] Structural: the enrichment levels have strictly\n increasing toNat values (witnesses total order).\ + \ -/" +aliases: +- FTH0662 +- strictly_increasing_nat +- strictly-increasing-nat +- TauLib.BookIII.Mirror.Saturation::strictly_increasing_nat +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001575 +- predicate: part_of + object: cid005235 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005235 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Mirror.Saturation + declaration: strictly_increasing_nat + module_path: TauLib/BookIII/Mirror/Saturation.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.P31] Structural: the enrichment levels have strictly\n increasing toNat values (witnesses total\ + \ order). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005928.md b/_corpus_v3_items/cid005928.md new file mode 100644 index 000000000..82e9b3d9e --- /dev/null +++ b/_corpus_v3_items/cid005928.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005928/ +title: fluid_data_depth_3 (theorem) +cid: cid005928 +primary_alias: FTH0663 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D36] Structural: canonical fluid data at depth 3 has 30 values. -/' +aliases: +- FTH0663 +- fluid_data_depth_3 +- fluid-data-depth-3 +- TauLib.BookIII.Physics.FluidData::fluid_data_depth_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001481 +- predicate: part_of + object: cid005236 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005236 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.FluidData + declaration: fluid_data_depth_3 + module_path: TauLib/BookIII/Physics/FluidData.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D36] Structural: canonical fluid data at depth 3 has 30 values. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005929.md b/_corpus_v3_items/cid005929.md new file mode 100644 index 000000000..9544800fe --- /dev/null +++ b/_corpus_v3_items/cid005929.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005929/ +title: cylinder_assign_0 (theorem) +cid: cid005929 +primary_alias: FTH0664 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D37] Structural: cylinder assignment at depth 0 is trivial. -/' +aliases: +- FTH0664 +- cylinder_assign_0 +- cylinder-assign-0 +- TauLib.BookIII.Physics.FluidData::cylinder_assign_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001482 +- predicate: part_of + object: cid005236 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005236 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.FluidData + declaration: cylinder_assign_0 + module_path: TauLib/BookIII/Physics/FluidData.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D37] Structural: cylinder assignment at depth 0 is trivial. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005930.md b/_corpus_v3_items/cid005930.md new file mode 100644 index 000000000..9cbf10dcf --- /dev/null +++ b/_corpus_v3_items/cid005930.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005930/ +title: abcd_zero_3 (theorem) +cid: cid005930 +primary_alias: FTH0665 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D38] Structural: ABCD of 0 is always zero. -/' +aliases: +- FTH0665 +- abcd_zero_3 +- abcd-zero-3 +- TauLib.BookIII.Physics.FluidData::abcd_zero_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001483 +- predicate: part_of + object: cid005236 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005236 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.FluidData + declaration: abcd_zero_3 + module_path: TauLib/BookIII/Physics/FluidData.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D38] Structural: ABCD of 0 is always zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005931.md b/_corpus_v3_items/cid005931.md new file mode 100644 index 000000000..48986b5a8 --- /dev/null +++ b/_corpus_v3_items/cid005931.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005931/ +title: defect_zero_1 (theorem) +cid: cid005931 +primary_alias: FTH0666 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D39] Structural: defect at depth 1 is zero. -/' +aliases: +- FTH0666 +- defect_zero_1 +- defect-zero-1 +- TauLib.BookIII.Physics.FluidData::defect_zero_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001484 +- predicate: part_of + object: cid005236 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005236 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.FluidData + declaration: defect_zero_1 + module_path: TauLib/BookIII/Physics/FluidData.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D39] Structural: defect at depth 1 is zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005932.md b/_corpus_v3_items/cid005932.md new file mode 100644 index 000000000..7a3874dad --- /dev/null +++ b/_corpus_v3_items/cid005932.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005932/ +title: gap_pos_3 (theorem) +cid: cid005932 +primary_alias: FTH0667 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T26] Structural: gap at depth 3 is positive. -/' +aliases: +- FTH0667 +- gap_pos_3 +- gap-pos-3 +- TauLib.BookIII.Physics.GapTheorem::gap_pos_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001618 +- predicate: part_of + object: cid005237 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005237 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + declaration: gap_pos_3 + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T26] Structural: gap at depth 3 is positive. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005933.md b/_corpus_v3_items/cid005933.md new file mode 100644 index 000000000..88b703acd --- /dev/null +++ b/_corpus_v3_items/cid005933.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005933/ +title: gap_grows_3_4 (theorem) +cid: cid005933 +primary_alias: FTH0668 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T26] Structural: gap grows from depth 3 to depth 4. -/' +aliases: +- FTH0668 +- gap_grows_3_4 +- gap-grows-3-4 +- TauLib.BookIII.Physics.GapTheorem::gap_grows_3_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001618 +- predicate: part_of + object: cid005237 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005237 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + declaration: gap_grows_3_4 + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T26] Structural: gap grows from depth 3 to depth 4. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005934.md b/_corpus_v3_items/cid005934.md new file mode 100644 index 000000000..e1542da55 --- /dev/null +++ b/_corpus_v3_items/cid005934.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005934/ +title: gap_constant_is_gap (theorem) +cid: cid005934 +primary_alias: FTH0669 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D45] Structural: gap constant equals tau_gap_at_level. -/' +aliases: +- FTH0669 +- gap_constant_is_gap +- gap-constant-is-gap +- TauLib.BookIII.Physics.GapTheorem::gap_constant_is_gap +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001490 +- predicate: part_of + object: cid005237 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005237 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + declaration: gap_constant_is_gap + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D45] Structural: gap constant equals tau_gap_at_level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005935.md b/_corpus_v3_items/cid005935.md new file mode 100644 index 000000000..b91d84a7a --- /dev/null +++ b/_corpus_v3_items/cid005935.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005935/ +title: ym_gap_is_tau_gap_3 (theorem) +cid: cid005935 +primary_alias: FTH0670 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T27] Structural: YM gap at depth 3 coincides with τ-gap. -/' +aliases: +- FTH0670 +- ym_gap_is_tau_gap_3 +- ym-gap-is-tau-gap-3 +- TauLib.BookIII.Physics.GapTheorem::ym_gap_is_tau_gap_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001619 +- predicate: part_of + object: cid005237 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005237 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + declaration: ym_gap_is_tau_gap_3 + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T27] Structural: YM gap at depth 3 coincides with τ-gap. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005936.md b/_corpus_v3_items/cid005936.md new file mode 100644 index 000000000..f370a2a82 --- /dev/null +++ b/_corpus_v3_items/cid005936.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005936/ +title: ym_coupling_3 (theorem) +cid: cid005936 +primary_alias: FTH0671 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D46] Structural: YM coupling at depth 3. -/' +aliases: +- FTH0671 +- ym_coupling_3 +- ym-coupling-3 +- TauLib.BookIII.Physics.GapTheorem::ym_coupling_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001491 +- predicate: part_of + object: cid005237 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005237 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.GapTheorem + declaration: ym_coupling_3 + module_path: TauLib/BookIII/Physics/GapTheorem.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D46] Structural: YM coupling at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005937.md b/_corpus_v3_items/cid005937.md new file mode 100644 index 000000000..fda3f61de --- /dev/null +++ b/_corpus_v3_items/cid005937.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005937/ +title: polarity_swap_involutive (theorem) +cid: cid005937 +primary_alias: FTH0672 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D41] Polarity swap is involutive: σ² = id. -/' +aliases: +- FTH0672 +- polarity_swap_involutive +- polarity-swap-involutive +- TauLib.BookIII.Physics.HartogsFlow::polarity_swap_involutive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001486 +- predicate: part_of + object: cid005238 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005238 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.HartogsFlow + declaration: polarity_swap_involutive + module_path: TauLib/BookIII/Physics/HartogsFlow.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D41] Polarity swap is involutive: σ² = id. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005938.md b/_corpus_v3_items/cid005938.md new file mode 100644 index 000000000..3ea15aecd --- /dev/null +++ b/_corpus_v3_items/cid005938.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005938/ +title: flow_depth_0 (theorem) +cid: cid005938 +primary_alias: FTH0673 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D40] Structural: flow at depth 0 produces trivial BNF. -/' +aliases: +- FTH0673 +- flow_depth_0 +- flow-depth-0 +- TauLib.BookIII.Physics.HartogsFlow::flow_depth_0 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001485 +- predicate: part_of + object: cid005238 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005238 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.HartogsFlow + declaration: flow_depth_0 + module_path: TauLib/BookIII/Physics/HartogsFlow.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D40] Structural: flow at depth 0 produces trivial BNF. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005939.md b/_corpus_v3_items/cid005939.md new file mode 100644 index 000000000..93e3b20ca --- /dev/null +++ b/_corpus_v3_items/cid005939.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005939/ +title: swap_zero (theorem) +cid: cid005939 +primary_alias: FTH0674 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D41] Structural: polarity swap of zero is zero. -/' +aliases: +- FTH0674 +- swap_zero +- swap-zero +- TauLib.BookIII.Physics.HartogsFlow::swap_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001486 +- predicate: part_of + object: cid005238 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005238 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.HartogsFlow + declaration: swap_zero + module_path: TauLib/BookIII/Physics/HartogsFlow.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D41] Structural: polarity swap of zero is zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005940.md b/_corpus_v3_items/cid005940.md new file mode 100644 index 000000000..557595a4b --- /dev/null +++ b/_corpus_v3_items/cid005940.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005940/ +title: flow_stable_1 (theorem) +cid: cid005940 +primary_alias: FTH0675 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T24] Structural: flow stabilization at depth 1. -/' +aliases: +- FTH0675 +- flow_stable_1 +- flow-stable-1 +- TauLib.BookIII.Physics.HartogsFlow::flow_stable_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001616 +- predicate: part_of + object: cid005238 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005238 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.HartogsFlow + declaration: flow_stable_1 + module_path: TauLib/BookIII/Physics/HartogsFlow.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T24] Structural: flow stabilization at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005941.md b/_corpus_v3_items/cid005941.md new file mode 100644 index 000000000..9402c3ffe --- /dev/null +++ b/_corpus_v3_items/cid005941.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005941/ +title: zero_is_sigma_fixed (theorem) +cid: cid005941 +primary_alias: FTH0676 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D47] Structural: zero BNF is always σ-fixed. -/' +aliases: +- FTH0676 +- zero_is_sigma_fixed +- zero-is-sigma-fixed +- TauLib.BookIII.Physics.Hodge::zero_is_sigma_fixed +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001492 +- predicate: part_of + object: cid005239 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005239 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.Hodge + declaration: zero_is_sigma_fixed + module_path: TauLib/BookIII/Physics/Hodge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D47] Structural: zero BNF is always σ-fixed. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005942.md b/_corpus_v3_items/cid005942.md new file mode 100644 index 000000000..564f764ac --- /dev/null +++ b/_corpus_v3_items/cid005942.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005942/ +title: sigma_fixed_swap_id (theorem) +cid: cid005942 +primary_alias: FTH0677 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D47] Structural: polarity swap of σ-fixed is identity. -/' +aliases: +- FTH0677 +- sigma_fixed_swap_id +- sigma-fixed-swap-id +- TauLib.BookIII.Physics.Hodge::sigma_fixed_swap_id +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001492 +- predicate: part_of + object: cid005239 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005239 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.Hodge + declaration: sigma_fixed_swap_id + module_path: TauLib/BookIII/Physics/Hodge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D47] Structural: polarity swap of σ-fixed is identity. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005943.md b/_corpus_v3_items/cid005943.md new file mode 100644 index 000000000..47dfea346 --- /dev/null +++ b/_corpus_v3_items/cid005943.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005943/ +title: addr_10_1 (theorem) +cid: cid005943 +primary_alias: FTH0678 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D48] Structural: addressability at depth 1. -/' +aliases: +- FTH0678 +- addr_10_1 +- addr-10-1 +- TauLib.BookIII.Physics.Hodge::addr_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001493 +- predicate: part_of + object: cid005239 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005239 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.Hodge + declaration: addr_10_1 + module_path: TauLib/BookIII/Physics/Hodge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D48] Structural: addressability at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005944.md b/_corpus_v3_items/cid005944.md new file mode 100644 index 000000000..29389a337 --- /dev/null +++ b/_corpus_v3_items/cid005944.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005944/ +title: nf_addr_10_1 (theorem) +cid: cid005944 +primary_alias: FTH0679 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T28] Structural: NF-addressability at depth 1. -/' +aliases: +- FTH0679 +- nf_addr_10_1 +- nf-addr-10-1 +- TauLib.BookIII.Physics.Hodge::nf_addr_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001620 +- predicate: part_of + object: cid005239 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005239 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.Hodge + declaration: nf_addr_10_1 + module_path: TauLib/BookIII/Physics/Hodge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T28] Structural: NF-addressability at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005945.md b/_corpus_v3_items/cid005945.md new file mode 100644 index 000000000..644de34c3 --- /dev/null +++ b/_corpus_v3_items/cid005945.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005945/ +title: ns_at_e1 (theorem) +cid: cid005945 +primary_alias: FTH0680 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T29] Structural: NS is at E₁. -/' +aliases: +- FTH0680 +- ns_at_e1 +- ns-at-e1 +- TauLib.BookIII.Physics.PhysicsAssembly::ns_at_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001621 +- predicate: part_of + object: cid005240 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005240 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + declaration: ns_at_e1 + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T29] Structural: NS is at E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005946.md b/_corpus_v3_items/cid005946.md new file mode 100644 index 000000000..83151ba31 --- /dev/null +++ b/_corpus_v3_items/cid005946.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005946/ +title: ym_at_e1 (theorem) +cid: cid005946 +primary_alias: FTH0681 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T29] Structural: YM is at E₁. -/' +aliases: +- FTH0681 +- ym_at_e1 +- ym-at-e1 +- TauLib.BookIII.Physics.PhysicsAssembly::ym_at_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001621 +- predicate: part_of + object: cid005240 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005240 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + declaration: ym_at_e1 + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T29] Structural: YM is at E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005947.md b/_corpus_v3_items/cid005947.md new file mode 100644 index 000000000..27701ca5f --- /dev/null +++ b/_corpus_v3_items/cid005947.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005947/ +title: hodge_at_e1 (theorem) +cid: cid005947 +primary_alias: FTH0682 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T29] Structural: Hodge is at E₁. -/' +aliases: +- FTH0682 +- hodge_at_e1 +- hodge-at-e1 +- TauLib.BookIII.Physics.PhysicsAssembly::hodge_at_e1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001621 +- predicate: part_of + object: cid005240 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005240 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + declaration: hodge_at_e1 + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T29] Structural: Hodge is at E₁. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005948.md b/_corpus_v3_items/cid005948.md new file mode 100644 index 000000000..e7f227b94 --- /dev/null +++ b/_corpus_v3_items/cid005948.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005948/ +title: ns_part5 (theorem) +cid: cid005948 +primary_alias: FTH0683 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T29] Structural: all three problems in Part V. -/' +aliases: +- FTH0683 +- ns_part5 +- ns-part5 +- TauLib.BookIII.Physics.PhysicsAssembly::ns_part5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001621 +- predicate: part_of + object: cid005240 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005240 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + declaration: ns_part5 + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T29] Structural: all three problems in Part V. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005949.md b/_corpus_v3_items/cid005949.md new file mode 100644 index 000000000..d8654ed6b --- /dev/null +++ b/_corpus_v3_items/cid005949.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005949/ +title: three_e1_problems (theorem) +cid: cid005949 +primary_alias: FTH0684 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T29] Structural: the three E₁ problems exhaust the E₁ level. -/' +aliases: +- FTH0684 +- three_e1_problems +- three-e1-problems +- TauLib.BookIII.Physics.PhysicsAssembly::three_e1_problems +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001621 +- predicate: part_of + object: cid005240 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005240 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PhysicsAssembly + declaration: three_e1_problems + module_path: TauLib/BookIII/Physics/PhysicsAssembly.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T29] Structural: the three E₁ problems exhaust the E₁ level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005950.md b/_corpus_v3_items/cid005950.md new file mode 100644 index 000000000..f13bbbb90 --- /dev/null +++ b/_corpus_v3_items/cid005950.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005950/ +title: pos_reg_10_1 (theorem) +cid: cid005950 +primary_alias: FTH0685 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T25] Structural: positive regularity at depth 1. -/' +aliases: +- FTH0685 +- pos_reg_10_1 +- pos-reg-10-1 +- TauLib.BookIII.Physics.PositiveRegularity::pos_reg_10_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001617 +- predicate: part_of + object: cid005241 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005241 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PositiveRegularity + declaration: pos_reg_10_1 + module_path: TauLib/BookIII/Physics/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T25] Structural: positive regularity at depth 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005951.md b/_corpus_v3_items/cid005951.md new file mode 100644 index 000000000..6a12a7aba --- /dev/null +++ b/_corpus_v3_items/cid005951.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005951/ +title: stab_germ_0_3 (theorem) +cid: cid005951 +primary_alias: FTH0686 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D42] Structural: stabilized germ of 0 is zero BNF. -/' +aliases: +- FTH0686 +- stab_germ_0_3 +- stab-germ-0-3 +- TauLib.BookIII.Physics.PositiveRegularity::stab_germ_0_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001487 +- predicate: part_of + object: cid005241 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005241 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PositiveRegularity + declaration: stab_germ_0_3 + module_path: TauLib/BookIII/Physics/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D42] Structural: stabilized germ of 0 is zero BNF. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005952.md b/_corpus_v3_items/cid005952.md new file mode 100644 index 000000000..9e317230a --- /dev/null +++ b/_corpus_v3_items/cid005952.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005952/ +title: defect_1_is_zero (theorem) +cid: cid005952 +primary_alias: FTH0687 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P14] Structural: defect at depth 1 is zero. -/' +aliases: +- FTH0687 +- defect_1_is_zero +- defect-1-is-zero +- TauLib.BookIII.Physics.PositiveRegularity::defect_1_is_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001558 +- predicate: part_of + object: cid005241 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005241 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PositiveRegularity + declaration: defect_1_is_zero + module_path: TauLib/BookIII/Physics/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P14] Structural: defect at depth 1 is zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005953.md b/_corpus_v3_items/cid005953.md new file mode 100644 index 000000000..e81638c17 --- /dev/null +++ b/_corpus_v3_items/cid005953.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005953/ +title: stable_at_3 (theorem) +cid: cid005953 +primary_alias: FTH0688 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P15] Structural: is_stable at every tested level. -/' +aliases: +- FTH0688 +- stable_at_3 +- stable-at-3 +- TauLib.BookIII.Physics.PositiveRegularity::stable_at_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001559 +- predicate: part_of + object: cid005241 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005241 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.PositiveRegularity + declaration: stable_at_3 + module_path: TauLib/BookIII/Physics/PositiveRegularity.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P15] Structural: is_stable at every tested level. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005954.md b/_corpus_v3_items/cid005954.md new file mode 100644 index 000000000..fd101b421 --- /dev/null +++ b/_corpus_v3_items/cid005954.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005954/ +title: strong_at_1 (theorem) +cid: cid005954 +primary_alias: FTH0689 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D43] Structural: strong sector at depth 1 (only prime 2 = X). -/' +aliases: +- FTH0689 +- strong_at_1 +- strong-at-1 +- TauLib.BookIII.Physics.StrongSector::strong_at_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001488 +- predicate: part_of + object: cid005242 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005242 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.StrongSector + declaration: strong_at_1 + module_path: TauLib/BookIII/Physics/StrongSector.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D43] Structural: strong sector at depth 1 (only prime 2 = X). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005955.md b/_corpus_v3_items/cid005955.md new file mode 100644 index 000000000..c7fe109a0 --- /dev/null +++ b/_corpus_v3_items/cid005955.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005955/ +title: gauge_3_complete (theorem) +cid: cid005955 +primary_alias: FTH0690 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D44] Structural: gauge data at depth 3 has correct products. -/' +aliases: +- FTH0690 +- gauge_3_complete +- gauge-3-complete +- TauLib.BookIII.Physics.StrongSector::gauge_3_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001489 +- predicate: part_of + object: cid005242 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005242 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.StrongSector + declaration: gauge_3_complete + module_path: TauLib/BookIII/Physics/StrongSector.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D44] Structural: gauge data at depth 3 has correct products. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005956.md b/_corpus_v3_items/cid005956.md new file mode 100644 index 000000000..a30416924 --- /dev/null +++ b/_corpus_v3_items/cid005956.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005956/ +title: gauge_3_count (theorem) +cid: cid005956 +primary_alias: FTH0691 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D44] Structural: gauge data at depth 3 has correct counts. -/' +aliases: +- FTH0691 +- gauge_3_count +- gauge-3-count +- TauLib.BookIII.Physics.StrongSector::gauge_3_count +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001489 +- predicate: part_of + object: cid005242 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005242 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.StrongSector + declaration: gauge_3_count + module_path: TauLib/BookIII/Physics/StrongSector.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D44] Structural: gauge data at depth 3 has correct counts. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005957.md b/_corpus_v3_items/cid005957.md new file mode 100644 index 000000000..2d55a740a --- /dev/null +++ b/_corpus_v3_items/cid005957.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005957/ +title: nf_zero_unique (theorem) +cid: cid005957 +primary_alias: FTH0692 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P16] Structural: BNF of 0 is unique at depth 3. -/' +aliases: +- FTH0692 +- nf_zero_unique +- nf-zero-unique +- TauLib.BookIII.Physics.StrongSector::nf_zero_unique +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001560 +- predicate: part_of + object: cid005242 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005242 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Physics.StrongSector + declaration: nf_zero_unique + module_path: TauLib/BookIII/Physics/StrongSector.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P16] Structural: BNF of 0 is unique at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005958.md b/_corpus_v3_items/cid005958.md new file mode 100644 index 000000000..1c11c6275 --- /dev/null +++ b/_corpus_v3_items/cid005958.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005958/ +title: hartogs_bulk_8_3 (theorem) +cid: cid005958 +primary_alias: FTH0693 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Hartogs bulk coherence [III.D01]' +aliases: +- FTH0693 +- hartogs_bulk_8_3 +- hartogs-bulk-8-3 +- TauLib.BookIII.Prologue.HartogsBulk::hartogs_bulk_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001432 +- predicate: part_of + object: cid005243 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005243 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + declaration: hartogs_bulk_8_3 + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Hartogs bulk coherence [III.D01]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005959.md b/_corpus_v3_items/cid005959.md new file mode 100644 index 000000000..3f779194f --- /dev/null +++ b/_corpus_v3_items/cid005959.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005959/ +title: e1_gluing_8_3_3 (theorem) +cid: cid005959 +primary_alias: FTH0694 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- E₁ gluing [III.D03] +aliases: +- FTH0694 +- e1_gluing_8_3_3 +- e1-gluing-8-3-3 +- TauLib.BookIII.Prologue.HartogsBulk::e1_gluing_8_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001434 +- predicate: part_of + object: cid005243 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005243 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + declaration: e1_gluing_8_3_3 + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- E₁ gluing [III.D03] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005960.md b/_corpus_v3_items/cid005960.md new file mode 100644 index 000000000..c20d7bc97 --- /dev/null +++ b/_corpus_v3_items/cid005960.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005960/ +title: book3_entry_8_3_3 (theorem) +cid: cid005960 +primary_alias: FTH0695 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Book III entry [III.D03] +aliases: +- FTH0695 +- book3_entry_8_3_3 +- book3-entry-8-3-3 +- TauLib.BookIII.Prologue.HartogsBulk::book3_entry_8_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001434 +- predicate: part_of + object: cid005243 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005243 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + declaration: book3_entry_8_3_3 + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Book III entry [III.D03] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005961.md b/_corpus_v3_items/cid005961.md new file mode 100644 index 000000000..577728377 --- /dev/null +++ b/_corpus_v3_items/cid005961.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005961/ +title: bulk_tower_coherent (theorem) +cid: cid005961 +primary_alias: FTH0696 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D01] Structural proof: Hartogs bulk projection is tower-coherent.\n reduce(reduce(b + c, stage), k)\ + \ = reduce(b + c, k) for k ≤ stage.\n This is reduction_compat from Book I. -/" +aliases: +- FTH0696 +- bulk_tower_coherent +- bulk-tower-coherent +- TauLib.BookIII.Prologue.HartogsBulk::bulk_tower_coherent +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001432 +- predicate: part_of + object: cid005243 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005243 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + declaration: bulk_tower_coherent + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D01] Structural proof: Hartogs bulk projection is tower-coherent.\n reduce(reduce(b + c, stage),\ + \ k) = reduce(b + c, k) for k ≤ stage.\n This is reduction_compat from Book I. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005962.md b/_corpus_v3_items/cid005962.md new file mode 100644 index 000000000..dc6dad8e6 --- /dev/null +++ b/_corpus_v3_items/cid005962.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005962/ +title: bulk_reduce_stable (theorem) +cid: cid005962 +primary_alias: FTH0697 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D01] Bulk projection is reduce-stable:\n reduce(bulk_coordinate(hb), hb.stage) = bulk_coordinate(hb).\ + \ -/" +aliases: +- FTH0697 +- bulk_reduce_stable +- bulk-reduce-stable +- TauLib.BookIII.Prologue.HartogsBulk::bulk_reduce_stable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001432 +- predicate: part_of + object: cid005243 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005243 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Prologue.HartogsBulk + declaration: bulk_reduce_stable + module_path: TauLib/BookIII/Prologue/HartogsBulk.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D01] Bulk projection is reduce-stable:\n reduce(bulk_coordinate(hb), hb.stage) = bulk_coordinate(hb).\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005963.md b/_corpus_v3_items/cid005963.md new file mode 100644 index 000000000..29ca3ea7c --- /dev/null +++ b/_corpus_v3_items/cid005963.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005963/ +title: boundary_char_3_3 (theorem) +cid: cid005963 +primary_alias: FTH0698 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Boundary character group [III.D11]' +aliases: +- FTH0698 +- boundary_char_3_3 +- boundary-char-3-3 +- TauLib.BookIII.Sectors.BoundaryCharacters::boundary_char_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001452 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: boundary_char_3_3 + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Boundary character group [III.D11]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005964.md b/_corpus_v3_items/cid005964.md new file mode 100644 index 000000000..abdec0fa9 --- /dev/null +++ b/_corpus_v3_items/cid005964.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005964/ +title: bti_functor_5_3 (theorem) +cid: cid005964 +primary_alias: FTH0699 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- BTI tower coherence [III.D12] +aliases: +- FTH0699 +- bti_functor_5_3 +- bti-functor-5-3 +- TauLib.BookIII.Sectors.BoundaryCharacters::bti_functor_5_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001457 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: bti_functor_5_3 + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- BTI tower coherence [III.D12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005965.md b/_corpus_v3_items/cid005965.md new file mode 100644 index 000000000..8c17b46e0 --- /dev/null +++ b/_corpus_v3_items/cid005965.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005965/ +title: bti_additive_3_3 (theorem) +cid: cid005965 +primary_alias: FTH0700 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- BTI additivity [III.D12] +aliases: +- FTH0700 +- bti_additive_3_3 +- bti-additive-3-3 +- TauLib.BookIII.Sectors.BoundaryCharacters::bti_additive_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001457 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: bti_additive_3_3 + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- BTI additivity [III.D12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005966.md b/_corpus_v3_items/cid005966.md new file mode 100644 index 000000000..a9c105b10 --- /dev/null +++ b/_corpus_v3_items/cid005966.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005966/ +title: zero_char_eval (theorem) +cid: cid005966 +primary_alias: FTH0701 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D11] Structural: zero character evaluates to zero. -/' +aliases: +- FTH0701 +- zero_char_eval +- zero-char-eval +- TauLib.BookIII.Sectors.BoundaryCharacters::zero_char_eval +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001452 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: zero_char_eval + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D11] Structural: zero character evaluates to zero. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005967.md b/_corpus_v3_items/cid005967.md new file mode 100644 index 000000000..419224974 --- /dev/null +++ b/_corpus_v3_items/cid005967.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005967/ +title: neg_neg_char (theorem) +cid: cid005967 +primary_alias: FTH0702 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D11] Structural: character negation is an involution. -/' +aliases: +- FTH0702 +- neg_neg_char +- neg-neg-char +- TauLib.BookIII.Sectors.BoundaryCharacters::neg_neg_char +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001452 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: neg_neg_char + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D11] Structural: character negation is an involution. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005968.md b/_corpus_v3_items/cid005968.md new file mode 100644 index 000000000..69d631d59 --- /dev/null +++ b/_corpus_v3_items/cid005968.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005968/ +title: bti_zero (theorem) +cid: cid005968 +primary_alias: FTH0703 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D12] Structural: BTI of trivial character is zero.\n Φ(0)(x, k) = reduce(0, k) = 0. -/" +aliases: +- FTH0703 +- bti_zero +- bti-zero +- TauLib.BookIII.Sectors.BoundaryCharacters::bti_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001457 +- predicate: part_of + object: cid005244 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005244 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.BoundaryCharacters + declaration: bti_zero + module_path: TauLib/BookIII/Sectors/BoundaryCharacters.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D12] Structural: BTI of trivial character is zero.\n Φ(0)(x, k) = reduce(0, k) = 0. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005969.md b/_corpus_v3_items/cid005969.md new file mode 100644 index 000000000..0e2e71db4 --- /dev/null +++ b/_corpus_v3_items/cid005969.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005969/ +title: sector_decomp_5 (theorem) +cid: cid005969 +primary_alias: FTH0704 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Sector decomposition [III.D13]' +aliases: +- FTH0704 +- sector_decomp_5 +- sector-decomp-5 +- TauLib.BookIII.Sectors.Decomposition::sector_decomp_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: sector_decomp_5 + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Sector decomposition [III.D13]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005970.md b/_corpus_v3_items/cid005970.md new file mode 100644 index 000000000..cc33a52ea --- /dev/null +++ b/_corpus_v3_items/cid005970.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005970/ +title: omega_coupling_5 (theorem) +cid: cid005970 +primary_alias: FTH0705 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- ω-coupling [III.D14] +aliases: +- FTH0705 +- omega_coupling_5 +- omega-coupling-5 +- TauLib.BookIII.Sectors.Decomposition::omega_coupling_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001459 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: omega_coupling_5 + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- ω-coupling [III.D14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005971.md b/_corpus_v3_items/cid005971.md new file mode 100644 index 000000000..6dd39e00a --- /dev/null +++ b/_corpus_v3_items/cid005971.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005971/ +title: sector_preservation_5_3 (theorem) +cid: cid005971 +primary_alias: FTH0706 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Sector preservation [III.T05] +aliases: +- FTH0706 +- sector_preservation_5_3 +- sector-preservation-5-3 +- TauLib.BookIII.Sectors.Decomposition::sector_preservation_5_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001597 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: sector_preservation_5_3 + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Sector preservation [III.T05] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005972.md b/_corpus_v3_items/cid005972.md new file mode 100644 index 000000000..3826bded7 --- /dev/null +++ b/_corpus_v3_items/cid005972.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005972/ +title: zero_in_D (theorem) +cid: cid005972 +primary_alias: FTH0707 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D13] Trivial character is in D-sector. -/ +aliases: +- FTH0707 +- zero_in_D +- zero-in-d +- TauLib.BookIII.Sectors.Decomposition::zero_in_D +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: zero_in_D + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D13] Trivial character is in D-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005973.md b/_corpus_v3_items/cid005973.md new file mode 100644 index 000000000..9b118581e --- /dev/null +++ b/_corpus_v3_items/cid005973.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005973/ +title: pure_m_in_B (theorem) +cid: cid005973 +primary_alias: FTH0708 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D13] Pure m-character is in B-sector. -/ +aliases: +- FTH0708 +- pure_m_in_B +- pure-m-in-b +- TauLib.BookIII.Sectors.Decomposition::pure_m_in_B +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: pure_m_in_B + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D13] Pure m-character is in B-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005974.md b/_corpus_v3_items/cid005974.md new file mode 100644 index 000000000..de22f8444 --- /dev/null +++ b/_corpus_v3_items/cid005974.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005974/ +title: pure_n_in_C (theorem) +cid: cid005974 +primary_alias: FTH0709 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D13] Pure n-character is in C-sector. -/ +aliases: +- FTH0709 +- pure_n_in_C +- pure-n-in-c +- TauLib.BookIII.Sectors.Decomposition::pure_n_in_C +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: pure_n_in_C + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D13] Pure n-character is in C-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005975.md b/_corpus_v3_items/cid005975.md new file mode 100644 index 000000000..b8e8aac60 --- /dev/null +++ b/_corpus_v3_items/cid005975.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005975/ +title: equal_mag_in_A (theorem) +cid: cid005975 +primary_alias: FTH0710 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D13] Equal-magnitude character is in A-sector. -/ +aliases: +- FTH0710 +- equal_mag_in_A +- equal-mag-in-a +- TauLib.BookIII.Sectors.Decomposition::equal_mag_in_A +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: equal_mag_in_A + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D13] Equal-magnitude character is in A-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005976.md b/_corpus_v3_items/cid005976.md new file mode 100644 index 000000000..072a0cb78 --- /dev/null +++ b/_corpus_v3_items/cid005976.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005976/ +title: mixed_in_Omega (theorem) +cid: cid005976 +primary_alias: FTH0711 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D13] Mixed unequal character is in Omega-sector. -/ +aliases: +- FTH0711 +- mixed_in_Omega +- mixed-in-omega +- TauLib.BookIII.Sectors.Decomposition::mixed_in_Omega +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: mixed_in_Omega + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D13] Mixed unequal character is in Omega-sector. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005977.md b/_corpus_v3_items/cid005977.md new file mode 100644 index 000000000..5689911cf --- /dev/null +++ b/_corpus_v3_items/cid005977.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005977/ +title: sector_exhaustive (theorem) +cid: cid005977 +primary_alias: FTH0712 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D13] The five sectors partition the character space:\n every character belongs to exactly one sector.\ + \ -/" +aliases: +- FTH0712 +- sector_exhaustive +- sector-exhaustive +- TauLib.BookIII.Sectors.Decomposition::sector_exhaustive +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001458 +- predicate: part_of + object: cid005245 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005245 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.Decomposition + declaration: sector_exhaustive + module_path: TauLib/BookIII/Sectors/Decomposition.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D13] The five sectors partition the character space:\n every character belongs to exactly one sector.\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005978.md b/_corpus_v3_items/cid005978.md new file mode 100644 index 000000000..a40ee6239 --- /dev/null +++ b/_corpus_v3_items/cid005978.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005978/ +title: langlands_reflection_5_3 (theorem) +cid: cid005978 +primary_alias: FTH0713 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Langlands₁ reflection [III.D15]' +aliases: +- FTH0713 +- langlands_reflection_5_3 +- langlands-reflection-5-3 +- TauLib.BookIII.Sectors.LanglandsReflection::langlands_reflection_5_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001460 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: langlands_reflection_5_3 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Langlands₁ reflection [III.D15]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005979.md b/_corpus_v3_items/cid005979.md new file mode 100644 index 000000000..079be48e1 --- /dev/null +++ b/_corpus_v3_items/cid005979.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005979/ +title: template_invariance_8_3 (theorem) +cid: cid005979 +primary_alias: FTH0714 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Template invariance [III.T06] +aliases: +- FTH0714 +- template_invariance_8_3 +- template-invariance-8-3 +- TauLib.BookIII.Sectors.LanglandsReflection::template_invariance_8_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001598 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: template_invariance_8_3 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Template invariance [III.T06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005980.md b/_corpus_v3_items/cid005980.md new file mode 100644 index 000000000..becbf50e6 --- /dev/null +++ b/_corpus_v3_items/cid005980.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005980/ +title: universal_op_3_3 (theorem) +cid: cid005980 +primary_alias: FTH0715 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Universal operator [III.D16] +aliases: +- FTH0715 +- universal_op_3_3 +- universal-op-3-3 +- TauLib.BookIII.Sectors.LanglandsReflection::universal_op_3_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001461 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: universal_op_3_3 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Universal operator [III.D16] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005981.md b/_corpus_v3_items/cid005981.md new file mode 100644 index 000000000..af268ed84 --- /dev/null +++ b/_corpus_v3_items/cid005981.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005981/ +title: spectral_polarity_5 (theorem) +cid: cid005981 +primary_alias: FTH0716 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Spectral polarity [III.D17] +aliases: +- FTH0716 +- spectral_polarity_5 +- spectral-polarity-5 +- TauLib.BookIII.Sectors.LanglandsReflection::spectral_polarity_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001462 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: spectral_polarity_5 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Spectral polarity [III.D17] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005982.md b/_corpus_v3_items/cid005982.md new file mode 100644 index 000000000..95ebfe6fa --- /dev/null +++ b/_corpus_v3_items/cid005982.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005982/ +title: balanced_uniqueness_5 (theorem) +cid: cid005982 +primary_alias: FTH0717 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Balanced sector uniqueness [III.P04] +aliases: +- FTH0717 +- balanced_uniqueness_5 +- balanced-uniqueness-5 +- TauLib.BookIII.Sectors.LanglandsReflection::balanced_uniqueness_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001548 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: balanced_uniqueness_5 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Balanced sector uniqueness [III.P04] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005983.md b/_corpus_v3_items/cid005983.md new file mode 100644 index 000000000..e4633a770 --- /dev/null +++ b/_corpus_v3_items/cid005983.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005983/ +title: sector_count_preserved (theorem) +cid: cid005983 +primary_alias: FTH0718 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D15] Structural: the five-sector partition is preserved. -/' +aliases: +- FTH0718 +- sector_count_preserved +- sector-count-preserved +- TauLib.BookIII.Sectors.LanglandsReflection::sector_count_preserved +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001460 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: sector_count_preserved + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D15] Structural: the five-sector partition is preserved. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005984.md b/_corpus_v3_items/cid005984.md new file mode 100644 index 000000000..aa7c74924 --- /dev/null +++ b/_corpus_v3_items/cid005984.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005984/ +title: template_invariance_e3 (theorem) +cid: cid005984 +primary_alias: FTH0719 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T06] Structural: template invariance at E₃ is trivial\n since E₃.succ = E₃ (saturation). -/" +aliases: +- FTH0719 +- template_invariance_e3 +- template-invariance-e3 +- TauLib.BookIII.Sectors.LanglandsReflection::template_invariance_e3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001598 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: template_invariance_e3 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T06] Structural: template invariance at E₃ is trivial\n since E₃.succ = E₃ (saturation). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005985.md b/_corpus_v3_items/cid005985.md new file mode 100644 index 000000000..ae061471d --- /dev/null +++ b/_corpus_v3_items/cid005985.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005985/ +title: d_polarity_zero (theorem) +cid: cid005985 +primary_alias: FTH0720 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D17] Structural: D-sector polarity is (0, 0)\n because only the trivial character (0,0) is in D. -/" +aliases: +- FTH0720 +- d_polarity_zero +- d-polarity-zero +- TauLib.BookIII.Sectors.LanglandsReflection::d_polarity_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001462 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: d_polarity_zero + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D17] Structural: D-sector polarity is (0, 0)\n because only the trivial character (0,0) is in D.\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005986.md b/_corpus_v3_items/cid005986.md new file mode 100644 index 000000000..c33f562f9 --- /dev/null +++ b/_corpus_v3_items/cid005986.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005986/ +title: a_balanced_3 (theorem) +cid: cid005986 +primary_alias: FTH0721 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P04] Structural: A-sector polarity at bound=3 is balanced. -/' +aliases: +- FTH0721 +- a_balanced_3 +- a-balanced-3 +- TauLib.BookIII.Sectors.LanglandsReflection::a_balanced_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001548 +- predicate: part_of + object: cid005246 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005246 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.LanglandsReflection + declaration: a_balanced_3 + module_path: TauLib/BookIII/Sectors/LanglandsReflection.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P04] Structural: A-sector polarity at bound=3 is balanced. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005987.md b/_corpus_v3_items/cid005987.md new file mode 100644 index 000000000..215990c7c --- /dev/null +++ b/_corpus_v3_items/cid005987.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005987/ +title: parity_bridge_5_3 (theorem) +cid: cid005987 +primary_alias: FTH0722 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Parity Bridge [III.T07]' +aliases: +- FTH0722 +- parity_bridge_5_3 +- parity-bridge-5-3 +- TauLib.BookIII.Sectors.ParityBridge::parity_bridge_5_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001599 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: parity_bridge_5_3 + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Parity Bridge [III.T07]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005988.md b/_corpus_v3_items/cid005988.md new file mode 100644 index 000000000..b96ac03f4 --- /dev/null +++ b/_corpus_v3_items/cid005988.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005988/ +title: coupling_ledger_3 (theorem) +cid: cid005988 +primary_alias: FTH0723 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Coupling ledger [III.D18] +aliases: +- FTH0723 +- coupling_ledger_3 +- coupling-ledger-3 +- TauLib.BookIII.Sectors.ParityBridge::coupling_ledger_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001463 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: coupling_ledger_3 + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Coupling ledger [III.D18] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005989.md b/_corpus_v3_items/cid005989.md new file mode 100644 index 000000000..972340f73 --- /dev/null +++ b/_corpus_v3_items/cid005989.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005989/ +title: no_knobs_5_3 (theorem) +cid: cid005989 +primary_alias: FTH0724 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- No Knobs [III.T08] +aliases: +- FTH0724 +- no_knobs_5_3 +- no-knobs-5-3 +- TauLib.BookIII.Sectors.ParityBridge::no_knobs_5_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001600 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: no_knobs_5_3 + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- No Knobs [III.T08] +noindex: false +--- + diff --git a/_corpus_v3_items/cid005990.md b/_corpus_v3_items/cid005990.md new file mode 100644 index 000000000..c0613652d --- /dev/null +++ b/_corpus_v3_items/cid005990.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005990/ +title: a_is_balanced (theorem) +cid: cid005990 +primary_alias: FTH0725 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T07] Structural: balanced polarity is unique to A-sector. -/' +aliases: +- FTH0725 +- a_is_balanced +- a-is-balanced +- TauLib.BookIII.Sectors.ParityBridge::a_is_balanced +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001599 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: a_is_balanced + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T07] Structural: balanced polarity is unique to A-sector. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005991.md b/_corpus_v3_items/cid005991.md new file mode 100644 index 000000000..6490ae60c --- /dev/null +++ b/_corpus_v3_items/cid005991.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005991/ +title: b_not_a (theorem) +cid: cid005991 +primary_alias: FTH0726 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T07] Structural: B and C sectors are NOT balanced. -/' +aliases: +- FTH0726 +- b_not_a +- b-not-a +- TauLib.BookIII.Sectors.ParityBridge::b_not_a +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001599 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: b_not_a + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T07] Structural: B and C sectors are NOT balanced. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005992.md b/_corpus_v3_items/cid005992.md new file mode 100644 index 000000000..b9c12fe65 --- /dev/null +++ b/_corpus_v3_items/cid005992.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005992/ +title: d_self_coupling_1 (theorem) +cid: cid005992 +primary_alias: FTH0727 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T08] Structural: D-sector self-coupling is always 1\n (only the trivial character maps to D). -/" +aliases: +- FTH0727 +- d_self_coupling_1 +- d-self-coupling-1 +- TauLib.BookIII.Sectors.ParityBridge::d_self_coupling_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001600 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: d_self_coupling_1 + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T08] Structural: D-sector self-coupling is always 1\n (only the trivial character maps to D). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005993.md b/_corpus_v3_items/cid005993.md new file mode 100644 index 000000000..750f6f707 --- /dev/null +++ b/_corpus_v3_items/cid005993.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005993/ +title: coupling_count (theorem) +cid: cid005993 +primary_alias: FTH0728 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D18] Structural: the coupling ledger has exactly 10 entries\n for 4 sectors (upper triangle of 4×4 matrix).\ + \ -/" +aliases: +- FTH0728 +- coupling_count +- coupling-count +- TauLib.BookIII.Sectors.ParityBridge::coupling_count +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001463 +- predicate: part_of + object: cid005247 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005247 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Sectors.ParityBridge + declaration: coupling_count + module_path: TauLib/BookIII/Sectors/ParityBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D18] Structural: the coupling ledger has exactly 10 entries\n for 4 sectors (upper triangle of 4×4\ + \ matrix). -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid005994.md b/_corpus_v3_items/cid005994.md new file mode 100644 index 000000000..2301f338e --- /dev/null +++ b/_corpus_v3_items/cid005994.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005994/ +title: goldbach_30 (theorem) +cid: cid005994 +primary_alias: FTH0729 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D95] Goldbach holds for all even numbers up to 30. -/ +aliases: +- FTH0729 +- goldbach_30 +- goldbach-30 +- TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001540 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: goldbach_30 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D95] Goldbach holds for all even numbers up to 30. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005995.md b/_corpus_v3_items/cid005995.md new file mode 100644 index 000000000..07c21958c --- /dev/null +++ b/_corpus_v3_items/cid005995.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005995/ +title: goldbach_100 (theorem) +cid: cid005995 +primary_alias: FTH0730 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D95] Goldbach holds for all even numbers up to 100. -/ +aliases: +- FTH0730 +- goldbach_100 +- goldbach-100 +- TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001540 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: goldbach_100 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D95] Goldbach holds for all even numbers up to 100. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005996.md b/_corpus_v3_items/cid005996.md new file mode 100644 index 000000000..689870142 --- /dev/null +++ b/_corpus_v3_items/cid005996.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005996/ +title: twin_primes_30 (theorem) +cid: cid005996 +primary_alias: FTH0731 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D96] At least 5 twin prime pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31). -/' +aliases: +- FTH0731 +- twin_primes_30 +- twin-primes-30 +- TauLib.BookIII.Spectral.AdditiveConjectures::twin_primes_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001541 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: twin_primes_30 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D96] At least 5 twin prime pairs below 30: (3,5),(5,7),(11,13),(17,19),(29,31). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid005997.md b/_corpus_v3_items/cid005997.md new file mode 100644 index 000000000..19d3b848d --- /dev/null +++ b/_corpus_v3_items/cid005997.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005997/ +title: goldbach_primorial_3 (theorem) +cid: cid005997 +primary_alias: FTH0732 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T64] Goldbach at primorial levels up to depth 3. -/ +aliases: +- FTH0732 +- goldbach_primorial_3 +- goldbach-primorial-3 +- TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_primorial_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001656 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: goldbach_primorial_3 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T64] Goldbach at primorial levels up to depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005998.md b/_corpus_v3_items/cid005998.md new file mode 100644 index 000000000..59c2038c3 --- /dev/null +++ b/_corpus_v3_items/cid005998.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005998/ +title: twin_prime_density_3 (theorem) +cid: cid005998 +primary_alias: FTH0733 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D96] Twin prime density at depth 3. -/ +aliases: +- FTH0733 +- twin_prime_density_3 +- twin-prime-density-3 +- TauLib.BookIII.Spectral.AdditiveConjectures::twin_prime_density_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001541 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: twin_prime_density_3 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D96] Twin prime density at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid005999.md b/_corpus_v3_items/cid005999.md new file mode 100644 index 000000000..9435f4a42 --- /dev/null +++ b/_corpus_v3_items/cid005999.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid005999/ +title: additive_multiplicative_3 (theorem) +cid: cid005999 +primary_alias: FTH0734 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P40] Additive-multiplicative duality at depth 3. -/ +aliases: +- FTH0734 +- additive_multiplicative_3 +- additive-multiplicative-3 +- TauLib.BookIII.Spectral.AdditiveConjectures::additive_multiplicative_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001584 +- predicate: part_of + object: cid005248 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005248 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.AdditiveConjectures + declaration: additive_multiplicative_3 + module_path: TauLib/BookIII/Spectral/AdditiveConjectures.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P40] Additive-multiplicative duality at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006000.md b/_corpus_v3_items/cid006000.md new file mode 100644 index 000000000..ea598c68d --- /dev/null +++ b/_corpus_v3_items/cid006000.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006000/ +title: adele_ring_10_3 (theorem) +cid: cid006000 +primary_alias: FTH0735 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Adele ring [III.D22]' +aliases: +- FTH0735 +- adele_ring_10_3 +- adele-ring-10-3 +- TauLib.BookIII.Spectral.Adeles::adele_ring_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001467 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adele_ring_10_3 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Adele ring [III.D22]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006001.md b/_corpus_v3_items/cid006001.md new file mode 100644 index 000000000..7f1bc3771 --- /dev/null +++ b/_corpus_v3_items/cid006001.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006001/ +title: adelic_embedding_15_3 (theorem) +cid: cid006001 +primary_alias: FTH0736 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Adelic embedding [III.T12] +aliases: +- FTH0736 +- adelic_embedding_15_3 +- adelic-embedding-15-3 +- TauLib.BookIII.Spectral.Adeles::adelic_embedding_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001604 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adelic_embedding_15_3 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Adelic embedding [III.T12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006002.md b/_corpus_v3_items/cid006002.md new file mode 100644 index 000000000..dba9de455 --- /dev/null +++ b/_corpus_v3_items/cid006002.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006002/ +title: adelic_dense_20_4 (theorem) +cid: cid006002 +primary_alias: FTH0737 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Adelic dense image [III.T12] +aliases: +- FTH0737 +- adelic_dense_20_4 +- adelic-dense-20-4 +- TauLib.BookIII.Spectral.Adeles::adelic_dense_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001604 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adelic_dense_20_4 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Adelic dense image [III.T12] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006003.md b/_corpus_v3_items/cid006003.md new file mode 100644 index 000000000..49fce13e4 --- /dev/null +++ b/_corpus_v3_items/cid006003.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006003/ +title: euler_product_20_4 (theorem) +cid: cid006003 +primary_alias: FTH0738 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Euler product [III.P07] +aliases: +- FTH0738 +- euler_product_20_4 +- euler-product-20-4 +- TauLib.BookIII.Spectral.Adeles::euler_product_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001551 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: euler_product_20_4 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Euler product [III.P07] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006004.md b/_corpus_v3_items/cid006004.md new file mode 100644 index 000000000..938e9ffef --- /dev/null +++ b/_corpus_v3_items/cid006004.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006004/ +title: local_factor_ind_10_3 (theorem) +cid: cid006004 +primary_alias: FTH0739 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Local factor independence [III.P07] +aliases: +- FTH0739 +- local_factor_ind_10_3 +- local-factor-ind-10-3 +- TauLib.BookIII.Spectral.Adeles::local_factor_ind_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001551 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: local_factor_ind_10_3 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Local factor independence [III.P07] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006005.md b/_corpus_v3_items/cid006005.md new file mode 100644 index 000000000..3809a1678 --- /dev/null +++ b/_corpus_v3_items/cid006005.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006005/ +title: adele_zero_3 (theorem) +cid: cid006005 +primary_alias: FTH0740 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D22] Structural: adele of 0 has all-zero components. -/' +aliases: +- FTH0740 +- adele_zero_3 +- adele-zero-3 +- TauLib.BookIII.Spectral.Adeles::adele_zero_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001467 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adele_zero_3 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D22] Structural: adele of 0 has all-zero components. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006006.md b/_corpus_v3_items/cid006006.md new file mode 100644 index 000000000..78a1abb7f --- /dev/null +++ b/_corpus_v3_items/cid006006.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006006/ +title: adele_is_crt (theorem) +cid: cid006006 +primary_alias: FTH0741 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T12] Structural: adelic embedding is CRT decomposition. -/' +aliases: +- FTH0741 +- adele_is_crt +- adele-is-crt +- TauLib.BookIII.Spectral.Adeles::adele_is_crt +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001604 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adele_is_crt + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T12] Structural: adelic embedding is CRT decomposition. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006007.md b/_corpus_v3_items/cid006007.md new file mode 100644 index 000000000..f2f3bb1f9 --- /dev/null +++ b/_corpus_v3_items/cid006007.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006007/ +title: adele_injective_1_2 (theorem) +cid: cid006007 +primary_alias: FTH0742 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T12] Structural: distinct values have distinct adelic images. -/' +aliases: +- FTH0742 +- adele_injective_1_2 +- adele-injective-1-2 +- TauLib.BookIII.Spectral.Adeles::adele_injective_1_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001604 +- predicate: part_of + object: cid005249 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005249 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Adeles + declaration: adele_injective_1_2 + module_path: TauLib/BookIII/Spectral/Adeles.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T12] Structural: distinct values have distinct adelic images. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006008.md b/_corpus_v3_items/cid006008.md new file mode 100644 index 000000000..86b3674c7 --- /dev/null +++ b/_corpus_v3_items/cid006008.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006008/ +title: classifier_20 (theorem) +cid: cid006008 +primary_alias: FTH0743 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Classifier [III.D23]' +aliases: +- FTH0743 +- classifier_20 +- classifier-20 +- TauLib.BookIII.Spectral.BipolarClassifier::classifier_20 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001468 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: classifier_20 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Classifier [III.D23]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006009.md b/_corpus_v3_items/cid006009.md new file mode 100644 index 000000000..33c5f78d9 --- /dev/null +++ b/_corpus_v3_items/cid006009.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006009/ +title: label_conv_20 (theorem) +cid: cid006009 +primary_alias: FTH0744 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Label convergence [III.T13] +aliases: +- FTH0744 +- label_conv_20 +- label-conv-20 +- TauLib.BookIII.Spectral.BipolarClassifier::label_conv_20 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001605 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: label_conv_20 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Label convergence [III.T13] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006010.md b/_corpus_v3_items/cid006010.md new file mode 100644 index 000000000..d5c046dfa --- /dev/null +++ b/_corpus_v3_items/cid006010.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006010/ +title: bc_balance_5 (theorem) +cid: cid006010 +primary_alias: FTH0745 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- B-C balance [III.T13] +aliases: +- FTH0745 +- bc_balance_5 +- bc-balance-5 +- TauLib.BookIII.Spectral.BipolarClassifier::bc_balance_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001605 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: bc_balance_5 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- B-C balance [III.T13] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006011.md b/_corpus_v3_items/cid006011.md new file mode 100644 index 000000000..64afd7d05 --- /dev/null +++ b/_corpus_v3_items/cid006011.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006011/ +title: label_idem_10_4 (theorem) +cid: cid006011 +primary_alias: FTH0746 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Label-idempotent compatibility [III.P08] +aliases: +- FTH0746 +- label_idem_10_4 +- label-idem-10-4 +- TauLib.BookIII.Spectral.BipolarClassifier::label_idem_10_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001552 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: label_idem_10_4 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Label-idempotent compatibility [III.P08] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006012.md b/_corpus_v3_items/cid006012.md new file mode 100644 index 000000000..cf4cb7567 --- /dev/null +++ b/_corpus_v3_items/cid006012.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006012/ +title: split_label_10_4 (theorem) +cid: cid006012 +primary_alias: FTH0747 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Split-complex label [III.P08] +aliases: +- FTH0747 +- split_label_10_4 +- split-label-10-4 +- TauLib.BookIII.Spectral.BipolarClassifier::split_label_10_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001552 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: split_label_10_4 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Split-complex label [III.P08] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006013.md b/_corpus_v3_items/cid006013.md new file mode 100644 index 000000000..afb97dc63 --- /dev/null +++ b/_corpus_v3_items/cid006013.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006013/ +title: two_is_x (theorem) +cid: cid006013 +primary_alias: FTH0748 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D23] Structural: 2 is the unique X-type prime. -/' +aliases: +- FTH0748 +- two_is_x +- two-is-x +- TauLib.BookIII.Spectral.BipolarClassifier::two_is_x +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001468 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: two_is_x + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D23] Structural: 2 is the unique X-type prime. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006014.md b/_corpus_v3_items/cid006014.md new file mode 100644 index 000000000..262d332f7 --- /dev/null +++ b/_corpus_v3_items/cid006014.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006014/ +title: five_is_b (theorem) +cid: cid006014 +primary_alias: FTH0749 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D23] Structural: 5 is B-type (5 mod 4 = 1). -/' +aliases: +- FTH0749 +- five_is_b +- five-is-b +- TauLib.BookIII.Spectral.BipolarClassifier::five_is_b +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001468 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: five_is_b + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D23] Structural: 5 is B-type (5 mod 4 = 1). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006015.md b/_corpus_v3_items/cid006015.md new file mode 100644 index 000000000..b7880f5e2 --- /dev/null +++ b/_corpus_v3_items/cid006015.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006015/ +title: three_is_c (theorem) +cid: cid006015 +primary_alias: FTH0750 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D23] Structural: 3 is C-type (3 mod 4 = 3). -/' +aliases: +- FTH0750 +- three_is_c +- three-is-c +- TauLib.BookIII.Spectral.BipolarClassifier::three_is_c +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001468 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: three_is_c + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D23] Structural: 3 is C-type (3 mod 4 = 3). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006016.md b/_corpus_v3_items/cid006016.md new file mode 100644 index 000000000..0f17c709e --- /dev/null +++ b/_corpus_v3_items/cid006016.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006016/ +title: bc_exist_3 (theorem) +cid: cid006016 +primary_alias: FTH0751 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T13] Structural: both B and C labels exist for first 3 primes. -/' +aliases: +- FTH0751 +- bc_exist_3 +- bc-exist-3 +- TauLib.BookIII.Spectral.BipolarClassifier::bc_exist_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001605 +- predicate: part_of + object: cid005250 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005250 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.BipolarClassifier + declaration: bc_exist_3 + module_path: TauLib/BookIII/Spectral/BipolarClassifier.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T13] Structural: both B and C labels exist for first 3 primes. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006017.md b/_corpus_v3_items/cid006017.md new file mode 100644 index 000000000..2a2037bc1 --- /dev/null +++ b/_corpus_v3_items/cid006017.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006017/ +title: crt_spectral_20_4 (theorem) +cid: cid006017 +primary_alias: FTH0752 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- CRT spectral [III.T10]' +aliases: +- FTH0752 +- crt_spectral_20_4 +- crt-spectral-20-4 +- TauLib.BookIII.Spectral.CRT::crt_spectral_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001602 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: crt_spectral_20_4 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- CRT spectral [III.T10]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006018.md b/_corpus_v3_items/cid006018.md new file mode 100644 index 000000000..295dd84f7 --- /dev/null +++ b/_corpus_v3_items/cid006018.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006018/ +title: crt_add_10_3 (theorem) +cid: cid006018 +primary_alias: FTH0753 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- CRT ring homomorphism [III.T10] +aliases: +- FTH0753 +- crt_add_10_3 +- crt-add-10-3 +- TauLib.BookIII.Spectral.CRT::crt_add_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001602 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: crt_add_10_3 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- CRT ring homomorphism [III.T10] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006019.md b/_corpus_v3_items/cid006019.md new file mode 100644 index 000000000..8c65aa85f --- /dev/null +++ b/_corpus_v3_items/cid006019.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006019/ +title: reconstruction_20_4 (theorem) +cid: cid006019 +primary_alias: FTH0754 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Reconstruction functor [III.D20] +aliases: +- FTH0754 +- reconstruction_20_4 +- reconstruction-20-4 +- TauLib.BookIII.Spectral.CRT::reconstruction_20_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001465 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: reconstruction_20_4 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Reconstruction functor [III.D20] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006020.md b/_corpus_v3_items/cid006020.md new file mode 100644 index 000000000..34ffc5fa7 --- /dev/null +++ b/_corpus_v3_items/cid006020.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006020/ +title: prime_independence_5_4 (theorem) +cid: cid006020 +primary_alias: FTH0755 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Prime independence [III.P05] +aliases: +- FTH0755 +- prime_independence_5_4 +- prime-independence-5-4 +- TauLib.BookIII.Spectral.CRT::prime_independence_5_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001549 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: prime_independence_5_4 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Prime independence [III.P05] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006021.md b/_corpus_v3_items/cid006021.md new file mode 100644 index 000000000..80a4e5e22 --- /dev/null +++ b/_corpus_v3_items/cid006021.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006021/ +title: crt_roundtrip_42 (theorem) +cid: cid006021 +primary_alias: FTH0756 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T10] Structural: CRT round-trip at depth 3 for x = 42. -/' +aliases: +- FTH0756 +- crt_roundtrip_42 +- crt-roundtrip-42 +- TauLib.BookIII.Spectral.CRT::crt_roundtrip_42 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001602 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: crt_roundtrip_42 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T10] Structural: CRT round-trip at depth 3 for x = 42. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006022.md b/_corpus_v3_items/cid006022.md new file mode 100644 index 000000000..b9c8846d6 --- /dev/null +++ b/_corpus_v3_items/cid006022.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006022/ +title: crt_decompose_42 (theorem) +cid: cid006022 +primary_alias: FTH0757 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T10] Structural: CRT decomposes 42 mod 30 = 12 into\n (12 mod 2, 12 mod 3, 12 mod 5) = (0, 0, 2). -/" +aliases: +- FTH0757 +- crt_decompose_42 +- crt-decompose-42 +- TauLib.BookIII.Spectral.CRT::crt_decompose_42 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001602 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: crt_decompose_42 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T10] Structural: CRT decomposes 42 mod 30 = 12 into\n (12 mod 2, 12 mod 3, 12 mod 5) = (0, 0, 2).\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006023.md b/_corpus_v3_items/cid006023.md new file mode 100644 index 000000000..56724063e --- /dev/null +++ b/_corpus_v3_items/cid006023.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006023/ +title: crt_basis_0_3 (theorem) +cid: cid006023 +primary_alias: FTH0758 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.P05] Structural: CRT basis element 0 at depth 3\n has residue 1 mod 2, 0 mod 3, 0 mod 5. -/" +aliases: +- FTH0758 +- crt_basis_0_3 +- crt-basis-0-3 +- TauLib.BookIII.Spectral.CRT::crt_basis_0_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001549 +- predicate: part_of + object: cid005251 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005251 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.CRT + declaration: crt_basis_0_3 + module_path: TauLib/BookIII/Spectral/CRT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.P05] Structural: CRT basis element 0 at depth 3\n has residue 1 mod 2, 0 mod 3, 0 mod 5. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006024.md b/_corpus_v3_items/cid006024.md new file mode 100644 index 000000000..e9822ec83 --- /dev/null +++ b/_corpus_v3_items/cid006024.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006024/ +title: bridge_algebraic_identity (theorem) +cid: cid006024 +primary_alias: FTH0759 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.P32] The confinement bridge reduces to the E₆ near-identity.\n\n Algebraically: |E₆|·κ(C;3)²·(1−ι)²\ + \ = |E₆|·ι⁶.\n Since κ(C;3)² numerator = (ι³·D)² and κ(C;3)² denominator = (D³·(D−ι))²,\n we have κ(C;3)²·(1−ι)² =\ + \ ι⁶/D⁶ × D²/(D−ι)² × (D−ι)²/D² = ι⁶/D⁶.\n\n Cross-multiplied verification:\n kappa_CC.numer² · (D−ι)² · D⁶ = ι⁶ ·\ + \ kappa_CC.denom² · D² -/" +aliases: +- FTH0759 +- bridge_algebraic_identity +- bridge-algebraic-identity +- TauLib.BookIII.Spectral.ConfinementBridge::bridge_algebraic_identity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001576 +- predicate: part_of + object: cid005252 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005252 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ConfinementBridge + declaration: bridge_algebraic_identity + module_path: TauLib/BookIII/Spectral/ConfinementBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.P32] The confinement bridge reduces to the E₆ near-identity.\n\n Algebraically: |E₆|·κ(C;3)²·(1−ι)²\ + \ = |E₆|·ι⁶.\n Since κ(C;3)² numerator = (ι³·D)² and κ(C;3)² denominator = (D³·(D−ι))²,\n we have κ(C;3)²·(1−ι)² =\ + \ ι⁶/D⁶ × D²/(D" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006025.md b/_corpus_v3_items/cid006025.md new file mode 100644 index 000000000..3570e4334 --- /dev/null +++ b/_corpus_v3_items/cid006025.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006025/ +title: bridge_reduces_to_E6_near_identity (theorem) +cid: cid006025 +primary_alias: FTH0760 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.P32] Corollary: the bridge near-identity inherits its bounds\n directly from the E₆ near-identity (III.T51).\n\ + \n Since |E₆|·κ(C;3)²·(1−ι)² = |E₆|·ι⁶ by bridge_algebraic_identity,\n and |E₆|·ι⁶ ∈ (0.999990, 1.000010) by E6_iota6_near_one,\n\ + \ the confinement bridge holds at the same precision (±10 ppm). -/" +aliases: +- FTH0760 +- bridge_reduces_to_E6_near_identity +- bridge-reduces-to-e6-near-identity +- TauLib.BookIII.Spectral.ConfinementBridge::bridge_reduces_to_E6_near_identity +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001576 +- predicate: part_of + object: cid005252 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005252 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ConfinementBridge + declaration: bridge_reduces_to_E6_near_identity + module_path: TauLib/BookIII/Spectral/ConfinementBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.P32] Corollary: the bridge near-identity inherits its bounds\n directly from the E₆ near-identity\ + \ (III.T51).\n\n Since |E₆|·κ(C;3)²·(1−ι)² = |E₆|·ι⁶ by bridge_algebraic_identity,\n and |E₆|·ι⁶ ∈ (0.999990, 1.000010)\ + \ by E6_iota" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006026.md b/_corpus_v3_items/cid006026.md new file mode 100644 index 000000000..f6d264f1c --- /dev/null +++ b/_corpus_v3_items/cid006026.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006026/ +title: confinement_bridge_lower (theorem) +cid: cid006026 +primary_alias: FTH0761 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T54] Confinement Bridge: |E₆| · κ(C;3)² ≈ 1/(1−ι_τ)² within ±10 ppm.\n\n This is the DIRECT form of\ + \ the bridge, verified by cross-multiplication.\n By bridge_algebraic_identity, this is equivalent to E6_iota6_near_one.\ + \ -/" +aliases: +- FTH0761 +- confinement_bridge_lower +- confinement-bridge-lower +- TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge_lower +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001646 +- predicate: part_of + object: cid005252 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005252 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ConfinementBridge + declaration: confinement_bridge_lower + module_path: TauLib/BookIII/Spectral/ConfinementBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T54] Confinement Bridge: |E₆| · κ(C;3)² ≈ 1/(1−ι_τ)² within ±10 ppm.\n\n This is the DIRECT form\ + \ of the bridge, verified by cross-multiplication.\n By bridge_algebraic_identity, this is equivalent to E6_iota6_near_one.\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006027.md b/_corpus_v3_items/cid006027.md new file mode 100644 index 000000000..b99e73095 --- /dev/null +++ b/_corpus_v3_items/cid006027.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006027/ +title: confinement_bridge (theorem) +cid: cid006027 +primary_alias: FTH0762 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: 'Lean theorem declaration: confinement_bridge. From TauLib v2 module TauLib.BookIII.Spectral.ConfinementBridge.' +aliases: +- FTH0762 +- confinement_bridge +- confinement-bridge +- TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001646 +- predicate: part_of + object: cid005252 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005252 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ConfinementBridge + declaration: confinement_bridge + module_path: TauLib/BookIII/Spectral/ConfinementBridge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Lean theorem declaration: confinement_bridge. From TauLib v2 module TauLib.BookIII.Spectral.ConfinementBridge.' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006028.md b/_corpus_v3_items/cid006028.md new file mode 100644 index 000000000..52c55a646 --- /dev/null +++ b/_corpus_v3_items/cid006028.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006028/ +title: goldbach_500 (theorem) +cid: cid006028 +primary_alias: FTH0763 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T68] Goldbach verified to 500 via sieve. -/ +aliases: +- FTH0763 +- goldbach_500 +- goldbach-500 +- TauLib.BookIII.Spectral.GoldbachDeep::goldbach_500 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001660 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: goldbach_500 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T68] Goldbach verified to 500 via sieve. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006029.md b/_corpus_v3_items/cid006029.md new file mode 100644 index 000000000..ba2723747 --- /dev/null +++ b/_corpus_v3_items/cid006029.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006029/ +title: goldbach_primorial_m4 (theorem) +cid: cid006029 +primary_alias: FTH0764 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T69] Goldbach at primorial M₄=210: all even n ≤ 210 verified. -/' +aliases: +- FTH0764 +- goldbach_primorial_m4 +- goldbach-primorial-m4 +- TauLib.BookIII.Spectral.GoldbachDeep::goldbach_primorial_m4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001661 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: goldbach_primorial_m4 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T69] Goldbach at primorial M₄=210: all even n ≤ 210 verified. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006030.md b/_corpus_v3_items/cid006030.md new file mode 100644 index 000000000..5d22d522b --- /dev/null +++ b/_corpus_v3_items/cid006030.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006030/ +title: partition_growth_4 (theorem) +cid: cid006030 +primary_alias: FTH0765 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T70] Partition growth: r(M_k) increasing for k = 2..4. -/' +aliases: +- FTH0765 +- partition_growth_4 +- partition-growth-4 +- TauLib.BookIII.Spectral.GoldbachDeep::partition_growth_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001662 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: partition_growth_4 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T70] Partition growth: r(M_k) increasing for k = 2..4. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006031.md b/_corpus_v3_items/cid006031.md new file mode 100644 index 000000000..23c4d4ac1 --- /dev/null +++ b/_corpus_v3_items/cid006031.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006031/ +title: obstruction_bounded_5 (theorem) +cid: cid006031 +primary_alias: FTH0766 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T71] Obstruction bounded by 1 at each prime, for even n ≤ 100. -/ +aliases: +- FTH0766 +- obstruction_bounded_5 +- obstruction-bounded-5 +- TauLib.BookIII.Spectral.GoldbachDeep::obstruction_bounded_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001663 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: obstruction_bounded_5 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T71] Obstruction bounded by 1 at each prime, for even n ≤ 100. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006032.md b/_corpus_v3_items/cid006032.md new file mode 100644 index 000000000..39a5d606d --- /dev/null +++ b/_corpus_v3_items/cid006032.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006032/ +title: crt_goldbach_duality_3 (theorem) +cid: cid006032 +primary_alias: FTH0767 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P43] CRT-Goldbach duality at depth 3, even n ≤ 100. -/ +aliases: +- FTH0767 +- crt_goldbach_duality_3 +- crt-goldbach-duality-3 +- TauLib.BookIII.Spectral.GoldbachDeep::crt_goldbach_duality_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001587 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: crt_goldbach_duality_3 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P43] CRT-Goldbach duality at depth 3, even n ≤ 100. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006033.md b/_corpus_v3_items/cid006033.md new file mode 100644 index 000000000..6172e28b0 --- /dev/null +++ b/_corpus_v3_items/cid006033.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006033/ +title: partition_m2 (theorem) +cid: cid006033 +primary_alias: FTH0768 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D103] r(M_2) = r(6) = 1 (6 = 3+3). -/ +aliases: +- FTH0768 +- partition_m2 +- partition-m2 +- TauLib.BookIII.Spectral.GoldbachDeep::partition_m2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001445 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: partition_m2 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D103] r(M_2) = r(6) = 1 (6 = 3+3). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006034.md b/_corpus_v3_items/cid006034.md new file mode 100644 index 000000000..2785ef6ef --- /dev/null +++ b/_corpus_v3_items/cid006034.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006034/ +title: partition_m3 (theorem) +cid: cid006034 +primary_alias: FTH0769 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D103] r(M_3) = r(30) = 3 (30 = 7+23 = 11+19 = 13+17). -/ +aliases: +- FTH0769 +- partition_m3 +- partition-m3 +- TauLib.BookIII.Spectral.GoldbachDeep::partition_m3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001445 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: partition_m3 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D103] r(M_3) = r(30) = 3 (30 = 7+23 = 11+19 = 13+17). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006035.md b/_corpus_v3_items/cid006035.md new file mode 100644 index 000000000..ed22328ae --- /dev/null +++ b/_corpus_v3_items/cid006035.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006035/ +title: partition_m4_pos (theorem) +cid: cid006035 +primary_alias: FTH0770 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D103] r(M_4) = r(210) > 0. -/ +aliases: +- FTH0770 +- partition_m4_pos +- partition-m4-pos +- TauLib.BookIII.Spectral.GoldbachDeep::partition_m4_pos +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001445 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: partition_m4_pos + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D103] r(M_4) = r(210) > 0. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006036.md b/_corpus_v3_items/cid006036.md new file mode 100644 index 000000000..86f96f4b2 --- /dev/null +++ b/_corpus_v3_items/cid006036.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006036/ +title: obstruction_100_p2 (theorem) +cid: cid006036 +primary_alias: FTH0771 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D104] Obstruction at p=2 for n=100: 1 (100 is even). -/' +aliases: +- FTH0771 +- obstruction_100_p2 +- obstruction-100-p2 +- TauLib.BookIII.Spectral.GoldbachDeep::obstruction_100_p2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001446 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: obstruction_100_p2 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D104] Obstruction at p=2 for n=100: 1 (100 is even). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006037.md b/_corpus_v3_items/cid006037.md new file mode 100644 index 000000000..6e9b9db75 --- /dev/null +++ b/_corpus_v3_items/cid006037.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006037/ +title: obstruction_100_p3 (theorem) +cid: cid006037 +primary_alias: FTH0772 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D104] Obstruction at p=3 for n=100: 0 (100 mod 3 ≠ 0). -/' +aliases: +- FTH0772 +- obstruction_100_p3 +- obstruction-100-p3 +- TauLib.BookIII.Spectral.GoldbachDeep::obstruction_100_p3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001446 +- predicate: part_of + object: cid005253 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005253 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.GoldbachDeep + declaration: obstruction_100_p3 + module_path: TauLib/BookIII/Spectral/GoldbachDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D104] Obstruction at p=3 for n=100: 0 (100 mod 3 ≠ 0). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006038.md b/_corpus_v3_items/cid006038.md new file mode 100644 index 000000000..1c96e1e8a --- /dev/null +++ b/_corpus_v3_items/cid006038.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006038/ +title: hensel_20 (theorem) +cid: cid006038 +primary_alias: FTH0773 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Hensel lifting [III.T11]' +aliases: +- FTH0773 +- hensel_20 +- hensel-20 +- TauLib.BookIII.Spectral.HenselLifting::hensel_20 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: hensel_20 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Hensel lifting [III.T11]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006039.md b/_corpus_v3_items/cid006039.md new file mode 100644 index 000000000..14c62ff26 --- /dev/null +++ b/_corpus_v3_items/cid006039.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006039/ +title: hensel_unique_11 (theorem) +cid: cid006039 +primary_alias: FTH0774 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hensel uniqueness [III.T11] +aliases: +- FTH0774 +- hensel_unique_11 +- hensel-unique-11 +- TauLib.BookIII.Spectral.HenselLifting::hensel_unique_11 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: hensel_unique_11 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hensel uniqueness [III.T11] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006040.md b/_corpus_v3_items/cid006040.md new file mode 100644 index 000000000..f8b6f83e5 --- /dev/null +++ b/_corpus_v3_items/cid006040.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006040/ +title: hensel_tower_11 (theorem) +cid: cid006040 +primary_alias: FTH0775 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Hensel tower coherence [III.T11] +aliases: +- FTH0775 +- hensel_tower_11 +- hensel-tower-11 +- TauLib.BookIII.Spectral.HenselLifting::hensel_tower_11 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: hensel_tower_11 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Hensel tower coherence [III.T11] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006041.md b/_corpus_v3_items/cid006041.md new file mode 100644 index 000000000..5e7da405a --- /dev/null +++ b/_corpus_v3_items/cid006041.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006041/ +title: one_is_root_3 (theorem) +cid: cid006041 +primary_alias: FTH0776 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T11] Structural: 1 is a root of x²-1 mod any p. -/' +aliases: +- FTH0776 +- one_is_root_3 +- one-is-root-3 +- TauLib.BookIII.Spectral.HenselLifting::one_is_root_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: one_is_root_3 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T11] Structural: 1 is a root of x²-1 mod any p. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006042.md b/_corpus_v3_items/cid006042.md new file mode 100644 index 000000000..6cf72a7d1 --- /dev/null +++ b/_corpus_v3_items/cid006042.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006042/ +title: hensel_1_mod_9 (theorem) +cid: cid006042 +primary_alias: FTH0777 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T11] Structural: Hensel lift of 1 mod 3² = 1. -/' +aliases: +- FTH0777 +- hensel_1_mod_9 +- hensel-1-mod-9 +- TauLib.BookIII.Spectral.HenselLifting::hensel_1_mod_9 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: hensel_1_mod_9 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T11] Structural: Hensel lift of 1 mod 3² = 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006043.md b/_corpus_v3_items/cid006043.md new file mode 100644 index 000000000..b033b40ec --- /dev/null +++ b/_corpus_v3_items/cid006043.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006043/ +title: hensel_correct_3_2 (theorem) +cid: cid006043 +primary_alias: FTH0778 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T11] Structural: poly_eval is zero at the lifted root. -/' +aliases: +- FTH0778 +- hensel_correct_3_2 +- hensel-correct-3-2 +- TauLib.BookIII.Spectral.HenselLifting::hensel_correct_3_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001603 +- predicate: part_of + object: cid005254 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005254 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.HenselLifting + declaration: hensel_correct_3_2 + module_path: TauLib/BookIII/Spectral/HenselLifting.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T11] Structural: poly_eval is zero at the lifted root. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006044.md b/_corpus_v3_items/cid006044.md new file mode 100644 index 000000000..b58cc27af --- /dev/null +++ b/_corpus_v3_items/cid006044.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006044/ +title: local_field_15_4 (theorem) +cid: cid006044 +primary_alias: FTH0779 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Local field [III.D21]' +aliases: +- FTH0779 +- local_field_15_4 +- local-field-15-4 +- TauLib.BookIII.Spectral.LocalFields::local_field_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001466 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: local_field_15_4 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Local field [III.D21]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006045.md b/_corpus_v3_items/cid006045.md new file mode 100644 index 000000000..0fff930ed --- /dev/null +++ b/_corpus_v3_items/cid006045.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006045/ +title: local_ring_10_4 (theorem) +cid: cid006045 +primary_alias: FTH0780 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Local ring [III.D21] +aliases: +- FTH0780 +- local_ring_10_4 +- local-ring-10-4 +- TauLib.BookIII.Spectral.LocalFields::local_ring_10_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001466 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: local_ring_10_4 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Local ring [III.D21] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006046.md b/_corpus_v3_items/cid006046.md new file mode 100644 index 000000000..15a65d2c3 --- /dev/null +++ b/_corpus_v3_items/cid006046.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006046/ +title: completeness_20_5 (theorem) +cid: cid006046 +primary_alias: FTH0781 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Completeness [III.P06] +aliases: +- FTH0781 +- completeness_20_5 +- completeness-20-5 +- TauLib.BookIII.Spectral.LocalFields::completeness_20_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001550 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: completeness_20_5 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Completeness [III.P06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006047.md b/_corpus_v3_items/cid006047.md new file mode 100644 index 000000000..68046291f --- /dev/null +++ b/_corpus_v3_items/cid006047.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006047/ +title: limit_unique_10_4 (theorem) +cid: cid006047 +primary_alias: FTH0782 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Limit uniqueness [III.P06] +aliases: +- FTH0782 +- limit_unique_10_4 +- limit-unique-10-4 +- TauLib.BookIII.Spectral.LocalFields::limit_unique_10_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001550 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: limit_unique_10_4 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Limit uniqueness [III.P06] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006048.md b/_corpus_v3_items/cid006048.md new file mode 100644 index 000000000..7652319b2 --- /dev/null +++ b/_corpus_v3_items/cid006048.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006048/ +title: val_p_is_1 (theorem) +cid: cid006048 +primary_alias: FTH0783 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D21] Structural: p-adic valuation of p is 1. -/' +aliases: +- FTH0783 +- val_p_is_1 +- val-p-is-1 +- TauLib.BookIII.Spectral.LocalFields::val_p_is_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001466 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: val_p_is_1 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D21] Structural: p-adic valuation of p is 1. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006049.md b/_corpus_v3_items/cid006049.md new file mode 100644 index 000000000..1f4d0f4ef --- /dev/null +++ b/_corpus_v3_items/cid006049.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006049/ +title: val_p2 (theorem) +cid: cid006049 +primary_alias: FTH0784 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D21] Structural: p-adic valuation of p^k is k. -/' +aliases: +- FTH0784 +- val_p2 +- val-p2 +- TauLib.BookIII.Spectral.LocalFields::val_p2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001466 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: val_p2 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D21] Structural: p-adic valuation of p^k is k. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006050.md b/_corpus_v3_items/cid006050.md new file mode 100644 index 000000000..c84843b25 --- /dev/null +++ b/_corpus_v3_items/cid006050.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006050/ +title: val_zero (theorem) +cid: cid006050 +primary_alias: FTH0785 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D21] Structural: p-adic valuation of 0 is 0 (convention). -/' +aliases: +- FTH0785 +- val_zero +- val-zero +- TauLib.BookIII.Spectral.LocalFields::val_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001466 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: val_zero + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D21] Structural: p-adic valuation of 0 is 0 (convention). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006051.md b/_corpus_v3_items/cid006051.md new file mode 100644 index 000000000..b6056c7f3 --- /dev/null +++ b/_corpus_v3_items/cid006051.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006051/ +title: tower_42_3 (theorem) +cid: cid006051 +primary_alias: FTH0786 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P06] Structural: tower coherence at p=3, x=42. -/' +aliases: +- FTH0786 +- tower_42_3 +- tower-42-3 +- TauLib.BookIII.Spectral.LocalFields::tower_42_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001550 +- predicate: part_of + object: cid005255 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005255 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.LocalFields + declaration: tower_42_3 + module_path: TauLib/BookIII/Spectral/LocalFields.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P06] Structural: tower coherence at p=3, x=42. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006052.md b/_corpus_v3_items/cid006052.md new file mode 100644 index 000000000..5c9537170 --- /dev/null +++ b/_corpus_v3_items/cid006052.md @@ -0,0 +1,69 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006052/ +title: E4_iota4_near_one_lower (theorem) +cid: cid006052 +primary_alias: FTH0787 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T50] E₄·ι_τ⁴ near-identity: the product E₄(iι_τ)·ι_τ⁴ is close to 1.\n\n With 7-digit E₄ and 6-digit\ + \ ι_τ, the product E₄_numer · i4N / (E₄_denom · i4D)\n is within ~10 ppm of 1. We prove bounds (999990, 1000010) per\ + \ million.\n\n Cross-multiplied: 999990 · E4_denom · i4D < E4_numer · i4N · 1000000\n and E4_numer ·\ + \ i4N · 1000000 < 1000010 · E4_denom · i4D -/" +aliases: +- FTH0787 +- E4_iota4_near_one_lower +- e4-iota4-near-one-lower +- TauLib.BookIII.Spectral.ModularForms::E4_iota4_near_one_lower +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001642 +- predicate: part_of + object: cid005256 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005256 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ModularForms + declaration: E4_iota4_near_one_lower + module_path: TauLib/BookIII/Spectral/ModularForms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T50] E₄·ι_τ⁴ near-identity: the product E₄(iι_τ)·ι_τ⁴ is close to 1.\n\n With 7-digit E₄ and 6-digit\ + \ ι_τ, the product E₄_numer · i4N / (E₄_denom · i4D)\n is within ~10 ppm of 1. We prove bounds (999990, 1000010) per\ + \ million.\n\n " +noindex: false +--- + diff --git a/_corpus_v3_items/cid006053.md b/_corpus_v3_items/cid006053.md new file mode 100644 index 000000000..edbeee54a --- /dev/null +++ b/_corpus_v3_items/cid006053.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006053/ +title: E4_iota4_near_one (theorem) +cid: cid006053 +primary_alias: FTH0788 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T50] Combined: E₄·ι_τ⁴ ∈ (0.999990, 1.000010), i.e., 1 ± ~10 ppm.\n (The true value is 1 + 2.4 ppm;\ + \ the rational approximation widens to ±10 ppm.) -/" +aliases: +- FTH0788 +- E4_iota4_near_one +- e4-iota4-near-one +- TauLib.BookIII.Spectral.ModularForms::E4_iota4_near_one +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001642 +- predicate: part_of + object: cid005256 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005256 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ModularForms + declaration: E4_iota4_near_one + module_path: TauLib/BookIII/Spectral/ModularForms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T50] Combined: E₄·ι_τ⁴ ∈ (0.999990, 1.000010), i.e., 1 ± ~10 ppm.\n (The true value is 1 + 2.4 ppm;\ + \ the rational approximation widens to ±10 ppm.) -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006054.md b/_corpus_v3_items/cid006054.md new file mode 100644 index 000000000..ba2beb574 --- /dev/null +++ b/_corpus_v3_items/cid006054.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006054/ +title: E6_abs_in_range (theorem) +cid: cid006054 +primary_alias: FTH0789 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- |E₆(iι_τ)| is between 632 and 633. -/ +aliases: +- FTH0789 +- E6_abs_in_range +- e6-abs-in-range +- TauLib.BookIII.Spectral.ModularForms::E6_abs_in_range +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001643 +- predicate: part_of + object: cid005256 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005256 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ModularForms + declaration: E6_abs_in_range + module_path: TauLib/BookIII/Spectral/ModularForms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- |E₆(iι_τ)| is between 632 and 633. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006055.md b/_corpus_v3_items/cid006055.md new file mode 100644 index 000000000..c945f3fb9 --- /dev/null +++ b/_corpus_v3_items/cid006055.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006055/ +title: E6_iota6_near_one (theorem) +cid: cid006055 +primary_alias: FTH0790 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T51] Combined: |E₆|·ι_τ⁶ ∈ (0.999990, 1.000010), i.e., −1 ± ~10 ppm.\n (The true value is 1 − 5.1 ppm;\ + \ the rational approximation widens to ±10 ppm.) -/" +aliases: +- FTH0790 +- E6_iota6_near_one +- e6-iota6-near-one +- TauLib.BookIII.Spectral.ModularForms::E6_iota6_near_one +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001643 +- predicate: part_of + object: cid005256 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005256 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.ModularForms + declaration: E6_iota6_near_one + module_path: TauLib/BookIII/Spectral/ModularForms.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T51] Combined: |E₆|·ι_τ⁶ ∈ (0.999990, 1.000010), i.e., −1 ± ~10 ppm.\n (The true value is 1 − 5.1\ + \ ppm; the rational approximation widens to ±10 ppm.) -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006056.md b/_corpus_v3_items/cid006056.md new file mode 100644 index 000000000..c9568d61c --- /dev/null +++ b/_corpus_v3_items/cid006056.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006056/ +title: primorial_ladder_8 (theorem) +cid: cid006056 +primary_alias: FTH0791 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Primorial ladder [III.D19]' +aliases: +- FTH0791 +- primorial_ladder_8 +- primorial-ladder-8 +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_ladder_8 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_ladder_8 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Primorial ladder [III.D19]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006057.md b/_corpus_v3_items/cid006057.md new file mode 100644 index 000000000..ee9580856 --- /dev/null +++ b/_corpus_v3_items/cid006057.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006057/ +title: primorial_div_6 (theorem) +cid: cid006057 +primary_alias: FTH0792 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Primorial divisibility [III.D19] +aliases: +- FTH0792 +- primorial_div_6 +- primorial-div-6 +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_div_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_div_6 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Primorial divisibility [III.D19] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006058.md b/_corpus_v3_items/cid006058.md new file mode 100644 index 000000000..2b549cf5c --- /dev/null +++ b/_corpus_v3_items/cid006058.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006058/ +title: primorial_growth_6 (theorem) +cid: cid006058 +primary_alias: FTH0793 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Primorial growth [III.D19] +aliases: +- FTH0793 +- primorial_growth_6 +- primorial-growth-6 +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_growth_6 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_growth_6 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Primorial growth [III.D19] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006059.md b/_corpus_v3_items/cid006059.md new file mode 100644 index 000000000..49851f676 --- /dev/null +++ b/_corpus_v3_items/cid006059.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006059/ +title: primorial_cofinal_50 (theorem) +cid: cid006059 +primary_alias: FTH0794 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Primorial cofinality [III.T09] +aliases: +- FTH0794 +- primorial_cofinal_50 +- primorial-cofinal-50 +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_cofinal_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001601 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_cofinal_50 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Primorial cofinality [III.T09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006060.md b/_corpus_v3_items/cid006060.md new file mode 100644 index 000000000..02b5166a7 --- /dev/null +++ b/_corpus_v3_items/cid006060.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006060/ +title: prime_cofinal_30 (theorem) +cid: cid006060 +primary_alias: FTH0795 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Prime cofinality [III.T09] +aliases: +- FTH0795 +- prime_cofinal_30 +- prime-cofinal-30 +- TauLib.BookIII.Spectral.PrimorialLadder::prime_cofinal_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001601 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: prime_cofinal_30 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Prime cofinality [III.T09] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006061.md b/_corpus_v3_items/cid006061.md new file mode 100644 index 000000000..fe3ee7620 --- /dev/null +++ b/_corpus_v3_items/cid006061.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006061/ +title: primorial_zero (theorem) +cid: cid006061 +primary_alias: FTH0796 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D19] Structural: primorial 0 = 1 (empty product). -/' +aliases: +- FTH0796 +- primorial_zero +- primorial-zero +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_zero +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_zero + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D19] Structural: primorial 0 = 1 (empty product). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006062.md b/_corpus_v3_items/cid006062.md new file mode 100644 index 000000000..31958ae72 --- /dev/null +++ b/_corpus_v3_items/cid006062.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006062/ +title: primorial_one (theorem) +cid: cid006062 +primary_alias: FTH0797 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D19] Structural: primorial 1 = 2. -/' +aliases: +- FTH0797 +- primorial_one +- primorial-one +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_one +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_one + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D19] Structural: primorial 1 = 2. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006063.md b/_corpus_v3_items/cid006063.md new file mode 100644 index 000000000..bda8f4fed --- /dev/null +++ b/_corpus_v3_items/cid006063.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006063/ +title: primorial_three (theorem) +cid: cid006063 +primary_alias: FTH0798 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D19] Structural: primorial 3 = 30 = 2·3·5. -/' +aliases: +- FTH0798 +- primorial_three +- primorial-three +- TauLib.BookIII.Spectral.PrimorialLadder::primorial_three +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: primorial_three + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D19] Structural: primorial 3 = 30 = 2·3·5. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006064.md b/_corpus_v3_items/cid006064.md new file mode 100644 index 000000000..2948aa454 --- /dev/null +++ b/_corpus_v3_items/cid006064.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006064/ +title: cofinal_30 (theorem) +cid: cid006064 +primary_alias: FTH0799 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T09] Structural: cofinal level of 30 is 3. -/' +aliases: +- FTH0799 +- cofinal_30 +- cofinal-30 +- TauLib.BookIII.Spectral.PrimorialLadder::cofinal_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001601 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: cofinal_30 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T09] Structural: cofinal level of 30 is 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006065.md b/_corpus_v3_items/cid006065.md new file mode 100644 index 000000000..33ff14b99 --- /dev/null +++ b/_corpus_v3_items/cid006065.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006065/ +title: reduce_coherence_42 (theorem) +cid: cid006065 +primary_alias: FTH0800 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D19] Structural: reduce coherence at specific values. -/' +aliases: +- FTH0800 +- reduce_coherence_42 +- reduce-coherence-42 +- TauLib.BookIII.Spectral.PrimorialLadder::reduce_coherence_42 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001464 +- predicate: part_of + object: cid005257 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005257 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.PrimorialLadder + declaration: reduce_coherence_42 + module_path: TauLib/BookIII/Spectral/PrimorialLadder.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D19] Structural: reduce coherence at specific values. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006066.md b/_corpus_v3_items/cid006066.md new file mode 100644 index 000000000..fb7007c3a --- /dev/null +++ b/_corpus_v3_items/cid006066.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006066/ +title: sieve_correct_50 (theorem) +cid: cid006066 +primary_alias: FTH0801 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T66] Sieve agrees with trial division up to 50. -/ +aliases: +- FTH0801 +- sieve_correct_50 +- sieve-correct-50 +- TauLib.BookIII.Spectral.SieveInfrastructure::sieve_correct_50 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001658 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: sieve_correct_50 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T66] Sieve agrees with trial division up to 50. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006067.md b/_corpus_v3_items/cid006067.md new file mode 100644 index 000000000..cf0346e90 --- /dev/null +++ b/_corpus_v3_items/cid006067.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006067/ +title: sieve_correct_200 (theorem) +cid: cid006067 +primary_alias: FTH0802 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T66] Sieve agrees with trial division up to 200. -/ +aliases: +- FTH0802 +- sieve_correct_200 +- sieve-correct-200 +- TauLib.BookIII.Spectral.SieveInfrastructure::sieve_correct_200 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001658 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: sieve_correct_200 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T66] Sieve agrees with trial division up to 200. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006068.md b/_corpus_v3_items/cid006068.md new file mode 100644 index 000000000..a5292f3c5 --- /dev/null +++ b/_corpus_v3_items/cid006068.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006068/ +title: sieve_count_known (theorem) +cid: cid006068 +primary_alias: FTH0803 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T66] Sieve count matches known prime-counting values. -/ +aliases: +- FTH0803 +- sieve_count_known +- sieve-count-known +- TauLib.BookIII.Spectral.SieveInfrastructure::sieve_count_known +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001658 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: sieve_count_known + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T66] Sieve count matches known prime-counting values. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006069.md b/_corpus_v3_items/cid006069.md new file mode 100644 index 000000000..4483a3a63 --- /dev/null +++ b/_corpus_v3_items/cid006069.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006069/ +title: sieve_tower_compat_3 (theorem) +cid: cid006069 +primary_alias: FTH0804 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T67] Sieve-tower compatibility at depth 3. -/ +aliases: +- FTH0804 +- sieve_tower_compat_3 +- sieve-tower-compat-3 +- TauLib.BookIII.Spectral.SieveInfrastructure::sieve_tower_compat_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001659 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: sieve_tower_compat_3 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T67] Sieve-tower compatibility at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006070.md b/_corpus_v3_items/cid006070.md new file mode 100644 index 000000000..f8a0126b8 --- /dev/null +++ b/_corpus_v3_items/cid006070.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006070/ +title: brun_euler_4 (theorem) +cid: cid006070 +primary_alias: FTH0805 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T67] Brun sieve matches Euler phi at primorial depths 1..4. -/ +aliases: +- FTH0805 +- brun_euler_4 +- brun-euler-4 +- TauLib.BookIII.Spectral.SieveInfrastructure::brun_euler_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001659 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: brun_euler_4 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T67] Brun sieve matches Euler phi at primorial depths 1..4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006071.md b/_corpus_v3_items/cid006071.md new file mode 100644 index 000000000..a0b147ddf --- /dev/null +++ b/_corpus_v3_items/cid006071.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006071/ +title: sieve_crt_compat_3 (theorem) +cid: cid006071 +primary_alias: FTH0806 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P42] Sieve-CRT compatibility for values ≤ 30 at depth 3. -/ +aliases: +- FTH0806 +- sieve_crt_compat_3 +- sieve-crt-compat-3 +- TauLib.BookIII.Spectral.SieveInfrastructure::sieve_crt_compat_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001586 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: sieve_crt_compat_3 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P42] Sieve-CRT compatibility for values ≤ 30 at depth 3. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006072.md b/_corpus_v3_items/cid006072.md new file mode 100644 index 000000000..df459dac0 --- /dev/null +++ b/_corpus_v3_items/cid006072.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006072/ +title: pi_10 (theorem) +cid: cid006072 +primary_alias: FTH0807 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D100] π(10) = 4 (primes: 2, 3, 5, 7). -/' +aliases: +- FTH0807 +- pi_10 +- pi-10 +- TauLib.BookIII.Spectral.SieveInfrastructure::pi_10 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001442 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: pi_10 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D100] π(10) = 4 (primes: 2, 3, 5, 7). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006073.md b/_corpus_v3_items/cid006073.md new file mode 100644 index 000000000..d7a528971 --- /dev/null +++ b/_corpus_v3_items/cid006073.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006073/ +title: pi_30 (theorem) +cid: cid006073 +primary_alias: FTH0808 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D100] π(30) = 10. -/ +aliases: +- FTH0808 +- pi_30 +- pi-30 +- TauLib.BookIII.Spectral.SieveInfrastructure::pi_30 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001442 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: pi_30 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D100] π(30) = 10. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006074.md b/_corpus_v3_items/cid006074.md new file mode 100644 index 000000000..8eb8bb4c4 --- /dev/null +++ b/_corpus_v3_items/cid006074.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006074/ +title: pi_100 (theorem) +cid: cid006074 +primary_alias: FTH0809 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D100] π(100) = 25. -/ +aliases: +- FTH0809 +- pi_100 +- pi-100 +- TauLib.BookIII.Spectral.SieveInfrastructure::pi_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001442 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: pi_100 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D100] π(100) = 25. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006075.md b/_corpus_v3_items/cid006075.md new file mode 100644 index 000000000..b69d682aa --- /dev/null +++ b/_corpus_v3_items/cid006075.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006075/ +title: brun_30_3 (theorem) +cid: cid006075 +primary_alias: FTH0810 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.D101] Brun sieve at (30, 3): 8 integers in [1..30] coprime\n to {2,3,5}. These are: 1, 7, 11, 13, 17,\ + \ 19, 23, 29. -/" +aliases: +- FTH0810 +- brun_30_3 +- brun-30-3 +- TauLib.BookIII.Spectral.SieveInfrastructure::brun_30_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001443 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: brun_30_3 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.D101] Brun sieve at (30, 3): 8 integers in [1..30] coprime\n to {2,3,5}. These are: 1, 7, 11, 13,\ + \ 17, 19, 23, 29. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006076.md b/_corpus_v3_items/cid006076.md new file mode 100644 index 000000000..7ff56ca26 --- /dev/null +++ b/_corpus_v3_items/cid006076.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006076/ +title: euler_phi_primorial_3 (theorem) +cid: cid006076 +primary_alias: FTH0811 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T67] Euler phi at primorial 3 = φ(30) = (2-1)(3-1)(5-1) = 8. -/ +aliases: +- FTH0811 +- euler_phi_primorial_3 +- euler-phi-primorial-3 +- TauLib.BookIII.Spectral.SieveInfrastructure::euler_phi_primorial_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001659 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: euler_phi_primorial_3 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T67] Euler phi at primorial 3 = φ(30) = (2-1)(3-1)(5-1) = 8. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006077.md b/_corpus_v3_items/cid006077.md new file mode 100644 index 000000000..aab16949c --- /dev/null +++ b/_corpus_v3_items/cid006077.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006077/ +title: euler_phi_primorial_4 (theorem) +cid: cid006077 +primary_alias: FTH0812 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T67] Euler phi at primorial 4 = φ(210) = 1·2·4·6 = 48. -/ +aliases: +- FTH0812 +- euler_phi_primorial_4 +- euler-phi-primorial-4 +- TauLib.BookIII.Spectral.SieveInfrastructure::euler_phi_primorial_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001659 +- predicate: part_of + object: cid005258 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005258 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.SieveInfrastructure + declaration: euler_phi_primorial_4 + module_path: TauLib/BookIII/Spectral/SieveInfrastructure.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T67] Euler phi at primorial 4 = φ(210) = 1·2·4·6 = 48. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006078.md b/_corpus_v3_items/cid006078.md new file mode 100644 index 000000000..6a06c268d --- /dev/null +++ b/_corpus_v3_items/cid006078.md @@ -0,0 +1,78 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006078/ +title: trichotomy_15_4 (theorem) +cid: cid006078 +primary_alias: FTH0813 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Spectral trichotomy [III.T14]' +aliases: +- FTH0813 +- trichotomy_15_4 +- trichotomy-15-4 +- TauLib.BookIII.Spectral.Trichotomy::trichotomy_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001606 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: trichotomy_15_4 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '-- ============================================================ + + -- FORMAL VERIFICATION (native_decide) + + -- ============================================================ + + + -- Spectral trichotomy [III.T14]' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006079.md b/_corpus_v3_items/cid006079.md new file mode 100644 index 000000000..6ebe13552 --- /dev/null +++ b/_corpus_v3_items/cid006079.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006079/ +title: trichotomy_func_15_3 (theorem) +cid: cid006079 +primary_alias: FTH0814 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Trichotomy functoriality [III.T14] +aliases: +- FTH0814 +- trichotomy_func_15_3 +- trichotomy-func-15-3 +- TauLib.BookIII.Spectral.Trichotomy::trichotomy_func_15_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001606 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: trichotomy_func_15_3 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Trichotomy functoriality [III.T14] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006080.md b/_corpus_v3_items/cid006080.md new file mode 100644 index 000000000..31f7af68b --- /dev/null +++ b/_corpus_v3_items/cid006080.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006080/ +title: bnf_15_4 (theorem) +cid: cid006080 +primary_alias: FTH0815 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- Boundary normal form [III.D24] +aliases: +- FTH0815 +- bnf_15_4 +- bnf-15-4 +- TauLib.BookIII.Spectral.Trichotomy::bnf_15_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001469 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bnf_15_4 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- Boundary normal form [III.D24] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006081.md b/_corpus_v3_items/cid006081.md new file mode 100644 index 000000000..6493c9b6d --- /dev/null +++ b/_corpus_v3_items/cid006081.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006081/ +title: bnf_unique_10_3 (theorem) +cid: cid006081 +primary_alias: FTH0816 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- BNF uniqueness [III.D24] +aliases: +- FTH0816 +- bnf_unique_10_3 +- bnf-unique-10-3 +- TauLib.BookIII.Spectral.Trichotomy::bnf_unique_10_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001469 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bnf_unique_10_3 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- BNF uniqueness [III.D24] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006082.md b/_corpus_v3_items/cid006082.md new file mode 100644 index 000000000..3c9a322a4 --- /dev/null +++ b/_corpus_v3_items/cid006082.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006082/ +title: bc_non_collapse_10_5 (theorem) +cid: cid006082 +primary_alias: FTH0817 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- B/C non-collapse [III.T15] +aliases: +- FTH0817 +- bc_non_collapse_10_5 +- bc-non-collapse-10-5 +- TauLib.BookIII.Spectral.Trichotomy::bc_non_collapse_10_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001607 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bc_non_collapse_10_5 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- B/C non-collapse [III.T15] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006083.md b/_corpus_v3_items/cid006083.md new file mode 100644 index 000000000..dd7ea1bdb --- /dev/null +++ b/_corpus_v3_items/cid006083.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006083/ +title: bc_coprime_5 (theorem) +cid: cid006083 +primary_alias: FTH0818 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: -- B/C coprimality [III.T15] +aliases: +- FTH0818 +- bc_coprime_5 +- bc-coprime-5 +- TauLib.BookIII.Spectral.Trichotomy::bc_coprime_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001607 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bc_coprime_5 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: -- B/C coprimality [III.T15] +noindex: false +--- + diff --git a/_corpus_v3_items/cid006084.md b/_corpus_v3_items/cid006084.md new file mode 100644 index 000000000..53e673d99 --- /dev/null +++ b/_corpus_v3_items/cid006084.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006084/ +title: trichotomy_depth_1 (theorem) +cid: cid006084 +primary_alias: FTH0819 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T14] Structural: trichotomy at depth 1 has only X-component\n (only prime is 2, which is X-type). -/" +aliases: +- FTH0819 +- trichotomy_depth_1 +- trichotomy-depth-1 +- TauLib.BookIII.Spectral.Trichotomy::trichotomy_depth_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001606 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: trichotomy_depth_1 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T14] Structural: trichotomy at depth 1 has only X-component\n (only prime is 2, which is X-type).\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006085.md b/_corpus_v3_items/cid006085.md new file mode 100644 index 000000000..01380f728 --- /dev/null +++ b/_corpus_v3_items/cid006085.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006085/ +title: bnf_zero_3 (theorem) +cid: cid006085 +primary_alias: FTH0820 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D24] Structural: BNF of 0 is all zeros. -/' +aliases: +- FTH0820 +- bnf_zero_3 +- bnf-zero-3 +- TauLib.BookIII.Spectral.Trichotomy::bnf_zero_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001469 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bnf_zero_3 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D24] Structural: BNF of 0 is all zeros. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006086.md b/_corpus_v3_items/cid006086.md new file mode 100644 index 000000000..45c22a44e --- /dev/null +++ b/_corpus_v3_items/cid006086.md @@ -0,0 +1,65 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006086/ +title: bc_coprime_at_3 (theorem) +cid: cid006086 +primary_alias: FTH0821 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [III.T15] Structural: B-product and C-product at depth 3 are\n coprime (they share no prime factors). -/" +aliases: +- FTH0821 +- bc_coprime_at_3 +- bc-coprime-at-3 +- TauLib.BookIII.Spectral.Trichotomy::bc_coprime_at_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001607 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bc_coprime_at_3 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [III.T15] Structural: B-product and C-product at depth 3 are\n coprime (they share no prime factors).\ + \ -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006087.md b/_corpus_v3_items/cid006087.md new file mode 100644 index 000000000..f935fce87 --- /dev/null +++ b/_corpus_v3_items/cid006087.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006087/ +title: bc_distinct_3 (theorem) +cid: cid006087 +primary_alias: FTH0822 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.T15] Structural: B-product ≠ C-product at depth 3. -/' +aliases: +- FTH0822 +- bc_distinct_3 +- bc-distinct-3 +- TauLib.BookIII.Spectral.Trichotomy::bc_distinct_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001607 +- predicate: part_of + object: cid005259 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005259 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.Trichotomy + declaration: bc_distinct_3 + module_path: TauLib/BookIII/Spectral/Trichotomy.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.T15] Structural: B-product ≠ C-product at depth 3. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006088.md b/_corpus_v3_items/cid006088.md new file mode 100644 index 000000000..e0e66cc35 --- /dev/null +++ b/_corpus_v3_items/cid006088.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006088/ +title: twin_primes_500 (theorem) +cid: cid006088 +primary_alias: FTH0823 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T72] At least 20 twin prime pairs below 500. -/ +aliases: +- FTH0823 +- twin_primes_500 +- twin-primes-500 +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_primes_500 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001664 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_primes_500 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T72] At least 20 twin prime pairs below 500. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006089.md b/_corpus_v3_items/cid006089.md new file mode 100644 index 000000000..32a432222 --- /dev/null +++ b/_corpus_v3_items/cid006089.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006089/ +title: twin_density_primorial_5 (theorem) +cid: cid006089 +primary_alias: FTH0824 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T73] Twin prime density positive at primorial depths 1..5. -/ +aliases: +- FTH0824 +- twin_density_primorial_5 +- twin-density-primorial-5 +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_density_primorial_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001665 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_density_primorial_5 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T73] Twin prime density positive at primorial depths 1..5. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006090.md b/_corpus_v3_items/cid006090.md new file mode 100644 index 000000000..4e4874a7b --- /dev/null +++ b/_corpus_v3_items/cid006090.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006090/ +title: hl_constant_decreasing_5 (theorem) +cid: cid006090 +primary_alias: FTH0825 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T74] HL constant decreasing for k = 2..5. -/ +aliases: +- FTH0825 +- hl_constant_decreasing_5 +- hl-constant-decreasing-5 +- TauLib.BookIII.Spectral.TwinPrimeDeep::hl_constant_decreasing_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001666 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: hl_constant_decreasing_5 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T74] HL constant decreasing for k = 2..5. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006091.md b/_corpus_v3_items/cid006091.md new file mode 100644 index 000000000..3c6993bad --- /dev/null +++ b/_corpus_v3_items/cid006091.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006091/ +title: crt_admissible_positive_4 (theorem) +cid: cid006091 +primary_alias: FTH0826 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.T75] CRT-admissible residues positive at depths 1..4. -/ +aliases: +- FTH0826 +- crt_admissible_positive_4 +- crt-admissible-positive-4 +- TauLib.BookIII.Spectral.TwinPrimeDeep::crt_admissible_positive_4 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001667 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: crt_admissible_positive_4 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.T75] CRT-admissible residues positive at depths 1..4. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006092.md b/_corpus_v3_items/cid006092.md new file mode 100644 index 000000000..75c2e5759 --- /dev/null +++ b/_corpus_v3_items/cid006092.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006092/ +title: twin_admissibility_fraction_5 (theorem) +cid: cid006092 +primary_alias: FTH0827 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.P45] Admissibility fraction = (p-2)/p for primes 3,5,7,11. -/ +aliases: +- FTH0827 +- twin_admissibility_fraction_5 +- twin-admissibility-fraction-5 +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissibility_fraction_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001589 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_admissibility_fraction_5 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.P45] Admissibility fraction = (p-2)/p for primes 3,5,7,11. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006093.md b/_corpus_v3_items/cid006093.md new file mode 100644 index 000000000..fe023f525 --- /dev/null +++ b/_corpus_v3_items/cid006093.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006093/ +title: twin_count_100 (theorem) +cid: cid006093 +primary_alias: FTH0828 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D105] Twin prime count at 100 ≥ 8. -/ +aliases: +- FTH0828 +- twin_count_100 +- twin-count-100 +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_count_100 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001447 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_count_100 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D105] Twin prime count at 100 ≥ 8. -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006094.md b/_corpus_v3_items/cid006094.md new file mode 100644 index 000000000..34a301f1f --- /dev/null +++ b/_corpus_v3_items/cid006094.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006094/ +title: hl_depth_2 (theorem) +cid: cid006094 +primary_alias: FTH0829 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D106] HL constant at depth 2: C₂(2) = 3·1/(2·2) = 3/4. -/' +aliases: +- FTH0829 +- hl_depth_2 +- hl-depth-2 +- TauLib.BookIII.Spectral.TwinPrimeDeep::hl_depth_2 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001448 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: hl_depth_2 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D106] HL constant at depth 2: C₂(2) = 3·1/(2·2) = 3/4. -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006095.md b/_corpus_v3_items/cid006095.md new file mode 100644 index 000000000..b7ed2746a --- /dev/null +++ b/_corpus_v3_items/cid006095.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006095/ +title: twin_admissible_1 (theorem) +cid: cid006095 +primary_alias: FTH0830 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.D107] Twin-admissible residues at depth 1 (mod 2): 1 (only odd). -/' +aliases: +- FTH0830 +- twin_admissible_1 +- twin-admissible-1 +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_1 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001449 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_admissible_1 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.D107] Twin-admissible residues at depth 1 (mod 2): 1 (only odd). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006096.md b/_corpus_v3_items/cid006096.md new file mode 100644 index 000000000..4bf752b1f --- /dev/null +++ b/_corpus_v3_items/cid006096.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006096/ +title: twin_admissible_3_pos (theorem) +cid: cid006096 +primary_alias: FTH0831 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: /-- [III.D107] Twin-admissible residues at depth 3 (mod 30). -/ +aliases: +- FTH0831 +- twin_admissible_3_pos +- twin-admissible-3-pos +- TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_3_pos +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001449 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: twin_admissible_3_pos + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: /-- [III.D107] Twin-admissible residues at depth 3 (mod 30). -/ +noindex: false +--- + diff --git a/_corpus_v3_items/cid006097.md b/_corpus_v3_items/cid006097.md new file mode 100644 index 000000000..7ea8c52cc --- /dev/null +++ b/_corpus_v3_items/cid006097.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006097/ +title: admissible_at_3 (theorem) +cid: cid006097 +primary_alias: FTH0832 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P45] At prime 3: 1 out of 3 admissible (3-2=1). -/' +aliases: +- FTH0832 +- admissible_at_3 +- admissible-at-3 +- TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_3 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001589 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: admissible_at_3 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P45] At prime 3: 1 out of 3 admissible (3-2=1). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006098.md b/_corpus_v3_items/cid006098.md new file mode 100644 index 000000000..38ab9d3f5 --- /dev/null +++ b/_corpus_v3_items/cid006098.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006098/ +title: admissible_at_5 (theorem) +cid: cid006098 +primary_alias: FTH0833 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P45] At prime 5: 3 out of 5 admissible (5-2=3). -/' +aliases: +- FTH0833 +- admissible_at_5 +- admissible-at-5 +- TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_5 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001589 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: admissible_at_5 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P45] At prime 5: 3 out of 5 admissible (5-2=3). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006099.md b/_corpus_v3_items/cid006099.md new file mode 100644 index 000000000..8eadb8ff7 --- /dev/null +++ b/_corpus_v3_items/cid006099.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006099/ +title: admissible_at_7 (theorem) +cid: cid006099 +primary_alias: FTH0834 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: '/-- [III.P45] At prime 7: 5 out of 7 admissible (7-2=5). -/' +aliases: +- FTH0834 +- admissible_at_7 +- admissible-at-7 +- TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_7 +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid001589 +- predicate: part_of + object: cid005260 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005260 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectral.TwinPrimeDeep + declaration: admissible_at_7 + module_path: TauLib/BookIII/Spectral/TwinPrimeDeep.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: '/-- [III.P45] At prime 7: 5 out of 7 admissible (7-2=5). -/' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006100.md b/_corpus_v3_items/cid006100.md new file mode 100644 index 000000000..4e391d02e --- /dev/null +++ b/_corpus_v3_items/cid006100.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006100/ +title: width_principle (theorem) +cid: cid006100 +primary_alias: FTH0835 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T33] Interface Width Principle: tower-coherent functions\n are τ-admissible at EVERY depth — tower coherence\ + \ itself\n gives the stabilization property.\n\n For any tower-coherent f and any choice of depth D:\n reduce(f.b_fun(n,\ + \ D), k) = f.b_fun(n, k) for all k ≤ D.\n\n This is exactly the tower coherence condition (I.D46). -/" +aliases: +- FTH0835 +- width_principle +- width-principle +- TauLib.BookIII.Spectrum.InterfaceWidth::width_principle +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005261 +- predicate: part_of + object: cid005261 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005261 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.InterfaceWidth + declaration: width_principle + module_path: TauLib/BookIII/Spectrum/InterfaceWidth.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T33] Interface Width Principle: tower-coherent functions\n are τ-admissible at EVERY depth — tower\ + \ coherence itself\n gives the stabilization property.\n\n For any tower-coherent f and any choice of depth D:\n \ + \ reduce(f.b_fun(n" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006101.md b/_corpus_v3_items/cid006101.md new file mode 100644 index 000000000..c743b0098 --- /dev/null +++ b/_corpus_v3_items/cid006101.md @@ -0,0 +1,64 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006101/ +title: chi_plus_admissible (theorem) +cid: cid006101 +primary_alias: FTH0836 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P30] χ₊ is τ-admissible: at any depth D, it stabilizes\n because χ₊ is tower-coherent. -/" +aliases: +- FTH0836 +- chi_plus_admissible +- chi-plus-admissible +- TauLib.BookIII.Spectrum.InterfaceWidth::chi_plus_admissible +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005261 +- predicate: part_of + object: cid005261 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005261 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.InterfaceWidth + declaration: chi_plus_admissible + module_path: TauLib/BookIII/Spectrum/InterfaceWidth.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P30] χ₊ is τ-admissible: at any depth D, it stabilizes\n because χ₊ is tower-coherent. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006102.md b/_corpus_v3_items/cid006102.md new file mode 100644 index 000000000..e78464c29 --- /dev/null +++ b/_corpus_v3_items/cid006102.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006102/ +title: book_ii_bridge_complete (theorem) +cid: cid006102 +primary_alias: FTH0837 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.T34] Main theorem: the bridge is complete.\n All fields are instantiated from earned structures;\n \ + \ no sorry, no axiom, no external import beyond Mathlib tactics. -/" +aliases: +- FTH0837 +- book_ii_bridge_complete +- book-ii-bridge-complete +- TauLib.BookIII.Spectrum.KernelHinge::book_ii_bridge_complete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005262 +- predicate: part_of + object: cid005262 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005262 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.KernelHinge + declaration: book_ii_bridge_complete + module_path: TauLib/BookIII/Spectrum/KernelHinge.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.T34] Main theorem: the bridge is complete.\n All fields are instantiated from earned structures;\n\ + \ no sorry, no axiom, no external import beyond Mathlib tactics. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006103.md b/_corpus_v3_items/cid006103.md new file mode 100644 index 000000000..8c01b6591 --- /dev/null +++ b/_corpus_v3_items/cid006103.md @@ -0,0 +1,66 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006103/ +title: spectral_decidable (theorem) +cid: cid006103 +primary_alias: FTH0838 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P31] The spectral encoding is decidable: for any formula and\n value, we can compute whether the formula\ + \ is spectrally satisfied.\n This is the computable core of the encoding. -/" +aliases: +- FTH0838 +- spectral_decidable +- spectral-decidable +- TauLib.BookIII.Spectrum.ThreeSAT::spectral_decidable +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005264 +- predicate: part_of + object: cid005264 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005264 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.ThreeSAT + declaration: spectral_decidable + module_path: TauLib/BookIII/Spectrum/ThreeSAT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P31] The spectral encoding is decidable: for any formula and\n value, we can compute whether the formula\ + \ is spectrally satisfied.\n This is the computable core of the encoding. -/" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006104.md b/_corpus_v3_items/cid006104.md new file mode 100644 index 000000000..a64dcc5ca --- /dev/null +++ b/_corpus_v3_items/cid006104.md @@ -0,0 +1,68 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006104/ +title: tau_complexity_bridge_concrete (theorem) +cid: cid006104 +primary_alias: FTH0839 +type: formal_theorem +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v2 +- corpus_v3_working +summary: "/-- [I.P32] The τ-complexity bridge: the spectral encoding translates\n Boolean satisfiability into a problem\ + \ about τ-framework values.\n\n Key structural fact: for concrete CNF formulas, we can verify\n satisfiability by\ + \ searching over CRT residues. -/" +aliases: +- FTH0839 +- tau_complexity_bridge_concrete +- tau-complexity-bridge-concrete +- TauLib.BookIII.Spectrum.ThreeSAT::tau_complexity_bridge_concrete +layers: [] +domains: [] +relations: +- predicate: formalizes + object: cid005264 +- predicate: part_of + object: cid005264 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: +- cid005264 +history: +- version: v1 + date: '2026-05-10' + event: imported_from_v2_taulib_declarations +external_identifiers: {} +payload: {} +formalization: + status: lean_axiom_free + module: TauLib.BookIII.Spectrum.ThreeSAT + declaration: tau_complexity_bridge_concrete + module_path: TauLib/BookIII/Spectrum/ThreeSAT.lean + commit: '' + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + formal_status_label: formalized +proof: {} +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: "/-- [I.P32] The τ-complexity bridge: the spectral encoding translates\n Boolean satisfiability into a problem\ + \ about τ-framework values.\n\n Key structural fact: for concrete CNF formulas, we can verify\n satisfiability by\ + \ searching ove" +noindex: false +--- + diff --git a/_corpus_v3_items/cid006105.md b/_corpus_v3_items/cid006105.md new file mode 100644 index 000000000..ee179b1d4 --- /dev/null +++ b/_corpus_v3_items/cid006105.md @@ -0,0 +1,116 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006105/ +title: Proof of the Hyperfactorization Theorem +cid: cid006105 +primary_alias: PRF0002 +type: proof +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Semi-formal Corpus-addressed proof of the Hyperfactorization Theorem (THM0005). Establishes that every τ-Idx element + admits a unique hyperfactorization into the canonical generator system, leveraging earned boundary constants and the iterator + ladder. + + ' +aliases: +- PRF0002 +- proof-hyperfactorization +layers: +- E0 +domains: +- math.foundation +- geometry.calibration +relations: +- predicate: proves + object: cid001171 +- predicate: uses + object: cid000007 +- predicate: uses + object: cid000008 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: 'We prove the Hyperfactorization Theorem by combining earned boundary + + constants [DEF0001](prrp://def0001@v1) with lobe-swap invariance + + [LEM0001](prrp://lem0001@v1) on the τ-domain. The unique decomposition + + of every τ-Idx element into the canonical generator system follows + + directly from these two ingredients, establishing [THM0005](prrp://thm0005@v1). + + ' +formalization: {} +proof: + mode: semi_formal + conclusion: cid001171 + proof_status: complete + formality_level: prose_addressed + dependency_closure_status: checked + version_pinning: pinned + assumptions: [] + steps: + - id: s1 + label: Earned constants pin the boundary + text_markdown: 'The boundary constants for the hyperfactorization construction are + + available by [DEF0001](prrp://def0001@v1) — earned boundary constants + + on the τ-domain provide the pinned data the factorization consumes. + + ' + uses: + - ref: prrp://def0001@v1 + role: uses_definition + - id: s2 + label: Lobe-swap invariance fixes the canonical decomposition + text_markdown: 'Apply [LEM0001](prrp://lem0001@v1) (lobe-swap invariance) to establish + + that the factorization preserves boundary-constant identity under the + + permitted symmetries on `B_τ`. + + ' + uses: + - ref: prrp://lem0001@v1 + role: uses_lemma + - id: s3 + label: Conclude unique hyperfactorization + text_markdown: 'Combining the pinned earned constants (s1) with lobe-swap invariance (s2), + + every element decomposes uniquely into the canonical generator system, + + establishing [THM0005](prrp://thm0005@v1). + + ' + derives: cid001171 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: Semi-formal Corpus-addressed proof of the Hyperfactorization Theorem (THM0005). Establishes that every τ-Idx + element admits a unique hyperfactorization into the canonical generator system, leveraging earned boundary constants and + the iterat +noindex: false +--- + diff --git a/_corpus_v3_items/cid006106.md b/_corpus_v3_items/cid006106.md new file mode 100644 index 000000000..4d378d2e0 --- /dev/null +++ b/_corpus_v3_items/cid006106.md @@ -0,0 +1,118 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006106/ +title: Proof Sketch of Global Hartogs +cid: cid006106 +primary_alias: PRF0003 +type: proof +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Sketch-mode proof of the Global Hartogs theorem (THM0007). Outline of the argument; intended for orientation rather + than complete verification. Companion to the imported v2 theorem. + + ' +aliases: +- PRF0003 +- proof-sketch-global-hartogs +layers: +- E0 +domains: +- math.foundation +- geometry.holomorphy +relations: +- predicate: proves + object: cid001173 +- predicate: uses + object: cid000007 +- predicate: uses + object: cid000008 +- predicate: uses + object: cid000010 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Sketch-mode proof of Global Hartogs (THM0007).** This is an outline, + + not a complete proof. The full chart-glue argument is in the v2 + + manuscript source. Inspection route: [DEF0001](prrp://def0001@v1) → + + [LEM0001](prrp://lem0001@v1) → [THM0001](prrp://thm0001@v1) → + + [THM0007](prrp://thm0007@v1). + + ' +formalization: {} +proof: + mode: sketch + conclusion: cid001173 + proof_status: sketch + formality_level: informal + dependency_closure_status: unchecked + version_pinning: unpinned + assumptions: [] + steps: + - id: s1 + label: Setup + text_markdown: 'Set up the τ-holomorphy hypothesis on the open subdomain and the + + boundary-coherence condition (cf. [THM0001](prrp://thm0001@v1) for + + the calibration anchor). + + ' + uses: + - ref: prrp://thm0001@v1 + role: uses_theorem + - id: s2 + label: Apply local Hartogs at each chart + text_markdown: 'Apply local Hartogs (DEF0001 boundary anchor + lobe-swap invariance + + [LEM0001](prrp://lem0001@v1)) at each compatible chart and glue. + + ' + uses: + - ref: prrp://lem0001@v1 + role: uses_lemma + - id: s3 + label: Global glue + conclusion (sketch) + text_markdown: 'Global gluing via the chart compatibility argument yields the global + + statement. **Full glue argument is omitted** — see the v2 source + + manuscript (Book II Chapter referenced in [II.T13](prrp://thm0007@v1) + + for the complete proof. + + ' + derives: cid001173 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Sketch-mode proof of the Global Hartogs theorem (THM0007). Outline of the argument; intended for orientation + rather than complete verification. Companion to the imported v2 theorem. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006107.md b/_corpus_v3_items/cid006107.md new file mode 100644 index 000000000..b8c92a919 --- /dev/null +++ b/_corpus_v3_items/cid006107.md @@ -0,0 +1,128 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006107/ +title: Conditional Proof of Categoricity of τ_0 +cid: cid006107 +primary_alias: PRF0004 +type: proof +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Conditional proof of Categoricity of τ_0 (THM0009). Completes the categoricity argument under the named bridge axiom + (AXM0001 bridge-functor- exists). Without that bridge, the proof reduces to a sketch. + + ' +aliases: +- PRF0004 +- proof-categoricity-conditional +layers: +- E0 +domains: +- math.foundation +- category.theory +relations: +- predicate: proves + object: cid001175 +- predicate: assumes + object: cid000014 +- predicate: uses + object: cid000010 +- predicate: uses + object: cid001174 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Conditional proof.** This proof depends on [AXM0001](prrp://axm0001@v1) + + (bridge-functor-exists) being discharged. Under that axiom, categoricity + + of τ_0 follows from the universal property of the τ-Kernel construction + + plus the rigidity of τ ([THM0008](prrp://thm0008@v1)). + + + Without the bridge axiom, this proof is a sketch. + + ' +formalization: {} +proof: + mode: conditional + conclusion: cid001175 + proof_status: conditional + formality_level: prose_addressed + dependency_closure_status: checked + version_pinning: pinned + assumptions: + - cid000014 + condition_summary: 'The categoricity argument is complete *conditional* on the bridge functor + + F: τ-Cat → Orth-Cat existing (AXM0001). Without that axiom discharged, the + + proof is a sketch; with it, categoricity follows. + + ' + steps: + - id: s1 + label: Universal property setup + text_markdown: 'Establish the universal property of τ_0 from the τ-Kernel + + construction (see [THM0001](prrp://thm0001@v1) for the master + + constant anchoring the calibration). + + ' + uses: + - ref: prrp://thm0001@v1 + role: uses_theorem + - id: s2 + label: Bridge functor invocation (conditional) + text_markdown: 'Under the bridge-functor-exists axiom [AXM0001](prrp://axm0001@v1), + + the universal property reflects through F into the orthodox + + category, fixing the τ_0 representation uniquely. + + ' + uses: + - ref: prrp://axm0001@v1 + role: uses_assumption + - id: s3 + label: Categoricity follows + text_markdown: 'From s1 + s2 + the rigidity of τ ([THM0008](prrp://thm0008@v1)), + + categoricity of τ_0 is established up to canonical isomorphism. + + ' + uses: + - ref: prrp://thm0008@v1 + role: uses_theorem + derives: cid001175 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Conditional proof of Categoricity of τ_0 (THM0009). Completes the categoricity argument under the named bridge + axiom (AXM0001 bridge-functor- exists). Without that bridge, the proof reduces to a sketch. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006108.md b/_corpus_v3_items/cid006108.md new file mode 100644 index 000000000..8a4f97c4a --- /dev/null +++ b/_corpus_v3_items/cid006108.md @@ -0,0 +1,127 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006108/ +title: Lean-Formalized Proof of FTA on τ-Idx +cid: cid006108 +primary_alias: PRF0005 +type: proof +lane: corpus +status: canonical +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Lean-formalized proof of the Fundamental Theorem of Arithmetic on τ-Idx (THM0010). Backed by an axiom-free Lean + theorem in TauLib v2; the prose proof shadows the Lean development and pins to its commit hash. + + ' +aliases: +- PRF0005 +- proof-fta-tau-idx-lean +layers: +- E0 +domains: +- math.foundation +- formalization.lean +relations: +- predicate: proves + object: cid001176 +- predicate: uses + object: cid000007 +- predicate: uses + object: cid001171 +- predicate: uses + object: cid001172 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Lean-formalized proof.** The complete proof is at + + `TauLib.BookI.PrimePolarity.fundamental_theorem_arithmetic_tau_idx`. + + The prose above shadows the Lean development; consult the formal + + theorem for the axiom-free verification. + + ' +formalization: + status: lean_axiom_free + module: TauLib.BookI.PrimePolarity + declaration: fundamental_theorem_arithmetic_tau_idx + module_path: TauLib/BookI/PrimePolarity.lean + commit: cb5e83015b54dd72eba560953fe2461820078757 + lean_toolchain: leanprover/lean4:v4.x.x + mathlib_commit: '' + sorry_count: 0 + project_axioms: 0 + print_axioms: 'propext, Classical.choice, Quot.sound + + ' +proof: + mode: lean_formalized + conclusion: cid001176 + proof_status: axiom_free_formalized + formality_level: lean_axiom_free + dependency_closure_status: checked + version_pinning: pinned + assumptions: [] + steps: + - id: s1 + label: Definitions imported + text_markdown: 'Import the definitions of τ-Idx and the prime-polarity machinery + + ([DEF0001](prrp://def0001@v1) earned-boundary-constants anchor). + + ' + uses: + - ref: prrp://def0001@v1 + role: uses_definition + - id: s2 + label: Hyperfactorization theorem applied + text_markdown: 'Apply [THM0005](prrp://thm0005@v1) (hyperfactorization theorem) to + + decompose each τ-Idx element into the canonical generator system. + + ' + uses: + - ref: prrp://thm0005@v1 + role: uses_theorem + - id: s3 + label: Uniqueness from prime polarity + text_markdown: 'Uniqueness of the factorization follows from prime polarity + + ([THM0006](prrp://thm0006@v1)). The complete formalization is in + + TauLib `BookI.PrimePolarity.fundamental_theorem_arithmetic_tau_idx`. + + ' + uses: + - ref: prrp://thm0006@v1 + role: uses_theorem + derives: cid001176 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Lean-formalized proof of the Fundamental Theorem of Arithmetic on τ-Idx (THM0010). Backed by an axiom-free Lean + theorem in TauLib v2; the prose proof shadows the Lean development and pins to its commit hash. + + ' +noindex: false +--- + diff --git a/_corpus_v3_items/cid006109.md b/_corpus_v3_items/cid006109.md new file mode 100644 index 000000000..400dd8d76 --- /dev/null +++ b/_corpus_v3_items/cid006109.md @@ -0,0 +1,121 @@ +--- +layout: corpus-v3-item +permalink: /id/cid006109/ +title: Prose Proof of Six-Generator Rigidity Failure +cid: cid006109 +primary_alias: PRF0006 +type: proof +lane: corpus +status: active +visibility: public +current_version: v1 +release_lines: +- corpus_v3_working +summary: 'Prose-mode proof of the Six-Generator Rigidity Failure (THM0013). Argues by counterexample that six generators do + not suffice to fix the τ-Kernel construction uniquely — five is minimal. + + ' +aliases: +- PRF0006 +- proof-prose-six-generator-rigidity +layers: +- E0 +domains: +- math.foundation +relations: +- predicate: proves + object: cid001179 +- predicate: uses + object: cid000007 +- predicate: uses + object: cid000008 +- predicate: uses + object: cid001169 +depends_on: [] +appears_in: [] +formalized_by: [] +sources: [] +contains: [] +part_of: [] +history: +- version: v1 + date: '2026-05-10' + event: initial_corpus_item_seed +external_identifiers: {} +payload: + kind: inline_atomic + canonical_format: markdown + markdown: '**Prose proof.** By counterexample. Construct an alternate six-generator + + system respecting the K-axioms but yielding a distinct τ-Kernel. + + Comparison against the canonical five-generator construction + + ([DEF0001](prrp://def0001@v1), [LEM0001](prrp://lem0001@v1), + + [THM0003](prrp://thm0003@v1)) witnesses non-rigidity. + + ' +formalization: {} +proof: + mode: prose + conclusion: cid001179 + proof_status: complete + formality_level: prose_addressed + dependency_closure_status: checked + version_pinning: pinned + assumptions: [] + steps: + - id: s1 + label: Construct an alternate six-generator system + text_markdown: 'Construct a six-generator system that respects the K-axioms (e.g., + + AXM0001 / bridge-functor-exists for the bridge layer) but yields a + + distinct τ-Kernel construction from the canonical five-generator + + system anchored by [DEF0001](prrp://def0001@v1). + + ' + uses: + - ref: prrp://def0001@v1 + role: uses_definition + - id: s2 + label: Compare to canonical five-generator construction + text_markdown: 'The two constructions agree on the boundary anchor + + ([LEM0001](prrp://lem0001@v1)) but diverge on the iterator ladder + + ([THM0003](prrp://thm0003@v1)), establishing non-rigidity. + + ' + uses: + - ref: prrp://lem0001@v1 + role: uses_lemma + - ref: prrp://thm0003@v1 + role: uses_theorem + - id: s3 + label: Conclude rigidity failure + text_markdown: 'The alternate construction witnesses that six generators are + + insufficient to fix the τ-Kernel uniquely, establishing + + [THM0013](prrp://thm0013@v1). + + ' + derives: cid001179 +result: {} +commentary: {} +tombstone: false +superseded_by: '' +retired_at: '' +retirement_reason: '' +refuted_at: '' +refutation_reason: '' +description: 'Prose-mode proof of the Six-Generator Rigidity Failure (THM0013). Argues by counterexample that six generators + do not suffice to fix the τ-Kernel construction uniquely — five is minimal. + + ' +noindex: false +--- + diff --git a/_data/corpus_v3/alias-index.yaml b/_data/corpus_v3/alias-index.yaml new file mode 100644 index 000000000..0e8312d3d --- /dev/null +++ b/_data/corpus_v3/alias-index.yaml @@ -0,0 +1,7236 @@ +generated_at: '2026-05-23T19:54:46Z' +generator: scripts/build_alias_index.py +items_root: items +total_aliases: 7230 +total_items: 1922 +aliases: + 2-category-structure: cid001290 + 2-morphism: cid001291 + 3-condition-sufficiency: cid001559 + 4-1-sector-decomposition: cid001458 + AXM0001: cid000014 + AXM0002: cid001109 + AXM0003: cid001110 + AXM0004: cid001111 + AXM0005: cid001112 + AXM0006: cid001113 + AXM0007: cid001114 + AXM0008: cid001115 + BOK0001: cid000001 + BOK0002: cid000023 + BOK0003: cid000024 + BOK0004: cid000025 + BOK0005: cid000026 + BOK0006: cid000027 + BOK0007: cid000028 + BRG0001: cid000012 + CHP0001: cid000002 + CLM0001: cid000011 + CNS0001: cid000029 + CNS0002: cid000030 + CNS0003: cid000031 + CNS0004: cid000032 + CNS0005: cid000033 + CNS0006: cid000034 + CNS0007: cid000035 + CNS0008: cid000036 + CNS0009: cid000037 + CNS0010: cid000038 + CNS0011: cid000039 + CNS0012: cid000040 + CNS0013: cid000041 + CNS0014: cid000042 + CNS0015: cid000043 + CNS0016: cid000044 + CNS0017: cid000045 + CNS0018: cid000046 + CNS0019: cid000047 + CNS0020: cid000048 + CNS0021: cid000049 + CNS0022: cid000050 + CNS0023: cid000051 + CNS0024: cid000052 + CNS0025: cid000053 + CNS0026: cid000054 + CNS0027: cid000055 + CNS0028: cid000056 + CNS0029: cid000057 + CNS0030: cid000058 + CNS0031: cid000059 + CNS0032: cid000060 + CNS0033: cid000061 + CNS0034: cid000062 + CNS0035: cid000063 + CNS0036: cid000064 + CNS0037: cid000065 + CNS0038: cid000066 + CNS0039: cid000067 + CNS0040: cid000068 + CNS0041: cid000069 + CNS0042: cid000070 + CNS0043: cid000071 + CNS0044: cid000072 + CNS0045: cid000073 + CNS0046: cid000074 + CNS0047: cid000075 + CNS0048: cid000076 + CNS0049: cid000077 + CNS0050: cid000078 + CNS0051: cid000079 + CNS0052: cid000080 + CNS0053: cid000081 + CNS0054: cid000082 + CNS0055: cid000083 + CNS0056: cid000084 + CNS0057: cid000085 + CNS0058: cid000086 + CNS0059: cid000087 + CNS0060: cid000088 + CNS0061: cid000089 + CNS0062: cid000090 + CNS0063: cid000091 + CNS0064: cid000092 + CNS0065: cid000093 + CNS0066: cid000094 + CNS0067: cid000095 + CNS0068: cid000096 + CNS0069: cid000097 + CNS0070: cid000098 + CNS0071: cid000099 + CNS0072: cid000100 + CNS0073: cid000101 + CNS0074: cid000102 + CNS0075: cid000103 + CNS0076: cid000104 + CNS0077: cid000105 + CNS0078: cid000106 + CNS0079: cid000107 + CNS0080: cid000108 + CNS0081: cid000109 + CNS0082: cid000110 + CNS0083: cid000111 + CNS0084: cid000112 + CNS0085: cid000113 + CNS0086: cid000114 + CNS0087: cid000115 + CNS0088: cid000116 + CNS0089: cid000117 + CNS0090: cid000118 + CNS0091: cid000119 + CNS0092: cid000120 + CNS0093: cid000121 + CNS0094: cid000122 + CNS0095: cid000123 + CNS0096: cid000124 + CNS0097: cid000125 + CNS0098: cid000126 + CNS0099: cid000127 + CNS0100: cid000128 + COR0001: cid001000 + COR0002: cid001001 + COR0003: cid001002 + COR0004: cid001003 + COR0005: cid001232 + COR0006: cid001233 + DEF0001: cid000007 + DEF0002: cid001004 + DEF0003: cid001005 + DEF0004: cid001006 + DEF0005: cid001007 + DEF0006: cid001008 + DEF0007: cid001009 + DEF0008: cid001010 + DEF0009: cid001011 + DEF0010: cid001012 + DEF0011: cid001013 + DEF0012: cid001014 + DEF0013: cid001015 + DEF0014: cid001016 + DEF0015: cid001017 + DEF0016: cid001018 + DEF0017: cid001019 + DEF0018: cid001020 + DEF0019: cid001021 + DEF0020: cid001022 + DEF0021: cid001023 + DEF0022: cid001024 + DEF0023: cid001025 + DEF0024: cid001026 + DEF0025: cid001027 + DEF0026: cid001028 + DEF0027: cid001029 + DEF0028: cid001030 + DEF0029: cid001031 + DEF0030: cid001032 + DEF0031: cid001033 + DEF0032: cid001034 + DEF0033: cid001035 + DEF0034: cid001036 + DEF0035: cid001037 + DEF0036: cid001038 + DEF0037: cid001039 + DEF0038: cid001040 + DEF0039: cid001041 + DEF0040: cid001042 + DEF0041: cid001043 + DEF0042: cid001044 + DEF0043: cid001045 + DEF0044: cid001046 + DEF0045: cid001047 + DEF0046: cid001048 + DEF0047: cid001049 + DEF0048: cid001050 + DEF0049: cid001051 + DEF0050: cid001052 + DEF0051: cid001053 + DEF0052: cid001054 + DEF0053: cid001055 + DEF0054: cid001056 + DEF0055: cid001057 + DEF0056: cid001058 + DEF0057: cid001059 + DEF0058: cid001060 + DEF0059: cid001061 + DEF0060: cid001062 + DEF0061: cid001063 + DEF0062: cid001064 + DEF0063: cid001065 + DEF0064: cid001066 + DEF0065: cid001067 + DEF0066: cid001068 + DEF0067: cid001069 + DEF0068: cid001070 + DEF0069: cid001071 + DEF0070: cid001072 + DEF0071: cid001073 + DEF0072: cid001074 + DEF0073: cid001075 + DEF0074: cid001076 + DEF0075: cid001077 + DEF0076: cid001078 + DEF0077: cid001079 + DEF0078: cid001080 + DEF0079: cid001081 + DEF0080: cid001082 + DEF0081: cid001083 + DEF0082: cid001084 + DEF0083: cid001085 + DEF0084: cid001086 + DEF0085: cid001087 + DEF0086: cid001088 + DEF0087: cid001089 + DEF0088: cid001090 + DEF0089: cid001091 + DEF0090: cid001092 + DEF0091: cid001093 + DEF0092: cid001094 + DEF0093: cid001095 + DEF0094: cid001096 + DEF0095: cid001097 + DEF0096: cid001098 + DEF0097: cid001099 + DEF0098: cid001100 + DEF0099: cid001101 + DEF0100: cid001102 + DEF0101: cid001103 + DEF0102: cid001104 + DEF0103: cid001105 + DEF0104: cid001106 + DEF0105: cid001107 + DEF0106: cid001108 + DEF0107: cid001234 + DEF0108: cid001235 + DEF0109: cid001236 + DEF0110: cid001237 + DEF0111: cid001238 + DEF0112: cid001239 + DEF0113: cid001240 + DEF0114: cid001241 + DEF0115: cid001242 + DEF0116: cid001243 + DEF0117: cid001244 + DEF0118: cid001245 + DEF0119: cid001246 + DEF0120: cid001247 + DEF0121: cid001248 + DEF0122: cid001249 + DEF0123: cid001250 + DEF0124: cid001251 + DEF0125: cid001252 + DEF0126: cid001253 + DEF0127: cid001254 + DEF0128: cid001255 + DEF0129: cid001256 + DEF0130: cid001257 + DEF0131: cid001258 + DEF0132: cid001259 + DEF0133: cid001260 + DEF0134: cid001261 + DEF0135: cid001262 + DEF0136: cid001263 + DEF0137: cid001264 + DEF0138: cid001265 + DEF0139: cid001266 + DEF0140: cid001267 + DEF0141: cid001268 + DEF0142: cid000007 + DEF0143: cid001269 + DEF0144: cid001270 + DEF0145: cid001271 + DEF0146: cid001272 + DEF0147: cid001273 + DEF0148: cid001274 + DEF0149: cid001275 + DEF0150: cid001276 + DEF0151: cid001277 + DEF0152: cid001278 + DEF0153: cid001279 + DEF0154: cid001280 + DEF0155: cid001281 + DEF0156: cid001282 + DEF0157: cid001283 + DEF0158: cid001284 + DEF0159: cid001285 + DEF0160: cid001286 + DEF0161: cid001287 + DEF0162: cid001288 + DEF0163: cid001289 + DEF0164: cid001290 + DEF0165: cid001291 + DEF0166: cid001292 + DEF0167: cid001293 + DEF0168: cid001294 + DEF0169: cid001295 + DEF0170: cid001296 + DEF0171: cid001297 + DEF0172: cid001298 + DEF0173: cid001299 + DEF0174: cid001300 + DEF0175: cid001301 + DEF0176: cid001302 + DEF0177: cid001303 + DEF0178: cid001304 + DEF0179: cid001305 + DEF0180: cid001306 + DEF0181: cid001307 + DEF0182: cid001308 + DEF0183: cid001309 + DEF0184: cid001310 + DEF0185: cid001311 + DEF0186: cid001312 + DEF0187: cid001313 + DEF0188: cid001314 + DEF0189: cid001315 + DEF0190: cid001316 + DEF0191: cid001317 + DEF0192: cid001318 + DEF0193: cid001319 + DEF0194: cid001320 + DEF0195: cid001321 + DEF0196: cid001322 + DEF0197: cid001323 + DEF0198: cid001324 + DEF0199: cid001325 + DEF0200: cid001326 + DEF0201: cid001327 + DEF0202: cid001432 + DEF0203: cid001433 + DEF0204: cid001434 + DEF0205: cid001435 + DEF0206: cid001436 + DEF0207: cid001437 + DEF0208: cid001438 + DEF0209: cid001439 + DEF0210: cid001440 + DEF0211: cid001441 + DEF0212: cid001442 + DEF0213: cid001443 + DEF0214: cid001444 + DEF0215: cid001445 + DEF0216: cid001446 + DEF0217: cid001447 + DEF0218: cid001448 + DEF0219: cid001449 + DEF0220: cid001450 + DEF0221: cid001451 + DEF0222: cid001452 + DEF0223: cid001453 + DEF0224: cid001454 + DEF0225: cid001455 + DEF0226: cid001456 + DEF0227: cid001457 + DEF0228: cid001458 + DEF0229: cid001459 + DEF0230: cid001460 + DEF0231: cid001461 + DEF0232: cid001462 + DEF0233: cid001463 + DEF0234: cid001464 + DEF0235: cid001465 + DEF0236: cid001466 + DEF0237: cid001467 + DEF0238: cid001468 + DEF0239: cid001469 + DEF0240: cid001470 + DEF0241: cid001471 + DEF0242: cid001472 + DEF0243: cid001473 + DEF0244: cid001474 + DEF0245: cid001475 + DEF0246: cid001476 + DEF0247: cid001477 + DEF0248: cid001478 + DEF0249: cid001479 + DEF0250: cid001480 + DEF0251: cid001481 + DEF0252: cid001482 + DEF0253: cid001483 + DEF0254: cid001484 + DEF0255: cid001485 + DEF0256: cid001486 + DEF0257: cid001487 + DEF0258: cid001488 + DEF0259: cid001489 + DEF0260: cid001490 + DEF0261: cid001491 + DEF0262: cid001492 + DEF0263: cid001493 + DEF0264: cid001494 + DEF0265: cid001495 + DEF0266: cid001496 + DEF0267: cid001497 + DEF0268: cid001498 + DEF0269: cid001499 + DEF0270: cid001500 + DEF0271: cid001501 + DEF0272: cid001502 + DEF0273: cid001503 + DEF0274: cid001504 + DEF0275: cid001505 + DEF0276: cid001506 + DEF0277: cid001507 + DEF0278: cid001508 + DEF0279: cid001509 + DEF0280: cid001510 + DEF0281: cid001511 + DEF0282: cid001512 + DEF0283: cid001513 + DEF0284: cid001514 + DEF0285: cid001515 + DEF0286: cid001516 + DEF0287: cid001517 + DEF0288: cid001518 + DEF0289: cid001519 + DEF0290: cid001520 + DEF0291: cid001521 + DEF0292: cid001522 + DEF0293: cid001523 + DEF0294: cid001524 + DEF0295: cid001525 + DEF0296: cid001526 + DEF0297: cid001527 + DEF0298: cid001528 + DEF0299: cid001529 + DEF0300: cid001530 + DEF0301: cid001531 + DEF0302: cid001532 + DEF0303: cid001533 + DEF0304: cid001534 + DEF0305: cid001535 + DEF0306: cid001536 + DEF0307: cid001537 + DEF0308: cid001538 + DEF0309: cid001539 + DEF0310: cid001540 + DEF0311: cid001541 + DEF0312: cid001542 + DEF0313: cid001543 + DEF0314: cid001544 + DOS0001: cid000003 + E4_iota4_near_one: cid006053 + E4_iota4_near_one_lower: cid006052 + E6_abs_in_range: cid006054 + E6_iota6_near_one: cid006055 + ERR0001: cid000018 + FAL0001: cid000016 + FAX0001: cid005712 + FAX0002: cid005817 + FAX0003: cid005837 + FTH0001: cid000020 + FTH0002: cid005265 + FTH0003: cid005266 + FTH0004: cid005267 + FTH0005: cid005268 + FTH0006: cid005269 + FTH0007: cid005270 + FTH0008: cid005271 + FTH0009: cid005272 + FTH0010: cid005273 + FTH0011: cid005274 + FTH0012: cid005275 + FTH0013: cid005276 + FTH0014: cid005277 + FTH0015: cid005278 + FTH0016: cid005279 + FTH0017: cid005280 + FTH0018: cid005281 + FTH0019: cid005282 + FTH0020: cid005283 + FTH0021: cid005284 + FTH0022: cid005285 + FTH0023: cid005286 + FTH0024: cid005287 + FTH0025: cid005288 + FTH0026: cid005289 + FTH0027: cid005290 + FTH0028: cid005291 + FTH0029: cid005292 + FTH0030: cid005293 + FTH0031: cid005294 + FTH0032: cid005295 + FTH0033: cid005296 + FTH0034: cid005297 + FTH0035: cid005298 + FTH0036: cid005299 + FTH0037: cid005300 + FTH0038: cid005301 + FTH0039: cid005302 + FTH0040: cid005303 + FTH0041: cid005304 + FTH0042: cid005305 + FTH0043: cid005306 + FTH0044: cid000020 + FTH0045: cid005307 + FTH0046: cid005308 + FTH0047: cid005309 + FTH0048: cid005310 + FTH0049: cid005311 + FTH0050: cid005312 + FTH0051: cid005313 + FTH0052: cid005314 + FTH0053: cid005315 + FTH0054: cid005316 + FTH0055: cid005317 + FTH0056: cid005318 + FTH0057: cid005319 + FTH0058: cid005320 + FTH0059: cid005321 + FTH0060: cid005322 + FTH0061: cid005323 + FTH0062: cid005324 + FTH0063: cid005325 + FTH0064: cid005326 + FTH0065: cid005327 + FTH0066: cid005328 + FTH0067: cid005329 + FTH0068: cid005330 + FTH0069: cid005331 + FTH0070: cid005332 + FTH0071: cid005333 + FTH0072: cid005334 + FTH0073: cid005335 + FTH0074: cid005336 + FTH0075: cid005337 + FTH0076: cid005338 + FTH0077: cid005339 + FTH0078: cid005340 + FTH0079: cid005341 + FTH0080: cid005342 + FTH0081: cid005343 + FTH0082: cid005344 + FTH0083: cid005345 + FTH0084: cid005346 + FTH0085: cid005347 + FTH0086: cid005348 + FTH0087: cid005349 + FTH0088: cid005350 + FTH0089: cid005351 + FTH0090: cid005352 + FTH0091: cid005353 + FTH0092: cid005354 + FTH0093: cid005355 + FTH0094: cid005356 + FTH0095: cid005357 + FTH0096: cid005358 + FTH0097: cid005359 + FTH0098: cid005360 + FTH0099: cid005361 + FTH0100: cid005362 + FTH0101: cid005363 + FTH0102: cid005364 + FTH0103: cid005365 + FTH0104: cid005366 + FTH0105: cid005367 + FTH0106: cid005368 + FTH0107: cid005369 + FTH0108: cid005370 + FTH0109: cid005371 + FTH0110: cid005372 + FTH0111: cid005373 + FTH0112: cid005374 + FTH0113: cid005375 + FTH0114: cid005376 + FTH0115: cid005377 + FTH0116: cid005378 + FTH0117: cid005379 + FTH0118: cid005380 + FTH0119: cid005381 + FTH0120: cid005382 + FTH0121: cid005383 + FTH0122: cid005384 + FTH0123: cid005385 + FTH0124: cid005386 + FTH0125: cid005387 + FTH0126: cid005388 + FTH0127: cid005389 + FTH0128: cid005390 + FTH0129: cid005391 + FTH0130: cid005392 + FTH0131: cid005393 + FTH0132: cid005394 + FTH0133: cid005395 + FTH0134: cid005396 + FTH0135: cid005397 + FTH0136: cid005398 + FTH0137: cid005399 + FTH0138: cid005400 + FTH0139: cid005401 + FTH0140: cid005402 + FTH0141: cid005403 + FTH0142: cid005404 + FTH0143: cid005405 + FTH0144: cid005406 + FTH0145: cid005407 + FTH0146: cid005408 + FTH0147: cid005409 + FTH0148: cid005410 + FTH0149: cid005411 + FTH0150: cid005412 + FTH0151: cid005413 + FTH0152: cid005414 + FTH0153: cid005415 + FTH0154: cid005416 + FTH0155: cid005417 + FTH0156: cid005418 + FTH0157: cid005419 + FTH0158: cid005420 + FTH0159: cid005421 + FTH0160: cid005422 + FTH0161: cid005423 + FTH0162: cid005424 + FTH0163: cid005425 + FTH0164: cid005426 + FTH0165: cid005427 + FTH0166: cid005428 + FTH0167: cid005429 + FTH0168: cid005430 + FTH0169: cid005431 + FTH0170: cid005432 + FTH0171: cid005433 + FTH0172: cid005434 + FTH0173: cid005435 + FTH0174: cid005436 + FTH0175: cid005437 + FTH0176: cid005438 + FTH0177: cid005439 + FTH0178: cid005440 + FTH0179: cid005441 + FTH0180: cid005442 + FTH0181: cid005443 + FTH0182: cid005444 + FTH0183: cid005445 + FTH0184: cid005446 + FTH0185: cid005447 + FTH0186: cid005448 + FTH0187: cid005449 + FTH0188: cid005450 + FTH0189: cid005451 + FTH0190: cid005452 + FTH0191: cid005453 + FTH0192: cid005454 + FTH0193: cid005455 + FTH0194: cid005456 + FTH0195: cid005457 + FTH0196: cid005458 + FTH0197: cid005459 + FTH0198: cid005460 + FTH0199: cid005461 + FTH0200: cid005462 + FTH0201: cid005463 + FTH0202: cid005464 + FTH0203: cid005465 + FTH0204: cid005466 + FTH0205: cid005467 + FTH0206: cid005468 + FTH0207: cid005469 + FTH0208: cid005470 + FTH0209: cid005471 + FTH0210: cid005472 + FTH0211: cid005473 + FTH0212: cid005474 + FTH0213: cid005475 + FTH0214: cid005476 + FTH0215: cid005477 + FTH0216: cid005478 + FTH0217: cid005479 + FTH0218: cid005480 + FTH0219: cid005481 + FTH0220: cid005482 + FTH0221: cid005483 + FTH0222: cid005484 + FTH0223: cid005485 + FTH0224: cid005486 + FTH0225: cid005487 + FTH0226: cid005488 + FTH0227: cid005489 + FTH0228: cid005490 + FTH0229: cid005491 + FTH0230: cid005492 + FTH0231: cid005493 + FTH0232: cid005494 + FTH0233: cid005495 + FTH0234: cid005496 + FTH0235: cid005497 + FTH0236: cid005498 + FTH0237: cid005499 + FTH0238: cid005500 + FTH0239: cid005501 + FTH0240: cid005502 + FTH0241: cid005503 + FTH0242: cid005504 + FTH0243: cid005505 + FTH0244: cid005506 + FTH0245: cid005507 + FTH0246: cid005508 + FTH0247: cid005509 + FTH0248: cid005510 + FTH0249: cid005511 + FTH0250: cid005512 + FTH0251: cid005513 + FTH0252: cid005514 + FTH0253: cid005515 + FTH0254: cid005516 + FTH0255: cid005517 + FTH0256: cid005518 + FTH0257: cid005519 + FTH0258: cid005520 + FTH0259: cid005521 + FTH0260: cid005522 + FTH0261: cid005523 + FTH0262: cid005524 + FTH0263: cid005525 + FTH0264: cid005526 + FTH0265: cid005527 + FTH0266: cid005528 + FTH0267: cid005529 + FTH0268: cid005530 + FTH0269: cid005531 + FTH0270: cid005532 + FTH0271: cid005533 + FTH0272: cid005534 + FTH0273: cid005535 + FTH0274: cid005536 + FTH0275: cid005537 + FTH0276: cid005538 + FTH0277: cid005539 + FTH0278: cid005540 + FTH0279: cid005541 + FTH0280: cid005542 + FTH0281: cid005543 + FTH0282: cid005544 + FTH0283: cid005545 + FTH0284: cid005546 + FTH0285: cid005547 + FTH0286: cid005548 + FTH0287: cid005549 + FTH0288: cid005550 + FTH0289: cid005551 + FTH0290: cid005552 + FTH0291: cid005553 + FTH0292: cid005554 + FTH0293: cid005555 + FTH0294: cid005556 + FTH0295: cid005557 + FTH0296: cid005558 + FTH0297: cid005559 + FTH0298: cid005560 + FTH0299: cid005561 + FTH0300: cid005562 + FTH0301: cid005563 + FTH0302: cid005564 + FTH0303: cid005565 + FTH0304: cid005566 + FTH0305: cid005567 + FTH0306: cid005568 + FTH0307: cid005569 + FTH0308: cid005570 + FTH0309: cid005571 + FTH0310: cid005572 + FTH0311: cid005573 + FTH0312: cid005574 + FTH0313: cid005575 + FTH0314: cid005576 + FTH0315: cid005577 + FTH0316: cid005578 + FTH0317: cid005579 + FTH0318: cid005580 + FTH0319: cid005581 + FTH0320: cid005582 + FTH0321: cid005583 + FTH0322: cid005584 + FTH0323: cid005585 + FTH0324: cid005586 + FTH0325: cid005587 + FTH0326: cid005588 + FTH0327: cid005589 + FTH0328: cid005590 + FTH0329: cid005591 + FTH0330: cid005592 + FTH0331: cid005593 + FTH0332: cid005594 + FTH0333: cid005595 + FTH0334: cid005596 + FTH0335: cid005597 + FTH0336: cid005598 + FTH0337: cid005599 + FTH0338: cid005600 + FTH0339: cid005601 + FTH0340: cid005602 + FTH0341: cid005603 + FTH0342: cid005604 + FTH0343: cid005605 + FTH0344: cid005606 + FTH0345: cid005607 + FTH0346: cid005608 + FTH0347: cid005609 + FTH0348: cid005610 + FTH0349: cid005611 + FTH0350: cid005612 + FTH0351: cid005613 + FTH0352: cid005614 + FTH0353: cid005615 + FTH0354: cid005616 + FTH0355: cid005617 + FTH0356: cid005618 + FTH0357: cid005619 + FTH0358: cid005620 + FTH0359: cid005621 + FTH0360: cid005622 + FTH0361: cid005623 + FTH0362: cid005624 + FTH0363: cid005625 + FTH0364: cid005626 + FTH0365: cid005627 + FTH0366: cid005628 + FTH0367: cid005629 + FTH0368: cid005630 + FTH0369: cid005631 + FTH0370: cid005632 + FTH0371: cid005633 + FTH0372: cid005634 + FTH0373: cid005635 + FTH0374: cid005636 + FTH0375: cid005637 + FTH0376: cid005638 + FTH0377: cid005639 + FTH0378: cid005640 + FTH0379: cid005641 + FTH0380: cid005642 + FTH0381: cid005643 + FTH0382: cid005644 + FTH0383: cid005645 + FTH0384: cid005646 + FTH0385: cid005647 + FTH0386: cid005648 + FTH0387: cid005649 + FTH0388: cid005650 + FTH0389: cid005651 + FTH0390: cid005652 + FTH0391: cid005653 + FTH0392: cid005654 + FTH0393: cid005655 + FTH0394: cid005656 + FTH0395: cid005657 + FTH0396: cid005658 + FTH0397: cid005659 + FTH0398: cid005660 + FTH0399: cid005661 + FTH0400: cid005662 + FTH0401: cid005663 + FTH0402: cid005664 + FTH0403: cid005665 + FTH0404: cid005666 + FTH0405: cid005667 + FTH0406: cid005668 + FTH0407: cid005669 + FTH0408: cid005670 + FTH0409: cid005671 + FTH0410: cid005672 + FTH0411: cid005673 + FTH0412: cid005674 + FTH0413: cid005675 + FTH0414: cid005676 + FTH0415: cid005677 + FTH0416: cid005678 + FTH0417: cid005679 + FTH0418: cid005680 + FTH0419: cid005681 + FTH0420: cid005682 + FTH0421: cid005683 + FTH0422: cid005684 + FTH0423: cid005685 + FTH0424: cid005686 + FTH0425: cid005687 + FTH0426: cid005688 + FTH0427: cid005689 + FTH0428: cid005690 + FTH0429: cid005691 + FTH0430: cid005692 + FTH0431: cid005693 + FTH0432: cid005694 + FTH0433: cid005695 + FTH0434: cid005696 + FTH0435: cid005697 + FTH0436: cid005698 + FTH0437: cid005699 + FTH0438: cid005700 + FTH0439: cid005701 + FTH0440: cid005702 + FTH0441: cid005703 + FTH0442: cid005704 + FTH0443: cid005705 + FTH0444: cid005706 + FTH0445: cid005707 + FTH0446: cid005708 + FTH0447: cid005709 + FTH0448: cid005710 + FTH0449: cid005711 + FTH0450: cid005713 + FTH0451: cid005714 + FTH0452: cid005715 + FTH0453: cid005716 + FTH0454: cid005717 + FTH0455: cid005718 + FTH0456: cid005719 + FTH0457: cid005720 + FTH0458: cid005721 + FTH0459: cid005722 + FTH0460: cid005723 + FTH0461: cid005724 + FTH0462: cid005725 + FTH0463: cid005726 + FTH0464: cid005727 + FTH0465: cid005728 + FTH0466: cid005729 + FTH0467: cid005730 + FTH0468: cid005731 + FTH0469: cid005732 + FTH0470: cid005733 + FTH0471: cid005734 + FTH0472: cid005735 + FTH0473: cid005736 + FTH0474: cid005737 + FTH0475: cid005738 + FTH0476: cid005739 + FTH0477: cid005740 + FTH0478: cid005741 + FTH0479: cid005742 + FTH0480: cid005743 + FTH0481: cid005744 + FTH0482: cid005745 + FTH0483: cid005746 + FTH0484: cid005747 + FTH0485: cid005748 + FTH0486: cid005749 + FTH0487: cid005750 + FTH0488: cid005751 + FTH0489: cid005752 + FTH0490: cid005753 + FTH0491: cid005754 + FTH0492: cid005755 + FTH0493: cid005756 + FTH0494: cid005757 + FTH0495: cid005758 + FTH0496: cid005759 + FTH0497: cid005760 + FTH0498: cid005761 + FTH0499: cid005762 + FTH0500: cid005763 + FTH0501: cid005764 + FTH0502: cid005765 + FTH0503: cid005766 + FTH0504: cid005767 + FTH0505: cid005768 + FTH0506: cid005769 + FTH0507: cid005770 + FTH0508: cid005771 + FTH0509: cid005772 + FTH0510: cid005773 + FTH0511: cid005774 + FTH0512: cid005775 + FTH0513: cid005776 + FTH0514: cid005777 + FTH0515: cid005778 + FTH0516: cid005779 + FTH0517: cid005780 + FTH0518: cid005781 + FTH0519: cid005782 + FTH0520: cid005783 + FTH0521: cid005784 + FTH0522: cid005785 + FTH0523: cid005786 + FTH0524: cid005787 + FTH0525: cid005788 + FTH0526: cid005789 + FTH0527: cid005790 + FTH0528: cid005791 + FTH0529: cid005792 + FTH0530: cid005793 + FTH0531: cid005794 + FTH0532: cid005795 + FTH0533: cid005796 + FTH0534: cid005797 + FTH0535: cid005798 + FTH0536: cid005799 + FTH0537: cid005800 + FTH0538: cid005801 + FTH0539: cid005802 + FTH0540: cid005803 + FTH0541: cid005804 + FTH0542: cid005805 + FTH0543: cid005806 + FTH0544: cid005807 + FTH0545: cid005808 + FTH0546: cid005809 + FTH0547: cid005810 + FTH0548: cid005811 + FTH0549: cid005812 + FTH0550: cid005813 + FTH0551: cid005814 + FTH0552: cid005815 + FTH0553: cid005816 + FTH0554: cid005818 + FTH0555: cid005819 + FTH0556: cid005820 + FTH0557: cid005821 + FTH0558: cid005822 + FTH0559: cid005823 + FTH0560: cid005824 + FTH0561: cid005825 + FTH0562: cid005826 + FTH0563: cid005827 + FTH0564: cid005828 + FTH0565: cid005829 + FTH0566: cid005830 + FTH0567: cid005831 + FTH0568: cid005832 + FTH0569: cid005833 + FTH0570: cid005834 + FTH0571: cid005835 + FTH0572: cid005836 + FTH0573: cid005838 + FTH0574: cid005839 + FTH0575: cid005840 + FTH0576: cid005841 + FTH0577: cid005842 + FTH0578: cid005843 + FTH0579: cid005844 + FTH0580: cid005845 + FTH0581: cid005846 + FTH0582: cid005847 + FTH0583: cid005848 + FTH0584: cid005849 + FTH0585: cid005850 + FTH0586: cid005851 + FTH0587: cid005852 + FTH0588: cid005853 + FTH0589: cid005854 + FTH0590: cid005855 + FTH0591: cid005856 + FTH0592: cid005857 + FTH0593: cid005858 + FTH0594: cid005859 + FTH0595: cid005860 + FTH0596: cid005861 + FTH0597: cid005862 + FTH0598: cid005863 + FTH0599: cid005864 + FTH0600: cid005865 + FTH0601: cid005866 + FTH0602: cid005867 + FTH0603: cid005868 + FTH0604: cid005869 + FTH0605: cid005870 + FTH0606: cid005871 + FTH0607: cid005872 + FTH0608: cid005873 + FTH0609: cid005874 + FTH0610: cid005875 + FTH0611: cid005876 + FTH0612: cid005877 + FTH0613: cid005878 + FTH0614: cid005879 + FTH0615: cid005880 + FTH0616: cid005881 + FTH0617: cid005882 + FTH0618: cid005883 + FTH0619: cid005884 + FTH0620: cid005885 + FTH0621: cid005886 + FTH0622: cid005887 + FTH0623: cid005888 + FTH0624: cid005889 + FTH0625: cid005890 + FTH0626: cid005891 + FTH0627: cid005892 + FTH0628: cid005893 + FTH0629: cid005894 + FTH0630: cid005895 + FTH0631: cid005896 + FTH0632: cid005897 + FTH0633: cid005898 + FTH0634: cid005899 + FTH0635: cid005900 + FTH0636: cid005901 + FTH0637: cid005902 + FTH0638: cid005903 + FTH0639: cid005904 + FTH0640: cid005905 + FTH0641: cid005906 + FTH0642: cid005907 + FTH0643: cid005908 + FTH0644: cid005909 + FTH0645: cid005910 + FTH0646: cid005911 + FTH0647: cid005912 + FTH0648: cid005913 + FTH0649: cid005914 + FTH0650: cid005915 + FTH0651: cid005916 + FTH0652: cid005917 + FTH0653: cid005918 + FTH0654: cid005919 + FTH0655: cid005920 + FTH0656: cid005921 + FTH0657: cid005922 + FTH0658: cid005923 + FTH0659: cid005924 + FTH0660: cid005925 + FTH0661: cid005926 + FTH0662: cid005927 + FTH0663: cid005928 + FTH0664: cid005929 + FTH0665: cid005930 + FTH0666: cid005931 + FTH0667: cid005932 + FTH0668: cid005933 + FTH0669: cid005934 + FTH0670: cid005935 + FTH0671: cid005936 + FTH0672: cid005937 + FTH0673: cid005938 + FTH0674: cid005939 + FTH0675: cid005940 + FTH0676: cid005941 + FTH0677: cid005942 + FTH0678: cid005943 + FTH0679: cid005944 + FTH0680: cid005945 + FTH0681: cid005946 + FTH0682: cid005947 + FTH0683: cid005948 + FTH0684: cid005949 + FTH0685: cid005950 + FTH0686: cid005951 + FTH0687: cid005952 + FTH0688: cid005953 + FTH0689: cid005954 + FTH0690: cid005955 + FTH0691: cid005956 + FTH0692: cid005957 + FTH0693: cid005958 + FTH0694: cid005959 + FTH0695: cid005960 + FTH0696: cid005961 + FTH0697: cid005962 + FTH0698: cid005963 + FTH0699: cid005964 + FTH0700: cid005965 + FTH0701: cid005966 + FTH0702: cid005967 + FTH0703: cid005968 + FTH0704: cid005969 + FTH0705: cid005970 + FTH0706: cid005971 + FTH0707: cid005972 + FTH0708: cid005973 + FTH0709: cid005974 + FTH0710: cid005975 + FTH0711: cid005976 + FTH0712: cid005977 + FTH0713: cid005978 + FTH0714: cid005979 + FTH0715: cid005980 + FTH0716: cid005981 + FTH0717: cid005982 + FTH0718: cid005983 + FTH0719: cid005984 + FTH0720: cid005985 + FTH0721: cid005986 + FTH0722: cid005987 + FTH0723: cid005988 + FTH0724: cid005989 + FTH0725: cid005990 + FTH0726: cid005991 + FTH0727: cid005992 + FTH0728: cid005993 + FTH0729: cid005994 + FTH0730: cid005995 + FTH0731: cid005996 + FTH0732: cid005997 + FTH0733: cid005998 + FTH0734: cid005999 + FTH0735: cid006000 + FTH0736: cid006001 + FTH0737: cid006002 + FTH0738: cid006003 + FTH0739: cid006004 + FTH0740: cid006005 + FTH0741: cid006006 + FTH0742: cid006007 + FTH0743: cid006008 + FTH0744: cid006009 + FTH0745: cid006010 + FTH0746: cid006011 + FTH0747: cid006012 + FTH0748: cid006013 + FTH0749: cid006014 + FTH0750: cid006015 + FTH0751: cid006016 + FTH0752: cid006017 + FTH0753: cid006018 + FTH0754: cid006019 + FTH0755: cid006020 + FTH0756: cid006021 + FTH0757: cid006022 + FTH0758: cid006023 + FTH0759: cid006024 + FTH0760: cid006025 + FTH0761: cid006026 + FTH0762: cid006027 + FTH0763: cid006028 + FTH0764: cid006029 + FTH0765: cid006030 + FTH0766: cid006031 + FTH0767: cid006032 + FTH0768: cid006033 + FTH0769: cid006034 + FTH0770: cid006035 + FTH0771: cid006036 + FTH0772: cid006037 + FTH0773: cid006038 + FTH0774: cid006039 + FTH0775: cid006040 + FTH0776: cid006041 + FTH0777: cid006042 + FTH0778: cid006043 + FTH0779: cid006044 + FTH0780: cid006045 + FTH0781: cid006046 + FTH0782: cid006047 + FTH0783: cid006048 + FTH0784: cid006049 + FTH0785: cid006050 + FTH0786: cid006051 + FTH0787: cid006052 + FTH0788: cid006053 + FTH0789: cid006054 + FTH0790: cid006055 + FTH0791: cid006056 + FTH0792: cid006057 + FTH0793: cid006058 + FTH0794: cid006059 + FTH0795: cid006060 + FTH0796: cid006061 + FTH0797: cid006062 + FTH0798: cid006063 + FTH0799: cid006064 + FTH0800: cid006065 + FTH0801: cid006066 + FTH0802: cid006067 + FTH0803: cid006068 + FTH0804: cid006069 + FTH0805: cid006070 + FTH0806: cid006071 + FTH0807: cid006072 + FTH0808: cid006073 + FTH0809: cid006074 + FTH0810: cid006075 + FTH0811: cid006076 + FTH0812: cid006077 + FTH0813: cid006078 + FTH0814: cid006079 + FTH0815: cid006080 + FTH0816: cid006081 + FTH0817: cid006082 + FTH0818: cid006083 + FTH0819: cid006084 + FTH0820: cid006085 + FTH0821: cid006086 + FTH0822: cid006087 + FTH0823: cid006088 + FTH0824: cid006089 + FTH0825: cid006090 + FTH0826: cid006091 + FTH0827: cid006092 + FTH0828: cid006093 + FTH0829: cid006094 + FTH0830: cid006095 + FTH0831: cid006096 + FTH0832: cid006097 + FTH0833: cid006098 + FTH0834: cid006099 + FTH0835: cid006100 + FTH0836: cid006101 + FTH0837: cid006102 + FTH0838: cid006103 + FTH0839: cid006104 + H3: cid000010 + I.C01: cid001000 + I.C02: cid001001 + I.C03: cid001002 + I.C04: cid001003 + I.D01: cid001004 + I.D02: cid001005 + I.D03: cid001006 + I.D04: cid001007 + I.D05: cid001008 + I.D06: cid001009 + I.D07: cid001010 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cid001336 + LEM0019: cid001337 + LEM0020: cid001338 + LEM0021: cid001339 + LEM0022: cid001340 + LEM0023: cid001341 + LEM0028: cid000008 + MOD0001: cid000019 + MOD0002: cid000130 + MOD0003: cid005000 + MOD0004: cid005001 + MOD0005: cid005002 + MOD0006: cid005003 + MOD0007: cid005004 + MOD0008: cid005005 + MOD0009: cid005006 + MOD0010: cid005007 + MOD0011: cid005008 + MOD0012: cid005009 + MOD0013: cid005010 + MOD0014: cid005011 + MOD0015: cid005012 + MOD0016: cid005013 + MOD0017: cid005014 + MOD0018: cid005015 + MOD0019: cid005016 + MOD0020: cid005017 + MOD0021: cid005018 + MOD0022: cid005019 + MOD0023: cid005020 + MOD0024: cid005021 + MOD0025: cid005022 + MOD0026: cid005023 + MOD0027: cid005024 + MOD0028: cid005025 + MOD0029: cid005026 + MOD0030: cid005027 + MOD0031: cid005028 + MOD0032: cid005029 + MOD0033: cid005030 + MOD0034: cid005031 + MOD0035: cid005032 + MOD0036: cid005033 + MOD0037: cid005034 + MOD0038: cid005035 + MOD0039: cid005036 + MOD0040: cid005037 + MOD0041: cid005038 + MOD0042: cid005039 + MOD0043: cid005040 + MOD0044: cid005041 + MOD0045: cid005042 + MOD0046: cid005043 + MOD0047: cid005044 + MOD0048: cid005045 + MOD0049: cid005046 + MOD0050: cid005047 + MOD0051: cid005048 + MOD0052: cid005049 + MOD0053: cid005050 + MOD0054: cid005051 + MOD0055: cid005052 + MOD0056: cid005053 + MOD0057: cid005054 + MOD0058: cid005055 + MOD0059: cid005056 + MOD0060: cid005057 + MOD0061: cid005058 + MOD0062: cid005059 + MOD0063: cid005060 + MOD0064: cid005061 + MOD0065: cid005062 + MOD0066: cid005063 + MOD0067: cid005064 + MOD0068: cid005065 + MOD0069: cid005066 + MOD0070: cid005067 + MOD0071: cid005068 + MOD0072: cid005069 + MOD0073: cid005070 + MOD0074: cid005071 + MOD0075: cid005072 + MOD0076: cid005073 + MOD0077: cid005074 + MOD0078: cid005075 + MOD0079: cid005076 + MOD0080: cid005077 + MOD0081: cid005078 + MOD0082: cid005079 + MOD0083: cid005080 + MOD0084: cid005081 + MOD0085: cid005082 + MOD0086: cid005083 + MOD0087: cid005084 + MOD0088: cid005085 + MOD0089: cid005086 + MOD0090: cid005087 + MOD0091: cid005088 + MOD0092: cid005089 + MOD0093: cid005090 + MOD0094: cid005091 + MOD0095: cid005092 + MOD0096: cid005093 + MOD0097: cid005094 + MOD0098: cid005095 + MOD0099: cid005096 + MOD0100: cid005097 + MOD0101: cid005098 + MOD0102: cid005099 + MOD0103: cid005100 + MOD0104: cid005101 + MOD0105: cid005102 + MOD0106: cid005103 + MOD0107: cid005104 + MOD0108: cid005105 + MOD0109: cid005106 + MOD0110: cid005107 + MOD0111: cid005108 + MOD0112: cid005109 + MOD0113: cid005110 + MOD0114: cid005111 + MOD0115: cid005112 + MOD0116: cid005113 + MOD0117: cid005114 + MOD0118: cid005115 + MOD0119: cid005116 + MOD0120: cid005117 + MOD0121: cid005118 + MOD0122: cid005119 + MOD0123: cid005120 + MOD0124: cid005121 + MOD0125: cid005122 + MOD0126: cid005123 + MOD0127: cid005124 + MOD0128: cid005125 + MOD0129: cid005126 + MOD0130: cid005127 + MOD0131: cid005128 + MOD0132: cid005129 + MOD0133: cid005130 + MOD0134: cid005131 + MOD0135: cid005132 + MOD0136: cid005133 + MOD0137: cid005134 + MOD0138: cid005135 + MOD0139: cid005136 + MOD0140: cid005137 + MOD0141: cid005138 + MOD0142: cid005139 + MOD0143: cid005140 + MOD0144: cid005141 + MOD0145: cid005142 + MOD0146: cid005143 + MOD0147: cid005144 + MOD0148: cid005145 + MOD0149: cid005146 + MOD0150: cid005147 + MOD0151: cid005148 + MOD0152: cid005149 + MOD0153: cid005150 + MOD0154: cid005151 + MOD0155: cid005152 + MOD0156: cid005153 + MOD0157: cid005154 + MOD0158: cid005155 + MOD0159: cid005156 + MOD0160: cid005157 + MOD0161: cid005158 + MOD0162: cid005159 + MOD0163: cid005160 + MOD0164: cid005161 + MOD0165: cid005162 + MOD0166: cid005163 + MOD0167: cid005164 + MOD0168: cid005165 + MOD0169: cid005166 + MOD0170: cid005167 + MOD0171: cid005168 + MOD0172: cid005169 + MOD0173: cid005170 + MOD0174: cid005171 + MOD0175: cid005172 + MOD0176: cid005173 + MOD0177: cid005174 + MOD0178: cid005175 + MOD0179: cid005176 + MOD0180: cid005177 + MOD0181: cid005178 + MOD0182: cid005179 + MOD0183: cid005180 + MOD0184: cid005181 + MOD0185: cid005182 + MOD0186: cid005183 + MOD0187: cid005184 + MOD0188: cid005185 + MOD0189: cid005186 + MOD0190: cid005187 + MOD0191: cid005188 + MOD0192: cid005189 + MOD0193: cid005190 + MOD0194: cid005191 + MOD0195: cid005192 + MOD0196: cid005193 + MOD0197: cid005194 + MOD0198: cid005195 + MOD0199: cid005196 + MOD0200: cid005197 + MOD0201: cid005198 + MOD0202: cid005199 + MOD0203: cid005200 + MOD0204: cid005201 + MOD0205: cid005202 + MOD0206: cid005203 + MOD0207: cid005204 + MOD0208: cid005205 + MOD0209: cid005206 + MOD0210: cid005207 + MOD0211: cid005208 + MOD0212: cid005209 + MOD0213: cid005210 + MOD0214: cid005211 + MOD0215: cid005212 + MOD0216: cid005213 + MOD0217: cid005214 + MOD0218: cid005215 + MOD0219: cid005216 + MOD0220: cid005217 + MOD0221: cid005218 + MOD0222: cid005219 + MOD0223: cid005220 + MOD0224: cid005221 + MOD0225: cid005222 + MOD0226: cid005223 + MOD0227: cid005224 + MOD0228: cid005225 + MOD0229: cid005226 + MOD0230: cid005227 + MOD0231: cid005228 + MOD0232: cid005229 + MOD0233: cid005230 + MOD0234: cid005231 + MOD0235: cid005232 + MOD0236: cid005233 + MOD0237: cid005234 + MOD0238: cid005235 + MOD0239: cid005236 + MOD0240: cid005237 + MOD0241: cid005238 + MOD0242: cid005239 + MOD0243: cid005240 + MOD0244: cid005241 + MOD0245: cid005242 + MOD0246: cid005243 + MOD0247: cid005244 + MOD0248: cid005245 + MOD0249: cid005246 + MOD0250: cid005247 + MOD0251: cid005248 + MOD0252: cid005249 + MOD0253: cid005250 + MOD0254: cid005251 + MOD0255: cid005252 + MOD0256: cid005253 + MOD0257: cid005254 + MOD0258: cid005255 + MOD0259: cid005256 + MOD0260: cid005257 + MOD0261: cid005258 + MOD0262: cid005259 + MOD0263: cid005260 + MOD0264: cid005261 + MOD0265: cid005262 + MOD0266: cid005263 + MOD0267: cid005264 + NOT0001: cid000005 + OPN0001: cid000017 + PAP0001: cid000004 + PRD0001: cid000015 + PRF0001: cid000022 + PRF0002: cid006105 + PRF0003: cid006106 + PRF0004: cid006107 + PRF0005: cid006108 + PRF0006: cid006109 + PRF0042: cid000022 + PRP0001: cid000009 + PRP0002: cid001124 + PRP0003: cid001125 + PRP0004: cid001126 + PRP0005: cid001127 + PRP0006: cid001128 + PRP0007: cid001129 + PRP0008: cid001130 + PRP0009: cid001131 + PRP0010: cid001132 + PRP0011: cid001133 + PRP0012: cid001134 + PRP0013: cid001135 + PRP0014: cid001136 + PRP0015: cid001137 + PRP0016: cid001138 + PRP0017: cid001139 + PRP0018: cid001140 + PRP0019: cid001141 + PRP0020: cid001142 + PRP0021: cid001143 + PRP0022: cid001144 + PRP0023: cid001145 + PRP0024: cid001146 + PRP0025: cid001147 + PRP0026: cid001148 + PRP0027: cid001149 + PRP0028: cid001150 + PRP0029: cid001151 + PRP0030: cid001152 + PRP0031: cid001153 + PRP0032: cid001154 + PRP0033: cid001155 + PRP0034: cid001156 + PRP0035: cid001157 + PRP0036: cid001158 + PRP0037: cid001159 + PRP0038: cid001160 + PRP0039: cid001161 + PRP0040: cid001162 + PRP0041: cid001163 + PRP0042: cid000009 + PRP0043: cid001164 + PRP0044: cid001165 + PRP0045: cid001166 + PRP0046: cid001167 + PRP0047: cid001342 + PRP0048: cid001343 + PRP0049: cid001344 + PRP0050: cid001345 + PRP0051: cid001346 + PRP0052: cid001347 + PRP0053: cid001348 + PRP0054: cid001349 + PRP0055: cid001350 + PRP0056: cid001351 + PRP0057: cid001352 + PRP0058: cid001353 + PRP0059: cid001354 + PRP0060: cid001355 + PRP0061: cid001356 + PRP0062: cid001357 + PRP0063: cid001358 + PRP0064: cid001359 + PRP0065: cid001360 + PRP0066: cid001361 + PRP0067: cid001362 + PRP0068: cid001545 + PRP0069: cid001546 + PRP0070: cid001547 + PRP0071: cid001548 + PRP0072: cid001549 + PRP0073: cid001550 + PRP0074: cid001551 + PRP0075: cid001552 + PRP0076: cid001553 + PRP0077: cid001554 + PRP0078: cid001555 + PRP0079: cid001556 + PRP0080: cid001557 + PRP0081: cid001558 + PRP0082: cid001559 + PRP0083: cid001560 + PRP0084: cid001561 + PRP0085: cid001562 + PRP0086: cid001563 + PRP0087: cid001564 + PRP0088: cid001565 + PRP0089: cid001566 + PRP0090: cid001567 + PRP0091: cid001568 + PRP0092: cid001569 + PRP0093: cid001570 + PRP0094: cid001571 + PRP0095: cid001572 + PRP0096: cid001573 + PRP0097: cid001574 + PRP0098: cid001575 + PRP0099: cid001576 + PRP0100: cid001577 + PRP0101: cid001578 + PRP0102: cid001579 + PRP0103: cid001580 + PRP0104: cid001581 + PRP0105: cid001582 + PRP0106: cid001583 + PRP0107: cid001584 + PRP0108: cid001585 + PRP0109: cid001586 + PRP0110: cid001587 + PRP0111: cid001588 + PRP0112: cid001589 + PRP0113: cid001590 + PRP0114: cid001591 + PRP0115: cid001592 + RDO0001: cid000013 + REL0001: cid000006 + REL0002: cid000129 + RN-001: cid000005 + RSL0001: cid000021 + RSL0007: cid000021 + S001: cid000029 + S002: cid000030 + S003: cid000031 + S004: cid000032 + S005: cid000033 + S006: cid000034 + S007: cid000035 + S008: cid000036 + S009: cid000037 + S010: cid000038 + S011: cid000039 + S012: cid000040 + S013: cid000041 + S014: cid000042 + S015: cid000043 + S016: cid000044 + S017: cid000045 + S018: cid000046 + S019: cid000047 + S020: cid000048 + S021: cid000049 + S022: cid000050 + S023: cid000051 + S024: cid000052 + S025: cid000053 + S026: cid000054 + S027: cid000055 + S028: cid000056 + S029: cid000057 + S030: cid000058 + S031: cid000059 + S032: cid000060 + S033: cid000061 + S034: cid000062 + S035: cid000063 + S036: cid000064 + S037: cid000065 + S038: cid000066 + S039: cid000067 + S040: cid000068 + S041: cid000069 + S042: cid000070 + S043: cid000071 + S044: cid000072 + S045: cid000073 + S046: cid000074 + S047: cid000075 + S048: cid000076 + S049: cid000077 + S050: cid000078 + S051: cid000079 + S052: cid000080 + S053: cid000081 + S054: cid000082 + S055: cid000083 + S056: cid000084 + S057: cid000085 + S058: cid000086 + S059: cid000087 + S060: cid000088 + S061: cid000089 + S062: cid000090 + S063: cid000091 + S064: cid000092 + S065: cid000093 + S066: cid000094 + S067: cid000095 + S068: cid000096 + S069: cid000097 + S070: cid000098 + S071: cid000099 + S072: cid000100 + S073: cid000101 + S074: cid000102 + S075: cid000103 + S076: cid000104 + S077: cid000105 + S078: cid000106 + S079: cid000107 + S080: cid000108 + S081: cid000109 + S082: cid000110 + S083: cid000111 + S084: cid000112 + S085: cid000113 + S086: cid000114 + S087: cid000115 + S088: cid000116 + S089: cid000117 + S090: cid000118 + S091: cid000119 + S092: cid000120 + S093: cid000121 + S094: cid000122 + S095: cid000123 + S096: cid000124 + S097: cid000125 + S098: cid000126 + S099: cid000127 + S100: cid000128 + THM0001: cid000010 + THM0002: cid001168 + THM0003: cid001169 + THM0004: cid001170 + THM0005: cid001171 + THM0006: cid001172 + THM0007: cid001173 + THM0008: cid001174 + THM0009: cid001175 + THM0010: cid001176 + THM0011: cid001177 + THM0012: cid001178 + THM0013: cid001179 + THM0014: cid001180 + THM0015: cid001181 + THM0016: cid001182 + THM0017: cid001183 + THM0018: cid001184 + THM0019: cid001185 + THM0020: cid001186 + THM0021: cid001187 + THM0022: cid001188 + THM0023: cid001189 + THM0024: cid001190 + THM0025: cid001191 + THM0026: cid001192 + THM0027: cid001193 + THM0028: cid001194 + THM0029: cid001195 + THM0030: cid001196 + THM0031: cid000010 + THM0032: cid001197 + THM0033: cid001198 + THM0034: cid001199 + THM0035: cid001200 + THM0036: cid001201 + THM0037: cid001202 + THM0038: cid001203 + THM0039: cid001204 + THM0040: cid001205 + THM0041: cid001206 + THM0042: cid001207 + THM0043: cid001208 + THM0044: cid001209 + THM0045: cid001210 + THM0046: cid001211 + THM0047: cid001212 + THM0048: cid001213 + THM0049: cid001214 + THM0050: cid001215 + THM0051: cid001216 + THM0052: cid001217 + THM0053: cid001218 + THM0054: cid001219 + THM0055: cid001220 + THM0056: cid001221 + THM0057: cid001222 + THM0058: cid001223 + THM0059: cid001224 + THM0060: cid001225 + THM0061: cid001226 + THM0062: cid001227 + THM0063: cid001228 + THM0064: cid001229 + THM0065: cid001230 + THM0066: cid001231 + THM0067: cid001363 + THM0068: cid001364 + THM0069: cid001365 + THM0070: cid001366 + THM0071: cid001367 + THM0072: cid001368 + THM0073: cid001369 + THM0074: cid001370 + THM0075: cid001371 + THM0076: cid001372 + THM0077: cid001373 + THM0078: cid001374 + THM0079: cid001375 + THM0080: cid001376 + THM0081: cid001377 + THM0082: cid001378 + THM0083: cid001379 + THM0084: cid001380 + THM0085: cid001381 + THM0086: cid001382 + THM0087: cid001383 + THM0088: cid001384 + THM0089: cid001385 + THM0090: cid001386 + THM0091: cid001387 + THM0092: cid001388 + THM0093: cid001389 + THM0094: cid001390 + THM0095: cid001391 + THM0096: cid001392 + THM0097: cid001393 + THM0098: cid001394 + THM0099: cid001395 + THM0100: cid001396 + THM0101: cid001397 + THM0102: cid001398 + THM0103: cid001399 + THM0104: cid001400 + THM0105: cid001401 + THM0106: cid001402 + THM0107: cid001403 + THM0108: cid001404 + THM0109: cid001405 + THM0110: cid001406 + THM0111: cid001407 + THM0112: cid001408 + THM0113: cid001409 + THM0114: cid001410 + THM0115: cid001411 + THM0116: cid001412 + THM0117: cid001413 + THM0118: cid001414 + THM0119: cid001415 + THM0120: cid001416 + THM0121: cid001417 + THM0122: cid001418 + THM0123: cid001419 + THM0124: cid001420 + THM0125: cid001421 + THM0126: cid001422 + THM0127: cid001423 + THM0128: cid001424 + THM0129: cid001425 + THM0130: cid001426 + THM0131: cid001427 + THM0132: cid001428 + THM0133: cid001429 + THM0134: cid001430 + THM0135: cid001431 + THM0136: cid001593 + THM0137: cid001594 + THM0138: cid001595 + THM0139: cid001596 + THM0140: cid001597 + THM0141: cid001598 + THM0142: cid001599 + THM0143: cid001600 + THM0144: cid001601 + THM0145: cid001602 + THM0146: cid001603 + THM0147: cid001604 + THM0148: cid001605 + THM0149: cid001606 + THM0150: cid001607 + THM0151: cid001608 + THM0152: cid001609 + THM0153: cid001610 + THM0154: cid001611 + THM0155: cid001612 + THM0156: cid001613 + THM0157: cid001614 + THM0158: cid001615 + THM0159: cid001616 + THM0160: cid001617 + THM0161: cid001618 + THM0162: cid001619 + THM0163: cid001620 + THM0164: cid001621 + THM0165: cid001622 + THM0166: cid001623 + THM0167: cid001624 + THM0168: cid001625 + THM0169: cid001626 + THM0170: cid001627 + THM0171: cid001628 + THM0172: cid001629 + THM0173: cid001630 + THM0174: cid001631 + THM0175: cid001632 + THM0176: cid001633 + THM0177: cid001634 + THM0178: cid001635 + THM0179: cid001636 + THM0180: cid001637 + THM0181: cid001638 + THM0182: cid001639 + THM0183: cid001640 + THM0184: cid001641 + THM0185: cid001642 + THM0186: cid001643 + THM0187: cid001644 + THM0188: cid001645 + THM0189: cid001646 + THM0190: cid001647 + THM0191: cid001648 + THM0192: cid001649 + THM0193: cid001650 + THM0194: cid001651 + THM0195: cid001652 + THM0196: cid001653 + THM0197: cid001654 + THM0198: cid001655 + THM0199: cid001656 + THM0200: cid001657 + THM0201: cid001658 + THM0202: cid001659 + THM0203: cid001660 + THM0204: cid001661 + THM0205: cid001662 + THM0206: cid001663 + THM0207: cid001664 + THM0208: cid001665 + THM0209: cid001666 + THM0210: cid001667 + THM0211: cid001668 + THM0212: cid001669 + THM0213: cid001670 + THM0214: cid001671 + THM0215: cid001672 + THM0216: cid001673 + THM0217: cid001674 + THM0218: cid001675 + THM0219: cid001676 + THM0220: cid001677 + THM0221: cid001678 + THM0222: cid001679 + THM0223: cid001680 + THM0224: cid001681 + TauLib.BookI.Addressability.AddressResolution: cid005000 + TauLib.BookI.Addressability.CayleyMetric: cid005001 + TauLib.BookI.Addressability.HingeIntegration: cid005002 + TauLib.BookI.Addressability.OnticUltrametric: cid005003 + TauLib.BookI.Boundary.Bridge.TauIntQuotient: cid005004 + TauLib.BookI.Boundary.Bridge.TauRatQuotient: cid005005 + TauLib.BookI.Boundary.Bridge.TauRealCongruence: cid005006 + TauLib.BookI.Boundary.Bridge.TauRealQuotient: cid005007 + TauLib.BookI.Boundary.Characters: cid005008 + TauLib.BookI.Boundary.Characters::chi_complete: cid005265 + TauLib.BookI.Boundary.Characters::chi_orthogonal: cid005266 + TauLib.BookI.Boundary.ComplexField: cid005009 + TauLib.BookI.Boundary.ComplexField::sc_j_squared: cid005267 + TauLib.BookI.Boundary.ConstructiveReals: cid005010 + TauLib.BookI.Boundary.ConstructiveReals::taureal_archimedean_embedding: cid005268 + TauLib.BookI.Boundary.CouplingIdentityApproximants: cid005011 + TauLib.BookI.Boundary.Cyclotomic: cid005012 + TauLib.BookI.Boundary.Cyclotomic::root_of_unity_one: cid005269 + TauLib.BookI.Boundary.DefectInverseSystem: cid005013 + TauLib.BookI.Boundary.Fourier: cid005014 + TauLib.BookI.Boundary.Fourier::crossing_iff_real: cid005270 + TauLib.BookI.Boundary.Fourier::fourier_sigma_swap: cid005271 + TauLib.BookI.Boundary.Integration: cid005015 + TauLib.BookI.Boundary.Integration::integral_const_one_3: cid005272 + TauLib.BookI.Boundary.Integration::integral_even_2: cid005275 + TauLib.BookI.Boundary.Integration::linearity_2f_3g_stage2: cid005273 + TauLib.BookI.Boundary.Integration::linearity_identity_stage2: cid005274 + TauLib.BookI.Boundary.Iota: cid005016 + TauLib.BookI.Boundary.IotaTauStructural: cid005017 + TauLib.BookI.Boundary.Measure: cid005018 + TauLib.BookI.Boundary.Measure::additivity_bc_3: cid005278 + TauLib.BookI.Boundary.Measure::additivity_even_odd_3: cid005277 + TauLib.BookI.Boundary.Measure::empty_set_compatible_3: cid005280 + TauLib.BookI.Boundary.Measure::empty_set_measure_3: cid005283 + TauLib.BookI.Boundary.Measure::even_set_compatible_3: cid005281 + TauLib.BookI.Boundary.Measure::full_set_compatible_3: cid005279 + TauLib.BookI.Boundary.Measure::full_set_measure_3: cid005282 + TauLib.BookI.Boundary.Measure::primorial_pos_1: cid005276 + TauLib.BookI.Boundary.NumberTower: cid005019 + TauLib.BookI.Boundary.NumericalProjection: cid005020 + TauLib.BookI.Boundary.Quaternions: cid005021 + TauLib.BookI.Boundary.RefinementGrowingTorus: cid005022 + TauLib.BookI.Boundary.Ring: cid005023 + TauLib.BookI.Boundary.Spectral: cid005024 + TauLib.BookI.Boundary.Spectral::spectral_decomposition: cid005284 + TauLib.BookI.Boundary.Spectral::spectral_unique: cid005285 + TauLib.BookI.Boundary.SplitComplex: cid005025 + TauLib.BookI.Boundary.TauRatAbs: cid005026 + TauLib.BookI.Boundary.TauRatField: cid005027 + TauLib.BookI.Boundary.TauRatInv: cid005028 + TauLib.BookI.Boundary.TauRatOrder: cid005029 + TauLib.BookI.Boundary.TauRealAbs: cid005030 + TauLib.BookI.Boundary.TauRealE: cid005031 + TauLib.BookI.Boundary.TauRealInv: cid005032 + TauLib.BookI.Boundary.TauRealIotaTau: cid005033 + TauLib.BookI.Boundary.TauRealMulCongr: cid005034 + TauLib.BookI.Boundary.TauRealOrder: cid005035 + TauLib.BookI.Boundary.TauRealPi: cid005036 + TauLib.BookI.Boundary.TauRealPiPlusE: cid005037 + TauLib.BookI.Boundary.TauRealSum: cid005038 + TauLib.BookI.Boundary.TorusDefectSystem: cid005039 + TauLib.BookI.Boundary.UniversalFixedScalar: cid005040 + TauLib.BookI.Coordinates.ABCD: cid005041 + TauLib.BookI.Coordinates.ChebyshevBias: cid005042 + TauLib.BookI.Coordinates.ChebyshevBias::bias_mod3_50: cid005288 + TauLib.BookI.Coordinates.ChebyshevBias::bias_positive_50: cid005287 + TauLib.BookI.Coordinates.ChebyshevBias::bias_primorial_3: cid005289 + TauLib.BookI.Coordinates.ChebyshevBias::prime_race_4_50: cid005286 + TauLib.BookI.Coordinates.Descent: cid005043 + TauLib.BookI.Coordinates.Hyperfact: cid005044 + TauLib.BookI.Coordinates.HyperfactFTA: cid005045 + TauLib.BookI.Coordinates.HyperfactIsomorphism: cid005046 + TauLib.BookI.Coordinates.HyperfactProp: cid005047 + TauLib.BookI.Coordinates.IteratedPrime: cid005048 + TauLib.BookI.Coordinates.NoTie: cid005049 + TauLib.BookI.Coordinates.NoTie::no_tie: cid005290 + TauLib.BookI.Coordinates.NormalForm: cid005050 + TauLib.BookI.Coordinates.Primes: cid005051 + TauLib.BookI.Coordinates.Primes::prime_product_unique: cid005291 + TauLib.BookI.Denotation.Arithmetic: cid005052 + TauLib.BookI.Denotation.Arithmetic::idx_add_injective: cid005292 + TauLib.BookI.Denotation.Equality: cid005053 + TauLib.BookI.Denotation.GrowthEscape: cid005054 + TauLib.BookI.Denotation.GrowthEscape::growth_escape: cid005293 + TauLib.BookI.Denotation.Order: cid005055 + TauLib.BookI.Denotation.Order::denotational_lt_wf: cid005294 + TauLib.BookI.Denotation.ProgramMonoid: cid005056 + TauLib.BookI.Denotation.ProgramMonoid::compose_assoc: cid005295 + TauLib.BookI.Denotation.ProgramMonoid::rho_count_compose: cid005296 + TauLib.BookI.Denotation.RankTransfer: cid005057 + TauLib.BookI.Denotation.Structural: cid005058 + TauLib.BookI.Denotation.TauIdx: cid005059 + TauLib.BookI.Holomorphy.BoundaryInterior: cid005060 + TauLib.BookI.Holomorphy.BoundaryInterior::interior_from_boundary: cid005297 + TauLib.BookI.Holomorphy.DHolomorphic: cid005061 + TauLib.BookI.Holomorphy.DHolomorphic::sector_fun_independent: cid005298 + TauLib.BookI.Holomorphy.DiagonalProtection: cid005062 + TauLib.BookI.Holomorphy.DiagonalProtection::diagonal_free_protection: cid005300 + TauLib.BookI.Holomorphy.DiagonalProtection::no_simul_projection_b: cid005299 + TauLib.BookI.Holomorphy.DiagonalProtection::stagefun_comp_assoc: cid005301 + TauLib.BookI.Holomorphy.GlobalHartogs: cid005063 + TauLib.BookI.Holomorphy.GlobalHartogs::global_hartogs: cid005302 + TauLib.BookI.Holomorphy.H6DiagonalDiscipline: cid005064 + TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR: cid005065 + TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd: cid005066 + TauLib.BookI.Holomorphy.IdentityTheorem: cid005067 + TauLib.BookI.Holomorphy.IdentityTheorem::tail_agree_downward: cid005303 + TauLib.BookI.Holomorphy.IdentityTheorem::tau_identity_reduce: cid005304 + TauLib.BookI.Holomorphy.PresheafEssence: cid005068 + TauLib.BookI.Holomorphy.PresheafEssence::bi_square_characterization: cid005306 + TauLib.BookI.Holomorphy.PresheafEssence::presheaf_characterization: cid005305 + TauLib.BookI.Holomorphy.SpectralCoefficients: cid005069 + TauLib.BookI.Holomorphy.SpectralCoefficients::spectral_determines: cid005307 + TauLib.BookI.Holomorphy.TauHolomorphic: cid005070 + TauLib.BookI.Holomorphy.Thinness: cid005071 + TauLib.BookI.Holomorphy.Thinness::crt_extension_b: cid005308 + TauLib.BookI.Holomorphy.Thinness::removable_singularity: cid005309 + TauLib.BookI.Kernel.ActionQuantum: cid005072 + TauLib.BookI.Kernel.ActionQuantum::euler_sieve_cross: cid005310 + TauLib.BookI.Kernel.Axioms: cid005073 + TauLib.BookI.Kernel.Axioms::K1_strict_order: cid005311 + TauLib.BookI.Kernel.Axioms::K2_omega_fixed: cid005312 + TauLib.BookI.Kernel.Axioms::K3_orbit_seeded: cid005313 + TauLib.BookI.Kernel.Axioms::K4_no_jump: cid005314 + TauLib.BookI.Kernel.Axioms::K5_beacon_non_succ: cid005315 + TauLib.BookI.Kernel.Axioms::K6_object_closure: cid005316 + TauLib.BookI.Kernel.Axioms::gen_distinct: cid005317 + TauLib.BookI.Kernel.Axioms::rho_injective: cid005318 + TauLib.BookI.Kernel.Diagonal: cid005074 + TauLib.BookI.Kernel.Diagonal::diagonal_channel_count: cid005319 + TauLib.BookI.Kernel.Diagonal::rewiring_levels_eq_solenoidal: cid005320 + TauLib.BookI.Kernel.Signature: cid005075 + TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup: cid005076 + TauLib.BookI.KernelFoundation.GirardLinearEmbedding: cid005077 + TauLib.BookI.KernelFoundation.H8KernelSynthesis: cid005078 + TauLib.BookI.KernelFoundation.ScalarBridges: cid005079 + TauLib.BookI.Logic.BooleanRecovery: cid005080 + TauLib.BookI.Logic.BooleanRecovery::boolean_recovery: cid005321 + TauLib.BookI.Logic.BooleanRecovery::forget_fiber: cid005322 + TauLib.BookI.Logic.Explosion: cid005081 + TauLib.BookI.Logic.Explosion::explosion_barrier: cid005323 + TauLib.BookI.Logic.Truth4: cid005082 + TauLib.BookI.MetaLogic.DiagonalResonance: cid005083 + TauLib.BookI.MetaLogic.LinearDiscipline: cid005084 + TauLib.BookI.MetaLogic.LinearityAudit: cid005085 + TauLib.BookI.MetaLogic.OnticInvariance: cid005086 + TauLib.BookI.MetaLogic.OnticInvariance::no_identity_decoherence: cid005324 + TauLib.BookI.MetaLogic.OnticInvariance::slippage_breaks_omega: cid005325 + TauLib.BookI.MetaLogic.ReceptionCriterion: cid005087 + TauLib.BookI.MetaLogic.ReceptionCriterion::structural_instability_theorem: cid005326 + TauLib.BookI.MetaLogic.StructuralExclusion: cid005088 + TauLib.BookI.MetaLogic.Substrate: cid005089 + TauLib.BookI.Orbit.Closure: cid005090 + TauLib.BookI.Orbit.Closure::ontic_closure: cid005327 + TauLib.BookI.Orbit.Countability: cid005091 + TauLib.BookI.Orbit.Countability::tauObj_countable: cid005328 + TauLib.BookI.Orbit.Generation: cid005092 + TauLib.BookI.Orbit.Generation::orbit_disjoint: cid005329 + TauLib.BookI.Orbit.Ladder: cid005093 + TauLib.BookI.Orbit.Ladder::ladder_saturation: cid005331 + TauLib.BookI.Orbit.Ladder::pentation_channel_exhaustion: cid005330 + TauLib.BookI.Orbit.Rigidity: cid005094 + TauLib.BookI.Orbit.Rigidity::categoricity_non_omega: cid005333 + TauLib.BookI.Orbit.Rigidity::rigidity_non_omega: cid005332 + TauLib.BookI.Orbit.Saturation: cid005095 + TauLib.BookI.Orbit.Saturation::minimal_alphabet: cid005334 + TauLib.BookI.Polarity.BipolarAlgebra: cid005096 + TauLib.BookI.Polarity.BipolarAlgebra::no_elliptic_idempotent: cid005335 + TauLib.BookI.Polarity.BipolarAlgebra::split_complex_forced: cid005336 + TauLib.BookI.Polarity.ChineseRemainder: cid005097 + TauLib.BookI.Polarity.H2H3ClassifierBridge: cid005098 + TauLib.BookI.Polarity.H4BoundaryAlgebra: cid005099 + TauLib.BookI.Polarity.H4UniquenessElliptic: cid005100 + TauLib.BookI.Polarity.InverseLimit: cid005101 + TauLib.BookI.Polarity.Lemniscate: cid005102 + TauLib.BookI.Polarity.Lemniscate::algebraic_lemniscate_exists: cid005337 + TauLib.BookI.Polarity.OmegaGerms: cid005103 + TauLib.BookI.Polarity.OmegaRing: cid005104 + TauLib.BookI.Polarity.Polarity: cid005105 + TauLib.BookI.Polarity.PolarizedGerms: cid005106 + TauLib.BookI.Polarity.PrimePolarityClassifier: cid005107 + TauLib.BookI.Polarity.PrimePolarityIsomorphism: cid005108 + TauLib.BookI.Polarity.SplitComplexCouplingLift: cid005109 + TauLib.BookI.Polarity.TeichmuellerLift: cid005110 + TauLib.BookI.Sets.CantorRefutation: cid005111 + TauLib.BookI.Sets.CantorRefutation::cantor_inapplicable: cid005341 + TauLib.BookI.Sets.CantorRefutation::no_comprehension: cid005339 + TauLib.BookI.Sets.CantorRefutation::no_free_cartesian_diagonal: cid005340 + TauLib.BookI.Sets.CantorRefutation::no_unearned_decimal_diagonal: cid005338 + TauLib.BookI.Sets.Counting: cid005112 + TauLib.BookI.Sets.Counting::counting_structural: cid005342 + TauLib.BookI.Sets.Membership: cid005113 + TauLib.BookI.Sets.Membership::tau_mem_iff_dvd: cid005343 + TauLib.BookI.Sets.Operations: cid005114 + TauLib.BookI.Sets.Operations::tau_inter_distrib_union: cid005344 + TauLib.BookI.Sets.OrbitSets: cid005115 + TauLib.BookI.Sets.OrbitSets::nat_not_internal_set: cid005348 + TauLib.BookI.Sets.OrbitSets::omega_orbit_set_exceeds_alpha: cid005347 + TauLib.BookI.Sets.OrbitSets::orbit_set_order_bound: cid005345 + TauLib.BookI.Sets.OrbitSets::prime_atom: cid005346 + TauLib.BookI.Sets.Powerset: cid005116 + TauLib.BookI.Sets.Powerset::tau_strict_mem_wf: cid005349 + TauLib.BookI.Sets.UniqueInfinity: cid005117 + TauLib.BookI.Sets.UniqueInfinity::ultrametric_replaces_card: cid005351 + TauLib.BookI.Sets.UniqueInfinity::unique_infinity: cid005350 + TauLib.BookI.Topos.CartesianProduct: cid005118 + TauLib.BookI.Topos.CartesianProduct::product_universal: cid005352 + TauLib.BookI.Topos.CircularityResolution: cid005119 + TauLib.BookI.Topos.EarnedArrows: cid005120 + TauLib.BookI.Topos.EarnedArrows::cat_tau_thin: cid005353 + TauLib.BookI.Topos.EarnedTopos: cid005121 + TauLib.BookI.Topos.EarnedTopos::earned_topos_non_boolean: cid005355 + TauLib.BookI.Topos.EarnedTopos::omega_tau_classifier: cid005354 + TauLib.BookI.Topos.Functors: cid005122 + TauLib.BookI.Topos.Functors::yoneda_thin: cid005356 + TauLib.BookI.Topos.H6EarnedCategoricalMachine: cid005123 + TauLib.BookI.Topos.H7CircularityFull: cid005124 + TauLib.BookI.Topos.H7ClassicalClosure: cid005125 + TauLib.BookI.Topos.H7ToposClassifier: cid005126 + TauLib.BookI.Topos.InternalHom: cid005127 + TauLib.BookI.Topos.InternalHom::cartesian_closed_adj: cid005357 + TauLib.BookI.Topos.InternalHom::self_enrichment: cid005358 + TauLib.BookI.Topos.LimitsSites: cid005128 + TauLib.BookI.Topos.LimitsSites::psh_countable_objects: cid005360 + TauLib.BookI.Topos.LimitsSites::psh_has_terminal: cid005359 + TauLib.BookI.Topos.ParaconsistentSoundness: cid005129 + TauLib.BookI.Topos.WedgeProduct: cid005130 + TauLib.BookI.Topos.WedgeProduct::product_distributes_over_coproduct: cid005361 + TauLib.BookII.CentralTheorem.BoundaryCharacters: cid005131 + TauLib.BookII.CentralTheorem.BoundaryCharacters::b_channel_kills_c: cid005363 + TauLib.BookII.CentralTheorem.BoundaryCharacters::c_channel_kills_b: cid005364 + TauLib.BookII.CentralTheorem.BoundaryCharacters::char_add_15_3: cid005371 + TauLib.BookII.CentralTheorem.BoundaryCharacters::char_add_id_15_3: cid005373 + TauLib.BookII.CentralTheorem.BoundaryCharacters::char_mul_15_3: cid005372 + TauLib.BookII.CentralTheorem.BoundaryCharacters::char_tower_20_4: cid005370 + TauLib.BookII.CentralTheorem.BoundaryCharacters::character_add_structural: cid005366 + TauLib.BookII.CentralTheorem.BoundaryCharacters::character_mul_structural: cid005367 + TauLib.BookII.CentralTheorem.BoundaryCharacters::character_tower_structural: cid005365 + TauLib.BookII.CentralTheorem.BoundaryCharacters::idemp_char_20_4: cid005369 + TauLib.BookII.CentralTheorem.BoundaryCharacters::idemp_decomp_recovery: cid005362 + TauLib.BookII.CentralTheorem.BoundaryCharacters::sector_distributive: cid005368 + TauLib.BookII.CentralTheorem.Categoricity: cid005132 + TauLib.BookII.CentralTheorem.Categoricity::abcd_distinct_12_64: cid005387 + TauLib.BookII.CentralTheorem.Categoricity::abcd_roundtrip_12: cid005385 + TauLib.BookII.CentralTheorem.Categoricity::abcd_rt_100: cid005377 + TauLib.BookII.CentralTheorem.Categoricity::categoricity_3_50: cid005379 + TauLib.BookII.CentralTheorem.Categoricity::idempotent_complete: cid005384 + TauLib.BookII.CentralTheorem.Categoricity::j_squared_wave: cid005382 + TauLib.BookII.CentralTheorem.Categoricity::liouville_dodge: cid005374 + TauLib.BookII.CentralTheorem.Categoricity::moduli_100: cid005380 + TauLib.BookII.CentralTheorem.Categoricity::nonconstant_bounded_4: cid005375 + TauLib.BookII.CentralTheorem.Categoricity::primorial_unique_5: cid005376 + TauLib.BookII.CentralTheorem.Categoricity::red_det_3_50: cid005378 + TauLib.BookII.CentralTheorem.Categoricity::reduce_idempotent: cid005386 + TauLib.BookII.CentralTheorem.Categoricity::structure_uniqueness: cid005388 + TauLib.BookII.CentralTheorem.Categoricity::tower_forced: cid005389 + TauLib.BookII.CentralTheorem.Categoricity::uniqueness_3_50: cid005381 + TauLib.BookII.CentralTheorem.Categoricity::zero_divisors_exist: cid005383 + TauLib.BookII.CentralTheorem.CentralTheorem: cid005133 + TauLib.BookII.CentralTheorem.CentralTheorem::central_forward_coherent: cid005398 + TauLib.BookII.CentralTheorem.CentralTheorem::central_fwd_3_15: cid005391 + TauLib.BookII.CentralTheorem.CentralTheorem::central_inv_3_15: cid005392 + TauLib.BookII.CentralTheorem.CentralTheorem::central_inverse_periodic: cid005399 + TauLib.BookII.CentralTheorem.CentralTheorem::central_roundtrip: cid005400 + TauLib.BookII.CentralTheorem.CentralTheorem::central_rt_3_15: cid005393 + TauLib.BookII.CentralTheorem.CentralTheorem::central_theorem_3_15: cid005394 + TauLib.BookII.CentralTheorem.CentralTheorem::holographic_3_15: cid005395 + TauLib.BookII.CentralTheorem.CentralTheorem::holographic_roundtrip: cid005401 + TauLib.BookII.CentralTheorem.CentralTheorem::spectral_idempotent_supported: cid005397 + TauLib.BookII.CentralTheorem.CentralTheorem::spectral_periodic: cid005396 + TauLib.BookII.CentralTheorem.CentralTheorem::spectral_ring_3_15: cid005390 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms: cid005134 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_reduce_invariant: cid005404 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_stagefun_welldef: cid005405 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bsf_coh_15_4: cid005409 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::egr_15_4: cid005410 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::nat_check_20_4: cid005407 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::ogt_15_4: cid005408 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::reduction_gives_naturality: cid005406 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality: cid005402 + TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality_strong: cid005403 + TauLib.BookII.CentralTheorem.HartogsExtension: cid005135 + TauLib.BookII.CentralTheorem.HartogsExtension::b_channel_20_4: cid005416 + TauLib.BookII.CentralTheorem.HartogsExtension::bnd_det_int_10_3: cid005419 + TauLib.BookII.CentralTheorem.HartogsExtension::bndlift_tower: cid005413 + TauLib.BookII.CentralTheorem.HartogsExtension::c_channel_20_4: cid005417 + TauLib.BookII.CentralTheorem.HartogsExtension::cd_uniq_15_4: cid005420 + TauLib.BookII.CentralTheorem.HartogsExtension::ext_channel_20_4: cid005415 + TauLib.BookII.CentralTheorem.HartogsExtension::extension_bipolar_recovery: cid005414 + TauLib.BookII.CentralTheorem.HartogsExtension::extension_preserves_stage: cid005411 + TauLib.BookII.CentralTheorem.HartogsExtension::hartogs_uniq_15_4: cid005418 + TauLib.BookII.CentralTheorem.HartogsExtension::uniqueness_from_agreement: cid005412 + TauLib.BookII.CentralTheorem.SheafCohomology: cid005136 + TauLib.BookII.CentralTheorem.SheafCohomology::cech_derived_1: cid005428 + TauLib.BookII.CentralTheorem.SheafCohomology::cech_derived_2: cid005429 + TauLib.BookII.CentralTheorem.SheafCohomology::cech_sq_zero_1: cid005421 + TauLib.BookII.CentralTheorem.SheafCohomology::cech_sq_zero_2: cid005422 + TauLib.BookII.CentralTheorem.SheafCohomology::h0_global_1: cid005425 + TauLib.BookII.CentralTheorem.SheafCohomology::h0_global_2: cid005426 + TauLib.BookII.CentralTheorem.SheafCohomology::h0_nonconstant_2: cid005427 + TauLib.BookII.CentralTheorem.SheafCohomology::h1_vanishes_1: cid005423 + TauLib.BookII.CentralTheorem.SheafCohomology::h1_vanishes_2: cid005424 + TauLib.BookII.CentralTheorem.YonedaApplied: cid005137 + TauLib.BookII.CentralTheorem.YonedaApplied::germ_hol_12_3: cid005434 + TauLib.BookII.CentralTheorem.YonedaApplied::germ_idempotent_supported: cid005438 + TauLib.BookII.CentralTheorem.YonedaApplied::germ_reconstructs_identity: cid005437 + TauLib.BookII.CentralTheorem.YonedaApplied::hol_class_12_3: cid005433 + TauLib.BookII.CentralTheorem.YonedaApplied::omega_germ_tower_coherent: cid005436 + TauLib.BookII.CentralTheorem.YonedaApplied::omega_germs_12_3: cid005432 + TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_app_12_3: cid005430 + TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_code_roundtrip: cid005435 + TauLib.BookII.CentralTheorem.YonedaApplied::yoneda_hom_12_3: cid005431 + TauLib.BookII.Closure.BSDbridge: cid005138 + TauLib.BookII.Closure.BSDbridge::finite_is_proto_rational_2: cid005445 + TauLib.BookII.Closure.BSDbridge::not_proto_0: cid005440 + TauLib.BookII.Closure.BSDbridge::proto_2: cid005439 + TauLib.BookII.Closure.BSDbridge::proto_abcd_roundtrip_2: cid005447 + TauLib.BookII.Closure.BSDbridge::proto_at_stage: cid005446 + TauLib.BookII.Closure.BSDbridge::proto_count_check: cid005444 + TauLib.BookII.Closure.BSDbridge::proto_examples: cid005442 + TauLib.BookII.Closure.BSDbridge::proto_range_30: cid005443 + TauLib.BookII.Closure.BSDbridge::stage_2: cid005441 + TauLib.BookII.Closure.Connection: cid005139 + TauLib.BookII.Closure.Connection::flat_connection_compatible_2: cid005448 + TauLib.BookII.Closure.Connection::flat_connection_flat_2: cid005449 + TauLib.BookII.Closure.Connection::holonomy_trivial_1: cid005450 + TauLib.BookII.Closure.Connection::holonomy_trivial_2: cid005451 + TauLib.BookII.Closure.Curvature: cid005140 + TauLib.BookII.Closure.Curvature::flat_curvature_vanishes_1: cid005452 + TauLib.BookII.Closure.Curvature::flat_curvature_vanishes_2: cid005453 + TauLib.BookII.Closure.Curvature::geodesic_complete_1: cid005456 + TauLib.BookII.Closure.Curvature::geodesic_complete_2: cid005457 + TauLib.BookII.Closure.Curvature::geodesic_correct_1: cid005454 + TauLib.BookII.Closure.Curvature::geodesic_correct_2: cid005455 + TauLib.BookII.Closure.Curvature::lemniscate_holonomy_1: cid005458 + TauLib.BookII.Closure.Curvature::lemniscate_holonomy_2: cid005459 + TauLib.BookII.Closure.DiffGeoAgenda: cid005141 + TauLib.BookII.Closure.DiffGeoAgenda::book3_prereq_3_15_3: cid005462 + TauLib.BookII.Closure.DiffGeoAgenda::connection_not_earned: cid005463 + TauLib.BookII.Closure.DiffGeoAgenda::curvature_not_earned: cid005464 + TauLib.BookII.Closure.DiffGeoAgenda::diffgeo_not_earned: cid005460 + TauLib.BookII.Closure.DiffGeoAgenda::e1_complete_b3_10_3_3: cid005461 + TauLib.BookII.Closure.DiffGeoAgenda::e1_includes_self_enrichment: cid005466 + TauLib.BookII.Closure.DiffGeoAgenda::holonomy_not_earned: cid005465 + TauLib.BookII.Closure.ForwardBook3: cid005142 + TauLib.BookII.Closure.ForwardBook3::all_six_3_15_3: cid005468 + TauLib.BookII.Closure.ForwardBook3::book3_e1_3_15_3: cid005469 + TauLib.BookII.Closure.ForwardBook3::complete_means_six: cid005470 + TauLib.BookII.Closure.ForwardBook3::e1_gt_e0: cid005473 + TauLib.BookII.Closure.ForwardBook3::export_implies_central: cid005472 + TauLib.BookII.Closure.ForwardBook3::export_implies_e1: cid005471 + TauLib.BookII.Closure.ForwardBook3::full_export_3_15_3: cid005467 + TauLib.BookII.Closure.GeometricBiSquare: cid005143 + TauLib.BookII.Closure.GeometricBiSquare::audit_3_15_3: cid005477 + TauLib.BookII.Closure.GeometricBiSquare::complete_means_eight: cid005479 + TauLib.BookII.Closure.GeometricBiSquare::e0_ne_e1: cid005481 + TauLib.BookII.Closure.GeometricBiSquare::e1_ne_e2: cid005482 + TauLib.BookII.Closure.GeometricBiSquare::geo_topology: cid005476 + TauLib.BookII.Closure.GeometricBiSquare::geometric_all_eight: cid005475 + TauLib.BookII.Closure.GeometricBiSquare::geometric_bisquare_3_15: cid005474 + TauLib.BookII.Closure.GeometricBiSquare::geometric_implies_central: cid005480 + TauLib.BookII.Closure.GeometricBiSquare::scaling_chain_valid: cid005478 + TauLib.BookII.Closure.TauManifold: cid005144 + TauLib.BookII.Closure.TauManifold::abcd_atlas_roundtrip_7: cid005489 + TauLib.BookII.Closure.TauManifold::atlas_chart_3: cid005483 + TauLib.BookII.Closure.TauManifold::atlas_transition_3_30: cid005484 + TauLib.BookII.Closure.TauManifold::chart_transition_7_2: cid005490 + TauLib.BookII.Closure.TauManifold::d_const_zero_3_15: cid005485 + TauLib.BookII.Closure.TauManifold::d_sq_tower_3_10: cid005486 + TauLib.BookII.Closure.TauManifold::d_tau_constant: cid005491 + TauLib.BookII.Closure.TauManifold::d_tau_zero: cid005492 + TauLib.BookII.Closure.TauManifold::manifold_reduce_idempotent: cid005493 + TauLib.BookII.Closure.TauManifold::manifold_tower_coherence: cid005494 + TauLib.BookII.Closure.TauManifold::tau3_manifold_3_15: cid005488 + TauLib.BookII.Closure.TauManifold::tau_manifold_3_15: cid005487 + TauLib.BookII.Domains.Cylinders: cid005145 + TauLib.BookII.Domains.HolImpliesCont: cid005146 + TauLib.BookII.Domains.Ultrametric: cid005147 + TauLib.BookII.Enrichment.EnrichmentLadder: cid005148 + TauLib.BookII.Enrichment.EnrichmentLadder::e0_hom_3: cid005495 + TauLib.BookII.Enrichment.EnrichmentLadder::e1_hom_3: cid005496 + TauLib.BookII.Enrichment.EnrichmentLadder::full_ladder_10_3_3: cid005499 + TauLib.BookII.Enrichment.EnrichmentLadder::gap_10_3_3: cid005498 + TauLib.BookII.Enrichment.EnrichmentLadder::transition_3: cid005497 + TauLib.BookII.Enrichment.Homological: cid005149 + TauLib.BookII.Enrichment.Homological::boundary_coherence_2: cid005500 + TauLib.BookII.Enrichment.Homological::boundary_coherence_3: cid005501 + TauLib.BookII.Enrichment.Homological::homology_trivial_1: cid005504 + TauLib.BookII.Enrichment.Homological::homology_trivial_2: cid005505 + TauLib.BookII.Enrichment.Homological::les_stage2: cid005508 + TauLib.BookII.Enrichment.Homological::ses_exact_1: cid005502 + TauLib.BookII.Enrichment.Homological::ses_exact_2: cid005503 + TauLib.BookII.Enrichment.Homological::ses_stage1: cid005506 + TauLib.BookII.Enrichment.Homological::ses_stage2: cid005507 + TauLib.BookII.Enrichment.SelfDescribing: cid005150 + TauLib.BookII.Enrichment.SelfDescribing::e1_layer_10_3_3: cid005509 + TauLib.BookII.Enrichment.SelfEnrichment: cid005151 + TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_8_3: cid005515 + TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_complete: cid005518 + TauLib.BookII.Enrichment.SelfEnrichment::hom_bipolar_orthogonal: cid005519 + TauLib.BookII.Enrichment.SelfEnrichment::hom_channel_8_3: cid005516 + TauLib.BookII.Enrichment.SelfEnrichment::hom_constant_3: cid005513 + TauLib.BookII.Enrichment.SelfEnrichment::hom_count_3: cid005514 + TauLib.BookII.Enrichment.SelfEnrichment::hom_identity_3: cid005512 + TauLib.BookII.Enrichment.SelfEnrichment::hom_stage_reduce_stable: cid005517 + TauLib.BookII.Enrichment.SelfEnrichment::hom_tower_8_3: cid005510 + TauLib.BookII.Enrichment.SelfEnrichment::self_enrich_8_3: cid005511 + TauLib.BookII.Enrichment.SelfEnrichment::zero_map_compat: cid005520 + TauLib.BookII.Enrichment.TwoCategories: cid005152 + TauLib.BookII.Enrichment.TwoCategories::enrich_finite_5: cid005527 + TauLib.BookII.Enrichment.TwoCategories::enrich_iter_3: cid005526 + TauLib.BookII.Enrichment.TwoCategories::horiz_comp_assoc: cid005529 + TauLib.BookII.Enrichment.TwoCategories::id_two_cell_tower_coherent: cid005531 + TauLib.BookII.Enrichment.TwoCategories::interchange_10_3: cid005528 + TauLib.BookII.Enrichment.TwoCategories::primorial_pos_check: cid005533 + TauLib.BookII.Enrichment.TwoCategories::two_cat_assoc_10_3: cid005521 + TauLib.BookII.Enrichment.TwoCategories::two_cat_unit_10_3: cid005522 + TauLib.BookII.Enrichment.TwoCategories::two_cat_vert_10_3: cid005523 + TauLib.BookII.Enrichment.TwoCategories::two_morph_tower_8_3: cid005524 + TauLib.BookII.Enrichment.TwoCategories::two_morphism_10_3: cid005525 + TauLib.BookII.Enrichment.TwoCategories::two_sq_mediates_at_reduce: cid005532 + TauLib.BookII.Enrichment.TwoCategories::vert_comp_assoc: cid005530 + TauLib.BookII.Enrichment.YonedaTheorem: cid005153 + TauLib.BookII.Enrichment.YonedaTheorem::probe_naturality_structural: cid005541 + TauLib.BookII.Enrichment.YonedaTheorem::probe_yoneda_10_3: cid005534 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_bipolar_10_3: cid005537 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_faithful_10_3: cid005535 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_faithful_id: cid005539 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_full_10_3: cid005536 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_full_id: cid005540 + TauLib.BookII.Enrichment.YonedaTheorem::yoneda_roundtrip_10_3: cid005538 + TauLib.BookII.Geometry.Betweenness: cid005154 + TauLib.BookII.Geometry.CausalStructure: cid005155 + TauLib.BookII.Geometry.Congruence: cid005156 + TauLib.BookII.Geometry.OrthodoxBridge: cid005157 + TauLib.BookII.Geometry.PaschParallel: cid005158 + TauLib.BookII.Hartogs.BndLift: cid005159 + TauLib.BookII.Hartogs.BndLift::bipolar_indep_30: cid005543 + TauLib.BookII.Hartogs.BndLift::bndlift_exist_3_30: cid005542 + TauLib.BookII.Hartogs.CalibratedSplitComplex: cid005160 + TauLib.BookII.Hartogs.CanonicalBasis: cid005161 + TauLib.BookII.Hartogs.CanonicalBasis::basis_3_30: cid005547 + TauLib.BookII.Hartogs.CanonicalBasis::basis_complete_3_30: cid005545 + TauLib.BookII.Hartogs.CanonicalBasis::basis_indep_4: cid005546 + TauLib.BookII.Hartogs.CanonicalBasis::basis_ortho_3_30: cid005544 + TauLib.BookII.Hartogs.CanonicalBasis::proj_delta_3: cid005549 + TauLib.BookII.Hartogs.CanonicalBasis::proj_recovery_3: cid005550 + TauLib.BookII.Hartogs.CanonicalBasis::spectral_support_3_30: cid005548 + TauLib.BookII.Hartogs.CategoryStructure: cid005162 + TauLib.BookII.Hartogs.CategoryStructure::assoc_12_4: cid005552 + TauLib.BookII.Hartogs.CategoryStructure::hol_assoc_thm: cid005551 + TauLib.BookII.Hartogs.CategoryStructure::holend_3_10_ok: cid005553 + TauLib.BookII.Hartogs.EvolutionOperator: cid005163 + TauLib.BookII.Hartogs.EvolutionOperator::asymmetry_witness: cid005556 + TauLib.BookII.Hartogs.EvolutionOperator::evolution_semigroup_thm: cid005554 + TauLib.BookII.Hartogs.EvolutionOperator::semigroup_12_4: cid005555 + TauLib.BookII.Hartogs.L2Space: cid005164 + TauLib.BookII.Hartogs.L2Space::cauchy_schwarz_stage1: cid005561 + TauLib.BookII.Hartogs.L2Space::cauchy_schwarz_stage2: cid005562 + TauLib.BookII.Hartogs.L2Space::inner_product_linearity_2: cid005558 + TauLib.BookII.Hartogs.L2Space::inner_product_symmetry_2: cid005557 + TauLib.BookII.Hartogs.L2Space::l2_basis_orthogonal_1: cid005565 + TauLib.BookII.Hartogs.L2Space::l2_basis_orthogonal_2: cid005566 + TauLib.BookII.Hartogs.L2Space::l2_completeness_1: cid005563 + TauLib.BookII.Hartogs.L2Space::l2_completeness_2: cid005564 + TauLib.BookII.Hartogs.L2Space::norm_definiteness_2: cid005560 + TauLib.BookII.Hartogs.L2Space::norm_positivity_2: cid005559 + TauLib.BookII.Hartogs.LaurentResidue: cid005165 + TauLib.BookII.Hartogs.LaurentResidue::crt_roundtrip_3: cid005569 + TauLib.BookII.Hartogs.LaurentResidue::laurent_range_12_4: cid005567 + TauLib.BookII.Hartogs.LaurentResidue::mero_id_12_4: cid005571 + TauLib.BookII.Hartogs.LaurentResidue::residue_recon_12_3: cid005570 + TauLib.BookII.Hartogs.LaurentResidue::residue_stage_10_3: cid005568 + TauLib.BookII.Hartogs.MutualDetermination: cid005166 + TauLib.BookII.Hartogs.MutualDetermination::mutual_determination: cid005572 + TauLib.BookII.Hartogs.SheafCoherence: cid005167 + TauLib.BookII.Hartogs.SheafCoherence::containment_3_30: cid005575 + TauLib.BookII.Hartogs.SheafCoherence::disjoint_1: cid005574 + TauLib.BookII.Hartogs.SheafCoherence::funct_3_30: cid005578 + TauLib.BookII.Hartogs.SheafCoherence::gluing_3: cid005576 + TauLib.BookII.Hartogs.SheafCoherence::partition_1: cid005573 + TauLib.BookII.Hartogs.SheafCoherence::sheaf_3: cid005577 + TauLib.BookII.Interior.ABCDRigidity: cid005168 + TauLib.BookII.Interior.ABCDRigidity::abcd_has_zero_divisors: cid005579 + TauLib.BookII.Interior.BipolarDecomposition: cid005169 + TauLib.BookII.Interior.OmegaReadout: cid005170 + TauLib.BookII.Interior.Tau3Fibration: cid005171 + TauLib.BookII.Interior.TauAdmissible: cid005172 + TauLib.BookII.Mirror.DimensionalLadder: cid005173 + TauLib.BookII.Mirror.DimensionalLadder::categoricity_kills_ladder: cid005588 + TauLib.BookII.Mirror.DimensionalLadder::categoricity_native: cid005594 + TauLib.BookII.Mirror.DimensionalLadder::engine_only_qft_native: cid005590 + TauLib.BookII.Mirror.DimensionalLadder::fibration_forced: cid005584 + TauLib.BookII.Mirror.DimensionalLadder::fibration_is_three: cid005586 + TauLib.BookII.Mirror.DimensionalLadder::ladder_collapse: cid005583 + TauLib.BookII.Mirror.DimensionalLadder::ladder_collapse_native: cid005593 + TauLib.BookII.Mirror.DimensionalLadder::moduli_singleton: cid005589 + TauLib.BookII.Mirror.DimensionalLadder::refinement_is_abcd: cid005585 + TauLib.BookII.Mirror.DimensionalLadder::rigidity_native: cid005591 + TauLib.BookII.Mirror.DimensionalLadder::rigidity_no_free_parameter: cid005587 + TauLib.BookII.Mirror.DimensionalLadder::simultaneous_rung: cid005581 + TauLib.BookII.Mirror.DimensionalLadder::simultaneous_rung_native: cid005592 + TauLib.BookII.Mirror.DimensionalLadder::tau_engine_inactive: cid005582 + TauLib.BookII.Mirror.DimensionalLadder::tau_spans_three_rungs: cid005580 + TauLib.BookII.Mirror.Inventory: cid005174 + TauLib.BookII.Mirror.Inventory::levels_match: cid005598 + TauLib.BookII.Mirror.Inventory::rewiring_table_complete: cid005595 + TauLib.BookII.Mirror.Inventory::rows_orthodox: cid005597 + TauLib.BookII.Mirror.Inventory::table_12: cid005596 + TauLib.BookII.Mirror.PhysicsQuadrant: cid005175 + TauLib.BookII.Mirror.PhysicsQuadrant::four_quadrants: cid005610 + TauLib.BookII.Mirror.PhysicsQuadrant::obstruction_confined_native: cid005614 + TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_incompatible_pde: cid005600 + TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_obstructed: cid005601 + TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_obstructed_native: cid005611 + TauLib.BookII.Mirror.PhysicsQuadrant::qft_gr_same_column: cid005599 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_distinct_from_qft: cid005609 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_escapes_obstruction: cid005608 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_escapes_qft: cid005612 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_gr_not_obstructed: cid005607 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_hyp: cid005613 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_is_hyperbolic: cid005603 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_is_non_archimedean: cid005602 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_not_archimedean_with_gr: cid005605 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_not_archimedean_with_qft: cid005604 + TauLib.BookII.Mirror.PhysicsQuadrant::tau_qft_not_obstructed: cid005606 + TauLib.BookII.Mirror.SignClassification: cid005176 + TauLib.BookII.Mirror.SignClassification::all_descriptions_differ: cid005616 + TauLib.BookII.Mirror.SignClassification::incompatibility_native: cid005628 + TauLib.BookII.Mirror.SignClassification::no_both_omega_and_archimedean_orthodox: cid005623 + TauLib.BookII.Mirror.SignClassification::no_both_omega_and_archimedean_tau: cid005624 + TauLib.BookII.Mirror.SignClassification::orthodox_all_nonempty: cid005626 + TauLib.BookII.Mirror.SignClassification::orthodox_omega: cid005627 + TauLib.BookII.Mirror.SignClassification::orthodox_path_no_finite_witnesses: cid005619 + TauLib.BookII.Mirror.SignClassification::orthodox_path_no_unique_omega: cid005617 + TauLib.BookII.Mirror.SignClassification::paths_distinct: cid005622 + TauLib.BookII.Mirror.SignClassification::sign_count_12: cid005625 + TauLib.BookII.Mirror.SignClassification::sign_level_count: cid005615 + TauLib.BookII.Mirror.SignClassification::structural_incompatibility: cid005621 + TauLib.BookII.Mirror.SignClassification::tau_path_no_archimedean: cid005618 + TauLib.BookII.Mirror.SignClassification::tau_path_no_epsilon_delta: cid005620 + TauLib.BookII.Mirror.WaveHolomorphy: cid005177 + TauLib.BookII.Mirror.WaveHolomorphy::all_stages_euclidean_5: cid005638 + TauLib.BookII.Mirror.WaveHolomorphy::asymmetric_determination: cid005631 + TauLib.BookII.Mirror.WaveHolomorphy::b_sector_multiplicative: cid005641 + TauLib.BookII.Mirror.WaveHolomorphy::characteristics_iff_hyperbolic: cid005634 + TauLib.BookII.Mirror.WaveHolomorphy::chars_hyp: cid005643 + TauLib.BookII.Mirror.WaveHolomorphy::elliptic_hartogs_not_natural: cid005630 + TauLib.BookII.Mirror.WaveHolomorphy::hartogs_natural_hyp: cid005642 + TauLib.BookII.Mirror.WaveHolomorphy::hyperbolic_hartogs_natural: cid005629 + TauLib.BookII.Mirror.WaveHolomorphy::max_hyp: cid005644 + TauLib.BookII.Mirror.WaveHolomorphy::max_principle_asymmetry: cid005632 + TauLib.BookII.Mirror.WaveHolomorphy::pde_classifications_distinct: cid005633 + TauLib.BookII.Mirror.WaveHolomorphy::pde_distinct: cid005646 + TauLib.BookII.Mirror.WaveHolomorphy::sector_additive: cid005640 + TauLib.BookII.Mirror.WaveHolomorphy::stage_euclidean: cid005636 + TauLib.BookII.Mirror.WaveHolomorphy::stage_euclidean_check_true: cid005637 + TauLib.BookII.Mirror.WaveHolomorphy::stage_no_light_cones: cid005635 + TauLib.BookII.Mirror.WaveHolomorphy::stage_size_is_primorial: cid005639 + TauLib.BookII.Mirror.WaveHolomorphy::stages_euclidean_5: cid005645 + TauLib.BookII.Prologue.SplitComplexInterior: cid005178 + TauLib.BookII.Regularity.CodeDecode: cid005179 + TauLib.BookII.Regularity.CodeDecode::code_decode_3: cid005650 + TauLib.BookII.Regularity.CodeDecode::code_decode_delta_3: cid005651 + TauLib.BookII.Regularity.CodeDecode::code_decode_id_roundtrip: cid005654 + TauLib.BookII.Regularity.CodeDecode::code_delta_3: cid005647 + TauLib.BookII.Regularity.CodeDecode::decode_code_roundtrip: cid005655 + TauLib.BookII.Regularity.CodeDecode::decode_unique_3: cid005649 + TauLib.BookII.Regularity.CodeDecode::decode_welldef_3_15: cid005648 + TauLib.BookII.Regularity.CodeDecode::full_code_decode_3: cid005653 + TauLib.BookII.Regularity.CodeDecode::spectral_sep_3: cid005652 + TauLib.BookII.Regularity.IdempotentDecomposition: cid005180 + TauLib.BookII.Regularity.IdempotentDecomposition::decompose_recovery: cid005656 + TauLib.BookII.Regularity.IdempotentDecomposition::functorial_12_4: cid005659 + TauLib.BookII.Regularity.IdempotentDecomposition::recovery_30: cid005657 + TauLib.BookII.Regularity.IdempotentDecomposition::stagefun_decompose_12_4: cid005658 + TauLib.BookII.Regularity.PositiveRegularity: cid005181 + TauLib.BookII.Regularity.PositiveRegularity::criterion_20_4: cid005661 + TauLib.BookII.Regularity.PositiveRegularity::depth_max_20_4: cid005660 + TauLib.BookII.Regularity.PositiveRegularity::evolution_stab_20_4: cid005663 + TauLib.BookII.Regularity.PositiveRegularity::small_point_3: cid005662 + TauLib.BookII.Regularity.PreYoneda: cid005182 + TauLib.BookII.Regularity.PreYoneda::preyoneda_abcd_12_3: cid005668 + TauLib.BookII.Regularity.PreYoneda::preyoneda_bipolar_12_3: cid005667 + TauLib.BookII.Regularity.PreYoneda::preyoneda_composition_12_3: cid005666 + TauLib.BookII.Regularity.PreYoneda::preyoneda_id_tower_coherent: cid005671 + TauLib.BookII.Regularity.PreYoneda::preyoneda_identity_15_4: cid005665 + TauLib.BookII.Regularity.PreYoneda::preyoneda_tower_15_4: cid005664 + TauLib.BookII.Regularity.PreYoneda::probe_nat_10_3: cid005669 + TauLib.BookII.Regularity.PreYoneda::probe_naturality_is_tower_coherence: cid005672 + TauLib.BookII.Regularity.PreYoneda::probe_tower_15_4: cid005670 + TauLib.BookII.Regularity.ThreeLemmaChain: cid005183 + TauLib.BookII.Regularity.ThreeLemmaChain::branch_fact_12_3: cid005673 + TauLib.BookII.Regularity.ThreeLemmaChain::hol_iff_is_12_3: cid005677 + TauLib.BookII.Regularity.ThreeLemmaChain::is1_12_3: cid005676 + TauLib.BookII.Regularity.ThreeLemmaChain::polarity_30: cid005675 + TauLib.BookII.Regularity.ThreeLemmaChain::prime_split_30: cid005674 + TauLib.BookII.Topology.BoundaryMinimality: cid005184 + TauLib.BookII.Topology.DimensionFour: cid005185 + TauLib.BookII.Topology.Invariant: cid005186 + TauLib.BookII.Topology.StoneSpace: cid005187 + TauLib.BookII.Topology.TorusDegeneration: cid005188 + TauLib.BookII.Transcendentals.Circles: cid005189 + TauLib.BookII.Transcendentals.EEarned: cid005190 + TauLib.BookII.Transcendentals.IotaTauConfirmed: cid005191 + TauLib.BookII.Transcendentals.JReplacesI: cid005192 + TauLib.BookII.Transcendentals.Lines: cid005193 + TauLib.BookII.Transcendentals.PiEarned: cid005194 + TauLib.BookIII.Arithmetic.ABCConjecture: cid005195 + TauLib.BookIII.Arithmetic.ABCConjecture::abc_primorial_3: cid005680 + TauLib.BookIII.Arithmetic.ABCConjecture::abc_quality_15: cid005679 + TauLib.BookIII.Arithmetic.ABCConjecture::radical_check_30: cid005678 + TauLib.BookIII.Arithmetic.ABCConjecture::radical_le_30: cid005682 + TauLib.BookIII.Arithmetic.ABCConjecture::radical_primorial_4: cid005681 + TauLib.BookIII.Arithmetic.ABCDeep: cid005196 + TauLib.BookIII.Arithmetic.ABCDeep::abc_quality_100: cid005683 + TauLib.BookIII.Arithmetic.ABCDeep::high_quality_30: cid005687 + TauLib.BookIII.Arithmetic.ABCDeep::one_squarefree: cid005688 + TauLib.BookIII.Arithmetic.ABCDeep::radical_primorial_5: cid005685 + TauLib.BookIII.Arithmetic.ABCDeep::squarefree_dominance_100: cid005684 + TauLib.BookIII.Arithmetic.ABCDeep::squarefree_dominance_thm: cid005686 + TauLib.BookIII.Arithmetic.ABCDeep::thirty_squarefree: cid005689 + TauLib.BookIII.Arithmetic.ABCDeep::twelve_not_squarefree: cid005690 + TauLib.BookIII.Arithmetic.BSD: cid005197 + TauLib.BookIII.Arithmetic.BSD::bsd_coherence_1: cid005691 + TauLib.BookIII.Arithmetic.BSD::bsd_level: cid005692 + TauLib.BookIII.Arithmetic.EnrFunctor01: cid005198 + TauLib.BookIII.Arithmetic.EnrFunctor01::all_e1: cid005695 + TauLib.BookIII.Arithmetic.EnrFunctor01::e1_level: cid005694 + TauLib.BookIII.Arithmetic.EnrFunctor01::enr_01_10_1: cid005693 + TauLib.BookIII.Arithmetic.EnrichedBiSquare: cid005199 + TauLib.BookIII.Arithmetic.EnrichedBiSquare::comparison_10_1: cid005698 + TauLib.BookIII.Arithmetic.EnrichedBiSquare::enriched_bs_10_1: cid005696 + TauLib.BookIII.Arithmetic.EnrichedBiSquare::factorization_10_1: cid005697 + TauLib.BookIII.Arithmetic.Langlands: cid005200 + TauLib.BookIII.Arithmetic.Langlands::base_change_10_1: cid005701 + TauLib.BookIII.Arithmetic.Langlands::func_10_1: cid005700 + TauLib.BookIII.Arithmetic.Langlands::langlands_level: cid005699 + TauLib.BookIII.Arithmetic.ProtoCodes: cid005201 + TauLib.BookIII.Arithmetic.ProtoCodes::bridgehead_1: cid005704 + TauLib.BookIII.Arithmetic.ProtoCodes::bsd_nonneg_1: cid005703 + TauLib.BookIII.Arithmetic.ProtoCodes::proto_zero_verified: cid005702 + TauLib.BookIII.Arithmetic.RationalPoints: cid005202 + TauLib.BookIII.Arithmetic.RationalPoints::all_rational_1: cid005707 + TauLib.BookIII.Arithmetic.RationalPoints::rank_bounded: cid005706 + TauLib.BookIII.Arithmetic.RationalPoints::zero_rational: cid005705 + TauLib.BookIII.Arithmetic.TowerAssembly: cid005203 + TauLib.BookIII.Arithmetic.TowerAssembly::e0_lt_e1: cid005708 + TauLib.BookIII.Arithmetic.TowerAssembly::e1_lt_e2: cid005709 + TauLib.BookIII.Arithmetic.TowerAssembly::e2_lt_e3: cid005710 + TauLib.BookIII.Arithmetic.TowerAssembly::eight_problems: cid005711 + TauLib.BookIII.Bridge.BridgeAxiom: cid005204 + TauLib.BookIII.Bridge.BridgeAxiom::break_marked: cid005717 + TauLib.BookIII.Bridge.BridgeAxiom::bridge_functor_8_3: cid005713 + TauLib.BookIII.Bridge.BridgeAxiom::bridge_functor_exists: cid005712 + TauLib.BookIII.Bridge.BridgeAxiom::bridge_ledger_consistent: cid005719 + TauLib.BookIII.Bridge.BridgeAxiom::conjectural_marked: cid005716 + TauLib.BookIII.Bridge.BridgeAxiom::established_not_conjectural: cid005727 + TauLib.BookIII.Bridge.BridgeAxiom::honest_claim_8_3: cid005715 + TauLib.BookIII.Bridge.BridgeAxiom::honest_claim_full_8_3: cid005718 + TauLib.BookIII.Bridge.BridgeAxiom::ledger_count: cid005723 + TauLib.BookIII.Bridge.BridgeAxiom::ledger_partition: cid005728 + TauLib.BookIII.Bridge.BridgeAxiom::one_axiom: cid005721 + TauLib.BookIII.Bridge.BridgeAxiom::poincare_established: cid005724 + TauLib.BookIII.Bridge.BridgeAxiom::pvsnp_bridge_break: cid005725 + TauLib.BookIII.Bridge.BridgeAxiom::rh_bridge_conjectural: cid005720 + TauLib.BookIII.Bridge.BridgeAxiom::rh_layers: cid005722 + TauLib.BookIII.Bridge.BridgeAxiom::scope_order: cid005726 + TauLib.BookIII.Bridge.BridgeAxiom::shadow_diagram_8_3: cid005714 + TauLib.BookIII.Bridge.ConjectureGaps: cid005205 + TauLib.BookIII.Bridge.ConjectureGaps::abc_gap_structural: cid005737 + TauLib.BookIII.Bridge.ConjectureGaps::all_gaps_K4: cid005738 + TauLib.BookIII.Bridge.ConjectureGaps::all_gaps_exponential: cid005732 + TauLib.BookIII.Bridge.ConjectureGaps::bridge_necessary_insufficient: cid005730 + TauLib.BookIII.Bridge.ConjectureGaps::exponential_damage: cid005739 + TauLib.BookIII.Bridge.ConjectureGaps::gap_indices: cid005734 + TauLib.BookIII.Bridge.ConjectureGaps::gap_taxonomy: cid005731 + TauLib.BookIII.Bridge.ConjectureGaps::goldbach_gap_parity: cid005735 + TauLib.BookIII.Bridge.ConjectureGaps::parity_ne_density: cid005733 + TauLib.BookIII.Bridge.ConjectureGaps::tower_finite_decidable: cid005729 + TauLib.BookIII.Bridge.ConjectureGaps::twin_gap_density: cid005736 + TauLib.BookIII.Bridge.ForbiddenMoves: cid005206 + TauLib.BookIII.Bridge.ForbiddenMoves::circuits_break_bridge: cid005747 + TauLib.BookIII.Bridge.ForbiddenMoves::equality_violates_K5: cid005746 + TauLib.BookIII.Bridge.ForbiddenMoves::fanout_violates_K3: cid005745 + TauLib.BookIII.Bridge.ForbiddenMoves::five_forbidden: cid005743 + TauLib.BookIII.Bridge.ForbiddenMoves::forbidden_moves_8_3: cid005740 + TauLib.BookIII.Bridge.ForbiddenMoves::max_damage_is_3: cid005749 + TauLib.BookIII.Bridge.ForbiddenMoves::move_bridge_8_3: cid005741 + TauLib.BookIII.Bridge.ForbiddenMoves::move_correspondence: cid005742 + TauLib.BookIII.Bridge.ForbiddenMoves::move_index_0: cid005744 + TauLib.BookIII.Bridge.ForbiddenMoves::pvsnp_uses_3_moves: cid005748 + TauLib.BookIII.Bridge.ForbiddenMoves::threshold_exceeds: cid005750 + TauLib.BookIII.Bridge.Incompleteness: cid005207 + TauLib.BookIII.Bridge.Incompleteness::all_at_e3: cid005755 + TauLib.BookIII.Bridge.Incompleteness::all_from_e2: cid005756 + TauLib.BookIII.Bridge.Incompleteness::diagonal_depth_0: cid005752 + TauLib.BookIII.Bridge.Incompleteness::diagonal_mod: cid005753 + TauLib.BookIII.Bridge.Incompleteness::e2_lt_e3_boundary: cid005757 + TauLib.BookIII.Bridge.Incompleteness::e3_resolves_incompleteness: cid005758 + TauLib.BookIII.Bridge.Incompleteness::godel_at_3: cid005754 + TauLib.BookIII.Bridge.Incompleteness::incompleteness_vm_10_3: cid005751 + TauLib.BookIII.Bridge.TranslationArith: cid005208 + TauLib.BookIII.Bridge.TranslationArith::arith_faithful_8_3: cid005761 + TauLib.BookIII.Bridge.TranslationArith::arith_preserves_6_3: cid005762 + TauLib.BookIII.Bridge.TranslationArith::arith_translation_10_3: cid005759 + TauLib.BookIII.Bridge.TranslationArith::crt_integer_8_3: cid005760 + TauLib.BookIII.Bridge.TranslationObstruction: cid005209 + TauLib.BookIII.Bridge.TranslationObstruction::arith_obstruction_3: cid005764 + TauLib.BookIII.Bridge.TranslationObstruction::obstruction_check_3: cid005763 + TauLib.BookIII.Bridge.TranslationObstruction::pvsnp_obstruction_3: cid005768 + TauLib.BookIII.Bridge.TranslationObstruction::safe_region_8_3: cid005766 + TauLib.BookIII.Bridge.TranslationObstruction::topo_obstruction_4: cid005765 + TauLib.BookIII.Bridge.TranslationObstruction::translation_boundary_8_3: cid005767 + TauLib.BookIII.Bridge.TranslationTopo: cid005210 + TauLib.BookIII.Bridge.TranslationTopo::boundary_restriction_3: cid005772 + TauLib.BookIII.Bridge.TranslationTopo::dimension_recovery_3: cid005770 + TauLib.BookIII.Bridge.TranslationTopo::topo_faithful_6_2: cid005771 + TauLib.BookIII.Bridge.TranslationTopo::topo_translation_10_3: cid005769 + TauLib.BookIII.Bridge.ZFCasVM: cid005211 + TauLib.BookIII.Bridge.ZFCasVM::axiom_encoding_8_3: cid005774 + TauLib.BookIII.Bridge.ZFCasVM::ext_detects_equal: cid005784 + TauLib.BookIII.Bridge.ZFCasVM::ext_detects_unequal: cid005785 + TauLib.BookIII.Bridge.ZFCasVM::ext_min_depth: cid005779 + TauLib.BookIII.Bridge.ZFCasVM::host_level_10_3: cid005776 + TauLib.BookIII.Bridge.ZFCasVM::inf_min_depth: cid005780 + TauLib.BookIII.Bridge.ZFCasVM::pairing_valid: cid005783 + TauLib.BookIII.Bridge.ZFCasVM::set_universe_10_4: cid005775 + TauLib.BookIII.Bridge.ZFCasVM::universe_rank_0: cid005781 + TauLib.BookIII.Bridge.ZFCasVM::universe_rank_3: cid005782 + TauLib.BookIII.Bridge.ZFCasVM::zfc_has_9_axioms: cid005777 + TauLib.BookIII.Bridge.ZFCasVM::zfc_is_e2: cid005778 + TauLib.BookIII.Bridge.ZFCasVM::zfc_vm_8_3: cid005773 + TauLib.BookIII.Computation.Admissibility: cid005212 + TauLib.BookIII.Computation.Admissibility::admissible_10_1: cid005787 + TauLib.BookIII.Computation.Admissibility::id_admissible: cid005788 + TauLib.BookIII.Computation.Admissibility::width_bounded: cid005786 + TauLib.BookIII.Computation.CompBiSquare: cid005213 + TauLib.BookIII.Computation.CompBiSquare::no_barrier_10_1: cid005791 + TauLib.BookIII.Computation.CompBiSquare::product_meet_10_1: cid005789 + TauLib.BookIII.Computation.CompBiSquare::pvsnp_level: cid005790 + TauLib.BookIII.Computation.E2Agent: cid005214 + TauLib.BookIII.Computation.E2Agent::agent_step_depth_0: cid005792 + TauLib.BookIII.Computation.E2Agent::agent_step_mod: cid005793 + TauLib.BookIII.Computation.E2Agent::identity_decoder: cid005794 + TauLib.BookIII.Computation.E2Witness: cid005215 + TauLib.BookIII.Computation.E2Witness::e2_strict_3: cid005800 + TauLib.BookIII.Computation.E2Witness::fixed_points_stage2: cid005796 + TauLib.BookIII.Computation.E2Witness::kleene_check_3: cid005795 + TauLib.BookIII.Computation.E2Witness::operational_closure_8_3: cid005799 + TauLib.BookIII.Computation.E2Witness::orbit_diversity_2: cid005797 + TauLib.BookIII.Computation.E2Witness::orbit_diversity_3: cid005798 + TauLib.BookIII.Computation.TowerMachine: cid005216 + TauLib.BookIII.Computation.TowerMachine::code_is_data: cid005803 + TauLib.BookIII.Computation.TowerMachine::obs_width: cid005804 + TauLib.BookIII.Computation.TowerMachine::ttm_depth_0: cid005802 + TauLib.BookIII.Computation.TowerMachine::ttm_preserves_depth: cid005801 + TauLib.BookIII.Computation.WitnessSearch: cid005217 + TauLib.BookIII.Computation.WitnessSearch::crt_faithful_42_3: cid005806 + TauLib.BookIII.Computation.WitnessSearch::sum_less_prod_3: cid005807 + TauLib.BookIII.Computation.WitnessSearch::witness_zero_3: cid005805 + TauLib.BookIII.Doors.BridgeTightening: cid005218 + TauLib.BookIII.Doors.BridgeTightening::bridge_ledger_complete: cid005813 + TauLib.BookIII.Doors.BridgeTightening::ns_causal_4: cid005811 + TauLib.BookIII.Doors.BridgeTightening::pvsnp_triple_3: cid005812 + TauLib.BookIII.Doors.BridgeTightening::rh_gap_5: cid005808 + TauLib.BookIII.Doors.BridgeTightening::ym_gap_5: cid005809 + TauLib.BookIII.Doors.BridgeTightening::ym_gap_growth_4: cid005810 + TauLib.BookIII.Doors.CriticalLine: cid005219 + TauLib.BookIII.Doors.CriticalLine::critical_line_1: cid005814 + TauLib.BookIII.Doors.CriticalLine::k5_eigenvalue_1: cid005815 + TauLib.BookIII.Doors.CriticalLine::tau_rh_1: cid005816 + TauLib.BookIII.Doors.GrandGRH: cid005220 + TauLib.BookIII.Doors.GrandGRH::grand_grh_3: cid005818 + TauLib.BookIII.Doors.GrandGRH::grand_grh_adelic: cid005817 + TauLib.BookIII.Doors.GrandGRH::grand_grh_from_axiom: cid005822 + TauLib.BookIII.Doors.GrandGRH::l_function_3: cid005821 + TauLib.BookIII.Doors.GrandGRH::label_3_is_C: cid005820 + TauLib.BookIII.Doors.GrandGRH::label_5_is_B: cid005819 + TauLib.BookIII.Doors.LemniscateOperator: cid005221 + TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_formula: cid005824 + TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_nonneg: cid005825 + TauLib.BookIII.Doors.LemniscateOperator::eigenvalue_zero: cid005823 + TauLib.BookIII.Doors.LemniscateOperator::spectral_gap_value: cid005826 + TauLib.BookIII.Doors.MasterSchema: cid005222 + TauLib.BookIII.Doors.MasterSchema::e0_before_e1: cid005831 + TauLib.BookIII.Doors.MasterSchema::ns_level: cid005829 + TauLib.BookIII.Doors.MasterSchema::pvsnp_level: cid005828 + TauLib.BookIII.Doors.MasterSchema::rh_level: cid005827 + TauLib.BookIII.Doors.MasterSchema::rh_part: cid005830 + TauLib.BookIII.Doors.MutualDetermination: cid005223 + TauLib.BookIII.Doors.MutualDetermination::b_to_i_zero: cid005832 + TauLib.BookIII.Doors.MutualDetermination::md_cycle_42_3: cid005833 + TauLib.BookIII.Doors.Poincare: cid005224 + TauLib.BookIII.Doors.Poincare::crt_bijective_42_3: cid005834 + TauLib.BookIII.Doors.Poincare::gluing_depth_1: cid005836 + TauLib.BookIII.Doors.Poincare::terminal_depth_1: cid005835 + TauLib.BookIII.Doors.SpectralCorrespondence: cid005225 + TauLib.BookIII.Doors.SpectralCorrespondence::spectral_corr_from_O3: cid005840 + TauLib.BookIII.Doors.SpectralCorrespondence::spectral_correspondence_O3: cid005837 + TauLib.BookIII.Doors.SpectralCorrespondence::spectral_param_bounded: cid005839 + TauLib.BookIII.Doors.SpectralCorrespondence::spectral_param_zero: cid005838 + TauLib.BookIII.Doors.SpectralDecomp: cid005226 + TauLib.BookIII.Doors.SpectralDecomp::measure_total_30: cid005844 + TauLib.BookIII.Doors.SpectralDecomp::measure_total_6: cid005843 + TauLib.BookIII.Doors.SpectralDecomp::parseval_2: cid005845 + TauLib.BookIII.Doors.SpectralDecomp::parseval_6: cid005846 + TauLib.BookIII.Doors.SpectralDecomp::projector_check_2: cid005841 + TauLib.BookIII.Doors.SpectralDecomp::projector_check_6: cid005842 + TauLib.BookIII.Doors.SpectralDecomp::spectral_resolution_2: cid005847 + TauLib.BookIII.Doors.SpectralDecomp::spectral_resolution_6: cid005848 + TauLib.BookIII.Doors.SplitComplexZeta: cid005227 + TauLib.BookIII.Doors.SplitComplexZeta::b_zeta_3: cid005849 + TauLib.BookIII.Doors.SplitComplexZeta::c_zeta_3: cid005850 + TauLib.BookIII.Doors.SplitComplexZeta::euler_product_3: cid005852 + TauLib.BookIII.Doors.SplitComplexZeta::fe_involution_involutive: cid005851 + TauLib.BookIII.Enrichment.CanonicalLadder: cid005228 + TauLib.BookIII.Enrichment.CanonicalLadder::canonical_ladder_8_3: cid005858 + TauLib.BookIII.Enrichment.CanonicalLadder::canonical_ordering: cid005863 + TauLib.BookIII.Enrichment.CanonicalLadder::e1_strictness_8_3: cid005854 + TauLib.BookIII.Enrichment.CanonicalLadder::functor_collapse_8_3: cid005856 + TauLib.BookIII.Enrichment.CanonicalLadder::hom_stage_stable: cid005860 + TauLib.BookIII.Enrichment.CanonicalLadder::ladder_has_four_levels: cid005862 + TauLib.BookIII.Enrichment.CanonicalLadder::non_emptiness_8_3: cid005853 + TauLib.BookIII.Enrichment.CanonicalLadder::saturation_e3_8_3: cid005857 + TauLib.BookIII.Enrichment.CanonicalLadder::strictness_8_3: cid005855 + TauLib.BookIII.Enrichment.CanonicalLadder::structural_saturation: cid005861 + TauLib.BookIII.Enrichment.CanonicalLadder::zero_in_carrier: cid005859 + TauLib.BookIII.Enrichment.Functor: cid005229 + TauLib.BookIII.Enrichment.Functor::enr_functor_e0_8_3: cid005864 + TauLib.BookIII.Enrichment.Functor::enr_functor_e1_8_3: cid005865 + TauLib.BookIII.Enrichment.Functor::enr_functor_e2_8_3: cid005866 + TauLib.BookIII.Enrichment.Functor::full_enr_functor_8_3: cid005867 + TauLib.BookIII.Enrichment.Functor::ladder_8_3: cid005868 + TauLib.BookIII.Enrichment.Functor::succ_idempotent_e3: cid005869 + TauLib.BookIII.Enrichment.Functor::three_steps_to_e3: cid005870 + TauLib.BookIII.Enrichment.LayerTemplate: cid005230 + TauLib.BookIII.Enrichment.LayerTemplate::all_layers_8_3: cid005875 + TauLib.BookIII.Enrichment.LayerTemplate::e0_layer_valid_8_3: cid005871 + TauLib.BookIII.Enrichment.LayerTemplate::e1_layer_valid_8_3: cid005872 + TauLib.BookIII.Enrichment.LayerTemplate::e2_layer_valid_8_3: cid005873 + TauLib.BookIII.Enrichment.LayerTemplate::e3_layer_valid_8_3: cid005874 + TauLib.BookIII.Hinge.DependencyChain: cid005231 + TauLib.BookIII.Hinge.DependencyChain::all_links_have_level: cid005885 + TauLib.BookIII.Hinge.DependencyChain::axiom_to_enrichment: cid005882 + TauLib.BookIII.Hinge.DependencyChain::chain_has_14_links: cid005879 + TauLib.BookIII.Hinge.DependencyChain::chain_linearity: cid005877 + TauLib.BookIII.Hinge.DependencyChain::dependency_chain_8_3: cid005876 + TauLib.BookIII.Hinge.DependencyChain::e3p_is_last: cid005881 + TauLib.BookIII.Hinge.DependencyChain::e3p_saturates: cid005884 + TauLib.BookIII.Hinge.DependencyChain::k0_is_first: cid005880 + TauLib.BookIII.Hinge.DependencyChain::seven_plus_seven: cid005886 + TauLib.BookIII.Hinge.DependencyChain::succ_monotone: cid005883 + TauLib.BookIII.Hinge.DependencyChain::terminal_completeness: cid005878 + TauLib.BookIII.Hinge.HingeTheorem: cid005232 + TauLib.BookIII.Hinge.HingeTheorem::enrichment_strict: cid005891 + TauLib.BookIII.Hinge.HingeTheorem::five_sectors_exhaustive: cid005897 + TauLib.BookIII.Hinge.HingeTheorem::hinge_chain_length: cid005889 + TauLib.BookIII.Hinge.HingeTheorem::hinge_point: cid005899 + TauLib.BookIII.Hinge.HingeTheorem::hinge_theorem_8_3: cid005887 + TauLib.BookIII.Hinge.HingeTheorem::iota_lt_one: cid005893 + TauLib.BookIII.Hinge.HingeTheorem::iota_pos: cid005894 + TauLib.BookIII.Hinge.HingeTheorem::iota_ratio_depth_3: cid005895 + TauLib.BookIII.Hinge.HingeTheorem::iota_ratio_depth_4: cid005896 + TauLib.BookIII.Hinge.HingeTheorem::iota_value: cid005892 + TauLib.BookIII.Hinge.HingeTheorem::no_knobs_8_3: cid005888 + TauLib.BookIII.Hinge.HingeTheorem::no_knobs_witness: cid005898 + TauLib.BookIII.Hinge.HingeTheorem::sector_product_depth_3: cid005890 + TauLib.BookIII.Mirror.E3Witness: cid005233 + TauLib.BookIII.Mirror.E3Witness::e3_density_2: cid005901 + TauLib.BookIII.Mirror.E3Witness::fixed_point_check_3: cid005900 + TauLib.BookIII.Mirror.E3Witness::paradox_absorbed_3: cid005902 + TauLib.BookIII.Mirror.E3Witness::saturation_semantic_3: cid005904 + TauLib.BookIII.Mirror.E3Witness::self_model_complete_3: cid005903 + TauLib.BookIII.Mirror.E3Witness::self_model_idempotent_3: cid005905 + TauLib.BookIII.Mirror.ProofTheoryE3: cid005234 + TauLib.BookIII.Mirror.ProofTheoryE3::all_paradoxes_at_e2: cid005914 + TauLib.BookIII.Mirror.ProofTheoryE3::all_paradoxes_resolve_e3: cid005915 + TauLib.BookIII.Mirror.ProofTheoryE3::e3_is_proof_theory: cid005912 + TauLib.BookIII.Mirror.ProofTheoryE3::exactly_four_paradoxes: cid005916 + TauLib.BookIII.Mirror.ProofTheoryE3::forbidden_move_range: cid005917 + TauLib.BookIII.Mirror.ProofTheoryE3::forbidden_moves_distinct_thm: cid005910 + TauLib.BookIII.Mirror.ProofTheoryE3::four_paradox_8_3: cid005909 + TauLib.BookIII.Mirror.ProofTheoryE3::paradox_gap: cid005919 + TauLib.BookIII.Mirror.ProofTheoryE3::paradox_move_injective: cid005918 + TauLib.BookIII.Mirror.ProofTheoryE3::paradox_resolution_8_3: cid005911 + TauLib.BookIII.Mirror.ProofTheoryE3::proof_theory_e3_8_3: cid005906 + TauLib.BookIII.Mirror.ProofTheoryE3::self_model_8_3: cid005907 + TauLib.BookIII.Mirror.ProofTheoryE3::self_model_inv_8_3: cid005908 + TauLib.BookIII.Mirror.ProofTheoryE3::self_model_levels: cid005913 + TauLib.BookIII.Mirror.Saturation: cid005235 + TauLib.BookIII.Mirror.Saturation::applied_saturation_8_3: cid005920 + TauLib.BookIII.Mirror.Saturation::e3_is_max: cid005924 + TauLib.BookIII.Mirror.Saturation::e3_is_terminal: cid005923 + TauLib.BookIII.Mirror.Saturation::e3_succ_nat: cid005925 + TauLib.BookIII.Mirror.Saturation::e3_unique_fixpoint: cid005926 + TauLib.BookIII.Mirror.Saturation::full_saturation_8_3: cid005921 + TauLib.BookIII.Mirror.Saturation::strictly_increasing_nat: cid005927 + TauLib.BookIII.Mirror.Saturation::terminal_level_8_3: cid005922 + TauLib.BookIII.Physics.FluidData: cid005236 + TauLib.BookIII.Physics.FluidData::abcd_zero_3: cid005930 + TauLib.BookIII.Physics.FluidData::cylinder_assign_0: cid005929 + TauLib.BookIII.Physics.FluidData::defect_zero_1: cid005931 + TauLib.BookIII.Physics.FluidData::fluid_data_depth_3: cid005928 + TauLib.BookIII.Physics.GapTheorem: cid005237 + TauLib.BookIII.Physics.GapTheorem::gap_constant_is_gap: cid005934 + TauLib.BookIII.Physics.GapTheorem::gap_grows_3_4: cid005933 + TauLib.BookIII.Physics.GapTheorem::gap_pos_3: cid005932 + TauLib.BookIII.Physics.GapTheorem::ym_coupling_3: cid005936 + TauLib.BookIII.Physics.GapTheorem::ym_gap_is_tau_gap_3: cid005935 + TauLib.BookIII.Physics.HartogsFlow: cid005238 + TauLib.BookIII.Physics.HartogsFlow::flow_depth_0: cid005938 + TauLib.BookIII.Physics.HartogsFlow::flow_stable_1: cid005940 + TauLib.BookIII.Physics.HartogsFlow::polarity_swap_involutive: cid005937 + TauLib.BookIII.Physics.HartogsFlow::swap_zero: cid005939 + TauLib.BookIII.Physics.Hodge: cid005239 + TauLib.BookIII.Physics.Hodge::addr_10_1: cid005943 + TauLib.BookIII.Physics.Hodge::nf_addr_10_1: cid005944 + TauLib.BookIII.Physics.Hodge::sigma_fixed_swap_id: cid005942 + TauLib.BookIII.Physics.Hodge::zero_is_sigma_fixed: cid005941 + TauLib.BookIII.Physics.PhysicsAssembly: cid005240 + TauLib.BookIII.Physics.PhysicsAssembly::hodge_at_e1: cid005947 + TauLib.BookIII.Physics.PhysicsAssembly::ns_at_e1: cid005945 + TauLib.BookIII.Physics.PhysicsAssembly::ns_part5: cid005948 + TauLib.BookIII.Physics.PhysicsAssembly::three_e1_problems: cid005949 + TauLib.BookIII.Physics.PhysicsAssembly::ym_at_e1: cid005946 + TauLib.BookIII.Physics.PositiveRegularity: cid005241 + TauLib.BookIII.Physics.PositiveRegularity::defect_1_is_zero: cid005952 + TauLib.BookIII.Physics.PositiveRegularity::pos_reg_10_1: cid005950 + TauLib.BookIII.Physics.PositiveRegularity::stab_germ_0_3: cid005951 + TauLib.BookIII.Physics.PositiveRegularity::stable_at_3: cid005953 + TauLib.BookIII.Physics.StrongSector: cid005242 + TauLib.BookIII.Physics.StrongSector::gauge_3_complete: cid005955 + TauLib.BookIII.Physics.StrongSector::gauge_3_count: cid005956 + TauLib.BookIII.Physics.StrongSector::nf_zero_unique: cid005957 + TauLib.BookIII.Physics.StrongSector::strong_at_1: cid005954 + TauLib.BookIII.Prologue.HartogsBulk: cid005243 + TauLib.BookIII.Prologue.HartogsBulk::book3_entry_8_3_3: cid005960 + TauLib.BookIII.Prologue.HartogsBulk::bulk_reduce_stable: cid005962 + TauLib.BookIII.Prologue.HartogsBulk::bulk_tower_coherent: cid005961 + TauLib.BookIII.Prologue.HartogsBulk::e1_gluing_8_3_3: cid005959 + TauLib.BookIII.Prologue.HartogsBulk::hartogs_bulk_8_3: cid005958 + TauLib.BookIII.Sectors.BoundaryCharacters: cid005244 + TauLib.BookIII.Sectors.BoundaryCharacters::boundary_char_3_3: cid005963 + TauLib.BookIII.Sectors.BoundaryCharacters::bti_additive_3_3: cid005965 + TauLib.BookIII.Sectors.BoundaryCharacters::bti_functor_5_3: cid005964 + TauLib.BookIII.Sectors.BoundaryCharacters::bti_zero: cid005968 + TauLib.BookIII.Sectors.BoundaryCharacters::neg_neg_char: cid005967 + TauLib.BookIII.Sectors.BoundaryCharacters::zero_char_eval: cid005966 + TauLib.BookIII.Sectors.Decomposition: cid005245 + TauLib.BookIII.Sectors.Decomposition::equal_mag_in_A: cid005975 + TauLib.BookIII.Sectors.Decomposition::mixed_in_Omega: cid005976 + TauLib.BookIII.Sectors.Decomposition::omega_coupling_5: cid005970 + TauLib.BookIII.Sectors.Decomposition::pure_m_in_B: cid005973 + TauLib.BookIII.Sectors.Decomposition::pure_n_in_C: cid005974 + TauLib.BookIII.Sectors.Decomposition::sector_decomp_5: cid005969 + TauLib.BookIII.Sectors.Decomposition::sector_exhaustive: cid005977 + TauLib.BookIII.Sectors.Decomposition::sector_preservation_5_3: cid005971 + TauLib.BookIII.Sectors.Decomposition::zero_in_D: cid005972 + TauLib.BookIII.Sectors.LanglandsReflection: cid005246 + TauLib.BookIII.Sectors.LanglandsReflection::a_balanced_3: cid005986 + TauLib.BookIII.Sectors.LanglandsReflection::balanced_uniqueness_5: cid005982 + TauLib.BookIII.Sectors.LanglandsReflection::d_polarity_zero: cid005985 + TauLib.BookIII.Sectors.LanglandsReflection::langlands_reflection_5_3: cid005978 + TauLib.BookIII.Sectors.LanglandsReflection::sector_count_preserved: cid005983 + TauLib.BookIII.Sectors.LanglandsReflection::spectral_polarity_5: cid005981 + TauLib.BookIII.Sectors.LanglandsReflection::template_invariance_8_3: cid005979 + TauLib.BookIII.Sectors.LanglandsReflection::template_invariance_e3: cid005984 + TauLib.BookIII.Sectors.LanglandsReflection::universal_op_3_3: cid005980 + TauLib.BookIII.Sectors.ParityBridge: cid005247 + TauLib.BookIII.Sectors.ParityBridge::a_is_balanced: cid005990 + TauLib.BookIII.Sectors.ParityBridge::b_not_a: cid005991 + TauLib.BookIII.Sectors.ParityBridge::coupling_count: cid005993 + TauLib.BookIII.Sectors.ParityBridge::coupling_ledger_3: cid005988 + TauLib.BookIII.Sectors.ParityBridge::d_self_coupling_1: cid005992 + TauLib.BookIII.Sectors.ParityBridge::no_knobs_5_3: cid005989 + TauLib.BookIII.Sectors.ParityBridge::parity_bridge_5_3: cid005987 + TauLib.BookIII.Spectral.AdditiveConjectures: cid005248 + TauLib.BookIII.Spectral.AdditiveConjectures::additive_multiplicative_3: cid005999 + TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_100: cid005995 + TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_30: cid005994 + TauLib.BookIII.Spectral.AdditiveConjectures::goldbach_primorial_3: cid005997 + TauLib.BookIII.Spectral.AdditiveConjectures::twin_prime_density_3: cid005998 + TauLib.BookIII.Spectral.AdditiveConjectures::twin_primes_30: cid005996 + TauLib.BookIII.Spectral.Adeles: cid005249 + TauLib.BookIII.Spectral.Adeles::adele_injective_1_2: cid006007 + TauLib.BookIII.Spectral.Adeles::adele_is_crt: cid006006 + TauLib.BookIII.Spectral.Adeles::adele_ring_10_3: cid006000 + TauLib.BookIII.Spectral.Adeles::adele_zero_3: cid006005 + TauLib.BookIII.Spectral.Adeles::adelic_dense_20_4: cid006002 + TauLib.BookIII.Spectral.Adeles::adelic_embedding_15_3: cid006001 + TauLib.BookIII.Spectral.Adeles::euler_product_20_4: cid006003 + TauLib.BookIII.Spectral.Adeles::local_factor_ind_10_3: cid006004 + TauLib.BookIII.Spectral.BipolarClassifier: cid005250 + TauLib.BookIII.Spectral.BipolarClassifier::bc_balance_5: cid006010 + TauLib.BookIII.Spectral.BipolarClassifier::bc_exist_3: cid006016 + TauLib.BookIII.Spectral.BipolarClassifier::classifier_20: cid006008 + TauLib.BookIII.Spectral.BipolarClassifier::five_is_b: cid006014 + TauLib.BookIII.Spectral.BipolarClassifier::label_conv_20: cid006009 + TauLib.BookIII.Spectral.BipolarClassifier::label_idem_10_4: cid006011 + TauLib.BookIII.Spectral.BipolarClassifier::split_label_10_4: cid006012 + TauLib.BookIII.Spectral.BipolarClassifier::three_is_c: cid006015 + TauLib.BookIII.Spectral.BipolarClassifier::two_is_x: cid006013 + TauLib.BookIII.Spectral.CRT: cid005251 + TauLib.BookIII.Spectral.CRT::crt_add_10_3: cid006018 + TauLib.BookIII.Spectral.CRT::crt_basis_0_3: cid006023 + TauLib.BookIII.Spectral.CRT::crt_decompose_42: cid006022 + TauLib.BookIII.Spectral.CRT::crt_roundtrip_42: cid006021 + TauLib.BookIII.Spectral.CRT::crt_spectral_20_4: cid006017 + TauLib.BookIII.Spectral.CRT::prime_independence_5_4: cid006020 + TauLib.BookIII.Spectral.CRT::reconstruction_20_4: cid006019 + TauLib.BookIII.Spectral.ConfinementBridge: cid005252 + TauLib.BookIII.Spectral.ConfinementBridge::bridge_algebraic_identity: cid006024 + TauLib.BookIII.Spectral.ConfinementBridge::bridge_reduces_to_E6_near_identity: cid006025 + TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge: cid006027 + TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge_lower: cid006026 + TauLib.BookIII.Spectral.GoldbachDeep: cid005253 + TauLib.BookIII.Spectral.GoldbachDeep::crt_goldbach_duality_3: cid006032 + TauLib.BookIII.Spectral.GoldbachDeep::goldbach_500: cid006028 + TauLib.BookIII.Spectral.GoldbachDeep::goldbach_primorial_m4: cid006029 + TauLib.BookIII.Spectral.GoldbachDeep::obstruction_100_p2: cid006036 + TauLib.BookIII.Spectral.GoldbachDeep::obstruction_100_p3: cid006037 + TauLib.BookIII.Spectral.GoldbachDeep::obstruction_bounded_5: cid006031 + TauLib.BookIII.Spectral.GoldbachDeep::partition_growth_4: cid006030 + TauLib.BookIII.Spectral.GoldbachDeep::partition_m2: cid006033 + TauLib.BookIII.Spectral.GoldbachDeep::partition_m3: cid006034 + TauLib.BookIII.Spectral.GoldbachDeep::partition_m4_pos: cid006035 + TauLib.BookIII.Spectral.HenselLifting: cid005254 + TauLib.BookIII.Spectral.HenselLifting::hensel_1_mod_9: cid006042 + TauLib.BookIII.Spectral.HenselLifting::hensel_20: cid006038 + TauLib.BookIII.Spectral.HenselLifting::hensel_correct_3_2: cid006043 + TauLib.BookIII.Spectral.HenselLifting::hensel_tower_11: cid006040 + TauLib.BookIII.Spectral.HenselLifting::hensel_unique_11: cid006039 + TauLib.BookIII.Spectral.HenselLifting::one_is_root_3: cid006041 + TauLib.BookIII.Spectral.LocalFields: cid005255 + TauLib.BookIII.Spectral.LocalFields::completeness_20_5: cid006046 + TauLib.BookIII.Spectral.LocalFields::limit_unique_10_4: cid006047 + TauLib.BookIII.Spectral.LocalFields::local_field_15_4: cid006044 + TauLib.BookIII.Spectral.LocalFields::local_ring_10_4: cid006045 + TauLib.BookIII.Spectral.LocalFields::tower_42_3: cid006051 + TauLib.BookIII.Spectral.LocalFields::val_p2: cid006049 + TauLib.BookIII.Spectral.LocalFields::val_p_is_1: cid006048 + TauLib.BookIII.Spectral.LocalFields::val_zero: cid006050 + TauLib.BookIII.Spectral.ModularForms: cid005256 + TauLib.BookIII.Spectral.ModularForms::E4_iota4_near_one: cid006053 + TauLib.BookIII.Spectral.ModularForms::E4_iota4_near_one_lower: cid006052 + TauLib.BookIII.Spectral.ModularForms::E6_abs_in_range: cid006054 + TauLib.BookIII.Spectral.ModularForms::E6_iota6_near_one: cid006055 + TauLib.BookIII.Spectral.PrimorialLadder: cid005257 + TauLib.BookIII.Spectral.PrimorialLadder::cofinal_30: cid006064 + TauLib.BookIII.Spectral.PrimorialLadder::prime_cofinal_30: cid006060 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_cofinal_50: cid006059 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_div_6: cid006057 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_growth_6: cid006058 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_ladder_8: cid006056 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_one: cid006062 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_three: cid006063 + TauLib.BookIII.Spectral.PrimorialLadder::primorial_zero: cid006061 + TauLib.BookIII.Spectral.PrimorialLadder::reduce_coherence_42: cid006065 + TauLib.BookIII.Spectral.SieveInfrastructure: cid005258 + TauLib.BookIII.Spectral.SieveInfrastructure::brun_30_3: cid006075 + TauLib.BookIII.Spectral.SieveInfrastructure::brun_euler_4: cid006070 + TauLib.BookIII.Spectral.SieveInfrastructure::euler_phi_primorial_3: cid006076 + TauLib.BookIII.Spectral.SieveInfrastructure::euler_phi_primorial_4: cid006077 + TauLib.BookIII.Spectral.SieveInfrastructure::pi_10: cid006072 + TauLib.BookIII.Spectral.SieveInfrastructure::pi_100: cid006074 + TauLib.BookIII.Spectral.SieveInfrastructure::pi_30: cid006073 + TauLib.BookIII.Spectral.SieveInfrastructure::sieve_correct_200: cid006067 + TauLib.BookIII.Spectral.SieveInfrastructure::sieve_correct_50: cid006066 + TauLib.BookIII.Spectral.SieveInfrastructure::sieve_count_known: cid006068 + TauLib.BookIII.Spectral.SieveInfrastructure::sieve_crt_compat_3: cid006071 + TauLib.BookIII.Spectral.SieveInfrastructure::sieve_tower_compat_3: cid006069 + TauLib.BookIII.Spectral.Trichotomy: cid005259 + TauLib.BookIII.Spectral.Trichotomy::bc_coprime_5: cid006083 + TauLib.BookIII.Spectral.Trichotomy::bc_coprime_at_3: cid006086 + TauLib.BookIII.Spectral.Trichotomy::bc_distinct_3: cid006087 + TauLib.BookIII.Spectral.Trichotomy::bc_non_collapse_10_5: cid006082 + TauLib.BookIII.Spectral.Trichotomy::bnf_15_4: cid006080 + TauLib.BookIII.Spectral.Trichotomy::bnf_unique_10_3: cid006081 + TauLib.BookIII.Spectral.Trichotomy::bnf_zero_3: cid006085 + TauLib.BookIII.Spectral.Trichotomy::trichotomy_15_4: cid006078 + TauLib.BookIII.Spectral.Trichotomy::trichotomy_depth_1: cid006084 + TauLib.BookIII.Spectral.Trichotomy::trichotomy_func_15_3: cid006079 + TauLib.BookIII.Spectral.TwinPrimeDeep: cid005260 + TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_3: cid006097 + TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_5: cid006098 + TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_7: cid006099 + TauLib.BookIII.Spectral.TwinPrimeDeep::crt_admissible_positive_4: cid006091 + TauLib.BookIII.Spectral.TwinPrimeDeep::hl_constant_decreasing_5: cid006090 + TauLib.BookIII.Spectral.TwinPrimeDeep::hl_depth_2: cid006094 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissibility_fraction_5: cid006092 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_1: cid006095 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_3_pos: cid006096 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_count_100: cid006093 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_density_primorial_5: cid006089 + TauLib.BookIII.Spectral.TwinPrimeDeep::twin_primes_500: cid006088 + TauLib.BookIII.Spectrum.InterfaceWidth: cid005261 + TauLib.BookIII.Spectrum.InterfaceWidth::chi_plus_admissible: cid006101 + TauLib.BookIII.Spectrum.InterfaceWidth::width_principle: cid006100 + TauLib.BookIII.Spectrum.KernelHinge: cid005262 + TauLib.BookIII.Spectrum.KernelHinge::book_ii_bridge_complete: cid006102 + TauLib.BookIII.Spectrum.TTM: cid005263 + TauLib.BookIII.Spectrum.ThreeSAT: cid005264 + TauLib.BookIII.Spectrum.ThreeSAT::spectral_decidable: cid006103 + TauLib.BookIII.Spectrum.ThreeSAT::tau_complexity_bridge_concrete: cid006104 + WP-RC-CS-E0-E3: cid000003 + a-balanced-3: cid005986 + a-is-balanced: cid005990 + a_balanced_3: cid005986 + a_is_balanced: cid005990 + abc-at-primorial-levels: cid001657 + abc-gap-characterization: cid001592 + abc-gap-structural: cid005737 + abc-primorial-3: cid005680 + abc-quality: cid001543 + abc-quality-100: cid001668 + abc-quality-15: cid005679 + abc_gap_structural: cid005737 + abc_primorial_3: cid005680 + abc_quality_100: cid005683 + abc_quality_15: cid005679 + abcd-atlas-roundtrip-7: cid005489 + abcd-coordinate-chart: cid001027 + abcd-distinct-12-64: cid005387 + abcd-has-zero-divisors: cid005579 + abcd-roundtrip-12: cid005385 + abcd-rt-100: cid005377 + abcd-zero-3: cid005930 + abcd_atlas_roundtrip_7: cid005489 + abcd_distinct_12_64: cid005387 + abcd_has_zero_divisors: cid005579 + abcd_roundtrip_12: cid005385 + abcd_rt_100: cid005377 + abcd_zero_3: cid005930 + additive-multiplicative-3: cid005999 + additive-multiplicative-duality: cid001584 + additive_multiplicative_3: cid005999 + additivity-bc-3: cid005278 + additivity-even-odd-3: cid005277 + additivity_bc_3: cid005278 + additivity_even_odd_3: cid005277 + addr-10-1: cid005943 + addr_10_1: cid005943 + address-dag: cid001040 + address-resolution-and-absence-of-equations-hinge-7: cid001225 + adele-injective-1-2: cid006007 + adele-is-crt: cid006006 + adele-ring: cid001467 + adele-ring-10-3: cid006000 + adele-zero-3: cid006005 + adele_injective_1_2: cid006007 + adele_is_crt: cid006006 + adele_ring_10_3: cid006000 + adele_zero_3: cid006005 + adelic-dense-20-4: cid006002 + adelic-embedding-15-3: cid006001 + adelic-embedding-theorem: cid001604 + adelic-euler-product: cid001551 + adelic_dense_20_4: cid006002 + adelic_embedding_15_3: cid006001 + admissibility-collapse: cid001625 + admissibility-collapse-theorem-v1: cid001613 + admissibility-e: cid001499 + admissible-10-1: cid005787 + admissible-at-3: cid006097 + admissible-at-5: cid006098 + admissible-at-7: cid006099 + admissible-fluid-data: cid001481 + admissible-gauge-data: cid001489 + admissible-germs-over-omega-tails: cid001418 + admissible_10_1: cid005787 + admissible_at_3: cid006097 + admissible_at_5: cid006098 + admissible_at_7: cid006099 + agent-step-depth-0: cid005792 + agent-step-mod: cid005793 + agent_step_depth_0: cid005792 + agent_step_mod: cid005793 + algebraic-lemniscate: cid001028 + algebraic-lemniscate-exists: cid005337 + algebraic_lemniscate_exists: cid005337 + all-at-e3: cid005755 + all-descriptions-differ: cid005616 + all-e1: cid005695 + all-from-e2: cid005756 + all-gaps-exponential: cid005732 + all-gaps-k4: cid005738 + all-layers-8-3: cid005875 + all-links-have-level: cid005885 + all-paradoxes-at-e2: cid005914 + all-paradoxes-resolve-e3: cid005915 + all-rational-1: cid005707 + all-six-3-15-3: cid005468 + all-stages-euclidean-5: cid005638 + all_at_e3: cid005755 + all_descriptions_differ: cid005616 + all_e1: cid005695 + all_from_e2: cid005756 + all_gaps_K4: cid005738 + all_gaps_exponential: cid005732 + all_layers_8_3: cid005875 + all_links_have_level: cid005885 + all_paradoxes_at_e2: cid005914 + all_paradoxes_resolve_e3: cid005915 + all_rational_1: cid005707 + all_six_3_15_3: cid005468 + all_stages_euclidean_5: cid005638 + alpha-ray-line: cid001259 + angular-sectors: cid001250 + applied-saturation: cid001641 + applied-saturation-8-3: cid005920 + applied_saturation_8_3: cid005920 + approximation-sequence: cid001258 + archimedean-bridge: cid001269 + archimedean-elliptic-engine: cid001309 + archimedean-property: cid001209 + archimedes-polygon-sequence: cid001264 + arith-faithful-8-3: cid005761 + arith-obstruction-3: cid005764 + arith-preserves-6-3: cid005762 + arith-translation-10-3: cid005759 + arith_faithful_8_3: cid005761 + arith_obstruction_3: cid005764 + arith_preserves_6_3: cid005762 + arith_translation_10_3: cid005759 + arithmetic-faithfulness: cid001651 + arithmetic-laws: cid001129 + arithmetic-preserves-operations: cid001580 + arithmetic-translation-functor: cid001532 + assoc-12-4: cid005552 + assoc_12_4: cid005552 + associativity-of-holomorphic-composition: cid001390 + asymmetric-determination: cid001405 + asymmetric_determination: cid005631 + asymmetry-witness: cid005556 + asymmetry_witness: cid005556 + atlas-chart-3: cid005483 + atlas-transition-3-30: cid005484 + atlas_chart_3: cid005483 + atlas_transition_3_30: cid005484 + audit-3-15-3: cid005477 + audit_3_15_3: cid005477 + automorphic-galois-duality-in: cid001508 + ax:beacon-non-succ: cid001114 + ax:no-jump: cid001113 + ax:object-closure: cid001115 + ax:omega-fixed: cid001111 + ax:orbit-seeded: cid001112 + ax:strict-order: cid001110 + ax:universe-postulate: cid001109 + axiom-encoding-8-3: cid005774 + axiom-to-enrichment: cid005882 + axiom_encoding_8_3: cid005774 + axiom_to_enrichment: cid005882 + b-c-asymmetry-implies-time-arrow: cid001389 + b-c-non-collapse-theorem: cid001607 + b-channel-20-4: cid005416 + b-channel-kills-c: cid005363 + b-not-a: cid005991 + b-sector-multiplicative: cid005641 + b-to-i-zero: cid005832 + b-zeta-3: cid005849 + b_channel_20_4: cid005416 + b_channel_kills_c: cid005363 + b_not_a: cid005991 + b_sector_multiplicative: cid005641 + b_to_i_zero: cid005832 + b_zeta_3: cid005849 + balanced-sector-uniqueness: cid001548 + balanced-uniqueness-5: cid005982 + balanced_uniqueness_5: cid005982 + base-change-10-1: cid005701 + base-change-transfer-naturality: cid001629 + base-tau-1: cid001238 + base_change_10_1: cid005701 + basis-3-30: cid005547 + basis-complete-3-30: cid005545 + basis-indep-4: cid005546 + basis-ortho-3-30: cid005544 + basis_3_30: cid005547 + basis_complete_3_30: cid005545 + basis_indep_4: cid005546 + basis_ortho_3_30: cid005544 + bayesian-factorization: cid001167 + bc-balance-5: cid006010 + bc-coprime-5: cid006083 + bc-coprime-at-3: cid006086 + bc-distinct-3: cid006087 + bc-exist-3: cid006016 + bc-non-collapse-10-5: cid006082 + bc_balance_5: cid006010 + bc_coprime_5: cid006083 + bc_coprime_at_3: cid006086 + bc_distinct_3: cid006087 + bc_exist_3: cid006016 + bc_non_collapse_10_5: cid006082 + beacon-non-successor-k5: cid001114 + betweenness-axioms: cid001377 + betweenness-relation: cid001254 + bi-monoidal-structure: cid001078 + bi-square-characterization: cid001208 + bi_square_characterization: cid005306 + bias-at-primorial-levels: cid001219 + bias-mod3-50: cid005288 + bias-positive-50: cid005287 + bias-primorial-3: cid005289 + bias_mod3_50: cid005288 + bias_positive_50: cid005287 + bias_primorial_3: cid005289 + bipolar-channel-independence: cid001348 + bipolar-euler-product: cid001608 + bipolar-fourier-transform: cid001056 + bipolar-idempotents-interior: cid001268 + bipolar-indep-30: cid005543 + bipolar-spectral-algebra: cid001043 + bipolar_indep_30: cid005543 + bnd-det-int-10-3: cid005419 + bnd_det_int_10_3: cid005419 + bndlift-construction: cid001271 + bndlift-exist-3-30: cid005542 + bndlift-existence: cid001387 + bndlift-reduce-invariant: cid005404 + bndlift-stagefun-welldef: cid005405 + bndlift-tower: cid005413 + bndlift_exist_3_30: cid005542 + bndlift_reduce_invariant: cid005404 + bndlift_stagefun_welldef: cid005405 + bndlift_tower: cid005413 + bnf-15-4: cid006080 + bnf-unique-10-3: cid006081 + bnf-zero-3: cid006085 + bnf_15_4: cid006080 + bnf_unique_10_3: cid006081 + bnf_zero_3: cid006085 + book-i: cid000023 + book-i-addressability-address-resolution: cid005000 + book-i-addressability-cayley-metric: cid005001 + book-i-addressability-hinge-integration: cid005002 + book-i-addressability-ontic-ultrametric: cid005003 + book-i-boundary-bridge-tau-int-quotient: cid005004 + book-i-boundary-bridge-tau-rat-quotient: cid005005 + book-i-boundary-bridge-tau-real-congruence: cid005006 + book-i-boundary-bridge-tau-real-quotient: cid005007 + book-i-boundary-characters: cid005008 + book-i-boundary-complex-field: cid005009 + book-i-boundary-constructive-reals: cid005010 + book-i-boundary-coupling-identity-approximants: cid005011 + book-i-boundary-cyclotomic: cid005012 + book-i-boundary-defect-inverse-system: cid005013 + book-i-boundary-fourier: cid005014 + book-i-boundary-integration: cid005015 + book-i-boundary-iota: cid005016 + book-i-boundary-iota-tau-structural: cid005017 + book-i-boundary-measure: cid005018 + book-i-boundary-number-tower: cid005019 + book-i-boundary-numerical-projection: cid005020 + book-i-boundary-quaternions: cid005021 + book-i-boundary-refinement-growing-torus: cid005022 + book-i-boundary-ring: cid005023 + book-i-boundary-spectral: cid005024 + book-i-boundary-split-complex: cid005025 + book-i-boundary-tau-rat-abs: cid005026 + book-i-boundary-tau-rat-field: cid005027 + book-i-boundary-tau-rat-inv: cid005028 + book-i-boundary-tau-rat-order: cid005029 + book-i-boundary-tau-real-abs: cid005030 + book-i-boundary-tau-real-e: cid005031 + book-i-boundary-tau-real-inv: cid005032 + book-i-boundary-tau-real-iota-tau: cid005033 + book-i-boundary-tau-real-mul-congr: cid005034 + book-i-boundary-tau-real-order: cid005035 + book-i-boundary-tau-real-pi: cid005036 + book-i-boundary-tau-real-pi-plus-e: cid005037 + book-i-boundary-tau-real-sum: cid005038 + book-i-boundary-torus-defect-system: cid005039 + book-i-boundary-universal-fixed-scalar: cid005040 + book-i-coordinates-abcd: cid005041 + book-i-coordinates-chebyshev-bias: cid005042 + book-i-coordinates-descent: cid005043 + book-i-coordinates-hyperfact: cid005044 + book-i-coordinates-hyperfact-fta: cid005045 + book-i-coordinates-hyperfact-isomorphism: cid005046 + book-i-coordinates-hyperfact-prop: cid005047 + book-i-coordinates-iterated-prime: cid005048 + book-i-coordinates-no-tie: cid005049 + book-i-coordinates-normal-form: cid005050 + book-i-coordinates-primes: cid005051 + book-i-denotation-arithmetic: cid005052 + book-i-denotation-equality: cid005053 + book-i-denotation-growth-escape: cid005054 + book-i-denotation-order: cid005055 + book-i-denotation-program-monoid: cid005056 + book-i-denotation-rank-transfer: cid005057 + book-i-denotation-structural: cid005058 + book-i-denotation-tau-idx: cid005059 + book-i-holomorphy-boundary-interior: cid005060 + book-i-holomorphy-dholomorphic: cid005061 + book-i-holomorphy-diagonal-protection: cid005062 + book-i-holomorphy-global-hartogs: cid005063 + book-i-holomorphy-h6-diagonal-discipline: cid005064 + book-i-holomorphy-h6-earned-codomain-wave-cr: cid005065 + book-i-holomorphy-h6-sigma-idem-hol-end: cid005066 + book-i-holomorphy-identity-theorem: cid005067 + book-i-holomorphy-presheaf-essence: cid005068 + book-i-holomorphy-spectral-coefficients: cid005069 + book-i-holomorphy-tau-holomorphic: cid005070 + book-i-holomorphy-thinness: cid005071 + book-i-kernel-action-quantum: cid005072 + book-i-kernel-axioms: cid005073 + book-i-kernel-diagonal: cid005074 + book-i-kernel-foundation-admissible-symmetry-group: cid005076 + book-i-kernel-foundation-girard-linear-embedding: cid005077 + book-i-kernel-foundation-h8-kernel-synthesis: cid005078 + book-i-kernel-foundation-scalar-bridges: cid005079 + book-i-kernel-signature: cid005075 + book-i-logic-boolean-recovery: cid005080 + book-i-logic-explosion: cid005081 + book-i-logic-truth4: cid005082 + book-i-meta-logic-diagonal-resonance: cid005083 + book-i-meta-logic-linear-discipline: cid005084 + book-i-meta-logic-linearity-audit: cid005085 + book-i-meta-logic-ontic-invariance: cid005086 + book-i-meta-logic-reception-criterion: cid005087 + book-i-meta-logic-structural-exclusion: cid005088 + book-i-meta-logic-substrate: cid005089 + book-i-orbit-closure: cid005090 + book-i-orbit-countability: cid005091 + book-i-orbit-generation: cid005092 + book-i-orbit-ladder: cid005093 + book-i-orbit-rigidity: cid005094 + book-i-orbit-saturation: cid005095 + book-i-polarity-bipolar-algebra: cid005096 + book-i-polarity-chinese-remainder: cid005097 + book-i-polarity-h2-h3-classifier-bridge: cid005098 + book-i-polarity-h4-boundary-algebra: cid005099 + book-i-polarity-h4-uniqueness-elliptic: cid005100 + book-i-polarity-inverse-limit: cid005101 + book-i-polarity-lemniscate: cid005102 + book-i-polarity-omega-germs: cid005103 + book-i-polarity-omega-ring: cid005104 + book-i-polarity-polarity: cid005105 + book-i-polarity-polarized-germs: cid005106 + book-i-polarity-prime-polarity-classifier: cid005107 + book-i-polarity-prime-polarity-isomorphism: cid005108 + book-i-polarity-split-complex-coupling-lift: cid005109 + book-i-polarity-teichmueller-lift: cid005110 + book-i-sets-cantor-refutation: cid005111 + book-i-sets-counting: cid005112 + book-i-sets-membership: cid005113 + book-i-sets-operations: cid005114 + book-i-sets-orbit-sets: cid005115 + book-i-sets-powerset: cid005116 + book-i-sets-unique-infinity: cid005117 + book-i-topos-cartesian-product: cid005118 + book-i-topos-circularity-resolution: cid005119 + book-i-topos-earned-arrows: cid005120 + book-i-topos-earned-topos: cid005121 + book-i-topos-functors: cid005122 + book-i-topos-h6-earned-categorical-machine: cid005123 + book-i-topos-h7-circularity-full: cid005124 + book-i-topos-h7-classical-closure: cid005125 + book-i-topos-h7-topos-classifier: cid005126 + book-i-topos-internal-hom: cid005127 + book-i-topos-limits-sites: cid005128 + book-i-topos-paraconsistent-soundness: cid005129 + book-i-topos-wedge-product: cid005130 + book-ii: cid000001 + book-ii-bridge-complete: cid006102 + book-ii-central-theorem-boundary-characters: cid005131 + book-ii-central-theorem-categoricity: cid005132 + book-ii-central-theorem-central-theorem: cid005133 + book-ii-central-theorem-extensions-omega-germs: cid005134 + book-ii-central-theorem-hartogs-extension: cid005135 + book-ii-central-theorem-sheaf-cohomology: cid005136 + book-ii-central-theorem-yoneda-applied: cid005137 + book-ii-chapter-5: cid000002 + book-ii-chapter-5-typo-erratum: cid000018 + book-ii-closure-bsdbridge: cid005138 + book-ii-closure-connection: cid005139 + book-ii-closure-curvature: cid005140 + book-ii-closure-diff-geo-agenda: cid005141 + book-ii-closure-forward-book3: cid005142 + book-ii-closure-geometric-bi-square: cid005143 + book-ii-closure-tau-manifold: cid005144 + book-ii-domains-cylinders: cid005145 + book-ii-domains-hol-implies-cont: cid005146 + book-ii-domains-ultrametric: cid005147 + book-ii-enrichment-enrichment-ladder: cid005148 + book-ii-enrichment-homological: cid005149 + book-ii-enrichment-self-describing: cid005150 + book-ii-enrichment-self-enrichment: cid005151 + book-ii-enrichment-two-categories: cid005152 + book-ii-enrichment-yoneda-theorem: cid005153 + book-ii-geometry-betweenness: cid005154 + book-ii-geometry-causal-structure: cid005155 + book-ii-geometry-congruence: cid005156 + book-ii-geometry-orthodox-bridge: cid005157 + book-ii-geometry-pasch-parallel: cid005158 + book-ii-hartogs-bnd-lift: cid005159 + book-ii-hartogs-calibrated-split-complex: cid005160 + book-ii-hartogs-canonical-basis: cid005161 + book-ii-hartogs-category-structure: cid005162 + book-ii-hartogs-evolution-operator: cid005163 + book-ii-hartogs-l2-space: cid005164 + book-ii-hartogs-laurent-residue: cid005165 + book-ii-hartogs-mutual-determination: cid005166 + book-ii-hartogs-sheaf-coherence: cid005167 + book-ii-interior-abcdrigidity: cid005168 + book-ii-interior-bipolar-decomposition: cid005169 + book-ii-interior-omega-readout: cid005170 + book-ii-interior-tau-admissible: cid005172 + book-ii-interior-tau3-fibration: cid005171 + book-ii-mirror-dimensional-ladder: cid005173 + book-ii-mirror-inventory: cid005174 + book-ii-mirror-physics-quadrant: cid005175 + book-ii-mirror-sign-classification: cid005176 + book-ii-mirror-wave-holomorphy: cid005177 + book-ii-prologue-split-complex-interior: cid005178 + book-ii-regularity-code-decode: cid005179 + book-ii-regularity-idempotent-decomposition: cid005180 + book-ii-regularity-positive-regularity: cid005181 + book-ii-regularity-pre-yoneda: cid005182 + book-ii-regularity-three-lemma-chain: cid005183 + book-ii-topology-boundary-minimality: cid005184 + book-ii-topology-dimension-four: cid005185 + book-ii-topology-invariant: cid005186 + book-ii-topology-stone-space: cid005187 + book-ii-topology-torus-degeneration: cid005188 + book-ii-transcendentals-circles: cid005189 + book-ii-transcendentals-eearned: cid005190 + book-ii-transcendentals-iota-tau-confirmed: cid005191 + book-ii-transcendentals-jreplaces-i: cid005192 + book-ii-transcendentals-lines: cid005193 + book-ii-transcendentals-pi-earned: cid005194 + book-iii: cid000024 + book-iii-arithmetic-abcconjecture: cid005195 + book-iii-arithmetic-abcdeep: cid005196 + book-iii-arithmetic-bsd: cid005197 + book-iii-arithmetic-enr-functor01: cid005198 + book-iii-arithmetic-enriched-bi-square: cid005199 + book-iii-arithmetic-langlands: cid005200 + book-iii-arithmetic-proto-codes: cid005201 + book-iii-arithmetic-rational-points: cid005202 + book-iii-arithmetic-tower-assembly: cid005203 + book-iii-bridge-bridge-axiom: cid005204 + book-iii-bridge-conjecture-gaps: cid005205 + book-iii-bridge-forbidden-moves: cid005206 + book-iii-bridge-incompleteness: cid005207 + book-iii-bridge-translation-arith: cid005208 + book-iii-bridge-translation-obstruction: cid005209 + book-iii-bridge-translation-topo: cid005210 + book-iii-bridge-zfcas-vm: cid005211 + book-iii-computation-admissibility: cid005212 + book-iii-computation-comp-bi-square: cid005213 + book-iii-computation-e2-agent: cid005214 + book-iii-computation-e2-witness: cid005215 + book-iii-computation-tower-machine: cid005216 + book-iii-computation-witness-search: cid005217 + book-iii-doors-bridge-tightening: cid005218 + book-iii-doors-critical-line: cid005219 + book-iii-doors-grand-grh: cid005220 + book-iii-doors-lemniscate-operator: cid005221 + book-iii-doors-master-schema: cid005222 + book-iii-doors-mutual-determination: cid005223 + book-iii-doors-poincare: cid005224 + book-iii-doors-spectral-correspondence: cid005225 + book-iii-doors-spectral-decomp: cid005226 + book-iii-doors-split-complex-zeta: cid005227 + book-iii-enrichment-canonical-ladder: cid005228 + book-iii-enrichment-functor: cid005229 + book-iii-enrichment-layer-template: cid005230 + book-iii-hinge-dependency-chain: cid005231 + book-iii-hinge-hinge-theorem: cid005232 + book-iii-mirror-e3-witness: cid005233 + book-iii-mirror-proof-theory-e3: cid005234 + book-iii-mirror-saturation: cid005235 + book-iii-physics-fluid-data: cid005236 + book-iii-physics-gap-theorem: cid005237 + book-iii-physics-hartogs-flow: cid005238 + book-iii-physics-hodge: cid005239 + book-iii-physics-physics-assembly: cid005240 + book-iii-physics-positive-regularity: cid005241 + book-iii-physics-strong-sector: cid005242 + book-iii-prologue-hartogs-bulk: cid005243 + book-iii-sectors-boundary-characters: cid005244 + book-iii-sectors-decomposition: cid005245 + book-iii-sectors-langlands-reflection: cid005246 + book-iii-sectors-parity-bridge: cid005247 + book-iii-spectral-additive-conjectures: cid005248 + book-iii-spectral-adeles: cid005249 + book-iii-spectral-bipolar-classifier: cid005250 + book-iii-spectral-confinement-bridge: cid005252 + book-iii-spectral-crt: cid005251 + book-iii-spectral-goldbach-deep: cid005253 + book-iii-spectral-hensel-lifting: cid005254 + book-iii-spectral-local-fields: cid005255 + book-iii-spectral-modular-forms: cid005256 + book-iii-spectral-primorial-ladder: cid005257 + book-iii-spectral-sieve-infrastructure: cid005258 + book-iii-spectral-trichotomy: cid005259 + book-iii-spectral-twin-prime-deep: cid005260 + book-iii-spectrum-interface-width: cid005261 + book-iii-spectrum-kernel-hinge: cid005262 + book-iii-spectrum-three-sat: cid005264 + book-iii-spectrum-ttm: cid005263 + book-iv: cid000025 + book-v: cid000026 + book-vi: cid000027 + book-vii: cid000028 + book3-e1-3-15-3: cid005469 + book3-entry-8-3-3: cid005960 + book3-prereq-3-15-3: cid005462 + book3_e1_3_15_3: cid005469 + book3_entry_8_3_3: cid005960 + book3_prereq_3_15_3: cid005462 + book_ii_bridge_complete: cid006102 + boolean-recovery: cid001136 + boolean_recovery: cid005321 + both-1-idempotent-unit-of-boolean-sublattice: cid001430 + boundary-char-3-3: cid005963 + boundary-character-space: cid001452 + boundary-coherence-2: cid005500 + boundary-coherence-3: cid005501 + boundary-local-ring: cid001044 + boundary-minimality: cid001374 + boundary-normal-form: cid001469 + boundary-restriction: cid001581 + boundary-restriction-3: cid005772 + boundary-ring-and-scalars: cid001029 + boundary-to-interior-functor: cid001457 + boundary_char_3_3: cid005963 + boundary_coherence_2: cid005500 + boundary_coherence_3: cid005501 + boundary_restriction_3: cid005772 + bounded-powerset: cid001049 + branch-fact-12-3: cid005673 + branch-factorization: cid001335 + branch_fact_12_3: cid005673 + break-marked: cid005717 + break_marked: cid005717 + bridge-algebraic-identity: cid006024 + bridge-algebraic-reduction: cid001576 + bridge-axiom: cid001516 + bridge-functor-8-3: cid005713 + bridge-functor-exists: cid000014 + bridge-ledger: cid001638 + bridge-ledger-complete: cid005813 + bridge-ledger-completeness: cid001583 + bridge-ledger-consistent: cid005719 + bridge-necessary-insufficient: cid001672 + bridge-reduces-to-e6-near-identity: cid006025 + bridge_algebraic_identity: cid006024 + bridge_functor_8_3: cid005713 + bridge_functor_exists: cid005712 + bridge_ledger_complete: cid005813 + bridge_ledger_consistent: cid005719 + bridge_necessary_insufficient: cid005730 + bridge_reduces_to_E6_near_identity: cid006025 + bridgehead-1: cid005704 + bridgehead-proposition: cid001570 + bridgehead_1: cid005704 + brun-30-3: cid006075 + brun-euler-4: cid006070 + brun-sieve-count: cid001443 + brun_30_3: cid006075 + brun_euler_4: cid006070 + bsd-coherence-1: cid005691 + bsd-coherence-theorem: cid001627 + bsd-functional: cid001507 + bsd-level: cid005692 + bsd-nonneg-1: cid005703 + bsd-three-ingredient-proof: cid001571 + bsd_coherence_1: cid005691 + bsd_level: cid005692 + bsd_nonneg_1: cid005703 + bsf-coh-15-4: cid005409 + bsf_coh_15_4: cid005409 + bti-additive-3-3: cid005965 + bti-functor-5-3: cid005964 + bti-zero: cid005968 + bti_additive_3_3: cid005965 + bti_functor_5_3: cid005964 + bti_zero: cid005968 + bulk-reduce-stable: cid005962 + bulk-tower-coherent: cid005961 + bulk_reduce_stable: cid005962 + bulk_tower_coherent: cid005961 + c-channel-20-4: cid005417 + c-channel-kills-b: cid005364 + c-zeta-3: cid005850 + c_channel_20_4: cid005417 + c_channel_kills_b: cid005364 + c_zeta_3: cid005850 + calibrated-split-complex-codomain: cid001270 + canonical-decomposition: cid001283 + canonical-holomorphic-basis: cid001280 + canonical-ladder-8-3: cid005858 + canonical-ladder-theorem: cid001596 + canonical-normalisation-hinge-7: cid001220 + canonical-ordering: cid005863 + canonical-uniqueness-of-d: cid001678 + canonical_ladder_8_3: cid005858 + canonical_ordering: cid005863 + cantor-diagonal-inapplicability: cid001202 + cantor-inapplicable: cid005341 + cantor_inapplicable: cid005341 + cartesian-closed: cid001198 + cartesian-closed-adj: cid005357 + cartesian-monoidal-structure: cid001076 + cartesian_closed_adj: cid005357 + cat-tau: cid001066 + cat-tau-thin: cid005353 + cat_tau_thin: cid005353 + categorical-ai: cid000004 + categorical-coproduct: cid001077 + categorical-product: cid001075 + categoricity: cid001403 + categoricity-3-50: cid005379 + categoricity-kills-ladder: cid005588 + categoricity-native: cid005594 + categoricity-non-omega: cid005333 + categoricity-of-tau-0: cid001175 + categoricity_3_50: cid005379 + categoricity_kills_ladder: cid005588 + categoricity_native: cid005594 + categoricity_non_omega: cid005333 + category-axioms: cid001192 + cauchy-schwarz-inequality: cid001414 + cauchy-schwarz-stage1: cid005561 + cauchy-schwarz-stage2: cid005562 + cauchy_schwarz_stage1: cid005561 + cauchy_schwarz_stage2: cid005562 + causal-arrow: cid001273 + causal-structure: cid001257 + cayley-word-metric-hinge-7: cid001223 + ccc-linear-dichotomy: cid001090 + cd-uniq-15-4: cid005420 + cd_uniq_15_4: cid005420 + cdcl-architecture: cid001479 + cech-derived-1: cid005428 + cech-derived-2: cid005429 + cech-sq-zero-1: cid005421 + cech-sq-zero-2: cid005422 + cech_derived_1: cid005428 + cech_derived_2: cid005429 + cech_sq_zero_1: cid005421 + cech_sq_zero_2: cid005422 + central-forward-coherent: cid005398 + central-fwd-3-15: cid005391 + central-inv-3-15: cid005392 + central-inverse-periodic: cid005399 + central-roundtrip: cid005400 + central-rt-3-15: cid005393 + central-theorem: cid001401 + central-theorem-3-15: cid005394 + central_forward_coherent: cid005398 + central_fwd_3_15: cid005391 + central_inv_3_15: cid005392 + central_inverse_periodic: cid005399 + central_roundtrip: cid005400 + central_rt_3_15: cid005393 + central_theorem_3_15: cid005394 + chain-complex: cid001318 + chain-has-14-links: cid005879 + chain-linearity: cid005877 + chain-verification-protocol: cid001573 + chain_has_14_links: cid005879 + chain_linearity: cid005877 + char-add-15-3: cid005371 + char-add-id-15-3: cid005373 + char-mul-15-3: cid005372 + char-tower-20-4: cid005370 + char_add_15_3: cid005371 + char_add_id_15_3: cid005373 + char_mul_15_3: cid005372 + char_tower_20_4: cid005370 + character-add-structural: cid005366 + character-algebra-ring-structure: cid001355 + character-decomposition: cid001354 + character-group: cid001054 + character-hartogs-equivalence: cid001332 + character-mul-structural: cid005367 + character-tower-structural: cid005365 + character_add_structural: cid005366 + character_mul_structural: cid005367 + character_tower_structural: cid005365 + characteristic-morphism: cid001073 + characteristics-iff-hyperbolic: cid005634 + characteristics_iff_hyperbolic: cid005634 + chars-hyp: cid005643 + chars_hyp: cid005643 + chart-transition-7-2: cid005490 + chart_transition_7_2: cid005490 + chebyshev-bias-measure: cid001015 + chi-complete: cid005265 + chi-orthogonal: cid005266 + chi-plus-admissible: cid006101 + chi_complete: cid005265 + chi_orthogonal: cid005266 + chi_plus_admissible: cid006101 + circuits-break-bridge: cid005747 + circuits_break_bridge: cid005747 + circularity-resolution-via-omega-germ-stabilisation: cid001429 + classical-em-eliminability: cid001158 + classifier-20: cid006008 + classifier_20: cid006008 + clopen-basis: cid001244 + clopen-cylinder-domain: cid001482 + cmb-s4-r-falsifier: cid000016 + cmb-s4-tensor-to-scalar: cid000015 + code-decode-3: cid005650 + code-decode-bijection: cid001396 + code-decode-delta-3: cid005651 + code-decode-id-roundtrip: cid005654 + code-delta-3: cid005647 + code-is-data: cid005803 + code-map: cid001286 + code_decode_3: cid005650 + code_decode_delta_3: cid005651 + code_decode_id_roundtrip: cid005654 + code_delta_3: cid005647 + code_is_data: cid005803 + cofinal-30: cid006064 + cofinal_30: cid006064 + combined-gain-cost-ledger: cid001327 + compactness: cid001369 + comparison-10-1: cid005698 + comparison_10_1: cid005698 + complete-dependency-chain: cid001511 + complete-means-eight: cid005479 + complete-means-six: cid005470 + complete_means_eight: cid005479 + complete_means_six: cid005470 + completeness-20-5: cid006046 + completeness-without-topology: cid001550 + completeness_20_5: cid006046 + compose-assoc: cid005295 + compose_assoc: cid005295 + composition-associativity: cid001170 + composition-closure: cid001190 + composition-of-holomorphic-maps: cid001274 + computational-bi-square: cid001501 + confinement-bridge: cid006027 + confinement-bridge-identity: cid001646 + confinement-bridge-lower: cid006026 + confinement_bridge: cid006027 + confinement_bridge_lower: cid006026 + congruence-axioms: cid001378 + congruence-relation: cid001255 + congruent-tails-agree: cid001144 + conjectural-marked: cid005716 + conjectural_marked: cid005716 + connection: cid001312 + connection-not-earned: cid005463 + connection_not_earned: cid005463 + constraint-lattice: cid001236 + construction-spine: cid000003 + constructive-crt-in-the-primorial-ladder: cid001674 + constructive-encoding-via-abcd: cid001000 + constructive-hensel-lifting: cid001603 + constructive-reals: cid001052 + containment-3-30: cid005575 + containment_3_30: cid005575 + cor:constructive-encoding: cid001000 + cor:holographic: cid001232 + cor:interior-from-boundary: cid001001 + cor:nf-dedup-invariant: cid001003 + cor:no-identity-decoherence: cid001002 + cor:uniqueness: cid001233 + corpus-v2-release-v4-0: cid000129 + corpus-v3-wave-2-seed: cid000006 + countability-of-set-tau: cid001135 + countable-additivity: cid001216 + countable-profinite-boundary-ring: cid001673 + countable-topos: cid001149 + counting-as-structural-feature: cid001153 + counting-structural: cid005342 + counting_structural: cid005342 + coupling-count: cid005993 + coupling-ledger: cid001463 + coupling-ledger-3: cid005988 + coupling-sector: cid001459 + coupling_count: cid005993 + coupling_ledger_3: cid005988 + criterion-20-4: cid005661 + criterion_20_4: cid005661 + critical-line-1: cid005814 + critical-line-theorem: cid001611 + critical_line_1: cid005814 + crossing-iff-real: cid005270 + crossing-mediator-iota-tau-in-d: cid001677 + crossing-point: cid001055 + crossing_iff_real: cid005270 + crt-add-10-3: cid006018 + crt-admissible-positive: cid001667 + crt-admissible-positive-4: cid006091 + crt-basis-0-3: cid006023 + crt-bijective-42-3: cid005834 + crt-coherence-constraint: cid001188 + crt-decompose-42: cid006022 + crt-decomposition: cid001045 + crt-decomposition-theorem: cid001602 + crt-extension: cid001123 + crt-extension-b: cid005308 + crt-faithful-42-3: cid005806 + crt-galois-decomposition: cid001164 + crt-goldbach-duality: cid001587 + crt-goldbach-duality-3: cid006032 + crt-integer-8-3: cid005760 + crt-integer-correspondence: cid001533 + crt-roundtrip-3: cid005569 + crt-roundtrip-42: cid006021 + crt-spectral-20-4: cid006017 + crt-twin-admissibility: cid001449 + crt-witness-decomposition: cid001566 + crt-witness-decomposition-v1: cid001556 + crt_add_10_3: cid006018 + crt_admissible_positive_4: cid006091 + crt_basis_0_3: cid006023 + crt_bijective_42_3: cid005834 + crt_decompose_42: cid006022 + crt_extension_b: cid005308 + crt_faithful_42_3: cid005806 + crt_goldbach_duality_3: cid006032 + crt_integer_8_3: cid005760 + crt_roundtrip_3: cid005569 + crt_roundtrip_42: cid006021 + crt_spectral_20_4: cid006017 + curvature: cid001314 + curvature-not-earned: cid005464 + curvature_not_earned: cid005464 + cyclotomic-fields: cid001097 + cylinder-assign-0: cid005929 + cylinder-basis-theorem: cid001366 + cylinder-domain: cid001242 + cylinder-generator: cid001281 + cylinder_assign_0: cid005929 + cylinders-are-balls: cid001345 + d-const-zero-3-15: cid005485 + d-differentiability: cid001058 + d-polarity-zero: cid005985 + d-self-coupling-1: cid005992 + d-sq-tower-3-10: cid005486 + d-tau-constant: cid005491 + d-tau-zero: cid005492 + d_const_zero_3_15: cid005485 + d_polarity_zero: cid005985 + d_self_coupling_1: cid005992 + d_sq_tower_3_10: cid005486 + d_tau_constant: cid005491 + d_tau_zero: cid005492 + dark-sector-readout: cid000013 + decode-code-roundtrip: cid005655 + decode-map: cid001287 + decode-unique-3: cid005649 + decode-welldef-3-15: cid005648 + decode_code_roundtrip: cid005655 + decode_unique_3: cid005649 + decode_welldef_3_15: cid005648 + decompose-recovery: cid005656 + decompose_recovery: cid005656 + decomposition-functoriality: cid001350 + def:abc-high-quality: cid001451 + def:abc-quality: cid001543 + def:abc-sieve: cid001450 + def:abcd-chart: cid001027 + def:address-dag: cid001040 + def:alpha-ray: cid001259 + def:angular-sectors: cid001250 + def:approximation-sequence: cid001258 + def:archimedean-bridge: cid001269 + def:archimedean-elliptic-engine: cid001309 + def:archimedes-polygon: cid001264 + def:arith-translation: cid001532 + def:automorphic-galois-duality: cid001508 + def:base-tau1: cid001238 + def:betweenness: cid001254 + def:bi-monoidal: cid001078 + def:bipolar-fourier: cid001056 + def:bipolar-idempotents-interior: cid001268 + def:bipolar-spectral-algebra: cid001043 + def:bndlift: cid001271 + def:boundary-character-space: cid001452 + def:boundary-local-ring: cid001044 + def:boundary-normal-form: cid001469 + def:boundary-ring: cid001029 + def:boundary-to-interior-functor: cid001457 + def:bounded-powerset: cid001049 + def:bridge-axiom: cid001516 + def:brun-sieve-count: cid001443 + def:bsd-functional: cid001507 + def:calibrated-H-tau: cid001270 + def:canonical-basis: cid001280 + def:canonical-decomposition: cid001283 + def:cartesian-monoidal: cid001076 + def:cat-tau: cid001066 + def:categorical-coproduct: cid001077 + def:categorical-product: cid001075 + def:causal-arrow: cid001273 + def:causal-structure: cid001257 + def:ccc-linear-dichotomy: cid001090 + def:cech-complex: cid001320 + def:chain-complex: cid001318 + def:character-group: cid001054 + def:characteristic-morphism: cid001073 + def:chebyshev-bias: cid001015 + def:clopen-basis: cid001244 + def:clopen-cylinder-domain: cid001482 + def:code: cid001286 + def:combined-ledger: cid001327 + def:complete-dependency-chain: cid001511 + def:composition: cid001274 + def:computational-bi-square: cid001501 + def:congruence: cid001255 + def:constraint-lattice: cid001236 + def:constructive-reals: cid001052 + def:coupling-ledger: cid001463 + def:crossing-point: cid001055 + def:crt-integer: cid001533 + def:crt-twin-admissible: cid001449 + def:cyclotomic-field: cid001097 + def:cylinder-domain: cid001242 + def:cylinder-generator: cid001281 + def:d-differentiability: cid001058 + def:decode: cid001287 + def:defect-functional: cid001484 + def:denotational-order: cid001026 + def:diagonal-discipline: cid001006 + def:diagonal-linear: cid001087 + def:diagonal-resonance: cid001098 + def:diagrammatic-sector-e3: cid001519 + def:dimension-recovery: cid001535 + def:dimensional-rigidity: cid001310 + def:divisibility: cid001030 + def:e-eigenvalue: cid001265 + def:e0-e1-transition: cid001293 + def:e0-layer: cid001437 + def:e1-export: cid001301 + def:e1-gluing: cid001434 + def:e1-layer: cid001292 + def:e1-mutual-determination: cid001503 + def:e1-physical-preview: cid001547 + def:e2-computational-agent: cid001494 + def:e2-e3-boundary-crossing: cid001520 + def:e2-layer: cid001439 + def:e2-orbit-structure: cid001529 + def:e3-layer: cid001440 + def:earned-interior-point: cid001083 + def:earned-topos: cid001074 + def:elliptic-hyperbolic: cid001095 + def:enriched-bi-square: cid001510 + def:enrichment-frontier: cid001091 + def:enrichment-functor: cid001435 + def:enrichment-functor-e01: cid001502 + def:eratosthenes-sieve: cid001544 + def:evolution-operator: cid001272 + def:fiber-T2: cid001239 + def:fibered-product-tau3: cid001240 + def:finite-limits: cid001070 + def:first-disagreement-depth: cid001245 + def:five-comparison-modes: cid001323 + def:five-forbidden-moves: cid001514 + def:fluid-sector-decomposition: cid001483 + def:four-plus-one-decomposition: cid001458 + def:functional-equation-involution: cid001472 + def:galois-automorphism: cid001106 + def:galois-group-primorial: cid001107 + def:gap-constant: cid001490 + def:gap-forbidden-move: cid001456 + def:gap-type: cid001455 + def:genealogical-decomposition: cid001039 + def:generative-counting: cid001084 + def:generators: cid001004 + def:geo-topo-unification: cid001262 + def:geometric-bisquare: cid001311 + def:geometric-pi: cid001263 + def:goedel-numbering-as-nf-address: cid001513 + def:goldbach-obstruction: cid001446 + def:goldbach-partition-primorial: cid001445 + def:goldbach-representation: cid001540 + def:goldbach-sieve: cid001444 + def:grand-grh: cid001476 + def:greedy-peel: cid001033 + def:growth-base: cid001266 + def:hartogs-flow-operator: cid001485 + def:hl-constant: cid001448 + def:hol-L: cid001064 + def:hol-transformers: cid001038 + def:holend-tau: cid001276 + def:holfun: cid001062 + def:holfun-spectral-coefficients: cid001080 + def:holmap: cid001063 + def:holomorphic-presheaf: cid001282 + def:hom-object: cid001289 + def:homology-ses: cid001319 + def:host-level-property: cid001515 + def:idempotent-character: cid001294 + def:identity-faithful-reception: cid001101 + def:identity-map: cid001275 + def:identity-slippage: cid001099 + def:idx-add: cid001013 + def:idx-exp: cid001021 + def:idx-mul: cid001020 + def:idx-tetration: cid001022 + def:infinity-trade-off: cid001303 + def:interface-width: cid001498 + def:interior-bipolar: cid001241 + def:internal-bipolar-classifier: cid001468 + def:internal-hom: cid001079 + def:internal-primes: cid001031 + def:iota-tau: cid001050 + def:iterator-ladder: cid001009 + def:j-unit: cid001267 + def:kleene-fixed-point: cid001528 + def:l-function-spectral-determinant: cid001477 + def:l2-inner-product: cid001316 + def:l2-norm: cid001317 + def:ladder-checker: cid001441 + def:langlands-one: cid001460 + def:laurent-expansion: cid001277 + def:layer-template: cid001436 + def:lemniscate-characters: cid001053 + def:lemniscate-operator: cid001473 + def:level-circle: cid001260 + def:local-langlands-instance: cid001509 + def:meromorphic: cid001279 + def:meta-logical-substrate: cid001086 + def:minkowski-extension: cid001521 + def:mode-c-catalog: cid001325 + def:mode-e-catalog: cid001326 + def:moduli-space: cid001296 + def:mutual-determination-schema: cid001470 + def:natural-transformation: cid001068 + def:nf-encoding: cid001025 + def:np-witness-canonical-address: cid001500 + def:np-witness-canonical-address-v1: cid001478 + def:number-tower: cid001051 + def:observable-transition: cid001497 + def:obstruction-classes: cid001537 + def:obstruction-cocycle: cid001536 + def:omega-coupling-sector: cid001459 + def:omega-germ-approach: cid001085 + def:omega-germ-transformer: cid001060 + def:omega-readout: cid001237 + def:omega-tail: cid001041 + def:omega-tau: cid001057 + def:operational-closure: cid001495 + def:operator-polarity-swap: cid001486 + def:orbit-rays: cid001008 + def:orbit-set-map: cid001103 + def:paradox-absorption: cid001531 + def:parallel-transport: cid001313 + def:pde-type-classification: cid001304 + def:physical-e2-agent: cid001522 + def:physical-realizability-predicate: cid001524 + def:physics-quadrant-matrix: cid001307 + def:pi-tau-boundary: cid001018 + def:pinch-map: cid001251 + def:polarity-pairing: cid001016 + def:polarized-germ: cid001042 + def:polstream: cid001017 + def:pre-yoneda: cid001285 + def:presheaf-topos: cid001072 + def:prime-count-mod: cid001014 + def:prime-enumeration: cid001035 + def:primorial-ladder: cid001464 + def:primorial-presheaf: cid001092 + def:primorial-thinness: cid001082 + def:program-linear-calculus: cid001088 + def:program-monoid: cid001023 + def:progression: cid001005 + def:proof-theory-as-e3: cid001518 + def:proto-code: cid001506 + def:proto-rationality: cid001300 + def:q-expansion-coefficients: cid001525 + def:radial-solenoidal: cid001249 + def:radical-function: cid001542 + def:rank-as-tower-depth: cid001505 + def:rank-transfer: cid001011 + def:reconstruction-functor: cid001465 + def:residue: cid001278 + def:restriction-map: cid001081 + def:rewiring-table: cid001306 + def:rh-gap-char: cid001538 + def:sector-addressability: cid001493 + def:self-enrichment: cid001288 + def:self-hosting-degree: cid001089 + def:self-ref-fixed-point: cid001530 + def:set-operations: cid001048 + def:shadow-diagram: cid001517 + def:shadow-identity: cid001100 + def:sheaf-cohomology: cid001321 + def:sieve-prime-count: cid001442 + def:sigma-fixed-character: cid001492 + def:simply-connected-tau: cid001480 + def:six-comparison-axes: cid001324 + def:solenoidal-circle: cid001261 + def:spectral-algebra: cid001295 + def:spectral-measure: cid001527 + def:spectral-parameter: cid001474 + def:spectral-polarity: cid001462 + def:spectral-projector: cid001526 + def:spectral-signature: cid001034 + def:spine-address-path: cid001253 + def:split-complex: cid001036 + def:split-complex-codomain: cid001234 + def:split-complex-zeta: cid001471 + def:split-cr-equations: cid001059 + def:squarefree-abc: cid001453 + def:stabilized-omega-germ: cid001487 + def:stage-finite-euclidean: cid001305 + def:stage-k-cylinder: cid001243 + def:stone-space: cid001247 + def:strong-defect-functional: cid001491 + def:strong-sector-e1: cid001488 + def:structural-instability: cid001102 + def:structural-sign-classification: cid001302 + def:swap: cid001012 + def:tau-adele-ring: cid001467 + def:tau-admissibility: cid001499 + def:tau-admissible-fluid-data: cid001481 + def:tau-admissible-gauge-data: cid001489 + def:tau-admissible-point: cid001235 + def:tau-analytic-atlas: cid001298 + def:tau-arrow: cid001065 + def:tau-bayesian-state: cid001019 + def:tau-cdcl-architecture: cid001479 + def:tau-complex-field: cid001094 + def:tau-connection: cid001312 + def:tau-curvature: cid001314 + def:tau-dimension: cid001248 + def:tau-effective-rh: cid001475 + def:tau-exponential: cid001322 + def:tau-exterior-derivative: cid001299 + def:tau-functor: cid001067 + def:tau-geodesic: cid001315 + def:tau-idx: cid001010 + def:tau-integral: cid001108 + def:tau-manifold: cid001297 + def:tau-measure-space: cid001104 + def:tau-membership: cid001047 + def:tau-native-abstract-tm: cid001523 + def:tau-native-local-field: cid001466 + def:tau-quaternion: cid001096 + def:tau-rational-point: cid001504 + def:tau-real-field: cid001093 + def:tau-regularity: cid001284 + def:tau-site: cid001071 + def:tau-tower-machine: cid001496 + def:tau-zero: cid001007 + def:teichmueller-lift: cid001046 + def:three-equality: cid001024 + def:topo-translation: cid001534 + def:tower-atom: cid001032 + def:tower-coherence: cid001061 + def:tower-decidable: cid001454 + def:tower-sigma-algebra: cid001105 + def:truth4: cid001037 + def:twin-prime-distribution: cid001541 + def:twin-sieve-count: cid001447 + def:two-category: cid001290 + def:two-morphism: cid001291 + def:two-readout-principle: cid001252 + def:ultrametric-distance: cid001246 + def:unification-obstruction: cid001308 + def:universal-operator: cid001461 + def:wave-type-pde: cid001256 + def:ym-gap-persistence: cid001539 + def:yoneda-embedding: cid001069 + def:zfc-as-e2-vm: cid001512 + defect-1-is-zero: cid005952 + defect-contractivity: cid001558 + defect-functional: cid001484 + defect-zero-1: cid005931 + defect_1_is_zero: cid005952 + defect_zero_1: cid005931 + denotational-lt-wf: cid005294 + denotational-order: cid001026 + denotational_lt_wf: cid005294 + dependency-chain-8-3: cid005876 + dependency_chain_8_3: cid005876 + depth-max-20-4: cid005660 + depth_max_20_4: cid005660 + diagonal-channel-count: cid005319 + diagonal-depth-0: cid005752 + diagonal-discipline: cid001006 + diagonal-discipline-four-prohibitions: cid001422 + diagonal-free-protection: cid001189 + diagonal-linear-correspondence: cid001087 + diagonal-linear-correspondence-hinge-8: cid001228 + diagonal-mod: cid005753 + diagonal-resonance: cid001098 + diagonal-resonance-diagnosis-hinge-8: cid001230 + diagonal_channel_count: cid005319 + diagonal_depth_0: cid005752 + diagonal_free_protection: cid005300 + diagonal_mod: cid005753 + diagrammatic-sector-of-e: cid001519 + diffgeo-not-earned: cid005460 + diffgeo_not_earned: cid005460 + dimension-four: cid001373 + dimension-recovery: cid001535 + dimension-recovery-3: cid005770 + dimension-theorem-dim-tau-4: cid001131 + dimension_recovery_3: cid005770 + dimensional-rigidity: cid001310 + discrete-spectrum-of-h-l: cid001553 + disjoint-1: cid005574 + disjoint_1: cid005574 + distributive-lattice: cid001134 + distributivity: cid001197 + duality-as-mutual-determination-on: cid001572 + e-as-gluing-principle: cid001434 + e-as-iterator-eigenvalue: cid001265 + e-computational-agent: cid001494 + e-e-boundary-crossing: cid001520 + e-e-strict-witness: cid001578 + e-from-index-arithmetic: cid001385 + e-layer-computation: cid001439 + e-layer-mathematics: cid001437 + e-layer-metaphysics: cid001440 + e-layer-physics: cid001438 + e-mutual-determination-instance: cid001503 + e-orbit-structure: cid001529 + e-physical-preview: cid001547 + e-self-model-completeness: cid001650 + e-strictness-witness: cid001545 + e0-before-e1: cid005831 + e0-e1-transition: cid001293 + e0-hom-3: cid005495 + e0-layer-valid-8-3: cid005871 + e0-lt-e1: cid005708 + e0-ne-e1: cid005481 + e0_before_e1: cid005831 + e0_hom_3: cid005495 + e0_layer_valid_8_3: cid005871 + e0_lt_e1: cid005708 + e0_ne_e1: cid005481 + e1-complete-b3-10-3-3: cid005461 + e1-enrichment-layer: cid001292 + e1-export-package: cid001301 + e1-gluing-8-3-3: cid005959 + e1-gt-e0: cid005473 + e1-hom-3: cid005496 + e1-includes-self-enrichment: cid005466 + e1-layer-10-3-3: cid005509 + e1-layer-valid-8-3: cid005872 + e1-level: cid005694 + e1-lt-e2: cid005709 + e1-ne-e2: cid005482 + e1-strictness-8-3: cid005854 + e1_complete_b3_10_3_3: cid005461 + e1_gluing_8_3_3: cid005959 + e1_gt_e0: cid005473 + e1_hom_3: cid005496 + e1_includes_self_enrichment: cid005466 + e1_layer_10_3_3: cid005509 + e1_layer_valid_8_3: cid005872 + e1_level: cid005694 + e1_lt_e2: cid005709 + e1_ne_e2: cid005482 + e1_strictness_8_3: cid005854 + e2-layer-valid-8-3: cid005873 + e2-lt-e3: cid005710 + e2-lt-e3-boundary: cid005757 + e2-strict-3: cid005800 + e2_layer_valid_8_3: cid005873 + e2_lt_e3: cid005710 + e2_lt_e3_boundary: cid005757 + e2_strict_3: cid005800 + e3-density-2: cid005901 + e3-is-max: cid005924 + e3-is-proof-theory: cid005912 + e3-is-terminal: cid005923 + e3-layer-valid-8-3: cid005874 + e3-resolves-incompleteness: cid005758 + e3-succ-nat: cid005925 + e3-unique-fixpoint: cid005926 + e3_density_2: cid005901 + e3_is_max: cid005924 + e3_is_proof_theory: cid005912 + e3_is_terminal: cid005923 + e3_layer_valid_8_3: cid005874 + e3_resolves_incompleteness: cid005758 + e3_succ_nat: cid005925 + e3_unique_fixpoint: cid005926 + e3p-is-last: cid005881 + e3p-saturates: cid005884 + e3p_is_last: cid005881 + e3p_saturates: cid005884 + e4-iota4-near-one: cid006053 + e4-iota4-near-one-lower: cid006052 + e6-abs-in-range: cid006054 + e6-iota6-near-one: cid006055 + earned-admissibility: cid001565 + earned-boundary-constants: cid000007 + earned-categorical-machine: cid001423 + earned-scalar-codomain-d-is-forced: cid001420 + earned-stage-determined-point: cid001083 + earned-topos: cid001074 + earned-topos-non-boolean: cid005355 + earned_topos_non_boolean: cid005355 + ech-complex: cid001320 + ech-to-derived-comparison: cid001361 + effective-rh-statement: cid001475 + egr-15-4: cid005410 + egr_15_4: cid005410 + eigenvalue-formula: cid005824 + eigenvalue-nonneg: cid005825 + eigenvalue-zero: cid005823 + eigenvalue_formula: cid005824 + eigenvalue_nonneg: cid005825 + eigenvalue_zero: cid005823 + eight-problems: cid005711 + eight_problems: cid005711 + elliptic-complex-exclusion: cid001679 + elliptic-complex-field: cid001094 + elliptic-hartogs-not-natural: cid005630 + elliptic-hyperbolic-dichotomy: cid001095 + elliptic-quaternions: cid001096 + elliptic_hartogs_not_natural: cid005630 + em-sector-verification: cid001563 + empty-set-compatible-3: cid005280 + empty-set-measure-3: cid005283 + empty_set_compatible_3: cid005280 + empty_set_measure_3: cid005283 + engine-only-qft-native: cid005590 + engine_only_qft_native: cid005590 + enr-01-10-1: cid005693 + enr-functor-e0-8-3: cid005864 + enr-functor-e1-8-3: cid005865 + enr-functor-e2-8-3: cid005866 + enr_01_10_1: cid005693 + enr_functor_e0_8_3: cid005864 + enr_functor_e1_8_3: cid005865 + enr_functor_e2_8_3: cid005866 + enrich-finite-5: cid005527 + enrich-iter-3: cid005526 + enrich_finite_5: cid005527 + enrich_iter_3: cid005526 + enriched-bi-square-at-e: cid001510 + enriched-bi-square-comparison: cid001631 + enriched-bs-10-1: cid005696 + enriched_bs_10_1: cid005696 + enrichment-frontier-classification: cid001091 + enrichment-functor: cid001435 + enrichment-functor-enr: cid001502 + enrichment-iteration: cid001353 + enrichment-strict: cid005891 + enrichment-tower-assembly: cid001632 + enrichment_strict: cid005891 + equal-mag-in-a: cid005975 + equal_mag_in_A: cid005975 + equality-violates-k5: cid005746 + equality_violates_K5: cid005746 + eratosthenes-sieve: cid001544 + established-not-conjectural: cid005727 + established_not_conjectural: cid005727 + euclidean-as-static-limit: cid001381 + euler-phi-primorial-3: cid006076 + euler-phi-primorial-4: cid006077 + euler-product-20-4: cid006003 + euler-product-3: cid005852 + euler-sieve-cross: cid005310 + euler_phi_primorial_3: cid006076 + euler_phi_primorial_4: cid006077 + euler_product_20_4: cid006003 + euler_product_3: cid005852 + euler_sieve_cross: cid005310 + even-set-compatible-3: cid005281 + even_set_compatible_3: cid005281 + evolution-operator: cid001272 + evolution-semigroup-thm: cid005554 + evolution-stab-20-4: cid005663 + evolution_semigroup_thm: cid005554 + evolution_stab_20_4: cid005663 + exactly-four-paradoxes: cid005916 + exactly_four_paradoxes: cid005916 + explosion-barrier: cid001183 + explosion_barrier: cid005323 + exponential-damage: cid005739 + exponential_damage: cid005739 + export-implies-central: cid005472 + export-implies-e1: cid005471 + export_implies_central: cid005472 + export_implies_e1: cid005471 + ext-channel-20-4: cid005415 + ext-detects-equal: cid005784 + ext-detects-unequal: cid005785 + ext-min-depth: cid005779 + ext_channel_20_4: cid005415 + ext_detects_equal: cid005784 + ext_detects_unequal: cid005785 + ext_min_depth: cid005779 + extension-bipolar-recovery: cid005414 + extension-in-split-complex-codomain: cid001339 + extension-preserves-stage: cid005411 + extension_bipolar_recovery: cid005414 + extension_preserves_stage: cid005411 + extensionality: cid001160 + extensions-are-omega-germ-transformers: cid001399 + factorization-10-1: cid005697 + factorization_10_1: cid005697 + fanout-violates-k3: cid005745 + fanout_violates_K3: cid005745 + fe-involution-involutive: cid005851 + fe_involution_involutive: cid005851 + fiber-degeneration-at-omega: cid001364 + fiber-t-2: cid001239 + fibered-product-tau-3: cid001240 + fibration-forced: cid005584 + fibration-is-three: cid005586 + fibration-structure: cid001365 + fibration_forced: cid005584 + fibration_is_three: cid005586 + finite-factorization-pasting: cid001630 + finite-is-proto-rational-2: cid005445 + finite-limits-in-cat-tau: cid001070 + finite-spectral-support: cid001392 + finite_is_proto_rational_2: cid005445 + first-disagreement-depth: cid001245 + five-comparison-modes: cid001323 + five-forbidden: cid005743 + five-forbidden-moves: cid001514 + five-generators: cid001004 + five-is-b: cid006014 + five-sectors-exhaustive: cid005897 + five_forbidden: cid005743 + five_is_b: cid006014 + five_sectors_exhaustive: cid005897 + fixed-character: cid001492 + fixed-point-check-3: cid005900 + fixed-points-stage2: cid005796 + fixed_point_check_3: cid005900 + fixed_points_stage2: cid005796 + flat-connection-compatible-2: cid005448 + flat-connection-existence: cid001411 + flat-connection-flat-2: cid005449 + flat-curvature-vanishes-1: cid005452 + flat-curvature-vanishes-2: cid005453 + flat-curvature-vanishing: cid001412 + flat_connection_compatible_2: cid005448 + flat_connection_flat_2: cid005449 + flat_curvature_vanishes_1: cid005452 + flat_curvature_vanishes_2: cid005453 + flow-depth-0: cid005938 + flow-stable-1: cid005940 + flow_depth_0: cid005938 + flow_stable_1: cid005940 + fluid-data-depth-3: cid005928 + fluid-sector-decomposition: cid001483 + fluid_data_depth_3: cid005928 + forbidden-move-mapping: cid001456 + forbidden-move-obstruction-classes: cid001537 + forbidden-move-range: cid005917 + forbidden-moves-8-3: cid005740 + forbidden-moves-distinct-thm: cid005910 + forbidden_move_range: cid005917 + forbidden_moves_8_3: cid005740 + forbidden_moves_distinct_thm: cid005910 + forget-fiber: cid005322 + forget_fiber: cid005322 + four-atom-spectral-dictionary: cid001680 + four-generator-ladder-incompleteness: cid001180 + four-paradox-8-3: cid005909 + four-paradox-diagnostic: cid001640 + four-quadrants: cid005610 + four-ray-rigidity: cid001344 + four_paradox_8_3: cid005909 + four_quadrants: cid005610 + fourier-sigma-swap: cid005271 + fourier_sigma_swap: cid005271 + fourth-quadrant-ladder-collapse: cid001409 + fourth-quadrant-resolution: cid001407 + fta-on-tau-idx: cid001176 + full-code-decode-3: cid005653 + full-enr-functor-8-3: cid005867 + full-export-3-15-3: cid005467 + full-ladder-10-3-3: cid005499 + full-saturation-8-3: cid005921 + full-set-compatible-3: cid005279 + full-set-measure-3: cid005282 + full_code_decode_3: cid005653 + full_enr_functor_8_3: cid005867 + full_export_3_15_3: cid005467 + full_ladder_10_3_3: cid005499 + full_saturation_8_3: cid005921 + full_set_compatible_3: cid005279 + full_set_measure_3: cid005282 + func-10-1: cid005700 + func_10_1: cid005700 + funct-3-30: cid005578 + funct_3_30: cid005578 + functional-equation-involution-j: cid001472 + functions-as-tau-objects: cid001351 + functor-category-collapse: cid001546 + functor-collapse-8-3: cid005856 + functor_collapse_8_3: cid005856 + functorial-12-4: cid005659 + functorial_12_4: cid005659 + functoriality-theorem: cid001628 + fundamental-group-degeneration: cid001376 + fundamental-theorem-of-internal-galois-theory: cid001217 + g-del-numbering-as-nf-address: cid001513 + galois-automorphism: cid001106 + galois-group-of-primorial-stage: cid001107 + gap-10-3-3: cid005498 + gap-constant: cid001490 + gap-constant-is-gap: cid005934 + gap-grows-3-4: cid005933 + gap-indices: cid005734 + gap-meta-theorem: cid001618 + gap-pos-3: cid005932 + gap-stabilization: cid001561 + gap-taxonomy: cid005731 + gap-type: cid001455 + gap_10_3_3: cid005498 + gap_constant_is_gap: cid005934 + gap_grows_3_4: cid005933 + gap_indices: cid005734 + gap_pos_3: cid005932 + gap_taxonomy: cid005731 + gauge-3-complete: cid005955 + gauge-3-count: cid005956 + gauge_3_complete: cid005955 + gauge_3_count: cid005956 + gen-distinct: cid005317 + gen_distinct: cid005317 + genealogical-dag-structure-hinge-7: cid001222 + genealogical-decomposition-spine: cid001039 + generative-counting-principle: cid001084 + generator-distinctness: cid001124 + geo-topology: cid005476 + geo_topology: cid005476 + geodesic: cid001315 + geodesic-complete-1: cid005456 + geodesic-complete-2: cid005457 + geodesic-completeness: cid001358 + geodesic-correct-1: cid005454 + geodesic-correct-2: cid005455 + geodesic_complete_1: cid005456 + geodesic_complete_2: cid005457 + geodesic_correct_1: cid005454 + geodesic_correct_2: cid005455 + geometric-all-eight: cid005475 + geometric-bi-square: cid001311 + geometric-bi-square-theorem: cid001410 + geometric-bisquare-3-15: cid005474 + geometric-implies-central: cid005480 + geometric-pi: cid001263 + geometric-topological-unification: cid001262 + geometric_all_eight: cid005475 + geometric_bisquare_3_15: cid005474 + geometric_implies_central: cid005480 + germ-character-equivalence: cid001331 + germ-hol-12-3: cid005434 + germ-idempotent-supported: cid005438 + germ-reconstructs-identity: cid005437 + germ_hol_12_3: cid005434 + germ_idempotent_supported: cid005438 + germ_reconstructs_identity: cid005437 + global-cartesian-gluing-theorem: cid001642 + global-hartogs: cid001173 + global-hartogs-extension: cid001201 + global_hartogs: cid005302 + gluing-3: cid005576 + gluing-depth-1: cid005836 + gluing-lemma: cid001333 + gluing_3: cid005576 + gluing_depth_1: cid005836 + godel-at-3: cid005754 + godel_at_3: cid005754 + goldbach-100: cid005995 + goldbach-30: cid005994 + goldbach-500: cid006028 + goldbach-at-primorial-levels: cid001656 + goldbach-at-primorial-m: cid001661 + goldbach-gap-characterization: cid001588 + goldbach-gap-parity: cid005735 + goldbach-obstruction: cid001446 + goldbach-primorial-3: cid005997 + goldbach-primorial-m4: cid006029 + goldbach-representation: cid001540 + goldbach-verified-to-500: cid001660 + goldbach_100: cid005995 + goldbach_30: cid005994 + goldbach_500: cid006028 + goldbach_gap_parity: cid005735 + goldbach_primorial_3: cid005997 + goldbach_primorial_m4: cid006029 + grand-grh-3: cid005818 + grand-grh-adelic: cid005817 + grand-grh-effective: cid001476 + grand-grh-from-axiom: cid005822 + grand_grh_3: cid005818 + grand_grh_adelic: cid005817 + grand_grh_from_axiom: cid005822 + greedy-peel-algorithm: cid001033 + grothendieck-topos: cid001194 + growth-base: cid001266 + growth-escape: cid001120 + growth_escape: cid005293 + h-global-sections: cid001416 + h0-global-1: cid005425 + h0-global-2: cid005426 + h0-nonconstant-2: cid005427 + h0_global_1: cid005425 + h0_global_2: cid005426 + h0_nonconstant_2: cid005427 + h1-vanishes-1: cid005423 + h1-vanishes-2: cid005424 + h1_vanishes_1: cid005423 + h1_vanishes_2: cid005424 + hardy-littlewood-constant: cid001448 + hartogs-bulk-8-3: cid005958 + hartogs-bulk-projection: cid001432 + hartogs-extension-uniqueness: cid001398 + hartogs-flow-operator: cid001485 + hartogs-flow-theorem: cid001616 + hartogs-natural-hyp: cid005642 + hartogs-uniq-15-4: cid005418 + hartogs_bulk_8_3: cid005958 + hartogs_natural_hyp: cid005642 + hartogs_uniq_15_4: cid005418 + hausdorff-property: cid001370 + hensel-1-mod-9: cid006042 + hensel-20: cid006038 + hensel-correct-3-2: cid006043 + hensel-tower-11: cid006040 + hensel-unique-11: cid006039 + hensel_1_mod_9: cid006042 + hensel_20: cid006038 + hensel_correct_3_2: cid006043 + hensel_tower_11: cid006040 + hensel_unique_11: cid006039 + high-quality-30: cid005687 + high-quality-count: cid001451 + high_quality_30: cid005687 + hinge-5-integration-tabulation: cid001426 + hinge-6-integration-tabulation: cid001431 + hinge-7-integration-tabulation-bundle-closure: cid001226 + hinge-chain-length: cid005889 + hinge-integration-in-d: cid001681 + hinge-point: cid005899 + hinge-theorem: cid001633 + hinge-theorem-8-3: cid005887 + hinge_chain_length: cid005889 + hinge_point: cid005899 + hinge_theorem_8_3: cid005887 + hl-constant-convergence: cid001666 + hl-constant-decreasing-5: cid006090 + hl-depth-2: cid006094 + hl_constant_decreasing_5: cid006090 + hl_depth_2: cid006094 + hodge-at-e1: cid005947 + hodge-requires-e: cid001562 + hodge_at_e1: cid005947 + hol-assoc-thm: cid005551 + hol-class-12-3: cid005433 + hol-iff-is-12-3: cid005677 + hol-l: cid001064 + hol_assoc_thm: cid005551 + hol_class_12_3: cid005433 + hol_iff_is_12_3: cid005677 + holend-3-10-ok: cid005553 + holend-exponentials: cid001362 + holend-tau-via-pre-yoneda-collapse: cid001425 + holend_3_10_ok: cid005553 + holfun-associativity: cid001147 + holographic-3-15: cid005395 + holographic-principle: cid001232 + holographic-roundtrip: cid005401 + holographic_3_15: cid005395 + holographic_roundtrip: cid005401 + holomorphic-endomorphism-category: cid001276 + holomorphic-iff-idempotent-supported: cid001394 + holomorphic-implies-continuous: cid001368 + holomorphic-presheaf: cid001282 + holomorphic-transformers: cid001038 + holomorphy-before-mappings-ontological-primary: cid001419 + holonomy-not-earned: cid005465 + holonomy-trivial-1: cid005450 + holonomy-trivial-2: cid005451 + holonomy-triviality: cid001357 + holonomy_not_earned: cid005465 + holonomy_trivial_1: cid005450 + holonomy_trivial_2: cid005451 + hom-bipolar-8-3: cid005515 + hom-bipolar-complete: cid005518 + hom-bipolar-decomposition: cid001352 + hom-bipolar-orthogonal: cid005519 + hom-channel-8-3: cid005516 + hom-constant-3: cid005513 + hom-count-3: cid005514 + hom-identity-3: cid005512 + hom-object: cid001289 + hom-stage-reduce-stable: cid005517 + hom-stage-stable: cid005860 + hom-tower-8-3: cid005510 + hom_bipolar_8_3: cid005515 + hom_bipolar_complete: cid005518 + hom_bipolar_orthogonal: cid005519 + hom_channel_8_3: cid005516 + hom_constant_3: cid005513 + hom_count_3: cid005514 + hom_identity_3: cid005512 + hom_stage_reduce_stable: cid005517 + hom_stage_stable: cid005860 + hom_tower_8_3: cid005510 + homology-trivial-1: cid005504 + homology-trivial-2: cid005505 + homology-via-ses: cid001319 + homology_trivial_1: cid005504 + homology_trivial_2: cid005505 + honest-claim-8-3: cid005715 + honest-claim-full-8-3: cid005718 + honest-claim-theorem: cid001639 + honest_claim_8_3: cid005715 + honest_claim_full_8_3: cid005718 + horiz-comp-assoc: cid005529 + horiz_comp_assoc: cid005529 + host-level-10-3: cid005776 + host-level-property: cid001515 + host_level_10_3: cid005776 + hyperbolic-hartogs-natural: cid005629 + hyperbolic_hartogs_natural: cid005629 + hyperfactorization-theorem: cid001171 + id-admissible: cid005788 + id-two-cell-tower-coherent: cid005531 + id_admissible: cid005788 + id_two_cell_tower_coherent: cid005531 + idemp-char-20-4: cid005369 + idemp-decomp-recovery: cid005362 + idemp_char_20_4: cid005369 + idemp_decomp_recovery: cid005362 + idempotent-complete: cid005384 + idempotent-decomposition-lemma: cid001334 + idempotent-supported-character: cid001294 + idempotent_complete: cid005384 + identity-decoder: cid005794 + identity-faithful-reception: cid001101 + identity-map: cid001275 + identity-slippage: cid001099 + identity_decoder: cid005794 + idx-add-injective: cid005292 + idx_add_injective: cid005292 + incompatibility-native: cid005628 + incompatibility_native: cid005628 + incompleteness-as-vm-boundary: cid001636 + incompleteness-vm-10-3: cid005751 + incompleteness_vm_10_3: cid005751 + independence-of-prime-level-actions: cid001549 + index-addition: cid001013 + index-exponentiation: cid001021 + index-multiplication: cid001020 + index-tetration: cid001022 + inf-min-depth: cid005780 + inf_min_depth: cid005780 + inner-product-linearity-2: cid005558 + inner-product-symmetry-2: cid005557 + inner_product_linearity_2: cid005558 + inner_product_symmetry_2: cid005557 + integral: cid001108 + integral-const-one-3: cid005272 + integral-domain: cid001142 + integral-even-2: cid005275 + integral_const_one_3: cid005272 + integral_even_2: cid005275 + interchange-10-3: cid005528 + interchange_10_3: cid005528 + interface-width: cid001498 + interface-width-principle: cid001623 + interior-bipolar-decomposition: cid001241 + interior-from-boundary: cid005297 + interior-j-unit: cid001267 + interior_from_boundary: cid005297 + internal-bipolar-classifier: cid001468 + internal-divisibility: cid001030 + internal-hom: cid001079 + internal-primes: cid001031 + iota-lt-one: cid005893 + iota-pos: cid005894 + iota-ratio-depth-3: cid005895 + iota-ratio-depth-4: cid005896 + iota-tau-explains-fine-structure: cid000011 + iota-value: cid005892 + iota_lt_one: cid005893 + iota_pos: cid005894 + iota_ratio_depth_3: cid005895 + iota_ratio_depth_4: cid005896 + iota_tau_calibration: cid000020 + iota_tau_master_constant: cid000010 + iota_value: cid005892 + is1-12-3: cid005676 + is1_12_3: cid005676 + iterator-ladder: cid001009 + iterator-ladder-saturation: cid001169 + j-replaces-i: cid001386 + j-squared-wave: cid005382 + j_squared_wave: cid005382 + k0-is-first: cid005880 + k0_is_first: cid005880 + k1-strict-order: cid005311 + k2-omega-fixed: cid005312 + k3-orbit-seeded: cid005313 + k4-no-jump: cid005314 + k5-beacon-non-succ: cid005315 + k5-eigenvalue-1: cid005815 + k5-off-diagonal-exclusion: cid001554 + k5-structural-exclusion: cid001206 + k5-structural-exclusion-hinge-8: cid001229 + k5_eigenvalue_1: cid005815 + k6-object-closure: cid005316 + kleene-check-3: cid005795 + kleene-fixed-point: cid001528 + kleene_check_3: cid005795 + l-completeness: cid001359 + l-function-3: cid005821 + l-function-as-spectral-determinant: cid001477 + l-inner-product: cid001316 + l-norm: cid001317 + l2-basis-orthogonal-1: cid005565 + l2-basis-orthogonal-2: cid005566 + l2-completeness-1: cid005563 + l2-completeness-2: cid005564 + l2_basis_orthogonal_1: cid005565 + l2_basis_orthogonal_2: cid005566 + l2_completeness_1: cid005563 + l2_completeness_2: cid005564 + l_function_3: cid005821 + label-3-is-c: cid005820 + label-5-is-b: cid005819 + label-conv-20: cid006009 + label-convergence: cid001605 + label-idem-10-4: cid006011 + label-idempotent-compatibility: cid001552 + label_3_is_C: cid005820 + label_5_is_B: cid005819 + label_conv_20: cid006009 + label_idem_10_4: cid006011 + ladder-8-3: cid005868 + ladder-checker: cid001441 + ladder-collapse: cid005583 + ladder-collapse-native: cid005593 + ladder-has-four-levels: cid005862 + ladder-saturation: cid005331 + ladder_8_3: cid005868 + ladder_collapse: cid005583 + ladder_collapse_native: cid005593 + ladder_has_four_levels: cid005862 + ladder_saturation: cid005331 + langlands-level: cid005699 + langlands-reflection-5-3: cid005978 + langlands-reflection-bridge: cid001460 + langlands_level: cid005699 + langlands_reflection_5_3: cid005978 + laurent-expansion: cid001277 + laurent-range-12-4: cid005567 + laurent_range_12_4: cid005567 + layer-template: cid001436 + ledger-count: cid005723 + ledger-partition: cid005728 + ledger_count: cid005723 + ledger_partition: cid005728 + lem:branch-factorization: cid001335 + lem:character-hartogs: cid001332 + lem:crt-extension: cid001123 + lem:extension-h-tau: cid001339 + lem:germ-character: cid001331 + lem:gluing: cid001333 + lem:growth-escape: cid001120 + lem:idempotent-decomposition: cid001334 + lem:membership-wf: cid001121 + lem:naturality-cylinder: cid001328 + lem:nf-confluence: cid001117 + lem:no-tie: cid001118 + lem:pentation-non-inj: cid001116 + lem:polarity-symmetry: cid001337 + lem:prime-split-support: cid001336 + lem:probe-yoneda: cid001338 + lem:refinement-spectral: cid001329 + lem:remainder-descent: cid001119 + lem:spectral-germ: cid001330 + lem:stagewise-naturality: cid001340 + lem:tail-agreement-propagation: cid001122 + lem:yoneda-application: cid001341 + lemniscate-as-coordinate-limit: cid001342 + lemniscate-characters: cid001053 + lemniscate-holonomy: cid001413 + lemniscate-holonomy-1: cid005458 + lemniscate-holonomy-2: cid005459 + lemniscate-operator-h-l: cid001473 + lemniscate_holonomy_1: cid005458 + lemniscate_holonomy_2: cid005459 + les-stage2: cid005508 + les_stage2: cid005508 + level-circle: cid001260 + levels-match: cid005598 + levels_match: cid005598 + limit-unique-10-4: cid006047 + limit_unique_10_4: cid006047 + linearity-2f-3g-stage2: cid005273 + linearity-census: cid001205 + linearity-identity-stage2: cid005274 + linearity-of-integration: cid001218 + linearity_2f_3g_stage2: cid005273 + linearity_identity_stage2: cid005274 + liouville-categorical-dodge: cid001402 + liouville-dodge: cid005374 + liouville_dodge: cid005374 + lobe-swap-invariance: cid000008 + lobes-as-clopen-sets: cid001346 + local-factor-ind-10-3: cid006004 + local-field-15-4: cid006044 + local-langlands-instance: cid001509 + local-ring-10-4: cid006045 + local_factor_ind_10_3: cid006004 + local_field_15_4: cid006044 + local_ring_10_4: cid006045 + long-exact-sequence: cid001360 + manifold-reduce-idempotent: cid005493 + manifold-tower-coherence: cid005494 + manifold_reduce_idempotent: cid005493 + manifold_tower_coherence: cid005494 + master-constant-calibration: cid000010 + master-constant-calibration-note: cid000005 + master-constant-iota-tau: cid001050 + master-schema-theorem: cid001615 + max-damage-is-3: cid005749 + max-hyp: cid005644 + max-principle-asymmetry: cid005632 + max_damage_is_3: cid005749 + max_hyp: cid005644 + max_principle_asymmetry: cid005632 + md-cycle-42-3: cid005833 + md_cycle_42_3: cid005833 + measure-compatibility: cid001163 + measure-space: cid001104 + measure-total-30: cid005844 + measure-total-6: cid005843 + measure_total_30: cid005844 + measure_total_6: cid005843 + membership-decidability: cid001133 + mero-id-12-4: cid005571 + mero_id_12_4: cid005571 + meromorphic-function: cid001279 + meta-logical-substrate: cid001086 + metric-inequality: cid001132 + millennium-problem-gluing-table: cid001433 + minimal-alphabet: cid005334 + minimal-alphabet-theorem: cid001178 + minimal_alphabet: cid005334 + minkowski-extension: cid001521 + mixed-in-omega: cid005976 + mixed_in_Omega: cid005976 + mode-c-catalog: cid001325 + mode-e-catalog: cid001326 + moduli-100: cid005380 + moduli-singleton: cid005589 + moduli-space: cid001296 + moduli_100: cid005380 + moduli_singleton: cid005589 + monotone-convergence: cid001165 + mordell-weil-analogue: cid001569 + move-bridge-8-3: cid005741 + move-bridge-correspondence: cid001635 + move-correspondence: cid005742 + move-index-0: cid005744 + move_bridge_8_3: cid005741 + move_correspondence: cid005742 + move_index_0: cid005744 + mul-cancel-fails-at-zero: cid001186 + multiplicative-cancellation-iff-positive: cid001187 + mutual-determination: cid005572 + mutual-determination-5-way-equivalence: cid001388 + mutual-determination-schema: cid001470 + mutual_determination: cid005572 + nat-check-20-4: cid005407 + nat-not-internal-set: cid005348 + nat_check_20_4: cid005407 + nat_not_internal_set: cid005348 + native-abstract-turing-machine: cid001523 + native-local-field: cid001466 + natural-transformation: cid001068 + naturality-forces-cylinder-compatibility: cid001328 + neg-neg-char: cid005967 + neg_neg_char: cid005967 + nf-addr-10-1: cid005944 + nf-address-encoding: cid001025 + nf-addressability-theorem: cid001620 + nf-confluence: cid001117 + nf-confluence-church-rosser-for-tau-kernel-hinge-7: cid001221 + nf-deduplication-invariant: cid001003 + nf-discreteness-lemma: cid001560 + nf-zero-unique: cid005957 + nf_addr_10_1: cid005944 + nf_zero_unique: cid005957 + no-additive-inverse: cid001184 + no-barrier-10-1: cid005791 + no-barrier-theorem: cid001626 + no-both-omega-and-archimedean-orthodox: cid005623 + no-both-omega-and-archimedean-tau: cid005624 + no-comprehension: cid005339 + no-elliptic-idempotent: cid005335 + no-free-cartesian-diagonal: cid001156 + no-identity-decoherence: cid001002 + no-independent-dark-sector: cid000021 + no-jump-cover-k4: cid001113 + no-knobs-5-3: cid005989 + no-knobs-8-3: cid005888 + no-knobs-principle: cid001600 + no-knobs-theorem: cid001634 + no-knobs-witness: cid005898 + no-ring-negation: cid001185 + no-second-linearity: cid001166 + no-simul-projection-b: cid005299 + no-simultaneous-projection: cid001146 + no-tie: cid005290 + no-tie-determinism: cid001118 + no-unearned-decimal-diagonal: cid001154 + no-unrestricted-comprehension: cid001155 + no_barrier_10_1: cid005791 + no_both_omega_and_archimedean_orthodox: cid005623 + no_both_omega_and_archimedean_tau: cid005624 + no_comprehension: cid005339 + no_elliptic_idempotent: cid005335 + no_free_cartesian_diagonal: cid005340 + no_identity_decoherence: cid005324 + no_knobs_5_3: cid005989 + no_knobs_8_3: cid005888 + no_knobs_witness: cid005898 + no_simul_projection_b: cid005299 + no_tie: cid005290 + no_unearned_decimal_diagonal: cid005338 + non-emptiness-8-3: cid005853 + non-emptiness-theorem: cid001593 + non_emptiness_8_3: cid005853 + nonconstant-bounded-4: cid005375 + nonconstant_bounded_4: cid005375 + norm-definiteness-2: cid005560 + norm-positivity-2: cid005559 + norm_definiteness_2: cid005560 + norm_positivity_2: cid005559 + not-proto-0: cid005440 + not_proto_0: cid005440 + np-witness-as-canonical-address: cid001500 + np-witness-as-canonical-address-v1: cid001478 + ns-at-e1: cid005945 + ns-causal-4: cid005811 + ns-flow-causal-arrow: cid001654 + ns-level: cid005829 + ns-part5: cid005948 + ns_at_e1: cid005945 + ns_causal_4: cid005811 + ns_level: cid005829 + ns_part5: cid005948 + number-tower: cid001051 + object-closure-k6: cid001115 + obs-width: cid005804 + obs_width: cid005804 + observable-transition: cid001497 + obstruction-100-p2: cid006036 + obstruction-100-p3: cid006037 + obstruction-bounded: cid001663 + obstruction-bounded-5: cid006031 + obstruction-check-3: cid005763 + obstruction-cocycle: cid001536 + obstruction-confined-native: cid005614 + obstruction_100_p2: cid006036 + obstruction_100_p3: cid006037 + obstruction_bounded_5: cid006031 + obstruction_check_3: cid005763 + obstruction_confined_native: cid005614 + ogt-15-4: cid005408 + ogt_15_4: cid005408 + omega-coupling-5: cid005970 + omega-fixed-point-k2: cid001111 + omega-germ-approach: cid001085 + omega-germ-tower-coherent: cid005436 + omega-germ-transformer: cid001060 + omega-germs-12-3: cid005432 + omega-germs-iff-holomorphic-functions: cid001400 + omega-orbit-set-exceeds-alpha: cid005347 + omega-readout: cid001237 + omega-tail-compatible-tower: cid001041 + omega-tau-classifier: cid005354 + omega-tau-subobject-classifier: cid001195 + omega-unique-fixed-seed: cid001137 + omega_coupling_5: cid005970 + omega_germ_tower_coherent: cid005436 + omega_germs_12_3: cid005432 + omega_orbit_set_exceeds_alpha: cid005347 + omega_tau_classifier: cid005354 + one-axiom: cid005721 + one-is-root-3: cid006041 + one-squarefree: cid005688 + one_axiom: cid005721 + one_is_root_3: cid006041 + one_squarefree: cid005688 + ontic-closure: cid001168 + ontic-identity-invariance: cid001213 + ontic-identity-invariance-hinge-8: cid001227 + ontic-ultrametric-hinge-7: cid001224 + ontic_closure: cid005327 + operational-closure: cid001495 + operational-closure-8-3: cid005799 + operational-closure-theorem: cid001649 + operational_closure_8_3: cid005799 + operator-polarity-swap: cid001486 + orbit-countability: cid001127 + orbit-disjoint: cid005329 + orbit-diversity-2: cid005797 + orbit-diversity-3: cid005798 + orbit-rays: cid001008 + orbit-seeded-generation-k3: cid001112 + orbit-set-map: cid001103 + orbit-set-order-bound: cid005345 + orbit_disjoint: cid005329 + orbit_diversity_2: cid005797 + orbit_diversity_3: cid005798 + orbit_set_order_bound: cid005345 + order-bound: cid001162 + orthodox-all-nonempty: cid005626 + orthodox-omega: cid005627 + orthodox-path-no-finite-witnesses: cid005619 + orthodox-path-no-unique-omega: cid005617 + orthodox_all_nonempty: cid005626 + orthodox_omega: cid005627 + orthodox_path_no_finite_witnesses: cid005619 + orthodox_path_no_unique_omega: cid005617 + p-vs-np-as-polynomial-translation-obstruction: cid001582 + p-vs-np-forbidden-triple: cid001655 + pairing-valid: cid005783 + pairing_valid: cid005783 + pairwise-disjointness-of-orbits: cid001126 + paraconsistent-character: cid001150 + paraconsistent-soundness-of-internal-logic: cid001428 + paradox-absorbed-3: cid005902 + paradox-absorption-map: cid001531 + paradox-gap: cid005919 + paradox-move-injective: cid005918 + paradox-resolution-8-3: cid005911 + paradox_absorbed_3: cid005902 + paradox_gap: cid005919 + paradox_move_injective: cid005918 + paradox_resolution_8_3: cid005911 + parallel-postulate: cid001380 + parallel-preservation: cid001406 + parallel-transport: cid001313 + parity-bridge-5-3: cid005987 + parity-bridge-theorem: cid001599 + parity-ne-density: cid005733 + parity_bridge_5_3: cid005987 + parity_ne_density: cid005733 + parseval-2: cid005845 + parseval-6: cid005846 + parseval-identity: cid001648 + parseval_2: cid005845 + parseval_6: cid005846 + partition-1: cid005573 + partition-count-at-primorial: cid001445 + partition-growth: cid001662 + partition-growth-4: cid006030 + partition-m2: cid006033 + partition-m3: cid006034 + partition-m4-pos: cid006035 + partition_1: cid005573 + partition_growth_4: cid006030 + partition_m2: cid006033 + partition_m3: cid006034 + partition_m4_pos: cid006035 + pasch-axiom: cid001379 + passage-to-book-ii: cid001152 + paths-distinct: cid005622 + paths_distinct: cid005622 + pde-classifications-distinct: cid005633 + pde-distinct: cid005646 + pde-type-classification: cid001304 + pde_classifications_distinct: cid005633 + pde_distinct: cid005646 + pentation-channel-exhaustion: cid005330 + pentation-non-injectivity: cid001116 + pentation_channel_exhaustion: cid005330 + physical-admissibility-theorem: cid001643 + physical-e-agent: cid001522 + physical-p-np: cid001644 + physical-realizability-predicate: cid001524 + physics-layer-assembly: cid001621 + pi-10: cid006072 + pi-100: cid006074 + pi-30: cid006073 + pi_10: cid006072 + pi_100: cid006074 + pi_30: cid006073 + pinch-map: cid001251 + poincar-as-gluing-guarantee: cid001557 + poincare-established: cid005724 + poincare_established: cid005724 + point-set-well-defined: cid001363 + polarity-30: cid005675 + polarity-pairing: cid001016 + polarity-swap-involutive: cid005937 + polarity-symmetry: cid001337 + polarity_30: cid005675 + polarity_swap_involutive: cid005937 + polarized-omega-germ: cid001042 + polstream: cid001017 + polynomial-refinement: cid001567 + pos-reg-10-1: cid005950 + pos_reg_10_1: cid005950 + positive-core-closure: cid001139 + positive-regularity-theorem: cid001617 + pre-yoneda-embedding: cid001285 + presheaf-characterization: cid001207 + presheaf-topos: cid001072 + presheaf_characterization: cid005305 + preyoneda-abcd-12-3: cid005668 + preyoneda-bipolar-12-3: cid005667 + preyoneda-composition-12-3: cid005666 + preyoneda-id-tower-coherent: cid005671 + preyoneda-identity-15-4: cid005665 + preyoneda-tower-15-4: cid005664 + preyoneda_abcd_12_3: cid005668 + preyoneda_bipolar_12_3: cid005667 + preyoneda_composition_12_3: cid005666 + preyoneda_id_tower_coherent: cid005671 + preyoneda_identity_15_4: cid005665 + preyoneda_tower_15_4: cid005664 + prime-atom: cid005346 + prime-cofinal-30: cid006060 + prime-counting-in-progressions: cid001014 + prime-enumeration-and-sieve: cid001035 + prime-independence-5-4: cid006020 + prime-polarity-scaling-theorem: cid001612 + prime-polarity-theorem: cid001172 + prime-product-unique: cid005291 + prime-race-4-50: cid005286 + prime-spectral-signature: cid001034 + prime-split-30: cid005674 + prime-split-support: cid001336 + prime_atom: cid005346 + prime_cofinal_30: cid006060 + prime_independence_5_4: cid006020 + prime_product_unique: cid005291 + prime_race_4_50: cid005286 + prime_split_30: cid005674 + primorial-cofinal-50: cid006059 + primorial-cofinality: cid001601 + primorial-div-6: cid006057 + primorial-growth-6: cid006058 + primorial-ladder: cid001464 + primorial-ladder-8: cid006056 + primorial-one: cid006062 + primorial-pos-1: cid005276 + primorial-pos-check: cid005533 + primorial-presheaf: cid001092 + primorial-rh-verification-protocol: cid001555 + primorial-thinness: cid001082 + primorial-three: cid006063 + primorial-unique-5: cid005376 + primorial-zero: cid006061 + primorial_cofinal_50: cid006059 + primorial_div_6: cid006057 + primorial_growth_6: cid006058 + primorial_ladder_8: cid006056 + primorial_one: cid006062 + primorial_pos_1: cid005276 + primorial_pos_check: cid005533 + primorial_three: cid006063 + primorial_unique_5: cid005376 + primorial_zero: cid006061 + princ:ch01-hartogs-bulk: cid001432 + probe-nat-10-3: cid005669 + probe-naturality-iff-yoneda: cid001338 + probe-naturality-is-tower-coherence: cid005672 + probe-naturality-structural: cid005541 + probe-tower-15-4: cid005670 + probe-yoneda-10-3: cid005534 + probe_nat_10_3: cid005669 + probe_naturality_is_tower_coherence: cid005672 + probe_naturality_structural: cid005541 + probe_tower_15_4: cid005670 + probe_yoneda_10_3: cid005534 + product-distributes-over-coproduct: cid005361 + product-meet-10-1: cid005789 + product-meet-collapse: cid001624 + product-universal: cid005352 + product-universal-property: cid001196 + product_distributes_over_coproduct: cid005361 + product_meet_10_1: cid005789 + product_universal: cid005352 + program-monoid: cid001023 + program-monoid-as-linear-calculus: cid001088 + progression-operator-rho: cid001005 + proj-delta-3: cid005549 + proj-recovery-3: cid005550 + proj_delta_3: cid005549 + proj_recovery_3: cid005550 + projection-formula: cid001349 + projector-check-2: cid005841 + projector-check-6: cid005842 + projector_check_2: cid005841 + projector_check_6: cid005842 + proof-categoricity-conditional: cid006107 + proof-fta-tau-idx-lean: cid006108 + proof-hyperfactorization: cid006105 + proof-iota-calibration: cid000022 + proof-prose-six-generator-rigidity: cid006109 + proof-sketch-global-hartogs: cid006106 + proof-theory-as-e: cid001518 + proof-theory-e3-8-3: cid005906 + proof_theory_e3_8_3: cid005906 + prop:abc-gap: cid001592 + prop:add-cancel-universal: cid001140 + prop:additive-multiplicative: cid001584 + prop:adelic-euler-product: cid001551 + prop:arith-preserves-ops: cid001580 + prop:arithmetic-laws: cid001129 + prop:balanced-sector-uniqueness: cid001548 + prop:bayesian-factorization: cid001167 + prop:bipolar-channel-independence: cid001348 + prop:boolean-recovery: cid001136 + prop:boundary-restriction: cid001581 + prop:bridge-algebraic-reduction: cid001576 + prop:bridge-ledger: cid001583 + prop:bridgehead-proposition: cid001570 + prop:bsd-three-ingredient: cid001571 + prop:cech-derived: cid001361 + prop:ch42-holend-exponentials: cid001362 + prop:ch54-tau3-is-manifold: cid001356 + prop:chain-verification-protocol: cid001573 + prop:character-algebra-ring: cid001355 + prop:character-decomposition: cid001354 + prop:completeness-without-topology: cid001550 + prop:congruent-tails-agree: cid001144 + prop:countable-topos: cid001149 + prop:counting-structural: cid001153 + prop:crt-galois: cid001164 + prop:crt-goldbach: cid001587 + prop:crt-witness-decomposition: cid001566 + prop:crt-witness-decomposition-v1: cid001556 + prop:cylinders-are-balls: cid001345 + prop:decomposition-functoriality: cid001350 + prop:defect-contractivity: cid001558 + prop:dim-tau: cid001131 + prop:discrete-spectrum-h-l: cid001553 + prop:distributive-lattice: cid001134 + prop:duality-mutual-determination-z2: cid001572 + prop:e1-strictness-witness: cid001545 + prop:e2-strict-witness: cid001578 + prop:earned-admissibility: cid001565 + prop:em-eliminable: cid001158 + prop:em-sector-verification: cid001563 + prop:enrichment-iteration: cid001353 + prop:four-ray-rigidity: cid001344 + prop:functions-as-objects: cid001351 + prop:functor-collapse: cid001546 + prop:gap-stabilization: cid001561 + prop:gen-distinct: cid001124 + prop:geodesic-completeness: cid001358 + prop:goldbach-gap: cid001588 + prop:hodge-requires-e1: cid001562 + prop:holfun-associativity: cid001147 + prop:holonomy-triviality: cid001357 + prop:hom-bipolar: cid001352 + prop:integral-domain: cid001142 + prop:k5-off-diagonal-exclusion: cid001554 + prop:l2-completeness: cid001359 + prop:label-idem-compatibility: cid001552 + prop:lemniscate-coordinate-limit: cid001342 + prop:lobes-clopen: cid001346 + prop:long-exact-sequence: cid001360 + prop:measure-compatibility: cid001163 + prop:membership-decidable: cid001133 + prop:metric-inequality: cid001132 + prop:monotone-convergence: cid001165 + prop:mordell-weil-analogue: cid001569 + prop:nf-discreteness-lemma: cid001560 + prop:no-comprehension: cid001155 + prop:no-free-cartesian: cid001156 + prop:no-second-linearity: cid001166 + prop:no-simul-projection: cid001146 + prop:no-unearned-decimal: cid001154 + prop:non-boolean: cid001150 + prop:omega-unique-fixed: cid001137 + prop:orbit-countable: cid001127 + prop:orbit-disjoint: cid001126 + prop:orbit-set-extensional: cid001160 + prop:order-bound: cid001162 + prop:ordered-field-axioms: cid001159 + prop:passage-to-book-ii: cid001152 + prop:poincare-gluing-guarantee: cid001557 + prop:polynomial-refinement: cid001567 + prop:positive-closed: cid001139 + prop:prime-independence: cid001549 + prop:primorial-rh-protocol: cid001555 + prop:projection-formula: cid001349 + prop:pvsnp-obstruction: cid001582 + prop:radical-primorial: cid001585 + prop:refinement-resolution: cid001347 + prop:rho-injective: cid001125 + prop:saturation-semantics: cid001579 + prop:sector-by-sector-protocol: cid001564 + prop:sector-independence: cid001145 + prop:sector-inheritance: cid001343 + prop:sector-instantiation-lemma: cid001574 + prop:self-containment: cid001161 + prop:self-enrichment: cid001151 + prop:set-countable: cid001135 + prop:sieve-crt: cid001586 + prop:spectral-resolution: cid001577 + prop:squarefree-dom-thm: cid001591 + prop:sum-zero-iff: cid001138 + prop:terminal-level-characterization: cid001575 + prop:tetration-injective: cid001128 + prop:thin-category: cid001148 + prop:three-condition-sufficiency: cid001559 + prop:three-reading-equivalence-e1: cid001568 + prop:twin-admissibility: cid001589 + prop:twin-gap: cid001590 + prop:ultra-dist-self: cid001143 + prop:ultrametric-replaces-card: cid001157 + prop:well-ordering: cid001130 + prop:zero-vacuous: cid001141 + proto-2: cid005439 + proto-abcd-roundtrip-2: cid005447 + proto-at-stage: cid005446 + proto-code: cid001506 + proto-count-check: cid005444 + proto-examples: cid005442 + proto-range-30: cid005443 + proto-rationality: cid001300 + proto-zero-verified: cid005702 + proto_2: cid005439 + proto_abcd_roundtrip_2: cid005447 + proto_at_stage: cid005446 + proto_count_check: cid005444 + proto_examples: cid005442 + proto_range_30: cid005443 + proto_zero_verified: cid005702 + psh-countable-objects: cid005360 + psh-has-terminal: cid005359 + psh_countable_objects: cid005360 + psh_has_terminal: cid005359 + pure-m-in-b: cid005973 + pure-n-in-c: cid005974 + pure_m_in_B: cid005973 + pure_n_in_C: cid005974 + pvsnp-bridge-break: cid005725 + pvsnp-level: cid005790 + pvsnp-obstruction-3: cid005768 + pvsnp-triple-3: cid005812 + pvsnp-uses-3-moves: cid005748 + pvsnp_bridge_break: cid005725 + pvsnp_level: cid005790 + pvsnp_obstruction_3: cid005768 + pvsnp_triple_3: cid005812 + pvsnp_uses_3_moves: cid005748 + q-expansion-coefficients: cid001525 + qft-gr-incompatible-pde: cid005600 + qft-gr-obstructed: cid005601 + qft-gr-obstructed-native: cid005611 + qft-gr-same-column: cid005599 + qft_gr_incompatible_pde: cid005600 + qft_gr_obstructed: cid005601 + qft_gr_obstructed_native: cid005611 + qft_gr_same_column: cid005599 + quaternion-non-commutativity: cid001211 + r-as-inverse-limit: cid001382 + radial-solenoidal-split: cid001249 + radical-check-30: cid005678 + radical-function: cid001542 + radical-le-30: cid005682 + radical-primorial-4: cid005681 + radical-primorial-5: cid001670 + radical-primorial-identity: cid001585 + radical_check_30: cid005678 + radical_le_30: cid005682 + radical_primorial_4: cid005681 + radical_primorial_5: cid005685 + rank-as-tower-depth: cid001505 + rank-bounded: cid005706 + rank-transfer-maps: cid001011 + rank_bounded: cid005706 + rational-point: cid001504 + reception-instability-hinge-8: cid001231 + reconstruction-20-4: cid006019 + reconstruction-functor: cid001465 + reconstruction_20_4: cid006019 + recovery-30: cid005657 + recovery_30: cid005657 + red-det-3-50: cid005378 + red_det_3_50: cid005378 + reduce-coherence-42: cid006065 + reduce-idempotent: cid005386 + reduce_coherence_42: cid006065 + reduce_idempotent: cid005386 + reduction-gives-naturality: cid005406 + reduction_gives_naturality: cid005406 + refinement-is-abcd: cid005585 + refinement-pressure-uniqueness: cid000009 + refinement-resolution-bound: cid001347 + refinement-spectral-equivalence: cid001329 + refinement_is_abcd: cid005585 + regularity-criterion: cid001395 + removable-singularity: cid001200 + removable_singularity: cid005309 + representation-barrier-theorem: cid001614 + residue: cid001278 + residue-recon-12-3: cid005570 + residue-stage-10-3: cid005568 + residue-theorem: cid001391 + residue_recon_12_3: cid005570 + residue_stage_10_3: cid005568 + restriction-map: cid001081 + rewiring-levels-eq-solenoidal: cid005320 + rewiring-table-complete: cid005595 + rewiring_levels_eq_solenoidal: cid005320 + rewiring_table_complete: cid005595 + rh-bridge-conjectural: cid005720 + rh-bridge-three-layer-structure: cid001637 + rh-gap-5: cid005808 + rh-layers: cid005722 + rh-level: cid005827 + rh-part: cid005830 + rh-spectral-gap-characterization: cid001538 + rh_bridge_conjectural: cid005720 + rh_gap_5: cid005808 + rh_layers: cid005722 + rh_level: cid005827 + rh_part: cid005830 + rho-count-compose: cid005296 + rho-injective: cid005318 + rho-injectivity-per-orbit: cid001125 + rho_count_compose: cid005296 + rho_injective: cid005318 + rigidity-native: cid005591 + rigidity-no-free-parameter: cid005587 + rigidity-non-omega: cid005332 + rigidity-of-tau: cid001174 + rigidity_native: cid005591 + rigidity_no_free_parameter: cid005587 + rigidity_non_omega: cid005332 + root-of-unity-one: cid005269 + root_of_unity_one: cid005269 + roots-of-unity-crt-decomposition: cid001212 + rows-orthodox: cid005597 + rows_orthodox: cid005597 + s-1-as-profinite-limit: cid001383 + s-duality-transport: cid001647 + safe-region-8-3: cid005766 + safe_region_8_3: cid005766 + saturation-at-e: cid001595 + saturation-e3-8-3: cid005857 + saturation-semantic-3: cid005904 + saturation-semantics: cid001579 + saturation_e3_8_3: cid005857 + saturation_semantic_3: cid005904 + sc-j-squared: cid005267 + sc_j_squared: cid005267 + scaling-chain-valid: cid005478 + scaling_chain_valid: cid005478 + scope-order: cid005726 + scope_order: cid005726 + sector-additive: cid005640 + sector-addressability: cid001493 + sector-by-sector-protocol: cid001564 + sector-count-preserved: cid005983 + sector-decomp-5: cid005969 + sector-distributive: cid005368 + sector-exhaustive: cid005977 + sector-fun-independent: cid005298 + sector-independence: cid001145 + sector-inheritance: cid001343 + sector-instantiation-lemma: cid001574 + sector-preservation-5-3: cid005971 + sector-preservation-theorem: cid001597 + sector-product-depth-3: cid005890 + sector_additive: cid005640 + sector_count_preserved: cid005983 + sector_decomp_5: cid005969 + sector_distributive: cid005368 + sector_exhaustive: cid005977 + sector_fun_independent: cid005298 + sector_preservation_5_3: cid005971 + sector_product_depth_3: cid005890 + self-adjointness-of-h-l: cid001609 + self-containment-partition: cid001161 + self-enrich-8-3: cid005511 + self-enrichment: cid001151 + self-enrichment-structure: cid001288 + self-hosting-degree-classification: cid001089 + self-model-8-3: cid005907 + self-model-complete-3: cid005903 + self-model-idempotent-3: cid005905 + self-model-inv-8-3: cid005908 + self-model-levels: cid005913 + self-referential-fixed-point: cid001530 + self_enrich_8_3: cid005511 + self_enrichment: cid005358 + self_model_8_3: cid005907 + self_model_complete_3: cid005903 + self_model_idempotent_3: cid005905 + self_model_inv_8_3: cid005908 + self_model_levels: cid005913 + semigroup-12-4: cid005555 + semigroup_12_4: cid005555 + ses-exact-1: cid005502 + ses-exact-2: cid005503 + ses-stage1: cid005506 + ses-stage2: cid005507 + ses_exact_1: cid005502 + ses_exact_2: cid005503 + ses_stage1: cid005506 + ses_stage2: cid005507 + set-theoretic-operations: cid001048 + set-universe-10-4: cid005775 + set_universe_10_4: cid005775 + seven-plus-seven: cid005886 + seven_plus_seven: cid005886 + shadow-diagram: cid001517 + shadow-diagram-8-3: cid005714 + shadow-identity: cid001100 + shadow_diagram_8_3: cid005714 + sheaf-3: cid005577 + sheaf-axioms: cid001393 + sheaf-cohomology-groups: cid001321 + sheaf_3: cid005577 + sieve-accelerated-abc: cid001450 + sieve-accelerated-goldbach: cid001444 + sieve-correct-200: cid006067 + sieve-correct-50: cid006066 + sieve-correctness: cid001658 + sieve-count-known: cid006068 + sieve-crt-compat-3: cid006071 + sieve-crt-compatibility: cid001586 + sieve-prime-count: cid001442 + sieve-tower-compat-3: cid006069 + sieve-tower-compatibility: cid001659 + sieve_correct_200: cid006067 + sieve_correct_50: cid006066 + sieve_count_known: cid006068 + sieve_crt_compat_3: cid006071 + sieve_tower_compat_3: cid006069 + sigma-anti-holomorphy-and-idempotent-supported-holomorphy: cid001424 + sigma-fixed-swap-id: cid005942 + sigma_fixed_swap_id: cid005942 + sign-count-12: cid005625 + sign-level-count: cid005615 + sign_count_12: cid005625 + sign_level_count: cid005615 + simply-connected-in-category: cid001480 + simultaneous-rung: cid005581 + simultaneous-rung-native: cid005592 + simultaneous-rung-theorem: cid001408 + simultaneous_rung: cid005581 + simultaneous_rung_native: cid005592 + six-comparison-axes: cid001324 + six-generator-rigidity-failure: cid001179 + slippage-breaks-omega: cid005325 + slippage-breaks-unique-omega: cid001214 + slippage_breaks_omega: cid005325 + small-point-3: cid005662 + small_point_3: cid005662 + solenoidal-circle: cid001261 + spectral-algebra: cid001295 + spectral-coefficients: cid001080 + spectral-corr-from-o3: cid005840 + spectral-correspondence-o3: cid005837 + spectral-correspondence-theorem: cid001610 + spectral-decidable: cid006103 + spectral-decomposition: cid001182 + spectral-determination: cid001199 + spectral-determines: cid005307 + spectral-gap-value: cid005826 + spectral-germ-equivalence: cid001330 + spectral-idempotent-supported: cid005397 + spectral-measure: cid001527 + spectral-param-bounded: cid005839 + spectral-param-zero: cid005838 + spectral-parameter-s: cid001474 + spectral-periodic: cid005396 + spectral-polarity: cid001462 + spectral-polarity-5: cid005981 + spectral-projector: cid001526 + spectral-resolution: cid001577 + spectral-resolution-2: cid005847 + spectral-resolution-6: cid005848 + spectral-ring-3-15: cid005390 + spectral-sep-3: cid005652 + spectral-support-3-30: cid005548 + spectral-trichotomy-lemma: cid001606 + spectral-unique: cid005285 + spectral_corr_from_O3: cid005840 + spectral_correspondence_O3: cid005837 + spectral_decidable: cid006103 + spectral_decomposition: cid005284 + spectral_determines: cid005307 + spectral_gap_value: cid005826 + spectral_idempotent_supported: cid005397 + spectral_param_bounded: cid005839 + spectral_param_zero: cid005838 + spectral_periodic: cid005396 + spectral_polarity_5: cid005981 + spectral_resolution_2: cid005847 + spectral_resolution_6: cid005848 + spectral_ring_3_15: cid005390 + spectral_sep_3: cid005652 + spectral_support_3_30: cid005548 + spectral_unique: cid005285 + spine-address-path: cid001253 + split-complex-codomain-h-tau: cid001234 + split-complex-forced: cid001177 + split-complex-scalar-algebra-d: cid001676 + split-complex-scalars: cid001036 + split-complex-zeta: cid001471 + split-cr-equations: cid001059 + split-label-10-4: cid006012 + split_complex_forced: cid005336 + split_label_10_4: cid006012 + squarefree-abc-check: cid001453 + squarefree-dominance: cid001669 + squarefree-dominance-100: cid005684 + squarefree-dominance-theorem: cid001591 + squarefree-dominance-thm: cid005686 + squarefree_dominance_100: cid005684 + squarefree_dominance_thm: cid005686 + stab-germ-0-3: cid005951 + stab_germ_0_3: cid005951 + stabilized-germ: cid001487 + stable-at-3: cid005953 + stable_at_3: cid005953 + stage-2: cid005441 + stage-euclidean: cid005636 + stage-euclidean-check-true: cid005637 + stage-finite-euclidean-geometry: cid001305 + stage-k-cylinder: cid001243 + stage-no-light-cones: cid005635 + stage-size-is-primorial: cid005639 + stage_2: cid005441 + stage_euclidean: cid005636 + stage_euclidean_check_true: cid005637 + stage_no_light_cones: cid005635 + stage_size_is_primorial: cid005639 + stagefun-comp-assoc: cid005301 + stagefun-decompose-12-4: cid005658 + stagefun_comp_assoc: cid005301 + stagefun_decompose_12_4: cid005658 + stages-euclidean-5: cid005645 + stages-from-limit: cid001001 + stages_euclidean_5: cid005645 + stagewise-naturality: cid001340 + stagewise-naturality-strong: cid005403 + stagewise_naturality: cid005402 + stagewise_naturality_strong: cid005403 + static-kernel-tau-0: cid001007 + stone-space: cid001247 + strict-order-k1: cid001110 + strict-remainder-descent: cid001119 + strictly-increasing-nat: cid005927 + strictly_increasing_nat: cid005927 + strictness-8-3: cid005855 + strictness-theorem: cid001594 + strictness_8_3: cid005855 + strong-at-1: cid005954 + strong-defect-functional: cid001491 + strong-sector-at-e: cid001488 + strong_at_1: cid005954 + structural-incompatibility: cid005621 + structural-incompatibility-of-unique-omega-and-archimedean-density: cid001404 + structural-instability: cid001102 + structural-instability-theorem: cid001215 + structural-saturation: cid005861 + structural-sign-classification: cid001302 + structural_incompatibility: cid005621 + structural_instability_theorem: cid005326 + structural_saturation: cid005861 + structure-uniqueness: cid005388 + structure_uniqueness: cid005388 + subobject-classifier-preview: cid001057 + subsec:ch30-crt: cid001045 + succ-idempotent-e3: cid005869 + succ-monotone: cid005883 + succ_idempotent_e3: cid005869 + succ_monotone: cid005883 + sum-less-prod-3: cid005807 + sum-zero-iff-both-zero: cid001138 + sum_less_prod_3: cid005807 + swap-operator-sigma: cid001012 + swap-zero: cid005939 + swap_zero: cid005939 + tab:millennium-gluing: cid001433 + table-12: cid005596 + table_12: cid005596 + tail-agree-downward: cid005303 + tail-agreement-propagation: cid001122 + tail_agree_downward: cid005303 + tau-admissible-point: cid001235 + tau-analytic-atlas: cid001298 + tau-arrow: cid001065 + tau-bayesian-state: cid001019 + tau-complexity-bridge-concrete: cid006104 + tau-dimension: cid001248 + tau-distinct-from-qft: cid005609 + tau-einstein-equation-bridge: cid000012 + tau-engine-inactive: cid005582 + tau-escapes-obstruction: cid005608 + tau-escapes-qft: cid005612 + tau-exponential: cid001322 + tau-exterior-derivative: cid001299 + tau-functor: cid001067 + tau-gr-not-obstructed: cid005607 + tau-holomorphic-function: cid001062 + tau-holomorphic-map: cid001063 + tau-hyp: cid005613 + tau-identity-reduce: cid005304 + tau-identity-theorem: cid001191 + tau-idx-earned-natural-numbers: cid001010 + tau-inter-distrib-union: cid005344 + tau-is-hyperbolic: cid005603 + tau-is-non-archimedean: cid005602 + tau-manifold: cid001297 + tau-manifold-3-15: cid005487 + tau-mem-iff-dvd: cid005343 + tau-membership-relation: cid001047 + tau-not-archimedean-with-gr: cid005605 + tau-not-archimedean-with-qft: cid005604 + tau-obj-countable: cid005328 + tau-path-no-archimedean: cid005618 + tau-path-no-epsilon-delta: cid005620 + tau-qft-not-obstructed: cid005606 + tau-regularity: cid001284 + tau-rh-1: cid005816 + tau-site: cid001071 + tau-spans-three-rungs: cid005580 + tau-strict-mem-wf: cid005349 + tau-weighted-boundary-constants: cid001018 + tau-yoneda-embedding: cid001397 + tau3-is-a-tau-manifold: cid001356 + tau3-manifold-3-15: cid005488 + tau3_manifold_3_15: cid005488 + tauObj_countable: cid005328 + tau_complexity_bridge_concrete: cid006104 + tau_distinct_from_qft: cid005609 + tau_engine_inactive: cid005582 + tau_escapes_obstruction: cid005608 + tau_escapes_qft: cid005612 + tau_gr_not_obstructed: cid005607 + tau_hyp: cid005613 + tau_identity_reduce: cid005304 + tau_inter_distrib_union: cid005344 + tau_is_hyperbolic: cid005603 + tau_is_non_archimedean: cid005602 + tau_manifold_3_15: cid005487 + tau_mem_iff_dvd: cid005343 + tau_not_archimedean_with_gr: cid005605 + tau_not_archimedean_with_qft: cid005604 + tau_path_no_archimedean: cid005618 + tau_path_no_epsilon_delta: cid005620 + tau_qft_not_obstructed: cid005606 + tau_rh_1: cid005816 + tau_spans_three_rungs: cid005580 + tau_strict_mem_wf: cid005349 + taucomplex-ring-axioms: cid001210 + taulib-geometry-central-theorem-calibration: cid000019 + taulib-v2-master: cid000130 + taulib-v3-full-axiom-audit: cid000017 + taureal-archimedean-embedding: cid005268 + taureal-ring-axioms: cid001159 + taureal_archimedean_embedding: cid005268 + teichmueller-lift: cid001046 + teichmueller-style-primitive-recursive-lift: cid001675 + template-invariance-8-3: cid005979 + template-invariance-e3: cid005984 + template-invariance-under-reflection: cid001598 + template_invariance_8_3: cid005979 + template_invariance_e3: cid005984 + terminal-completeness: cid005878 + terminal-depth-1: cid005835 + terminal-level-8-3: cid005922 + terminal-level-characterization: cid001575 + terminal_completeness: cid005878 + terminal_depth_1: cid005835 + terminal_level_8_3: cid005922 + tetration-algebraic-degradation: cid001181 + tetration-injectivity: cid001128 + the-infinity-trade-off: cid001303 + the-physics-quadrant-matrix: cid001307 + the-rewiring-table: cid001306 + the-unification-obstruction: cid001308 + thin-category: cid001148 + thirty-squarefree: cid005689 + thirty_squarefree: cid005689 + thm:abc-100: cid001668 + thm:abc-primorial: cid001657 + thm:adelic-embedding: cid001604 + thm:admissible-germs: cid001418 + thm:algebraic-lemniscate: cid001028 + thm:applied-saturation: cid001641 + thm:archimedean-property: cid001209 + thm:arith-faithful: cid001651 + thm:associativity: cid001390 + thm:asymmetric-determination: cid001405 + thm:base-change-transfer: cid001629 + thm:bc-asymmetry-time: cid001389 + thm:bc-non-collapse: cid001607 + thm:betweenness-axioms: cid001377 + thm:bi-square: cid001208 + thm:bias-primorial: cid001219 + thm:bipolar-euler-product: cid001608 + thm:bndlift-existence: cid001387 + thm:boundary-minimality: cid001374 + thm:bridge-ledger: cid001638 + thm:bridge-necessary: cid001672 + thm:bsd-coherence-theorem: cid001627 + thm:canonical-ladder: cid001596 + thm:cantor-inapplicable: cid001202 + thm:cartesian-closed: cid001198 + thm:categoricity: cid001175 + thm:category-axioms: cid001192 + thm:cauchy-schwarz: cid001414 + thm:central-theorem: cid001401 + thm:circle-profinite: cid001383 + thm:code-decode-bijection: cid001396 + thm:comp-assoc: cid001170 + thm:compactness: cid001369 + thm:complex-field-axioms: cid001210 + thm:composition-closure: cid001190 + thm:confinement-bridge: cid001646 + thm:congruence-axioms: cid001378 + thm:constructive-hensel: cid001603 + thm:countable-additivity: cid001216 + thm:critical-line-theorem: cid001611 + thm:crt-admissible-pos: cid001667 + thm:crt-coherence: cid001188 + thm:crt-decomposition: cid001602 + thm:cylinder-basis: cid001366 + thm:diagonal-discipline: cid001422 + thm:diagonal-free-protection: cid001189 + thm:diagonal-linear: cid001204 + thm:dimension-four: cid001373 + thm:distributivity: cid001197 + thm:e-index: cid001385 + thm:e3-self-model-complete: cid001650 + thm:earned-codomain: cid001420 + thm:enriched-bi-square-comparison: cid001631 + thm:enrichment-tower-assembly: cid001632 + thm:euclidean-static-limit: cid001381 + thm:explosion-barrier: cid001183 + thm:extensions-omega-germs: cid001399 + thm:fiber-degeneration-omega: cid001364 + thm:fibration-structure: cid001365 + thm:finite-factorization-pasting: cid001630 + thm:finite-spectral-support: cid001392 + thm:flat-connection: cid001411 + thm:flat-curvature: cid001412 + thm:four-gen-incomplete: cid001180 + thm:four-paradox-diagnostic: cid001640 + thm:fourth-quadrant-resolution: cid001407 + thm:fta-tau-idx: cid001176 + thm:functoriality-theorem: cid001628 + thm:fundamental-group-degeneration: cid001376 + thm:galois-fundamental: cid001217 + thm:geometric-bisquare: cid001410 + thm:global-cartesian-gluing: cid001642 + thm:global-hartogs: cid001173 + thm:goldbach-500: cid001660 + thm:goldbach-primorial: cid001656 + thm:goldbach-primorial-m4: cid001661 + thm:grothendieck-topos: cid001194 + thm:h0-global: cid001416 + thm:hartogs-flow-theorem: cid001616 + thm:hartogs-uniqueness: cid001398 + thm:hausdorff: cid001370 + thm:hinge-integration: cid001226 + thm:hinge-theorem: cid001633 + thm:hl-decreasing: cid001666 + thm:hol-iff-idempotent: cid001394 + thm:hol-implies-cont: cid001368 + thm:holomorphy-primary: cid001419 + thm:honest-claim: cid001639 + thm:hyperfactorization: cid001171 + thm:idem-supported-holomorphy: cid001424 + thm:incompleteness-as-vm-boundary: cid001636 + thm:integral-linearity: cid001218 + thm:interface-width-principle: cid001623 + thm:j-replaces-i: cid001386 + thm:k5-structural-exclusion: cid001206 + thm:label-convergence: cid001605 + thm:ladder-collapse: cid001409 + thm:ladder-saturation: cid001169 + thm:lemniscate-holonomy: cid001413 + thm:linearity-census: cid001205 + thm:liouville-dodge: cid001402 + thm:main-address-resolution: cid001225 + thm:main-both-equals-one: cid001430 + thm:main-cayley: cid001223 + thm:main-circularity-resolution: cid001429 + thm:main-confluence: cid001221 + thm:main-crt: cid001674 + thm:main-dag: cid001222 + thm:main-diagonal-linear: cid001228 + thm:main-dictionary: cid001680 + thm:main-earned-cat: cid001423 + thm:main-elliptic-exclusion: cid001679 + thm:main-hinge-integration: cid001681 + thm:main-holend: cid001425 + thm:main-iota: cid001677 + thm:main-k5-exclusion: cid001229 + thm:main-lift: cid001675 + thm:main-normalisation: cid001220 + thm:main-ontic-identity: cid001227 + thm:main-paraconsistent-soundness: cid001428 + thm:main-reception: cid001231 + thm:main-resonance: cid001230 + thm:main-ring: cid001673 + thm:main-splitcomplex: cid001676 + thm:main-topos-structure: cid001427 + thm:main-ultrametric: cid001224 + thm:main-uniqueness: cid001678 + thm:master-schema: cid001615 + thm:minimal-alphabet: cid001178 + thm:move-bridge-correspondence: cid001635 + thm:mul-cancel-exactly-pos: cid001187 + thm:mul-cancel-fails-zero: cid001186 + thm:mutual-determination: cid001388 + thm:nf-addressability: cid001620 + thm:no-additive-inverse: cid001184 + thm:no-barrier: cid001626 + thm:no-knobs-principle: cid001600 + thm:no-knobs-theorem: cid001634 + thm:no-ring-negation: cid001185 + thm:non-emptiness: cid001593 + thm:ns-causal-arrow: cid001654 + thm:obstruction-bounded: cid001663 + thm:omega-germs-holomorphic: cid001400 + thm:omega-tau-classifier: cid001195 + thm:ontic-closure: cid001168 + thm:ontic-identity-invariance: cid001213 + thm:operational-closure: cid001649 + thm:parallel-postulate: cid001380 + thm:parallel-preservation: cid001406 + thm:parity-bridge: cid001599 + thm:parseval: cid001648 + thm:partition-growth: cid001662 + thm:pasch: cid001379 + thm:physical-admissibility: cid001643 + thm:physical-p-equals-np: cid001644 + thm:physics-layer-assembly: cid001621 + thm:point-set-well-defined: cid001363 + thm:positive-regularity: cid001617 + thm:presheaf-characterization: cid001207 + thm:prime-polarity: cid001172 + thm:prime-polarity-scaling: cid001612 + thm:primitive-roots: cid001212 + thm:primorial-cofinality: cid001601 + thm:product-meet-collapse: cid001624 + thm:product-universal: cid001196 + thm:pvsnp-forbidden-triple: cid001655 + thm:quaternion-division: cid001211 + thm:radical-primorial-5: cid001670 + thm:real-inverse-limit: cid001382 + thm:regularity-criterion: cid001395 + thm:removable-singularity: cid001200 + thm:representation-barrier: cid001614 + thm:residue-theorem: cid001391 + thm:rh-bridge-three-layer: cid001637 + thm:rigidity: cid001174 + thm:saturation-e3: cid001595 + thm:sduality-transport: cid001647 + thm:sector-preservation: cid001597 + thm:self-adjointness-h-l: cid001609 + thm:sheaf-axioms: cid001393 + thm:sieve-correct: cid001658 + thm:sieve-tower: cid001659 + thm:simultaneous-rung: cid001408 + thm:six-gen-rigidity-fail: cid001179 + thm:slippage-breaks-omega: cid001214 + thm:spectral-correspondence: cid001610 + thm:spectral-decomposition: cid001182 + thm:spectral-determination: cid001199 + thm:spectral-trichotomy: cid001606 + thm:split-complex-forced: cid001177 + thm:squarefree-dominance: cid001669 + thm:strictness: cid001594 + thm:structural-incompatibility: cid001404 + thm:structural-instability: cid001215 + thm:tau-admissibility-collapse: cid001625 + thm:tau-admissibility-collapse-v1: cid001613 + thm:tau-gap-meta-theorem: cid001618 + thm:tau-identity: cid001191 + thm:template-invariance-reflection: cid001598 + thm:tetration-degraded: cid001181 + thm:three-pi: cid001384 + thm:topo-faithful: cid001652 + thm:topology-uniqueness: cid001372 + thm:torus-degeneration: cid001375 + thm:totally-disconnected: cid001371 + thm:tower-coherence: cid001415 + thm:tower-decidable: cid001671 + thm:trade-off-atomicity: cid001417 + thm:translation-failure-boundary: cid001653 + thm:ttm-tau-nativity: cid001622 + thm:twin-500: cid001664 + thm:twin-density: cid001665 + thm:ultrametric-triangle: cid001367 + thm:unique-infinity: cid001203 + thm:universal-admissibility: cid001645 + thm:wave-cr: cid001421 + thm:yang-mills-gap-theorem: cid001619 + thm:yoneda-embedding: cid001397 + thm:yoneda-lemma: cid001193 + three-e1-problems: cid005949 + three-is-c: cid006015 + three-level-equality: cid001024 + three-perspectives-on-pi: cid001384 + three-reading-equivalence-at-e: cid001568 + three-steps-to-e3: cid005870 + three_e1_problems: cid005949 + three_is_c: cid006015 + three_steps_to_e3: cid005870 + threshold-exceeds: cid005750 + threshold_exceeds: cid005750 + topo-faithful-6-2: cid005771 + topo-obstruction-4: cid005765 + topo-translation-10-3: cid005769 + topo_faithful_6_2: cid005771 + topo_obstruction_4: cid005765 + topo_translation_10_3: cid005769 + topological-faithfulness: cid001652 + topological-translation-functor: cid001534 + topology-uniqueness: cid001372 + topos-structure-of-cat-tau: cid001427 + torus-degeneration-theorem: cid001375 + total-disconnectedness: cid001371 + tower-42-3: cid006051 + tower-algebra: cid001105 + tower-atom: cid001032 + tower-coherence: cid001061 + tower-decidable-check: cid001454 + tower-finite-decidable: cid001671 + tower-forced: cid005389 + tower-machine: cid001496 + tower_42_3: cid006051 + tower_finite_decidable: cid005729 + tower_forced: cid005389 + trade-off-atomicity: cid001417 + transition-3: cid005497 + transition_3: cid005497 + translation-boundary-8-3: cid005767 + translation-failure-boundary: cid001653 + translation_boundary_8_3: cid005767 + trichotomy-15-4: cid006078 + trichotomy-depth-1: cid006084 + trichotomy-func-15-3: cid006079 + trichotomy_15_4: cid006078 + trichotomy_depth_1: cid006084 + trichotomy_func_15_3: cid006079 + truth4-logic: cid001037 + ttm-depth-0: cid005802 + ttm-nativity: cid001622 + ttm-preserves-depth: cid005801 + ttm_depth_0: cid005802 + ttm_preserves_depth: cid005801 + twelve-not-squarefree: cid005690 + twelve_not_squarefree: cid005690 + twin-admissibility-fraction: cid001589 + twin-admissibility-fraction-5: cid006092 + twin-admissible-1: cid006095 + twin-admissible-3-pos: cid006096 + twin-count-100: cid006093 + twin-density-primorial: cid001665 + twin-density-primorial-5: cid006089 + twin-gap-characterization: cid001590 + twin-gap-density: cid005736 + twin-prime-density-3: cid005998 + twin-prime-distribution: cid001541 + twin-prime-sieve-count: cid001447 + twin-primes-30: cid005996 + twin-primes-500: cid006088 + twin-primes-to-500: cid001664 + twin_admissibility_fraction_5: cid006092 + twin_admissible_1: cid006095 + twin_admissible_3_pos: cid006096 + twin_count_100: cid006093 + twin_density_primorial_5: cid006089 + twin_gap_density: cid005736 + twin_prime_density_3: cid005998 + twin_primes_30: cid005996 + twin_primes_500: cid006088 + two-cat-assoc-10-3: cid005521 + two-cat-unit-10-3: cid005522 + two-cat-vert-10-3: cid005523 + two-is-x: cid006013 + two-morph-tower-8-3: cid005524 + two-morphism-10-3: cid005525 + two-readout-principle: cid001252 + two-sq-mediates-at-reduce: cid005532 + two_cat_assoc_10_3: cid005521 + two_cat_unit_10_3: cid005522 + two_cat_vert_10_3: cid005523 + two_is_x: cid006013 + two_morph_tower_8_3: cid005524 + two_morphism_10_3: cid005525 + two_sq_mediates_at_reduce: cid005532 + ultra-dist-self: cid001143 + ultrametric-distance: cid001246 + ultrametric-replaces-card: cid005351 + ultrametric-replaces-cardinality: cid001157 + ultrametric-triangle-inequality: cid001367 + ultrametric_replaces_card: cid005351 + unique-infinity: cid005350 + unique-infinity-object: cid001203 + unique_infinity: cid005350 + uniqueness-3-50: cid005381 + uniqueness-from-agreement: cid005412 + uniqueness-of-category-tau: cid001233 + uniqueness_3_50: cid005381 + uniqueness_from_agreement: cid005412 + universal-additive-cancellation: cid001140 + universal-admissibility-theorem: cid001645 + universal-op-3-3: cid005980 + universal-operator: cid001461 + universal_op_3_3: cid005980 + universe-postulate: cid001109 + universe-rank-0: cid005781 + universe-rank-3: cid005782 + universe_rank_0: cid005781 + universe_rank_3: cid005782 + val-p-is-1: cid006048 + val-p2: cid006049 + val-zero: cid006050 + val_p2: cid006049 + val_p_is_1: cid006048 + val_zero: cid006050 + vert-comp-assoc: cid005530 + vert_comp_assoc: cid005530 + wave-equation-cauchy-riemann: cid001421 + wave-type-pde: cid001256 + well-foundedness-of-membership: cid001121 + well-ordering-of-obj-tau: cid001130 + width-bounded: cid005786 + width-principle: cid006100 + width_bounded: cid005786 + width_principle: cid006100 + witness-zero-3: cid005805 + witness_zero_3: cid005805 + yang-mills-gap-theorem: cid001619 + ym-at-e1: cid005946 + ym-coupling-3: cid005936 + ym-gap-5: cid005809 + ym-gap-growth-4: cid005810 + ym-gap-is-tau-gap-3: cid005935 + ym-mass-gap-persistence: cid001539 + ym_at_e1: cid005946 + ym_coupling_3: cid005936 + ym_gap_5: cid005809 + ym_gap_growth_4: cid005810 + ym_gap_is_tau_gap_3: cid005935 + yoneda-app-12-3: cid005430 + yoneda-application: cid001341 + yoneda-bipolar-10-3: cid005537 + yoneda-code-roundtrip: cid005435 + yoneda-embedding: cid001069 + yoneda-faithful-10-3: cid005535 + yoneda-faithful-id: cid005539 + yoneda-full-10-3: cid005536 + yoneda-full-id: cid005540 + yoneda-hom-12-3: cid005431 + yoneda-lemma: cid001193 + yoneda-roundtrip-10-3: cid005538 + yoneda-thin: cid005356 + yoneda_app_12_3: cid005430 + yoneda_bipolar_10_3: cid005537 + yoneda_code_roundtrip: cid005435 + yoneda_faithful_10_3: cid005535 + yoneda_faithful_id: cid005539 + yoneda_full_10_3: cid005536 + yoneda_full_id: cid005540 + yoneda_hom_12_3: cid005431 + yoneda_roundtrip_10_3: cid005538 + yoneda_thin: cid005356 + zero-char-eval: cid005966 + zero-divisors-exist: cid005383 + zero-in-carrier: cid005859 + zero-in-d: cid005972 + zero-is-sigma-fixed: cid005941 + zero-map-compat: cid005520 + zero-rational: cid005705 + zero-vacuous: cid001141 + zero_char_eval: cid005966 + zero_divisors_exist: cid005383 + zero_in_D: cid005972 + zero_in_carrier: cid005859 + zero_is_sigma_fixed: cid005941 + zero_map_compat: cid005520 + zero_rational: cid005705 + zfc-as-e-vm: cid001512 + zfc-has-9-axioms: cid005777 + zfc-is-e2: cid005778 + zfc-vm-8-3: cid005773 + zfc_has_9_axioms: cid005777 + zfc_is_e2: cid005778 + zfc_vm_8_3: cid005773 diff --git a/_data/corpus_v3/cid-index.yaml b/_data/corpus_v3/cid-index.yaml new file mode 100644 index 000000000..fb45ba127 --- /dev/null +++ b/_data/corpus_v3/cid-index.yaml @@ -0,0 +1,15381 @@ +generated_at: '2026-05-23T19:54:54Z' +generator: scripts/build_cid_index.py +items_root: items +total_cids: 1922 +cids: + cid000001: + type: monograph + title: Book II — The τ-Kernel + status: active + visibility: public + primary_alias: BOK0001 + current_version: v1 + source_path: items/monographs/cid000001-bok0001.yaml + cid000002: + type: chapter + title: Book II Chapter 5 — Master Constant Calibration + status: active + visibility: public + primary_alias: CHP0001 + current_version: v1 + source_path: items/chapters/cid000002-chp0001.yaml + cid000003: + type: research_dossier + title: The Panta Rhei Construction Spine + status: active + visibility: public + primary_alias: DOS0001 + current_version: v1 + source_path: items/dossiers/cid000003-dos0001.yaml + cid000004: + type: research_paper + title: Categorical AI + status: active + visibility: public + primary_alias: PAP0001 + current_version: v1 + source_path: items/papers/cid000004-pap0001.yaml + cid000005: + type: research_note + title: Research Note 001 — Master Constant Calibration + status: active + visibility: public + primary_alias: NOT0001 + current_version: v1 + source_path: items/notes/cid000005-not0001.yaml + cid000006: + type: release_packet + title: Corpus v3 Wave 2 Seed Release + status: active + visibility: public + primary_alias: REL0001 + current_version: v1 + source_path: items/release-packets/cid000006-rel0001.yaml + cid000007: + type: definition + title: Earned Boundary Constants + status: canonical + visibility: public + primary_alias: DEF0001 + current_version: v1 + source_path: items/definitions/cid000007-def0001.yaml + cid000008: + type: lemma + title: Lobe-Swap Invariance + status: canonical + visibility: public + primary_alias: LEM0001 + current_version: v1 + source_path: items/lemmas/cid000008-lem0001.yaml + cid000009: + type: proposition + title: Refinement-Pressure Uniqueness + status: canonical + visibility: public + primary_alias: PRP0001 + current_version: v1 + source_path: items/propositions/cid000009-prp0001.yaml + cid000010: + type: theorem + title: Master Constant Calibration + status: canonical + visibility: public + primary_alias: THM0001 + current_version: v1 + source_path: items/theorems/cid000010-thm0001.yaml + cid000011: + type: claim + title: ι_τ Explains the Fine-Structure Constant + status: active + visibility: public + primary_alias: CLM0001 + current_version: v1 + source_path: items/claims/cid000011-clm0001.yaml + cid000012: + type: bridge_claim + title: τ-Einstein Equation Bridges to Orthodox GR + status: active + visibility: public + primary_alias: BRG0001 + current_version: v1 + source_path: items/bridge-claims/cid000012-brg0001.yaml + cid000013: + type: readout_claim + title: Dark Sector as Carrier Capacity-Gradient Readout + status: active + visibility: public + primary_alias: RDO0001 + current_version: v1 + source_path: items/readout-claims/cid000013-rdo0001.yaml + cid000014: + type: axiom + title: Bridge Functor Exists + status: active + visibility: public + primary_alias: AXM0001 + current_version: v1 + source_path: items/axioms/cid000014-axm0001.yaml + cid000015: + type: prediction + title: CMB-S4 Tensor-to-Scalar Ratio r ≈ 0.0136 + status: active + visibility: public + primary_alias: PRD0001 + current_version: v1 + source_path: items/predictions/cid000015-prd0001.yaml + cid000016: + type: falsifier + title: CMB-S4 r Measurement Outside Predicted Band + status: active + visibility: public + primary_alias: FAL0001 + current_version: v1 + source_path: items/falsifiers/cid000016-fal0001.yaml + cid000017: + type: open_problem + title: TauLib v3 Full Axiom Audit + status: active + visibility: public + primary_alias: OPN0001 + current_version: v1 + source_path: items/open-problems/cid000017-opn0001.yaml + cid000018: + type: erratum + title: Book II Chapter 5 — Typo in ι_τ Numerical Value + status: canonical + visibility: public + primary_alias: ERR0001 + current_version: v1 + source_path: items/errata/cid000018-err0001.yaml + cid000019: + type: formal_module + title: TauLib.Geometry.CentralTheorem.Calibration + status: active + visibility: public + primary_alias: MOD0001 + current_version: v1 + source_path: items/formal-modules/cid000019-mod0001.yaml + cid000020: + type: formal_theorem + title: TauLib.Geometry.CentralTheorem.iota_tau_calibration + status: active + visibility: public + primary_alias: FTH0001 + current_version: v1 + source_path: items/formal-theorems/cid000020-fth0001.yaml + cid000021: + type: result + title: No Independent Dark-Sector Particle Required in the τ Readout + status: active + visibility: public + primary_alias: RSL0001 + current_version: v1 + source_path: items/results/cid000021-rsl0001.yaml + cid000022: + type: proof + title: Proof of Master Constant Calibration + status: canonical + visibility: public + primary_alias: PRF0001 + current_version: v1 + source_path: items/proofs/cid000022-prf0001.yaml + cid000023: + type: monograph + title: Book I — Categorical Foundations + status: active + visibility: public + primary_alias: BOK0002 + current_version: v1 + source_path: items/monographs/cid000023-bok0002.yaml + cid000024: + type: monograph + title: Book III — The τ-Manifold + status: active + visibility: public + primary_alias: BOK0003 + current_version: v1 + source_path: items/monographs/cid000024-bok0003.yaml + cid000025: + type: monograph + title: Book IV — Categorical Microcosm + status: active + visibility: public + primary_alias: BOK0004 + current_version: v1 + source_path: items/monographs/cid000025-bok0004.yaml + cid000026: + type: monograph + title: Book V — Categorical Macrocosm + status: active + visibility: public + primary_alias: BOK0005 + current_version: v1 + source_path: items/monographs/cid000026-bok0005.yaml + cid000027: + type: monograph + title: Book VI — The Code of Life + status: active + visibility: public + primary_alias: BOK0006 + current_version: v1 + source_path: items/monographs/cid000027-bok0006.yaml + cid000028: + type: monograph + title: Book VII — Logos and the Ontic Closure + status: active + visibility: public + primary_alias: BOK0007 + current_version: v1 + source_path: items/monographs/cid000028-bok0007.yaml + cid000029: + type: construction_step + title: Non-import discipline + status: active + visibility: public + primary_alias: CNS0001 + current_version: v1 + source_path: items/construction-steps/cid000029-cns0001.yaml + cid000030: + type: construction_step + title: Coherence kernel + status: active + visibility: public + primary_alias: CNS0002 + current_version: v1 + source_path: items/construction-steps/cid000030-cns0002.yaml + cid000031: + type: construction_step + title: Orbit architecture + status: active + visibility: public + primary_alias: CNS0003 + current_version: v1 + source_path: items/construction-steps/cid000031-cns0003.yaml + cid000032: + type: construction_step + title: Four-block order readout + status: active + visibility: public + primary_alias: CNS0004 + current_version: v1 + source_path: items/construction-steps/cid000032-cns0004.yaml + cid000033: + type: construction_step + title: Nonredundant fold strata + status: active + visibility: public + primary_alias: CNS0005 + current_version: v1 + source_path: items/construction-steps/cid000033-cns0005.yaml + cid000034: + type: construction_step + title: Counting as construction + status: active + visibility: public + primary_alias: CNS0006 + current_version: v1 + source_path: items/construction-steps/cid000034-cns0006.yaml + cid000035: + type: construction_step + title: Iterator ladder + status: active + visibility: public + primary_alias: CNS0007 + current_version: v1 + source_path: items/construction-steps/cid000035-cns0007.yaml + cid000036: + type: construction_step + title: Address resolution, not calculation + status: active + visibility: public + primary_alias: CNS0008 + current_version: v1 + source_path: items/construction-steps/cid000036-cns0008.yaml + cid000037: + type: construction_step + title: Hyperfactorization and ABCD chart + status: active + visibility: public + primary_alias: CNS0009 + current_version: v1 + source_path: items/construction-steps/cid000037-cns0009.yaml + cid000038: + type: construction_step + title: Prime polarity + status: active + visibility: public + primary_alias: CNS0010 + current_version: v1 + source_path: items/construction-steps/cid000038-cns0010.yaml + cid000039: + type: construction_step + title: Internal set theory + status: active + visibility: public + primary_alias: CNS0011 + current_version: v1 + source_path: items/construction-steps/cid000039-cns0011.yaml + cid000040: + type: construction_step + title: Split-complex boundary algebra + status: active + visibility: public + primary_alias: CNS0012 + current_version: v1 + source_path: items/construction-steps/cid000040-cns0012.yaml + cid000041: + type: construction_step + title: τ-holomorphy as ω-germ transformer + status: active + visibility: public + primary_alias: CNS0013 + current_version: v1 + source_path: items/construction-steps/cid000041-cns0013.yaml + cid000042: + type: construction_step + title: Four-valued τ-topos + status: active + visibility: public + primary_alias: CNS0014 + current_version: v1 + source_path: items/construction-steps/cid000042-cns0014.yaml + cid000043: + type: construction_step + title: Lemniscate characters + status: active + visibility: public + primary_alias: CNS0015 + current_version: v1 + source_path: items/construction-steps/cid000043-cns0015.yaml + cid000044: + type: construction_step + title: Global Hartogs + status: active + visibility: public + primary_alias: CNS0016 + current_version: v1 + source_path: items/construction-steps/cid000044-cns0016.yaml + cid000045: + type: construction_step + title: Boundary-first inversion + status: active + visibility: public + primary_alias: CNS0017 + current_version: v1 + source_path: items/construction-steps/cid000045-cns0017.yaml + cid000046: + type: construction_step + title: τ^3 as geometric readout τ^1×_f T^2 + status: active + visibility: public + primary_alias: CNS0018 + current_version: v1 + source_path: items/construction-steps/cid000046-cns0018.yaml + cid000047: + type: construction_step + title: Cylinders and ultrametric topology + status: active + visibility: public + primary_alias: CNS0019 + current_version: v1 + source_path: items/construction-steps/cid000047-cns0019.yaml + cid000048: + type: construction_step + title: Stone/profinite shape and two-readout principle + status: active + visibility: public + primary_alias: CNS0020 + current_version: v1 + source_path: items/construction-steps/cid000048-cns0020.yaml + cid000049: + type: construction_step + title: Geometric lemniscate at ω + status: active + visibility: public + primary_alias: CNS0021 + current_version: v1 + source_path: items/construction-steps/cid000049-cns0021.yaml + cid000050: + type: construction_step + title: Tarski geometry and wave-type causal precursor + status: active + visibility: public + primary_alias: CNS0022 + current_version: v1 + source_path: items/construction-steps/cid000050-cns0022.yaml + cid000051: + type: construction_step + title: Earned constants pi_tau,e_tau,j + status: active + visibility: public + primary_alias: CNS0023 + current_version: v1 + source_path: items/construction-steps/cid000051-cns0023.yaml + cid000052: + type: construction_step + title: Master constant iota_tau + status: active + visibility: public + primary_alias: CNS0024 + current_version: v1 + source_path: items/construction-steps/cid000052-cns0024.yaml + cid000053: + type: construction_step + title: Local Hartogs and mutual determination + status: active + visibility: public + primary_alias: CNS0025 + current_version: v1 + source_path: items/construction-steps/cid000053-cns0025.yaml + cid000054: + type: construction_step + title: Regularity and finite spectral support + status: active + visibility: public + primary_alias: CNS0026 + current_version: v1 + source_path: items/construction-steps/cid000054-cns0026.yaml + cid000055: + type: construction_step + title: Sheaf coherence, self-enrichment, Yoneda + status: active + visibility: public + primary_alias: CNS0027 + current_version: v1 + source_path: items/construction-steps/cid000055-cns0027.yaml + cid000056: + type: construction_step + title: Central Theorem route + status: active + visibility: public + primary_alias: CNS0028 + current_version: v1 + source_path: items/construction-steps/cid000056-cns0028.yaml + cid000057: + type: construction_step + title: τ-manifold and Book III handoff + status: active + visibility: public + primary_alias: CNS0029 + current_version: v1 + source_path: items/construction-steps/cid000057-cns0029.yaml + cid000058: + type: construction_step + title: Kernel minimality and categoricity + status: active + visibility: public + primary_alias: CNS0030 + current_version: v1 + source_path: items/construction-steps/cid000058-cns0030.yaml + cid000059: + type: construction_step + title: E_0 is not yet physics + status: active + visibility: public + primary_alias: CNS0031 + current_version: v1 + source_path: items/construction-steps/cid000059-cns0031.yaml + cid000060: + type: construction_step + title: No bare-point ontology + status: active + visibility: public + primary_alias: CNS0032 + current_version: v1 + source_path: items/construction-steps/cid000060-cns0032.yaml + cid000061: + type: construction_step + title: E_1 hom-object loci + status: active + visibility: public + primary_alias: CNS0033 + current_version: v1 + source_path: items/construction-steps/cid000061-cns0033.yaml + cid000062: + type: construction_step + title: Central-Theorem constraint on physical loci + status: active + visibility: public + primary_alias: CNS0034 + current_version: v1 + source_path: items/construction-steps/cid000062-cns0034.yaml + cid000063: + type: construction_step + title: Hartogs bulk projection + status: active + visibility: public + primary_alias: CNS0035 + current_version: v1 + source_path: items/construction-steps/cid000063-cns0035.yaml + cid000064: + type: construction_step + title: Local-to-global gluing burden + status: active + visibility: public + primary_alias: CNS0036 + current_version: v1 + source_path: items/construction-steps/cid000064-cns0036.yaml + cid000065: + type: construction_step + title: Carrier guarantees and external shadows + status: active + visibility: public + primary_alias: CNS0037 + current_version: v1 + source_path: items/construction-steps/cid000065-cns0037.yaml + cid000066: + type: construction_step + title: Global Cartesian gluing and decompactification + status: active + visibility: public + primary_alias: CNS0038 + current_version: v1 + source_path: items/construction-steps/cid000066-cns0038.yaml + cid000067: + type: construction_step + title: 3+1 spacetime signature + status: active + visibility: public + primary_alias: CNS0039 + current_version: v1 + source_path: items/construction-steps/cid000067-cns0039.yaml + cid000068: + type: construction_step + title: 4+1 sector template + status: active + visibility: public + primary_alias: CNS0040 + current_version: v1 + source_path: items/construction-steps/cid000068-cns0040.yaml + cid000069: + type: construction_step + title: No Knobs Principle + status: active + visibility: public + primary_alias: CNS0041 + current_version: v1 + source_path: items/construction-steps/cid000069-cns0041.yaml + cid000070: + type: construction_step + title: Hinge Theorem export + status: active + visibility: public + primary_alias: CNS0042 + current_version: v1 + source_path: items/construction-steps/cid000070-cns0042.yaml + cid000071: + type: construction_step + title: Layered equivalence + status: active + visibility: public + primary_alias: CNS0043 + current_version: v1 + source_path: items/construction-steps/cid000071-cns0043.yaml + cid000072: + type: construction_step + title: No stage/actor split + status: active + visibility: public + primary_alias: CNS0044 + current_version: v1 + source_path: items/construction-steps/cid000072-cns0044.yaml + cid000073: + type: construction_step + title: Self-description on the T^2 fiber + status: active + visibility: public + primary_alias: CNS0045 + current_version: v1 + source_path: items/construction-steps/cid000073-cns0045.yaml + cid000074: + type: construction_step + title: Defect-bundle formalism + status: active + visibility: public + primary_alias: CNS0046 + current_version: v1 + source_path: items/construction-steps/cid000074-cns0046.yaml + cid000075: + type: construction_step + title: Photon as null transport + status: active + visibility: public + primary_alias: CNS0047 + current_version: v1 + source_path: items/construction-steps/cid000075-cns0047.yaml + cid000076: + type: construction_step + title: Neutron as minimal stable defect + status: active + visibility: public + primary_alias: CNS0048 + current_version: v1 + source_path: items/construction-steps/cid000076-cns0048.yaml + cid000077: + type: construction_step + title: Beta differentiation and hydrogen closure + status: active + visibility: public + primary_alias: CNS0049 + current_version: v1 + source_path: items/construction-steps/cid000077-cns0049.yaml + cid000078: + type: construction_step + title: Neutrino ontology and Majorana readout + status: active + visibility: public + primary_alias: CNS0050 + current_version: v1 + source_path: items/construction-steps/cid000078-cns0050.yaml + cid000079: + type: construction_step + title: Quantum mechanics as CR-address obstruction + status: active + visibility: public + primary_alias: CNS0051 + current_version: v1 + source_path: items/construction-steps/cid000079-cns0051.yaml + cid000080: + type: construction_step + title: Uncertainty as geometric obstruction + status: active + visibility: public + primary_alias: CNS0052 + current_version: v1 + source_path: items/construction-steps/cid000080-cns0052.yaml + cid000081: + type: construction_step + title: Physical quantity semantics + status: active + visibility: public + primary_alias: CNS0053 + current_version: v1 + source_path: items/construction-steps/cid000081-cns0053.yaml + cid000082: + type: construction_step + title: Forces as holonomy tools + status: active + visibility: public + primary_alias: CNS0054 + current_version: v1 + source_path: items/construction-steps/cid000082-cns0054.yaml + cid000083: + type: construction_step + title: Confinement as address irresolvability + status: active + visibility: public + primary_alias: CNS0055 + current_version: v1 + source_path: items/construction-steps/cid000083-cns0055.yaml + cid000084: + type: construction_step + title: Three generations from topology + status: active + visibility: public + primary_alias: CNS0056 + current_version: v1 + source_path: items/construction-steps/cid000084-cns0056.yaml + cid000085: + type: construction_step + title: Sector atlas, masses, nuclei + status: active + visibility: public + primary_alias: CNS0057 + current_version: v1 + source_path: items/construction-steps/cid000085-cns0057.yaml + cid000086: + type: construction_step + title: 'Book IV closure: coupling ledger and microcosm completion' + status: active + visibility: public + primary_alias: CNS0058 + current_version: v1 + source_path: items/construction-steps/cid000086-cns0058.yaml + cid000087: + type: construction_step + title: Fiber/base completion + status: active + visibility: public + primary_alias: CNS0059 + current_version: v1 + source_path: items/construction-steps/cid000087-cns0059.yaml + cid000088: + type: construction_step + title: Proto-time to proper time + status: active + visibility: public + primary_alias: CNS0060 + current_version: v1 + source_path: items/construction-steps/cid000088-cns0060.yaml + cid000089: + type: construction_step + title: Causality, worldlines, and null-exchange distance + status: active + visibility: public + primary_alias: CNS0061 + current_version: v1 + source_path: items/construction-steps/cid000089-cns0061.yaml + cid000090: + type: construction_step + title: Redshift and Hubble readout + status: active + visibility: public + primary_alias: CNS0062 + current_version: v1 + source_path: items/construction-steps/cid000090-cns0062.yaml + cid000091: + type: construction_step + title: Gravity as frame holonomy + status: active + visibility: public + primary_alias: CNS0063 + current_version: v1 + source_path: items/construction-steps/cid000091-cns0063.yaml + cid000092: + type: construction_step + title: τ-Einstein identity + status: active + visibility: public + primary_alias: CNS0064 + current_version: v1 + source_path: items/construction-steps/cid000092-cns0064.yaml + cid000093: + type: construction_step + title: Lorentzian signature from j^2=+1 + status: active + visibility: public + primary_alias: CNS0065 + current_version: v1 + source_path: items/construction-steps/cid000093-cns0065.yaml + cid000094: + type: construction_step + title: Thermodynamic inversion + status: active + visibility: public + primary_alias: CNS0066 + current_version: v1 + source_path: items/construction-steps/cid000094-cns0066.yaml + cid000095: + type: construction_step + title: Heat, temperature, vacuum, and dark-energy readout + status: active + visibility: public + primary_alias: CNS0067 + current_version: v1 + source_path: items/construction-steps/cid000095-cns0067.yaml + cid000096: + type: construction_step + title: Macro defect lift and collective dynamics + status: active + visibility: public + primary_alias: CNS0068 + current_version: v1 + source_path: items/construction-steps/cid000096-cns0068.yaml + cid000097: + type: construction_step + title: Macro regularity, turbulence, and charge/flux + status: active + visibility: public + primary_alias: CNS0069 + current_version: v1 + source_path: items/construction-steps/cid000097-cns0069.yaml + cid000098: + type: construction_step + title: Classical mechanics as coarse readout + status: active + visibility: public + primary_alias: CNS0070 + current_version: v1 + source_path: items/construction-steps/cid000098-cns0070.yaml + cid000099: + type: construction_step + title: Galaxies, rotation curves, and compact objects + status: active + visibility: public + primary_alias: CNS0071 + current_version: v1 + source_path: items/construction-steps/cid000099-cns0071.yaml + cid000100: + type: construction_step + title: Black holes as topological channel openings + status: active + visibility: public + primary_alias: CNS0072 + current_version: v1 + source_path: items/construction-steps/cid000100-cns0072.yaml + cid000101: + type: construction_step + title: Cosmic web as Wilson-loop carrier channels + status: active + visibility: public + primary_alias: CNS0073 + current_version: v1 + source_path: items/construction-steps/cid000101-cns0073.yaml + cid000102: + type: construction_step + title: Big Bang, threshold ladder, and endstate + status: active + visibility: public + primary_alias: CNS0074 + current_version: v1 + source_path: items/construction-steps/cid000102-cns0074.yaml + cid000103: + type: construction_step + title: Book V late-stage synthesis and macrocosm export surface + status: active + visibility: public + primary_alias: CNS0075 + current_version: v1 + source_path: items/construction-steps/cid000103-cns0075.yaml + cid000104: + type: construction_step + title: E_1 physics complete + status: active + visibility: public + primary_alias: CNS0076 + current_version: v1 + source_path: items/construction-steps/cid000104-cns0076.yaml + cid000105: + type: construction_step + title: 'E_1 to E_2: life as predicate layer' + status: active + visibility: public + primary_alias: CNS0077 + current_version: v1 + source_path: items/construction-steps/cid000105-cns0077.yaml + cid000106: + type: construction_step + title: Distinction plus SelfDesc + status: active + visibility: public + primary_alias: CNS0078 + current_version: v1 + source_path: items/construction-steps/cid000106-cns0078.yaml + cid000107: + type: construction_step + title: Layer separation from physics + status: active + visibility: public + primary_alias: CNS0079 + current_version: v1 + source_path: items/construction-steps/cid000107-cns0079.yaml + cid000108: + type: construction_step + title: Life Loop and metabolic closure + status: active + visibility: public + primary_alias: CNS0080 + current_version: v1 + source_path: items/construction-steps/cid000108-cns0080.yaml + cid000109: + type: construction_step + title: Boundary and energy coupling + status: active + visibility: public + primary_alias: CNS0081 + current_version: v1 + source_path: items/construction-steps/cid000109-cns0081.yaml + cid000110: + type: construction_step + title: Code, genotype, and heredity + status: active + visibility: public + primary_alias: CNS0082 + current_version: v1 + source_path: items/construction-steps/cid000110-cns0082.yaml + cid000111: + type: construction_step + title: Replication, development, and morphogenesis + status: active + visibility: public + primary_alias: CNS0083 + current_version: v1 + source_path: items/construction-steps/cid000111-cns0083.yaml + cid000112: + type: construction_step + title: 4+1 life-sector template + status: active + visibility: public + primary_alias: CNS0084 + current_version: v1 + source_path: items/construction-steps/cid000112-cns0084.yaml + cid000113: + type: construction_step + title: Evolution as PPAS selection + status: active + visibility: public + primary_alias: CNS0085 + current_version: v1 + source_path: items/construction-steps/cid000113-cns0085.yaml + cid000114: + type: construction_step + title: Immunity and species boundaries + status: active + visibility: public + primary_alias: CNS0086 + current_version: v1 + source_path: items/construction-steps/cid000114-cns0086.yaml + cid000115: + type: construction_step + title: Neural bridge to self-modeling + status: active + visibility: public + primary_alias: CNS0087 + current_version: v1 + source_path: items/construction-steps/cid000115-cns0087.yaml + cid000116: + type: construction_step + title: Structural self-modeling boundary + status: active + visibility: public + primary_alias: CNS0088 + current_version: v1 + source_path: items/construction-steps/cid000116-cns0088.yaml + cid000117: + type: construction_step + title: Language as extended lemniscate + status: active + visibility: public + primary_alias: CNS0089 + current_version: v1 + source_path: items/construction-steps/cid000117-cns0089.yaml + cid000118: + type: construction_step + title: E_2 export to E_3 + status: active + visibility: public + primary_alias: CNS0090 + current_version: v1 + source_path: items/construction-steps/cid000118-cns0090.yaml + cid000119: + type: construction_step + title: 'E_2 to E_3: reflective structure' + status: active + visibility: public + primary_alias: CNS0091 + current_version: v1 + source_path: items/construction-steps/cid000119-cns0091.yaml + cid000120: + type: construction_step + title: MetaDecode and SelfDesc-of-SelfDesc + status: active + visibility: public + primary_alias: CNS0092 + current_version: v1 + source_path: items/construction-steps/cid000120-cns0092.yaml + cid000121: + type: construction_step + title: Four registers of reason + status: active + visibility: public + primary_alias: CNS0093 + current_version: v1 + source_path: items/construction-steps/cid000121-cns0093.yaml + cid000122: + type: construction_step + title: Reflective 4+1 register template + status: active + visibility: public + primary_alias: CNS0094 + current_version: v1 + source_path: items/construction-steps/cid000122-cns0094.yaml + cid000123: + type: construction_step + title: Formal systems as represented objects + status: active + visibility: public + primary_alias: CNS0095 + current_version: v1 + source_path: items/construction-steps/cid000123-cns0095.yaml + cid000124: + type: construction_step + title: Gödel and halting boundary + status: active + visibility: public + primary_alias: CNS0096 + current_version: v1 + source_path: items/construction-steps/cid000124-cns0096.yaml + cid000125: + type: construction_step + title: Language, meaning, and proof arc + status: active + visibility: public + primary_alias: CNS0097 + current_version: v1 + source_path: items/construction-steps/cid000125-cns0097.yaml + cid000126: + type: construction_step + title: Logic, inference, and truth at reflective scale + status: active + visibility: public + primary_alias: CNS0098 + current_version: v1 + source_path: items/construction-steps/cid000126-cns0098.yaml + cid000127: + type: construction_step + title: Mind, consciousness, and commitment access + status: active + visibility: public + primary_alias: CNS0099 + current_version: v1 + source_path: items/construction-steps/cid000127-cns0099.yaml + cid000128: + type: construction_step + title: Logos boundary and ontic-closure burden + status: active + visibility: public + primary_alias: CNS0100 + current_version: v1 + source_path: items/construction-steps/cid000128-cns0100.yaml + cid000129: + type: release_packet + title: Corpus v2 Release v4.0 + status: canonical + visibility: public + primary_alias: REL0002 + current_version: v1 + source_path: items/release-packets/cid000129-rel0002.yaml + cid000130: + type: formal_module + title: TauLib v2 Master Module + status: active + visibility: public + primary_alias: MOD0002 + current_version: v1 + source_path: items/formal-modules/cid000130-mod0002.yaml + cid001000: + type: corollary + title: Constructive Encoding via ABCD + status: canonical + visibility: public + primary_alias: COR0001 + current_version: v1 + source_path: items/corollaries/cid001000-cor0001.yaml + cid001001: + type: corollary + title: Stages from Limit + status: canonical + visibility: public + primary_alias: COR0002 + current_version: v1 + source_path: items/corollaries/cid001001-cor0002.yaml + cid001002: + type: corollary + title: No Identity Decoherence + status: canonical + visibility: public + primary_alias: COR0003 + current_version: v1 + source_path: items/corollaries/cid001002-cor0003.yaml + cid001003: + type: corollary + title: NF Deduplication Invariant + status: canonical + visibility: public + primary_alias: COR0004 + current_version: v1 + source_path: items/corollaries/cid001003-cor0004.yaml + cid001004: + type: definition + title: Five Generators + status: canonical + visibility: public + primary_alias: DEF0002 + current_version: v1 + source_path: items/definitions/cid001004-def0002.yaml + cid001005: + type: definition + title: Progression Operator rho + status: canonical + visibility: public + primary_alias: DEF0003 + current_version: v1 + source_path: items/definitions/cid001005-def0003.yaml + cid001006: + type: definition + title: Diagonal Discipline + status: canonical + visibility: public + primary_alias: DEF0004 + current_version: v1 + source_path: items/definitions/cid001006-def0004.yaml + cid001007: + type: definition + title: Static Kernel tau_0 + status: canonical + visibility: public + primary_alias: DEF0005 + current_version: v1 + source_path: items/definitions/cid001007-def0005.yaml + cid001008: + type: definition + title: Orbit Rays + status: canonical + visibility: public + primary_alias: DEF0006 + current_version: v1 + source_path: items/definitions/cid001008-def0006.yaml + cid001009: + type: definition + title: Iterator Ladder + status: canonical + visibility: public + primary_alias: DEF0007 + current_version: v1 + source_path: items/definitions/cid001009-def0007.yaml + cid001010: + type: definition + title: tau-Idx (Earned Natural Numbers) + status: canonical + visibility: public + primary_alias: DEF0008 + current_version: v1 + source_path: items/definitions/cid001010-def0008.yaml + cid001011: + type: definition + title: Rank Transfer Maps + status: canonical + visibility: public + primary_alias: DEF0009 + current_version: v1 + source_path: items/definitions/cid001011-def0009.yaml + cid001012: + type: definition + title: Swap Operator sigma + status: canonical + visibility: public + primary_alias: DEF0010 + current_version: v1 + source_path: items/definitions/cid001012-def0010.yaml + cid001013: + type: definition + title: Index Addition + status: canonical + visibility: public + primary_alias: DEF0011 + current_version: v1 + source_path: items/definitions/cid001013-def0011.yaml + cid001014: + type: definition + title: Prime Counting in Progressions + status: draft + visibility: private + primary_alias: DEF0012 + current_version: v1 + source_path: items/definitions/cid001014-def0012.yaml + cid001015: + type: definition + title: Chebyshev Bias Measure + status: draft + visibility: private + primary_alias: DEF0013 + current_version: v1 + source_path: items/definitions/cid001015-def0013.yaml + cid001016: + type: definition + title: Polarity Pairing + status: canonical + visibility: public + primary_alias: DEF0014 + current_version: v1 + source_path: items/definitions/cid001016-def0014.yaml + cid001017: + type: definition + title: PolStream + status: canonical + visibility: public + primary_alias: DEF0015 + current_version: v1 + source_path: items/definitions/cid001017-def0015.yaml + cid001018: + type: definition + title: Tau-Weighted Boundary Constants + status: canonical + visibility: public + primary_alias: DEF0016 + current_version: v1 + source_path: items/definitions/cid001018-def0016.yaml + cid001019: + type: definition + title: Tau-Bayesian State + status: canonical + visibility: public + primary_alias: DEF0017 + current_version: v1 + source_path: items/definitions/cid001019-def0017.yaml + cid001020: + type: definition + title: Index Multiplication + status: canonical + visibility: public + primary_alias: DEF0018 + current_version: v1 + source_path: items/definitions/cid001020-def0018.yaml + cid001021: + type: definition + title: Index Exponentiation + status: canonical + visibility: public + primary_alias: DEF0019 + current_version: v1 + source_path: items/definitions/cid001021-def0019.yaml + cid001022: + type: definition + title: Index Tetration + status: canonical + visibility: public + primary_alias: DEF0020 + current_version: v1 + source_path: items/definitions/cid001022-def0020.yaml + cid001023: + type: definition + title: Program Monoid + status: canonical + visibility: public + primary_alias: DEF0021 + current_version: v1 + source_path: items/definitions/cid001023-def0021.yaml + cid001024: + type: definition + title: Three-Level Equality + status: canonical + visibility: public + primary_alias: DEF0022 + current_version: v1 + source_path: items/definitions/cid001024-def0022.yaml + cid001025: + type: definition + title: NF Address Encoding + status: canonical + visibility: public + primary_alias: DEF0023 + current_version: v1 + source_path: items/definitions/cid001025-def0023.yaml + cid001026: + type: definition + title: Denotational Order + status: canonical + visibility: public + primary_alias: DEF0024 + current_version: v1 + source_path: items/definitions/cid001026-def0024.yaml + cid001027: + type: definition + title: ABCD Coordinate Chart + status: canonical + visibility: public + primary_alias: DEF0025 + current_version: v1 + source_path: items/definitions/cid001027-def0025.yaml + cid001028: + type: definition + title: Algebraic Lemniscate + status: canonical + visibility: public + primary_alias: DEF0026 + current_version: v1 + source_path: items/definitions/cid001028-def0026.yaml + cid001029: + type: definition + title: Boundary Ring and Scalars + status: canonical + visibility: public + primary_alias: DEF0027 + current_version: v1 + source_path: items/definitions/cid001029-def0027.yaml + cid001030: + type: definition + title: Internal Divisibility + status: canonical + visibility: public + primary_alias: DEF0028 + current_version: v1 + source_path: items/definitions/cid001030-def0028.yaml + cid001031: + type: definition + title: Internal Primes + status: canonical + visibility: public + primary_alias: DEF0029 + current_version: v1 + source_path: items/definitions/cid001031-def0029.yaml + cid001032: + type: definition + title: Tower Atom + status: canonical + visibility: public + primary_alias: DEF0030 + current_version: v1 + source_path: items/definitions/cid001032-def0030.yaml + cid001033: + type: definition + title: Greedy Peel Algorithm + status: canonical + visibility: public + primary_alias: DEF0031 + current_version: v1 + source_path: items/definitions/cid001033-def0031.yaml + cid001034: + type: definition + title: Prime Spectral Signature + status: canonical + visibility: public + primary_alias: DEF0032 + current_version: v1 + source_path: items/definitions/cid001034-def0032.yaml + cid001035: + type: definition + title: Prime Enumeration and Sieve + status: canonical + visibility: public + primary_alias: DEF0033 + current_version: v1 + source_path: items/definitions/cid001035-def0033.yaml + cid001036: + type: definition + title: Split-Complex Scalars + status: canonical + visibility: public + primary_alias: DEF0034 + current_version: v1 + source_path: items/definitions/cid001036-def0034.yaml + cid001037: + type: definition + title: Truth4 Logic + status: canonical + visibility: public + primary_alias: DEF0035 + current_version: v1 + source_path: items/definitions/cid001037-def0035.yaml + cid001038: + type: definition + title: Holomorphic Transformers + status: canonical + visibility: public + primary_alias: DEF0036 + current_version: v1 + source_path: items/definitions/cid001038-def0036.yaml + cid001039: + type: definition + title: Genealogical Decomposition (Spine) + status: canonical + visibility: public + primary_alias: DEF0037 + current_version: v1 + source_path: items/definitions/cid001039-def0037.yaml + cid001040: + type: definition + title: Address DAG + status: canonical + visibility: public + primary_alias: DEF0038 + current_version: v1 + source_path: items/definitions/cid001040-def0038.yaml + cid001041: + type: definition + title: Omega-Tail (Compatible Tower) + status: canonical + visibility: public + primary_alias: DEF0039 + current_version: v1 + source_path: items/definitions/cid001041-def0039.yaml + cid001042: + type: definition + title: Polarized Omega-Germ + status: canonical + visibility: public + primary_alias: DEF0040 + current_version: v1 + source_path: items/definitions/cid001042-def0040.yaml + cid001043: + type: definition + title: Bipolar Spectral Algebra + status: canonical + visibility: public + primary_alias: DEF0041 + current_version: v1 + source_path: items/definitions/cid001043-def0041.yaml + cid001044: + type: definition + title: Boundary Local Ring + status: canonical + visibility: public + primary_alias: DEF0042 + current_version: v1 + source_path: items/definitions/cid001044-def0042.yaml + cid001045: + type: definition + title: CRT Decomposition + status: canonical + visibility: public + primary_alias: DEF0043 + current_version: v1 + source_path: items/definitions/cid001045-def0043.yaml + cid001046: + type: definition + title: Teichmueller Lift + status: canonical + visibility: public + primary_alias: DEF0044 + current_version: v1 + source_path: items/definitions/cid001046-def0044.yaml + cid001047: + type: definition + title: tau-Membership Relation + status: canonical + visibility: public + primary_alias: DEF0045 + current_version: v1 + source_path: items/definitions/cid001047-def0045.yaml + cid001048: + type: definition + title: Set-Theoretic Operations + status: canonical + visibility: public + primary_alias: DEF0046 + current_version: v1 + source_path: items/definitions/cid001048-def0046.yaml + cid001049: + type: definition + title: Bounded Powerset + status: canonical + visibility: public + primary_alias: DEF0047 + current_version: v1 + source_path: items/definitions/cid001049-def0047.yaml + cid001050: + type: definition + title: Master Constant iota_tau + status: canonical + visibility: public + primary_alias: DEF0048 + current_version: v1 + source_path: items/definitions/cid001050-def0048.yaml + cid001051: + type: definition + title: Number Tower + status: canonical + visibility: public + primary_alias: DEF0049 + current_version: v1 + source_path: items/definitions/cid001051-def0049.yaml + cid001052: + type: definition + title: Constructive Reals + status: canonical + visibility: public + primary_alias: DEF0050 + current_version: v1 + source_path: items/definitions/cid001052-def0050.yaml + cid001053: + type: definition + title: Lemniscate Characters + status: canonical + visibility: public + primary_alias: DEF0051 + current_version: v1 + source_path: items/definitions/cid001053-def0051.yaml + cid001054: + type: definition + title: Character Group + status: canonical + visibility: public + primary_alias: DEF0052 + current_version: v1 + source_path: items/definitions/cid001054-def0052.yaml + cid001055: + type: definition + title: Crossing Point + status: canonical + visibility: public + primary_alias: DEF0053 + current_version: v1 + source_path: items/definitions/cid001055-def0053.yaml + cid001056: + type: definition + title: Bipolar Fourier Transform + status: canonical + visibility: public + primary_alias: DEF0054 + current_version: v1 + source_path: items/definitions/cid001056-def0054.yaml + cid001057: + type: definition + title: Subobject Classifier Preview + status: canonical + visibility: public + primary_alias: DEF0055 + current_version: v1 + source_path: items/definitions/cid001057-def0055.yaml + cid001058: + type: definition + title: D-Differentiability + status: canonical + visibility: public + primary_alias: DEF0056 + current_version: v1 + source_path: items/definitions/cid001058-def0056.yaml + cid001059: + type: definition + title: Split-CR Equations + status: canonical + visibility: public + primary_alias: DEF0057 + current_version: v1 + source_path: items/definitions/cid001059-def0057.yaml + cid001060: + type: definition + title: Omega-Germ Transformer + status: canonical + visibility: public + primary_alias: DEF0058 + current_version: v1 + source_path: items/definitions/cid001060-def0058.yaml + cid001061: + type: definition + title: Tower Coherence + status: canonical + visibility: public + primary_alias: DEF0059 + current_version: v1 + source_path: items/definitions/cid001061-def0059.yaml + cid001062: + type: definition + title: Tau-Holomorphic Function + status: canonical + visibility: public + primary_alias: DEF0060 + current_version: v1 + source_path: items/definitions/cid001062-def0060.yaml + cid001063: + type: definition + title: Tau-Holomorphic Map + status: canonical + visibility: public + primary_alias: DEF0061 + current_version: v1 + source_path: items/definitions/cid001063-def0061.yaml + cid001064: + type: definition + title: Hol(L) + status: canonical + visibility: public + primary_alias: DEF0062 + current_version: v1 + source_path: items/definitions/cid001064-def0062.yaml + cid001065: + type: definition + title: Tau-Arrow + status: canonical + visibility: public + primary_alias: DEF0063 + current_version: v1 + source_path: items/definitions/cid001065-def0063.yaml + cid001066: + type: definition + title: Cat_tau + status: canonical + visibility: public + primary_alias: DEF0064 + current_version: v1 + source_path: items/definitions/cid001066-def0064.yaml + cid001067: + type: definition + title: Tau-Functor + status: canonical + visibility: public + primary_alias: DEF0065 + current_version: v1 + source_path: items/definitions/cid001067-def0065.yaml + cid001068: + type: definition + title: Natural Transformation + status: canonical + visibility: public + primary_alias: DEF0066 + current_version: v1 + source_path: items/definitions/cid001068-def0066.yaml + cid001069: + type: definition + title: Yoneda Embedding + status: canonical + visibility: public + primary_alias: DEF0067 + current_version: v1 + source_path: items/definitions/cid001069-def0067.yaml + cid001070: + type: definition + title: Finite Limits in Cat_tau + status: canonical + visibility: public + primary_alias: DEF0068 + current_version: v1 + source_path: items/definitions/cid001070-def0068.yaml + cid001071: + type: definition + title: Tau-Site + status: canonical + visibility: public + primary_alias: DEF0069 + current_version: v1 + source_path: items/definitions/cid001071-def0069.yaml + cid001072: + type: definition + title: Presheaf Topos + status: canonical + visibility: public + primary_alias: DEF0070 + current_version: v1 + source_path: items/definitions/cid001072-def0070.yaml + cid001073: + type: definition + title: Characteristic Morphism + status: canonical + visibility: public + primary_alias: DEF0071 + current_version: v1 + source_path: items/definitions/cid001073-def0071.yaml + cid001074: + type: definition + title: Earned Topos + status: canonical + visibility: public + primary_alias: DEF0072 + current_version: v1 + source_path: items/definitions/cid001074-def0072.yaml + cid001075: + type: definition + title: Categorical Product + status: canonical + visibility: public + primary_alias: DEF0073 + current_version: v1 + source_path: items/definitions/cid001075-def0073.yaml + cid001076: + type: definition + title: Cartesian Monoidal Structure + status: canonical + visibility: public + primary_alias: DEF0074 + current_version: v1 + source_path: items/definitions/cid001076-def0074.yaml + cid001077: + type: definition + title: Categorical Coproduct + status: canonical + visibility: public + primary_alias: DEF0075 + current_version: v1 + source_path: items/definitions/cid001077-def0075.yaml + cid001078: + type: definition + title: Bi-Monoidal Structure + status: canonical + visibility: public + primary_alias: DEF0076 + current_version: v1 + source_path: items/definitions/cid001078-def0076.yaml + cid001079: + type: definition + title: Internal Hom + status: canonical + visibility: public + primary_alias: DEF0077 + current_version: v1 + source_path: items/definitions/cid001079-def0077.yaml + cid001080: + type: definition + title: Spectral Coefficients + status: canonical + visibility: public + primary_alias: DEF0078 + current_version: v1 + source_path: items/definitions/cid001080-def0078.yaml + cid001081: + type: definition + title: Restriction Map + status: canonical + visibility: public + primary_alias: DEF0079 + current_version: v1 + source_path: items/definitions/cid001081-def0079.yaml + cid001082: + type: definition + title: Primorial Thinness + status: canonical + visibility: public + primary_alias: DEF0080 + current_version: v1 + source_path: items/definitions/cid001082-def0080.yaml + cid001083: + type: definition + title: Earned Stage-Determined Point + status: canonical + visibility: public + primary_alias: DEF0081 + current_version: v1 + source_path: items/definitions/cid001083-def0081.yaml + cid001084: + type: definition + title: Generative Counting Principle + status: canonical + visibility: public + primary_alias: DEF0082 + current_version: v1 + source_path: items/definitions/cid001084-def0082.yaml + cid001085: + type: definition + title: Omega-Germ Approach + status: canonical + visibility: public + primary_alias: DEF0083 + current_version: v1 + source_path: items/definitions/cid001085-def0083.yaml + cid001086: + type: definition + title: Meta-Logical Substrate + status: canonical + visibility: public + primary_alias: DEF0084 + current_version: v1 + source_path: items/definitions/cid001086-def0084.yaml + cid001087: + type: definition + title: Diagonal-Linear Correspondence + status: canonical + visibility: public + primary_alias: DEF0085 + current_version: v1 + source_path: items/definitions/cid001087-def0085.yaml + cid001088: + type: definition + title: Program Monoid as Linear Calculus + status: canonical + visibility: public + primary_alias: DEF0086 + current_version: v1 + source_path: items/definitions/cid001088-def0086.yaml + cid001089: + type: definition + title: Self-Hosting Degree Classification + status: canonical + visibility: public + primary_alias: DEF0087 + current_version: v1 + source_path: items/definitions/cid001089-def0087.yaml + cid001090: + type: definition + title: CCC-Linear Dichotomy + status: canonical + visibility: public + primary_alias: DEF0088 + current_version: v1 + source_path: items/definitions/cid001090-def0088.yaml + cid001091: + type: definition + title: Enrichment Frontier Classification + status: canonical + visibility: public + primary_alias: DEF0089 + current_version: v1 + source_path: items/definitions/cid001091-def0089.yaml + cid001092: + type: definition + title: Primorial Presheaf + status: canonical + visibility: public + primary_alias: DEF0090 + current_version: v1 + source_path: items/definitions/cid001092-def0090.yaml + cid001093: + type: definition + title: Constructive Reals + status: canonical + visibility: public + primary_alias: DEF0091 + current_version: v1 + source_path: items/definitions/cid001093-def0091.yaml + cid001094: + type: definition + title: Elliptic Complex Field + status: canonical + visibility: public + primary_alias: DEF0092 + current_version: v1 + source_path: items/definitions/cid001094-def0092.yaml + cid001095: + type: definition + title: Elliptic-Hyperbolic Dichotomy + status: canonical + visibility: public + primary_alias: DEF0093 + current_version: v1 + source_path: items/definitions/cid001095-def0093.yaml + cid001096: + type: definition + title: Elliptic Quaternions + status: canonical + visibility: public + primary_alias: DEF0094 + current_version: v1 + source_path: items/definitions/cid001096-def0094.yaml + cid001097: + type: definition + title: Cyclotomic Fields + status: canonical + visibility: public + primary_alias: DEF0095 + current_version: v1 + source_path: items/definitions/cid001097-def0095.yaml + cid001098: + type: definition + title: Diagonal Resonance + status: canonical + visibility: public + primary_alias: DEF0096 + current_version: v1 + source_path: items/definitions/cid001098-def0096.yaml + cid001099: + type: definition + title: Identity Slippage + status: canonical + visibility: public + primary_alias: DEF0097 + current_version: v1 + source_path: items/definitions/cid001099-def0097.yaml + cid001100: + type: definition + title: Shadow Identity + status: canonical + visibility: public + primary_alias: DEF0098 + current_version: v1 + source_path: items/definitions/cid001100-def0098.yaml + cid001101: + type: definition + title: Identity-Faithful Reception + status: canonical + visibility: public + primary_alias: DEF0099 + current_version: v1 + source_path: items/definitions/cid001101-def0099.yaml + cid001102: + type: definition + title: Structural Instability + status: canonical + visibility: public + primary_alias: DEF0100 + current_version: v1 + source_path: items/definitions/cid001102-def0100.yaml + cid001103: + type: definition + title: Orbit-Set Map + status: canonical + visibility: public + primary_alias: DEF0101 + current_version: v1 + source_path: items/definitions/cid001103-def0101.yaml + cid001104: + type: definition + title: τ-Measure Space + status: canonical + visibility: public + primary_alias: DEF0102 + current_version: v1 + source_path: items/definitions/cid001104-def0102.yaml + cid001105: + type: definition + title: Tower σ-Algebra + status: canonical + visibility: public + primary_alias: DEF0103 + current_version: v1 + source_path: items/definitions/cid001105-def0103.yaml + cid001106: + type: definition + title: Galois Automorphism + status: canonical + visibility: public + primary_alias: DEF0104 + current_version: v1 + source_path: items/definitions/cid001106-def0104.yaml + cid001107: + type: definition + title: Galois Group of Primorial Stage + status: canonical + visibility: public + primary_alias: DEF0105 + current_version: v1 + source_path: items/definitions/cid001107-def0105.yaml + cid001108: + type: definition + title: τ-Integral + status: canonical + visibility: public + primary_alias: DEF0106 + current_version: v1 + source_path: items/definitions/cid001108-def0106.yaml + cid001109: + type: axiom + title: Universe Postulate + status: canonical + visibility: public + primary_alias: AXM0002 + current_version: v1 + source_path: items/axioms/cid001109-axm0002.yaml + cid001110: + type: axiom + title: Strict Order (K1) + status: canonical + visibility: public + primary_alias: AXM0003 + current_version: v1 + source_path: items/axioms/cid001110-axm0003.yaml + cid001111: + type: axiom + title: Omega Fixed Point (K2) + status: canonical + visibility: public + primary_alias: AXM0004 + current_version: v1 + source_path: items/axioms/cid001111-axm0004.yaml + cid001112: + type: axiom + title: Orbit-Seeded Generation (K3) + status: canonical + visibility: public + primary_alias: AXM0005 + current_version: v1 + source_path: items/axioms/cid001112-axm0005.yaml + cid001113: + type: axiom + title: No-Jump / Cover (K4) + status: canonical + visibility: public + primary_alias: AXM0006 + current_version: v1 + source_path: items/axioms/cid001113-axm0006.yaml + cid001114: + type: axiom + title: Beacon Non-Successor (K5) + status: canonical + visibility: public + primary_alias: AXM0007 + current_version: v1 + source_path: items/axioms/cid001114-axm0007.yaml + cid001115: + type: axiom + title: Object Closure (K6) + status: canonical + visibility: public + primary_alias: AXM0008 + current_version: v1 + source_path: items/axioms/cid001115-axm0008.yaml + cid001116: + type: lemma + title: Pentation Non-Injectivity + status: canonical + visibility: public + primary_alias: LEM0002 + current_version: v1 + source_path: items/lemmas/cid001116-lem0002.yaml + cid001117: + type: lemma + title: NF-Confluence + status: canonical + visibility: public + primary_alias: LEM0003 + current_version: v1 + source_path: items/lemmas/cid001117-lem0003.yaml + cid001118: + type: lemma + title: No-Tie Determinism + status: canonical + visibility: public + primary_alias: LEM0004 + current_version: v1 + source_path: items/lemmas/cid001118-lem0004.yaml + cid001119: + type: lemma + title: Strict Remainder Descent + status: canonical + visibility: public + primary_alias: LEM0005 + current_version: v1 + source_path: items/lemmas/cid001119-lem0005.yaml + cid001120: + type: lemma + title: Growth Escape + status: canonical + visibility: public + primary_alias: LEM0006 + current_version: v1 + source_path: items/lemmas/cid001120-lem0006.yaml + cid001121: + type: lemma + title: Well-Foundedness of Membership + status: canonical + visibility: public + primary_alias: LEM0007 + current_version: v1 + source_path: items/lemmas/cid001121-lem0007.yaml + cid001122: + type: lemma + title: Tail Agreement Propagation + status: canonical + visibility: public + primary_alias: LEM0008 + current_version: v1 + source_path: items/lemmas/cid001122-lem0008.yaml + cid001123: + type: lemma + title: CRT Extension + status: canonical + visibility: public + primary_alias: LEM0009 + current_version: v1 + source_path: items/lemmas/cid001123-lem0009.yaml + cid001124: + type: proposition + title: Generator Distinctness + status: canonical + visibility: public + primary_alias: PRP0002 + current_version: v1 + source_path: items/propositions/cid001124-prp0002.yaml + cid001125: + type: proposition + title: rho Injectivity Per Orbit + status: canonical + visibility: public + primary_alias: PRP0003 + current_version: v1 + source_path: items/propositions/cid001125-prp0003.yaml + cid001126: + type: proposition + title: Pairwise Disjointness of Orbits + status: canonical + visibility: public + primary_alias: PRP0004 + current_version: v1 + source_path: items/propositions/cid001126-prp0004.yaml + cid001127: + type: proposition + title: Orbit Countability + status: canonical + visibility: public + primary_alias: PRP0005 + current_version: v1 + source_path: items/propositions/cid001127-prp0005.yaml + cid001128: + type: proposition + title: Tetration Injectivity + status: canonical + visibility: public + primary_alias: PRP0006 + current_version: v1 + source_path: items/propositions/cid001128-prp0006.yaml + cid001129: + type: proposition + title: Arithmetic Laws + status: canonical + visibility: public + primary_alias: PRP0007 + current_version: v1 + source_path: items/propositions/cid001129-prp0007.yaml + cid001130: + type: proposition + title: Well-Ordering of Obj(tau) + status: canonical + visibility: public + primary_alias: PRP0008 + current_version: v1 + source_path: items/propositions/cid001130-prp0008.yaml + cid001131: + type: proposition + title: Dimension Theorem (dim_tau = 4) + status: canonical + visibility: public + primary_alias: PRP0009 + current_version: v1 + source_path: items/propositions/cid001131-prp0009.yaml + cid001132: + type: proposition + title: Metric Inequality + status: canonical + visibility: public + primary_alias: PRP0010 + current_version: v1 + source_path: items/propositions/cid001132-prp0010.yaml + cid001133: + type: proposition + title: Membership Decidability + status: canonical + visibility: public + primary_alias: PRP0011 + current_version: v1 + source_path: items/propositions/cid001133-prp0011.yaml + cid001134: + type: proposition + title: Distributive Lattice + status: canonical + visibility: public + primary_alias: PRP0012 + current_version: v1 + source_path: items/propositions/cid001134-prp0012.yaml + cid001135: + type: proposition + title: Countability of Set(tau) + status: canonical + visibility: public + primary_alias: PRP0013 + current_version: v1 + source_path: items/propositions/cid001135-prp0013.yaml + cid001136: + type: proposition + title: Boolean Recovery + status: canonical + visibility: public + primary_alias: PRP0014 + current_version: v1 + source_path: items/propositions/cid001136-prp0014.yaml + cid001137: + type: proposition + title: Omega Unique Fixed Seed + status: canonical + visibility: public + primary_alias: PRP0015 + current_version: v1 + source_path: items/propositions/cid001137-prp0015.yaml + cid001138: + type: proposition + title: Sum Zero Iff Both Zero + status: canonical + visibility: public + primary_alias: PRP0016 + current_version: v1 + source_path: items/propositions/cid001138-prp0016.yaml + cid001139: + type: proposition + title: Positive Core Closure + status: canonical + visibility: public + primary_alias: PRP0017 + current_version: v1 + source_path: items/propositions/cid001139-prp0017.yaml + cid001140: + type: proposition + title: Universal Additive Cancellation + status: canonical + visibility: public + primary_alias: PRP0018 + current_version: v1 + source_path: items/propositions/cid001140-prp0018.yaml + cid001141: + type: proposition + title: Zero Vacuous + status: canonical + visibility: public + primary_alias: PRP0019 + current_version: v1 + source_path: items/propositions/cid001141-prp0019.yaml + cid001142: + type: proposition + title: Integral Domain + status: canonical + visibility: public + primary_alias: PRP0020 + current_version: v1 + source_path: items/propositions/cid001142-prp0020.yaml + cid001143: + type: proposition + title: Ultra Dist Self + status: canonical + visibility: public + primary_alias: PRP0021 + current_version: v1 + source_path: items/propositions/cid001143-prp0021.yaml + cid001144: + type: proposition + title: Congruent Tails Agree + status: canonical + visibility: public + primary_alias: PRP0022 + current_version: v1 + source_path: items/propositions/cid001144-prp0022.yaml + cid001145: + type: proposition + title: Sector Independence + status: canonical + visibility: public + primary_alias: PRP0023 + current_version: v1 + source_path: items/propositions/cid001145-prp0023.yaml + cid001146: + type: proposition + title: No Simultaneous Projection + status: canonical + visibility: public + primary_alias: PRP0024 + current_version: v1 + source_path: items/propositions/cid001146-prp0024.yaml + cid001147: + type: proposition + title: HolFun Associativity + status: canonical + visibility: public + primary_alias: PRP0025 + current_version: v1 + source_path: items/propositions/cid001147-prp0025.yaml + cid001148: + type: proposition + title: Thin Category + status: canonical + visibility: public + primary_alias: PRP0026 + current_version: v1 + source_path: items/propositions/cid001148-prp0026.yaml + cid001149: + type: proposition + title: Countable Topos + status: canonical + visibility: public + primary_alias: PRP0027 + current_version: v1 + source_path: items/propositions/cid001149-prp0027.yaml + cid001150: + type: proposition + title: Paraconsistent Character + status: canonical + visibility: public + primary_alias: PRP0028 + current_version: v1 + source_path: items/propositions/cid001150-prp0028.yaml + cid001151: + type: proposition + title: Self-Enrichment + status: canonical + visibility: public + primary_alias: PRP0029 + current_version: v1 + source_path: items/propositions/cid001151-prp0029.yaml + cid001152: + type: proposition + title: Passage to Book II + status: canonical + visibility: public + primary_alias: PRP0030 + current_version: v1 + source_path: items/propositions/cid001152-prp0030.yaml + cid001153: + type: proposition + title: Counting as Structural Feature + status: canonical + visibility: public + primary_alias: PRP0031 + current_version: v1 + source_path: items/propositions/cid001153-prp0031.yaml + cid001154: + type: proposition + title: No Unearned Decimal Diagonal + status: canonical + visibility: public + primary_alias: PRP0032 + current_version: v1 + source_path: items/propositions/cid001154-prp0032.yaml + cid001155: + type: proposition + title: No Unrestricted Comprehension + status: canonical + visibility: public + primary_alias: PRP0033 + current_version: v1 + source_path: items/propositions/cid001155-prp0033.yaml + cid001156: + type: proposition + title: No Free Cartesian Diagonal + status: canonical + visibility: public + primary_alias: PRP0034 + current_version: v1 + source_path: items/propositions/cid001156-prp0034.yaml + cid001157: + type: proposition + title: Ultrametric Replaces Cardinality + status: canonical + visibility: public + primary_alias: PRP0035 + current_version: v1 + source_path: items/propositions/cid001157-prp0035.yaml + cid001158: + type: proposition + title: Classical.em Eliminability + status: canonical + visibility: public + primary_alias: PRP0036 + current_version: v1 + source_path: items/propositions/cid001158-prp0036.yaml + cid001159: + type: proposition + title: TauReal Ring Axioms + status: canonical + visibility: public + primary_alias: PRP0037 + current_version: v1 + source_path: items/propositions/cid001159-prp0037.yaml + cid001160: + type: proposition + title: Extensionality + status: canonical + visibility: public + primary_alias: PRP0038 + current_version: v1 + source_path: items/propositions/cid001160-prp0038.yaml + cid001161: + type: proposition + title: Self-Containment Partition + status: canonical + visibility: public + primary_alias: PRP0039 + current_version: v1 + source_path: items/propositions/cid001161-prp0039.yaml + cid001162: + type: proposition + title: Order Bound + status: canonical + visibility: public + primary_alias: PRP0040 + current_version: v1 + source_path: items/propositions/cid001162-prp0040.yaml + cid001163: + type: proposition + title: Measure Compatibility + status: canonical + visibility: public + primary_alias: PRP0041 + current_version: v1 + source_path: items/propositions/cid001163-prp0041.yaml + cid001164: + type: proposition + title: CRT-Galois Decomposition + status: canonical + visibility: public + primary_alias: PRP0043 + current_version: v1 + source_path: items/propositions/cid001164-prp0043.yaml + cid001165: + type: proposition + title: Monotone Convergence + status: canonical + visibility: public + primary_alias: PRP0044 + current_version: v1 + source_path: items/propositions/cid001165-prp0044.yaml + cid001166: + type: proposition + title: No Second Linearity + status: canonical + visibility: public + primary_alias: PRP0045 + current_version: v1 + source_path: items/propositions/cid001166-prp0045.yaml + cid001167: + type: proposition + title: Bayesian Factorization + status: canonical + visibility: public + primary_alias: PRP0046 + current_version: v1 + source_path: items/propositions/cid001167-prp0046.yaml + cid001168: + type: theorem + title: Ontic Closure + status: canonical + visibility: public + primary_alias: THM0002 + current_version: v1 + source_path: items/theorems/cid001168-thm0002.yaml + cid001169: + type: theorem + title: Iterator Ladder Saturation + status: canonical + visibility: public + primary_alias: THM0003 + current_version: v1 + source_path: items/theorems/cid001169-thm0003.yaml + cid001170: + type: theorem + title: Composition Associativity + status: canonical + visibility: public + primary_alias: THM0004 + current_version: v1 + source_path: items/theorems/cid001170-thm0004.yaml + cid001171: + type: theorem + title: Hyperfactorization Theorem + status: canonical + visibility: public + primary_alias: THM0005 + current_version: v1 + source_path: items/theorems/cid001171-thm0005.yaml + cid001172: + type: theorem + title: Prime Polarity Theorem + status: canonical + visibility: public + primary_alias: THM0006 + current_version: v1 + source_path: items/theorems/cid001172-thm0006.yaml + cid001173: + type: theorem + title: Global Hartogs + status: canonical + visibility: public + primary_alias: THM0007 + current_version: v1 + source_path: items/theorems/cid001173-thm0007.yaml + cid001174: + type: theorem + title: Rigidity of tau + status: canonical + visibility: public + primary_alias: THM0008 + current_version: v1 + source_path: items/theorems/cid001174-thm0008.yaml + cid001175: + type: theorem + title: Categoricity of tau_0 + status: canonical + visibility: public + primary_alias: THM0009 + current_version: v1 + source_path: items/theorems/cid001175-thm0009.yaml + cid001176: + type: theorem + title: FTA on tau-Idx + status: canonical + visibility: public + primary_alias: THM0010 + current_version: v1 + source_path: items/theorems/cid001176-thm0010.yaml + cid001177: + type: theorem + title: Split-Complex Forced + status: canonical + visibility: public + primary_alias: THM0011 + current_version: v1 + source_path: items/theorems/cid001177-thm0011.yaml + cid001178: + type: theorem + title: Minimal Alphabet Theorem + status: canonical + visibility: public + primary_alias: THM0012 + current_version: v1 + source_path: items/theorems/cid001178-thm0012.yaml + cid001179: + type: theorem + title: Six-Generator Rigidity Failure + status: canonical + visibility: public + primary_alias: THM0013 + current_version: v1 + source_path: items/theorems/cid001179-thm0013.yaml + cid001180: + type: theorem + title: Four-Generator Ladder Incompleteness + status: canonical + visibility: public + primary_alias: THM0014 + current_version: v1 + source_path: items/theorems/cid001180-thm0014.yaml + cid001181: + type: theorem + title: Tetration Algebraic Degradation + status: canonical + visibility: public + primary_alias: THM0015 + current_version: v1 + source_path: items/theorems/cid001181-thm0015.yaml + cid001182: + type: theorem + title: Spectral Decomposition + status: canonical + visibility: public + primary_alias: THM0016 + current_version: v1 + source_path: items/theorems/cid001182-thm0016.yaml + cid001183: + type: theorem + title: Explosion Barrier + status: canonical + visibility: public + primary_alias: THM0017 + current_version: v1 + source_path: items/theorems/cid001183-thm0017.yaml + cid001184: + type: theorem + title: No Additive Inverse + status: canonical + visibility: public + primary_alias: THM0018 + current_version: v1 + source_path: items/theorems/cid001184-thm0018.yaml + cid001185: + type: theorem + title: No Ring Negation + status: canonical + visibility: public + primary_alias: THM0019 + current_version: v1 + source_path: items/theorems/cid001185-thm0019.yaml + cid001186: + type: theorem + title: Mul Cancel Fails at Zero + status: canonical + visibility: public + primary_alias: THM0020 + current_version: v1 + source_path: items/theorems/cid001186-thm0020.yaml + cid001187: + type: theorem + title: Multiplicative Cancellation iff Positive + status: canonical + visibility: public + primary_alias: THM0021 + current_version: v1 + source_path: items/theorems/cid001187-thm0021.yaml + cid001188: + type: theorem + title: CRT Coherence Constraint + status: canonical + visibility: public + primary_alias: THM0022 + current_version: v1 + source_path: items/theorems/cid001188-thm0022.yaml + cid001189: + type: theorem + title: Diagonal-Free Protection + status: canonical + visibility: public + primary_alias: THM0023 + current_version: v1 + source_path: items/theorems/cid001189-thm0023.yaml + cid001190: + type: theorem + title: Composition Closure + status: canonical + visibility: public + primary_alias: THM0024 + current_version: v1 + source_path: items/theorems/cid001190-thm0024.yaml + cid001191: + type: theorem + title: Tau-Identity Theorem + status: canonical + visibility: public + primary_alias: THM0025 + current_version: v1 + source_path: items/theorems/cid001191-thm0025.yaml + cid001192: + type: theorem + title: Category Axioms + status: canonical + visibility: public + primary_alias: THM0026 + current_version: v1 + source_path: items/theorems/cid001192-thm0026.yaml + cid001193: + type: theorem + title: Yoneda Lemma + status: canonical + visibility: public + primary_alias: THM0027 + current_version: v1 + source_path: items/theorems/cid001193-thm0027.yaml + cid001194: + type: theorem + title: Grothendieck Topos + status: canonical + visibility: public + primary_alias: THM0028 + current_version: v1 + source_path: items/theorems/cid001194-thm0028.yaml + cid001195: + type: theorem + title: Omega_tau Subobject Classifier + status: canonical + visibility: public + primary_alias: THM0029 + current_version: v1 + source_path: items/theorems/cid001195-thm0029.yaml + cid001196: + type: theorem + title: Product Universal Property + status: canonical + visibility: public + primary_alias: THM0030 + current_version: v1 + source_path: items/theorems/cid001196-thm0030.yaml + cid001197: + type: theorem + title: Distributivity + status: canonical + visibility: public + primary_alias: THM0032 + current_version: v1 + source_path: items/theorems/cid001197-thm0032.yaml + cid001198: + type: theorem + title: Cartesian Closed + status: canonical + visibility: public + primary_alias: THM0033 + current_version: v1 + source_path: items/theorems/cid001198-thm0033.yaml + cid001199: + type: theorem + title: Spectral Determination + status: canonical + visibility: public + primary_alias: THM0034 + current_version: v1 + source_path: items/theorems/cid001199-thm0034.yaml + cid001200: + type: theorem + title: Removable Singularity + status: canonical + visibility: public + primary_alias: THM0035 + current_version: v1 + source_path: items/theorems/cid001200-thm0035.yaml + cid001201: + type: theorem + title: Global Hartogs Extension + status: canonical + visibility: public + primary_alias: THM0036 + current_version: v1 + source_path: items/theorems/cid001201-thm0036.yaml + cid001202: + type: theorem + title: Cantor Diagonal Inapplicability + status: canonical + visibility: public + primary_alias: THM0037 + current_version: v1 + source_path: items/theorems/cid001202-thm0037.yaml + cid001203: + type: theorem + title: Unique Infinity Object + status: canonical + visibility: public + primary_alias: THM0038 + current_version: v1 + source_path: items/theorems/cid001203-thm0038.yaml + cid001204: + type: theorem + title: Diagonal-Linear Correspondence + status: canonical + visibility: public + primary_alias: THM0039 + current_version: v1 + source_path: items/theorems/cid001204-thm0039.yaml + cid001205: + type: theorem + title: Linearity Census + status: canonical + visibility: public + primary_alias: THM0040 + current_version: v1 + source_path: items/theorems/cid001205-thm0040.yaml + cid001206: + type: theorem + title: K5 Structural Exclusion + status: canonical + visibility: public + primary_alias: THM0041 + current_version: v1 + source_path: items/theorems/cid001206-thm0041.yaml + cid001207: + type: theorem + title: Presheaf Characterization + status: canonical + visibility: public + primary_alias: THM0042 + current_version: v1 + source_path: items/theorems/cid001207-thm0042.yaml + cid001208: + type: theorem + title: Bi-Square Characterization + status: canonical + visibility: public + primary_alias: THM0043 + current_version: v1 + source_path: items/theorems/cid001208-thm0043.yaml + cid001209: + type: theorem + title: Archimedean Property + status: canonical + visibility: public + primary_alias: THM0044 + current_version: v1 + source_path: items/theorems/cid001209-thm0044.yaml + cid001210: + type: theorem + title: TauComplex Ring Axioms + status: canonical + visibility: public + primary_alias: THM0045 + current_version: v1 + source_path: items/theorems/cid001210-thm0045.yaml + cid001211: + type: theorem + title: Quaternion Non-Commutativity + status: canonical + visibility: public + primary_alias: THM0046 + current_version: v1 + source_path: items/theorems/cid001211-thm0046.yaml + cid001212: + type: theorem + title: Roots of Unity CRT Decomposition + status: canonical + visibility: public + primary_alias: THM0047 + current_version: v1 + source_path: items/theorems/cid001212-thm0047.yaml + cid001213: + type: theorem + title: Ontic Identity Invariance + status: canonical + visibility: public + primary_alias: THM0048 + current_version: v1 + source_path: items/theorems/cid001213-thm0048.yaml + cid001214: + type: theorem + title: Slippage Breaks Unique Omega + status: canonical + visibility: public + primary_alias: THM0049 + current_version: v1 + source_path: items/theorems/cid001214-thm0049.yaml + cid001215: + type: theorem + title: Structural Instability Theorem + status: canonical + visibility: public + primary_alias: THM0050 + current_version: v1 + source_path: items/theorems/cid001215-thm0050.yaml + cid001216: + type: theorem + title: Countable Additivity + status: canonical + visibility: public + primary_alias: THM0051 + current_version: v1 + source_path: items/theorems/cid001216-thm0051.yaml + cid001217: + type: theorem + title: Fundamental Theorem of Internal Galois Theory + status: canonical + visibility: public + primary_alias: THM0052 + current_version: v1 + source_path: items/theorems/cid001217-thm0052.yaml + cid001218: + type: theorem + title: Linearity of Integration + status: canonical + visibility: public + primary_alias: THM0053 + current_version: v1 + source_path: items/theorems/cid001218-thm0053.yaml + cid001219: + type: theorem + title: Bias at Primorial Levels + status: draft + visibility: private + primary_alias: THM0054 + current_version: v1 + source_path: items/theorems/cid001219-thm0054.yaml + cid001220: + type: theorem + title: Canonical Normalisation (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0055 + current_version: v1 + source_path: items/theorems/cid001220-thm0055.yaml + cid001221: + type: theorem + title: NF Confluence / Church-Rosser for tau-kernel (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0056 + current_version: v1 + source_path: items/theorems/cid001221-thm0056.yaml + cid001222: + type: theorem + title: Genealogical DAG Structure (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0057 + current_version: v1 + source_path: items/theorems/cid001222-thm0057.yaml + cid001223: + type: theorem + title: Cayley Word Metric (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0058 + current_version: v1 + source_path: items/theorems/cid001223-thm0058.yaml + cid001224: + type: theorem + title: Ontic Ultrametric (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0059 + current_version: v1 + source_path: items/theorems/cid001224-thm0059.yaml + cid001225: + type: theorem + title: Address Resolution and Absence of Equations (Hinge 7) + status: canonical + visibility: public + primary_alias: THM0060 + current_version: v1 + source_path: items/theorems/cid001225-thm0060.yaml + cid001226: + type: theorem + title: Hinge 7 Integration Tabulation (Bundle Closure) + status: canonical + visibility: public + primary_alias: THM0061 + current_version: v1 + source_path: items/theorems/cid001226-thm0061.yaml + cid001227: + type: theorem + title: Ontic Identity Invariance (Hinge 8) + status: canonical + visibility: public + primary_alias: THM0062 + current_version: v1 + source_path: items/theorems/cid001227-thm0062.yaml + cid001228: + type: theorem + title: Diagonal-Linear Correspondence (Hinge 8) + status: canonical + visibility: internal + primary_alias: THM0063 + current_version: v1 + source_path: items/theorems/cid001228-thm0063.yaml + cid001229: + type: theorem + title: K5 Structural Exclusion (Hinge 8) + status: canonical + visibility: internal + primary_alias: THM0064 + current_version: v1 + source_path: items/theorems/cid001229-thm0064.yaml + cid001230: + type: theorem + title: Diagonal Resonance Diagnosis (Hinge 8) + status: canonical + visibility: internal + primary_alias: THM0065 + current_version: v1 + source_path: items/theorems/cid001230-thm0065.yaml + cid001231: + type: theorem + title: Reception Instability (Hinge 8) + status: canonical + visibility: internal + primary_alias: THM0066 + current_version: v1 + source_path: items/theorems/cid001231-thm0066.yaml + cid001232: + type: corollary + title: Holographic Principle + status: canonical + visibility: public + primary_alias: COR0005 + current_version: v1 + source_path: items/corollaries/cid001232-cor0005.yaml + cid001233: + type: corollary + title: Uniqueness of Category Tau + status: canonical + visibility: public + primary_alias: COR0006 + current_version: v1 + source_path: items/corollaries/cid001233-cor0006.yaml + cid001234: + type: definition + title: Split-Complex Codomain H_tau + status: canonical + visibility: public + primary_alias: DEF0107 + current_version: v1 + source_path: items/definitions/cid001234-def0107.yaml + cid001235: + type: definition + title: Tau-Admissible Point + status: canonical + visibility: public + primary_alias: DEF0108 + current_version: v1 + source_path: items/definitions/cid001235-def0108.yaml + cid001236: + type: definition + title: Constraint Lattice + status: canonical + visibility: public + primary_alias: DEF0109 + current_version: v1 + source_path: items/definitions/cid001236-def0109.yaml + cid001237: + type: definition + title: Omega Readout + status: canonical + visibility: public + primary_alias: DEF0110 + current_version: v1 + source_path: items/definitions/cid001237-def0110.yaml + cid001238: + type: definition + title: Base tau^1 + status: canonical + visibility: public + primary_alias: DEF0111 + current_version: v1 + source_path: items/definitions/cid001238-def0111.yaml + cid001239: + type: definition + title: Fiber T^2 + status: canonical + visibility: public + primary_alias: DEF0112 + current_version: v1 + source_path: items/definitions/cid001239-def0112.yaml + cid001240: + type: definition + title: Fibered Product tau^3 + status: canonical + visibility: public + primary_alias: DEF0113 + current_version: v1 + source_path: items/definitions/cid001240-def0113.yaml + cid001241: + type: definition + title: Interior Bipolar Decomposition + status: canonical + visibility: public + primary_alias: DEF0114 + current_version: v1 + source_path: items/definitions/cid001241-def0114.yaml + cid001242: + type: definition + title: Cylinder Domain + status: canonical + visibility: public + primary_alias: DEF0115 + current_version: v1 + source_path: items/definitions/cid001242-def0115.yaml + cid001243: + type: definition + title: Stage-k Cylinder + status: canonical + visibility: public + primary_alias: DEF0116 + current_version: v1 + source_path: items/definitions/cid001243-def0116.yaml + cid001244: + type: definition + title: Clopen Basis + status: canonical + visibility: public + primary_alias: DEF0117 + current_version: v1 + source_path: items/definitions/cid001244-def0117.yaml + cid001245: + type: definition + title: First Disagreement Depth + status: canonical + visibility: public + primary_alias: DEF0118 + current_version: v1 + source_path: items/definitions/cid001245-def0118.yaml + cid001246: + type: definition + title: Ultrametric Distance + status: canonical + visibility: public + primary_alias: DEF0119 + current_version: v1 + source_path: items/definitions/cid001246-def0119.yaml + cid001247: + type: definition + title: Stone Space + status: canonical + visibility: public + primary_alias: DEF0120 + current_version: v1 + source_path: items/definitions/cid001247-def0120.yaml + cid001248: + type: definition + title: Tau-Dimension + status: canonical + visibility: public + primary_alias: DEF0121 + current_version: v1 + source_path: items/definitions/cid001248-def0121.yaml + cid001249: + type: definition + title: Radial-Solenoidal Split + status: canonical + visibility: public + primary_alias: DEF0122 + current_version: v1 + source_path: items/definitions/cid001249-def0122.yaml + cid001250: + type: definition + title: Angular Sectors + status: canonical + visibility: public + primary_alias: DEF0123 + current_version: v1 + source_path: items/definitions/cid001250-def0123.yaml + cid001251: + type: definition + title: Pinch Map + status: canonical + visibility: public + primary_alias: DEF0124 + current_version: v1 + source_path: items/definitions/cid001251-def0124.yaml + cid001252: + type: definition + title: Two-Readout Principle + status: canonical + visibility: public + primary_alias: DEF0125 + current_version: v1 + source_path: items/definitions/cid001252-def0125.yaml + cid001253: + type: definition + title: Spine Address Path + status: canonical + visibility: public + primary_alias: DEF0126 + current_version: v1 + source_path: items/definitions/cid001253-def0126.yaml + cid001254: + type: definition + title: Betweenness Relation + status: canonical + visibility: public + primary_alias: DEF0127 + current_version: v1 + source_path: items/definitions/cid001254-def0127.yaml + cid001255: + type: definition + title: Congruence Relation + status: canonical + visibility: public + primary_alias: DEF0128 + current_version: v1 + source_path: items/definitions/cid001255-def0128.yaml + cid001256: + type: definition + title: Wave-Type PDE + status: canonical + visibility: public + primary_alias: DEF0129 + current_version: v1 + source_path: items/definitions/cid001256-def0129.yaml + cid001257: + type: definition + title: Causal Structure + status: canonical + visibility: public + primary_alias: DEF0130 + current_version: v1 + source_path: items/definitions/cid001257-def0130.yaml + cid001258: + type: definition + title: Approximation Sequence + status: canonical + visibility: public + primary_alias: DEF0131 + current_version: v1 + source_path: items/definitions/cid001258-def0131.yaml + cid001259: + type: definition + title: Alpha-Ray Line + status: canonical + visibility: public + primary_alias: DEF0132 + current_version: v1 + source_path: items/definitions/cid001259-def0132.yaml + cid001260: + type: definition + title: Level Circle + status: canonical + visibility: public + primary_alias: DEF0133 + current_version: v1 + source_path: items/definitions/cid001260-def0133.yaml + cid001261: + type: definition + title: Solenoidal Circle + status: canonical + visibility: public + primary_alias: DEF0134 + current_version: v1 + source_path: items/definitions/cid001261-def0134.yaml + cid001262: + type: definition + title: Geometric-Topological Unification + status: canonical + visibility: public + primary_alias: DEF0135 + current_version: v1 + source_path: items/definitions/cid001262-def0135.yaml + cid001263: + type: definition + title: Geometric Pi + status: canonical + visibility: public + primary_alias: DEF0136 + current_version: v1 + source_path: items/definitions/cid001263-def0136.yaml + cid001264: + type: definition + title: Archimedes Polygon Sequence + status: canonical + visibility: public + primary_alias: DEF0137 + current_version: v1 + source_path: items/definitions/cid001264-def0137.yaml + cid001265: + type: definition + title: e as Iterator Eigenvalue + status: canonical + visibility: public + primary_alias: DEF0138 + current_version: v1 + source_path: items/definitions/cid001265-def0138.yaml + cid001266: + type: definition + title: Growth Base + status: canonical + visibility: public + primary_alias: DEF0139 + current_version: v1 + source_path: items/definitions/cid001266-def0139.yaml + cid001267: + type: definition + title: Interior j-Unit + status: canonical + visibility: public + primary_alias: DEF0140 + current_version: v1 + source_path: items/definitions/cid001267-def0140.yaml + cid001268: + type: definition + title: Bipolar Idempotents Interior + status: canonical + visibility: public + primary_alias: DEF0141 + current_version: v1 + source_path: items/definitions/cid001268-def0141.yaml + cid001269: + type: definition + title: Archimedean Bridge + status: canonical + visibility: public + primary_alias: DEF0143 + current_version: v1 + source_path: items/definitions/cid001269-def0143.yaml + cid001270: + type: definition + title: Calibrated Split-Complex Codomain + status: canonical + visibility: public + primary_alias: DEF0144 + current_version: v1 + source_path: items/definitions/cid001270-def0144.yaml + cid001271: + type: definition + title: BndLift Construction + status: canonical + visibility: public + primary_alias: DEF0145 + current_version: v1 + source_path: items/definitions/cid001271-def0145.yaml + cid001272: + type: definition + title: Evolution Operator + status: canonical + visibility: public + primary_alias: DEF0146 + current_version: v1 + source_path: items/definitions/cid001272-def0146.yaml + cid001273: + type: definition + title: Causal Arrow + status: canonical + visibility: public + primary_alias: DEF0147 + current_version: v1 + source_path: items/definitions/cid001273-def0147.yaml + cid001274: + type: definition + title: Composition of Holomorphic Maps + status: canonical + visibility: public + primary_alias: DEF0148 + current_version: v1 + source_path: items/definitions/cid001274-def0148.yaml + cid001275: + type: definition + title: Identity Map + status: canonical + visibility: public + primary_alias: DEF0149 + current_version: v1 + source_path: items/definitions/cid001275-def0149.yaml + cid001276: + type: definition + title: Holomorphic Endomorphism Category + status: canonical + visibility: public + primary_alias: DEF0150 + current_version: v1 + source_path: items/definitions/cid001276-def0150.yaml + cid001277: + type: definition + title: Laurent Expansion + status: canonical + visibility: public + primary_alias: DEF0151 + current_version: v1 + source_path: items/definitions/cid001277-def0151.yaml + cid001278: + type: definition + title: Residue + status: canonical + visibility: public + primary_alias: DEF0152 + current_version: v1 + source_path: items/definitions/cid001278-def0152.yaml + cid001279: + type: definition + title: Meromorphic Function + status: canonical + visibility: public + primary_alias: DEF0153 + current_version: v1 + source_path: items/definitions/cid001279-def0153.yaml + cid001280: + type: definition + title: Canonical Holomorphic Basis + status: canonical + visibility: public + primary_alias: DEF0154 + current_version: v1 + source_path: items/definitions/cid001280-def0154.yaml + cid001281: + type: definition + title: Cylinder Generator + status: canonical + visibility: public + primary_alias: DEF0155 + current_version: v1 + source_path: items/definitions/cid001281-def0155.yaml + cid001282: + type: definition + title: Holomorphic Presheaf + status: canonical + visibility: public + primary_alias: DEF0156 + current_version: v1 + source_path: items/definitions/cid001282-def0156.yaml + cid001283: + type: definition + title: Canonical Decomposition + status: canonical + visibility: public + primary_alias: DEF0157 + current_version: v1 + source_path: items/definitions/cid001283-def0157.yaml + cid001284: + type: definition + title: Tau-Regularity + status: canonical + visibility: public + primary_alias: DEF0158 + current_version: v1 + source_path: items/definitions/cid001284-def0158.yaml + cid001285: + type: definition + title: Pre-Yoneda Embedding + status: canonical + visibility: public + primary_alias: DEF0159 + current_version: v1 + source_path: items/definitions/cid001285-def0159.yaml + cid001286: + type: definition + title: Code Map + status: canonical + visibility: public + primary_alias: DEF0160 + current_version: v1 + source_path: items/definitions/cid001286-def0160.yaml + cid001287: + type: definition + title: Decode Map + status: canonical + visibility: public + primary_alias: DEF0161 + current_version: v1 + source_path: items/definitions/cid001287-def0161.yaml + cid001288: + type: definition + title: Self-Enrichment Structure + status: canonical + visibility: public + primary_alias: DEF0162 + current_version: v1 + source_path: items/definitions/cid001288-def0162.yaml + cid001289: + type: definition + title: Hom Object + status: canonical + visibility: public + primary_alias: DEF0163 + current_version: v1 + source_path: items/definitions/cid001289-def0163.yaml + cid001290: + type: definition + title: 2-Category Structure + status: canonical + visibility: public + primary_alias: DEF0164 + current_version: v1 + source_path: items/definitions/cid001290-def0164.yaml + cid001291: + type: definition + title: 2-Morphism + status: canonical + visibility: public + primary_alias: DEF0165 + current_version: v1 + source_path: items/definitions/cid001291-def0165.yaml + cid001292: + type: definition + title: E1 Enrichment Layer + status: canonical + visibility: public + primary_alias: DEF0166 + current_version: v1 + source_path: items/definitions/cid001292-def0166.yaml + cid001293: + type: definition + title: E0/E1 Transition + status: canonical + visibility: public + primary_alias: DEF0167 + current_version: v1 + source_path: items/definitions/cid001293-def0167.yaml + cid001294: + type: definition + title: Idempotent-Supported Character + status: canonical + visibility: public + primary_alias: DEF0168 + current_version: v1 + source_path: items/definitions/cid001294-def0168.yaml + cid001295: + type: definition + title: Spectral Algebra + status: canonical + visibility: public + primary_alias: DEF0169 + current_version: v1 + source_path: items/definitions/cid001295-def0169.yaml + cid001296: + type: definition + title: Moduli Space + status: canonical + visibility: public + primary_alias: DEF0170 + current_version: v1 + source_path: items/definitions/cid001296-def0170.yaml + cid001297: + type: definition + title: Tau-Manifold + status: canonical + visibility: public + primary_alias: DEF0171 + current_version: v1 + source_path: items/definitions/cid001297-def0171.yaml + cid001298: + type: definition + title: Tau-Analytic Atlas + status: canonical + visibility: public + primary_alias: DEF0172 + current_version: v1 + source_path: items/definitions/cid001298-def0172.yaml + cid001299: + type: definition + title: Tau-Exterior Derivative + status: canonical + visibility: public + primary_alias: DEF0173 + current_version: v1 + source_path: items/definitions/cid001299-def0173.yaml + cid001300: + type: definition + title: Proto-Rationality + status: canonical + visibility: public + primary_alias: DEF0174 + current_version: v1 + source_path: items/definitions/cid001300-def0174.yaml + cid001301: + type: definition + title: E1 Export Package + status: canonical + visibility: public + primary_alias: DEF0175 + current_version: v1 + source_path: items/definitions/cid001301-def0175.yaml + cid001302: + type: definition + title: Structural Sign Classification + status: canonical + visibility: public + primary_alias: DEF0176 + current_version: v1 + source_path: items/definitions/cid001302-def0176.yaml + cid001303: + type: definition + title: The Infinity Trade-Off + status: canonical + visibility: public + primary_alias: DEF0177 + current_version: v1 + source_path: items/definitions/cid001303-def0177.yaml + cid001304: + type: definition + title: PDE Type Classification + status: canonical + visibility: public + primary_alias: DEF0178 + current_version: v1 + source_path: items/definitions/cid001304-def0178.yaml + cid001305: + type: definition + title: Stage-Finite Euclidean Geometry + status: canonical + visibility: public + primary_alias: DEF0179 + current_version: v1 + source_path: items/definitions/cid001305-def0179.yaml + cid001306: + type: definition + title: The Rewiring Table + status: canonical + visibility: public + primary_alias: DEF0180 + current_version: v1 + source_path: items/definitions/cid001306-def0180.yaml + cid001307: + type: definition + title: The Physics Quadrant Matrix + status: canonical + visibility: public + primary_alias: DEF0181 + current_version: v1 + source_path: items/definitions/cid001307-def0181.yaml + cid001308: + type: definition + title: The Unification Obstruction + status: canonical + visibility: public + primary_alias: DEF0182 + current_version: v1 + source_path: items/definitions/cid001308-def0182.yaml + cid001309: + type: definition + title: Archimedean-Elliptic Engine + status: canonical + visibility: public + primary_alias: DEF0183 + current_version: v1 + source_path: items/definitions/cid001309-def0183.yaml + cid001310: + type: definition + title: Dimensional Rigidity + status: canonical + visibility: public + primary_alias: DEF0184 + current_version: v1 + source_path: items/definitions/cid001310-def0184.yaml + cid001311: + type: definition + title: Geometric Bi-Square + status: canonical + visibility: public + primary_alias: DEF0185 + current_version: v1 + source_path: items/definitions/cid001311-def0185.yaml + cid001312: + type: definition + title: τ-Connection + status: canonical + visibility: public + primary_alias: DEF0186 + current_version: v1 + source_path: items/definitions/cid001312-def0186.yaml + cid001313: + type: definition + title: Parallel Transport + status: canonical + visibility: public + primary_alias: DEF0187 + current_version: v1 + source_path: items/definitions/cid001313-def0187.yaml + cid001314: + type: definition + title: τ-Curvature + status: canonical + visibility: public + primary_alias: DEF0188 + current_version: v1 + source_path: items/definitions/cid001314-def0188.yaml + cid001315: + type: definition + title: τ-Geodesic + status: canonical + visibility: public + primary_alias: DEF0189 + current_version: v1 + source_path: items/definitions/cid001315-def0189.yaml + cid001316: + type: definition + title: L² Inner Product + status: canonical + visibility: public + primary_alias: DEF0190 + current_version: v1 + source_path: items/definitions/cid001316-def0190.yaml + cid001317: + type: definition + title: L² Norm + status: canonical + visibility: public + primary_alias: DEF0191 + current_version: v1 + source_path: items/definitions/cid001317-def0191.yaml + cid001318: + type: definition + title: Chain Complex + status: canonical + visibility: public + primary_alias: DEF0192 + current_version: v1 + source_path: items/definitions/cid001318-def0192.yaml + cid001319: + type: definition + title: Homology via SES + status: canonical + visibility: public + primary_alias: DEF0193 + current_version: v1 + source_path: items/definitions/cid001319-def0193.yaml + cid001320: + type: definition + title: Čech Complex + status: canonical + visibility: public + primary_alias: DEF0194 + current_version: v1 + source_path: items/definitions/cid001320-def0194.yaml + cid001321: + type: definition + title: Sheaf Cohomology Groups + status: canonical + visibility: public + primary_alias: DEF0195 + current_version: v1 + source_path: items/definitions/cid001321-def0195.yaml + cid001322: + type: definition + title: Tau-Exponential + status: canonical + visibility: public + primary_alias: DEF0196 + current_version: v1 + source_path: items/definitions/cid001322-def0196.yaml + cid001323: + type: definition + title: Five Comparison Modes + status: canonical + visibility: public + primary_alias: DEF0197 + current_version: v1 + source_path: items/definitions/cid001323-def0197.yaml + cid001324: + type: definition + title: Six Comparison Axes + status: canonical + visibility: public + primary_alias: DEF0198 + current_version: v1 + source_path: items/definitions/cid001324-def0198.yaml + cid001325: + type: definition + title: Mode C Catalog + status: canonical + visibility: public + primary_alias: DEF0199 + current_version: v1 + source_path: items/definitions/cid001325-def0199.yaml + cid001326: + type: definition + title: Mode E Catalog + status: canonical + visibility: public + primary_alias: DEF0200 + current_version: v1 + source_path: items/definitions/cid001326-def0200.yaml + cid001327: + type: definition + title: Combined Gain-Cost Ledger + status: canonical + visibility: public + primary_alias: DEF0201 + current_version: v1 + source_path: items/definitions/cid001327-def0201.yaml + cid001328: + type: lemma + title: Naturality Forces Cylinder Compatibility + status: canonical + visibility: public + primary_alias: LEM0010 + current_version: v1 + source_path: items/lemmas/cid001328-lem0010.yaml + cid001329: + type: lemma + title: Refinement-Spectral Equivalence + status: canonical + visibility: public + primary_alias: LEM0011 + current_version: v1 + source_path: items/lemmas/cid001329-lem0011.yaml + cid001330: + type: lemma + title: Spectral-Germ Equivalence + status: canonical + visibility: public + primary_alias: LEM0012 + current_version: v1 + source_path: items/lemmas/cid001330-lem0012.yaml + cid001331: + type: lemma + title: Germ-Character Equivalence + status: canonical + visibility: public + primary_alias: LEM0013 + current_version: v1 + source_path: items/lemmas/cid001331-lem0013.yaml + cid001332: + type: lemma + title: Character-Hartogs Equivalence + status: canonical + visibility: public + primary_alias: LEM0014 + current_version: v1 + source_path: items/lemmas/cid001332-lem0014.yaml + cid001333: + type: lemma + title: Gluing Lemma + status: canonical + visibility: public + primary_alias: LEM0015 + current_version: v1 + source_path: items/lemmas/cid001333-lem0015.yaml + cid001334: + type: lemma + title: Idempotent Decomposition Lemma + status: canonical + visibility: public + primary_alias: LEM0016 + current_version: v1 + source_path: items/lemmas/cid001334-lem0016.yaml + cid001335: + type: lemma + title: Branch Factorization + status: canonical + visibility: public + primary_alias: LEM0017 + current_version: v1 + source_path: items/lemmas/cid001335-lem0017.yaml + cid001336: + type: lemma + title: Prime-Split Support + status: canonical + visibility: public + primary_alias: LEM0018 + current_version: v1 + source_path: items/lemmas/cid001336-lem0018.yaml + cid001337: + type: lemma + title: Polarity Symmetry + status: canonical + visibility: public + primary_alias: LEM0019 + current_version: v1 + source_path: items/lemmas/cid001337-lem0019.yaml + cid001338: + type: lemma + title: Probe Naturality iff Yoneda + status: canonical + visibility: public + primary_alias: LEM0020 + current_version: v1 + source_path: items/lemmas/cid001338-lem0020.yaml + cid001339: + type: lemma + title: Extension in Split-Complex Codomain + status: canonical + visibility: public + primary_alias: LEM0021 + current_version: v1 + source_path: items/lemmas/cid001339-lem0021.yaml + cid001340: + type: lemma + title: Stagewise Naturality + status: canonical + visibility: public + primary_alias: LEM0022 + current_version: v1 + source_path: items/lemmas/cid001340-lem0022.yaml + cid001341: + type: lemma + title: Yoneda Application + status: canonical + visibility: public + primary_alias: LEM0023 + current_version: v1 + source_path: items/lemmas/cid001341-lem0023.yaml + cid001342: + type: proposition + title: Lemniscate as Coordinate Limit + status: canonical + visibility: public + primary_alias: PRP0047 + current_version: v1 + source_path: items/propositions/cid001342-prp0047.yaml + cid001343: + type: proposition + title: Sector Inheritance + status: canonical + visibility: public + primary_alias: PRP0048 + current_version: v1 + source_path: items/propositions/cid001343-prp0048.yaml + cid001344: + type: proposition + title: Four-Ray Rigidity + status: canonical + visibility: public + primary_alias: PRP0049 + current_version: v1 + source_path: items/propositions/cid001344-prp0049.yaml + cid001345: + type: proposition + title: Cylinders Are Balls + status: canonical + visibility: public + primary_alias: PRP0050 + current_version: v1 + source_path: items/propositions/cid001345-prp0050.yaml + cid001346: + type: proposition + title: Lobes as Clopen Sets + status: canonical + visibility: public + primary_alias: PRP0051 + current_version: v1 + source_path: items/propositions/cid001346-prp0051.yaml + cid001347: + type: proposition + title: Refinement Resolution Bound + status: canonical + visibility: public + primary_alias: PRP0052 + current_version: v1 + source_path: items/propositions/cid001347-prp0052.yaml + cid001348: + type: proposition + title: Bipolar Channel Independence + status: canonical + visibility: public + primary_alias: PRP0053 + current_version: v1 + source_path: items/propositions/cid001348-prp0053.yaml + cid001349: + type: proposition + title: Projection Formula + status: canonical + visibility: public + primary_alias: PRP0054 + current_version: v1 + source_path: items/propositions/cid001349-prp0054.yaml + cid001350: + type: proposition + title: Decomposition Functoriality + status: canonical + visibility: public + primary_alias: PRP0055 + current_version: v1 + source_path: items/propositions/cid001350-prp0055.yaml + cid001351: + type: proposition + title: Functions as Tau-Objects + status: canonical + visibility: public + primary_alias: PRP0056 + current_version: v1 + source_path: items/propositions/cid001351-prp0056.yaml + cid001352: + type: proposition + title: Hom Bipolar Decomposition + status: canonical + visibility: public + primary_alias: PRP0057 + current_version: v1 + source_path: items/propositions/cid001352-prp0057.yaml + cid001353: + type: proposition + title: Enrichment Iteration + status: canonical + visibility: public + primary_alias: PRP0058 + current_version: v1 + source_path: items/propositions/cid001353-prp0058.yaml + cid001354: + type: proposition + title: Character Decomposition + status: canonical + visibility: public + primary_alias: PRP0059 + current_version: v1 + source_path: items/propositions/cid001354-prp0059.yaml + cid001355: + type: proposition + title: Character Algebra Ring Structure + status: canonical + visibility: public + primary_alias: PRP0060 + current_version: v1 + source_path: items/propositions/cid001355-prp0060.yaml + cid001356: + type: proposition + title: Tau3 Is a Tau-Manifold + status: canonical + visibility: public + primary_alias: PRP0061 + current_version: v1 + source_path: items/propositions/cid001356-prp0061.yaml + cid001357: + type: proposition + title: Holonomy Triviality + status: canonical + visibility: public + primary_alias: PRP0062 + current_version: v1 + source_path: items/propositions/cid001357-prp0062.yaml + cid001358: + type: proposition + title: Geodesic Completeness + status: canonical + visibility: public + primary_alias: PRP0063 + current_version: v1 + source_path: items/propositions/cid001358-prp0063.yaml + cid001359: + type: proposition + title: L² Completeness + status: canonical + visibility: public + primary_alias: PRP0064 + current_version: v1 + source_path: items/propositions/cid001359-prp0064.yaml + cid001360: + type: proposition + title: Long Exact Sequence + status: canonical + visibility: public + primary_alias: PRP0065 + current_version: v1 + source_path: items/propositions/cid001360-prp0065.yaml + cid001361: + type: proposition + title: Čech-to-Derived Comparison + status: canonical + visibility: public + primary_alias: PRP0066 + current_version: v1 + source_path: items/propositions/cid001361-prp0066.yaml + cid001362: + type: proposition + title: HolEnd Exponentials + status: canonical + visibility: public + primary_alias: PRP0067 + current_version: v1 + source_path: items/propositions/cid001362-prp0067.yaml + cid001363: + type: theorem + title: Point Set Well-Defined + status: canonical + visibility: public + primary_alias: THM0067 + current_version: v1 + source_path: items/theorems/cid001363-thm0067.yaml + cid001364: + type: theorem + title: Fiber Degeneration at Omega + status: canonical + visibility: public + primary_alias: THM0068 + current_version: v1 + source_path: items/theorems/cid001364-thm0068.yaml + cid001365: + type: theorem + title: Fibration Structure + status: canonical + visibility: public + primary_alias: THM0069 + current_version: v1 + source_path: items/theorems/cid001365-thm0069.yaml + cid001366: + type: theorem + title: Cylinder Basis Theorem + status: canonical + visibility: public + primary_alias: THM0070 + current_version: v1 + source_path: items/theorems/cid001366-thm0070.yaml + cid001367: + type: theorem + title: Ultrametric Triangle Inequality + status: canonical + visibility: public + primary_alias: THM0071 + current_version: v1 + source_path: items/theorems/cid001367-thm0071.yaml + cid001368: + type: theorem + title: Holomorphic Implies Continuous + status: canonical + visibility: public + primary_alias: THM0072 + current_version: v1 + source_path: items/theorems/cid001368-thm0072.yaml + cid001369: + type: theorem + title: Compactness + status: canonical + visibility: public + primary_alias: THM0073 + current_version: v1 + source_path: items/theorems/cid001369-thm0073.yaml + cid001370: + type: theorem + title: Hausdorff Property + status: canonical + visibility: public + primary_alias: THM0074 + current_version: v1 + source_path: items/theorems/cid001370-thm0074.yaml + cid001371: + type: theorem + title: Total Disconnectedness + status: canonical + visibility: public + primary_alias: THM0075 + current_version: v1 + source_path: items/theorems/cid001371-thm0075.yaml + cid001372: + type: theorem + title: Topology Uniqueness + status: canonical + visibility: public + primary_alias: THM0076 + current_version: v1 + source_path: items/theorems/cid001372-thm0076.yaml + cid001373: + type: theorem + title: Dimension Four + status: canonical + visibility: public + primary_alias: THM0077 + current_version: v1 + source_path: items/theorems/cid001373-thm0077.yaml + cid001374: + type: theorem + title: Boundary Minimality + status: canonical + visibility: public + primary_alias: THM0078 + current_version: v1 + source_path: items/theorems/cid001374-thm0078.yaml + cid001375: + type: theorem + title: Torus Degeneration Theorem + status: canonical + visibility: public + primary_alias: THM0079 + current_version: v1 + source_path: items/theorems/cid001375-thm0079.yaml + cid001376: + type: theorem + title: Fundamental Group Degeneration + status: canonical + visibility: public + primary_alias: THM0080 + current_version: v1 + source_path: items/theorems/cid001376-thm0080.yaml + cid001377: + type: theorem + title: Betweenness Axioms + status: canonical + visibility: public + primary_alias: THM0081 + current_version: v1 + source_path: items/theorems/cid001377-thm0081.yaml + cid001378: + type: theorem + title: Congruence Axioms + status: canonical + visibility: public + primary_alias: THM0082 + current_version: v1 + source_path: items/theorems/cid001378-thm0082.yaml + cid001379: + type: theorem + title: Pasch Axiom + status: canonical + visibility: public + primary_alias: THM0083 + current_version: v1 + source_path: items/theorems/cid001379-thm0083.yaml + cid001380: + type: theorem + title: Parallel Postulate + status: canonical + visibility: public + primary_alias: THM0084 + current_version: v1 + source_path: items/theorems/cid001380-thm0084.yaml + cid001381: + type: theorem + title: Euclidean as Static Limit + status: canonical + visibility: public + primary_alias: THM0085 + current_version: v1 + source_path: items/theorems/cid001381-thm0085.yaml + cid001382: + type: theorem + title: R as Inverse Limit + status: canonical + visibility: public + primary_alias: THM0086 + current_version: v1 + source_path: items/theorems/cid001382-thm0086.yaml + cid001383: + type: theorem + title: S^1 as Profinite Limit + status: canonical + visibility: public + primary_alias: THM0087 + current_version: v1 + source_path: items/theorems/cid001383-thm0087.yaml + cid001384: + type: theorem + title: Three Perspectives on Pi + status: canonical + visibility: public + primary_alias: THM0088 + current_version: v1 + source_path: items/theorems/cid001384-thm0088.yaml + cid001385: + type: theorem + title: e from Index Arithmetic + status: canonical + visibility: public + primary_alias: THM0089 + current_version: v1 + source_path: items/theorems/cid001385-thm0089.yaml + cid001386: + type: theorem + title: j Replaces i + status: canonical + visibility: public + primary_alias: THM0090 + current_version: v1 + source_path: items/theorems/cid001386-thm0090.yaml + cid001387: + type: theorem + title: BndLift Existence + status: canonical + visibility: public + primary_alias: THM0091 + current_version: v1 + source_path: items/theorems/cid001387-thm0091.yaml + cid001388: + type: theorem + title: Mutual Determination (5-Way Equivalence) + status: canonical + visibility: public + primary_alias: THM0092 + current_version: v1 + source_path: items/theorems/cid001388-thm0092.yaml + cid001389: + type: theorem + title: B/C Asymmetry Implies Time Arrow + status: canonical + visibility: public + primary_alias: THM0093 + current_version: v1 + source_path: items/theorems/cid001389-thm0093.yaml + cid001390: + type: theorem + title: Associativity of Holomorphic Composition + status: canonical + visibility: public + primary_alias: THM0094 + current_version: v1 + source_path: items/theorems/cid001390-thm0094.yaml + cid001391: + type: theorem + title: Residue Theorem + status: canonical + visibility: public + primary_alias: THM0095 + current_version: v1 + source_path: items/theorems/cid001391-thm0095.yaml + cid001392: + type: theorem + title: Finite Spectral Support + status: canonical + visibility: public + primary_alias: THM0096 + current_version: v1 + source_path: items/theorems/cid001392-thm0096.yaml + cid001393: + type: theorem + title: Sheaf Axioms + status: canonical + visibility: public + primary_alias: THM0097 + current_version: v1 + source_path: items/theorems/cid001393-thm0097.yaml + cid001394: + type: theorem + title: Holomorphic iff Idempotent-Supported + status: canonical + visibility: public + primary_alias: THM0098 + current_version: v1 + source_path: items/theorems/cid001394-thm0098.yaml + cid001395: + type: theorem + title: Regularity Criterion + status: canonical + visibility: public + primary_alias: THM0099 + current_version: v1 + source_path: items/theorems/cid001395-thm0099.yaml + cid001396: + type: theorem + title: Code/Decode Bijection + status: canonical + visibility: public + primary_alias: THM0100 + current_version: v1 + source_path: items/theorems/cid001396-thm0100.yaml + cid001397: + type: theorem + title: Tau-Yoneda Embedding + status: canonical + visibility: public + primary_alias: THM0101 + current_version: v1 + source_path: items/theorems/cid001397-thm0101.yaml + cid001398: + type: theorem + title: Hartogs Extension Uniqueness + status: canonical + visibility: public + primary_alias: THM0102 + current_version: v1 + source_path: items/theorems/cid001398-thm0102.yaml + cid001399: + type: theorem + title: Extensions Are Omega-Germ Transformers + status: canonical + visibility: public + primary_alias: THM0103 + current_version: v1 + source_path: items/theorems/cid001399-thm0103.yaml + cid001400: + type: theorem + title: Omega-Germs iff Holomorphic Functions + status: canonical + visibility: public + primary_alias: THM0104 + current_version: v1 + source_path: items/theorems/cid001400-thm0104.yaml + cid001401: + type: theorem + title: Central Theorem + status: canonical + visibility: public + primary_alias: THM0105 + current_version: v1 + source_path: items/theorems/cid001401-thm0105.yaml + cid001402: + type: theorem + title: Liouville Categorical Dodge + status: canonical + visibility: public + primary_alias: THM0106 + current_version: v1 + source_path: items/theorems/cid001402-thm0106.yaml + cid001403: + type: theorem + title: Categoricity + status: canonical + visibility: public + primary_alias: THM0107 + current_version: v1 + source_path: items/theorems/cid001403-thm0107.yaml + cid001404: + type: theorem + title: Structural Incompatibility of Unique Omega and Archimedean Density + status: canonical + visibility: public + primary_alias: THM0108 + current_version: v1 + source_path: items/theorems/cid001404-thm0108.yaml + cid001405: + type: theorem + title: Asymmetric Determination + status: canonical + visibility: public + primary_alias: THM0109 + current_version: v1 + source_path: items/theorems/cid001405-thm0109.yaml + cid001406: + type: theorem + title: Parallel Preservation + status: canonical + visibility: public + primary_alias: THM0110 + current_version: v1 + source_path: items/theorems/cid001406-thm0110.yaml + cid001407: + type: theorem + title: Fourth Quadrant Resolution + status: canonical + visibility: public + primary_alias: THM0111 + current_version: v1 + source_path: items/theorems/cid001407-thm0111.yaml + cid001408: + type: theorem + title: Simultaneous Rung Theorem + status: canonical + visibility: public + primary_alias: THM0112 + current_version: v1 + source_path: items/theorems/cid001408-thm0112.yaml + cid001409: + type: theorem + title: Fourth Quadrant Ladder Collapse + status: canonical + visibility: public + primary_alias: THM0113 + current_version: v1 + source_path: items/theorems/cid001409-thm0113.yaml + cid001410: + type: theorem + title: Geometric Bi-Square Theorem + status: canonical + visibility: public + primary_alias: THM0114 + current_version: v1 + source_path: items/theorems/cid001410-thm0114.yaml + cid001411: + type: theorem + title: Flat Connection Existence + status: canonical + visibility: public + primary_alias: THM0115 + current_version: v1 + source_path: items/theorems/cid001411-thm0115.yaml + cid001412: + type: theorem + title: Flat Curvature Vanishing + status: canonical + visibility: public + primary_alias: THM0116 + current_version: v1 + source_path: items/theorems/cid001412-thm0116.yaml + cid001413: + type: theorem + title: Lemniscate Holonomy + status: canonical + visibility: public + primary_alias: THM0117 + current_version: v1 + source_path: items/theorems/cid001413-thm0117.yaml + cid001414: + type: theorem + title: Cauchy-Schwarz Inequality + status: canonical + visibility: public + primary_alias: THM0118 + current_version: v1 + source_path: items/theorems/cid001414-thm0118.yaml + cid001415: + type: theorem + title: Tower Coherence + status: canonical + visibility: public + primary_alias: THM0119 + current_version: v1 + source_path: items/theorems/cid001415-thm0119.yaml + cid001416: + type: theorem + title: H⁰ = Global Sections + status: canonical + visibility: public + primary_alias: THM0120 + current_version: v1 + source_path: items/theorems/cid001416-thm0120.yaml + cid001417: + type: theorem + title: Trade-Off Atomicity + status: canonical + visibility: public + primary_alias: THM0121 + current_version: v1 + source_path: items/theorems/cid001417-thm0121.yaml + cid001418: + type: theorem + title: Admissible Germs Over Omega-Tails + status: canonical + visibility: internal + primary_alias: THM0122 + current_version: v1 + source_path: items/theorems/cid001418-thm0122.yaml + cid001419: + type: theorem + title: Holomorphy Before Mappings (Ontological Primary) + status: canonical + visibility: internal + primary_alias: THM0123 + current_version: v1 + source_path: items/theorems/cid001419-thm0123.yaml + cid001420: + type: theorem + title: Earned Scalar Codomain (D Is Forced) + status: canonical + visibility: internal + primary_alias: THM0124 + current_version: v1 + source_path: items/theorems/cid001420-thm0124.yaml + cid001421: + type: theorem + title: Wave-Equation Cauchy-Riemann + status: canonical + visibility: internal + primary_alias: THM0125 + current_version: v1 + source_path: items/theorems/cid001421-thm0125.yaml + cid001422: + type: theorem + title: Diagonal Discipline (Four Prohibitions) + status: canonical + visibility: internal + primary_alias: THM0126 + current_version: v1 + source_path: items/theorems/cid001422-thm0126.yaml + cid001423: + type: theorem + title: Earned Categorical Machine + status: canonical + visibility: internal + primary_alias: THM0127 + current_version: v1 + source_path: items/theorems/cid001423-thm0127.yaml + cid001424: + type: theorem + title: Sigma-Anti-Holomorphy and Idempotent-Supported Holomorphy + status: canonical + visibility: internal + primary_alias: THM0128 + current_version: v1 + source_path: items/theorems/cid001424-thm0128.yaml + cid001425: + type: theorem + title: HolEnd_tau via Pre-Yoneda Collapse + status: canonical + visibility: internal + primary_alias: THM0129 + current_version: v1 + source_path: items/theorems/cid001425-thm0129.yaml + cid001426: + type: theorem + title: Hinge 5 Integration Tabulation + status: canonical + visibility: internal + primary_alias: THM0130 + current_version: v1 + source_path: items/theorems/cid001426-thm0130.yaml + cid001427: + type: theorem + title: Topos Structure of Cat_tau + status: canonical + visibility: internal + primary_alias: THM0131 + current_version: v1 + source_path: items/theorems/cid001427-thm0131.yaml + cid001428: + type: theorem + title: Paraconsistent Soundness of Internal Logic + status: canonical + visibility: internal + primary_alias: THM0132 + current_version: v1 + source_path: items/theorems/cid001428-thm0132.yaml + cid001429: + type: theorem + title: Circularity Resolution via Omega-Germ Stabilisation + status: canonical + visibility: internal + primary_alias: THM0133 + current_version: v1 + source_path: items/theorems/cid001429-thm0133.yaml + cid001430: + type: theorem + title: 'Both = 1: Idempotent Unit of Boolean Sublattice' + status: canonical + visibility: internal + primary_alias: THM0134 + current_version: v1 + source_path: items/theorems/cid001430-thm0134.yaml + cid001431: + type: theorem + title: Hinge 6 Integration Tabulation + status: canonical + visibility: internal + primary_alias: THM0135 + current_version: v1 + source_path: items/theorems/cid001431-thm0135.yaml + cid001432: + type: definition + title: Hartogs Bulk Projection + status: canonical + visibility: public + primary_alias: DEF0202 + current_version: v1 + source_path: items/definitions/cid001432-def0202.yaml + cid001433: + type: definition + title: Millennium-Problem Gluing Table + status: canonical + visibility: public + primary_alias: DEF0203 + current_version: v1 + source_path: items/definitions/cid001433-def0203.yaml + cid001434: + type: definition + title: E₁ as Gluing Principle + status: canonical + visibility: public + primary_alias: DEF0204 + current_version: v1 + source_path: items/definitions/cid001434-def0204.yaml + cid001435: + type: definition + title: Enrichment Functor + status: canonical + visibility: public + primary_alias: DEF0205 + current_version: v1 + source_path: items/definitions/cid001435-def0205.yaml + cid001436: + type: definition + title: Layer Template + status: canonical + visibility: public + primary_alias: DEF0206 + current_version: v1 + source_path: items/definitions/cid001436-def0206.yaml + cid001437: + type: definition + title: E₀ Layer (Mathematics) + status: canonical + visibility: public + primary_alias: DEF0207 + current_version: v1 + source_path: items/definitions/cid001437-def0207.yaml + cid001438: + type: definition + title: E₁ Layer (Physics) + status: canonical + visibility: public + primary_alias: DEF0208 + current_version: v1 + source_path: items/definitions/cid001438-def0208.yaml + cid001439: + type: definition + title: E₂ Layer (Computation) + status: canonical + visibility: public + primary_alias: DEF0209 + current_version: v1 + source_path: items/definitions/cid001439-def0209.yaml + cid001440: + type: definition + title: E₃ Layer (Metaphysics) + status: canonical + visibility: public + primary_alias: DEF0210 + current_version: v1 + source_path: items/definitions/cid001440-def0210.yaml + cid001441: + type: definition + title: Ladder Checker + status: canonical + visibility: public + primary_alias: DEF0211 + current_version: v1 + source_path: items/definitions/cid001441-def0211.yaml + cid001442: + type: definition + title: Sieve Prime Count + status: canonical + visibility: public + primary_alias: DEF0212 + current_version: v1 + source_path: items/definitions/cid001442-def0212.yaml + cid001443: + type: definition + title: Brun Sieve Count + status: canonical + visibility: public + primary_alias: DEF0213 + current_version: v1 + source_path: items/definitions/cid001443-def0213.yaml + cid001444: + type: definition + title: Sieve-Accelerated Goldbach + status: canonical + visibility: public + primary_alias: DEF0214 + current_version: v1 + source_path: items/definitions/cid001444-def0214.yaml + cid001445: + type: definition + title: Partition Count at Primorial + status: canonical + visibility: public + primary_alias: DEF0215 + current_version: v1 + source_path: items/definitions/cid001445-def0215.yaml + cid001446: + type: definition + title: Goldbach Obstruction + status: canonical + visibility: public + primary_alias: DEF0216 + current_version: v1 + source_path: items/definitions/cid001446-def0216.yaml + cid001447: + type: definition + title: Twin Prime Sieve Count + status: canonical + visibility: public + primary_alias: DEF0217 + current_version: v1 + source_path: items/definitions/cid001447-def0217.yaml + cid001448: + type: definition + title: Hardy-Littlewood Constant + status: canonical + visibility: public + primary_alias: DEF0218 + current_version: v1 + source_path: items/definitions/cid001448-def0218.yaml + cid001449: + type: definition + title: CRT Twin Admissibility + status: canonical + visibility: public + primary_alias: DEF0219 + current_version: v1 + source_path: items/definitions/cid001449-def0219.yaml + cid001450: + type: definition + title: Sieve-Accelerated ABC + status: canonical + visibility: public + primary_alias: DEF0220 + current_version: v1 + source_path: items/definitions/cid001450-def0220.yaml + cid001451: + type: definition + title: High Quality Count + status: canonical + visibility: public + primary_alias: DEF0221 + current_version: v1 + source_path: items/definitions/cid001451-def0221.yaml + cid001452: + type: definition + title: Boundary Character Space + status: canonical + visibility: public + primary_alias: DEF0222 + current_version: v1 + source_path: items/definitions/cid001452-def0222.yaml + cid001453: + type: definition + title: Squarefree ABC Check + status: canonical + visibility: public + primary_alias: DEF0223 + current_version: v1 + source_path: items/definitions/cid001453-def0223.yaml + cid001454: + type: definition + title: Tower Decidable Check + status: canonical + visibility: public + primary_alias: DEF0224 + current_version: v1 + source_path: items/definitions/cid001454-def0224.yaml + cid001455: + type: definition + title: Gap Type + status: canonical + visibility: public + primary_alias: DEF0225 + current_version: v1 + source_path: items/definitions/cid001455-def0225.yaml + cid001456: + type: definition + title: Forbidden Move Mapping + status: canonical + visibility: public + primary_alias: DEF0226 + current_version: v1 + source_path: items/definitions/cid001456-def0226.yaml + cid001457: + type: definition + title: Boundary-to-Interior Functor + status: canonical + visibility: public + primary_alias: DEF0227 + current_version: v1 + source_path: items/definitions/cid001457-def0227.yaml + cid001458: + type: definition + title: 4+1 Sector Decomposition + status: canonical + visibility: public + primary_alias: DEF0228 + current_version: v1 + source_path: items/definitions/cid001458-def0228.yaml + cid001459: + type: definition + title: ω-Coupling Sector + status: canonical + visibility: public + primary_alias: DEF0229 + current_version: v1 + source_path: items/definitions/cid001459-def0229.yaml + cid001460: + type: definition + title: Langlands₁ Reflection Bridge + status: canonical + visibility: public + primary_alias: DEF0230 + current_version: v1 + source_path: items/definitions/cid001460-def0230.yaml + cid001461: + type: definition + title: Universal Operator + status: canonical + visibility: public + primary_alias: DEF0231 + current_version: v1 + source_path: items/definitions/cid001461-def0231.yaml + cid001462: + type: definition + title: Spectral Polarity + status: canonical + visibility: public + primary_alias: DEF0232 + current_version: v1 + source_path: items/definitions/cid001462-def0232.yaml + cid001463: + type: definition + title: Coupling Ledger + status: canonical + visibility: public + primary_alias: DEF0233 + current_version: v1 + source_path: items/definitions/cid001463-def0233.yaml + cid001464: + type: definition + title: Primorial Ladder + status: canonical + visibility: public + primary_alias: DEF0234 + current_version: v1 + source_path: items/definitions/cid001464-def0234.yaml + cid001465: + type: definition + title: Reconstruction Functor + status: canonical + visibility: public + primary_alias: DEF0235 + current_version: v1 + source_path: items/definitions/cid001465-def0235.yaml + cid001466: + type: definition + title: τ-Native Local Field + status: canonical + visibility: public + primary_alias: DEF0236 + current_version: v1 + source_path: items/definitions/cid001466-def0236.yaml + cid001467: + type: definition + title: τ-Adele Ring + status: canonical + visibility: public + primary_alias: DEF0237 + current_version: v1 + source_path: items/definitions/cid001467-def0237.yaml + cid001468: + type: definition + title: Internal Bipolar Classifier + status: canonical + visibility: public + primary_alias: DEF0238 + current_version: v1 + source_path: items/definitions/cid001468-def0238.yaml + cid001469: + type: definition + title: Boundary Normal Form + status: canonical + visibility: public + primary_alias: DEF0239 + current_version: v1 + source_path: items/definitions/cid001469-def0239.yaml + cid001470: + type: definition + title: Mutual Determination Schema + status: canonical + visibility: public + primary_alias: DEF0240 + current_version: v1 + source_path: items/definitions/cid001470-def0240.yaml + cid001471: + type: definition + title: Split-Complex Zeta ζ_τ + status: canonical + visibility: public + primary_alias: DEF0241 + current_version: v1 + source_path: items/definitions/cid001471-def0241.yaml + cid001472: + type: definition + title: Functional Equation Involution J + status: canonical + visibility: public + primary_alias: DEF0242 + current_version: v1 + source_path: items/definitions/cid001472-def0242.yaml + cid001473: + type: definition + title: Lemniscate Operator H_L + status: canonical + visibility: public + primary_alias: DEF0243 + current_version: v1 + source_path: items/definitions/cid001473-def0243.yaml + cid001474: + type: definition + title: Spectral Parameter Λ(s) + status: canonical + visibility: public + primary_alias: DEF0244 + current_version: v1 + source_path: items/definitions/cid001474-def0244.yaml + cid001475: + type: definition + title: τ-Effective RH Statement + status: canonical + visibility: public + primary_alias: DEF0245 + current_version: v1 + source_path: items/definitions/cid001475-def0245.yaml + cid001476: + type: definition + title: Grand GRH (τ-effective) + status: canonical + visibility: public + primary_alias: DEF0246 + current_version: v1 + source_path: items/definitions/cid001476-def0246.yaml + cid001477: + type: definition + title: L-Function as Spectral Determinant + status: canonical + visibility: public + primary_alias: DEF0247 + current_version: v1 + source_path: items/definitions/cid001477-def0247.yaml + cid001478: + type: definition + title: NP Witness as Canonical Address (v1) + status: canonical + visibility: internal + primary_alias: DEF0248 + current_version: v1 + source_path: items/definitions/cid001478-def0248.yaml + cid001479: + type: definition + title: τ-CDCL Architecture + status: canonical + visibility: internal + primary_alias: DEF0249 + current_version: v1 + source_path: items/definitions/cid001479-def0249.yaml + cid001480: + type: definition + title: Simply Connected in Category τ + status: canonical + visibility: public + primary_alias: DEF0250 + current_version: v1 + source_path: items/definitions/cid001480-def0250.yaml + cid001481: + type: definition + title: τ-Admissible Fluid Data + status: canonical + visibility: public + primary_alias: DEF0251 + current_version: v1 + source_path: items/definitions/cid001481-def0251.yaml + cid001482: + type: definition + title: Clopen Cylinder Domain + status: canonical + visibility: public + primary_alias: DEF0252 + current_version: v1 + source_path: items/definitions/cid001482-def0252.yaml + cid001483: + type: definition + title: Fluid Sector Decomposition + status: canonical + visibility: public + primary_alias: DEF0253 + current_version: v1 + source_path: items/definitions/cid001483-def0253.yaml + cid001484: + type: definition + title: Defect Functional Δ + status: canonical + visibility: public + primary_alias: DEF0254 + current_version: v1 + source_path: items/definitions/cid001484-def0254.yaml + cid001485: + type: definition + title: Hartogs Flow Operator + status: canonical + visibility: public + primary_alias: DEF0255 + current_version: v1 + source_path: items/definitions/cid001485-def0255.yaml + cid001486: + type: definition + title: Operator Polarity Swap + status: canonical + visibility: public + primary_alias: DEF0256 + current_version: v1 + source_path: items/definitions/cid001486-def0256.yaml + cid001487: + type: definition + title: Stabilized ω-Germ + status: canonical + visibility: public + primary_alias: DEF0257 + current_version: v1 + source_path: items/definitions/cid001487-def0257.yaml + cid001488: + type: definition + title: Strong Sector at E₁ + status: canonical + visibility: public + primary_alias: DEF0258 + current_version: v1 + source_path: items/definitions/cid001488-def0258.yaml + cid001489: + type: definition + title: τ-Admissible Gauge Data + status: canonical + visibility: public + primary_alias: DEF0259 + current_version: v1 + source_path: items/definitions/cid001489-def0259.yaml + cid001490: + type: definition + title: Gap Constant Γ* + status: canonical + visibility: public + primary_alias: DEF0260 + current_version: v1 + source_path: items/definitions/cid001490-def0260.yaml + cid001491: + type: definition + title: Strong Defect Functional + status: canonical + visibility: public + primary_alias: DEF0261 + current_version: v1 + source_path: items/definitions/cid001491-def0261.yaml + cid001492: + type: definition + title: σ-Fixed Character + status: canonical + visibility: public + primary_alias: DEF0262 + current_version: v1 + source_path: items/definitions/cid001492-def0262.yaml + cid001493: + type: definition + title: Sector Addressability + status: canonical + visibility: public + primary_alias: DEF0263 + current_version: v1 + source_path: items/definitions/cid001493-def0263.yaml + cid001494: + type: definition + title: E₂ Computational Agent + status: canonical + visibility: public + primary_alias: DEF0264 + current_version: v1 + source_path: items/definitions/cid001494-def0264.yaml + cid001495: + type: definition + title: Operational Closure + status: canonical + visibility: public + primary_alias: DEF0265 + current_version: v1 + source_path: items/definitions/cid001495-def0265.yaml + cid001496: + type: definition + title: τ-Tower Machine + status: canonical + visibility: public + primary_alias: DEF0266 + current_version: v1 + source_path: items/definitions/cid001496-def0266.yaml + cid001497: + type: definition + title: Observable Transition + status: canonical + visibility: public + primary_alias: DEF0267 + current_version: v1 + source_path: items/definitions/cid001497-def0267.yaml + cid001498: + type: definition + title: Interface Width + status: canonical + visibility: public + primary_alias: DEF0268 + current_version: v1 + source_path: items/definitions/cid001498-def0268.yaml + cid001499: + type: definition + title: τ-Admissibility (E₂) + status: canonical + visibility: public + primary_alias: DEF0269 + current_version: v1 + source_path: items/definitions/cid001499-def0269.yaml + cid001500: + type: definition + title: NP Witness as Canonical Address + status: canonical + visibility: public + primary_alias: DEF0270 + current_version: v1 + source_path: items/definitions/cid001500-def0270.yaml + cid001501: + type: definition + title: Computational Bi-Square + status: canonical + visibility: public + primary_alias: DEF0271 + current_version: v1 + source_path: items/definitions/cid001501-def0271.yaml + cid001502: + type: definition + title: Enrichment Functor Enr₀₁ + status: canonical + visibility: public + primary_alias: DEF0272 + current_version: v1 + source_path: items/definitions/cid001502-def0272.yaml + cid001503: + type: definition + title: E₁ Mutual Determination Instance + status: canonical + visibility: public + primary_alias: DEF0273 + current_version: v1 + source_path: items/definitions/cid001503-def0273.yaml + cid001504: + type: definition + title: τ-Rational Point + status: canonical + visibility: public + primary_alias: DEF0274 + current_version: v1 + source_path: items/definitions/cid001504-def0274.yaml + cid001505: + type: definition + title: Rank as Tower Depth + status: canonical + visibility: public + primary_alias: DEF0275 + current_version: v1 + source_path: items/definitions/cid001505-def0275.yaml + cid001506: + type: definition + title: Proto-Code + status: canonical + visibility: public + primary_alias: DEF0276 + current_version: v1 + source_path: items/definitions/cid001506-def0276.yaml + cid001507: + type: definition + title: BSD Functional + status: canonical + visibility: public + primary_alias: DEF0277 + current_version: v1 + source_path: items/definitions/cid001507-def0277.yaml + cid001508: + type: definition + title: Automorphic-Galois Duality in τ + status: canonical + visibility: public + primary_alias: DEF0278 + current_version: v1 + source_path: items/definitions/cid001508-def0278.yaml + cid001509: + type: definition + title: Local Langlands Instance + status: canonical + visibility: public + primary_alias: DEF0279 + current_version: v1 + source_path: items/definitions/cid001509-def0279.yaml + cid001510: + type: definition + title: Enriched Bi-Square at E₁⁺ + status: canonical + visibility: public + primary_alias: DEF0280 + current_version: v1 + source_path: items/definitions/cid001510-def0280.yaml + cid001511: + type: definition + title: Complete Dependency Chain + status: canonical + visibility: public + primary_alias: DEF0281 + current_version: v1 + source_path: items/definitions/cid001511-def0281.yaml + cid001512: + type: definition + title: ZFC as E₂ VM + status: canonical + visibility: public + primary_alias: DEF0282 + current_version: v1 + source_path: items/definitions/cid001512-def0282.yaml + cid001513: + type: definition + title: Gödel Numbering as NF Address + status: canonical + visibility: public + primary_alias: DEF0283 + current_version: v1 + source_path: items/definitions/cid001513-def0283.yaml + cid001514: + type: definition + title: Five Forbidden Moves + status: canonical + visibility: public + primary_alias: DEF0284 + current_version: v1 + source_path: items/definitions/cid001514-def0284.yaml + cid001515: + type: definition + title: Host-Level Property + status: canonical + visibility: public + primary_alias: DEF0285 + current_version: v1 + source_path: items/definitions/cid001515-def0285.yaml + cid001516: + type: definition + title: Bridge Axiom + status: canonical + visibility: public + primary_alias: DEF0286 + current_version: v1 + source_path: items/definitions/cid001516-def0286.yaml + cid001517: + type: definition + title: Shadow Diagram + status: canonical + visibility: public + primary_alias: DEF0287 + current_version: v1 + source_path: items/definitions/cid001517-def0287.yaml + cid001518: + type: definition + title: Proof Theory as E₃ + status: canonical + visibility: public + primary_alias: DEF0288 + current_version: v1 + source_path: items/definitions/cid001518-def0288.yaml + cid001519: + type: definition + title: Diagrammatic Sector of E₃ + status: canonical + visibility: public + primary_alias: DEF0289 + current_version: v1 + source_path: items/definitions/cid001519-def0289.yaml + cid001520: + type: definition + title: E₂→E₃ Boundary Crossing + status: canonical + visibility: public + primary_alias: DEF0290 + current_version: v1 + source_path: items/definitions/cid001520-def0290.yaml + cid001521: + type: definition + title: Minkowski Extension + status: canonical + visibility: public + primary_alias: DEF0291 + current_version: v1 + source_path: items/definitions/cid001521-def0291.yaml + cid001522: + type: definition + title: Physical E₂ Agent + status: canonical + visibility: public + primary_alias: DEF0292 + current_version: v1 + source_path: items/definitions/cid001522-def0292.yaml + cid001523: + type: definition + title: τ-Native Abstract Turing Machine + status: canonical + visibility: public + primary_alias: DEF0293 + current_version: v1 + source_path: items/definitions/cid001523-def0293.yaml + cid001524: + type: definition + title: Physical Realizability Predicate + status: canonical + visibility: public + primary_alias: DEF0294 + current_version: v1 + source_path: items/definitions/cid001524-def0294.yaml + cid001525: + type: definition + title: q-Expansion Coefficients + status: canonical + visibility: public + primary_alias: DEF0295 + current_version: v1 + source_path: items/definitions/cid001525-def0295.yaml + cid001526: + type: definition + title: Spectral Projector + status: canonical + visibility: public + primary_alias: DEF0296 + current_version: v1 + source_path: items/definitions/cid001526-def0296.yaml + cid001527: + type: definition + title: Spectral Measure + status: canonical + visibility: public + primary_alias: DEF0297 + current_version: v1 + source_path: items/definitions/cid001527-def0297.yaml + cid001528: + type: definition + title: Kleene Fixed Point + status: canonical + visibility: public + primary_alias: DEF0298 + current_version: v1 + source_path: items/definitions/cid001528-def0298.yaml + cid001529: + type: definition + title: E₂ Orbit Structure + status: canonical + visibility: public + primary_alias: DEF0299 + current_version: v1 + source_path: items/definitions/cid001529-def0299.yaml + cid001530: + type: definition + title: Self-Referential Fixed Point + status: canonical + visibility: public + primary_alias: DEF0300 + current_version: v1 + source_path: items/definitions/cid001530-def0300.yaml + cid001531: + type: definition + title: Paradox Absorption Map + status: canonical + visibility: public + primary_alias: DEF0301 + current_version: v1 + source_path: items/definitions/cid001531-def0301.yaml + cid001532: + type: definition + title: Arithmetic Translation Functor + status: canonical + visibility: public + primary_alias: DEF0302 + current_version: v1 + source_path: items/definitions/cid001532-def0302.yaml + cid001533: + type: definition + title: CRT-Integer Correspondence + status: canonical + visibility: public + primary_alias: DEF0303 + current_version: v1 + source_path: items/definitions/cid001533-def0303.yaml + cid001534: + type: definition + title: Topological Translation Functor + status: canonical + visibility: public + primary_alias: DEF0304 + current_version: v1 + source_path: items/definitions/cid001534-def0304.yaml + cid001535: + type: definition + title: Dimension Recovery + status: canonical + visibility: public + primary_alias: DEF0305 + current_version: v1 + source_path: items/definitions/cid001535-def0305.yaml + cid001536: + type: definition + title: Obstruction Cocycle + status: canonical + visibility: public + primary_alias: DEF0306 + current_version: v1 + source_path: items/definitions/cid001536-def0306.yaml + cid001537: + type: definition + title: Forbidden Move Obstruction Classes + status: canonical + visibility: public + primary_alias: DEF0307 + current_version: v1 + source_path: items/definitions/cid001537-def0307.yaml + cid001538: + type: definition + title: RH Spectral Gap Characterization + status: canonical + visibility: public + primary_alias: DEF0308 + current_version: v1 + source_path: items/definitions/cid001538-def0308.yaml + cid001539: + type: definition + title: YM Mass Gap Persistence + status: canonical + visibility: public + primary_alias: DEF0309 + current_version: v1 + source_path: items/definitions/cid001539-def0309.yaml + cid001540: + type: definition + title: Goldbach Representation + status: canonical + visibility: public + primary_alias: DEF0310 + current_version: v1 + source_path: items/definitions/cid001540-def0310.yaml + cid001541: + type: definition + title: Twin Prime Distribution + status: canonical + visibility: public + primary_alias: DEF0311 + current_version: v1 + source_path: items/definitions/cid001541-def0311.yaml + cid001542: + type: definition + title: Radical Function + status: canonical + visibility: public + primary_alias: DEF0312 + current_version: v1 + source_path: items/definitions/cid001542-def0312.yaml + cid001543: + type: definition + title: ABC Quality + status: canonical + visibility: public + primary_alias: DEF0313 + current_version: v1 + source_path: items/definitions/cid001543-def0313.yaml + cid001544: + type: definition + title: Eratosthenes Sieve + status: canonical + visibility: public + primary_alias: DEF0314 + current_version: v1 + source_path: items/definitions/cid001544-def0314.yaml + cid001545: + type: proposition + title: E₁ Strictness Witness + status: canonical + visibility: public + primary_alias: PRP0068 + current_version: v1 + source_path: items/propositions/cid001545-prp0068.yaml + cid001546: + type: proposition + title: Functor Category Collapse + status: canonical + visibility: public + primary_alias: PRP0069 + current_version: v1 + source_path: items/propositions/cid001546-prp0069.yaml + cid001547: + type: proposition + title: E₁ Physical Preview + status: canonical + visibility: public + primary_alias: PRP0070 + current_version: v1 + source_path: items/propositions/cid001547-prp0070.yaml + cid001548: + type: proposition + title: Balanced Sector Uniqueness + status: canonical + visibility: public + primary_alias: PRP0071 + current_version: v1 + source_path: items/propositions/cid001548-prp0071.yaml + cid001549: + type: proposition + title: Independence of Prime-Level Actions + status: canonical + visibility: public + primary_alias: PRP0072 + current_version: v1 + source_path: items/propositions/cid001549-prp0072.yaml + cid001550: + type: proposition + title: Completeness Without Topology + status: canonical + visibility: public + primary_alias: PRP0073 + current_version: v1 + source_path: items/propositions/cid001550-prp0073.yaml + cid001551: + type: proposition + title: Adelic Euler Product + status: canonical + visibility: public + primary_alias: PRP0074 + current_version: v1 + source_path: items/propositions/cid001551-prp0074.yaml + cid001552: + type: proposition + title: Label-Idempotent Compatibility + status: canonical + visibility: public + primary_alias: PRP0075 + current_version: v1 + source_path: items/propositions/cid001552-prp0075.yaml + cid001553: + type: proposition + title: Discrete Spectrum of H_L + status: canonical + visibility: public + primary_alias: PRP0076 + current_version: v1 + source_path: items/propositions/cid001553-prp0076.yaml + cid001554: + type: proposition + title: K5 Off-Diagonal Exclusion + status: canonical + visibility: public + primary_alias: PRP0077 + current_version: v1 + source_path: items/propositions/cid001554-prp0077.yaml + cid001555: + type: proposition + title: Primorial RH Verification Protocol + status: canonical + visibility: public + primary_alias: PRP0078 + current_version: v1 + source_path: items/propositions/cid001555-prp0078.yaml + cid001556: + type: proposition + title: CRT Witness Decomposition (v1) + status: canonical + visibility: internal + primary_alias: PRP0079 + current_version: v1 + source_path: items/propositions/cid001556-prp0079.yaml + cid001557: + type: proposition + title: Poincaré as Gluing Guarantee + status: canonical + visibility: public + primary_alias: PRP0080 + current_version: v1 + source_path: items/propositions/cid001557-prp0080.yaml + cid001558: + type: proposition + title: Defect Contractivity + status: canonical + visibility: public + primary_alias: PRP0081 + current_version: v1 + source_path: items/propositions/cid001558-prp0081.yaml + cid001559: + type: proposition + title: 3-Condition Sufficiency + status: canonical + visibility: public + primary_alias: PRP0082 + current_version: v1 + source_path: items/propositions/cid001559-prp0082.yaml + cid001560: + type: proposition + title: NF Discreteness Lemma + status: canonical + visibility: public + primary_alias: PRP0083 + current_version: v1 + source_path: items/propositions/cid001560-prp0083.yaml + cid001561: + type: proposition + title: Gap Stabilization + status: canonical + visibility: public + primary_alias: PRP0084 + current_version: v1 + source_path: items/propositions/cid001561-prp0084.yaml + cid001562: + type: proposition + title: Hodge Requires E₁ + status: canonical + visibility: public + primary_alias: PRP0085 + current_version: v1 + source_path: items/propositions/cid001562-prp0085.yaml + cid001563: + type: proposition + title: EM Sector Verification + status: canonical + visibility: public + primary_alias: PRP0086 + current_version: v1 + source_path: items/propositions/cid001563-prp0086.yaml + cid001564: + type: proposition + title: Sector-by-Sector Protocol + status: canonical + visibility: public + primary_alias: PRP0087 + current_version: v1 + source_path: items/propositions/cid001564-prp0087.yaml + cid001565: + type: proposition + title: Earned Admissibility + status: canonical + visibility: public + primary_alias: PRP0088 + current_version: v1 + source_path: items/propositions/cid001565-prp0088.yaml + cid001566: + type: proposition + title: CRT Witness Decomposition + status: canonical + visibility: public + primary_alias: PRP0089 + current_version: v1 + source_path: items/propositions/cid001566-prp0089.yaml + cid001567: + type: proposition + title: Polynomial Refinement + status: canonical + visibility: public + primary_alias: PRP0090 + current_version: v1 + source_path: items/propositions/cid001567-prp0090.yaml + cid001568: + type: proposition + title: Three-Reading Equivalence at E₁ + status: canonical + visibility: public + primary_alias: PRP0091 + current_version: v1 + source_path: items/propositions/cid001568-prp0091.yaml + cid001569: + type: proposition + title: Mordell-Weil Analogue + status: canonical + visibility: public + primary_alias: PRP0092 + current_version: v1 + source_path: items/propositions/cid001569-prp0092.yaml + cid001570: + type: proposition + title: Bridgehead Proposition + status: canonical + visibility: public + primary_alias: PRP0093 + current_version: v1 + source_path: items/propositions/cid001570-prp0093.yaml + cid001571: + type: proposition + title: BSD Three-Ingredient Proof + status: canonical + visibility: public + primary_alias: PRP0094 + current_version: v1 + source_path: items/propositions/cid001571-prp0094.yaml + cid001572: + type: proposition + title: Duality as Mutual Determination on ℤ² + status: canonical + visibility: public + primary_alias: PRP0095 + current_version: v1 + source_path: items/propositions/cid001572-prp0095.yaml + cid001573: + type: proposition + title: Chain Verification Protocol + status: canonical + visibility: public + primary_alias: PRP0096 + current_version: v1 + source_path: items/propositions/cid001573-prp0096.yaml + cid001574: + type: proposition + title: Sector Instantiation Lemma + status: canonical + visibility: public + primary_alias: PRP0097 + current_version: v1 + source_path: items/propositions/cid001574-prp0097.yaml + cid001575: + type: proposition + title: Terminal Level Characterization + status: canonical + visibility: public + primary_alias: PRP0098 + current_version: v1 + source_path: items/propositions/cid001575-prp0098.yaml + cid001576: + type: proposition + title: Bridge Algebraic Reduction + status: canonical + visibility: public + primary_alias: PRP0099 + current_version: v1 + source_path: items/propositions/cid001576-prp0099.yaml + cid001577: + type: proposition + title: Spectral Resolution + status: canonical + visibility: internal + primary_alias: PRP0100 + current_version: v1 + source_path: items/propositions/cid001577-prp0100.yaml + cid001578: + type: proposition + title: E₂ ⊋ E₁ Strict Witness + status: canonical + visibility: public + primary_alias: PRP0101 + current_version: v1 + source_path: items/propositions/cid001578-prp0101.yaml + cid001579: + type: proposition + title: Saturation Semantics + status: canonical + visibility: public + primary_alias: PRP0102 + current_version: v1 + source_path: items/propositions/cid001579-prp0102.yaml + cid001580: + type: proposition + title: Arithmetic Preserves Operations + status: canonical + visibility: public + primary_alias: PRP0103 + current_version: v1 + source_path: items/propositions/cid001580-prp0103.yaml + cid001581: + type: proposition + title: Boundary Restriction + status: canonical + visibility: public + primary_alias: PRP0104 + current_version: v1 + source_path: items/propositions/cid001581-prp0104.yaml + cid001582: + type: proposition + title: P vs NP as Polynomial Translation Obstruction + status: canonical + visibility: public + primary_alias: PRP0105 + current_version: v1 + source_path: items/propositions/cid001582-prp0105.yaml + cid001583: + type: proposition + title: Bridge Ledger Completeness + status: canonical + visibility: public + primary_alias: PRP0106 + current_version: v1 + source_path: items/propositions/cid001583-prp0106.yaml + cid001584: + type: proposition + title: Additive-Multiplicative Duality + status: canonical + visibility: public + primary_alias: PRP0107 + current_version: v1 + source_path: items/propositions/cid001584-prp0107.yaml + cid001585: + type: proposition + title: Radical-Primorial Identity + status: canonical + visibility: public + primary_alias: PRP0108 + current_version: v1 + source_path: items/propositions/cid001585-prp0108.yaml + cid001586: + type: proposition + title: Sieve-CRT Compatibility + status: canonical + visibility: public + primary_alias: PRP0109 + current_version: v1 + source_path: items/propositions/cid001586-prp0109.yaml + cid001587: + type: proposition + title: CRT-Goldbach Duality + status: canonical + visibility: public + primary_alias: PRP0110 + current_version: v1 + source_path: items/propositions/cid001587-prp0110.yaml + cid001588: + type: proposition + title: Goldbach Gap Characterization + status: canonical + visibility: public + primary_alias: PRP0111 + current_version: v1 + source_path: items/propositions/cid001588-prp0111.yaml + cid001589: + type: proposition + title: Twin Admissibility Fraction + status: canonical + visibility: public + primary_alias: PRP0112 + current_version: v1 + source_path: items/propositions/cid001589-prp0112.yaml + cid001590: + type: proposition + title: Twin Gap Characterization + status: canonical + visibility: public + primary_alias: PRP0113 + current_version: v1 + source_path: items/propositions/cid001590-prp0113.yaml + cid001591: + type: proposition + title: Squarefree Dominance Theorem + status: canonical + visibility: public + primary_alias: PRP0114 + current_version: v1 + source_path: items/propositions/cid001591-prp0114.yaml + cid001592: + type: proposition + title: ABC Gap Characterization + status: canonical + visibility: public + primary_alias: PRP0115 + current_version: v1 + source_path: items/propositions/cid001592-prp0115.yaml + cid001593: + type: theorem + title: Non-Emptiness Theorem + status: canonical + visibility: public + primary_alias: THM0136 + current_version: v1 + source_path: items/theorems/cid001593-thm0136.yaml + cid001594: + type: theorem + title: Strictness Theorem + status: canonical + visibility: public + primary_alias: THM0137 + current_version: v1 + source_path: items/theorems/cid001594-thm0137.yaml + cid001595: + type: theorem + title: Saturation at E₃ + status: canonical + visibility: public + primary_alias: THM0138 + current_version: v1 + source_path: items/theorems/cid001595-thm0138.yaml + cid001596: + type: theorem + title: Canonical Ladder Theorem + status: canonical + visibility: public + primary_alias: THM0139 + current_version: v1 + source_path: items/theorems/cid001596-thm0139.yaml + cid001597: + type: theorem + title: Sector Preservation Theorem + status: canonical + visibility: public + primary_alias: THM0140 + current_version: v1 + source_path: items/theorems/cid001597-thm0140.yaml + cid001598: + type: theorem + title: Template Invariance Under Reflection + status: canonical + visibility: public + primary_alias: THM0141 + current_version: v1 + source_path: items/theorems/cid001598-thm0141.yaml + cid001599: + type: theorem + title: Parity Bridge Theorem + status: canonical + visibility: public + primary_alias: THM0142 + current_version: v1 + source_path: items/theorems/cid001599-thm0142.yaml + cid001600: + type: theorem + title: No Knobs Principle + status: canonical + visibility: public + primary_alias: THM0143 + current_version: v1 + source_path: items/theorems/cid001600-thm0143.yaml + cid001601: + type: theorem + title: Primorial Cofinality + status: canonical + visibility: public + primary_alias: THM0144 + current_version: v1 + source_path: items/theorems/cid001601-thm0144.yaml + cid001602: + type: theorem + title: CRT Decomposition Theorem + status: canonical + visibility: public + primary_alias: THM0145 + current_version: v1 + source_path: items/theorems/cid001602-thm0145.yaml + cid001603: + type: theorem + title: Constructive Hensel Lifting + status: canonical + visibility: public + primary_alias: THM0146 + current_version: v1 + source_path: items/theorems/cid001603-thm0146.yaml + cid001604: + type: theorem + title: Adelic Embedding Theorem + status: canonical + visibility: public + primary_alias: THM0147 + current_version: v1 + source_path: items/theorems/cid001604-thm0147.yaml + cid001605: + type: theorem + title: Label Convergence + status: canonical + visibility: public + primary_alias: THM0148 + current_version: v1 + source_path: items/theorems/cid001605-thm0148.yaml + cid001606: + type: theorem + title: Spectral Trichotomy Lemma + status: canonical + visibility: public + primary_alias: THM0149 + current_version: v1 + source_path: items/theorems/cid001606-thm0149.yaml + cid001607: + type: theorem + title: B/C Non-Collapse Theorem + status: canonical + visibility: public + primary_alias: THM0150 + current_version: v1 + source_path: items/theorems/cid001607-thm0150.yaml + cid001608: + type: theorem + title: Bipolar Euler Product + status: canonical + visibility: public + primary_alias: THM0151 + current_version: v1 + source_path: items/theorems/cid001608-thm0151.yaml + cid001609: + type: theorem + title: Self-Adjointness of H_L + status: canonical + visibility: public + primary_alias: THM0152 + current_version: v1 + source_path: items/theorems/cid001609-thm0152.yaml + cid001610: + type: theorem + title: Spectral Correspondence Theorem + status: canonical + visibility: public + primary_alias: THM0153 + current_version: v1 + source_path: items/theorems/cid001610-thm0153.yaml + cid001611: + type: theorem + title: Critical Line Theorem + status: canonical + visibility: public + primary_alias: THM0154 + current_version: v1 + source_path: items/theorems/cid001611-thm0154.yaml + cid001612: + type: theorem + title: Prime Polarity Scaling Theorem + status: canonical + visibility: public + primary_alias: THM0155 + current_version: v1 + source_path: items/theorems/cid001612-thm0155.yaml + cid001613: + type: theorem + title: τ-Admissibility Collapse Theorem (v1) + status: canonical + visibility: internal + primary_alias: THM0156 + current_version: v1 + source_path: items/theorems/cid001613-thm0156.yaml + cid001614: + type: theorem + title: Representation Barrier Theorem + status: canonical + visibility: internal + primary_alias: THM0157 + current_version: v1 + source_path: items/theorems/cid001614-thm0157.yaml + cid001615: + type: theorem + title: Master Schema Theorem + status: canonical + visibility: public + primary_alias: THM0158 + current_version: v1 + source_path: items/theorems/cid001615-thm0158.yaml + cid001616: + type: theorem + title: Hartogs Flow Theorem + status: canonical + visibility: public + primary_alias: THM0159 + current_version: v1 + source_path: items/theorems/cid001616-thm0159.yaml + cid001617: + type: theorem + title: Positive Regularity Theorem + status: canonical + visibility: public + primary_alias: THM0160 + current_version: v1 + source_path: items/theorems/cid001617-thm0160.yaml + cid001618: + type: theorem + title: τ-Gap Meta-Theorem + status: canonical + visibility: public + primary_alias: THM0161 + current_version: v1 + source_path: items/theorems/cid001618-thm0161.yaml + cid001619: + type: theorem + title: Yang-Mills Gap Theorem + status: canonical + visibility: public + primary_alias: THM0162 + current_version: v1 + source_path: items/theorems/cid001619-thm0162.yaml + cid001620: + type: theorem + title: NF-Addressability Theorem + status: canonical + visibility: public + primary_alias: THM0163 + current_version: v1 + source_path: items/theorems/cid001620-thm0163.yaml + cid001621: + type: theorem + title: Physics Layer Assembly + status: canonical + visibility: public + primary_alias: THM0164 + current_version: v1 + source_path: items/theorems/cid001621-thm0164.yaml + cid001622: + type: theorem + title: TTM τ-Nativity + status: canonical + visibility: public + primary_alias: THM0165 + current_version: v1 + source_path: items/theorems/cid001622-thm0165.yaml + cid001623: + type: theorem + title: Interface Width Principle + status: canonical + visibility: public + primary_alias: THM0166 + current_version: v1 + source_path: items/theorems/cid001623-thm0166.yaml + cid001624: + type: theorem + title: Product-Meet Collapse + status: canonical + visibility: public + primary_alias: THM0167 + current_version: v1 + source_path: items/theorems/cid001624-thm0167.yaml + cid001625: + type: theorem + title: τ-Admissibility Collapse + status: canonical + visibility: public + primary_alias: THM0168 + current_version: v1 + source_path: items/theorems/cid001625-thm0168.yaml + cid001626: + type: theorem + title: No Barrier Theorem + status: canonical + visibility: public + primary_alias: THM0169 + current_version: v1 + source_path: items/theorems/cid001626-thm0169.yaml + cid001627: + type: theorem + title: BSD Coherence Theorem + status: canonical + visibility: public + primary_alias: THM0170 + current_version: v1 + source_path: items/theorems/cid001627-thm0170.yaml + cid001628: + type: theorem + title: Functoriality Theorem + status: canonical + visibility: public + primary_alias: THM0171 + current_version: v1 + source_path: items/theorems/cid001628-thm0171.yaml + cid001629: + type: theorem + title: Base Change-Transfer Naturality + status: canonical + visibility: public + primary_alias: THM0172 + current_version: v1 + source_path: items/theorems/cid001629-thm0172.yaml + cid001630: + type: theorem + title: Finite Factorization Pasting + status: canonical + visibility: public + primary_alias: THM0173 + current_version: v1 + source_path: items/theorems/cid001630-thm0173.yaml + cid001631: + type: theorem + title: Enriched Bi-Square Comparison + status: canonical + visibility: public + primary_alias: THM0174 + current_version: v1 + source_path: items/theorems/cid001631-thm0174.yaml + cid001632: + type: theorem + title: Enrichment Tower Assembly + status: canonical + visibility: public + primary_alias: THM0175 + current_version: v1 + source_path: items/theorems/cid001632-thm0175.yaml + cid001633: + type: theorem + title: Hinge Theorem + status: canonical + visibility: public + primary_alias: THM0176 + current_version: v1 + source_path: items/theorems/cid001633-thm0176.yaml + cid001634: + type: theorem + title: No Knobs Theorem + status: canonical + visibility: public + primary_alias: THM0177 + current_version: v1 + source_path: items/theorems/cid001634-thm0177.yaml + cid001635: + type: theorem + title: Move-Bridge Correspondence + status: canonical + visibility: public + primary_alias: THM0178 + current_version: v1 + source_path: items/theorems/cid001635-thm0178.yaml + cid001636: + type: theorem + title: Incompleteness as VM Boundary + status: canonical + visibility: public + primary_alias: THM0179 + current_version: v1 + source_path: items/theorems/cid001636-thm0179.yaml + cid001637: + type: theorem + title: RH Bridge Three-Layer Structure + status: canonical + visibility: public + primary_alias: THM0180 + current_version: v1 + source_path: items/theorems/cid001637-thm0180.yaml + cid001638: + type: theorem + title: Bridge Ledger + status: canonical + visibility: public + primary_alias: THM0181 + current_version: v1 + source_path: items/theorems/cid001638-thm0181.yaml + cid001639: + type: theorem + title: Honest Claim Theorem + status: canonical + visibility: public + primary_alias: THM0182 + current_version: v1 + source_path: items/theorems/cid001639-thm0182.yaml + cid001640: + type: theorem + title: Four Paradox Diagnostic + status: canonical + visibility: public + primary_alias: THM0183 + current_version: v1 + source_path: items/theorems/cid001640-thm0183.yaml + cid001641: + type: theorem + title: Applied Saturation + status: canonical + visibility: public + primary_alias: THM0184 + current_version: v1 + source_path: items/theorems/cid001641-thm0184.yaml + cid001642: + type: theorem + title: Global Cartesian Gluing Theorem + status: canonical + visibility: public + primary_alias: THM0185 + current_version: v1 + source_path: items/theorems/cid001642-thm0185.yaml + cid001643: + type: theorem + title: Physical Admissibility Theorem + status: canonical + visibility: public + primary_alias: THM0186 + current_version: v1 + source_path: items/theorems/cid001643-thm0186.yaml + cid001644: + type: theorem + title: Physical P=NP + status: canonical + visibility: public + primary_alias: THM0187 + current_version: v1 + source_path: items/theorems/cid001644-thm0187.yaml + cid001645: + type: theorem + title: Universal Admissibility Theorem + status: canonical + visibility: public + primary_alias: THM0188 + current_version: v1 + source_path: items/theorems/cid001645-thm0188.yaml + cid001646: + type: theorem + title: Confinement Bridge Identity + status: canonical + visibility: public + primary_alias: THM0189 + current_version: v1 + source_path: items/theorems/cid001646-thm0189.yaml + cid001647: + type: theorem + title: S-Duality Transport + status: canonical + visibility: public + primary_alias: THM0190 + current_version: v1 + source_path: items/theorems/cid001647-thm0190.yaml + cid001648: + type: theorem + title: Parseval Identity + status: canonical + visibility: public + primary_alias: THM0191 + current_version: v1 + source_path: items/theorems/cid001648-thm0191.yaml + cid001649: + type: theorem + title: Operational Closure Theorem + status: canonical + visibility: public + primary_alias: THM0192 + current_version: v1 + source_path: items/theorems/cid001649-thm0192.yaml + cid001650: + type: theorem + title: E₃ Self-Model Completeness + status: canonical + visibility: public + primary_alias: THM0193 + current_version: v1 + source_path: items/theorems/cid001650-thm0193.yaml + cid001651: + type: theorem + title: Arithmetic Faithfulness + status: canonical + visibility: public + primary_alias: THM0194 + current_version: v1 + source_path: items/theorems/cid001651-thm0194.yaml + cid001652: + type: theorem + title: Topological Faithfulness + status: canonical + visibility: public + primary_alias: THM0195 + current_version: v1 + source_path: items/theorems/cid001652-thm0195.yaml + cid001653: + type: theorem + title: Translation Failure Boundary + status: canonical + visibility: public + primary_alias: THM0196 + current_version: v1 + source_path: items/theorems/cid001653-thm0196.yaml + cid001654: + type: theorem + title: NS Flow Causal Arrow + status: canonical + visibility: public + primary_alias: THM0197 + current_version: v1 + source_path: items/theorems/cid001654-thm0197.yaml + cid001655: + type: theorem + title: P vs NP Forbidden Triple + status: canonical + visibility: public + primary_alias: THM0198 + current_version: v1 + source_path: items/theorems/cid001655-thm0198.yaml + cid001656: + type: theorem + title: Goldbach at Primorial Levels + status: canonical + visibility: public + primary_alias: THM0199 + current_version: v1 + source_path: items/theorems/cid001656-thm0199.yaml + cid001657: + type: theorem + title: ABC at Primorial Levels + status: canonical + visibility: public + primary_alias: THM0200 + current_version: v1 + source_path: items/theorems/cid001657-thm0200.yaml + cid001658: + type: theorem + title: Sieve Correctness + status: canonical + visibility: public + primary_alias: THM0201 + current_version: v1 + source_path: items/theorems/cid001658-thm0201.yaml + cid001659: + type: theorem + title: Sieve-Tower Compatibility + status: canonical + visibility: public + primary_alias: THM0202 + current_version: v1 + source_path: items/theorems/cid001659-thm0202.yaml + cid001660: + type: theorem + title: Goldbach Verified to 500 + status: canonical + visibility: public + primary_alias: THM0203 + current_version: v1 + source_path: items/theorems/cid001660-thm0203.yaml + cid001661: + type: theorem + title: Goldbach at Primorial M₄ + status: canonical + visibility: public + primary_alias: THM0204 + current_version: v1 + source_path: items/theorems/cid001661-thm0204.yaml + cid001662: + type: theorem + title: Partition Growth + status: canonical + visibility: public + primary_alias: THM0205 + current_version: v1 + source_path: items/theorems/cid001662-thm0205.yaml + cid001663: + type: theorem + title: Obstruction Bounded + status: canonical + visibility: public + primary_alias: THM0206 + current_version: v1 + source_path: items/theorems/cid001663-thm0206.yaml + cid001664: + type: theorem + title: Twin Primes to 500 + status: canonical + visibility: public + primary_alias: THM0207 + current_version: v1 + source_path: items/theorems/cid001664-thm0207.yaml + cid001665: + type: theorem + title: Twin Density Primorial + status: canonical + visibility: public + primary_alias: THM0208 + current_version: v1 + source_path: items/theorems/cid001665-thm0208.yaml + cid001666: + type: theorem + title: HL Constant Convergence + status: canonical + visibility: public + primary_alias: THM0209 + current_version: v1 + source_path: items/theorems/cid001666-thm0209.yaml + cid001667: + type: theorem + title: CRT Admissible Positive + status: canonical + visibility: public + primary_alias: THM0210 + current_version: v1 + source_path: items/theorems/cid001667-thm0210.yaml + cid001668: + type: theorem + title: ABC Quality 100 + status: canonical + visibility: public + primary_alias: THM0211 + current_version: v1 + source_path: items/theorems/cid001668-thm0211.yaml + cid001669: + type: theorem + title: Squarefree Dominance + status: canonical + visibility: public + primary_alias: THM0212 + current_version: v1 + source_path: items/theorems/cid001669-thm0212.yaml + cid001670: + type: theorem + title: Radical Primorial 5 + status: canonical + visibility: public + primary_alias: THM0213 + current_version: v1 + source_path: items/theorems/cid001670-thm0213.yaml + cid001671: + type: theorem + title: Tower Finite Decidable + status: canonical + visibility: public + primary_alias: THM0214 + current_version: v1 + source_path: items/theorems/cid001671-thm0214.yaml + cid001672: + type: theorem + title: Bridge Necessary Insufficient + status: canonical + visibility: public + primary_alias: THM0215 + current_version: v1 + source_path: items/theorems/cid001672-thm0215.yaml + cid001673: + type: theorem + title: Countable Profinite Boundary Ring + status: canonical + visibility: internal + primary_alias: THM0216 + current_version: v1 + source_path: items/theorems/cid001673-thm0216.yaml + cid001674: + type: theorem + title: Constructive CRT in the Primorial Ladder + status: canonical + visibility: internal + primary_alias: THM0217 + current_version: v1 + source_path: items/theorems/cid001674-thm0217.yaml + cid001675: + type: theorem + title: Teichmueller-Style Primitive-Recursive Lift + status: canonical + visibility: internal + primary_alias: THM0218 + current_version: v1 + source_path: items/theorems/cid001675-thm0218.yaml + cid001676: + type: theorem + title: Split-Complex Scalar Algebra D + status: canonical + visibility: internal + primary_alias: THM0219 + current_version: v1 + source_path: items/theorems/cid001676-thm0219.yaml + cid001677: + type: theorem + title: Crossing Mediator iota_tau in D + status: canonical + visibility: internal + primary_alias: THM0220 + current_version: v1 + source_path: items/theorems/cid001677-thm0220.yaml + cid001678: + type: theorem + title: Canonical Uniqueness of D + status: canonical + visibility: internal + primary_alias: THM0221 + current_version: v1 + source_path: items/theorems/cid001678-thm0221.yaml + cid001679: + type: theorem + title: Elliptic Complex Exclusion + status: canonical + visibility: internal + primary_alias: THM0222 + current_version: v1 + source_path: items/theorems/cid001679-thm0222.yaml + cid001680: + type: theorem + title: Four-Atom Spectral Dictionary + status: canonical + visibility: internal + primary_alias: THM0223 + current_version: v1 + source_path: items/theorems/cid001680-thm0223.yaml + cid001681: + type: theorem + title: Hinge Integration in D + status: canonical + visibility: internal + primary_alias: THM0224 + current_version: v1 + source_path: items/theorems/cid001681-thm0224.yaml + cid005000: + type: formal_module + title: TauLib.BookI.Addressability.AddressResolution + status: canonical + visibility: public + primary_alias: MOD0003 + current_version: v1 + source_path: items/formal-modules/cid005000-mod0003.yaml + cid005001: + type: formal_module + title: TauLib.BookI.Addressability.CayleyMetric + status: canonical + visibility: public + primary_alias: MOD0004 + current_version: v1 + source_path: items/formal-modules/cid005001-mod0004.yaml + cid005002: + type: formal_module + title: TauLib.BookI.Addressability.HingeIntegration + status: canonical + visibility: public + primary_alias: MOD0005 + current_version: v1 + source_path: items/formal-modules/cid005002-mod0005.yaml + cid005003: + type: formal_module + title: TauLib.BookI.Addressability.OnticUltrametric + status: canonical + visibility: public + primary_alias: MOD0006 + current_version: v1 + source_path: items/formal-modules/cid005003-mod0006.yaml + cid005004: + type: formal_module + title: TauLib.BookI.Boundary.Bridge.TauIntQuotient + status: canonical + visibility: public + primary_alias: MOD0007 + current_version: v1 + source_path: items/formal-modules/cid005004-mod0007.yaml + cid005005: + type: formal_module + title: TauLib.BookI.Boundary.Bridge.TauRatQuotient + status: canonical + visibility: public + primary_alias: MOD0008 + current_version: v1 + source_path: items/formal-modules/cid005005-mod0008.yaml + cid005006: + type: formal_module + title: TauLib.BookI.Boundary.Bridge.TauRealCongruence + status: canonical + visibility: public + primary_alias: MOD0009 + current_version: v1 + source_path: items/formal-modules/cid005006-mod0009.yaml + cid005007: + type: formal_module + title: TauLib.BookI.Boundary.Bridge.TauRealQuotient + status: canonical + visibility: public + primary_alias: MOD0010 + current_version: v1 + source_path: items/formal-modules/cid005007-mod0010.yaml + cid005008: + type: formal_module + title: TauLib.BookI.Boundary.Characters + status: canonical + visibility: public + primary_alias: MOD0011 + current_version: v1 + source_path: items/formal-modules/cid005008-mod0011.yaml + cid005009: + type: formal_module + title: TauLib.BookI.Boundary.ComplexField + status: canonical + visibility: public + primary_alias: MOD0012 + current_version: v1 + source_path: items/formal-modules/cid005009-mod0012.yaml + cid005010: + type: formal_module + title: TauLib.BookI.Boundary.ConstructiveReals + status: canonical + visibility: public + primary_alias: MOD0013 + current_version: v1 + source_path: items/formal-modules/cid005010-mod0013.yaml + cid005011: + type: formal_module + title: TauLib.BookI.Boundary.CouplingIdentityApproximants + status: canonical + visibility: public + primary_alias: MOD0014 + current_version: v1 + source_path: items/formal-modules/cid005011-mod0014.yaml + cid005012: + type: formal_module + title: TauLib.BookI.Boundary.Cyclotomic + status: canonical + visibility: public + primary_alias: MOD0015 + current_version: v1 + source_path: items/formal-modules/cid005012-mod0015.yaml + cid005013: + type: formal_module + title: TauLib.BookI.Boundary.DefectInverseSystem + status: canonical + visibility: public + primary_alias: MOD0016 + current_version: v1 + source_path: items/formal-modules/cid005013-mod0016.yaml + cid005014: + type: formal_module + title: TauLib.BookI.Boundary.Fourier + status: canonical + visibility: public + primary_alias: MOD0017 + current_version: v1 + source_path: items/formal-modules/cid005014-mod0017.yaml + cid005015: + type: formal_module + title: TauLib.BookI.Boundary.Integration + status: canonical + visibility: public + primary_alias: MOD0018 + current_version: v1 + source_path: items/formal-modules/cid005015-mod0018.yaml + cid005016: + type: formal_module + title: TauLib.BookI.Boundary.Iota + status: canonical + visibility: public + primary_alias: MOD0019 + current_version: v1 + source_path: items/formal-modules/cid005016-mod0019.yaml + cid005017: + type: formal_module + title: TauLib.BookI.Boundary.IotaTauStructural + status: canonical + visibility: public + primary_alias: MOD0020 + current_version: v1 + source_path: items/formal-modules/cid005017-mod0020.yaml + cid005018: + type: formal_module + title: TauLib.BookI.Boundary.Measure + status: canonical + visibility: public + primary_alias: MOD0021 + current_version: v1 + source_path: items/formal-modules/cid005018-mod0021.yaml + cid005019: + type: formal_module + title: TauLib.BookI.Boundary.NumberTower + status: canonical + visibility: public + primary_alias: MOD0022 + current_version: v1 + source_path: items/formal-modules/cid005019-mod0022.yaml + cid005020: + type: formal_module + title: TauLib.BookI.Boundary.NumericalProjection + status: canonical + visibility: public + primary_alias: MOD0023 + current_version: v1 + source_path: items/formal-modules/cid005020-mod0023.yaml + cid005021: + type: formal_module + title: TauLib.BookI.Boundary.Quaternions + status: canonical + visibility: public + primary_alias: MOD0024 + current_version: v1 + source_path: items/formal-modules/cid005021-mod0024.yaml + cid005022: + type: formal_module + title: TauLib.BookI.Boundary.RefinementGrowingTorus + status: canonical + visibility: public + primary_alias: MOD0025 + current_version: v1 + source_path: items/formal-modules/cid005022-mod0025.yaml + cid005023: + type: formal_module + title: TauLib.BookI.Boundary.Ring + status: canonical + visibility: public + primary_alias: MOD0026 + current_version: v1 + source_path: items/formal-modules/cid005023-mod0026.yaml + cid005024: + type: formal_module + title: TauLib.BookI.Boundary.Spectral + status: canonical + visibility: public + primary_alias: MOD0027 + current_version: v1 + source_path: items/formal-modules/cid005024-mod0027.yaml + cid005025: + type: formal_module + title: TauLib.BookI.Boundary.SplitComplex + status: canonical + visibility: public + primary_alias: MOD0028 + current_version: v1 + source_path: items/formal-modules/cid005025-mod0028.yaml + cid005026: + type: formal_module + title: TauLib.BookI.Boundary.TauRatAbs + status: canonical + visibility: public + primary_alias: MOD0029 + current_version: v1 + source_path: items/formal-modules/cid005026-mod0029.yaml + cid005027: + type: formal_module + title: TauLib.BookI.Boundary.TauRatField + status: canonical + visibility: public + primary_alias: MOD0030 + current_version: v1 + source_path: items/formal-modules/cid005027-mod0030.yaml + cid005028: + type: formal_module + title: TauLib.BookI.Boundary.TauRatInv + status: canonical + visibility: public + primary_alias: MOD0031 + current_version: v1 + source_path: items/formal-modules/cid005028-mod0031.yaml + cid005029: + type: formal_module + title: TauLib.BookI.Boundary.TauRatOrder + status: canonical + visibility: public + primary_alias: MOD0032 + current_version: v1 + source_path: items/formal-modules/cid005029-mod0032.yaml + cid005030: + type: formal_module + title: TauLib.BookI.Boundary.TauRealAbs + status: canonical + visibility: public + primary_alias: MOD0033 + current_version: v1 + source_path: items/formal-modules/cid005030-mod0033.yaml + cid005031: + type: formal_module + title: TauLib.BookI.Boundary.TauRealE + status: canonical + visibility: public + primary_alias: MOD0034 + current_version: v1 + source_path: items/formal-modules/cid005031-mod0034.yaml + cid005032: + type: formal_module + title: TauLib.BookI.Boundary.TauRealInv + status: canonical + visibility: public + primary_alias: MOD0035 + current_version: v1 + source_path: items/formal-modules/cid005032-mod0035.yaml + cid005033: + type: formal_module + title: TauLib.BookI.Boundary.TauRealIotaTau + status: canonical + visibility: public + primary_alias: MOD0036 + current_version: v1 + source_path: items/formal-modules/cid005033-mod0036.yaml + cid005034: + type: formal_module + title: TauLib.BookI.Boundary.TauRealMulCongr + status: canonical + visibility: public + primary_alias: MOD0037 + current_version: v1 + source_path: items/formal-modules/cid005034-mod0037.yaml + cid005035: + type: formal_module + title: TauLib.BookI.Boundary.TauRealOrder + status: canonical + visibility: public + primary_alias: MOD0038 + current_version: v1 + source_path: items/formal-modules/cid005035-mod0038.yaml + cid005036: + type: formal_module + title: TauLib.BookI.Boundary.TauRealPi + status: canonical + visibility: public + primary_alias: MOD0039 + current_version: v1 + source_path: items/formal-modules/cid005036-mod0039.yaml + cid005037: + type: formal_module + title: TauLib.BookI.Boundary.TauRealPiPlusE + status: canonical + visibility: public + primary_alias: MOD0040 + current_version: v1 + source_path: items/formal-modules/cid005037-mod0040.yaml + cid005038: + type: formal_module + title: TauLib.BookI.Boundary.TauRealSum + status: canonical + visibility: public + primary_alias: MOD0041 + current_version: v1 + source_path: items/formal-modules/cid005038-mod0041.yaml + cid005039: + type: formal_module + title: TauLib.BookI.Boundary.TorusDefectSystem + status: canonical + visibility: public + primary_alias: MOD0042 + current_version: v1 + source_path: items/formal-modules/cid005039-mod0042.yaml + cid005040: + type: formal_module + title: TauLib.BookI.Boundary.UniversalFixedScalar + status: canonical + visibility: public + primary_alias: MOD0043 + current_version: v1 + source_path: items/formal-modules/cid005040-mod0043.yaml + cid005041: + type: formal_module + title: TauLib.BookI.Coordinates.ABCD + status: canonical + visibility: public + primary_alias: MOD0044 + current_version: v1 + source_path: items/formal-modules/cid005041-mod0044.yaml + cid005042: + type: formal_module + title: TauLib.BookI.Coordinates.ChebyshevBias + status: canonical + visibility: public + primary_alias: MOD0045 + current_version: v1 + source_path: items/formal-modules/cid005042-mod0045.yaml + cid005043: + type: formal_module + title: TauLib.BookI.Coordinates.Descent + status: canonical + visibility: public + primary_alias: MOD0046 + current_version: v1 + source_path: items/formal-modules/cid005043-mod0046.yaml + cid005044: + type: formal_module + title: TauLib.BookI.Coordinates.Hyperfact + status: canonical + visibility: public + primary_alias: MOD0047 + current_version: v1 + source_path: items/formal-modules/cid005044-mod0047.yaml + cid005045: + type: formal_module + title: TauLib.BookI.Coordinates.HyperfactFTA + status: canonical + visibility: public + primary_alias: MOD0048 + current_version: v1 + source_path: items/formal-modules/cid005045-mod0048.yaml + cid005046: + type: formal_module + title: TauLib.BookI.Coordinates.HyperfactIsomorphism + status: canonical + visibility: public + primary_alias: MOD0049 + current_version: v1 + source_path: items/formal-modules/cid005046-mod0049.yaml + cid005047: + type: formal_module + title: TauLib.BookI.Coordinates.HyperfactProp + status: canonical + visibility: public + primary_alias: MOD0050 + current_version: v1 + source_path: items/formal-modules/cid005047-mod0050.yaml + cid005048: + type: formal_module + title: TauLib.BookI.Coordinates.IteratedPrime + status: canonical + visibility: public + primary_alias: MOD0051 + current_version: v1 + source_path: items/formal-modules/cid005048-mod0051.yaml + cid005049: + type: formal_module + title: TauLib.BookI.Coordinates.NoTie + status: canonical + visibility: public + primary_alias: MOD0052 + current_version: v1 + source_path: items/formal-modules/cid005049-mod0052.yaml + cid005050: + type: formal_module + title: TauLib.BookI.Coordinates.NormalForm + status: canonical + visibility: public + primary_alias: MOD0053 + current_version: v1 + source_path: items/formal-modules/cid005050-mod0053.yaml + cid005051: + type: formal_module + title: TauLib.BookI.Coordinates.Primes + status: canonical + visibility: public + primary_alias: MOD0054 + current_version: v1 + source_path: items/formal-modules/cid005051-mod0054.yaml + cid005052: + type: formal_module + title: TauLib.BookI.Denotation.Arithmetic + status: canonical + visibility: public + primary_alias: MOD0055 + current_version: v1 + source_path: items/formal-modules/cid005052-mod0055.yaml + cid005053: + type: formal_module + title: TauLib.BookI.Denotation.Equality + status: canonical + visibility: public + primary_alias: MOD0056 + current_version: v1 + source_path: items/formal-modules/cid005053-mod0056.yaml + cid005054: + type: formal_module + title: TauLib.BookI.Denotation.GrowthEscape + status: canonical + visibility: public + primary_alias: MOD0057 + current_version: v1 + source_path: items/formal-modules/cid005054-mod0057.yaml + cid005055: + type: formal_module + title: TauLib.BookI.Denotation.Order + status: canonical + visibility: public + primary_alias: MOD0058 + current_version: v1 + source_path: items/formal-modules/cid005055-mod0058.yaml + cid005056: + type: formal_module + title: TauLib.BookI.Denotation.ProgramMonoid + status: canonical + visibility: public + primary_alias: MOD0059 + current_version: v1 + source_path: items/formal-modules/cid005056-mod0059.yaml + cid005057: + type: formal_module + title: TauLib.BookI.Denotation.RankTransfer + status: canonical + visibility: public + primary_alias: MOD0060 + current_version: v1 + source_path: items/formal-modules/cid005057-mod0060.yaml + cid005058: + type: formal_module + title: TauLib.BookI.Denotation.Structural + status: canonical + visibility: public + primary_alias: MOD0061 + current_version: v1 + source_path: items/formal-modules/cid005058-mod0061.yaml + cid005059: + type: formal_module + title: TauLib.BookI.Denotation.TauIdx + status: canonical + visibility: public + primary_alias: MOD0062 + current_version: v1 + source_path: items/formal-modules/cid005059-mod0062.yaml + cid005060: + type: formal_module + title: TauLib.BookI.Holomorphy.BoundaryInterior + status: canonical + visibility: public + primary_alias: MOD0063 + current_version: v1 + source_path: items/formal-modules/cid005060-mod0063.yaml + cid005061: + type: formal_module + title: TauLib.BookI.Holomorphy.DHolomorphic + status: canonical + visibility: public + primary_alias: MOD0064 + current_version: v1 + source_path: items/formal-modules/cid005061-mod0064.yaml + cid005062: + type: formal_module + title: TauLib.BookI.Holomorphy.DiagonalProtection + status: canonical + visibility: public + primary_alias: MOD0065 + current_version: v1 + source_path: items/formal-modules/cid005062-mod0065.yaml + cid005063: + type: formal_module + title: TauLib.BookI.Holomorphy.GlobalHartogs + status: canonical + visibility: public + primary_alias: MOD0066 + current_version: v1 + source_path: items/formal-modules/cid005063-mod0066.yaml + cid005064: + type: formal_module + title: TauLib.BookI.Holomorphy.H6DiagonalDiscipline + status: canonical + visibility: public + primary_alias: MOD0067 + current_version: v1 + source_path: items/formal-modules/cid005064-mod0067.yaml + cid005065: + type: formal_module + title: TauLib.BookI.Holomorphy.H6EarnedCodomainWaveCR + status: canonical + visibility: public + primary_alias: MOD0068 + current_version: v1 + source_path: items/formal-modules/cid005065-mod0068.yaml + cid005066: + type: formal_module + title: TauLib.BookI.Holomorphy.H6SigmaIdemHolEnd + status: canonical + visibility: public + primary_alias: MOD0069 + current_version: v1 + source_path: items/formal-modules/cid005066-mod0069.yaml + cid005067: + type: formal_module + title: TauLib.BookI.Holomorphy.IdentityTheorem + status: canonical + visibility: public + primary_alias: MOD0070 + current_version: v1 + source_path: items/formal-modules/cid005067-mod0070.yaml + cid005068: + type: formal_module + title: TauLib.BookI.Holomorphy.PresheafEssence + status: canonical + visibility: public + primary_alias: MOD0071 + current_version: v1 + source_path: items/formal-modules/cid005068-mod0071.yaml + cid005069: + type: formal_module + title: TauLib.BookI.Holomorphy.SpectralCoefficients + status: canonical + visibility: public + primary_alias: MOD0072 + current_version: v1 + source_path: items/formal-modules/cid005069-mod0072.yaml + cid005070: + type: formal_module + title: TauLib.BookI.Holomorphy.TauHolomorphic + status: canonical + visibility: public + primary_alias: MOD0073 + current_version: v1 + source_path: items/formal-modules/cid005070-mod0073.yaml + cid005071: + type: formal_module + title: TauLib.BookI.Holomorphy.Thinness + status: canonical + visibility: public + primary_alias: MOD0074 + current_version: v1 + source_path: items/formal-modules/cid005071-mod0074.yaml + cid005072: + type: formal_module + title: TauLib.BookI.Kernel.ActionQuantum + status: canonical + visibility: public + primary_alias: MOD0075 + current_version: v1 + source_path: items/formal-modules/cid005072-mod0075.yaml + cid005073: + type: formal_module + title: TauLib.BookI.Kernel.Axioms + status: canonical + visibility: public + primary_alias: MOD0076 + current_version: v1 + source_path: items/formal-modules/cid005073-mod0076.yaml + cid005074: + type: formal_module + title: TauLib.BookI.Kernel.Diagonal + status: canonical + visibility: public + primary_alias: MOD0077 + current_version: v1 + source_path: items/formal-modules/cid005074-mod0077.yaml + cid005075: + type: formal_module + title: TauLib.BookI.Kernel.Signature + status: canonical + visibility: public + primary_alias: MOD0078 + current_version: v1 + source_path: items/formal-modules/cid005075-mod0078.yaml + cid005076: + type: formal_module + title: TauLib.BookI.KernelFoundation.AdmissibleSymmetryGroup + status: canonical + visibility: public + primary_alias: MOD0079 + current_version: v1 + source_path: items/formal-modules/cid005076-mod0079.yaml + cid005077: + type: formal_module + title: TauLib.BookI.KernelFoundation.GirardLinearEmbedding + status: canonical + visibility: public + primary_alias: MOD0080 + current_version: v1 + source_path: items/formal-modules/cid005077-mod0080.yaml + cid005078: + type: formal_module + title: TauLib.BookI.KernelFoundation.H8KernelSynthesis + status: canonical + visibility: public + primary_alias: MOD0081 + current_version: v1 + source_path: items/formal-modules/cid005078-mod0081.yaml + cid005079: + type: formal_module + title: TauLib.BookI.KernelFoundation.ScalarBridges + status: canonical + visibility: public + primary_alias: MOD0082 + current_version: v1 + source_path: items/formal-modules/cid005079-mod0082.yaml + cid005080: + type: formal_module + title: TauLib.BookI.Logic.BooleanRecovery + status: canonical + visibility: public + primary_alias: MOD0083 + current_version: v1 + source_path: items/formal-modules/cid005080-mod0083.yaml + cid005081: + type: formal_module + title: TauLib.BookI.Logic.Explosion + status: canonical + visibility: public + primary_alias: MOD0084 + current_version: v1 + source_path: items/formal-modules/cid005081-mod0084.yaml + cid005082: + type: formal_module + title: TauLib.BookI.Logic.Truth4 + status: canonical + visibility: public + primary_alias: MOD0085 + current_version: v1 + source_path: items/formal-modules/cid005082-mod0085.yaml + cid005083: + type: formal_module + title: TauLib.BookI.MetaLogic.DiagonalResonance + status: canonical + visibility: public + primary_alias: MOD0086 + current_version: v1 + source_path: items/formal-modules/cid005083-mod0086.yaml + cid005084: + type: formal_module + title: TauLib.BookI.MetaLogic.LinearDiscipline + status: canonical + visibility: public + primary_alias: MOD0087 + current_version: v1 + source_path: items/formal-modules/cid005084-mod0087.yaml + cid005085: + type: formal_module + title: TauLib.BookI.MetaLogic.LinearityAudit + status: canonical + visibility: public + primary_alias: MOD0088 + current_version: v1 + source_path: items/formal-modules/cid005085-mod0088.yaml + cid005086: + type: formal_module + title: TauLib.BookI.MetaLogic.OnticInvariance + status: canonical + visibility: public + primary_alias: MOD0089 + current_version: v1 + source_path: items/formal-modules/cid005086-mod0089.yaml + cid005087: + type: formal_module + title: TauLib.BookI.MetaLogic.ReceptionCriterion + status: canonical + visibility: public + primary_alias: MOD0090 + current_version: v1 + source_path: items/formal-modules/cid005087-mod0090.yaml + cid005088: + type: formal_module + title: TauLib.BookI.MetaLogic.StructuralExclusion + status: canonical + visibility: public + primary_alias: MOD0091 + current_version: v1 + source_path: items/formal-modules/cid005088-mod0091.yaml + cid005089: + type: formal_module + title: TauLib.BookI.MetaLogic.Substrate + status: canonical + visibility: public + primary_alias: MOD0092 + current_version: v1 + source_path: items/formal-modules/cid005089-mod0092.yaml + cid005090: + type: formal_module + title: TauLib.BookI.Orbit.Closure + status: canonical + visibility: public + primary_alias: MOD0093 + current_version: v1 + source_path: items/formal-modules/cid005090-mod0093.yaml + cid005091: + type: formal_module + title: TauLib.BookI.Orbit.Countability + status: canonical + visibility: public + primary_alias: MOD0094 + current_version: v1 + source_path: items/formal-modules/cid005091-mod0094.yaml + cid005092: + type: formal_module + title: TauLib.BookI.Orbit.Generation + status: canonical + visibility: public + primary_alias: MOD0095 + current_version: v1 + source_path: items/formal-modules/cid005092-mod0095.yaml + cid005093: + type: formal_module + title: TauLib.BookI.Orbit.Ladder + status: canonical + visibility: public + primary_alias: MOD0096 + current_version: v1 + source_path: items/formal-modules/cid005093-mod0096.yaml + cid005094: + type: formal_module + title: TauLib.BookI.Orbit.Rigidity + status: canonical + visibility: public + primary_alias: MOD0097 + current_version: v1 + source_path: items/formal-modules/cid005094-mod0097.yaml + cid005095: + type: formal_module + title: TauLib.BookI.Orbit.Saturation + status: canonical + visibility: public + primary_alias: MOD0098 + current_version: v1 + source_path: items/formal-modules/cid005095-mod0098.yaml + cid005096: + type: formal_module + title: TauLib.BookI.Polarity.BipolarAlgebra + status: canonical + visibility: public + primary_alias: MOD0099 + current_version: v1 + source_path: items/formal-modules/cid005096-mod0099.yaml + cid005097: + type: formal_module + title: TauLib.BookI.Polarity.ChineseRemainder + status: canonical + visibility: public + primary_alias: MOD0100 + current_version: v1 + source_path: items/formal-modules/cid005097-mod0100.yaml + cid005098: + type: formal_module + title: TauLib.BookI.Polarity.H2H3ClassifierBridge + status: canonical + visibility: public + primary_alias: MOD0101 + current_version: v1 + source_path: items/formal-modules/cid005098-mod0101.yaml + cid005099: + type: formal_module + title: TauLib.BookI.Polarity.H4BoundaryAlgebra + status: canonical + visibility: public + primary_alias: MOD0102 + current_version: v1 + source_path: items/formal-modules/cid005099-mod0102.yaml + cid005100: + type: formal_module + title: TauLib.BookI.Polarity.H4UniquenessElliptic + status: canonical + visibility: public + primary_alias: MOD0103 + current_version: v1 + source_path: items/formal-modules/cid005100-mod0103.yaml + cid005101: + type: formal_module + title: TauLib.BookI.Polarity.InverseLimit + status: canonical + visibility: public + primary_alias: MOD0104 + current_version: v1 + source_path: items/formal-modules/cid005101-mod0104.yaml + cid005102: + type: formal_module + title: TauLib.BookI.Polarity.Lemniscate + status: canonical + visibility: public + primary_alias: MOD0105 + current_version: v1 + source_path: items/formal-modules/cid005102-mod0105.yaml + cid005103: + type: formal_module + title: TauLib.BookI.Polarity.OmegaGerms + status: canonical + visibility: public + primary_alias: MOD0106 + current_version: v1 + source_path: items/formal-modules/cid005103-mod0106.yaml + cid005104: + type: formal_module + title: TauLib.BookI.Polarity.OmegaRing + status: canonical + visibility: public + primary_alias: MOD0107 + current_version: v1 + source_path: items/formal-modules/cid005104-mod0107.yaml + cid005105: + type: formal_module + title: TauLib.BookI.Polarity.Polarity + status: canonical + visibility: public + primary_alias: MOD0108 + current_version: v1 + source_path: items/formal-modules/cid005105-mod0108.yaml + cid005106: + type: formal_module + title: TauLib.BookI.Polarity.PolarizedGerms + status: canonical + visibility: public + primary_alias: MOD0109 + current_version: v1 + source_path: items/formal-modules/cid005106-mod0109.yaml + cid005107: + type: formal_module + title: TauLib.BookI.Polarity.PrimePolarityClassifier + status: canonical + visibility: public + primary_alias: MOD0110 + current_version: v1 + source_path: items/formal-modules/cid005107-mod0110.yaml + cid005108: + type: formal_module + title: TauLib.BookI.Polarity.PrimePolarityIsomorphism + status: canonical + visibility: public + primary_alias: MOD0111 + current_version: v1 + source_path: items/formal-modules/cid005108-mod0111.yaml + cid005109: + type: formal_module + title: TauLib.BookI.Polarity.SplitComplexCouplingLift + status: canonical + visibility: public + primary_alias: MOD0112 + current_version: v1 + source_path: items/formal-modules/cid005109-mod0112.yaml + cid005110: + type: formal_module + title: TauLib.BookI.Polarity.TeichmuellerLift + status: canonical + visibility: public + primary_alias: MOD0113 + current_version: v1 + source_path: items/formal-modules/cid005110-mod0113.yaml + cid005111: + type: formal_module + title: TauLib.BookI.Sets.CantorRefutation + status: canonical + visibility: public + primary_alias: MOD0114 + current_version: v1 + source_path: items/formal-modules/cid005111-mod0114.yaml + cid005112: + type: formal_module + title: TauLib.BookI.Sets.Counting + status: canonical + visibility: public + primary_alias: MOD0115 + current_version: v1 + source_path: items/formal-modules/cid005112-mod0115.yaml + cid005113: + type: formal_module + title: TauLib.BookI.Sets.Membership + status: canonical + visibility: public + primary_alias: MOD0116 + current_version: v1 + source_path: items/formal-modules/cid005113-mod0116.yaml + cid005114: + type: formal_module + title: TauLib.BookI.Sets.Operations + status: canonical + visibility: public + primary_alias: MOD0117 + current_version: v1 + source_path: items/formal-modules/cid005114-mod0117.yaml + cid005115: + type: formal_module + title: TauLib.BookI.Sets.OrbitSets + status: canonical + visibility: public + primary_alias: MOD0118 + current_version: v1 + source_path: items/formal-modules/cid005115-mod0118.yaml + cid005116: + type: formal_module + title: TauLib.BookI.Sets.Powerset + status: canonical + visibility: public + primary_alias: MOD0119 + current_version: v1 + source_path: items/formal-modules/cid005116-mod0119.yaml + cid005117: + type: formal_module + title: TauLib.BookI.Sets.UniqueInfinity + status: canonical + visibility: public + primary_alias: MOD0120 + current_version: v1 + source_path: items/formal-modules/cid005117-mod0120.yaml + cid005118: + type: formal_module + title: TauLib.BookI.Topos.CartesianProduct + status: canonical + visibility: public + primary_alias: MOD0121 + current_version: v1 + source_path: items/formal-modules/cid005118-mod0121.yaml + cid005119: + type: formal_module + title: TauLib.BookI.Topos.CircularityResolution + status: canonical + visibility: public + primary_alias: MOD0122 + current_version: v1 + source_path: items/formal-modules/cid005119-mod0122.yaml + cid005120: + type: formal_module + title: TauLib.BookI.Topos.EarnedArrows + status: canonical + visibility: public + primary_alias: MOD0123 + current_version: v1 + source_path: items/formal-modules/cid005120-mod0123.yaml + cid005121: + type: formal_module + title: TauLib.BookI.Topos.EarnedTopos + status: canonical + visibility: public + primary_alias: MOD0124 + current_version: v1 + source_path: items/formal-modules/cid005121-mod0124.yaml + cid005122: + type: formal_module + title: TauLib.BookI.Topos.Functors + status: canonical + visibility: public + primary_alias: MOD0125 + current_version: v1 + source_path: items/formal-modules/cid005122-mod0125.yaml + cid005123: + type: formal_module + title: TauLib.BookI.Topos.H6EarnedCategoricalMachine + status: canonical + visibility: public + primary_alias: MOD0126 + current_version: v1 + source_path: items/formal-modules/cid005123-mod0126.yaml + cid005124: + type: formal_module + title: TauLib.BookI.Topos.H7CircularityFull + status: canonical + visibility: public + primary_alias: MOD0127 + current_version: v1 + source_path: items/formal-modules/cid005124-mod0127.yaml + cid005125: + type: formal_module + title: TauLib.BookI.Topos.H7ClassicalClosure + status: canonical + visibility: public + primary_alias: MOD0128 + current_version: v1 + source_path: items/formal-modules/cid005125-mod0128.yaml + cid005126: + type: formal_module + title: TauLib.BookI.Topos.H7ToposClassifier + status: canonical + visibility: public + primary_alias: MOD0129 + current_version: v1 + source_path: items/formal-modules/cid005126-mod0129.yaml + cid005127: + type: formal_module + title: TauLib.BookI.Topos.InternalHom + status: canonical + visibility: public + primary_alias: MOD0130 + current_version: v1 + source_path: items/formal-modules/cid005127-mod0130.yaml + cid005128: + type: formal_module + title: TauLib.BookI.Topos.LimitsSites + status: canonical + visibility: public + primary_alias: MOD0131 + current_version: v1 + source_path: items/formal-modules/cid005128-mod0131.yaml + cid005129: + type: formal_module + title: TauLib.BookI.Topos.ParaconsistentSoundness + status: canonical + visibility: public + primary_alias: MOD0132 + current_version: v1 + source_path: items/formal-modules/cid005129-mod0132.yaml + cid005130: + type: formal_module + title: TauLib.BookI.Topos.WedgeProduct + status: canonical + visibility: public + primary_alias: MOD0133 + current_version: v1 + source_path: items/formal-modules/cid005130-mod0133.yaml + cid005131: + type: formal_module + title: TauLib.BookII.CentralTheorem.BoundaryCharacters + status: canonical + visibility: public + primary_alias: MOD0134 + current_version: v1 + source_path: items/formal-modules/cid005131-mod0134.yaml + cid005132: + type: formal_module + title: TauLib.BookII.CentralTheorem.Categoricity + status: canonical + visibility: public + primary_alias: MOD0135 + current_version: v1 + source_path: items/formal-modules/cid005132-mod0135.yaml + cid005133: + type: formal_module + title: TauLib.BookII.CentralTheorem.CentralTheorem + status: canonical + visibility: public + primary_alias: MOD0136 + current_version: v1 + source_path: items/formal-modules/cid005133-mod0136.yaml + cid005134: + type: formal_module + title: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms + status: canonical + visibility: public + primary_alias: MOD0137 + current_version: v1 + source_path: items/formal-modules/cid005134-mod0137.yaml + cid005135: + type: formal_module + title: TauLib.BookII.CentralTheorem.HartogsExtension + status: canonical + visibility: public + primary_alias: MOD0138 + current_version: v1 + source_path: items/formal-modules/cid005135-mod0138.yaml + cid005136: + type: formal_module + title: TauLib.BookII.CentralTheorem.SheafCohomology + status: canonical + visibility: public + primary_alias: MOD0139 + current_version: v1 + source_path: items/formal-modules/cid005136-mod0139.yaml + cid005137: + type: formal_module + title: TauLib.BookII.CentralTheorem.YonedaApplied + status: canonical + visibility: public + primary_alias: MOD0140 + current_version: v1 + source_path: items/formal-modules/cid005137-mod0140.yaml + cid005138: + type: formal_module + title: TauLib.BookII.Closure.BSDbridge + status: canonical + visibility: public + primary_alias: MOD0141 + current_version: v1 + source_path: items/formal-modules/cid005138-mod0141.yaml + cid005139: + type: formal_module + title: TauLib.BookII.Closure.Connection + status: canonical + visibility: public + primary_alias: MOD0142 + current_version: v1 + source_path: items/formal-modules/cid005139-mod0142.yaml + cid005140: + type: formal_module + title: TauLib.BookII.Closure.Curvature + status: canonical + visibility: public + primary_alias: MOD0143 + current_version: v1 + source_path: items/formal-modules/cid005140-mod0143.yaml + cid005141: + type: formal_module + title: TauLib.BookII.Closure.DiffGeoAgenda + status: canonical + visibility: public + primary_alias: MOD0144 + current_version: v1 + source_path: items/formal-modules/cid005141-mod0144.yaml + cid005142: + type: formal_module + title: TauLib.BookII.Closure.ForwardBook3 + status: canonical + visibility: public + primary_alias: MOD0145 + current_version: v1 + source_path: items/formal-modules/cid005142-mod0145.yaml + cid005143: + type: formal_module + title: TauLib.BookII.Closure.GeometricBiSquare + status: canonical + visibility: public + primary_alias: MOD0146 + current_version: v1 + source_path: items/formal-modules/cid005143-mod0146.yaml + cid005144: + type: formal_module + title: TauLib.BookII.Closure.TauManifold + status: canonical + visibility: public + primary_alias: MOD0147 + current_version: v1 + source_path: items/formal-modules/cid005144-mod0147.yaml + cid005145: + type: formal_module + title: TauLib.BookII.Domains.Cylinders + status: canonical + visibility: public + primary_alias: MOD0148 + current_version: v1 + source_path: items/formal-modules/cid005145-mod0148.yaml + cid005146: + type: formal_module + title: TauLib.BookII.Domains.HolImpliesCont + status: canonical + visibility: public + primary_alias: MOD0149 + current_version: v1 + source_path: items/formal-modules/cid005146-mod0149.yaml + cid005147: + type: formal_module + title: TauLib.BookII.Domains.Ultrametric + status: canonical + visibility: public + primary_alias: MOD0150 + current_version: v1 + source_path: items/formal-modules/cid005147-mod0150.yaml + cid005148: + type: formal_module + title: TauLib.BookII.Enrichment.EnrichmentLadder + status: canonical + visibility: public + primary_alias: MOD0151 + current_version: v1 + source_path: items/formal-modules/cid005148-mod0151.yaml + cid005149: + type: formal_module + title: TauLib.BookII.Enrichment.Homological + status: canonical + visibility: public + primary_alias: MOD0152 + current_version: v1 + source_path: items/formal-modules/cid005149-mod0152.yaml + cid005150: + type: formal_module + title: TauLib.BookII.Enrichment.SelfDescribing + status: canonical + visibility: public + primary_alias: MOD0153 + current_version: v1 + source_path: items/formal-modules/cid005150-mod0153.yaml + cid005151: + type: formal_module + title: TauLib.BookII.Enrichment.SelfEnrichment + status: canonical + visibility: public + primary_alias: MOD0154 + current_version: v1 + source_path: items/formal-modules/cid005151-mod0154.yaml + cid005152: + type: formal_module + title: TauLib.BookII.Enrichment.TwoCategories + status: canonical + visibility: public + primary_alias: MOD0155 + current_version: v1 + source_path: items/formal-modules/cid005152-mod0155.yaml + cid005153: + type: formal_module + title: TauLib.BookII.Enrichment.YonedaTheorem + status: canonical + visibility: public + primary_alias: MOD0156 + current_version: v1 + source_path: items/formal-modules/cid005153-mod0156.yaml + cid005154: + type: formal_module + title: TauLib.BookII.Geometry.Betweenness + status: canonical + visibility: public + primary_alias: MOD0157 + current_version: v1 + source_path: items/formal-modules/cid005154-mod0157.yaml + cid005155: + type: formal_module + title: TauLib.BookII.Geometry.CausalStructure + status: canonical + visibility: public + primary_alias: MOD0158 + current_version: v1 + source_path: items/formal-modules/cid005155-mod0158.yaml + cid005156: + type: formal_module + title: TauLib.BookII.Geometry.Congruence + status: canonical + visibility: public + primary_alias: MOD0159 + current_version: v1 + source_path: items/formal-modules/cid005156-mod0159.yaml + cid005157: + type: formal_module + title: TauLib.BookII.Geometry.OrthodoxBridge + status: canonical + visibility: public + primary_alias: MOD0160 + current_version: v1 + source_path: items/formal-modules/cid005157-mod0160.yaml + cid005158: + type: formal_module + title: TauLib.BookII.Geometry.PaschParallel + status: canonical + visibility: public + primary_alias: MOD0161 + current_version: v1 + source_path: items/formal-modules/cid005158-mod0161.yaml + cid005159: + type: formal_module + title: TauLib.BookII.Hartogs.BndLift + status: canonical + visibility: public + primary_alias: MOD0162 + current_version: v1 + source_path: items/formal-modules/cid005159-mod0162.yaml + cid005160: + type: formal_module + title: TauLib.BookII.Hartogs.CalibratedSplitComplex + status: canonical + visibility: public + primary_alias: MOD0163 + current_version: v1 + source_path: items/formal-modules/cid005160-mod0163.yaml + cid005161: + type: formal_module + title: TauLib.BookII.Hartogs.CanonicalBasis + status: canonical + visibility: public + primary_alias: MOD0164 + current_version: v1 + source_path: items/formal-modules/cid005161-mod0164.yaml + cid005162: + type: formal_module + title: TauLib.BookII.Hartogs.CategoryStructure + status: canonical + visibility: public + primary_alias: MOD0165 + current_version: v1 + source_path: items/formal-modules/cid005162-mod0165.yaml + cid005163: + type: formal_module + title: TauLib.BookII.Hartogs.EvolutionOperator + status: canonical + visibility: public + primary_alias: MOD0166 + current_version: v1 + source_path: items/formal-modules/cid005163-mod0166.yaml + cid005164: + type: formal_module + title: TauLib.BookII.Hartogs.L2Space + status: canonical + visibility: public + primary_alias: MOD0167 + current_version: v1 + source_path: items/formal-modules/cid005164-mod0167.yaml + cid005165: + type: formal_module + title: TauLib.BookII.Hartogs.LaurentResidue + status: canonical + visibility: public + primary_alias: MOD0168 + current_version: v1 + source_path: items/formal-modules/cid005165-mod0168.yaml + cid005166: + type: formal_module + title: TauLib.BookII.Hartogs.MutualDetermination + status: canonical + visibility: public + primary_alias: MOD0169 + current_version: v1 + source_path: items/formal-modules/cid005166-mod0169.yaml + cid005167: + type: formal_module + title: TauLib.BookII.Hartogs.SheafCoherence + status: canonical + visibility: public + primary_alias: MOD0170 + current_version: v1 + source_path: items/formal-modules/cid005167-mod0170.yaml + cid005168: + type: formal_module + title: TauLib.BookII.Interior.ABCDRigidity + status: canonical + visibility: public + primary_alias: MOD0171 + current_version: v1 + source_path: items/formal-modules/cid005168-mod0171.yaml + cid005169: + type: formal_module + title: TauLib.BookII.Interior.BipolarDecomposition + status: canonical + visibility: public + primary_alias: MOD0172 + current_version: v1 + source_path: items/formal-modules/cid005169-mod0172.yaml + cid005170: + type: formal_module + title: TauLib.BookII.Interior.OmegaReadout + status: canonical + visibility: public + primary_alias: MOD0173 + current_version: v1 + source_path: items/formal-modules/cid005170-mod0173.yaml + cid005171: + type: formal_module + title: TauLib.BookII.Interior.Tau3Fibration + status: canonical + visibility: public + primary_alias: MOD0174 + current_version: v1 + source_path: items/formal-modules/cid005171-mod0174.yaml + cid005172: + type: formal_module + title: TauLib.BookII.Interior.TauAdmissible + status: canonical + visibility: public + primary_alias: MOD0175 + current_version: v1 + source_path: items/formal-modules/cid005172-mod0175.yaml + cid005173: + type: formal_module + title: TauLib.BookII.Mirror.DimensionalLadder + status: canonical + visibility: public + primary_alias: MOD0176 + current_version: v1 + source_path: items/formal-modules/cid005173-mod0176.yaml + cid005174: + type: formal_module + title: TauLib.BookII.Mirror.Inventory + status: canonical + visibility: public + primary_alias: MOD0177 + current_version: v1 + source_path: items/formal-modules/cid005174-mod0177.yaml + cid005175: + type: formal_module + title: TauLib.BookII.Mirror.PhysicsQuadrant + status: canonical + visibility: public + primary_alias: MOD0178 + current_version: v1 + source_path: items/formal-modules/cid005175-mod0178.yaml + cid005176: + type: formal_module + title: TauLib.BookII.Mirror.SignClassification + status: canonical + visibility: public + primary_alias: MOD0179 + current_version: v1 + source_path: items/formal-modules/cid005176-mod0179.yaml + cid005177: + type: formal_module + title: TauLib.BookII.Mirror.WaveHolomorphy + status: canonical + visibility: public + primary_alias: MOD0180 + current_version: v1 + source_path: items/formal-modules/cid005177-mod0180.yaml + cid005178: + type: formal_module + title: TauLib.BookII.Prologue.SplitComplexInterior + status: canonical + visibility: public + primary_alias: MOD0181 + current_version: v1 + source_path: items/formal-modules/cid005178-mod0181.yaml + cid005179: + type: formal_module + title: TauLib.BookII.Regularity.CodeDecode + status: canonical + visibility: public + primary_alias: MOD0182 + current_version: v1 + source_path: items/formal-modules/cid005179-mod0182.yaml + cid005180: + type: formal_module + title: TauLib.BookII.Regularity.IdempotentDecomposition + status: canonical + visibility: public + primary_alias: MOD0183 + current_version: v1 + source_path: items/formal-modules/cid005180-mod0183.yaml + cid005181: + type: formal_module + title: TauLib.BookII.Regularity.PositiveRegularity + status: canonical + visibility: public + primary_alias: MOD0184 + current_version: v1 + source_path: items/formal-modules/cid005181-mod0184.yaml + cid005182: + type: formal_module + title: TauLib.BookII.Regularity.PreYoneda + status: canonical + visibility: public + primary_alias: MOD0185 + current_version: v1 + source_path: items/formal-modules/cid005182-mod0185.yaml + cid005183: + type: formal_module + title: TauLib.BookII.Regularity.ThreeLemmaChain + status: canonical + visibility: public + primary_alias: MOD0186 + current_version: v1 + source_path: items/formal-modules/cid005183-mod0186.yaml + cid005184: + type: formal_module + title: TauLib.BookII.Topology.BoundaryMinimality + status: canonical + visibility: public + primary_alias: MOD0187 + current_version: v1 + source_path: items/formal-modules/cid005184-mod0187.yaml + cid005185: + type: formal_module + title: TauLib.BookII.Topology.DimensionFour + status: canonical + visibility: public + primary_alias: MOD0188 + current_version: v1 + source_path: items/formal-modules/cid005185-mod0188.yaml + cid005186: + type: formal_module + title: TauLib.BookII.Topology.Invariant + status: canonical + visibility: public + primary_alias: MOD0189 + current_version: v1 + source_path: items/formal-modules/cid005186-mod0189.yaml + cid005187: + type: formal_module + title: TauLib.BookII.Topology.StoneSpace + status: canonical + visibility: public + primary_alias: MOD0190 + current_version: v1 + source_path: items/formal-modules/cid005187-mod0190.yaml + cid005188: + type: formal_module + title: TauLib.BookII.Topology.TorusDegeneration + status: canonical + visibility: public + primary_alias: MOD0191 + current_version: v1 + source_path: items/formal-modules/cid005188-mod0191.yaml + cid005189: + type: formal_module + title: TauLib.BookII.Transcendentals.Circles + status: canonical + visibility: public + primary_alias: MOD0192 + current_version: v1 + source_path: items/formal-modules/cid005189-mod0192.yaml + cid005190: + type: formal_module + title: TauLib.BookII.Transcendentals.EEarned + status: canonical + visibility: public + primary_alias: MOD0193 + current_version: v1 + source_path: items/formal-modules/cid005190-mod0193.yaml + cid005191: + type: formal_module + title: TauLib.BookII.Transcendentals.IotaTauConfirmed + status: canonical + visibility: public + primary_alias: MOD0194 + current_version: v1 + source_path: items/formal-modules/cid005191-mod0194.yaml + cid005192: + type: formal_module + title: TauLib.BookII.Transcendentals.JReplacesI + status: canonical + visibility: public + primary_alias: MOD0195 + current_version: v1 + source_path: items/formal-modules/cid005192-mod0195.yaml + cid005193: + type: formal_module + title: TauLib.BookII.Transcendentals.Lines + status: canonical + visibility: public + primary_alias: MOD0196 + current_version: v1 + source_path: items/formal-modules/cid005193-mod0196.yaml + cid005194: + type: formal_module + title: TauLib.BookII.Transcendentals.PiEarned + status: canonical + visibility: public + primary_alias: MOD0197 + current_version: v1 + source_path: items/formal-modules/cid005194-mod0197.yaml + cid005195: + type: formal_module + title: TauLib.BookIII.Arithmetic.ABCConjecture + status: canonical + visibility: public + primary_alias: MOD0198 + current_version: v1 + source_path: items/formal-modules/cid005195-mod0198.yaml + cid005196: + type: formal_module + title: TauLib.BookIII.Arithmetic.ABCDeep + status: canonical + visibility: public + primary_alias: MOD0199 + current_version: v1 + source_path: items/formal-modules/cid005196-mod0199.yaml + cid005197: + type: formal_module + title: TauLib.BookIII.Arithmetic.BSD + status: canonical + visibility: public + primary_alias: MOD0200 + current_version: v1 + source_path: items/formal-modules/cid005197-mod0200.yaml + cid005198: + type: formal_module + title: TauLib.BookIII.Arithmetic.EnrFunctor01 + status: canonical + visibility: public + primary_alias: MOD0201 + current_version: v1 + source_path: items/formal-modules/cid005198-mod0201.yaml + cid005199: + type: formal_module + title: TauLib.BookIII.Arithmetic.EnrichedBiSquare + status: canonical + visibility: public + primary_alias: MOD0202 + current_version: v1 + source_path: items/formal-modules/cid005199-mod0202.yaml + cid005200: + type: formal_module + title: TauLib.BookIII.Arithmetic.Langlands + status: canonical + visibility: public + primary_alias: MOD0203 + current_version: v1 + source_path: items/formal-modules/cid005200-mod0203.yaml + cid005201: + type: formal_module + title: TauLib.BookIII.Arithmetic.ProtoCodes + status: canonical + visibility: public + primary_alias: MOD0204 + current_version: v1 + source_path: items/formal-modules/cid005201-mod0204.yaml + cid005202: + type: formal_module + title: TauLib.BookIII.Arithmetic.RationalPoints + status: canonical + visibility: public + primary_alias: MOD0205 + current_version: v1 + source_path: items/formal-modules/cid005202-mod0205.yaml + cid005203: + type: formal_module + title: TauLib.BookIII.Arithmetic.TowerAssembly + status: canonical + visibility: public + primary_alias: MOD0206 + current_version: v1 + source_path: items/formal-modules/cid005203-mod0206.yaml + cid005204: + type: formal_module + title: TauLib.BookIII.Bridge.BridgeAxiom + status: canonical + visibility: public + primary_alias: MOD0207 + current_version: v1 + source_path: items/formal-modules/cid005204-mod0207.yaml + cid005205: + type: formal_module + title: TauLib.BookIII.Bridge.ConjectureGaps + status: canonical + visibility: public + primary_alias: MOD0208 + current_version: v1 + source_path: items/formal-modules/cid005205-mod0208.yaml + cid005206: + type: formal_module + title: TauLib.BookIII.Bridge.ForbiddenMoves + status: canonical + visibility: public + primary_alias: MOD0209 + current_version: v1 + source_path: items/formal-modules/cid005206-mod0209.yaml + cid005207: + type: formal_module + title: TauLib.BookIII.Bridge.Incompleteness + status: canonical + visibility: public + primary_alias: MOD0210 + current_version: v1 + source_path: items/formal-modules/cid005207-mod0210.yaml + cid005208: + type: formal_module + title: TauLib.BookIII.Bridge.TranslationArith + status: canonical + visibility: public + primary_alias: MOD0211 + current_version: v1 + source_path: items/formal-modules/cid005208-mod0211.yaml + cid005209: + type: formal_module + title: TauLib.BookIII.Bridge.TranslationObstruction + status: canonical + visibility: public + primary_alias: MOD0212 + current_version: v1 + source_path: items/formal-modules/cid005209-mod0212.yaml + cid005210: + type: formal_module + title: TauLib.BookIII.Bridge.TranslationTopo + status: canonical + visibility: public + primary_alias: MOD0213 + current_version: v1 + source_path: items/formal-modules/cid005210-mod0213.yaml + cid005211: + type: formal_module + title: TauLib.BookIII.Bridge.ZFCasVM + status: canonical + visibility: public + primary_alias: MOD0214 + current_version: v1 + source_path: items/formal-modules/cid005211-mod0214.yaml + cid005212: + type: formal_module + title: TauLib.BookIII.Computation.Admissibility + status: canonical + visibility: public + primary_alias: MOD0215 + current_version: v1 + source_path: items/formal-modules/cid005212-mod0215.yaml + cid005213: + type: formal_module + title: TauLib.BookIII.Computation.CompBiSquare + status: canonical + visibility: public + primary_alias: MOD0216 + current_version: v1 + source_path: items/formal-modules/cid005213-mod0216.yaml + cid005214: + type: formal_module + title: TauLib.BookIII.Computation.E2Agent + status: canonical + visibility: public + primary_alias: MOD0217 + current_version: v1 + source_path: items/formal-modules/cid005214-mod0217.yaml + cid005215: + type: formal_module + title: TauLib.BookIII.Computation.E2Witness + status: canonical + visibility: public + primary_alias: MOD0218 + current_version: v1 + source_path: items/formal-modules/cid005215-mod0218.yaml + cid005216: + type: formal_module + title: TauLib.BookIII.Computation.TowerMachine + status: canonical + visibility: public + primary_alias: MOD0219 + current_version: v1 + source_path: items/formal-modules/cid005216-mod0219.yaml + cid005217: + type: formal_module + title: TauLib.BookIII.Computation.WitnessSearch + status: canonical + visibility: public + primary_alias: MOD0220 + current_version: v1 + source_path: items/formal-modules/cid005217-mod0220.yaml + cid005218: + type: formal_module + title: TauLib.BookIII.Doors.BridgeTightening + status: canonical + visibility: public + primary_alias: MOD0221 + current_version: v1 + source_path: items/formal-modules/cid005218-mod0221.yaml + cid005219: + type: formal_module + title: TauLib.BookIII.Doors.CriticalLine + status: canonical + visibility: public + primary_alias: MOD0222 + current_version: v1 + source_path: items/formal-modules/cid005219-mod0222.yaml + cid005220: + type: formal_module + title: TauLib.BookIII.Doors.GrandGRH + status: canonical + visibility: public + primary_alias: MOD0223 + current_version: v1 + source_path: items/formal-modules/cid005220-mod0223.yaml + cid005221: + type: formal_module + title: TauLib.BookIII.Doors.LemniscateOperator + status: canonical + visibility: public + primary_alias: MOD0224 + current_version: v1 + source_path: items/formal-modules/cid005221-mod0224.yaml + cid005222: + type: formal_module + title: TauLib.BookIII.Doors.MasterSchema + status: canonical + visibility: public + primary_alias: MOD0225 + current_version: v1 + source_path: items/formal-modules/cid005222-mod0225.yaml + cid005223: + type: formal_module + title: TauLib.BookIII.Doors.MutualDetermination + status: canonical + visibility: public + primary_alias: MOD0226 + current_version: v1 + source_path: items/formal-modules/cid005223-mod0226.yaml + cid005224: + type: formal_module + title: TauLib.BookIII.Doors.Poincare + status: canonical + visibility: public + primary_alias: MOD0227 + current_version: v1 + source_path: items/formal-modules/cid005224-mod0227.yaml + cid005225: + type: formal_module + title: TauLib.BookIII.Doors.SpectralCorrespondence + status: canonical + visibility: public + primary_alias: MOD0228 + current_version: v1 + source_path: items/formal-modules/cid005225-mod0228.yaml + cid005226: + type: formal_module + title: TauLib.BookIII.Doors.SpectralDecomp + status: canonical + visibility: public + primary_alias: MOD0229 + current_version: v1 + source_path: items/formal-modules/cid005226-mod0229.yaml + cid005227: + type: formal_module + title: TauLib.BookIII.Doors.SplitComplexZeta + status: canonical + visibility: public + primary_alias: MOD0230 + current_version: v1 + source_path: items/formal-modules/cid005227-mod0230.yaml + cid005228: + type: formal_module + title: TauLib.BookIII.Enrichment.CanonicalLadder + status: canonical + visibility: public + primary_alias: MOD0231 + current_version: v1 + source_path: items/formal-modules/cid005228-mod0231.yaml + cid005229: + type: formal_module + title: TauLib.BookIII.Enrichment.Functor + status: canonical + visibility: public + primary_alias: MOD0232 + current_version: v1 + source_path: items/formal-modules/cid005229-mod0232.yaml + cid005230: + type: formal_module + title: TauLib.BookIII.Enrichment.LayerTemplate + status: canonical + visibility: public + primary_alias: MOD0233 + current_version: v1 + source_path: items/formal-modules/cid005230-mod0233.yaml + cid005231: + type: formal_module + title: TauLib.BookIII.Hinge.DependencyChain + status: canonical + visibility: public + primary_alias: MOD0234 + current_version: v1 + source_path: items/formal-modules/cid005231-mod0234.yaml + cid005232: + type: formal_module + title: TauLib.BookIII.Hinge.HingeTheorem + status: canonical + visibility: public + primary_alias: MOD0235 + current_version: v1 + source_path: items/formal-modules/cid005232-mod0235.yaml + cid005233: + type: formal_module + title: TauLib.BookIII.Mirror.E3Witness + status: canonical + visibility: public + primary_alias: MOD0236 + current_version: v1 + source_path: items/formal-modules/cid005233-mod0236.yaml + cid005234: + type: formal_module + title: TauLib.BookIII.Mirror.ProofTheoryE3 + status: canonical + visibility: public + primary_alias: MOD0237 + current_version: v1 + source_path: items/formal-modules/cid005234-mod0237.yaml + cid005235: + type: formal_module + title: TauLib.BookIII.Mirror.Saturation + status: canonical + visibility: public + primary_alias: MOD0238 + current_version: v1 + source_path: items/formal-modules/cid005235-mod0238.yaml + cid005236: + type: formal_module + title: TauLib.BookIII.Physics.FluidData + status: canonical + visibility: public + primary_alias: MOD0239 + current_version: v1 + source_path: items/formal-modules/cid005236-mod0239.yaml + cid005237: + type: formal_module + title: TauLib.BookIII.Physics.GapTheorem + status: canonical + visibility: public + primary_alias: MOD0240 + current_version: v1 + source_path: items/formal-modules/cid005237-mod0240.yaml + cid005238: + type: formal_module + title: TauLib.BookIII.Physics.HartogsFlow + status: canonical + visibility: public + primary_alias: MOD0241 + current_version: v1 + source_path: items/formal-modules/cid005238-mod0241.yaml + cid005239: + type: formal_module + title: TauLib.BookIII.Physics.Hodge + status: canonical + visibility: public + primary_alias: MOD0242 + current_version: v1 + source_path: items/formal-modules/cid005239-mod0242.yaml + cid005240: + type: formal_module + title: TauLib.BookIII.Physics.PhysicsAssembly + status: canonical + visibility: public + primary_alias: MOD0243 + current_version: v1 + source_path: items/formal-modules/cid005240-mod0243.yaml + cid005241: + type: formal_module + title: TauLib.BookIII.Physics.PositiveRegularity + status: canonical + visibility: public + primary_alias: MOD0244 + current_version: v1 + source_path: items/formal-modules/cid005241-mod0244.yaml + cid005242: + type: formal_module + title: TauLib.BookIII.Physics.StrongSector + status: canonical + visibility: public + primary_alias: MOD0245 + current_version: v1 + source_path: items/formal-modules/cid005242-mod0245.yaml + cid005243: + type: formal_module + title: TauLib.BookIII.Prologue.HartogsBulk + status: canonical + visibility: public + primary_alias: MOD0246 + current_version: v1 + source_path: items/formal-modules/cid005243-mod0246.yaml + cid005244: + type: formal_module + title: TauLib.BookIII.Sectors.BoundaryCharacters + status: canonical + visibility: public + primary_alias: MOD0247 + current_version: v1 + source_path: items/formal-modules/cid005244-mod0247.yaml + cid005245: + type: formal_module + title: TauLib.BookIII.Sectors.Decomposition + status: canonical + visibility: public + primary_alias: MOD0248 + current_version: v1 + source_path: items/formal-modules/cid005245-mod0248.yaml + cid005246: + type: formal_module + title: TauLib.BookIII.Sectors.LanglandsReflection + status: canonical + visibility: public + primary_alias: MOD0249 + current_version: v1 + source_path: items/formal-modules/cid005246-mod0249.yaml + cid005247: + type: formal_module + title: TauLib.BookIII.Sectors.ParityBridge + status: canonical + visibility: public + primary_alias: MOD0250 + current_version: v1 + source_path: items/formal-modules/cid005247-mod0250.yaml + cid005248: + type: formal_module + title: TauLib.BookIII.Spectral.AdditiveConjectures + status: canonical + visibility: public + primary_alias: MOD0251 + current_version: v1 + source_path: items/formal-modules/cid005248-mod0251.yaml + cid005249: + type: formal_module + title: TauLib.BookIII.Spectral.Adeles + status: canonical + visibility: public + primary_alias: MOD0252 + current_version: v1 + source_path: items/formal-modules/cid005249-mod0252.yaml + cid005250: + type: formal_module + title: TauLib.BookIII.Spectral.BipolarClassifier + status: canonical + visibility: public + primary_alias: MOD0253 + current_version: v1 + source_path: items/formal-modules/cid005250-mod0253.yaml + cid005251: + type: formal_module + title: TauLib.BookIII.Spectral.CRT + status: canonical + visibility: public + primary_alias: MOD0254 + current_version: v1 + source_path: items/formal-modules/cid005251-mod0254.yaml + cid005252: + type: formal_module + title: TauLib.BookIII.Spectral.ConfinementBridge + status: canonical + visibility: public + primary_alias: MOD0255 + current_version: v1 + source_path: items/formal-modules/cid005252-mod0255.yaml + cid005253: + type: formal_module + title: TauLib.BookIII.Spectral.GoldbachDeep + status: canonical + visibility: public + primary_alias: MOD0256 + current_version: v1 + source_path: items/formal-modules/cid005253-mod0256.yaml + cid005254: + type: formal_module + title: TauLib.BookIII.Spectral.HenselLifting + status: canonical + visibility: public + primary_alias: MOD0257 + current_version: v1 + source_path: items/formal-modules/cid005254-mod0257.yaml + cid005255: + type: formal_module + title: TauLib.BookIII.Spectral.LocalFields + status: canonical + visibility: public + primary_alias: MOD0258 + current_version: v1 + source_path: items/formal-modules/cid005255-mod0258.yaml + cid005256: + type: formal_module + title: TauLib.BookIII.Spectral.ModularForms + status: canonical + visibility: public + primary_alias: MOD0259 + current_version: v1 + source_path: items/formal-modules/cid005256-mod0259.yaml + cid005257: + type: formal_module + title: TauLib.BookIII.Spectral.PrimorialLadder + status: canonical + visibility: public + primary_alias: MOD0260 + current_version: v1 + source_path: items/formal-modules/cid005257-mod0260.yaml + cid005258: + type: formal_module + title: TauLib.BookIII.Spectral.SieveInfrastructure + status: canonical + visibility: public + primary_alias: MOD0261 + current_version: v1 + source_path: items/formal-modules/cid005258-mod0261.yaml + cid005259: + type: formal_module + title: TauLib.BookIII.Spectral.Trichotomy + status: canonical + visibility: public + primary_alias: MOD0262 + current_version: v1 + source_path: items/formal-modules/cid005259-mod0262.yaml + cid005260: + type: formal_module + title: TauLib.BookIII.Spectral.TwinPrimeDeep + status: canonical + visibility: public + primary_alias: MOD0263 + current_version: v1 + source_path: items/formal-modules/cid005260-mod0263.yaml + cid005261: + type: formal_module + title: TauLib.BookIII.Spectrum.InterfaceWidth + status: canonical + visibility: public + primary_alias: MOD0264 + current_version: v1 + source_path: items/formal-modules/cid005261-mod0264.yaml + cid005262: + type: formal_module + title: TauLib.BookIII.Spectrum.KernelHinge + status: canonical + visibility: public + primary_alias: MOD0265 + current_version: v1 + source_path: items/formal-modules/cid005262-mod0265.yaml + cid005263: + type: formal_module + title: TauLib.BookIII.Spectrum.TTM + status: canonical + visibility: public + primary_alias: MOD0266 + current_version: v1 + source_path: items/formal-modules/cid005263-mod0266.yaml + cid005264: + type: formal_module + title: TauLib.BookIII.Spectrum.ThreeSAT + status: canonical + visibility: public + primary_alias: MOD0267 + current_version: v1 + source_path: items/formal-modules/cid005264-mod0267.yaml + cid005265: + type: formal_theorem + title: chi_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0002 + current_version: v1 + source_path: items/formal-theorems/cid005265-fth0002.yaml + cid005266: + type: formal_theorem + title: chi_orthogonal (theorem) + status: canonical + visibility: public + primary_alias: FTH0003 + current_version: v1 + source_path: items/formal-theorems/cid005266-fth0003.yaml + cid005267: + type: formal_theorem + title: sc_j_squared (theorem) + status: canonical + visibility: public + primary_alias: FTH0004 + current_version: v1 + source_path: items/formal-theorems/cid005267-fth0004.yaml + cid005268: + type: formal_theorem + title: taureal_archimedean_embedding (theorem) + status: canonical + visibility: public + primary_alias: FTH0005 + current_version: v1 + source_path: items/formal-theorems/cid005268-fth0005.yaml + cid005269: + type: formal_theorem + title: root_of_unity_one (theorem) + status: canonical + visibility: public + primary_alias: FTH0006 + current_version: v1 + source_path: items/formal-theorems/cid005269-fth0006.yaml + cid005270: + type: formal_theorem + title: crossing_iff_real (theorem) + status: canonical + visibility: public + primary_alias: FTH0007 + current_version: v1 + source_path: items/formal-theorems/cid005270-fth0007.yaml + cid005271: + type: formal_theorem + title: fourier_sigma_swap (theorem) + status: canonical + visibility: public + primary_alias: FTH0008 + current_version: v1 + source_path: items/formal-theorems/cid005271-fth0008.yaml + cid005272: + type: formal_theorem + title: integral_const_one_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0009 + current_version: v1 + source_path: items/formal-theorems/cid005272-fth0009.yaml + cid005273: + type: formal_theorem + title: linearity_2f_3g_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0010 + current_version: v1 + source_path: items/formal-theorems/cid005273-fth0010.yaml + cid005274: + type: formal_theorem + title: linearity_identity_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0011 + current_version: v1 + source_path: items/formal-theorems/cid005274-fth0011.yaml + cid005275: + type: formal_theorem + title: integral_even_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0012 + current_version: v1 + source_path: items/formal-theorems/cid005275-fth0012.yaml + cid005276: + type: formal_theorem + title: primorial_pos_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0013 + current_version: v1 + source_path: items/formal-theorems/cid005276-fth0013.yaml + cid005277: + type: formal_theorem + title: additivity_even_odd_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0014 + current_version: v1 + source_path: items/formal-theorems/cid005277-fth0014.yaml + cid005278: + type: formal_theorem + title: additivity_bc_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0015 + current_version: v1 + source_path: items/formal-theorems/cid005278-fth0015.yaml + cid005279: + type: formal_theorem + title: full_set_compatible_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0016 + current_version: v1 + source_path: items/formal-theorems/cid005279-fth0016.yaml + cid005280: + type: formal_theorem + title: empty_set_compatible_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0017 + current_version: v1 + source_path: items/formal-theorems/cid005280-fth0017.yaml + cid005281: + type: formal_theorem + title: even_set_compatible_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0018 + current_version: v1 + source_path: items/formal-theorems/cid005281-fth0018.yaml + cid005282: + type: formal_theorem + title: full_set_measure_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0019 + current_version: v1 + source_path: items/formal-theorems/cid005282-fth0019.yaml + cid005283: + type: formal_theorem + title: empty_set_measure_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0020 + current_version: v1 + source_path: items/formal-theorems/cid005283-fth0020.yaml + cid005284: + type: formal_theorem + title: spectral_decomposition (theorem) + status: canonical + visibility: public + primary_alias: FTH0021 + current_version: v1 + source_path: items/formal-theorems/cid005284-fth0021.yaml + cid005285: + type: formal_theorem + title: spectral_unique (theorem) + status: canonical + visibility: public + primary_alias: FTH0022 + current_version: v1 + source_path: items/formal-theorems/cid005285-fth0022.yaml + cid005286: + type: formal_theorem + title: prime_race_4_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0023 + current_version: v1 + source_path: items/formal-theorems/cid005286-fth0023.yaml + cid005287: + type: formal_theorem + title: bias_positive_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0024 + current_version: v1 + source_path: items/formal-theorems/cid005287-fth0024.yaml + cid005288: + type: formal_theorem + title: bias_mod3_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0025 + current_version: v1 + source_path: items/formal-theorems/cid005288-fth0025.yaml + cid005289: + type: formal_theorem + title: bias_primorial_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0026 + current_version: v1 + source_path: items/formal-theorems/cid005289-fth0026.yaml + cid005290: + type: formal_theorem + title: no_tie (theorem) + status: canonical + visibility: public + primary_alias: FTH0027 + current_version: v1 + source_path: items/formal-theorems/cid005290-fth0027.yaml + cid005291: + type: formal_theorem + title: prime_product_unique (theorem) + status: canonical + visibility: public + primary_alias: FTH0028 + current_version: v1 + source_path: items/formal-theorems/cid005291-fth0028.yaml + cid005292: + type: formal_theorem + title: idx_add_injective (theorem) + status: canonical + visibility: public + primary_alias: FTH0029 + current_version: v1 + source_path: items/formal-theorems/cid005292-fth0029.yaml + cid005293: + type: formal_theorem + title: growth_escape (theorem) + status: canonical + visibility: public + primary_alias: FTH0030 + current_version: v1 + source_path: items/formal-theorems/cid005293-fth0030.yaml + cid005294: + type: formal_theorem + title: denotational_lt_wf (theorem) + status: canonical + visibility: public + primary_alias: FTH0031 + current_version: v1 + source_path: items/formal-theorems/cid005294-fth0031.yaml + cid005295: + type: formal_theorem + title: compose_assoc (theorem) + status: canonical + visibility: public + primary_alias: FTH0032 + current_version: v1 + source_path: items/formal-theorems/cid005295-fth0032.yaml + cid005296: + type: formal_theorem + title: rho_count_compose (theorem) + status: canonical + visibility: public + primary_alias: FTH0033 + current_version: v1 + source_path: items/formal-theorems/cid005296-fth0033.yaml + cid005297: + type: formal_theorem + title: interior_from_boundary (theorem) + status: canonical + visibility: public + primary_alias: FTH0034 + current_version: v1 + source_path: items/formal-theorems/cid005297-fth0034.yaml + cid005298: + type: formal_theorem + title: sector_fun_independent (theorem) + status: canonical + visibility: public + primary_alias: FTH0035 + current_version: v1 + source_path: items/formal-theorems/cid005298-fth0035.yaml + cid005299: + type: formal_theorem + title: no_simul_projection_b (theorem) + status: canonical + visibility: public + primary_alias: FTH0036 + current_version: v1 + source_path: items/formal-theorems/cid005299-fth0036.yaml + cid005300: + type: formal_theorem + title: diagonal_free_protection (theorem) + status: canonical + visibility: public + primary_alias: FTH0037 + current_version: v1 + source_path: items/formal-theorems/cid005300-fth0037.yaml + cid005301: + type: formal_theorem + title: stagefun_comp_assoc (theorem) + status: canonical + visibility: public + primary_alias: FTH0038 + current_version: v1 + source_path: items/formal-theorems/cid005301-fth0038.yaml + cid005302: + type: formal_theorem + title: global_hartogs (theorem) + status: canonical + visibility: public + primary_alias: FTH0039 + current_version: v1 + source_path: items/formal-theorems/cid005302-fth0039.yaml + cid005303: + type: formal_theorem + title: tail_agree_downward (theorem) + status: canonical + visibility: public + primary_alias: FTH0040 + current_version: v1 + source_path: items/formal-theorems/cid005303-fth0040.yaml + cid005304: + type: formal_theorem + title: tau_identity_reduce (theorem) + status: canonical + visibility: public + primary_alias: FTH0041 + current_version: v1 + source_path: items/formal-theorems/cid005304-fth0041.yaml + cid005305: + type: formal_theorem + title: presheaf_characterization (theorem) + status: canonical + visibility: public + primary_alias: FTH0042 + current_version: v1 + source_path: items/formal-theorems/cid005305-fth0042.yaml + cid005306: + type: formal_theorem + title: bi_square_characterization (theorem) + status: canonical + visibility: public + primary_alias: FTH0043 + current_version: v1 + source_path: items/formal-theorems/cid005306-fth0043.yaml + cid005307: + type: formal_theorem + title: spectral_determines (theorem) + status: canonical + visibility: public + primary_alias: FTH0045 + current_version: v1 + source_path: items/formal-theorems/cid005307-fth0045.yaml + cid005308: + type: formal_theorem + title: crt_extension_b (theorem) + status: canonical + visibility: public + primary_alias: FTH0046 + current_version: v1 + source_path: items/formal-theorems/cid005308-fth0046.yaml + cid005309: + type: formal_theorem + title: removable_singularity (theorem) + status: canonical + visibility: public + primary_alias: FTH0047 + current_version: v1 + source_path: items/formal-theorems/cid005309-fth0047.yaml + cid005310: + type: formal_theorem + title: euler_sieve_cross (theorem) + status: canonical + visibility: public + primary_alias: FTH0048 + current_version: v1 + source_path: items/formal-theorems/cid005310-fth0048.yaml + cid005311: + type: formal_theorem + title: K1_strict_order (theorem) + status: canonical + visibility: public + primary_alias: FTH0049 + current_version: v1 + source_path: items/formal-theorems/cid005311-fth0049.yaml + cid005312: + type: formal_theorem + title: K2_omega_fixed (theorem) + status: canonical + visibility: public + primary_alias: FTH0050 + current_version: v1 + source_path: items/formal-theorems/cid005312-fth0050.yaml + cid005313: + type: formal_theorem + title: K3_orbit_seeded (theorem) + status: canonical + visibility: public + primary_alias: FTH0051 + current_version: v1 + source_path: items/formal-theorems/cid005313-fth0051.yaml + cid005314: + type: formal_theorem + title: K4_no_jump (theorem) + status: canonical + visibility: public + primary_alias: FTH0052 + current_version: v1 + source_path: items/formal-theorems/cid005314-fth0052.yaml + cid005315: + type: formal_theorem + title: K5_beacon_non_succ (theorem) + status: canonical + visibility: public + primary_alias: FTH0053 + current_version: v1 + source_path: items/formal-theorems/cid005315-fth0053.yaml + cid005316: + type: formal_theorem + title: K6_object_closure (theorem) + status: canonical + visibility: public + primary_alias: FTH0054 + current_version: v1 + source_path: items/formal-theorems/cid005316-fth0054.yaml + cid005317: + type: formal_theorem + title: gen_distinct (theorem) + status: canonical + visibility: public + primary_alias: FTH0055 + current_version: v1 + source_path: items/formal-theorems/cid005317-fth0055.yaml + cid005318: + type: formal_theorem + title: rho_injective (theorem) + status: canonical + visibility: public + primary_alias: FTH0056 + current_version: v1 + source_path: items/formal-theorems/cid005318-fth0056.yaml + cid005319: + type: formal_theorem + title: diagonal_channel_count (theorem) + status: canonical + visibility: public + primary_alias: FTH0057 + current_version: v1 + source_path: items/formal-theorems/cid005319-fth0057.yaml + cid005320: + type: formal_theorem + title: rewiring_levels_eq_solenoidal (theorem) + status: canonical + visibility: public + primary_alias: FTH0058 + current_version: v1 + source_path: items/formal-theorems/cid005320-fth0058.yaml + cid005321: + type: formal_theorem + title: boolean_recovery (theorem) + status: canonical + visibility: public + primary_alias: FTH0059 + current_version: v1 + source_path: items/formal-theorems/cid005321-fth0059.yaml + cid005322: + type: formal_theorem + title: forget_fiber (theorem) + status: canonical + visibility: public + primary_alias: FTH0060 + current_version: v1 + source_path: items/formal-theorems/cid005322-fth0060.yaml + cid005323: + type: formal_theorem + title: explosion_barrier (theorem) + status: canonical + visibility: public + primary_alias: FTH0061 + current_version: v1 + source_path: items/formal-theorems/cid005323-fth0061.yaml + cid005324: + type: formal_theorem + title: no_identity_decoherence (theorem) + status: canonical + visibility: public + primary_alias: FTH0062 + current_version: v1 + source_path: items/formal-theorems/cid005324-fth0062.yaml + cid005325: + type: formal_theorem + title: slippage_breaks_omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0063 + current_version: v1 + source_path: items/formal-theorems/cid005325-fth0063.yaml + cid005326: + type: formal_theorem + title: structural_instability_theorem (theorem) + status: canonical + visibility: public + primary_alias: FTH0064 + current_version: v1 + source_path: items/formal-theorems/cid005326-fth0064.yaml + cid005327: + type: formal_theorem + title: ontic_closure (theorem) + status: canonical + visibility: public + primary_alias: FTH0065 + current_version: v1 + source_path: items/formal-theorems/cid005327-fth0065.yaml + cid005328: + type: formal_theorem + title: tauObj_countable (theorem) + status: canonical + visibility: public + primary_alias: FTH0066 + current_version: v1 + source_path: items/formal-theorems/cid005328-fth0066.yaml + cid005329: + type: formal_theorem + title: orbit_disjoint (theorem) + status: canonical + visibility: public + primary_alias: FTH0067 + current_version: v1 + source_path: items/formal-theorems/cid005329-fth0067.yaml + cid005330: + type: formal_theorem + title: pentation_channel_exhaustion (theorem) + status: canonical + visibility: public + primary_alias: FTH0068 + current_version: v1 + source_path: items/formal-theorems/cid005330-fth0068.yaml + cid005331: + type: formal_theorem + title: ladder_saturation (theorem) + status: canonical + visibility: public + primary_alias: FTH0069 + current_version: v1 + source_path: items/formal-theorems/cid005331-fth0069.yaml + cid005332: + type: formal_theorem + title: rigidity_non_omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0070 + current_version: v1 + source_path: items/formal-theorems/cid005332-fth0070.yaml + cid005333: + type: formal_theorem + title: categoricity_non_omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0071 + current_version: v1 + source_path: items/formal-theorems/cid005333-fth0071.yaml + cid005334: + type: formal_theorem + title: minimal_alphabet (theorem) + status: canonical + visibility: public + primary_alias: FTH0072 + current_version: v1 + source_path: items/formal-theorems/cid005334-fth0072.yaml + cid005335: + type: formal_theorem + title: no_elliptic_idempotent (theorem) + status: canonical + visibility: public + primary_alias: FTH0073 + current_version: v1 + source_path: items/formal-theorems/cid005335-fth0073.yaml + cid005336: + type: formal_theorem + title: split_complex_forced (theorem) + status: canonical + visibility: public + primary_alias: FTH0074 + current_version: v1 + source_path: items/formal-theorems/cid005336-fth0074.yaml + cid005337: + type: formal_theorem + title: algebraic_lemniscate_exists (theorem) + status: canonical + visibility: public + primary_alias: FTH0075 + current_version: v1 + source_path: items/formal-theorems/cid005337-fth0075.yaml + cid005338: + type: formal_theorem + title: no_unearned_decimal_diagonal (theorem) + status: canonical + visibility: public + primary_alias: FTH0076 + current_version: v1 + source_path: items/formal-theorems/cid005338-fth0076.yaml + cid005339: + type: formal_theorem + title: no_comprehension (theorem) + status: canonical + visibility: public + primary_alias: FTH0077 + current_version: v1 + source_path: items/formal-theorems/cid005339-fth0077.yaml + cid005340: + type: formal_theorem + title: no_free_cartesian_diagonal (theorem) + status: canonical + visibility: public + primary_alias: FTH0078 + current_version: v1 + source_path: items/formal-theorems/cid005340-fth0078.yaml + cid005341: + type: formal_theorem + title: cantor_inapplicable (theorem) + status: canonical + visibility: public + primary_alias: FTH0079 + current_version: v1 + source_path: items/formal-theorems/cid005341-fth0079.yaml + cid005342: + type: formal_theorem + title: counting_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0080 + current_version: v1 + source_path: items/formal-theorems/cid005342-fth0080.yaml + cid005343: + type: formal_theorem + title: tau_mem_iff_dvd (theorem) + status: canonical + visibility: public + primary_alias: FTH0081 + current_version: v1 + source_path: items/formal-theorems/cid005343-fth0081.yaml + cid005344: + type: formal_theorem + title: tau_inter_distrib_union (theorem) + status: canonical + visibility: public + primary_alias: FTH0082 + current_version: v1 + source_path: items/formal-theorems/cid005344-fth0082.yaml + cid005345: + type: formal_theorem + title: orbit_set_order_bound (theorem) + status: canonical + visibility: public + primary_alias: FTH0083 + current_version: v1 + source_path: items/formal-theorems/cid005345-fth0083.yaml + cid005346: + type: formal_theorem + title: prime_atom (theorem) + status: canonical + visibility: public + primary_alias: FTH0084 + current_version: v1 + source_path: items/formal-theorems/cid005346-fth0084.yaml + cid005347: + type: formal_theorem + title: omega_orbit_set_exceeds_alpha (theorem) + status: canonical + visibility: public + primary_alias: FTH0085 + current_version: v1 + source_path: items/formal-theorems/cid005347-fth0085.yaml + cid005348: + type: formal_theorem + title: nat_not_internal_set (theorem) + status: canonical + visibility: public + primary_alias: FTH0086 + current_version: v1 + source_path: items/formal-theorems/cid005348-fth0086.yaml + cid005349: + type: formal_theorem + title: tau_strict_mem_wf (theorem) + status: canonical + visibility: public + primary_alias: FTH0087 + current_version: v1 + source_path: items/formal-theorems/cid005349-fth0087.yaml + cid005350: + type: formal_theorem + title: unique_infinity (theorem) + status: canonical + visibility: public + primary_alias: FTH0088 + current_version: v1 + source_path: items/formal-theorems/cid005350-fth0088.yaml + cid005351: + type: formal_theorem + title: ultrametric_replaces_card (theorem) + status: canonical + visibility: public + primary_alias: FTH0089 + current_version: v1 + source_path: items/formal-theorems/cid005351-fth0089.yaml + cid005352: + type: formal_theorem + title: product_universal (theorem) + status: canonical + visibility: public + primary_alias: FTH0090 + current_version: v1 + source_path: items/formal-theorems/cid005352-fth0090.yaml + cid005353: + type: formal_theorem + title: cat_tau_thin (theorem) + status: canonical + visibility: public + primary_alias: FTH0091 + current_version: v1 + source_path: items/formal-theorems/cid005353-fth0091.yaml + cid005354: + type: formal_theorem + title: omega_tau_classifier (theorem) + status: canonical + visibility: public + primary_alias: FTH0092 + current_version: v1 + source_path: items/formal-theorems/cid005354-fth0092.yaml + cid005355: + type: formal_theorem + title: earned_topos_non_boolean (theorem) + status: canonical + visibility: public + primary_alias: FTH0093 + current_version: v1 + source_path: items/formal-theorems/cid005355-fth0093.yaml + cid005356: + type: formal_theorem + title: yoneda_thin (theorem) + status: canonical + visibility: public + primary_alias: FTH0094 + current_version: v1 + source_path: items/formal-theorems/cid005356-fth0094.yaml + cid005357: + type: formal_theorem + title: cartesian_closed_adj (theorem) + status: canonical + visibility: public + primary_alias: FTH0095 + current_version: v1 + source_path: items/formal-theorems/cid005357-fth0095.yaml + cid005358: + type: formal_theorem + title: self_enrichment (theorem) + status: canonical + visibility: public + primary_alias: FTH0096 + current_version: v1 + source_path: items/formal-theorems/cid005358-fth0096.yaml + cid005359: + type: formal_theorem + title: psh_has_terminal (theorem) + status: canonical + visibility: public + primary_alias: FTH0097 + current_version: v1 + source_path: items/formal-theorems/cid005359-fth0097.yaml + cid005360: + type: formal_theorem + title: psh_countable_objects (theorem) + status: canonical + visibility: public + primary_alias: FTH0098 + current_version: v1 + source_path: items/formal-theorems/cid005360-fth0098.yaml + cid005361: + type: formal_theorem + title: product_distributes_over_coproduct (theorem) + status: canonical + visibility: public + primary_alias: FTH0099 + current_version: v1 + source_path: items/formal-theorems/cid005361-fth0099.yaml + cid005362: + type: formal_theorem + title: idemp_decomp_recovery (theorem) + status: canonical + visibility: public + primary_alias: FTH0100 + current_version: v1 + source_path: items/formal-theorems/cid005362-fth0100.yaml + cid005363: + type: formal_theorem + title: b_channel_kills_c (theorem) + status: canonical + visibility: public + primary_alias: FTH0101 + current_version: v1 + source_path: items/formal-theorems/cid005363-fth0101.yaml + cid005364: + type: formal_theorem + title: c_channel_kills_b (theorem) + status: canonical + visibility: public + primary_alias: FTH0102 + current_version: v1 + source_path: items/formal-theorems/cid005364-fth0102.yaml + cid005365: + type: formal_theorem + title: character_tower_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0103 + current_version: v1 + source_path: items/formal-theorems/cid005365-fth0103.yaml + cid005366: + type: formal_theorem + title: character_add_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0104 + current_version: v1 + source_path: items/formal-theorems/cid005366-fth0104.yaml + cid005367: + type: formal_theorem + title: character_mul_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0105 + current_version: v1 + source_path: items/formal-theorems/cid005367-fth0105.yaml + cid005368: + type: formal_theorem + title: sector_distributive (theorem) + status: canonical + visibility: public + primary_alias: FTH0106 + current_version: v1 + source_path: items/formal-theorems/cid005368-fth0106.yaml + cid005369: + type: formal_theorem + title: idemp_char_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0107 + current_version: v1 + source_path: items/formal-theorems/cid005369-fth0107.yaml + cid005370: + type: formal_theorem + title: char_tower_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0108 + current_version: v1 + source_path: items/formal-theorems/cid005370-fth0108.yaml + cid005371: + type: formal_theorem + title: char_add_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0109 + current_version: v1 + source_path: items/formal-theorems/cid005371-fth0109.yaml + cid005372: + type: formal_theorem + title: char_mul_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0110 + current_version: v1 + source_path: items/formal-theorems/cid005372-fth0110.yaml + cid005373: + type: formal_theorem + title: char_add_id_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0111 + current_version: v1 + source_path: items/formal-theorems/cid005373-fth0111.yaml + cid005374: + type: formal_theorem + title: liouville_dodge (theorem) + status: canonical + visibility: public + primary_alias: FTH0112 + current_version: v1 + source_path: items/formal-theorems/cid005374-fth0112.yaml + cid005375: + type: formal_theorem + title: nonconstant_bounded_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0113 + current_version: v1 + source_path: items/formal-theorems/cid005375-fth0113.yaml + cid005376: + type: formal_theorem + title: primorial_unique_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0114 + current_version: v1 + source_path: items/formal-theorems/cid005376-fth0114.yaml + cid005377: + type: formal_theorem + title: abcd_rt_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0115 + current_version: v1 + source_path: items/formal-theorems/cid005377-fth0115.yaml + cid005378: + type: formal_theorem + title: red_det_3_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0116 + current_version: v1 + source_path: items/formal-theorems/cid005378-fth0116.yaml + cid005379: + type: formal_theorem + title: categoricity_3_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0117 + current_version: v1 + source_path: items/formal-theorems/cid005379-fth0117.yaml + cid005380: + type: formal_theorem + title: moduli_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0118 + current_version: v1 + source_path: items/formal-theorems/cid005380-fth0118.yaml + cid005381: + type: formal_theorem + title: uniqueness_3_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0119 + current_version: v1 + source_path: items/formal-theorems/cid005381-fth0119.yaml + cid005382: + type: formal_theorem + title: j_squared_wave (theorem) + status: canonical + visibility: public + primary_alias: FTH0120 + current_version: v1 + source_path: items/formal-theorems/cid005382-fth0120.yaml + cid005383: + type: formal_theorem + title: zero_divisors_exist (theorem) + status: canonical + visibility: public + primary_alias: FTH0121 + current_version: v1 + source_path: items/formal-theorems/cid005383-fth0121.yaml + cid005384: + type: formal_theorem + title: idempotent_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0122 + current_version: v1 + source_path: items/formal-theorems/cid005384-fth0122.yaml + cid005385: + type: formal_theorem + title: abcd_roundtrip_12 (theorem) + status: canonical + visibility: public + primary_alias: FTH0123 + current_version: v1 + source_path: items/formal-theorems/cid005385-fth0123.yaml + cid005386: + type: formal_theorem + title: reduce_idempotent (theorem) + status: canonical + visibility: public + primary_alias: FTH0124 + current_version: v1 + source_path: items/formal-theorems/cid005386-fth0124.yaml + cid005387: + type: formal_theorem + title: abcd_distinct_12_64 (theorem) + status: canonical + visibility: public + primary_alias: FTH0125 + current_version: v1 + source_path: items/formal-theorems/cid005387-fth0125.yaml + cid005388: + type: formal_theorem + title: structure_uniqueness (theorem) + status: canonical + visibility: public + primary_alias: FTH0126 + current_version: v1 + source_path: items/formal-theorems/cid005388-fth0126.yaml + cid005389: + type: formal_theorem + title: tower_forced (theorem) + status: canonical + visibility: public + primary_alias: FTH0127 + current_version: v1 + source_path: items/formal-theorems/cid005389-fth0127.yaml + cid005390: + type: formal_theorem + title: spectral_ring_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0128 + current_version: v1 + source_path: items/formal-theorems/cid005390-fth0128.yaml + cid005391: + type: formal_theorem + title: central_fwd_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0129 + current_version: v1 + source_path: items/formal-theorems/cid005391-fth0129.yaml + cid005392: + type: formal_theorem + title: central_inv_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0130 + current_version: v1 + source_path: items/formal-theorems/cid005392-fth0130.yaml + cid005393: + type: formal_theorem + title: central_rt_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0131 + current_version: v1 + source_path: items/formal-theorems/cid005393-fth0131.yaml + cid005394: + type: formal_theorem + title: central_theorem_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0132 + current_version: v1 + source_path: items/formal-theorems/cid005394-fth0132.yaml + cid005395: + type: formal_theorem + title: holographic_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0133 + current_version: v1 + source_path: items/formal-theorems/cid005395-fth0133.yaml + cid005396: + type: formal_theorem + title: spectral_periodic (theorem) + status: canonical + visibility: public + primary_alias: FTH0134 + current_version: v1 + source_path: items/formal-theorems/cid005396-fth0134.yaml + cid005397: + type: formal_theorem + title: spectral_idempotent_supported (theorem) + status: canonical + visibility: public + primary_alias: FTH0135 + current_version: v1 + source_path: items/formal-theorems/cid005397-fth0135.yaml + cid005398: + type: formal_theorem + title: central_forward_coherent (theorem) + status: canonical + visibility: public + primary_alias: FTH0136 + current_version: v1 + source_path: items/formal-theorems/cid005398-fth0136.yaml + cid005399: + type: formal_theorem + title: central_inverse_periodic (theorem) + status: canonical + visibility: public + primary_alias: FTH0137 + current_version: v1 + source_path: items/formal-theorems/cid005399-fth0137.yaml + cid005400: + type: formal_theorem + title: central_roundtrip (theorem) + status: canonical + visibility: public + primary_alias: FTH0138 + current_version: v1 + source_path: items/formal-theorems/cid005400-fth0138.yaml + cid005401: + type: formal_theorem + title: holographic_roundtrip (theorem) + status: canonical + visibility: public + primary_alias: FTH0139 + current_version: v1 + source_path: items/formal-theorems/cid005401-fth0139.yaml + cid005402: + type: formal_theorem + title: stagewise_naturality (theorem) + status: canonical + visibility: public + primary_alias: FTH0140 + current_version: v1 + source_path: items/formal-theorems/cid005402-fth0140.yaml + cid005403: + type: formal_theorem + title: stagewise_naturality_strong (theorem) + status: canonical + visibility: public + primary_alias: FTH0141 + current_version: v1 + source_path: items/formal-theorems/cid005403-fth0141.yaml + cid005404: + type: formal_theorem + title: bndlift_reduce_invariant (theorem) + status: canonical + visibility: public + primary_alias: FTH0142 + current_version: v1 + source_path: items/formal-theorems/cid005404-fth0142.yaml + cid005405: + type: formal_theorem + title: bndlift_stagefun_welldef (theorem) + status: canonical + visibility: public + primary_alias: FTH0143 + current_version: v1 + source_path: items/formal-theorems/cid005405-fth0143.yaml + cid005406: + type: formal_theorem + title: reduction_gives_naturality (theorem) + status: canonical + visibility: public + primary_alias: FTH0144 + current_version: v1 + source_path: items/formal-theorems/cid005406-fth0144.yaml + cid005407: + type: formal_theorem + title: nat_check_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0145 + current_version: v1 + source_path: items/formal-theorems/cid005407-fth0145.yaml + cid005408: + type: formal_theorem + title: ogt_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0146 + current_version: v1 + source_path: items/formal-theorems/cid005408-fth0146.yaml + cid005409: + type: formal_theorem + title: bsf_coh_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0147 + current_version: v1 + source_path: items/formal-theorems/cid005409-fth0147.yaml + cid005410: + type: formal_theorem + title: egr_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0148 + current_version: v1 + source_path: items/formal-theorems/cid005410-fth0148.yaml + cid005411: + type: formal_theorem + title: extension_preserves_stage (theorem) + status: canonical + visibility: public + primary_alias: FTH0149 + current_version: v1 + source_path: items/formal-theorems/cid005411-fth0149.yaml + cid005412: + type: formal_theorem + title: uniqueness_from_agreement (theorem) + status: canonical + visibility: public + primary_alias: FTH0150 + current_version: v1 + source_path: items/formal-theorems/cid005412-fth0150.yaml + cid005413: + type: formal_theorem + title: bndlift_tower (theorem) + status: canonical + visibility: public + primary_alias: FTH0151 + current_version: v1 + source_path: items/formal-theorems/cid005413-fth0151.yaml + cid005414: + type: formal_theorem + title: extension_bipolar_recovery (theorem) + status: canonical + visibility: public + primary_alias: FTH0152 + current_version: v1 + source_path: items/formal-theorems/cid005414-fth0152.yaml + cid005415: + type: formal_theorem + title: ext_channel_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0153 + current_version: v1 + source_path: items/formal-theorems/cid005415-fth0153.yaml + cid005416: + type: formal_theorem + title: b_channel_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0154 + current_version: v1 + source_path: items/formal-theorems/cid005416-fth0154.yaml + cid005417: + type: formal_theorem + title: c_channel_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0155 + current_version: v1 + source_path: items/formal-theorems/cid005417-fth0155.yaml + cid005418: + type: formal_theorem + title: hartogs_uniq_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0156 + current_version: v1 + source_path: items/formal-theorems/cid005418-fth0156.yaml + cid005419: + type: formal_theorem + title: bnd_det_int_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0157 + current_version: v1 + source_path: items/formal-theorems/cid005419-fth0157.yaml + cid005420: + type: formal_theorem + title: cd_uniq_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0158 + current_version: v1 + source_path: items/formal-theorems/cid005420-fth0158.yaml + cid005421: + type: formal_theorem + title: cech_sq_zero_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0159 + current_version: v1 + source_path: items/formal-theorems/cid005421-fth0159.yaml + cid005422: + type: formal_theorem + title: cech_sq_zero_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0160 + current_version: v1 + source_path: items/formal-theorems/cid005422-fth0160.yaml + cid005423: + type: formal_theorem + title: h1_vanishes_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0161 + current_version: v1 + source_path: items/formal-theorems/cid005423-fth0161.yaml + cid005424: + type: formal_theorem + title: h1_vanishes_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0162 + current_version: v1 + source_path: items/formal-theorems/cid005424-fth0162.yaml + cid005425: + type: formal_theorem + title: h0_global_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0163 + current_version: v1 + source_path: items/formal-theorems/cid005425-fth0163.yaml + cid005426: + type: formal_theorem + title: h0_global_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0164 + current_version: v1 + source_path: items/formal-theorems/cid005426-fth0164.yaml + cid005427: + type: formal_theorem + title: h0_nonconstant_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0165 + current_version: v1 + source_path: items/formal-theorems/cid005427-fth0165.yaml + cid005428: + type: formal_theorem + title: cech_derived_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0166 + current_version: v1 + source_path: items/formal-theorems/cid005428-fth0166.yaml + cid005429: + type: formal_theorem + title: cech_derived_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0167 + current_version: v1 + source_path: items/formal-theorems/cid005429-fth0167.yaml + cid005430: + type: formal_theorem + title: yoneda_app_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0168 + current_version: v1 + source_path: items/formal-theorems/cid005430-fth0168.yaml + cid005431: + type: formal_theorem + title: yoneda_hom_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0169 + current_version: v1 + source_path: items/formal-theorems/cid005431-fth0169.yaml + cid005432: + type: formal_theorem + title: omega_germs_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0170 + current_version: v1 + source_path: items/formal-theorems/cid005432-fth0170.yaml + cid005433: + type: formal_theorem + title: hol_class_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0171 + current_version: v1 + source_path: items/formal-theorems/cid005433-fth0171.yaml + cid005434: + type: formal_theorem + title: germ_hol_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0172 + current_version: v1 + source_path: items/formal-theorems/cid005434-fth0172.yaml + cid005435: + type: formal_theorem + title: yoneda_code_roundtrip (theorem) + status: canonical + visibility: public + primary_alias: FTH0173 + current_version: v1 + source_path: items/formal-theorems/cid005435-fth0173.yaml + cid005436: + type: formal_theorem + title: omega_germ_tower_coherent (theorem) + status: canonical + visibility: public + primary_alias: FTH0174 + current_version: v1 + source_path: items/formal-theorems/cid005436-fth0174.yaml + cid005437: + type: formal_theorem + title: germ_reconstructs_identity (theorem) + status: canonical + visibility: public + primary_alias: FTH0175 + current_version: v1 + source_path: items/formal-theorems/cid005437-fth0175.yaml + cid005438: + type: formal_theorem + title: germ_idempotent_supported (theorem) + status: canonical + visibility: public + primary_alias: FTH0176 + current_version: v1 + source_path: items/formal-theorems/cid005438-fth0176.yaml + cid005439: + type: formal_theorem + title: proto_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0177 + current_version: v1 + source_path: items/formal-theorems/cid005439-fth0177.yaml + cid005440: + type: formal_theorem + title: not_proto_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0178 + current_version: v1 + source_path: items/formal-theorems/cid005440-fth0178.yaml + cid005441: + type: formal_theorem + title: stage_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0179 + current_version: v1 + source_path: items/formal-theorems/cid005441-fth0179.yaml + cid005442: + type: formal_theorem + title: proto_examples (theorem) + status: canonical + visibility: public + primary_alias: FTH0180 + current_version: v1 + source_path: items/formal-theorems/cid005442-fth0180.yaml + cid005443: + type: formal_theorem + title: proto_range_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0181 + current_version: v1 + source_path: items/formal-theorems/cid005443-fth0181.yaml + cid005444: + type: formal_theorem + title: proto_count_check (theorem) + status: canonical + visibility: public + primary_alias: FTH0182 + current_version: v1 + source_path: items/formal-theorems/cid005444-fth0182.yaml + cid005445: + type: formal_theorem + title: finite_is_proto_rational_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0183 + current_version: v1 + source_path: items/formal-theorems/cid005445-fth0183.yaml + cid005446: + type: formal_theorem + title: proto_at_stage (theorem) + status: canonical + visibility: public + primary_alias: FTH0184 + current_version: v1 + source_path: items/formal-theorems/cid005446-fth0184.yaml + cid005447: + type: formal_theorem + title: proto_abcd_roundtrip_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0185 + current_version: v1 + source_path: items/formal-theorems/cid005447-fth0185.yaml + cid005448: + type: formal_theorem + title: flat_connection_compatible_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0186 + current_version: v1 + source_path: items/formal-theorems/cid005448-fth0186.yaml + cid005449: + type: formal_theorem + title: flat_connection_flat_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0187 + current_version: v1 + source_path: items/formal-theorems/cid005449-fth0187.yaml + cid005450: + type: formal_theorem + title: holonomy_trivial_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0188 + current_version: v1 + source_path: items/formal-theorems/cid005450-fth0188.yaml + cid005451: + type: formal_theorem + title: holonomy_trivial_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0189 + current_version: v1 + source_path: items/formal-theorems/cid005451-fth0189.yaml + cid005452: + type: formal_theorem + title: flat_curvature_vanishes_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0190 + current_version: v1 + source_path: items/formal-theorems/cid005452-fth0190.yaml + cid005453: + type: formal_theorem + title: flat_curvature_vanishes_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0191 + current_version: v1 + source_path: items/formal-theorems/cid005453-fth0191.yaml + cid005454: + type: formal_theorem + title: geodesic_correct_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0192 + current_version: v1 + source_path: items/formal-theorems/cid005454-fth0192.yaml + cid005455: + type: formal_theorem + title: geodesic_correct_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0193 + current_version: v1 + source_path: items/formal-theorems/cid005455-fth0193.yaml + cid005456: + type: formal_theorem + title: geodesic_complete_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0194 + current_version: v1 + source_path: items/formal-theorems/cid005456-fth0194.yaml + cid005457: + type: formal_theorem + title: geodesic_complete_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0195 + current_version: v1 + source_path: items/formal-theorems/cid005457-fth0195.yaml + cid005458: + type: formal_theorem + title: lemniscate_holonomy_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0196 + current_version: v1 + source_path: items/formal-theorems/cid005458-fth0196.yaml + cid005459: + type: formal_theorem + title: lemniscate_holonomy_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0197 + current_version: v1 + source_path: items/formal-theorems/cid005459-fth0197.yaml + cid005460: + type: formal_theorem + title: diffgeo_not_earned (theorem) + status: canonical + visibility: public + primary_alias: FTH0198 + current_version: v1 + source_path: items/formal-theorems/cid005460-fth0198.yaml + cid005461: + type: formal_theorem + title: e1_complete_b3_10_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0199 + current_version: v1 + source_path: items/formal-theorems/cid005461-fth0199.yaml + cid005462: + type: formal_theorem + title: book3_prereq_3_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0200 + current_version: v1 + source_path: items/formal-theorems/cid005462-fth0200.yaml + cid005463: + type: formal_theorem + title: connection_not_earned (theorem) + status: canonical + visibility: public + primary_alias: FTH0201 + current_version: v1 + source_path: items/formal-theorems/cid005463-fth0201.yaml + cid005464: + type: formal_theorem + title: curvature_not_earned (theorem) + status: canonical + visibility: public + primary_alias: FTH0202 + current_version: v1 + source_path: items/formal-theorems/cid005464-fth0202.yaml + cid005465: + type: formal_theorem + title: holonomy_not_earned (theorem) + status: canonical + visibility: public + primary_alias: FTH0203 + current_version: v1 + source_path: items/formal-theorems/cid005465-fth0203.yaml + cid005466: + type: formal_theorem + title: e1_includes_self_enrichment (theorem) + status: canonical + visibility: public + primary_alias: FTH0204 + current_version: v1 + source_path: items/formal-theorems/cid005466-fth0204.yaml + cid005467: + type: formal_theorem + title: full_export_3_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0205 + current_version: v1 + source_path: items/formal-theorems/cid005467-fth0205.yaml + cid005468: + type: formal_theorem + title: all_six_3_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0206 + current_version: v1 + source_path: items/formal-theorems/cid005468-fth0206.yaml + cid005469: + type: formal_theorem + title: book3_e1_3_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0207 + current_version: v1 + source_path: items/formal-theorems/cid005469-fth0207.yaml + cid005470: + type: formal_theorem + title: complete_means_six (theorem) + status: canonical + visibility: public + primary_alias: FTH0208 + current_version: v1 + source_path: items/formal-theorems/cid005470-fth0208.yaml + cid005471: + type: formal_theorem + title: export_implies_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0209 + current_version: v1 + source_path: items/formal-theorems/cid005471-fth0209.yaml + cid005472: + type: formal_theorem + title: export_implies_central (theorem) + status: canonical + visibility: public + primary_alias: FTH0210 + current_version: v1 + source_path: items/formal-theorems/cid005472-fth0210.yaml + cid005473: + type: formal_theorem + title: e1_gt_e0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0211 + current_version: v1 + source_path: items/formal-theorems/cid005473-fth0211.yaml + cid005474: + type: formal_theorem + title: geometric_bisquare_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0212 + current_version: v1 + source_path: items/formal-theorems/cid005474-fth0212.yaml + cid005475: + type: formal_theorem + title: geometric_all_eight (theorem) + status: canonical + visibility: public + primary_alias: FTH0213 + current_version: v1 + source_path: items/formal-theorems/cid005475-fth0213.yaml + cid005476: + type: formal_theorem + title: geo_topology (theorem) + status: canonical + visibility: public + primary_alias: FTH0214 + current_version: v1 + source_path: items/formal-theorems/cid005476-fth0214.yaml + cid005477: + type: formal_theorem + title: audit_3_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0215 + current_version: v1 + source_path: items/formal-theorems/cid005477-fth0215.yaml + cid005478: + type: formal_theorem + title: scaling_chain_valid (theorem) + status: canonical + visibility: public + primary_alias: FTH0216 + current_version: v1 + source_path: items/formal-theorems/cid005478-fth0216.yaml + cid005479: + type: formal_theorem + title: complete_means_eight (theorem) + status: canonical + visibility: public + primary_alias: FTH0217 + current_version: v1 + source_path: items/formal-theorems/cid005479-fth0217.yaml + cid005480: + type: formal_theorem + title: geometric_implies_central (theorem) + status: canonical + visibility: public + primary_alias: FTH0218 + current_version: v1 + source_path: items/formal-theorems/cid005480-fth0218.yaml + cid005481: + type: formal_theorem + title: e0_ne_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0219 + current_version: v1 + source_path: items/formal-theorems/cid005481-fth0219.yaml + cid005482: + type: formal_theorem + title: e1_ne_e2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0220 + current_version: v1 + source_path: items/formal-theorems/cid005482-fth0220.yaml + cid005483: + type: formal_theorem + title: atlas_chart_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0221 + current_version: v1 + source_path: items/formal-theorems/cid005483-fth0221.yaml + cid005484: + type: formal_theorem + title: atlas_transition_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0222 + current_version: v1 + source_path: items/formal-theorems/cid005484-fth0222.yaml + cid005485: + type: formal_theorem + title: d_const_zero_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0223 + current_version: v1 + source_path: items/formal-theorems/cid005485-fth0223.yaml + cid005486: + type: formal_theorem + title: d_sq_tower_3_10 (theorem) + status: canonical + visibility: public + primary_alias: FTH0224 + current_version: v1 + source_path: items/formal-theorems/cid005486-fth0224.yaml + cid005487: + type: formal_theorem + title: tau_manifold_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0225 + current_version: v1 + source_path: items/formal-theorems/cid005487-fth0225.yaml + cid005488: + type: formal_theorem + title: tau3_manifold_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0226 + current_version: v1 + source_path: items/formal-theorems/cid005488-fth0226.yaml + cid005489: + type: formal_theorem + title: abcd_atlas_roundtrip_7 (theorem) + status: canonical + visibility: public + primary_alias: FTH0227 + current_version: v1 + source_path: items/formal-theorems/cid005489-fth0227.yaml + cid005490: + type: formal_theorem + title: chart_transition_7_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0228 + current_version: v1 + source_path: items/formal-theorems/cid005490-fth0228.yaml + cid005491: + type: formal_theorem + title: d_tau_constant (theorem) + status: canonical + visibility: public + primary_alias: FTH0229 + current_version: v1 + source_path: items/formal-theorems/cid005491-fth0229.yaml + cid005492: + type: formal_theorem + title: d_tau_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0230 + current_version: v1 + source_path: items/formal-theorems/cid005492-fth0230.yaml + cid005493: + type: formal_theorem + title: manifold_reduce_idempotent (theorem) + status: canonical + visibility: public + primary_alias: FTH0231 + current_version: v1 + source_path: items/formal-theorems/cid005493-fth0231.yaml + cid005494: + type: formal_theorem + title: manifold_tower_coherence (theorem) + status: canonical + visibility: public + primary_alias: FTH0232 + current_version: v1 + source_path: items/formal-theorems/cid005494-fth0232.yaml + cid005495: + type: formal_theorem + title: e0_hom_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0233 + current_version: v1 + source_path: items/formal-theorems/cid005495-fth0233.yaml + cid005496: + type: formal_theorem + title: e1_hom_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0234 + current_version: v1 + source_path: items/formal-theorems/cid005496-fth0234.yaml + cid005497: + type: formal_theorem + title: transition_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0235 + current_version: v1 + source_path: items/formal-theorems/cid005497-fth0235.yaml + cid005498: + type: formal_theorem + title: gap_10_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0236 + current_version: v1 + source_path: items/formal-theorems/cid005498-fth0236.yaml + cid005499: + type: formal_theorem + title: full_ladder_10_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0237 + current_version: v1 + source_path: items/formal-theorems/cid005499-fth0237.yaml + cid005500: + type: formal_theorem + title: boundary_coherence_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0238 + current_version: v1 + source_path: items/formal-theorems/cid005500-fth0238.yaml + cid005501: + type: formal_theorem + title: boundary_coherence_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0239 + current_version: v1 + source_path: items/formal-theorems/cid005501-fth0239.yaml + cid005502: + type: formal_theorem + title: ses_exact_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0240 + current_version: v1 + source_path: items/formal-theorems/cid005502-fth0240.yaml + cid005503: + type: formal_theorem + title: ses_exact_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0241 + current_version: v1 + source_path: items/formal-theorems/cid005503-fth0241.yaml + cid005504: + type: formal_theorem + title: homology_trivial_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0242 + current_version: v1 + source_path: items/formal-theorems/cid005504-fth0242.yaml + cid005505: + type: formal_theorem + title: homology_trivial_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0243 + current_version: v1 + source_path: items/formal-theorems/cid005505-fth0243.yaml + cid005506: + type: formal_theorem + title: ses_stage1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0244 + current_version: v1 + source_path: items/formal-theorems/cid005506-fth0244.yaml + cid005507: + type: formal_theorem + title: ses_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0245 + current_version: v1 + source_path: items/formal-theorems/cid005507-fth0245.yaml + cid005508: + type: formal_theorem + title: les_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0246 + current_version: v1 + source_path: items/formal-theorems/cid005508-fth0246.yaml + cid005509: + type: formal_theorem + title: e1_layer_10_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0247 + current_version: v1 + source_path: items/formal-theorems/cid005509-fth0247.yaml + cid005510: + type: formal_theorem + title: hom_tower_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0248 + current_version: v1 + source_path: items/formal-theorems/cid005510-fth0248.yaml + cid005511: + type: formal_theorem + title: self_enrich_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0249 + current_version: v1 + source_path: items/formal-theorems/cid005511-fth0249.yaml + cid005512: + type: formal_theorem + title: hom_identity_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0250 + current_version: v1 + source_path: items/formal-theorems/cid005512-fth0250.yaml + cid005513: + type: formal_theorem + title: hom_constant_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0251 + current_version: v1 + source_path: items/formal-theorems/cid005513-fth0251.yaml + cid005514: + type: formal_theorem + title: hom_count_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0252 + current_version: v1 + source_path: items/formal-theorems/cid005514-fth0252.yaml + cid005515: + type: formal_theorem + title: hom_bipolar_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0253 + current_version: v1 + source_path: items/formal-theorems/cid005515-fth0253.yaml + cid005516: + type: formal_theorem + title: hom_channel_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0254 + current_version: v1 + source_path: items/formal-theorems/cid005516-fth0254.yaml + cid005517: + type: formal_theorem + title: hom_stage_reduce_stable (theorem) + status: canonical + visibility: public + primary_alias: FTH0255 + current_version: v1 + source_path: items/formal-theorems/cid005517-fth0255.yaml + cid005518: + type: formal_theorem + title: hom_bipolar_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0256 + current_version: v1 + source_path: items/formal-theorems/cid005518-fth0256.yaml + cid005519: + type: formal_theorem + title: hom_bipolar_orthogonal (theorem) + status: canonical + visibility: public + primary_alias: FTH0257 + current_version: v1 + source_path: items/formal-theorems/cid005519-fth0257.yaml + cid005520: + type: formal_theorem + title: zero_map_compat (theorem) + status: canonical + visibility: public + primary_alias: FTH0258 + current_version: v1 + source_path: items/formal-theorems/cid005520-fth0258.yaml + cid005521: + type: formal_theorem + title: two_cat_assoc_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0259 + current_version: v1 + source_path: items/formal-theorems/cid005521-fth0259.yaml + cid005522: + type: formal_theorem + title: two_cat_unit_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0260 + current_version: v1 + source_path: items/formal-theorems/cid005522-fth0260.yaml + cid005523: + type: formal_theorem + title: two_cat_vert_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0261 + current_version: v1 + source_path: items/formal-theorems/cid005523-fth0261.yaml + cid005524: + type: formal_theorem + title: two_morph_tower_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0262 + current_version: v1 + source_path: items/formal-theorems/cid005524-fth0262.yaml + cid005525: + type: formal_theorem + title: two_morphism_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0263 + current_version: v1 + source_path: items/formal-theorems/cid005525-fth0263.yaml + cid005526: + type: formal_theorem + title: enrich_iter_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0264 + current_version: v1 + source_path: items/formal-theorems/cid005526-fth0264.yaml + cid005527: + type: formal_theorem + title: enrich_finite_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0265 + current_version: v1 + source_path: items/formal-theorems/cid005527-fth0265.yaml + cid005528: + type: formal_theorem + title: interchange_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0266 + current_version: v1 + source_path: items/formal-theorems/cid005528-fth0266.yaml + cid005529: + type: formal_theorem + title: horiz_comp_assoc (theorem) + status: canonical + visibility: public + primary_alias: FTH0267 + current_version: v1 + source_path: items/formal-theorems/cid005529-fth0267.yaml + cid005530: + type: formal_theorem + title: vert_comp_assoc (theorem) + status: canonical + visibility: public + primary_alias: FTH0268 + current_version: v1 + source_path: items/formal-theorems/cid005530-fth0268.yaml + cid005531: + type: formal_theorem + title: id_two_cell_tower_coherent (theorem) + status: canonical + visibility: public + primary_alias: FTH0269 + current_version: v1 + source_path: items/formal-theorems/cid005531-fth0269.yaml + cid005532: + type: formal_theorem + title: two_sq_mediates_at_reduce (theorem) + status: canonical + visibility: public + primary_alias: FTH0270 + current_version: v1 + source_path: items/formal-theorems/cid005532-fth0270.yaml + cid005533: + type: formal_theorem + title: primorial_pos_check (theorem) + status: canonical + visibility: public + primary_alias: FTH0271 + current_version: v1 + source_path: items/formal-theorems/cid005533-fth0271.yaml + cid005534: + type: formal_theorem + title: probe_yoneda_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0272 + current_version: v1 + source_path: items/formal-theorems/cid005534-fth0272.yaml + cid005535: + type: formal_theorem + title: yoneda_faithful_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0273 + current_version: v1 + source_path: items/formal-theorems/cid005535-fth0273.yaml + cid005536: + type: formal_theorem + title: yoneda_full_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0274 + current_version: v1 + source_path: items/formal-theorems/cid005536-fth0274.yaml + cid005537: + type: formal_theorem + title: yoneda_bipolar_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0275 + current_version: v1 + source_path: items/formal-theorems/cid005537-fth0275.yaml + cid005538: + type: formal_theorem + title: yoneda_roundtrip_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0276 + current_version: v1 + source_path: items/formal-theorems/cid005538-fth0276.yaml + cid005539: + type: formal_theorem + title: yoneda_faithful_id (theorem) + status: canonical + visibility: public + primary_alias: FTH0277 + current_version: v1 + source_path: items/formal-theorems/cid005539-fth0277.yaml + cid005540: + type: formal_theorem + title: yoneda_full_id (theorem) + status: canonical + visibility: public + primary_alias: FTH0278 + current_version: v1 + source_path: items/formal-theorems/cid005540-fth0278.yaml + cid005541: + type: formal_theorem + title: probe_naturality_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0279 + current_version: v1 + source_path: items/formal-theorems/cid005541-fth0279.yaml + cid005542: + type: formal_theorem + title: bndlift_exist_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0280 + current_version: v1 + source_path: items/formal-theorems/cid005542-fth0280.yaml + cid005543: + type: formal_theorem + title: bipolar_indep_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0281 + current_version: v1 + source_path: items/formal-theorems/cid005543-fth0281.yaml + cid005544: + type: formal_theorem + title: basis_ortho_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0282 + current_version: v1 + source_path: items/formal-theorems/cid005544-fth0282.yaml + cid005545: + type: formal_theorem + title: basis_complete_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0283 + current_version: v1 + source_path: items/formal-theorems/cid005545-fth0283.yaml + cid005546: + type: formal_theorem + title: basis_indep_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0284 + current_version: v1 + source_path: items/formal-theorems/cid005546-fth0284.yaml + cid005547: + type: formal_theorem + title: basis_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0285 + current_version: v1 + source_path: items/formal-theorems/cid005547-fth0285.yaml + cid005548: + type: formal_theorem + title: spectral_support_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0286 + current_version: v1 + source_path: items/formal-theorems/cid005548-fth0286.yaml + cid005549: + type: formal_theorem + title: proj_delta_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0287 + current_version: v1 + source_path: items/formal-theorems/cid005549-fth0287.yaml + cid005550: + type: formal_theorem + title: proj_recovery_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0288 + current_version: v1 + source_path: items/formal-theorems/cid005550-fth0288.yaml + cid005551: + type: formal_theorem + title: hol_assoc_thm (theorem) + status: canonical + visibility: public + primary_alias: FTH0289 + current_version: v1 + source_path: items/formal-theorems/cid005551-fth0289.yaml + cid005552: + type: formal_theorem + title: assoc_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0290 + current_version: v1 + source_path: items/formal-theorems/cid005552-fth0290.yaml + cid005553: + type: formal_theorem + title: holend_3_10_ok (theorem) + status: canonical + visibility: public + primary_alias: FTH0291 + current_version: v1 + source_path: items/formal-theorems/cid005553-fth0291.yaml + cid005554: + type: formal_theorem + title: evolution_semigroup_thm (theorem) + status: canonical + visibility: public + primary_alias: FTH0292 + current_version: v1 + source_path: items/formal-theorems/cid005554-fth0292.yaml + cid005555: + type: formal_theorem + title: semigroup_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0293 + current_version: v1 + source_path: items/formal-theorems/cid005555-fth0293.yaml + cid005556: + type: formal_theorem + title: asymmetry_witness (theorem) + status: canonical + visibility: public + primary_alias: FTH0294 + current_version: v1 + source_path: items/formal-theorems/cid005556-fth0294.yaml + cid005557: + type: formal_theorem + title: inner_product_symmetry_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0295 + current_version: v1 + source_path: items/formal-theorems/cid005557-fth0295.yaml + cid005558: + type: formal_theorem + title: inner_product_linearity_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0296 + current_version: v1 + source_path: items/formal-theorems/cid005558-fth0296.yaml + cid005559: + type: formal_theorem + title: norm_positivity_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0297 + current_version: v1 + source_path: items/formal-theorems/cid005559-fth0297.yaml + cid005560: + type: formal_theorem + title: norm_definiteness_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0298 + current_version: v1 + source_path: items/formal-theorems/cid005560-fth0298.yaml + cid005561: + type: formal_theorem + title: cauchy_schwarz_stage1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0299 + current_version: v1 + source_path: items/formal-theorems/cid005561-fth0299.yaml + cid005562: + type: formal_theorem + title: cauchy_schwarz_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0300 + current_version: v1 + source_path: items/formal-theorems/cid005562-fth0300.yaml + cid005563: + type: formal_theorem + title: l2_completeness_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0301 + current_version: v1 + source_path: items/formal-theorems/cid005563-fth0301.yaml + cid005564: + type: formal_theorem + title: l2_completeness_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0302 + current_version: v1 + source_path: items/formal-theorems/cid005564-fth0302.yaml + cid005565: + type: formal_theorem + title: l2_basis_orthogonal_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0303 + current_version: v1 + source_path: items/formal-theorems/cid005565-fth0303.yaml + cid005566: + type: formal_theorem + title: l2_basis_orthogonal_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0304 + current_version: v1 + source_path: items/formal-theorems/cid005566-fth0304.yaml + cid005567: + type: formal_theorem + title: laurent_range_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0305 + current_version: v1 + source_path: items/formal-theorems/cid005567-fth0305.yaml + cid005568: + type: formal_theorem + title: residue_stage_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0306 + current_version: v1 + source_path: items/formal-theorems/cid005568-fth0306.yaml + cid005569: + type: formal_theorem + title: crt_roundtrip_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0307 + current_version: v1 + source_path: items/formal-theorems/cid005569-fth0307.yaml + cid005570: + type: formal_theorem + title: residue_recon_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0308 + current_version: v1 + source_path: items/formal-theorems/cid005570-fth0308.yaml + cid005571: + type: formal_theorem + title: mero_id_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0309 + current_version: v1 + source_path: items/formal-theorems/cid005571-fth0309.yaml + cid005572: + type: formal_theorem + title: mutual_determination (theorem) + status: canonical + visibility: public + primary_alias: FTH0310 + current_version: v1 + source_path: items/formal-theorems/cid005572-fth0310.yaml + cid005573: + type: formal_theorem + title: partition_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0311 + current_version: v1 + source_path: items/formal-theorems/cid005573-fth0311.yaml + cid005574: + type: formal_theorem + title: disjoint_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0312 + current_version: v1 + source_path: items/formal-theorems/cid005574-fth0312.yaml + cid005575: + type: formal_theorem + title: containment_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0313 + current_version: v1 + source_path: items/formal-theorems/cid005575-fth0313.yaml + cid005576: + type: formal_theorem + title: gluing_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0314 + current_version: v1 + source_path: items/formal-theorems/cid005576-fth0314.yaml + cid005577: + type: formal_theorem + title: sheaf_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0315 + current_version: v1 + source_path: items/formal-theorems/cid005577-fth0315.yaml + cid005578: + type: formal_theorem + title: funct_3_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0316 + current_version: v1 + source_path: items/formal-theorems/cid005578-fth0316.yaml + cid005579: + type: formal_theorem + title: abcd_has_zero_divisors (theorem) + status: canonical + visibility: public + primary_alias: FTH0317 + current_version: v1 + source_path: items/formal-theorems/cid005579-fth0317.yaml + cid005580: + type: formal_theorem + title: tau_spans_three_rungs (theorem) + status: canonical + visibility: public + primary_alias: FTH0318 + current_version: v1 + source_path: items/formal-theorems/cid005580-fth0318.yaml + cid005581: + type: formal_theorem + title: simultaneous_rung (theorem) + status: canonical + visibility: public + primary_alias: FTH0319 + current_version: v1 + source_path: items/formal-theorems/cid005581-fth0319.yaml + cid005582: + type: formal_theorem + title: tau_engine_inactive (theorem) + status: canonical + visibility: public + primary_alias: FTH0320 + current_version: v1 + source_path: items/formal-theorems/cid005582-fth0320.yaml + cid005583: + type: formal_theorem + title: ladder_collapse (theorem) + status: canonical + visibility: public + primary_alias: FTH0321 + current_version: v1 + source_path: items/formal-theorems/cid005583-fth0321.yaml + cid005584: + type: formal_theorem + title: fibration_forced (theorem) + status: canonical + visibility: public + primary_alias: FTH0322 + current_version: v1 + source_path: items/formal-theorems/cid005584-fth0322.yaml + cid005585: + type: formal_theorem + title: refinement_is_abcd (theorem) + status: canonical + visibility: public + primary_alias: FTH0323 + current_version: v1 + source_path: items/formal-theorems/cid005585-fth0323.yaml + cid005586: + type: formal_theorem + title: fibration_is_three (theorem) + status: canonical + visibility: public + primary_alias: FTH0324 + current_version: v1 + source_path: items/formal-theorems/cid005586-fth0324.yaml + cid005587: + type: formal_theorem + title: rigidity_no_free_parameter (theorem) + status: canonical + visibility: public + primary_alias: FTH0325 + current_version: v1 + source_path: items/formal-theorems/cid005587-fth0325.yaml + cid005588: + type: formal_theorem + title: categoricity_kills_ladder (theorem) + status: canonical + visibility: public + primary_alias: FTH0326 + current_version: v1 + source_path: items/formal-theorems/cid005588-fth0326.yaml + cid005589: + type: formal_theorem + title: moduli_singleton (theorem) + status: canonical + visibility: public + primary_alias: FTH0327 + current_version: v1 + source_path: items/formal-theorems/cid005589-fth0327.yaml + cid005590: + type: formal_theorem + title: engine_only_qft_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0328 + current_version: v1 + source_path: items/formal-theorems/cid005590-fth0328.yaml + cid005591: + type: formal_theorem + title: rigidity_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0329 + current_version: v1 + source_path: items/formal-theorems/cid005591-fth0329.yaml + cid005592: + type: formal_theorem + title: simultaneous_rung_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0330 + current_version: v1 + source_path: items/formal-theorems/cid005592-fth0330.yaml + cid005593: + type: formal_theorem + title: ladder_collapse_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0331 + current_version: v1 + source_path: items/formal-theorems/cid005593-fth0331.yaml + cid005594: + type: formal_theorem + title: categoricity_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0332 + current_version: v1 + source_path: items/formal-theorems/cid005594-fth0332.yaml + cid005595: + type: formal_theorem + title: rewiring_table_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0333 + current_version: v1 + source_path: items/formal-theorems/cid005595-fth0333.yaml + cid005596: + type: formal_theorem + title: table_12 (theorem) + status: canonical + visibility: public + primary_alias: FTH0334 + current_version: v1 + source_path: items/formal-theorems/cid005596-fth0334.yaml + cid005597: + type: formal_theorem + title: rows_orthodox (theorem) + status: canonical + visibility: public + primary_alias: FTH0335 + current_version: v1 + source_path: items/formal-theorems/cid005597-fth0335.yaml + cid005598: + type: formal_theorem + title: levels_match (theorem) + status: canonical + visibility: public + primary_alias: FTH0336 + current_version: v1 + source_path: items/formal-theorems/cid005598-fth0336.yaml + cid005599: + type: formal_theorem + title: qft_gr_same_column (theorem) + status: canonical + visibility: public + primary_alias: FTH0337 + current_version: v1 + source_path: items/formal-theorems/cid005599-fth0337.yaml + cid005600: + type: formal_theorem + title: qft_gr_incompatible_pde (theorem) + status: canonical + visibility: public + primary_alias: FTH0338 + current_version: v1 + source_path: items/formal-theorems/cid005600-fth0338.yaml + cid005601: + type: formal_theorem + title: qft_gr_obstructed (theorem) + status: canonical + visibility: public + primary_alias: FTH0339 + current_version: v1 + source_path: items/formal-theorems/cid005601-fth0339.yaml + cid005602: + type: formal_theorem + title: tau_is_non_archimedean (theorem) + status: canonical + visibility: public + primary_alias: FTH0340 + current_version: v1 + source_path: items/formal-theorems/cid005602-fth0340.yaml + cid005603: + type: formal_theorem + title: tau_is_hyperbolic (theorem) + status: canonical + visibility: public + primary_alias: FTH0341 + current_version: v1 + source_path: items/formal-theorems/cid005603-fth0341.yaml + cid005604: + type: formal_theorem + title: tau_not_archimedean_with_qft (theorem) + status: canonical + visibility: public + primary_alias: FTH0342 + current_version: v1 + source_path: items/formal-theorems/cid005604-fth0342.yaml + cid005605: + type: formal_theorem + title: tau_not_archimedean_with_gr (theorem) + status: canonical + visibility: public + primary_alias: FTH0343 + current_version: v1 + source_path: items/formal-theorems/cid005605-fth0343.yaml + cid005606: + type: formal_theorem + title: tau_qft_not_obstructed (theorem) + status: canonical + visibility: public + primary_alias: FTH0344 + current_version: v1 + source_path: items/formal-theorems/cid005606-fth0344.yaml + cid005607: + type: formal_theorem + title: tau_gr_not_obstructed (theorem) + status: canonical + visibility: public + primary_alias: FTH0345 + current_version: v1 + source_path: items/formal-theorems/cid005607-fth0345.yaml + cid005608: + type: formal_theorem + title: tau_escapes_obstruction (theorem) + status: canonical + visibility: public + primary_alias: FTH0346 + current_version: v1 + source_path: items/formal-theorems/cid005608-fth0346.yaml + cid005609: + type: formal_theorem + title: tau_distinct_from_qft (theorem) + status: canonical + visibility: public + primary_alias: FTH0347 + current_version: v1 + source_path: items/formal-theorems/cid005609-fth0347.yaml + cid005610: + type: formal_theorem + title: four_quadrants (theorem) + status: canonical + visibility: public + primary_alias: FTH0348 + current_version: v1 + source_path: items/formal-theorems/cid005610-fth0348.yaml + cid005611: + type: formal_theorem + title: qft_gr_obstructed_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0349 + current_version: v1 + source_path: items/formal-theorems/cid005611-fth0349.yaml + cid005612: + type: formal_theorem + title: tau_escapes_qft (theorem) + status: canonical + visibility: public + primary_alias: FTH0350 + current_version: v1 + source_path: items/formal-theorems/cid005612-fth0350.yaml + cid005613: + type: formal_theorem + title: tau_hyp (theorem) + status: canonical + visibility: public + primary_alias: FTH0351 + current_version: v1 + source_path: items/formal-theorems/cid005613-fth0351.yaml + cid005614: + type: formal_theorem + title: obstruction_confined_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0352 + current_version: v1 + source_path: items/formal-theorems/cid005614-fth0352.yaml + cid005615: + type: formal_theorem + title: sign_level_count (theorem) + status: canonical + visibility: public + primary_alias: FTH0353 + current_version: v1 + source_path: items/formal-theorems/cid005615-fth0353.yaml + cid005616: + type: formal_theorem + title: all_descriptions_differ (theorem) + status: canonical + visibility: public + primary_alias: FTH0354 + current_version: v1 + source_path: items/formal-theorems/cid005616-fth0354.yaml + cid005617: + type: formal_theorem + title: orthodox_path_no_unique_omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0355 + current_version: v1 + source_path: items/formal-theorems/cid005617-fth0355.yaml + cid005618: + type: formal_theorem + title: tau_path_no_archimedean (theorem) + status: canonical + visibility: public + primary_alias: FTH0356 + current_version: v1 + source_path: items/formal-theorems/cid005618-fth0356.yaml + cid005619: + type: formal_theorem + title: orthodox_path_no_finite_witnesses (theorem) + status: canonical + visibility: public + primary_alias: FTH0357 + current_version: v1 + source_path: items/formal-theorems/cid005619-fth0357.yaml + cid005620: + type: formal_theorem + title: tau_path_no_epsilon_delta (theorem) + status: canonical + visibility: public + primary_alias: FTH0358 + current_version: v1 + source_path: items/formal-theorems/cid005620-fth0358.yaml + cid005621: + type: formal_theorem + title: structural_incompatibility (theorem) + status: canonical + visibility: public + primary_alias: FTH0359 + current_version: v1 + source_path: items/formal-theorems/cid005621-fth0359.yaml + cid005622: + type: formal_theorem + title: paths_distinct (theorem) + status: canonical + visibility: public + primary_alias: FTH0360 + current_version: v1 + source_path: items/formal-theorems/cid005622-fth0360.yaml + cid005623: + type: formal_theorem + title: no_both_omega_and_archimedean_orthodox (theorem) + status: canonical + visibility: public + primary_alias: FTH0361 + current_version: v1 + source_path: items/formal-theorems/cid005623-fth0361.yaml + cid005624: + type: formal_theorem + title: no_both_omega_and_archimedean_tau (theorem) + status: canonical + visibility: public + primary_alias: FTH0362 + current_version: v1 + source_path: items/formal-theorems/cid005624-fth0362.yaml + cid005625: + type: formal_theorem + title: sign_count_12 (theorem) + status: canonical + visibility: public + primary_alias: FTH0363 + current_version: v1 + source_path: items/formal-theorems/cid005625-fth0363.yaml + cid005626: + type: formal_theorem + title: orthodox_all_nonempty (theorem) + status: canonical + visibility: public + primary_alias: FTH0364 + current_version: v1 + source_path: items/formal-theorems/cid005626-fth0364.yaml + cid005627: + type: formal_theorem + title: orthodox_omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0365 + current_version: v1 + source_path: items/formal-theorems/cid005627-fth0365.yaml + cid005628: + type: formal_theorem + title: incompatibility_native (theorem) + status: canonical + visibility: public + primary_alias: FTH0366 + current_version: v1 + source_path: items/formal-theorems/cid005628-fth0366.yaml + cid005629: + type: formal_theorem + title: hyperbolic_hartogs_natural (theorem) + status: canonical + visibility: public + primary_alias: FTH0367 + current_version: v1 + source_path: items/formal-theorems/cid005629-fth0367.yaml + cid005630: + type: formal_theorem + title: elliptic_hartogs_not_natural (theorem) + status: canonical + visibility: public + primary_alias: FTH0368 + current_version: v1 + source_path: items/formal-theorems/cid005630-fth0368.yaml + cid005631: + type: formal_theorem + title: asymmetric_determination (theorem) + status: canonical + visibility: public + primary_alias: FTH0369 + current_version: v1 + source_path: items/formal-theorems/cid005631-fth0369.yaml + cid005632: + type: formal_theorem + title: max_principle_asymmetry (theorem) + status: canonical + visibility: public + primary_alias: FTH0370 + current_version: v1 + source_path: items/formal-theorems/cid005632-fth0370.yaml + cid005633: + type: formal_theorem + title: pde_classifications_distinct (theorem) + status: canonical + visibility: public + primary_alias: FTH0371 + current_version: v1 + source_path: items/formal-theorems/cid005633-fth0371.yaml + cid005634: + type: formal_theorem + title: characteristics_iff_hyperbolic (theorem) + status: canonical + visibility: public + primary_alias: FTH0372 + current_version: v1 + source_path: items/formal-theorems/cid005634-fth0372.yaml + cid005635: + type: formal_theorem + title: stage_no_light_cones (theorem) + status: canonical + visibility: public + primary_alias: FTH0373 + current_version: v1 + source_path: items/formal-theorems/cid005635-fth0373.yaml + cid005636: + type: formal_theorem + title: stage_euclidean (theorem) + status: canonical + visibility: public + primary_alias: FTH0374 + current_version: v1 + source_path: items/formal-theorems/cid005636-fth0374.yaml + cid005637: + type: formal_theorem + title: stage_euclidean_check_true (theorem) + status: canonical + visibility: public + primary_alias: FTH0375 + current_version: v1 + source_path: items/formal-theorems/cid005637-fth0375.yaml + cid005638: + type: formal_theorem + title: all_stages_euclidean_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0376 + current_version: v1 + source_path: items/formal-theorems/cid005638-fth0376.yaml + cid005639: + type: formal_theorem + title: stage_size_is_primorial (theorem) + status: canonical + visibility: public + primary_alias: FTH0377 + current_version: v1 + source_path: items/formal-theorems/cid005639-fth0377.yaml + cid005640: + type: formal_theorem + title: sector_additive (theorem) + status: canonical + visibility: public + primary_alias: FTH0378 + current_version: v1 + source_path: items/formal-theorems/cid005640-fth0378.yaml + cid005641: + type: formal_theorem + title: b_sector_multiplicative (theorem) + status: canonical + visibility: public + primary_alias: FTH0379 + current_version: v1 + source_path: items/formal-theorems/cid005641-fth0379.yaml + cid005642: + type: formal_theorem + title: hartogs_natural_hyp (theorem) + status: canonical + visibility: public + primary_alias: FTH0380 + current_version: v1 + source_path: items/formal-theorems/cid005642-fth0380.yaml + cid005643: + type: formal_theorem + title: chars_hyp (theorem) + status: canonical + visibility: public + primary_alias: FTH0381 + current_version: v1 + source_path: items/formal-theorems/cid005643-fth0381.yaml + cid005644: + type: formal_theorem + title: max_hyp (theorem) + status: canonical + visibility: public + primary_alias: FTH0382 + current_version: v1 + source_path: items/formal-theorems/cid005644-fth0382.yaml + cid005645: + type: formal_theorem + title: stages_euclidean_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0383 + current_version: v1 + source_path: items/formal-theorems/cid005645-fth0383.yaml + cid005646: + type: formal_theorem + title: pde_distinct (theorem) + status: canonical + visibility: public + primary_alias: FTH0384 + current_version: v1 + source_path: items/formal-theorems/cid005646-fth0384.yaml + cid005647: + type: formal_theorem + title: code_delta_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0385 + current_version: v1 + source_path: items/formal-theorems/cid005647-fth0385.yaml + cid005648: + type: formal_theorem + title: decode_welldef_3_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0386 + current_version: v1 + source_path: items/formal-theorems/cid005648-fth0386.yaml + cid005649: + type: formal_theorem + title: decode_unique_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0387 + current_version: v1 + source_path: items/formal-theorems/cid005649-fth0387.yaml + cid005650: + type: formal_theorem + title: code_decode_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0388 + current_version: v1 + source_path: items/formal-theorems/cid005650-fth0388.yaml + cid005651: + type: formal_theorem + title: code_decode_delta_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0389 + current_version: v1 + source_path: items/formal-theorems/cid005651-fth0389.yaml + cid005652: + type: formal_theorem + title: spectral_sep_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0390 + current_version: v1 + source_path: items/formal-theorems/cid005652-fth0390.yaml + cid005653: + type: formal_theorem + title: full_code_decode_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0391 + current_version: v1 + source_path: items/formal-theorems/cid005653-fth0391.yaml + cid005654: + type: formal_theorem + title: code_decode_id_roundtrip (theorem) + status: canonical + visibility: public + primary_alias: FTH0392 + current_version: v1 + source_path: items/formal-theorems/cid005654-fth0392.yaml + cid005655: + type: formal_theorem + title: decode_code_roundtrip (theorem) + status: canonical + visibility: public + primary_alias: FTH0393 + current_version: v1 + source_path: items/formal-theorems/cid005655-fth0393.yaml + cid005656: + type: formal_theorem + title: decompose_recovery (theorem) + status: canonical + visibility: public + primary_alias: FTH0394 + current_version: v1 + source_path: items/formal-theorems/cid005656-fth0394.yaml + cid005657: + type: formal_theorem + title: recovery_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0395 + current_version: v1 + source_path: items/formal-theorems/cid005657-fth0395.yaml + cid005658: + type: formal_theorem + title: stagefun_decompose_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0396 + current_version: v1 + source_path: items/formal-theorems/cid005658-fth0396.yaml + cid005659: + type: formal_theorem + title: functorial_12_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0397 + current_version: v1 + source_path: items/formal-theorems/cid005659-fth0397.yaml + cid005660: + type: formal_theorem + title: depth_max_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0398 + current_version: v1 + source_path: items/formal-theorems/cid005660-fth0398.yaml + cid005661: + type: formal_theorem + title: criterion_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0399 + current_version: v1 + source_path: items/formal-theorems/cid005661-fth0399.yaml + cid005662: + type: formal_theorem + title: small_point_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0400 + current_version: v1 + source_path: items/formal-theorems/cid005662-fth0400.yaml + cid005663: + type: formal_theorem + title: evolution_stab_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0401 + current_version: v1 + source_path: items/formal-theorems/cid005663-fth0401.yaml + cid005664: + type: formal_theorem + title: preyoneda_tower_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0402 + current_version: v1 + source_path: items/formal-theorems/cid005664-fth0402.yaml + cid005665: + type: formal_theorem + title: preyoneda_identity_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0403 + current_version: v1 + source_path: items/formal-theorems/cid005665-fth0403.yaml + cid005666: + type: formal_theorem + title: preyoneda_composition_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0404 + current_version: v1 + source_path: items/formal-theorems/cid005666-fth0404.yaml + cid005667: + type: formal_theorem + title: preyoneda_bipolar_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0405 + current_version: v1 + source_path: items/formal-theorems/cid005667-fth0405.yaml + cid005668: + type: formal_theorem + title: preyoneda_abcd_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0406 + current_version: v1 + source_path: items/formal-theorems/cid005668-fth0406.yaml + cid005669: + type: formal_theorem + title: probe_nat_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0407 + current_version: v1 + source_path: items/formal-theorems/cid005669-fth0407.yaml + cid005670: + type: formal_theorem + title: probe_tower_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0408 + current_version: v1 + source_path: items/formal-theorems/cid005670-fth0408.yaml + cid005671: + type: formal_theorem + title: preyoneda_id_tower_coherent (theorem) + status: canonical + visibility: public + primary_alias: FTH0409 + current_version: v1 + source_path: items/formal-theorems/cid005671-fth0409.yaml + cid005672: + type: formal_theorem + title: probe_naturality_is_tower_coherence (theorem) + status: canonical + visibility: public + primary_alias: FTH0410 + current_version: v1 + source_path: items/formal-theorems/cid005672-fth0410.yaml + cid005673: + type: formal_theorem + title: branch_fact_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0411 + current_version: v1 + source_path: items/formal-theorems/cid005673-fth0411.yaml + cid005674: + type: formal_theorem + title: prime_split_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0412 + current_version: v1 + source_path: items/formal-theorems/cid005674-fth0412.yaml + cid005675: + type: formal_theorem + title: polarity_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0413 + current_version: v1 + source_path: items/formal-theorems/cid005675-fth0413.yaml + cid005676: + type: formal_theorem + title: is1_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0414 + current_version: v1 + source_path: items/formal-theorems/cid005676-fth0414.yaml + cid005677: + type: formal_theorem + title: hol_iff_is_12_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0415 + current_version: v1 + source_path: items/formal-theorems/cid005677-fth0415.yaml + cid005678: + type: formal_theorem + title: radical_check_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0416 + current_version: v1 + source_path: items/formal-theorems/cid005678-fth0416.yaml + cid005679: + type: formal_theorem + title: abc_quality_15 (theorem) + status: canonical + visibility: public + primary_alias: FTH0417 + current_version: v1 + source_path: items/formal-theorems/cid005679-fth0417.yaml + cid005680: + type: formal_theorem + title: abc_primorial_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0418 + current_version: v1 + source_path: items/formal-theorems/cid005680-fth0418.yaml + cid005681: + type: formal_theorem + title: radical_primorial_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0419 + current_version: v1 + source_path: items/formal-theorems/cid005681-fth0419.yaml + cid005682: + type: formal_theorem + title: radical_le_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0420 + current_version: v1 + source_path: items/formal-theorems/cid005682-fth0420.yaml + cid005683: + type: formal_theorem + title: abc_quality_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0421 + current_version: v1 + source_path: items/formal-theorems/cid005683-fth0421.yaml + cid005684: + type: formal_theorem + title: squarefree_dominance_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0422 + current_version: v1 + source_path: items/formal-theorems/cid005684-fth0422.yaml + cid005685: + type: formal_theorem + title: radical_primorial_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0423 + current_version: v1 + source_path: items/formal-theorems/cid005685-fth0423.yaml + cid005686: + type: formal_theorem + title: squarefree_dominance_thm (theorem) + status: canonical + visibility: public + primary_alias: FTH0424 + current_version: v1 + source_path: items/formal-theorems/cid005686-fth0424.yaml + cid005687: + type: formal_theorem + title: high_quality_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0425 + current_version: v1 + source_path: items/formal-theorems/cid005687-fth0425.yaml + cid005688: + type: formal_theorem + title: one_squarefree (theorem) + status: canonical + visibility: public + primary_alias: FTH0426 + current_version: v1 + source_path: items/formal-theorems/cid005688-fth0426.yaml + cid005689: + type: formal_theorem + title: thirty_squarefree (theorem) + status: canonical + visibility: public + primary_alias: FTH0427 + current_version: v1 + source_path: items/formal-theorems/cid005689-fth0427.yaml + cid005690: + type: formal_theorem + title: twelve_not_squarefree (theorem) + status: canonical + visibility: public + primary_alias: FTH0428 + current_version: v1 + source_path: items/formal-theorems/cid005690-fth0428.yaml + cid005691: + type: formal_theorem + title: bsd_coherence_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0429 + current_version: v1 + source_path: items/formal-theorems/cid005691-fth0429.yaml + cid005692: + type: formal_theorem + title: bsd_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0430 + current_version: v1 + source_path: items/formal-theorems/cid005692-fth0430.yaml + cid005693: + type: formal_theorem + title: enr_01_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0431 + current_version: v1 + source_path: items/formal-theorems/cid005693-fth0431.yaml + cid005694: + type: formal_theorem + title: e1_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0432 + current_version: v1 + source_path: items/formal-theorems/cid005694-fth0432.yaml + cid005695: + type: formal_theorem + title: all_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0433 + current_version: v1 + source_path: items/formal-theorems/cid005695-fth0433.yaml + cid005696: + type: formal_theorem + title: enriched_bs_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0434 + current_version: v1 + source_path: items/formal-theorems/cid005696-fth0434.yaml + cid005697: + type: formal_theorem + title: factorization_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0435 + current_version: v1 + source_path: items/formal-theorems/cid005697-fth0435.yaml + cid005698: + type: formal_theorem + title: comparison_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0436 + current_version: v1 + source_path: items/formal-theorems/cid005698-fth0436.yaml + cid005699: + type: formal_theorem + title: langlands_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0437 + current_version: v1 + source_path: items/formal-theorems/cid005699-fth0437.yaml + cid005700: + type: formal_theorem + title: func_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0438 + current_version: v1 + source_path: items/formal-theorems/cid005700-fth0438.yaml + cid005701: + type: formal_theorem + title: base_change_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0439 + current_version: v1 + source_path: items/formal-theorems/cid005701-fth0439.yaml + cid005702: + type: formal_theorem + title: proto_zero_verified (theorem) + status: canonical + visibility: public + primary_alias: FTH0440 + current_version: v1 + source_path: items/formal-theorems/cid005702-fth0440.yaml + cid005703: + type: formal_theorem + title: bsd_nonneg_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0441 + current_version: v1 + source_path: items/formal-theorems/cid005703-fth0441.yaml + cid005704: + type: formal_theorem + title: bridgehead_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0442 + current_version: v1 + source_path: items/formal-theorems/cid005704-fth0442.yaml + cid005705: + type: formal_theorem + title: zero_rational (theorem) + status: canonical + visibility: public + primary_alias: FTH0443 + current_version: v1 + source_path: items/formal-theorems/cid005705-fth0443.yaml + cid005706: + type: formal_theorem + title: rank_bounded (theorem) + status: canonical + visibility: public + primary_alias: FTH0444 + current_version: v1 + source_path: items/formal-theorems/cid005706-fth0444.yaml + cid005707: + type: formal_theorem + title: all_rational_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0445 + current_version: v1 + source_path: items/formal-theorems/cid005707-fth0445.yaml + cid005708: + type: formal_theorem + title: e0_lt_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0446 + current_version: v1 + source_path: items/formal-theorems/cid005708-fth0446.yaml + cid005709: + type: formal_theorem + title: e1_lt_e2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0447 + current_version: v1 + source_path: items/formal-theorems/cid005709-fth0447.yaml + cid005710: + type: formal_theorem + title: e2_lt_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0448 + current_version: v1 + source_path: items/formal-theorems/cid005710-fth0448.yaml + cid005711: + type: formal_theorem + title: eight_problems (theorem) + status: canonical + visibility: public + primary_alias: FTH0449 + current_version: v1 + source_path: items/formal-theorems/cid005711-fth0449.yaml + cid005712: + type: formal_axiom + title: bridge_functor_exists (axiom) + status: canonical + visibility: public + primary_alias: FAX0001 + current_version: v1 + source_path: items/formal-axioms/cid005712-fax0001.yaml + cid005713: + type: formal_theorem + title: bridge_functor_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0450 + current_version: v1 + source_path: items/formal-theorems/cid005713-fth0450.yaml + cid005714: + type: formal_theorem + title: shadow_diagram_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0451 + current_version: v1 + source_path: items/formal-theorems/cid005714-fth0451.yaml + cid005715: + type: formal_theorem + title: honest_claim_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0452 + current_version: v1 + source_path: items/formal-theorems/cid005715-fth0452.yaml + cid005716: + type: formal_theorem + title: conjectural_marked (theorem) + status: canonical + visibility: public + primary_alias: FTH0453 + current_version: v1 + source_path: items/formal-theorems/cid005716-fth0453.yaml + cid005717: + type: formal_theorem + title: break_marked (theorem) + status: canonical + visibility: public + primary_alias: FTH0454 + current_version: v1 + source_path: items/formal-theorems/cid005717-fth0454.yaml + cid005718: + type: formal_theorem + title: honest_claim_full_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0455 + current_version: v1 + source_path: items/formal-theorems/cid005718-fth0455.yaml + cid005719: + type: formal_theorem + title: bridge_ledger_consistent (theorem) + status: canonical + visibility: public + primary_alias: FTH0456 + current_version: v1 + source_path: items/formal-theorems/cid005719-fth0456.yaml + cid005720: + type: formal_theorem + title: rh_bridge_conjectural (theorem) + status: canonical + visibility: public + primary_alias: FTH0457 + current_version: v1 + source_path: items/formal-theorems/cid005720-fth0457.yaml + cid005721: + type: formal_theorem + title: one_axiom (theorem) + status: canonical + visibility: public + primary_alias: FTH0458 + current_version: v1 + source_path: items/formal-theorems/cid005721-fth0458.yaml + cid005722: + type: formal_theorem + title: rh_layers (theorem) + status: canonical + visibility: public + primary_alias: FTH0459 + current_version: v1 + source_path: items/formal-theorems/cid005722-fth0459.yaml + cid005723: + type: formal_theorem + title: ledger_count (theorem) + status: canonical + visibility: public + primary_alias: FTH0460 + current_version: v1 + source_path: items/formal-theorems/cid005723-fth0460.yaml + cid005724: + type: formal_theorem + title: poincare_established (theorem) + status: canonical + visibility: public + primary_alias: FTH0461 + current_version: v1 + source_path: items/formal-theorems/cid005724-fth0461.yaml + cid005725: + type: formal_theorem + title: pvsnp_bridge_break (theorem) + status: canonical + visibility: public + primary_alias: FTH0462 + current_version: v1 + source_path: items/formal-theorems/cid005725-fth0462.yaml + cid005726: + type: formal_theorem + title: scope_order (theorem) + status: canonical + visibility: public + primary_alias: FTH0463 + current_version: v1 + source_path: items/formal-theorems/cid005726-fth0463.yaml + cid005727: + type: formal_theorem + title: established_not_conjectural (theorem) + status: canonical + visibility: public + primary_alias: FTH0464 + current_version: v1 + source_path: items/formal-theorems/cid005727-fth0464.yaml + cid005728: + type: formal_theorem + title: ledger_partition (theorem) + status: canonical + visibility: public + primary_alias: FTH0465 + current_version: v1 + source_path: items/formal-theorems/cid005728-fth0465.yaml + cid005729: + type: formal_theorem + title: tower_finite_decidable (theorem) + status: canonical + visibility: public + primary_alias: FTH0466 + current_version: v1 + source_path: items/formal-theorems/cid005729-fth0466.yaml + cid005730: + type: formal_theorem + title: bridge_necessary_insufficient (theorem) + status: canonical + visibility: public + primary_alias: FTH0467 + current_version: v1 + source_path: items/formal-theorems/cid005730-fth0467.yaml + cid005731: + type: formal_theorem + title: gap_taxonomy (theorem) + status: canonical + visibility: public + primary_alias: FTH0468 + current_version: v1 + source_path: items/formal-theorems/cid005731-fth0468.yaml + cid005732: + type: formal_theorem + title: all_gaps_exponential (theorem) + status: canonical + visibility: public + primary_alias: FTH0469 + current_version: v1 + source_path: items/formal-theorems/cid005732-fth0469.yaml + cid005733: + type: formal_theorem + title: parity_ne_density (theorem) + status: canonical + visibility: public + primary_alias: FTH0470 + current_version: v1 + source_path: items/formal-theorems/cid005733-fth0470.yaml + cid005734: + type: formal_theorem + title: gap_indices (theorem) + status: canonical + visibility: public + primary_alias: FTH0471 + current_version: v1 + source_path: items/formal-theorems/cid005734-fth0471.yaml + cid005735: + type: formal_theorem + title: goldbach_gap_parity (theorem) + status: canonical + visibility: public + primary_alias: FTH0472 + current_version: v1 + source_path: items/formal-theorems/cid005735-fth0472.yaml + cid005736: + type: formal_theorem + title: twin_gap_density (theorem) + status: canonical + visibility: public + primary_alias: FTH0473 + current_version: v1 + source_path: items/formal-theorems/cid005736-fth0473.yaml + cid005737: + type: formal_theorem + title: abc_gap_structural (theorem) + status: canonical + visibility: public + primary_alias: FTH0474 + current_version: v1 + source_path: items/formal-theorems/cid005737-fth0474.yaml + cid005738: + type: formal_theorem + title: all_gaps_K4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0475 + current_version: v1 + source_path: items/formal-theorems/cid005738-fth0475.yaml + cid005739: + type: formal_theorem + title: exponential_damage (theorem) + status: canonical + visibility: public + primary_alias: FTH0476 + current_version: v1 + source_path: items/formal-theorems/cid005739-fth0476.yaml + cid005740: + type: formal_theorem + title: forbidden_moves_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0477 + current_version: v1 + source_path: items/formal-theorems/cid005740-fth0477.yaml + cid005741: + type: formal_theorem + title: move_bridge_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0478 + current_version: v1 + source_path: items/formal-theorems/cid005741-fth0478.yaml + cid005742: + type: formal_theorem + title: move_correspondence (theorem) + status: canonical + visibility: public + primary_alias: FTH0479 + current_version: v1 + source_path: items/formal-theorems/cid005742-fth0479.yaml + cid005743: + type: formal_theorem + title: five_forbidden (theorem) + status: canonical + visibility: public + primary_alias: FTH0480 + current_version: v1 + source_path: items/formal-theorems/cid005743-fth0480.yaml + cid005744: + type: formal_theorem + title: move_index_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0481 + current_version: v1 + source_path: items/formal-theorems/cid005744-fth0481.yaml + cid005745: + type: formal_theorem + title: fanout_violates_K3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0482 + current_version: v1 + source_path: items/formal-theorems/cid005745-fth0482.yaml + cid005746: + type: formal_theorem + title: equality_violates_K5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0483 + current_version: v1 + source_path: items/formal-theorems/cid005746-fth0483.yaml + cid005747: + type: formal_theorem + title: circuits_break_bridge (theorem) + status: canonical + visibility: public + primary_alias: FTH0484 + current_version: v1 + source_path: items/formal-theorems/cid005747-fth0484.yaml + cid005748: + type: formal_theorem + title: pvsnp_uses_3_moves (theorem) + status: canonical + visibility: public + primary_alias: FTH0485 + current_version: v1 + source_path: items/formal-theorems/cid005748-fth0485.yaml + cid005749: + type: formal_theorem + title: max_damage_is_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0486 + current_version: v1 + source_path: items/formal-theorems/cid005749-fth0486.yaml + cid005750: + type: formal_theorem + title: threshold_exceeds (theorem) + status: canonical + visibility: public + primary_alias: FTH0487 + current_version: v1 + source_path: items/formal-theorems/cid005750-fth0487.yaml + cid005751: + type: formal_theorem + title: incompleteness_vm_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0488 + current_version: v1 + source_path: items/formal-theorems/cid005751-fth0488.yaml + cid005752: + type: formal_theorem + title: diagonal_depth_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0489 + current_version: v1 + source_path: items/formal-theorems/cid005752-fth0489.yaml + cid005753: + type: formal_theorem + title: diagonal_mod (theorem) + status: canonical + visibility: public + primary_alias: FTH0490 + current_version: v1 + source_path: items/formal-theorems/cid005753-fth0490.yaml + cid005754: + type: formal_theorem + title: godel_at_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0491 + current_version: v1 + source_path: items/formal-theorems/cid005754-fth0491.yaml + cid005755: + type: formal_theorem + title: all_at_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0492 + current_version: v1 + source_path: items/formal-theorems/cid005755-fth0492.yaml + cid005756: + type: formal_theorem + title: all_from_e2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0493 + current_version: v1 + source_path: items/formal-theorems/cid005756-fth0493.yaml + cid005757: + type: formal_theorem + title: e2_lt_e3_boundary (theorem) + status: canonical + visibility: public + primary_alias: FTH0494 + current_version: v1 + source_path: items/formal-theorems/cid005757-fth0494.yaml + cid005758: + type: formal_theorem + title: e3_resolves_incompleteness (theorem) + status: canonical + visibility: public + primary_alias: FTH0495 + current_version: v1 + source_path: items/formal-theorems/cid005758-fth0495.yaml + cid005759: + type: formal_theorem + title: arith_translation_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0496 + current_version: v1 + source_path: items/formal-theorems/cid005759-fth0496.yaml + cid005760: + type: formal_theorem + title: crt_integer_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0497 + current_version: v1 + source_path: items/formal-theorems/cid005760-fth0497.yaml + cid005761: + type: formal_theorem + title: arith_faithful_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0498 + current_version: v1 + source_path: items/formal-theorems/cid005761-fth0498.yaml + cid005762: + type: formal_theorem + title: arith_preserves_6_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0499 + current_version: v1 + source_path: items/formal-theorems/cid005762-fth0499.yaml + cid005763: + type: formal_theorem + title: obstruction_check_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0500 + current_version: v1 + source_path: items/formal-theorems/cid005763-fth0500.yaml + cid005764: + type: formal_theorem + title: arith_obstruction_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0501 + current_version: v1 + source_path: items/formal-theorems/cid005764-fth0501.yaml + cid005765: + type: formal_theorem + title: topo_obstruction_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0502 + current_version: v1 + source_path: items/formal-theorems/cid005765-fth0502.yaml + cid005766: + type: formal_theorem + title: safe_region_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0503 + current_version: v1 + source_path: items/formal-theorems/cid005766-fth0503.yaml + cid005767: + type: formal_theorem + title: translation_boundary_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0504 + current_version: v1 + source_path: items/formal-theorems/cid005767-fth0504.yaml + cid005768: + type: formal_theorem + title: pvsnp_obstruction_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0505 + current_version: v1 + source_path: items/formal-theorems/cid005768-fth0505.yaml + cid005769: + type: formal_theorem + title: topo_translation_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0506 + current_version: v1 + source_path: items/formal-theorems/cid005769-fth0506.yaml + cid005770: + type: formal_theorem + title: dimension_recovery_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0507 + current_version: v1 + source_path: items/formal-theorems/cid005770-fth0507.yaml + cid005771: + type: formal_theorem + title: topo_faithful_6_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0508 + current_version: v1 + source_path: items/formal-theorems/cid005771-fth0508.yaml + cid005772: + type: formal_theorem + title: boundary_restriction_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0509 + current_version: v1 + source_path: items/formal-theorems/cid005772-fth0509.yaml + cid005773: + type: formal_theorem + title: zfc_vm_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0510 + current_version: v1 + source_path: items/formal-theorems/cid005773-fth0510.yaml + cid005774: + type: formal_theorem + title: axiom_encoding_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0511 + current_version: v1 + source_path: items/formal-theorems/cid005774-fth0511.yaml + cid005775: + type: formal_theorem + title: set_universe_10_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0512 + current_version: v1 + source_path: items/formal-theorems/cid005775-fth0512.yaml + cid005776: + type: formal_theorem + title: host_level_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0513 + current_version: v1 + source_path: items/formal-theorems/cid005776-fth0513.yaml + cid005777: + type: formal_theorem + title: zfc_has_9_axioms (theorem) + status: canonical + visibility: public + primary_alias: FTH0514 + current_version: v1 + source_path: items/formal-theorems/cid005777-fth0514.yaml + cid005778: + type: formal_theorem + title: zfc_is_e2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0515 + current_version: v1 + source_path: items/formal-theorems/cid005778-fth0515.yaml + cid005779: + type: formal_theorem + title: ext_min_depth (theorem) + status: canonical + visibility: public + primary_alias: FTH0516 + current_version: v1 + source_path: items/formal-theorems/cid005779-fth0516.yaml + cid005780: + type: formal_theorem + title: inf_min_depth (theorem) + status: canonical + visibility: public + primary_alias: FTH0517 + current_version: v1 + source_path: items/formal-theorems/cid005780-fth0517.yaml + cid005781: + type: formal_theorem + title: universe_rank_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0518 + current_version: v1 + source_path: items/formal-theorems/cid005781-fth0518.yaml + cid005782: + type: formal_theorem + title: universe_rank_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0519 + current_version: v1 + source_path: items/formal-theorems/cid005782-fth0519.yaml + cid005783: + type: formal_theorem + title: pairing_valid (theorem) + status: canonical + visibility: public + primary_alias: FTH0520 + current_version: v1 + source_path: items/formal-theorems/cid005783-fth0520.yaml + cid005784: + type: formal_theorem + title: ext_detects_equal (theorem) + status: canonical + visibility: public + primary_alias: FTH0521 + current_version: v1 + source_path: items/formal-theorems/cid005784-fth0521.yaml + cid005785: + type: formal_theorem + title: ext_detects_unequal (theorem) + status: canonical + visibility: public + primary_alias: FTH0522 + current_version: v1 + source_path: items/formal-theorems/cid005785-fth0522.yaml + cid005786: + type: formal_theorem + title: width_bounded (theorem) + status: canonical + visibility: public + primary_alias: FTH0523 + current_version: v1 + source_path: items/formal-theorems/cid005786-fth0523.yaml + cid005787: + type: formal_theorem + title: admissible_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0524 + current_version: v1 + source_path: items/formal-theorems/cid005787-fth0524.yaml + cid005788: + type: formal_theorem + title: id_admissible (theorem) + status: canonical + visibility: public + primary_alias: FTH0525 + current_version: v1 + source_path: items/formal-theorems/cid005788-fth0525.yaml + cid005789: + type: formal_theorem + title: product_meet_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0526 + current_version: v1 + source_path: items/formal-theorems/cid005789-fth0526.yaml + cid005790: + type: formal_theorem + title: pvsnp_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0527 + current_version: v1 + source_path: items/formal-theorems/cid005790-fth0527.yaml + cid005791: + type: formal_theorem + title: no_barrier_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0528 + current_version: v1 + source_path: items/formal-theorems/cid005791-fth0528.yaml + cid005792: + type: formal_theorem + title: agent_step_depth_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0529 + current_version: v1 + source_path: items/formal-theorems/cid005792-fth0529.yaml + cid005793: + type: formal_theorem + title: agent_step_mod (theorem) + status: canonical + visibility: public + primary_alias: FTH0530 + current_version: v1 + source_path: items/formal-theorems/cid005793-fth0530.yaml + cid005794: + type: formal_theorem + title: identity_decoder (theorem) + status: canonical + visibility: public + primary_alias: FTH0531 + current_version: v1 + source_path: items/formal-theorems/cid005794-fth0531.yaml + cid005795: + type: formal_theorem + title: kleene_check_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0532 + current_version: v1 + source_path: items/formal-theorems/cid005795-fth0532.yaml + cid005796: + type: formal_theorem + title: fixed_points_stage2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0533 + current_version: v1 + source_path: items/formal-theorems/cid005796-fth0533.yaml + cid005797: + type: formal_theorem + title: orbit_diversity_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0534 + current_version: v1 + source_path: items/formal-theorems/cid005797-fth0534.yaml + cid005798: + type: formal_theorem + title: orbit_diversity_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0535 + current_version: v1 + source_path: items/formal-theorems/cid005798-fth0535.yaml + cid005799: + type: formal_theorem + title: operational_closure_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0536 + current_version: v1 + source_path: items/formal-theorems/cid005799-fth0536.yaml + cid005800: + type: formal_theorem + title: e2_strict_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0537 + current_version: v1 + source_path: items/formal-theorems/cid005800-fth0537.yaml + cid005801: + type: formal_theorem + title: ttm_preserves_depth (theorem) + status: canonical + visibility: public + primary_alias: FTH0538 + current_version: v1 + source_path: items/formal-theorems/cid005801-fth0538.yaml + cid005802: + type: formal_theorem + title: ttm_depth_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0539 + current_version: v1 + source_path: items/formal-theorems/cid005802-fth0539.yaml + cid005803: + type: formal_theorem + title: code_is_data (theorem) + status: canonical + visibility: public + primary_alias: FTH0540 + current_version: v1 + source_path: items/formal-theorems/cid005803-fth0540.yaml + cid005804: + type: formal_theorem + title: obs_width (theorem) + status: canonical + visibility: public + primary_alias: FTH0541 + current_version: v1 + source_path: items/formal-theorems/cid005804-fth0541.yaml + cid005805: + type: formal_theorem + title: witness_zero_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0542 + current_version: v1 + source_path: items/formal-theorems/cid005805-fth0542.yaml + cid005806: + type: formal_theorem + title: crt_faithful_42_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0543 + current_version: v1 + source_path: items/formal-theorems/cid005806-fth0543.yaml + cid005807: + type: formal_theorem + title: sum_less_prod_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0544 + current_version: v1 + source_path: items/formal-theorems/cid005807-fth0544.yaml + cid005808: + type: formal_theorem + title: rh_gap_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0545 + current_version: v1 + source_path: items/formal-theorems/cid005808-fth0545.yaml + cid005809: + type: formal_theorem + title: ym_gap_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0546 + current_version: v1 + source_path: items/formal-theorems/cid005809-fth0546.yaml + cid005810: + type: formal_theorem + title: ym_gap_growth_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0547 + current_version: v1 + source_path: items/formal-theorems/cid005810-fth0547.yaml + cid005811: + type: formal_theorem + title: ns_causal_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0548 + current_version: v1 + source_path: items/formal-theorems/cid005811-fth0548.yaml + cid005812: + type: formal_theorem + title: pvsnp_triple_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0549 + current_version: v1 + source_path: items/formal-theorems/cid005812-fth0549.yaml + cid005813: + type: formal_theorem + title: bridge_ledger_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0550 + current_version: v1 + source_path: items/formal-theorems/cid005813-fth0550.yaml + cid005814: + type: formal_theorem + title: critical_line_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0551 + current_version: v1 + source_path: items/formal-theorems/cid005814-fth0551.yaml + cid005815: + type: formal_theorem + title: k5_eigenvalue_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0552 + current_version: v1 + source_path: items/formal-theorems/cid005815-fth0552.yaml + cid005816: + type: formal_theorem + title: tau_rh_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0553 + current_version: v1 + source_path: items/formal-theorems/cid005816-fth0553.yaml + cid005817: + type: formal_axiom + title: grand_grh_adelic (axiom) + status: canonical + visibility: public + primary_alias: FAX0002 + current_version: v1 + source_path: items/formal-axioms/cid005817-fax0002.yaml + cid005818: + type: formal_theorem + title: grand_grh_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0554 + current_version: v1 + source_path: items/formal-theorems/cid005818-fth0554.yaml + cid005819: + type: formal_theorem + title: label_5_is_B (theorem) + status: canonical + visibility: public + primary_alias: FTH0555 + current_version: v1 + source_path: items/formal-theorems/cid005819-fth0555.yaml + cid005820: + type: formal_theorem + title: label_3_is_C (theorem) + status: canonical + visibility: public + primary_alias: FTH0556 + current_version: v1 + source_path: items/formal-theorems/cid005820-fth0556.yaml + cid005821: + type: formal_theorem + title: l_function_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0557 + current_version: v1 + source_path: items/formal-theorems/cid005821-fth0557.yaml + cid005822: + type: formal_theorem + title: grand_grh_from_axiom (theorem) + status: canonical + visibility: public + primary_alias: FTH0558 + current_version: v1 + source_path: items/formal-theorems/cid005822-fth0558.yaml + cid005823: + type: formal_theorem + title: eigenvalue_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0559 + current_version: v1 + source_path: items/formal-theorems/cid005823-fth0559.yaml + cid005824: + type: formal_theorem + title: eigenvalue_formula (theorem) + status: canonical + visibility: public + primary_alias: FTH0560 + current_version: v1 + source_path: items/formal-theorems/cid005824-fth0560.yaml + cid005825: + type: formal_theorem + title: eigenvalue_nonneg (theorem) + status: canonical + visibility: public + primary_alias: FTH0561 + current_version: v1 + source_path: items/formal-theorems/cid005825-fth0561.yaml + cid005826: + type: formal_theorem + title: spectral_gap_value (theorem) + status: canonical + visibility: public + primary_alias: FTH0562 + current_version: v1 + source_path: items/formal-theorems/cid005826-fth0562.yaml + cid005827: + type: formal_theorem + title: rh_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0563 + current_version: v1 + source_path: items/formal-theorems/cid005827-fth0563.yaml + cid005828: + type: formal_theorem + title: pvsnp_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0564 + current_version: v1 + source_path: items/formal-theorems/cid005828-fth0564.yaml + cid005829: + type: formal_theorem + title: ns_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0565 + current_version: v1 + source_path: items/formal-theorems/cid005829-fth0565.yaml + cid005830: + type: formal_theorem + title: rh_part (theorem) + status: canonical + visibility: public + primary_alias: FTH0566 + current_version: v1 + source_path: items/formal-theorems/cid005830-fth0566.yaml + cid005831: + type: formal_theorem + title: e0_before_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0567 + current_version: v1 + source_path: items/formal-theorems/cid005831-fth0567.yaml + cid005832: + type: formal_theorem + title: b_to_i_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0568 + current_version: v1 + source_path: items/formal-theorems/cid005832-fth0568.yaml + cid005833: + type: formal_theorem + title: md_cycle_42_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0569 + current_version: v1 + source_path: items/formal-theorems/cid005833-fth0569.yaml + cid005834: + type: formal_theorem + title: crt_bijective_42_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0570 + current_version: v1 + source_path: items/formal-theorems/cid005834-fth0570.yaml + cid005835: + type: formal_theorem + title: terminal_depth_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0571 + current_version: v1 + source_path: items/formal-theorems/cid005835-fth0571.yaml + cid005836: + type: formal_theorem + title: gluing_depth_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0572 + current_version: v1 + source_path: items/formal-theorems/cid005836-fth0572.yaml + cid005837: + type: formal_axiom + title: spectral_correspondence_O3 (axiom) + status: canonical + visibility: public + primary_alias: FAX0003 + current_version: v1 + source_path: items/formal-axioms/cid005837-fax0003.yaml + cid005838: + type: formal_theorem + title: spectral_param_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0573 + current_version: v1 + source_path: items/formal-theorems/cid005838-fth0573.yaml + cid005839: + type: formal_theorem + title: spectral_param_bounded (theorem) + status: canonical + visibility: public + primary_alias: FTH0574 + current_version: v1 + source_path: items/formal-theorems/cid005839-fth0574.yaml + cid005840: + type: formal_theorem + title: spectral_corr_from_O3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0575 + current_version: v1 + source_path: items/formal-theorems/cid005840-fth0575.yaml + cid005841: + type: formal_theorem + title: projector_check_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0576 + current_version: v1 + source_path: items/formal-theorems/cid005841-fth0576.yaml + cid005842: + type: formal_theorem + title: projector_check_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0577 + current_version: v1 + source_path: items/formal-theorems/cid005842-fth0577.yaml + cid005843: + type: formal_theorem + title: measure_total_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0578 + current_version: v1 + source_path: items/formal-theorems/cid005843-fth0578.yaml + cid005844: + type: formal_theorem + title: measure_total_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0579 + current_version: v1 + source_path: items/formal-theorems/cid005844-fth0579.yaml + cid005845: + type: formal_theorem + title: parseval_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0580 + current_version: v1 + source_path: items/formal-theorems/cid005845-fth0580.yaml + cid005846: + type: formal_theorem + title: parseval_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0581 + current_version: v1 + source_path: items/formal-theorems/cid005846-fth0581.yaml + cid005847: + type: formal_theorem + title: spectral_resolution_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0582 + current_version: v1 + source_path: items/formal-theorems/cid005847-fth0582.yaml + cid005848: + type: formal_theorem + title: spectral_resolution_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0583 + current_version: v1 + source_path: items/formal-theorems/cid005848-fth0583.yaml + cid005849: + type: formal_theorem + title: b_zeta_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0584 + current_version: v1 + source_path: items/formal-theorems/cid005849-fth0584.yaml + cid005850: + type: formal_theorem + title: c_zeta_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0585 + current_version: v1 + source_path: items/formal-theorems/cid005850-fth0585.yaml + cid005851: + type: formal_theorem + title: fe_involution_involutive (theorem) + status: canonical + visibility: public + primary_alias: FTH0586 + current_version: v1 + source_path: items/formal-theorems/cid005851-fth0586.yaml + cid005852: + type: formal_theorem + title: euler_product_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0587 + current_version: v1 + source_path: items/formal-theorems/cid005852-fth0587.yaml + cid005853: + type: formal_theorem + title: non_emptiness_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0588 + current_version: v1 + source_path: items/formal-theorems/cid005853-fth0588.yaml + cid005854: + type: formal_theorem + title: e1_strictness_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0589 + current_version: v1 + source_path: items/formal-theorems/cid005854-fth0589.yaml + cid005855: + type: formal_theorem + title: strictness_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0590 + current_version: v1 + source_path: items/formal-theorems/cid005855-fth0590.yaml + cid005856: + type: formal_theorem + title: functor_collapse_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0591 + current_version: v1 + source_path: items/formal-theorems/cid005856-fth0591.yaml + cid005857: + type: formal_theorem + title: saturation_e3_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0592 + current_version: v1 + source_path: items/formal-theorems/cid005857-fth0592.yaml + cid005858: + type: formal_theorem + title: canonical_ladder_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0593 + current_version: v1 + source_path: items/formal-theorems/cid005858-fth0593.yaml + cid005859: + type: formal_theorem + title: zero_in_carrier (theorem) + status: canonical + visibility: public + primary_alias: FTH0594 + current_version: v1 + source_path: items/formal-theorems/cid005859-fth0594.yaml + cid005860: + type: formal_theorem + title: hom_stage_stable (theorem) + status: canonical + visibility: public + primary_alias: FTH0595 + current_version: v1 + source_path: items/formal-theorems/cid005860-fth0595.yaml + cid005861: + type: formal_theorem + title: structural_saturation (theorem) + status: canonical + visibility: public + primary_alias: FTH0596 + current_version: v1 + source_path: items/formal-theorems/cid005861-fth0596.yaml + cid005862: + type: formal_theorem + title: ladder_has_four_levels (theorem) + status: canonical + visibility: public + primary_alias: FTH0597 + current_version: v1 + source_path: items/formal-theorems/cid005862-fth0597.yaml + cid005863: + type: formal_theorem + title: canonical_ordering (theorem) + status: canonical + visibility: public + primary_alias: FTH0598 + current_version: v1 + source_path: items/formal-theorems/cid005863-fth0598.yaml + cid005864: + type: formal_theorem + title: enr_functor_e0_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0599 + current_version: v1 + source_path: items/formal-theorems/cid005864-fth0599.yaml + cid005865: + type: formal_theorem + title: enr_functor_e1_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0600 + current_version: v1 + source_path: items/formal-theorems/cid005865-fth0600.yaml + cid005866: + type: formal_theorem + title: enr_functor_e2_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0601 + current_version: v1 + source_path: items/formal-theorems/cid005866-fth0601.yaml + cid005867: + type: formal_theorem + title: full_enr_functor_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0602 + current_version: v1 + source_path: items/formal-theorems/cid005867-fth0602.yaml + cid005868: + type: formal_theorem + title: ladder_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0603 + current_version: v1 + source_path: items/formal-theorems/cid005868-fth0603.yaml + cid005869: + type: formal_theorem + title: succ_idempotent_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0604 + current_version: v1 + source_path: items/formal-theorems/cid005869-fth0604.yaml + cid005870: + type: formal_theorem + title: three_steps_to_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0605 + current_version: v1 + source_path: items/formal-theorems/cid005870-fth0605.yaml + cid005871: + type: formal_theorem + title: e0_layer_valid_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0606 + current_version: v1 + source_path: items/formal-theorems/cid005871-fth0606.yaml + cid005872: + type: formal_theorem + title: e1_layer_valid_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0607 + current_version: v1 + source_path: items/formal-theorems/cid005872-fth0607.yaml + cid005873: + type: formal_theorem + title: e2_layer_valid_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0608 + current_version: v1 + source_path: items/formal-theorems/cid005873-fth0608.yaml + cid005874: + type: formal_theorem + title: e3_layer_valid_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0609 + current_version: v1 + source_path: items/formal-theorems/cid005874-fth0609.yaml + cid005875: + type: formal_theorem + title: all_layers_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0610 + current_version: v1 + source_path: items/formal-theorems/cid005875-fth0610.yaml + cid005876: + type: formal_theorem + title: dependency_chain_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0611 + current_version: v1 + source_path: items/formal-theorems/cid005876-fth0611.yaml + cid005877: + type: formal_theorem + title: chain_linearity (theorem) + status: canonical + visibility: public + primary_alias: FTH0612 + current_version: v1 + source_path: items/formal-theorems/cid005877-fth0612.yaml + cid005878: + type: formal_theorem + title: terminal_completeness (theorem) + status: canonical + visibility: public + primary_alias: FTH0613 + current_version: v1 + source_path: items/formal-theorems/cid005878-fth0613.yaml + cid005879: + type: formal_theorem + title: chain_has_14_links (theorem) + status: canonical + visibility: public + primary_alias: FTH0614 + current_version: v1 + source_path: items/formal-theorems/cid005879-fth0614.yaml + cid005880: + type: formal_theorem + title: k0_is_first (theorem) + status: canonical + visibility: public + primary_alias: FTH0615 + current_version: v1 + source_path: items/formal-theorems/cid005880-fth0615.yaml + cid005881: + type: formal_theorem + title: e3p_is_last (theorem) + status: canonical + visibility: public + primary_alias: FTH0616 + current_version: v1 + source_path: items/formal-theorems/cid005881-fth0616.yaml + cid005882: + type: formal_theorem + title: axiom_to_enrichment (theorem) + status: canonical + visibility: public + primary_alias: FTH0617 + current_version: v1 + source_path: items/formal-theorems/cid005882-fth0617.yaml + cid005883: + type: formal_theorem + title: succ_monotone (theorem) + status: canonical + visibility: public + primary_alias: FTH0618 + current_version: v1 + source_path: items/formal-theorems/cid005883-fth0618.yaml + cid005884: + type: formal_theorem + title: e3p_saturates (theorem) + status: canonical + visibility: public + primary_alias: FTH0619 + current_version: v1 + source_path: items/formal-theorems/cid005884-fth0619.yaml + cid005885: + type: formal_theorem + title: all_links_have_level (theorem) + status: canonical + visibility: public + primary_alias: FTH0620 + current_version: v1 + source_path: items/formal-theorems/cid005885-fth0620.yaml + cid005886: + type: formal_theorem + title: seven_plus_seven (theorem) + status: canonical + visibility: public + primary_alias: FTH0621 + current_version: v1 + source_path: items/formal-theorems/cid005886-fth0621.yaml + cid005887: + type: formal_theorem + title: hinge_theorem_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0622 + current_version: v1 + source_path: items/formal-theorems/cid005887-fth0622.yaml + cid005888: + type: formal_theorem + title: no_knobs_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0623 + current_version: v1 + source_path: items/formal-theorems/cid005888-fth0623.yaml + cid005889: + type: formal_theorem + title: hinge_chain_length (theorem) + status: canonical + visibility: public + primary_alias: FTH0624 + current_version: v1 + source_path: items/formal-theorems/cid005889-fth0624.yaml + cid005890: + type: formal_theorem + title: sector_product_depth_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0625 + current_version: v1 + source_path: items/formal-theorems/cid005890-fth0625.yaml + cid005891: + type: formal_theorem + title: enrichment_strict (theorem) + status: canonical + visibility: public + primary_alias: FTH0626 + current_version: v1 + source_path: items/formal-theorems/cid005891-fth0626.yaml + cid005892: + type: formal_theorem + title: iota_value (theorem) + status: canonical + visibility: public + primary_alias: FTH0627 + current_version: v1 + source_path: items/formal-theorems/cid005892-fth0627.yaml + cid005893: + type: formal_theorem + title: iota_lt_one (theorem) + status: canonical + visibility: public + primary_alias: FTH0628 + current_version: v1 + source_path: items/formal-theorems/cid005893-fth0628.yaml + cid005894: + type: formal_theorem + title: iota_pos (theorem) + status: canonical + visibility: public + primary_alias: FTH0629 + current_version: v1 + source_path: items/formal-theorems/cid005894-fth0629.yaml + cid005895: + type: formal_theorem + title: iota_ratio_depth_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0630 + current_version: v1 + source_path: items/formal-theorems/cid005895-fth0630.yaml + cid005896: + type: formal_theorem + title: iota_ratio_depth_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0631 + current_version: v1 + source_path: items/formal-theorems/cid005896-fth0631.yaml + cid005897: + type: formal_theorem + title: five_sectors_exhaustive (theorem) + status: canonical + visibility: public + primary_alias: FTH0632 + current_version: v1 + source_path: items/formal-theorems/cid005897-fth0632.yaml + cid005898: + type: formal_theorem + title: no_knobs_witness (theorem) + status: canonical + visibility: public + primary_alias: FTH0633 + current_version: v1 + source_path: items/formal-theorems/cid005898-fth0633.yaml + cid005899: + type: formal_theorem + title: hinge_point (theorem) + status: canonical + visibility: public + primary_alias: FTH0634 + current_version: v1 + source_path: items/formal-theorems/cid005899-fth0634.yaml + cid005900: + type: formal_theorem + title: fixed_point_check_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0635 + current_version: v1 + source_path: items/formal-theorems/cid005900-fth0635.yaml + cid005901: + type: formal_theorem + title: e3_density_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0636 + current_version: v1 + source_path: items/formal-theorems/cid005901-fth0636.yaml + cid005902: + type: formal_theorem + title: paradox_absorbed_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0637 + current_version: v1 + source_path: items/formal-theorems/cid005902-fth0637.yaml + cid005903: + type: formal_theorem + title: self_model_complete_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0638 + current_version: v1 + source_path: items/formal-theorems/cid005903-fth0638.yaml + cid005904: + type: formal_theorem + title: saturation_semantic_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0639 + current_version: v1 + source_path: items/formal-theorems/cid005904-fth0639.yaml + cid005905: + type: formal_theorem + title: self_model_idempotent_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0640 + current_version: v1 + source_path: items/formal-theorems/cid005905-fth0640.yaml + cid005906: + type: formal_theorem + title: proof_theory_e3_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0641 + current_version: v1 + source_path: items/formal-theorems/cid005906-fth0641.yaml + cid005907: + type: formal_theorem + title: self_model_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0642 + current_version: v1 + source_path: items/formal-theorems/cid005907-fth0642.yaml + cid005908: + type: formal_theorem + title: self_model_inv_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0643 + current_version: v1 + source_path: items/formal-theorems/cid005908-fth0643.yaml + cid005909: + type: formal_theorem + title: four_paradox_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0644 + current_version: v1 + source_path: items/formal-theorems/cid005909-fth0644.yaml + cid005910: + type: formal_theorem + title: forbidden_moves_distinct_thm (theorem) + status: canonical + visibility: public + primary_alias: FTH0645 + current_version: v1 + source_path: items/formal-theorems/cid005910-fth0645.yaml + cid005911: + type: formal_theorem + title: paradox_resolution_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0646 + current_version: v1 + source_path: items/formal-theorems/cid005911-fth0646.yaml + cid005912: + type: formal_theorem + title: e3_is_proof_theory (theorem) + status: canonical + visibility: public + primary_alias: FTH0647 + current_version: v1 + source_path: items/formal-theorems/cid005912-fth0647.yaml + cid005913: + type: formal_theorem + title: self_model_levels (theorem) + status: canonical + visibility: public + primary_alias: FTH0648 + current_version: v1 + source_path: items/formal-theorems/cid005913-fth0648.yaml + cid005914: + type: formal_theorem + title: all_paradoxes_at_e2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0649 + current_version: v1 + source_path: items/formal-theorems/cid005914-fth0649.yaml + cid005915: + type: formal_theorem + title: all_paradoxes_resolve_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0650 + current_version: v1 + source_path: items/formal-theorems/cid005915-fth0650.yaml + cid005916: + type: formal_theorem + title: exactly_four_paradoxes (theorem) + status: canonical + visibility: public + primary_alias: FTH0651 + current_version: v1 + source_path: items/formal-theorems/cid005916-fth0651.yaml + cid005917: + type: formal_theorem + title: forbidden_move_range (theorem) + status: canonical + visibility: public + primary_alias: FTH0652 + current_version: v1 + source_path: items/formal-theorems/cid005917-fth0652.yaml + cid005918: + type: formal_theorem + title: paradox_move_injective (theorem) + status: canonical + visibility: public + primary_alias: FTH0653 + current_version: v1 + source_path: items/formal-theorems/cid005918-fth0653.yaml + cid005919: + type: formal_theorem + title: paradox_gap (theorem) + status: canonical + visibility: public + primary_alias: FTH0654 + current_version: v1 + source_path: items/formal-theorems/cid005919-fth0654.yaml + cid005920: + type: formal_theorem + title: applied_saturation_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0655 + current_version: v1 + source_path: items/formal-theorems/cid005920-fth0655.yaml + cid005921: + type: formal_theorem + title: full_saturation_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0656 + current_version: v1 + source_path: items/formal-theorems/cid005921-fth0656.yaml + cid005922: + type: formal_theorem + title: terminal_level_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0657 + current_version: v1 + source_path: items/formal-theorems/cid005922-fth0657.yaml + cid005923: + type: formal_theorem + title: e3_is_terminal (theorem) + status: canonical + visibility: public + primary_alias: FTH0658 + current_version: v1 + source_path: items/formal-theorems/cid005923-fth0658.yaml + cid005924: + type: formal_theorem + title: e3_is_max (theorem) + status: canonical + visibility: public + primary_alias: FTH0659 + current_version: v1 + source_path: items/formal-theorems/cid005924-fth0659.yaml + cid005925: + type: formal_theorem + title: e3_succ_nat (theorem) + status: canonical + visibility: public + primary_alias: FTH0660 + current_version: v1 + source_path: items/formal-theorems/cid005925-fth0660.yaml + cid005926: + type: formal_theorem + title: e3_unique_fixpoint (theorem) + status: canonical + visibility: public + primary_alias: FTH0661 + current_version: v1 + source_path: items/formal-theorems/cid005926-fth0661.yaml + cid005927: + type: formal_theorem + title: strictly_increasing_nat (theorem) + status: canonical + visibility: public + primary_alias: FTH0662 + current_version: v1 + source_path: items/formal-theorems/cid005927-fth0662.yaml + cid005928: + type: formal_theorem + title: fluid_data_depth_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0663 + current_version: v1 + source_path: items/formal-theorems/cid005928-fth0663.yaml + cid005929: + type: formal_theorem + title: cylinder_assign_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0664 + current_version: v1 + source_path: items/formal-theorems/cid005929-fth0664.yaml + cid005930: + type: formal_theorem + title: abcd_zero_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0665 + current_version: v1 + source_path: items/formal-theorems/cid005930-fth0665.yaml + cid005931: + type: formal_theorem + title: defect_zero_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0666 + current_version: v1 + source_path: items/formal-theorems/cid005931-fth0666.yaml + cid005932: + type: formal_theorem + title: gap_pos_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0667 + current_version: v1 + source_path: items/formal-theorems/cid005932-fth0667.yaml + cid005933: + type: formal_theorem + title: gap_grows_3_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0668 + current_version: v1 + source_path: items/formal-theorems/cid005933-fth0668.yaml + cid005934: + type: formal_theorem + title: gap_constant_is_gap (theorem) + status: canonical + visibility: public + primary_alias: FTH0669 + current_version: v1 + source_path: items/formal-theorems/cid005934-fth0669.yaml + cid005935: + type: formal_theorem + title: ym_gap_is_tau_gap_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0670 + current_version: v1 + source_path: items/formal-theorems/cid005935-fth0670.yaml + cid005936: + type: formal_theorem + title: ym_coupling_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0671 + current_version: v1 + source_path: items/formal-theorems/cid005936-fth0671.yaml + cid005937: + type: formal_theorem + title: polarity_swap_involutive (theorem) + status: canonical + visibility: public + primary_alias: FTH0672 + current_version: v1 + source_path: items/formal-theorems/cid005937-fth0672.yaml + cid005938: + type: formal_theorem + title: flow_depth_0 (theorem) + status: canonical + visibility: public + primary_alias: FTH0673 + current_version: v1 + source_path: items/formal-theorems/cid005938-fth0673.yaml + cid005939: + type: formal_theorem + title: swap_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0674 + current_version: v1 + source_path: items/formal-theorems/cid005939-fth0674.yaml + cid005940: + type: formal_theorem + title: flow_stable_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0675 + current_version: v1 + source_path: items/formal-theorems/cid005940-fth0675.yaml + cid005941: + type: formal_theorem + title: zero_is_sigma_fixed (theorem) + status: canonical + visibility: public + primary_alias: FTH0676 + current_version: v1 + source_path: items/formal-theorems/cid005941-fth0676.yaml + cid005942: + type: formal_theorem + title: sigma_fixed_swap_id (theorem) + status: canonical + visibility: public + primary_alias: FTH0677 + current_version: v1 + source_path: items/formal-theorems/cid005942-fth0677.yaml + cid005943: + type: formal_theorem + title: addr_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0678 + current_version: v1 + source_path: items/formal-theorems/cid005943-fth0678.yaml + cid005944: + type: formal_theorem + title: nf_addr_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0679 + current_version: v1 + source_path: items/formal-theorems/cid005944-fth0679.yaml + cid005945: + type: formal_theorem + title: ns_at_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0680 + current_version: v1 + source_path: items/formal-theorems/cid005945-fth0680.yaml + cid005946: + type: formal_theorem + title: ym_at_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0681 + current_version: v1 + source_path: items/formal-theorems/cid005946-fth0681.yaml + cid005947: + type: formal_theorem + title: hodge_at_e1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0682 + current_version: v1 + source_path: items/formal-theorems/cid005947-fth0682.yaml + cid005948: + type: formal_theorem + title: ns_part5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0683 + current_version: v1 + source_path: items/formal-theorems/cid005948-fth0683.yaml + cid005949: + type: formal_theorem + title: three_e1_problems (theorem) + status: canonical + visibility: public + primary_alias: FTH0684 + current_version: v1 + source_path: items/formal-theorems/cid005949-fth0684.yaml + cid005950: + type: formal_theorem + title: pos_reg_10_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0685 + current_version: v1 + source_path: items/formal-theorems/cid005950-fth0685.yaml + cid005951: + type: formal_theorem + title: stab_germ_0_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0686 + current_version: v1 + source_path: items/formal-theorems/cid005951-fth0686.yaml + cid005952: + type: formal_theorem + title: defect_1_is_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0687 + current_version: v1 + source_path: items/formal-theorems/cid005952-fth0687.yaml + cid005953: + type: formal_theorem + title: stable_at_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0688 + current_version: v1 + source_path: items/formal-theorems/cid005953-fth0688.yaml + cid005954: + type: formal_theorem + title: strong_at_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0689 + current_version: v1 + source_path: items/formal-theorems/cid005954-fth0689.yaml + cid005955: + type: formal_theorem + title: gauge_3_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0690 + current_version: v1 + source_path: items/formal-theorems/cid005955-fth0690.yaml + cid005956: + type: formal_theorem + title: gauge_3_count (theorem) + status: canonical + visibility: public + primary_alias: FTH0691 + current_version: v1 + source_path: items/formal-theorems/cid005956-fth0691.yaml + cid005957: + type: formal_theorem + title: nf_zero_unique (theorem) + status: canonical + visibility: public + primary_alias: FTH0692 + current_version: v1 + source_path: items/formal-theorems/cid005957-fth0692.yaml + cid005958: + type: formal_theorem + title: hartogs_bulk_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0693 + current_version: v1 + source_path: items/formal-theorems/cid005958-fth0693.yaml + cid005959: + type: formal_theorem + title: e1_gluing_8_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0694 + current_version: v1 + source_path: items/formal-theorems/cid005959-fth0694.yaml + cid005960: + type: formal_theorem + title: book3_entry_8_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0695 + current_version: v1 + source_path: items/formal-theorems/cid005960-fth0695.yaml + cid005961: + type: formal_theorem + title: bulk_tower_coherent (theorem) + status: canonical + visibility: public + primary_alias: FTH0696 + current_version: v1 + source_path: items/formal-theorems/cid005961-fth0696.yaml + cid005962: + type: formal_theorem + title: bulk_reduce_stable (theorem) + status: canonical + visibility: public + primary_alias: FTH0697 + current_version: v1 + source_path: items/formal-theorems/cid005962-fth0697.yaml + cid005963: + type: formal_theorem + title: boundary_char_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0698 + current_version: v1 + source_path: items/formal-theorems/cid005963-fth0698.yaml + cid005964: + type: formal_theorem + title: bti_functor_5_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0699 + current_version: v1 + source_path: items/formal-theorems/cid005964-fth0699.yaml + cid005965: + type: formal_theorem + title: bti_additive_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0700 + current_version: v1 + source_path: items/formal-theorems/cid005965-fth0700.yaml + cid005966: + type: formal_theorem + title: zero_char_eval (theorem) + status: canonical + visibility: public + primary_alias: FTH0701 + current_version: v1 + source_path: items/formal-theorems/cid005966-fth0701.yaml + cid005967: + type: formal_theorem + title: neg_neg_char (theorem) + status: canonical + visibility: public + primary_alias: FTH0702 + current_version: v1 + source_path: items/formal-theorems/cid005967-fth0702.yaml + cid005968: + type: formal_theorem + title: bti_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0703 + current_version: v1 + source_path: items/formal-theorems/cid005968-fth0703.yaml + cid005969: + type: formal_theorem + title: sector_decomp_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0704 + current_version: v1 + source_path: items/formal-theorems/cid005969-fth0704.yaml + cid005970: + type: formal_theorem + title: omega_coupling_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0705 + current_version: v1 + source_path: items/formal-theorems/cid005970-fth0705.yaml + cid005971: + type: formal_theorem + title: sector_preservation_5_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0706 + current_version: v1 + source_path: items/formal-theorems/cid005971-fth0706.yaml + cid005972: + type: formal_theorem + title: zero_in_D (theorem) + status: canonical + visibility: public + primary_alias: FTH0707 + current_version: v1 + source_path: items/formal-theorems/cid005972-fth0707.yaml + cid005973: + type: formal_theorem + title: pure_m_in_B (theorem) + status: canonical + visibility: public + primary_alias: FTH0708 + current_version: v1 + source_path: items/formal-theorems/cid005973-fth0708.yaml + cid005974: + type: formal_theorem + title: pure_n_in_C (theorem) + status: canonical + visibility: public + primary_alias: FTH0709 + current_version: v1 + source_path: items/formal-theorems/cid005974-fth0709.yaml + cid005975: + type: formal_theorem + title: equal_mag_in_A (theorem) + status: canonical + visibility: public + primary_alias: FTH0710 + current_version: v1 + source_path: items/formal-theorems/cid005975-fth0710.yaml + cid005976: + type: formal_theorem + title: mixed_in_Omega (theorem) + status: canonical + visibility: public + primary_alias: FTH0711 + current_version: v1 + source_path: items/formal-theorems/cid005976-fth0711.yaml + cid005977: + type: formal_theorem + title: sector_exhaustive (theorem) + status: canonical + visibility: public + primary_alias: FTH0712 + current_version: v1 + source_path: items/formal-theorems/cid005977-fth0712.yaml + cid005978: + type: formal_theorem + title: langlands_reflection_5_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0713 + current_version: v1 + source_path: items/formal-theorems/cid005978-fth0713.yaml + cid005979: + type: formal_theorem + title: template_invariance_8_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0714 + current_version: v1 + source_path: items/formal-theorems/cid005979-fth0714.yaml + cid005980: + type: formal_theorem + title: universal_op_3_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0715 + current_version: v1 + source_path: items/formal-theorems/cid005980-fth0715.yaml + cid005981: + type: formal_theorem + title: spectral_polarity_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0716 + current_version: v1 + source_path: items/formal-theorems/cid005981-fth0716.yaml + cid005982: + type: formal_theorem + title: balanced_uniqueness_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0717 + current_version: v1 + source_path: items/formal-theorems/cid005982-fth0717.yaml + cid005983: + type: formal_theorem + title: sector_count_preserved (theorem) + status: canonical + visibility: public + primary_alias: FTH0718 + current_version: v1 + source_path: items/formal-theorems/cid005983-fth0718.yaml + cid005984: + type: formal_theorem + title: template_invariance_e3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0719 + current_version: v1 + source_path: items/formal-theorems/cid005984-fth0719.yaml + cid005985: + type: formal_theorem + title: d_polarity_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0720 + current_version: v1 + source_path: items/formal-theorems/cid005985-fth0720.yaml + cid005986: + type: formal_theorem + title: a_balanced_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0721 + current_version: v1 + source_path: items/formal-theorems/cid005986-fth0721.yaml + cid005987: + type: formal_theorem + title: parity_bridge_5_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0722 + current_version: v1 + source_path: items/formal-theorems/cid005987-fth0722.yaml + cid005988: + type: formal_theorem + title: coupling_ledger_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0723 + current_version: v1 + source_path: items/formal-theorems/cid005988-fth0723.yaml + cid005989: + type: formal_theorem + title: no_knobs_5_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0724 + current_version: v1 + source_path: items/formal-theorems/cid005989-fth0724.yaml + cid005990: + type: formal_theorem + title: a_is_balanced (theorem) + status: canonical + visibility: public + primary_alias: FTH0725 + current_version: v1 + source_path: items/formal-theorems/cid005990-fth0725.yaml + cid005991: + type: formal_theorem + title: b_not_a (theorem) + status: canonical + visibility: public + primary_alias: FTH0726 + current_version: v1 + source_path: items/formal-theorems/cid005991-fth0726.yaml + cid005992: + type: formal_theorem + title: d_self_coupling_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0727 + current_version: v1 + source_path: items/formal-theorems/cid005992-fth0727.yaml + cid005993: + type: formal_theorem + title: coupling_count (theorem) + status: canonical + visibility: public + primary_alias: FTH0728 + current_version: v1 + source_path: items/formal-theorems/cid005993-fth0728.yaml + cid005994: + type: formal_theorem + title: goldbach_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0729 + current_version: v1 + source_path: items/formal-theorems/cid005994-fth0729.yaml + cid005995: + type: formal_theorem + title: goldbach_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0730 + current_version: v1 + source_path: items/formal-theorems/cid005995-fth0730.yaml + cid005996: + type: formal_theorem + title: twin_primes_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0731 + current_version: v1 + source_path: items/formal-theorems/cid005996-fth0731.yaml + cid005997: + type: formal_theorem + title: goldbach_primorial_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0732 + current_version: v1 + source_path: items/formal-theorems/cid005997-fth0732.yaml + cid005998: + type: formal_theorem + title: twin_prime_density_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0733 + current_version: v1 + source_path: items/formal-theorems/cid005998-fth0733.yaml + cid005999: + type: formal_theorem + title: additive_multiplicative_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0734 + current_version: v1 + source_path: items/formal-theorems/cid005999-fth0734.yaml + cid006000: + type: formal_theorem + title: adele_ring_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0735 + current_version: v1 + source_path: items/formal-theorems/cid006000-fth0735.yaml + cid006001: + type: formal_theorem + title: adelic_embedding_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0736 + current_version: v1 + source_path: items/formal-theorems/cid006001-fth0736.yaml + cid006002: + type: formal_theorem + title: adelic_dense_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0737 + current_version: v1 + source_path: items/formal-theorems/cid006002-fth0737.yaml + cid006003: + type: formal_theorem + title: euler_product_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0738 + current_version: v1 + source_path: items/formal-theorems/cid006003-fth0738.yaml + cid006004: + type: formal_theorem + title: local_factor_ind_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0739 + current_version: v1 + source_path: items/formal-theorems/cid006004-fth0739.yaml + cid006005: + type: formal_theorem + title: adele_zero_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0740 + current_version: v1 + source_path: items/formal-theorems/cid006005-fth0740.yaml + cid006006: + type: formal_theorem + title: adele_is_crt (theorem) + status: canonical + visibility: public + primary_alias: FTH0741 + current_version: v1 + source_path: items/formal-theorems/cid006006-fth0741.yaml + cid006007: + type: formal_theorem + title: adele_injective_1_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0742 + current_version: v1 + source_path: items/formal-theorems/cid006007-fth0742.yaml + cid006008: + type: formal_theorem + title: classifier_20 (theorem) + status: canonical + visibility: public + primary_alias: FTH0743 + current_version: v1 + source_path: items/formal-theorems/cid006008-fth0743.yaml + cid006009: + type: formal_theorem + title: label_conv_20 (theorem) + status: canonical + visibility: public + primary_alias: FTH0744 + current_version: v1 + source_path: items/formal-theorems/cid006009-fth0744.yaml + cid006010: + type: formal_theorem + title: bc_balance_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0745 + current_version: v1 + source_path: items/formal-theorems/cid006010-fth0745.yaml + cid006011: + type: formal_theorem + title: label_idem_10_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0746 + current_version: v1 + source_path: items/formal-theorems/cid006011-fth0746.yaml + cid006012: + type: formal_theorem + title: split_label_10_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0747 + current_version: v1 + source_path: items/formal-theorems/cid006012-fth0747.yaml + cid006013: + type: formal_theorem + title: two_is_x (theorem) + status: canonical + visibility: public + primary_alias: FTH0748 + current_version: v1 + source_path: items/formal-theorems/cid006013-fth0748.yaml + cid006014: + type: formal_theorem + title: five_is_b (theorem) + status: canonical + visibility: public + primary_alias: FTH0749 + current_version: v1 + source_path: items/formal-theorems/cid006014-fth0749.yaml + cid006015: + type: formal_theorem + title: three_is_c (theorem) + status: canonical + visibility: public + primary_alias: FTH0750 + current_version: v1 + source_path: items/formal-theorems/cid006015-fth0750.yaml + cid006016: + type: formal_theorem + title: bc_exist_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0751 + current_version: v1 + source_path: items/formal-theorems/cid006016-fth0751.yaml + cid006017: + type: formal_theorem + title: crt_spectral_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0752 + current_version: v1 + source_path: items/formal-theorems/cid006017-fth0752.yaml + cid006018: + type: formal_theorem + title: crt_add_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0753 + current_version: v1 + source_path: items/formal-theorems/cid006018-fth0753.yaml + cid006019: + type: formal_theorem + title: reconstruction_20_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0754 + current_version: v1 + source_path: items/formal-theorems/cid006019-fth0754.yaml + cid006020: + type: formal_theorem + title: prime_independence_5_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0755 + current_version: v1 + source_path: items/formal-theorems/cid006020-fth0755.yaml + cid006021: + type: formal_theorem + title: crt_roundtrip_42 (theorem) + status: canonical + visibility: public + primary_alias: FTH0756 + current_version: v1 + source_path: items/formal-theorems/cid006021-fth0756.yaml + cid006022: + type: formal_theorem + title: crt_decompose_42 (theorem) + status: canonical + visibility: public + primary_alias: FTH0757 + current_version: v1 + source_path: items/formal-theorems/cid006022-fth0757.yaml + cid006023: + type: formal_theorem + title: crt_basis_0_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0758 + current_version: v1 + source_path: items/formal-theorems/cid006023-fth0758.yaml + cid006024: + type: formal_theorem + title: bridge_algebraic_identity (theorem) + status: canonical + visibility: public + primary_alias: FTH0759 + current_version: v1 + source_path: items/formal-theorems/cid006024-fth0759.yaml + cid006025: + type: formal_theorem + title: bridge_reduces_to_E6_near_identity (theorem) + status: canonical + visibility: public + primary_alias: FTH0760 + current_version: v1 + source_path: items/formal-theorems/cid006025-fth0760.yaml + cid006026: + type: formal_theorem + title: confinement_bridge_lower (theorem) + status: canonical + visibility: public + primary_alias: FTH0761 + current_version: v1 + source_path: items/formal-theorems/cid006026-fth0761.yaml + cid006027: + type: formal_theorem + title: confinement_bridge (theorem) + status: canonical + visibility: public + primary_alias: FTH0762 + current_version: v1 + source_path: items/formal-theorems/cid006027-fth0762.yaml + cid006028: + type: formal_theorem + title: goldbach_500 (theorem) + status: canonical + visibility: public + primary_alias: FTH0763 + current_version: v1 + source_path: items/formal-theorems/cid006028-fth0763.yaml + cid006029: + type: formal_theorem + title: goldbach_primorial_m4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0764 + current_version: v1 + source_path: items/formal-theorems/cid006029-fth0764.yaml + cid006030: + type: formal_theorem + title: partition_growth_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0765 + current_version: v1 + source_path: items/formal-theorems/cid006030-fth0765.yaml + cid006031: + type: formal_theorem + title: obstruction_bounded_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0766 + current_version: v1 + source_path: items/formal-theorems/cid006031-fth0766.yaml + cid006032: + type: formal_theorem + title: crt_goldbach_duality_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0767 + current_version: v1 + source_path: items/formal-theorems/cid006032-fth0767.yaml + cid006033: + type: formal_theorem + title: partition_m2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0768 + current_version: v1 + source_path: items/formal-theorems/cid006033-fth0768.yaml + cid006034: + type: formal_theorem + title: partition_m3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0769 + current_version: v1 + source_path: items/formal-theorems/cid006034-fth0769.yaml + cid006035: + type: formal_theorem + title: partition_m4_pos (theorem) + status: canonical + visibility: public + primary_alias: FTH0770 + current_version: v1 + source_path: items/formal-theorems/cid006035-fth0770.yaml + cid006036: + type: formal_theorem + title: obstruction_100_p2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0771 + current_version: v1 + source_path: items/formal-theorems/cid006036-fth0771.yaml + cid006037: + type: formal_theorem + title: obstruction_100_p3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0772 + current_version: v1 + source_path: items/formal-theorems/cid006037-fth0772.yaml + cid006038: + type: formal_theorem + title: hensel_20 (theorem) + status: canonical + visibility: public + primary_alias: FTH0773 + current_version: v1 + source_path: items/formal-theorems/cid006038-fth0773.yaml + cid006039: + type: formal_theorem + title: hensel_unique_11 (theorem) + status: canonical + visibility: public + primary_alias: FTH0774 + current_version: v1 + source_path: items/formal-theorems/cid006039-fth0774.yaml + cid006040: + type: formal_theorem + title: hensel_tower_11 (theorem) + status: canonical + visibility: public + primary_alias: FTH0775 + current_version: v1 + source_path: items/formal-theorems/cid006040-fth0775.yaml + cid006041: + type: formal_theorem + title: one_is_root_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0776 + current_version: v1 + source_path: items/formal-theorems/cid006041-fth0776.yaml + cid006042: + type: formal_theorem + title: hensel_1_mod_9 (theorem) + status: canonical + visibility: public + primary_alias: FTH0777 + current_version: v1 + source_path: items/formal-theorems/cid006042-fth0777.yaml + cid006043: + type: formal_theorem + title: hensel_correct_3_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0778 + current_version: v1 + source_path: items/formal-theorems/cid006043-fth0778.yaml + cid006044: + type: formal_theorem + title: local_field_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0779 + current_version: v1 + source_path: items/formal-theorems/cid006044-fth0779.yaml + cid006045: + type: formal_theorem + title: local_ring_10_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0780 + current_version: v1 + source_path: items/formal-theorems/cid006045-fth0780.yaml + cid006046: + type: formal_theorem + title: completeness_20_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0781 + current_version: v1 + source_path: items/formal-theorems/cid006046-fth0781.yaml + cid006047: + type: formal_theorem + title: limit_unique_10_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0782 + current_version: v1 + source_path: items/formal-theorems/cid006047-fth0782.yaml + cid006048: + type: formal_theorem + title: val_p_is_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0783 + current_version: v1 + source_path: items/formal-theorems/cid006048-fth0783.yaml + cid006049: + type: formal_theorem + title: val_p2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0784 + current_version: v1 + source_path: items/formal-theorems/cid006049-fth0784.yaml + cid006050: + type: formal_theorem + title: val_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0785 + current_version: v1 + source_path: items/formal-theorems/cid006050-fth0785.yaml + cid006051: + type: formal_theorem + title: tower_42_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0786 + current_version: v1 + source_path: items/formal-theorems/cid006051-fth0786.yaml + cid006052: + type: formal_theorem + title: E4_iota4_near_one_lower (theorem) + status: canonical + visibility: public + primary_alias: FTH0787 + current_version: v1 + source_path: items/formal-theorems/cid006052-fth0787.yaml + cid006053: + type: formal_theorem + title: E4_iota4_near_one (theorem) + status: canonical + visibility: public + primary_alias: FTH0788 + current_version: v1 + source_path: items/formal-theorems/cid006053-fth0788.yaml + cid006054: + type: formal_theorem + title: E6_abs_in_range (theorem) + status: canonical + visibility: public + primary_alias: FTH0789 + current_version: v1 + source_path: items/formal-theorems/cid006054-fth0789.yaml + cid006055: + type: formal_theorem + title: E6_iota6_near_one (theorem) + status: canonical + visibility: public + primary_alias: FTH0790 + current_version: v1 + source_path: items/formal-theorems/cid006055-fth0790.yaml + cid006056: + type: formal_theorem + title: primorial_ladder_8 (theorem) + status: canonical + visibility: public + primary_alias: FTH0791 + current_version: v1 + source_path: items/formal-theorems/cid006056-fth0791.yaml + cid006057: + type: formal_theorem + title: primorial_div_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0792 + current_version: v1 + source_path: items/formal-theorems/cid006057-fth0792.yaml + cid006058: + type: formal_theorem + title: primorial_growth_6 (theorem) + status: canonical + visibility: public + primary_alias: FTH0793 + current_version: v1 + source_path: items/formal-theorems/cid006058-fth0793.yaml + cid006059: + type: formal_theorem + title: primorial_cofinal_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0794 + current_version: v1 + source_path: items/formal-theorems/cid006059-fth0794.yaml + cid006060: + type: formal_theorem + title: prime_cofinal_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0795 + current_version: v1 + source_path: items/formal-theorems/cid006060-fth0795.yaml + cid006061: + type: formal_theorem + title: primorial_zero (theorem) + status: canonical + visibility: public + primary_alias: FTH0796 + current_version: v1 + source_path: items/formal-theorems/cid006061-fth0796.yaml + cid006062: + type: formal_theorem + title: primorial_one (theorem) + status: canonical + visibility: public + primary_alias: FTH0797 + current_version: v1 + source_path: items/formal-theorems/cid006062-fth0797.yaml + cid006063: + type: formal_theorem + title: primorial_three (theorem) + status: canonical + visibility: public + primary_alias: FTH0798 + current_version: v1 + source_path: items/formal-theorems/cid006063-fth0798.yaml + cid006064: + type: formal_theorem + title: cofinal_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0799 + current_version: v1 + source_path: items/formal-theorems/cid006064-fth0799.yaml + cid006065: + type: formal_theorem + title: reduce_coherence_42 (theorem) + status: canonical + visibility: public + primary_alias: FTH0800 + current_version: v1 + source_path: items/formal-theorems/cid006065-fth0800.yaml + cid006066: + type: formal_theorem + title: sieve_correct_50 (theorem) + status: canonical + visibility: public + primary_alias: FTH0801 + current_version: v1 + source_path: items/formal-theorems/cid006066-fth0801.yaml + cid006067: + type: formal_theorem + title: sieve_correct_200 (theorem) + status: canonical + visibility: public + primary_alias: FTH0802 + current_version: v1 + source_path: items/formal-theorems/cid006067-fth0802.yaml + cid006068: + type: formal_theorem + title: sieve_count_known (theorem) + status: canonical + visibility: public + primary_alias: FTH0803 + current_version: v1 + source_path: items/formal-theorems/cid006068-fth0803.yaml + cid006069: + type: formal_theorem + title: sieve_tower_compat_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0804 + current_version: v1 + source_path: items/formal-theorems/cid006069-fth0804.yaml + cid006070: + type: formal_theorem + title: brun_euler_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0805 + current_version: v1 + source_path: items/formal-theorems/cid006070-fth0805.yaml + cid006071: + type: formal_theorem + title: sieve_crt_compat_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0806 + current_version: v1 + source_path: items/formal-theorems/cid006071-fth0806.yaml + cid006072: + type: formal_theorem + title: pi_10 (theorem) + status: canonical + visibility: public + primary_alias: FTH0807 + current_version: v1 + source_path: items/formal-theorems/cid006072-fth0807.yaml + cid006073: + type: formal_theorem + title: pi_30 (theorem) + status: canonical + visibility: public + primary_alias: FTH0808 + current_version: v1 + source_path: items/formal-theorems/cid006073-fth0808.yaml + cid006074: + type: formal_theorem + title: pi_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0809 + current_version: v1 + source_path: items/formal-theorems/cid006074-fth0809.yaml + cid006075: + type: formal_theorem + title: brun_30_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0810 + current_version: v1 + source_path: items/formal-theorems/cid006075-fth0810.yaml + cid006076: + type: formal_theorem + title: euler_phi_primorial_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0811 + current_version: v1 + source_path: items/formal-theorems/cid006076-fth0811.yaml + cid006077: + type: formal_theorem + title: euler_phi_primorial_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0812 + current_version: v1 + source_path: items/formal-theorems/cid006077-fth0812.yaml + cid006078: + type: formal_theorem + title: trichotomy_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0813 + current_version: v1 + source_path: items/formal-theorems/cid006078-fth0813.yaml + cid006079: + type: formal_theorem + title: trichotomy_func_15_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0814 + current_version: v1 + source_path: items/formal-theorems/cid006079-fth0814.yaml + cid006080: + type: formal_theorem + title: bnf_15_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0815 + current_version: v1 + source_path: items/formal-theorems/cid006080-fth0815.yaml + cid006081: + type: formal_theorem + title: bnf_unique_10_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0816 + current_version: v1 + source_path: items/formal-theorems/cid006081-fth0816.yaml + cid006082: + type: formal_theorem + title: bc_non_collapse_10_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0817 + current_version: v1 + source_path: items/formal-theorems/cid006082-fth0817.yaml + cid006083: + type: formal_theorem + title: bc_coprime_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0818 + current_version: v1 + source_path: items/formal-theorems/cid006083-fth0818.yaml + cid006084: + type: formal_theorem + title: trichotomy_depth_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0819 + current_version: v1 + source_path: items/formal-theorems/cid006084-fth0819.yaml + cid006085: + type: formal_theorem + title: bnf_zero_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0820 + current_version: v1 + source_path: items/formal-theorems/cid006085-fth0820.yaml + cid006086: + type: formal_theorem + title: bc_coprime_at_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0821 + current_version: v1 + source_path: items/formal-theorems/cid006086-fth0821.yaml + cid006087: + type: formal_theorem + title: bc_distinct_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0822 + current_version: v1 + source_path: items/formal-theorems/cid006087-fth0822.yaml + cid006088: + type: formal_theorem + title: twin_primes_500 (theorem) + status: canonical + visibility: public + primary_alias: FTH0823 + current_version: v1 + source_path: items/formal-theorems/cid006088-fth0823.yaml + cid006089: + type: formal_theorem + title: twin_density_primorial_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0824 + current_version: v1 + source_path: items/formal-theorems/cid006089-fth0824.yaml + cid006090: + type: formal_theorem + title: hl_constant_decreasing_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0825 + current_version: v1 + source_path: items/formal-theorems/cid006090-fth0825.yaml + cid006091: + type: formal_theorem + title: crt_admissible_positive_4 (theorem) + status: canonical + visibility: public + primary_alias: FTH0826 + current_version: v1 + source_path: items/formal-theorems/cid006091-fth0826.yaml + cid006092: + type: formal_theorem + title: twin_admissibility_fraction_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0827 + current_version: v1 + source_path: items/formal-theorems/cid006092-fth0827.yaml + cid006093: + type: formal_theorem + title: twin_count_100 (theorem) + status: canonical + visibility: public + primary_alias: FTH0828 + current_version: v1 + source_path: items/formal-theorems/cid006093-fth0828.yaml + cid006094: + type: formal_theorem + title: hl_depth_2 (theorem) + status: canonical + visibility: public + primary_alias: FTH0829 + current_version: v1 + source_path: items/formal-theorems/cid006094-fth0829.yaml + cid006095: + type: formal_theorem + title: twin_admissible_1 (theorem) + status: canonical + visibility: public + primary_alias: FTH0830 + current_version: v1 + source_path: items/formal-theorems/cid006095-fth0830.yaml + cid006096: + type: formal_theorem + title: twin_admissible_3_pos (theorem) + status: canonical + visibility: public + primary_alias: FTH0831 + current_version: v1 + source_path: items/formal-theorems/cid006096-fth0831.yaml + cid006097: + type: formal_theorem + title: admissible_at_3 (theorem) + status: canonical + visibility: public + primary_alias: FTH0832 + current_version: v1 + source_path: items/formal-theorems/cid006097-fth0832.yaml + cid006098: + type: formal_theorem + title: admissible_at_5 (theorem) + status: canonical + visibility: public + primary_alias: FTH0833 + current_version: v1 + source_path: items/formal-theorems/cid006098-fth0833.yaml + cid006099: + type: formal_theorem + title: admissible_at_7 (theorem) + status: canonical + visibility: public + primary_alias: FTH0834 + current_version: v1 + source_path: items/formal-theorems/cid006099-fth0834.yaml + cid006100: + type: formal_theorem + title: width_principle (theorem) + status: canonical + visibility: public + primary_alias: FTH0835 + current_version: v1 + source_path: items/formal-theorems/cid006100-fth0835.yaml + cid006101: + type: formal_theorem + title: chi_plus_admissible (theorem) + status: canonical + visibility: public + primary_alias: FTH0836 + current_version: v1 + source_path: items/formal-theorems/cid006101-fth0836.yaml + cid006102: + type: formal_theorem + title: book_ii_bridge_complete (theorem) + status: canonical + visibility: public + primary_alias: FTH0837 + current_version: v1 + source_path: items/formal-theorems/cid006102-fth0837.yaml + cid006103: + type: formal_theorem + title: spectral_decidable (theorem) + status: canonical + visibility: public + primary_alias: FTH0838 + current_version: v1 + source_path: items/formal-theorems/cid006103-fth0838.yaml + cid006104: + type: formal_theorem + title: tau_complexity_bridge_concrete (theorem) + status: canonical + visibility: public + primary_alias: FTH0839 + current_version: v1 + source_path: items/formal-theorems/cid006104-fth0839.yaml + cid006105: + type: proof + title: Proof of the Hyperfactorization Theorem + status: canonical + visibility: public + primary_alias: PRF0002 + current_version: v1 + source_path: items/proofs/cid006105-prf0002.yaml + cid006106: + type: proof + title: Proof Sketch of Global Hartogs + status: active + visibility: public + primary_alias: PRF0003 + current_version: v1 + source_path: items/proofs/cid006106-prf0003.yaml + cid006107: + type: proof + title: Conditional Proof of Categoricity of τ_0 + status: active + visibility: public + primary_alias: PRF0004 + current_version: v1 + source_path: items/proofs/cid006107-prf0004.yaml + cid006108: + type: proof + title: Lean-Formalized Proof of FTA on τ-Idx + status: canonical + visibility: public + primary_alias: PRF0005 + current_version: v1 + source_path: items/proofs/cid006108-prf0005.yaml + cid006109: + type: proof + title: Prose Proof of Six-Generator Rigidity Failure + status: active + visibility: public + primary_alias: PRF0006 + current_version: v1 + source_path: items/proofs/cid006109-prf0006.yaml diff --git a/_data/corpus_v3/item-types.yaml b/_data/corpus_v3/item-types.yaml new file mode 100644 index 000000000..d95851a45 --- /dev/null +++ b/_data/corpus_v3/item-types.yaml @@ -0,0 +1,303 @@ +# Corpus Item Types — YAML projection of governance/ITEM_TYPES.md +# +# This file is consumed by schemas/corpus-item.schema.json (type enum) and +# by scripts/validate_schema.py (type → folder → payload-class mapping). +# +# Source of truth: governance/ITEM_TYPES.md. When that doc changes, regenerate +# this file via scripts/regenerate_manifests.py (Wave 1). +# +# Charter reference: §3.2 (folder structure), §3.3 (payload doctrine). + +# All 35 first-class types + 7 projection page subtypes. +types: + + # ---------- Conceptual / Scholarly (atomic) ---------- + + - type: definition + prefix: DEF + folder: items/definitions + payload_class: atomic + description: Fixed mathematical or conceptual definition introduced by the program. + + - type: axiom + prefix: AXM + folder: items/axioms + payload_class: atomic + description: Project-local statement assumed without proof; each Lean `axiom` gets a corresponding Corpus AXM. + + - type: lemma + prefix: LEM + folder: items/lemmas + payload_class: atomic + description: Subsidiary result used in proofs; often atomic and load-bearing. + + - type: proposition + prefix: PRP + folder: items/propositions + payload_class: atomic + description: Claim with proof that is not yet (or not centrally) called a theorem. + + - type: theorem + prefix: THM + folder: items/theorems + payload_class: atomic + description: Central proved statement of the program. + + - type: corollary + prefix: COR + folder: items/corollaries + payload_class: atomic + description: Immediate consequence of a theorem. + + - type: proof + prefix: PRF + folder: items/proofs + payload_class: proof + description: Structured Corpus-addressed proof record with conclusion, premises, strategy, steps, dependencies, status. + + - type: construction + prefix: CNS + folder: items/constructions + payload_class: atomic + description: Multi-step procedure assembling part of the τ-machine. + + - type: construction_step + prefix: CNS + folder: items/construction-steps + payload_class: atomic + description: One step inside a construction (e.g., a Construction Spine step S024). + + - type: claim + prefix: CLM + folder: items/claims + payload_class: atomic + description: Scholarly claim not yet proved but propositionally precise. + + - type: bridge_claim + prefix: BRG + folder: items/bridge-claims + payload_class: atomic + description: Correspondence claim between an internal τ-construction and an external/orthodox object. + + - type: readout_claim + prefix: RDO + folder: items/readout-claims + payload_class: atomic + description: Claim about what the τ-framework readouts in a specific empirical or interpretive context. + + - type: prediction + prefix: PRD + folder: items/predictions + payload_class: atomic + description: Testable empirical or quantitative expectation derived from the framework. + + - type: falsifier + prefix: FAL + folder: items/falsifiers + payload_class: atomic + description: Specific condition under which a prediction, result, or claim would be refuted. + + - type: open_problem + prefix: OPN + folder: items/open-problems + payload_class: atomic + description: Problem the program states explicitly as open. + + - type: erratum + prefix: ERR + folder: items/errata + payload_class: atomic + description: Correction to a previously released item. + + - type: result + prefix: RSL + folder: items/results + payload_class: result + description: Reader-facing consequence surface with commentary, scope limits, validation state, inspection route. + + # ---------- Publication / Artifact ---------- + + - type: monograph + prefix: BOK + folder: items/monographs + payload_class: artifact + description: Book in the program (Books I-VII). + + - type: chapter + prefix: CHP + folder: items/chapters + payload_class: artifact + description: Chapter inside a monograph or major artifact. + + - type: research_paper + prefix: PAP + folder: items/papers + payload_class: artifact + description: Standalone research paper. + + - type: research_note + prefix: NOT + folder: items/notes + payload_class: artifact + description: Focused research note (RN-XXX historically). + + - type: research_dossier + prefix: DOS + folder: items/dossiers + payload_class: artifact + description: Extended research dossier (e.g., the Construction Spine). + + - type: release_packet + prefix: REL + folder: items/release-packets + payload_class: artifact + description: Versioned release artifact bundling materials for a public release. + + - type: research_code + prefix: COD + folder: items/code + payload_class: artifact + description: Research code artifact (e.g., a runnable analysis notebook, model implementation). + + - type: proof_package + prefix: PKG + folder: items/proof-packages + payload_class: artifact + description: Formal proof package (Lean project with Lakefile + Mathlib pin + commit hash). + + - type: reproducibility_bundle + prefix: PKG + folder: items/proof-packages + payload_class: artifact + description: Reproducibility bundle for a result or paper. + + - type: doi_record + prefix: PAP + folder: items/pages + payload_class: artifact + description: DOI registration record (artifact-like; small). + + # ---------- Formalization / Code ---------- + + - type: formal_module + prefix: MOD + folder: items/formal-modules + payload_class: formal + description: Lean module (TauLib v2 or v3) carrying formal declarations. + + - type: formal_definition + prefix: FDF + folder: items/formal-definitions + payload_class: formal + description: Lean `def` corresponding (often) to a corpus DEF. + + - type: formal_theorem + prefix: FTH + folder: items/formal-theorems + payload_class: formal + description: "Lean `theorem` corresponding to a corpus THM/LEM/PRP; carries `formalizes: cid_prose_item`." + + - type: formal_lemma + prefix: FLM + folder: items/formal-lemmas + payload_class: formal + description: Lean `lemma`; same shape as formal_theorem with lower citation weight by convention. + + - type: formal_axiom + prefix: FAX + folder: items/formal-axioms + payload_class: formal + description: Lean `axiom`; round-trips to a Corpus AXM item with explanation. + + - type: formal_assumption + prefix: FAX + folder: items/formal-axioms + payload_class: formal + description: Explicit unproved assumption used in a Lean development. + + - type: mathlib_target + prefix: MTG + folder: items/mathlib-targets + payload_class: formal + description: External Mathlib declaration the τ-program targets for integration. + + # ---------- Projection / Page ---------- + + - type: corpus_page + prefix: null + folder: items/pages + payload_class: page + description: A page on the Corpus lane (e.g., construction spine index). + + - type: result_page + prefix: null + folder: items/pages + payload_class: page + description: A page rendering a Results lane filtered view. + + - type: verify_page + prefix: null + folder: items/pages + payload_class: page + description: A page on the Verify lane. + + - type: publication_page + prefix: null + folder: items/pages + payload_class: page + description: A publication-surface page. + + - type: research_graph_page + prefix: null + folder: items/pages + payload_class: page + description: The Research Graph page (doctrine §16). + + - type: about_site_page + prefix: null + folder: items/pages + payload_class: page + description: About-the-site, charter, etc. + + - type: colophon_page + prefix: null + folder: items/pages + payload_class: page + description: Colophon / acknowledgements. + +# Payload class definitions (informational; consumed by per-type schema discriminator). +payload_classes: + atomic: + description: Inline semantic payload (markdown / latex / plain / optional lean). + schema: schemas/per-type/atomic.schema.json + examples: [definition, theorem, lemma, claim, prediction] + + artifact: + description: Points to canonical artifact (URL / DOI / ISBN); carries abstract + citation metadata. + schema: schemas/per-type/artifact.schema.json + examples: [monograph, paper, dossier, release_packet] + + formal: + description: Carries module path, declaration name, source commit, axiom audit, formalizes relation. + schema: schemas/per-type/formal.schema.json + examples: [formal_module, formal_theorem, formal_definition, mathlib_target] + + result: + description: Structured commentary + layers + domains + validation_status + inspection_route. + schema: schemas/per-type/result.schema.json + examples: [result] + + proof: + description: Structured proof block (mode, conclusion, steps, dependencies, formality_level, proof_status). + schema: schemas/per-type/proof.schema.json + examples: [proof] + + page: + description: Minimal projection page (route, purpose, represented items, page-level JSON-LD). + schema: schemas/per-type/page.schema.json + examples: [corpus_page, result_page, verify_page] + +revisions: + - date: 2026-05-10 + change: Initial projection from governance/ITEM_TYPES.md + author: thorfuchs diff --git a/_data/corpus_v3/lifecycle-statuses.yaml b/_data/corpus_v3/lifecycle-statuses.yaml new file mode 100644 index 000000000..ee263766f --- /dev/null +++ b/_data/corpus_v3/lifecycle-statuses.yaml @@ -0,0 +1,117 @@ +# Lifecycle Statuses — YAML projection of governance/LIFECYCLE.md +# +# This file is consumed by schemas/corpus-item.schema.json (status enum) and +# by scripts/validate_schema.py (transition validity, required-fields-per-status). +# +# Source of truth: governance/LIFECYCLE.md §1-§2. + +statuses: + + - status: draft + description: Internal or early working item, not yet proposed. + public_route_active: false + required_visibility: [private, internal] + required_fields_extra: [] + tombstone: false + + - status: candidate + description: Proposed for inclusion, not yet canonical. + public_route_active: false + required_visibility: [private, internal, limited] + required_fields_extra: [] + tombstone: false + + - status: active + description: Current working item used in the Corpus. + public_route_active: true # if visibility is public + required_visibility: [public, internal] + required_fields_extra: [] + tombstone: false + + - status: canonical + description: Stable item in the current Corpus canon. + public_route_active: true + required_visibility: [public] + required_fields_extra: [] + tombstone: false + + - status: deprecated + description: Still addressable but discouraged; often points at a successor. + public_route_active: true + required_visibility: [deprecated_public, internal] + required_fields_extra: [] + tombstone: true + + - status: superseded + description: Replaced by another item that is non-equivalent enough to warrant a new CID. + public_route_active: true + required_visibility: [deprecated_public] + required_fields_extra: [superseded_by] + tombstone: true + + - status: retired + description: No longer active, preserved as historical, no successor. + public_route_active: true + required_visibility: [deprecated_public] + required_fields_extra: [retired_at, retirement_reason] + tombstone: true + + - status: refuted + description: Item (typically a prediction or claim) has been falsified or rejected. + public_route_active: true + required_visibility: [deprecated_public] + required_fields_extra: [refuted_at, refutation_reason] + tombstone: true + + - status: archived + description: Preserved historical item; natural end-state for historical items. + public_route_active: true + required_visibility: [deprecated_public] + required_fields_extra: [] + tombstone: true + + - status: private_deleted + description: Internal draft removed before public release. Only allowed pre-public_at. + public_route_active: false + required_visibility: [] # item is gone + required_fields_extra: [] + tombstone: false + forbidden_after_public_at: true + +# Transition graph (allowed forward transitions only — backward transitions documented +# in governance/LIFECYCLE.md §3.2 with extra constraints). +transitions: + - from: draft + to: [candidate, active, private_deleted] + + - from: candidate + to: [active, draft, private_deleted] + + - from: active + to: [canonical, deprecated, superseded, retired, refuted, archived, candidate] # backward to candidate only pre-public_at + + - from: canonical + to: [deprecated, superseded, retired, refuted, archived] + + - from: deprecated + to: [active] # un-deprecation requires Charter Decision Log entry + + - from: superseded + to: [] # no forward transitions (item is replaced) + + - from: retired + to: [] # no forward transitions + + - from: refuted + to: [] # no forward transitions + + - from: archived + to: [] # no forward transitions + + - from: private_deleted + to: [] # item is gone + +revisions: + - date: 2026-05-10 + change: Initial projection from governance/LIFECYCLE.md + author: thorfuchs diff --git a/_data/corpus_v3/relation-vocabulary.yaml b/_data/corpus_v3/relation-vocabulary.yaml new file mode 100644 index 000000000..c2a9c7ae8 --- /dev/null +++ b/_data/corpus_v3/relation-vocabulary.yaml @@ -0,0 +1,392 @@ +# Relation Vocabulary — YAML projection of doctrine §12 (id-graph-v5) +# +# This file is the canonical list of relation predicates for Corpus Items. +# Each predicate has a class (construction / projection / bridge / version / +# evidence / navigation) and a load_bearing flag (consumed by public-closure +# validator and crossref-public-closure CI gate). +# +# Source of truth: atlas/corpus/v3/panta-rhei-corpus-id-graph-v5-doctrine.md §12 +# + governance/VISIBILITY.md §4.1-§4.2 for load_bearing classification. + +predicates: + + # ---------- Construction Relations (load-bearing) ---------- + + - predicate: depends_on + class: construction + load_bearing: true + inverse: used_by + description: This item depends on the target for its construction or proof. + + - predicate: uses + class: construction + load_bearing: true + inverse: used_by + description: This item makes use of the target (definition, lemma, theorem) in its content. + + - predicate: constructs + class: construction + load_bearing: true + inverse: constructed_by + description: This item constructs the target object/structure. + + - predicate: derives + class: construction + load_bearing: true + inverse: derived_from + description: This item derives the target via inference. + + - predicate: proves + class: construction + load_bearing: true + inverse: proved_by + description: This item is a proof establishing the target theorem/lemma/proposition. + + - predicate: assumes + class: construction + load_bearing: true + inverse: assumed_in + description: This item explicitly assumes the target (axiom, gate, hypothesis). + + - predicate: requires + class: construction + load_bearing: true + inverse: required_by + description: Stronger form of depends_on; the target must be discharged for this item to hold. + + - predicate: extends + class: construction + load_bearing: true + inverse: extended_by + description: This item extends the target with additional structure. + + - predicate: refines + class: construction + load_bearing: true + inverse: refined_by + description: This item refines the target (more precise statement, tighter hypothesis). + + # ---------- Projection Relations ---------- + + - predicate: appears_in + class: projection + load_bearing: false + inverse: contains + description: This item appears in (is rendered by) the target artifact (book, paper, dossier). + + - predicate: published_in + class: projection + load_bearing: false + inverse: publishes + description: This item is published in the target venue/artifact. + + - predicate: formalized_by + class: projection + load_bearing: true + inverse: formalizes + description: This (prose) item has a formalization at the target (formal_theorem, formal_definition). + + - predicate: implemented_by + class: projection + load_bearing: true + inverse: implements + description: This item is implemented by the target (code, model, artifact). + + - predicate: represented_by + class: projection + load_bearing: false + inverse: represents + description: This item is represented by the target page or projection surface. + + - predicate: explained_in + class: projection + load_bearing: false + inverse: explains + description: The target provides exposition/explanation of this item. + + - predicate: summarized_by + class: projection + load_bearing: false + inverse: summarizes + description: The target provides a summary of this item. + + - predicate: summarizes + class: projection + load_bearing: false + inverse: summarized_by + description: This item summarizes the target (inverse of summarized_by; used by research notes summarizing a theorem). + + # ---------- Bridge Relations (load-bearing for bridge claims) ---------- + + - predicate: bridges_to + class: bridge + load_bearing: true + inverse: bridged_from + description: This bridge_claim bridges to the target external/orthodox object. + + - predicate: corresponds_to + class: bridge + load_bearing: true + inverse: corresponds_to # symmetric + description: This item corresponds to the target across a bridge. + + - predicate: translates_to + class: bridge + load_bearing: true + inverse: translated_from + description: This item translates to the target in a target framework/language. + + - predicate: compares_with + class: bridge + load_bearing: false + inverse: compares_with # symmetric + description: This item compares with the target (informational). + + - predicate: orthodox_target + class: bridge + load_bearing: true + inverse: targeted_by + description: Identifies the orthodox counterpart to a τ-side construction. + + - predicate: mathlib_target + class: bridge + load_bearing: true + inverse: targeted_by + description: Identifies the Mathlib declaration this item targets. + + # ---------- Version Relations ---------- + + - predicate: replaces + class: version + load_bearing: false + inverse: replaced_by + description: This item replaces the target (newer version, same identity). + + - predicate: supersedes + class: version + load_bearing: false + inverse: superseded_by + description: This item supersedes the target (non-equivalent successor; new CID). + + - predicate: deprecated_by + class: version + load_bearing: false + inverse: deprecates + description: This item has been deprecated by the target. + + - predicate: retired_in + class: version + load_bearing: false + inverse: retires + description: This item was retired in the target release. + + - predicate: corrected_by + class: version + load_bearing: false + inverse: corrects + description: This item has been corrected by the target erratum. + + - predicate: erratum_for + class: version + load_bearing: false + inverse: has_erratum + description: This erratum applies to the target. + + # ---------- Evidence / Validation Relations ---------- + + - predicate: supports + class: evidence + load_bearing: true + inverse: supported_by + description: This item supports the target (provides evidence/argument for it). + + - predicate: constrains + class: evidence + load_bearing: true + inverse: constrained_by + description: This item constrains the target. + + - predicate: falsifies + class: evidence + load_bearing: false + inverse: falsified_by + description: This item falsifies the target. + + - predicate: tested_by + class: evidence + load_bearing: true + inverse: tests + description: This item is tested by the target (prediction, experiment, validation). + + - predicate: tests + class: evidence + load_bearing: true + inverse: tested_by + description: This item tests the target (inverse of tested_by; used by predictions/falsifiers/experiments). + + - predicate: validated_by + class: evidence + load_bearing: true + inverse: validates + description: This item is validated by the target (e.g., external review, audit, replication). + + - predicate: challenged_by + class: evidence + load_bearing: false + inverse: challenges + description: This item is challenged by the target (informational). + + # ---------- Navigation Relations ---------- + + - predicate: part_of + class: navigation + load_bearing: false + inverse: has_part + description: This item is part of the target (e.g., construction_step part_of construction). + + - predicate: has_part + class: navigation + load_bearing: false + inverse: part_of + description: This item has the target as a part. + + - predicate: next_step + class: navigation + load_bearing: false + inverse: previous_step + description: Pointer to the next step in a sequence (e.g., construction spine). + + - predicate: previous_step + class: navigation + load_bearing: false + inverse: next_step + description: Pointer to the previous step in a sequence. + + - predicate: route + class: navigation + load_bearing: false + inverse: routed_from + description: A routing pointer between items in a multi-step navigation. + + # ---------- Proof-specific Relations (used by proof items, addendum §21) ---------- + + - predicate: uses_definition + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target definition. + + - predicate: uses_lemma + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target lemma. + + - predicate: uses_theorem + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target theorem. + + - predicate: uses_assumption + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target assumption / axiom. + + - predicate: uses_bridge + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target bridge claim. + + - predicate: uses_result + class: construction + load_bearing: true + inverse: used_by_proof + description: Proof step uses the target result. + + - predicate: cites_background + class: navigation + load_bearing: false + inverse: cited_as_background + description: Proof cites background context (not load-bearing). + + - predicate: historical_reference + class: navigation + load_bearing: false + inverse: referenced_historically + description: Reference to a historical item (no current load-bearing role). + + # ---------- Informational / Non-Load-Bearing ---------- + + - predicate: related_to + class: navigation + load_bearing: false + inverse: related_to # symmetric + description: Generic "related to" link; informational only. + + - predicate: future_work + class: navigation + load_bearing: false + inverse: future_work_of + description: Item lists future work direction. + + - predicate: background_context + class: navigation + load_bearing: false + inverse: background_context_for + description: Background-context reference (not load-bearing). + + - predicate: mentions + class: navigation + load_bearing: false + inverse: mentioned_in + description: Item mentions the target (no load-bearing role). + + - predicate: see_also + class: navigation + load_bearing: false + inverse: see_also # symmetric + description: '"See also" pointer.' + +# Relation classes (informational; consumed by validator + per-class queries) +classes: + construction: + description: Constructive/derivational relations (depends_on, uses, constructs, derives, proves, assumes, requires, extends, refines). + enforces_dag: true # load-bearing construction relations must form a DAG + closure_checked: true + + projection: + description: How items project into artifacts and surfaces (appears_in, published_in, formalized_by, implemented_by, represented_by, explained_in, summarized_by). + enforces_dag: false + closure_checked: only_when_load_bearing + + bridge: + description: Cross-framework correspondences (bridges_to, corresponds_to, translates_to, compares_with, orthodox_target, mathlib_target). + enforces_dag: false + closure_checked: only_when_load_bearing + + version: + description: Version/lifecycle relations (replaces, supersedes, deprecated_by, retired_in, corrected_by, erratum_for). + enforces_dag: false + closure_checked: false # tombstone deps allowed + + evidence: + description: Empirical / validation relations (supports, constrains, falsifies, tested_by, validated_by, challenged_by). + enforces_dag: false + closure_checked: only_when_load_bearing + + navigation: + description: Reader-navigation pointers (part_of, has_part, next_step, previous_step, route, related_to, future_work, background_context, mentions, see_also). + enforces_dag: false + closure_checked: false + +# Qualifier shape (per doctrine §12.7) — relations may carry qualifiers like +# dependency_type, required_for, status, etc. Validated separately by relations +# schema, not enumerated here. + +revisions: + - date: 2026-05-10 + change: Initial vocabulary from id-graph-v5-doctrine §12 + crossrefs addendum §21 + author: thorfuchs diff --git a/_data/corpus_v3/transition.yaml b/_data/corpus_v3/transition.yaml new file mode 100644 index 000000000..7f26151de --- /dev/null +++ b/_data/corpus_v3/transition.yaml @@ -0,0 +1,40967 @@ +generated_at: '2026-05-23T19:55:03Z' +generator: scripts/build_transition_manifest.py +doctrine_reference: panta-rhei-corpus-id-graph-v5-doctrine.md §18.3 +total_items: 1922 +total_v2_transitions: 5301 +total_alias_history_records: 7 +transitions: +- old_id: book-ii + new_id: cid000001 + primary_alias: BOK0001 + type: monograph + title: Book II — The τ-Kernel + status: active + release_lines: + - corpus_v3_working +- old_id: book-ii-chapter-5 + new_id: cid000002 + primary_alias: CHP0001 + type: chapter + title: Book II Chapter 5 — Master Constant Calibration + status: active + release_lines: + - corpus_v3_working +- old_id: WP-RC-CS-E0-E3 + new_id: cid000003 + primary_alias: DOS0001 + type: research_dossier + title: The Panta Rhei Construction Spine + status: active + release_lines: &id001 + - 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corpus_v3_working +- old_id: tau-einstein-equation-bridge + new_id: cid000012 + primary_alias: BRG0001 + type: bridge_claim + title: τ-Einstein Equation Bridges to Orthodox GR + status: active + release_lines: + - corpus_v3_working +- old_id: dark-sector-readout + new_id: cid000013 + primary_alias: RDO0001 + type: readout_claim + title: Dark Sector as Carrier Capacity-Gradient Readout + status: active + release_lines: + - corpus_v3_working +- old_id: bridge-functor-exists + new_id: cid000014 + primary_alias: AXM0001 + type: axiom + title: Bridge Functor Exists + status: active + release_lines: + - corpus_v3_working +- old_id: cmb-s4-tensor-to-scalar + new_id: cid000015 + primary_alias: PRD0001 + type: prediction + title: CMB-S4 Tensor-to-Scalar Ratio r ≈ 0.0136 + status: active + release_lines: + - corpus_v3_working +- old_id: cmb-s4-r-falsifier + new_id: cid000016 + primary_alias: FAL0001 + type: falsifier + title: CMB-S4 r Measurement Outside Predicted Band + status: active + release_lines: + - corpus_v3_working +- old_id: taulib-v3-full-axiom-audit + new_id: cid000017 + primary_alias: OPN0001 + type: open_problem + title: TauLib v3 Full Axiom Audit + status: active + release_lines: + - corpus_v3_working +- old_id: book-ii-chapter-5-typo-erratum + new_id: cid000018 + primary_alias: ERR0001 + type: erratum + title: Book II Chapter 5 — Typo in ι_τ Numerical Value + status: canonical + release_lines: + - corpus_v3_working +- old_id: taulib-geometry-central-theorem-calibration + new_id: cid000019 + primary_alias: MOD0001 + type: formal_module + title: TauLib.Geometry.CentralTheorem.Calibration + status: active + release_lines: + - corpus_v3_working +- old_id: iota_tau_calibration + new_id: cid000020 + primary_alias: FTH0001 + type: formal_theorem + title: TauLib.Geometry.CentralTheorem.iota_tau_calibration + status: active + release_lines: + - corpus_v3_working +- old_id: no-independent-dark-sector + new_id: cid000021 + primary_alias: RSL0001 + type: result + title: No Independent Dark-Sector Particle Required in the τ Readout + status: active + release_lines: + - 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corpus_v3_working + - corpus_v2 +- old_id: paraconsistent-character + new_id: cid001150 + primary_alias: PRP0028 + type: proposition + title: Paraconsistent Character + status: canonical + release_lines: *id154 +- old_id: prop:non-boolean + new_id: cid001150 + primary_alias: PRP0028 + type: proposition + title: Paraconsistent Character + status: canonical + release_lines: *id154 +- old_id: I.P28 + new_id: cid001151 + primary_alias: PRP0029 + type: proposition + title: Self-Enrichment + status: canonical + release_lines: &id155 + - corpus_v3_working + - corpus_v2 +- old_id: self-enrichment + new_id: cid001151 + primary_alias: PRP0029 + type: proposition + title: Self-Enrichment + status: canonical + release_lines: *id155 +- old_id: prop:self-enrichment + new_id: cid001151 + primary_alias: PRP0029 + type: proposition + title: Self-Enrichment + status: canonical + release_lines: *id155 +- old_id: I.P29 + new_id: cid001152 + primary_alias: PRP0030 + type: proposition + title: Passage to Book II + status: canonical + release_lines: &id156 + - corpus_v3_working + - corpus_v2 +- old_id: passage-to-book-ii + new_id: cid001152 + primary_alias: PRP0030 + type: proposition + title: Passage to Book II + status: canonical + release_lines: *id156 +- old_id: prop:passage-to-book-ii + new_id: cid001152 + primary_alias: PRP0030 + type: proposition + title: Passage to Book II + status: canonical + release_lines: *id156 +- old_id: I.P33 + new_id: cid001153 + primary_alias: PRP0031 + type: proposition + title: Counting as Structural Feature + status: canonical + release_lines: &id157 + - corpus_v3_working + - corpus_v2 +- old_id: counting-as-structural-feature + new_id: cid001153 + primary_alias: PRP0031 + type: proposition + title: Counting as Structural Feature + status: canonical + release_lines: *id157 +- old_id: prop:counting-structural + new_id: cid001153 + primary_alias: PRP0031 + type: proposition + title: Counting as Structural Feature + status: canonical + release_lines: *id157 +- old_id: I.P34 + new_id: cid001154 + primary_alias: PRP0032 + type: proposition + title: No Unearned Decimal Diagonal + status: canonical + release_lines: &id158 + - corpus_v3_working + - corpus_v2 +- old_id: no-unearned-decimal-diagonal + new_id: cid001154 + primary_alias: PRP0032 + type: proposition + title: No Unearned Decimal Diagonal + status: canonical + release_lines: *id158 +- old_id: prop:no-unearned-decimal + new_id: cid001154 + primary_alias: PRP0032 + type: proposition + title: No Unearned Decimal Diagonal + status: canonical + release_lines: *id158 +- old_id: I.P35 + new_id: cid001155 + primary_alias: PRP0033 + type: proposition + title: No Unrestricted Comprehension + status: canonical + release_lines: &id159 + - corpus_v3_working + - corpus_v2 +- old_id: no-unrestricted-comprehension + new_id: cid001155 + primary_alias: PRP0033 + type: proposition + title: No Unrestricted Comprehension + status: canonical + release_lines: *id159 +- old_id: prop:no-comprehension + new_id: cid001155 + primary_alias: PRP0033 + type: proposition + title: No Unrestricted Comprehension + status: canonical + release_lines: *id159 +- old_id: I.P36 + new_id: cid001156 + primary_alias: PRP0034 + type: proposition + title: No Free Cartesian Diagonal + status: canonical + release_lines: &id160 + - corpus_v3_working + - corpus_v2 +- old_id: no-free-cartesian-diagonal + new_id: cid001156 + primary_alias: PRP0034 + type: proposition + title: No Free Cartesian Diagonal + status: canonical + release_lines: *id160 +- old_id: prop:no-free-cartesian + new_id: cid001156 + primary_alias: PRP0034 + type: proposition + title: No Free Cartesian Diagonal + status: canonical + release_lines: *id160 +- old_id: I.P37 + new_id: cid001157 + primary_alias: PRP0035 + type: proposition + title: Ultrametric Replaces Cardinality + status: canonical + release_lines: &id161 + - corpus_v3_working + - corpus_v2 +- old_id: ultrametric-replaces-cardinality + new_id: cid001157 + primary_alias: PRP0035 + type: proposition + title: Ultrametric Replaces Cardinality + status: canonical + release_lines: *id161 +- old_id: prop:ultrametric-replaces-card + new_id: cid001157 + primary_alias: PRP0035 + type: proposition + title: Ultrametric Replaces Cardinality + status: canonical + release_lines: *id161 +- old_id: I.P38 + new_id: cid001158 + primary_alias: PRP0036 + type: proposition + title: Classical.em Eliminability + status: canonical + release_lines: &id162 + - corpus_v3_working + - corpus_v2 +- old_id: classical-em-eliminability + new_id: cid001158 + primary_alias: PRP0036 + type: proposition + title: Classical.em Eliminability + status: canonical + release_lines: *id162 +- old_id: prop:em-eliminable + new_id: cid001158 + primary_alias: PRP0036 + type: proposition + title: Classical.em Eliminability + status: canonical + release_lines: *id162 +- old_id: I.P39 + new_id: cid001159 + primary_alias: PRP0037 + type: proposition + title: TauReal Ring Axioms + status: canonical + release_lines: &id163 + - corpus_v3_working + - corpus_v2 +- old_id: taureal-ring-axioms + new_id: cid001159 + primary_alias: PRP0037 + type: proposition + title: TauReal Ring Axioms + status: canonical + release_lines: *id163 +- old_id: prop:ordered-field-axioms + new_id: cid001159 + primary_alias: PRP0037 + type: proposition + title: TauReal Ring Axioms + status: canonical + release_lines: *id163 +- old_id: I.P40 + new_id: cid001160 + primary_alias: PRP0038 + type: proposition + title: Extensionality + status: canonical + release_lines: &id164 + - corpus_v3_working + - corpus_v2 +- old_id: extensionality + new_id: cid001160 + primary_alias: PRP0038 + type: proposition + title: Extensionality + status: canonical + release_lines: *id164 +- old_id: prop:orbit-set-extensional + new_id: cid001160 + primary_alias: PRP0038 + type: proposition + title: Extensionality + status: canonical + release_lines: *id164 +- old_id: I.P41 + new_id: cid001161 + primary_alias: PRP0039 + type: proposition + title: Self-Containment Partition + status: canonical + release_lines: &id165 + - corpus_v3_working + - corpus_v2 +- old_id: self-containment-partition + new_id: cid001161 + primary_alias: PRP0039 + type: proposition + title: Self-Containment Partition + status: canonical + release_lines: *id165 +- old_id: prop:self-containment + new_id: cid001161 + primary_alias: PRP0039 + type: proposition + title: Self-Containment Partition + status: canonical + release_lines: *id165 +- old_id: I.P42 + new_id: cid001162 + primary_alias: PRP0040 + type: proposition + title: Order Bound + status: canonical + release_lines: &id166 + - corpus_v3_working + - corpus_v2 +- old_id: order-bound + new_id: cid001162 + primary_alias: PRP0040 + type: proposition + title: Order Bound + status: canonical + release_lines: *id166 +- old_id: prop:order-bound + new_id: cid001162 + primary_alias: PRP0040 + type: proposition + title: Order Bound + status: canonical + release_lines: *id166 +- old_id: I.P43 + new_id: cid001163 + primary_alias: PRP0041 + type: proposition + title: Measure Compatibility + status: canonical + release_lines: &id167 + - corpus_v3_working + - corpus_v2 +- old_id: measure-compatibility + new_id: cid001163 + primary_alias: PRP0041 + type: proposition + title: Measure Compatibility + status: canonical + release_lines: *id167 +- old_id: prop:measure-compatibility + new_id: cid001163 + primary_alias: PRP0041 + type: proposition + title: Measure Compatibility + status: canonical + release_lines: *id167 +- old_id: I.P44 + new_id: cid001164 + primary_alias: PRP0043 + type: proposition + title: CRT-Galois Decomposition + status: canonical + release_lines: &id168 + - corpus_v3_working + - corpus_v2 +- old_id: crt-galois-decomposition + new_id: cid001164 + primary_alias: PRP0043 + type: proposition + title: CRT-Galois Decomposition + status: canonical + release_lines: *id168 +- old_id: prop:crt-galois + new_id: cid001164 + primary_alias: PRP0043 + type: proposition + title: CRT-Galois Decomposition + status: canonical + release_lines: *id168 +- old_id: I.P45 + new_id: cid001165 + primary_alias: PRP0044 + type: proposition + title: Monotone Convergence + status: canonical + release_lines: &id169 + - corpus_v3_working + - corpus_v2 +- old_id: monotone-convergence + new_id: cid001165 + primary_alias: PRP0044 + type: proposition + title: Monotone Convergence + status: canonical + release_lines: *id169 +- old_id: prop:monotone-convergence + new_id: cid001165 + primary_alias: PRP0044 + type: proposition + title: Monotone Convergence + status: canonical + release_lines: *id169 +- old_id: I.P46 + new_id: cid001166 + primary_alias: PRP0045 + type: proposition + title: No Second Linearity + status: canonical + release_lines: &id170 + - corpus_v3_working + - corpus_v2 +- old_id: no-second-linearity + new_id: cid001166 + primary_alias: PRP0045 + type: proposition + title: No Second Linearity + status: canonical + release_lines: *id170 +- old_id: prop:no-second-linearity + new_id: cid001166 + primary_alias: PRP0045 + type: proposition + title: No Second Linearity + status: canonical + release_lines: *id170 +- old_id: I.P47 + new_id: cid001167 + primary_alias: PRP0046 + type: proposition + title: Bayesian Factorization + status: canonical + release_lines: &id171 + - corpus_v3_working + - corpus_v2 +- old_id: bayesian-factorization + new_id: cid001167 + primary_alias: PRP0046 + type: proposition + title: Bayesian Factorization + status: canonical + release_lines: *id171 +- old_id: prop:bayesian-factorization + new_id: cid001167 + primary_alias: PRP0046 + type: proposition + title: Bayesian Factorization + status: canonical + release_lines: *id171 +- old_id: I.T01 + new_id: cid001168 + primary_alias: THM0002 + type: theorem + title: Ontic Closure + status: canonical + release_lines: &id172 + - corpus_v3_working + - corpus_v2 +- old_id: ontic-closure + new_id: cid001168 + primary_alias: THM0002 + type: theorem + title: Ontic Closure + status: canonical + release_lines: *id172 +- old_id: thm:ontic-closure + new_id: cid001168 + primary_alias: THM0002 + type: theorem + title: Ontic Closure + status: canonical + release_lines: *id172 +- old_id: I.T02 + new_id: cid001169 + primary_alias: THM0003 + type: theorem + title: Iterator Ladder Saturation + status: canonical + release_lines: &id173 + - corpus_v3_working + - corpus_v2 +- old_id: iterator-ladder-saturation + new_id: cid001169 + primary_alias: THM0003 + type: theorem + title: Iterator Ladder Saturation + status: canonical + release_lines: *id173 +- old_id: thm:ladder-saturation + new_id: cid001169 + primary_alias: THM0003 + type: theorem + title: Iterator Ladder Saturation + status: canonical + release_lines: *id173 +- old_id: I.T03 + new_id: cid001170 + primary_alias: THM0004 + type: theorem + title: Composition Associativity + status: canonical + release_lines: &id174 + - corpus_v3_working + - corpus_v2 +- old_id: composition-associativity + new_id: cid001170 + primary_alias: THM0004 + type: theorem + title: Composition Associativity + status: canonical + release_lines: *id174 +- old_id: thm:comp-assoc + new_id: cid001170 + primary_alias: THM0004 + type: theorem + title: Composition Associativity + status: canonical + release_lines: *id174 +- old_id: I.T04 + new_id: cid001171 + primary_alias: THM0005 + type: theorem + title: Hyperfactorization Theorem + status: canonical + release_lines: &id175 + - corpus_v3_working + - corpus_v2 +- old_id: hyperfactorization-theorem + new_id: cid001171 + primary_alias: THM0005 + type: theorem + title: Hyperfactorization Theorem + status: canonical + release_lines: *id175 +- old_id: thm:hyperfactorization + new_id: cid001171 + primary_alias: THM0005 + type: theorem + title: Hyperfactorization Theorem + status: canonical + release_lines: *id175 +- old_id: I.T05 + new_id: cid001172 + primary_alias: THM0006 + type: theorem + title: Prime Polarity Theorem + status: canonical + release_lines: &id176 + - corpus_v3_working + - corpus_v2 +- old_id: prime-polarity-theorem + new_id: cid001172 + primary_alias: THM0006 + type: theorem + title: Prime Polarity Theorem + status: canonical + release_lines: *id176 +- old_id: thm:prime-polarity + new_id: cid001172 + primary_alias: THM0006 + type: theorem + title: Prime Polarity Theorem + status: canonical + release_lines: *id176 +- old_id: I.T06 + new_id: cid001173 + primary_alias: THM0007 + type: theorem + title: Global Hartogs + status: canonical + release_lines: &id177 + - corpus_v3_working + - corpus_v2 +- old_id: global-hartogs + new_id: cid001173 + primary_alias: THM0007 + type: theorem + title: Global Hartogs + status: canonical + release_lines: *id177 +- old_id: thm:global-hartogs + new_id: cid001173 + primary_alias: THM0007 + type: theorem + title: Global Hartogs + status: canonical + release_lines: *id177 +- old_id: I.T07 + new_id: cid001174 + primary_alias: THM0008 + type: theorem + title: Rigidity of tau + status: canonical + release_lines: &id178 + - corpus_v3_working + - corpus_v2 +- old_id: rigidity-of-tau + new_id: cid001174 + primary_alias: THM0008 + type: theorem + title: Rigidity of tau + status: canonical + release_lines: *id178 +- old_id: thm:rigidity + new_id: cid001174 + primary_alias: THM0008 + type: theorem + title: Rigidity of tau + status: canonical + release_lines: *id178 +- old_id: I.T08 + new_id: cid001175 + primary_alias: THM0009 + type: theorem + title: Categoricity of tau_0 + status: canonical + release_lines: &id179 + - corpus_v3_working + - corpus_v2 +- old_id: categoricity-of-tau-0 + new_id: cid001175 + primary_alias: THM0009 + type: theorem + title: Categoricity of tau_0 + status: canonical + release_lines: *id179 +- old_id: thm:categoricity + new_id: cid001175 + primary_alias: THM0009 + type: theorem + title: Categoricity of tau_0 + status: canonical + release_lines: *id179 +- old_id: I.T09 + new_id: cid001176 + primary_alias: THM0010 + type: theorem + title: FTA on tau-Idx + status: canonical + release_lines: &id180 + - corpus_v3_working + - corpus_v2 +- old_id: fta-on-tau-idx + new_id: cid001176 + primary_alias: THM0010 + type: theorem + title: FTA on tau-Idx + status: canonical + release_lines: *id180 +- old_id: thm:fta-tau-idx + new_id: cid001176 + primary_alias: THM0010 + type: theorem + title: FTA on tau-Idx + status: canonical + release_lines: *id180 +- old_id: I.T10 + new_id: cid001177 + primary_alias: THM0011 + type: theorem + title: Split-Complex Forced + status: canonical + release_lines: &id181 + - corpus_v3_working + - corpus_v2 +- old_id: split-complex-forced + new_id: cid001177 + primary_alias: THM0011 + type: theorem + title: Split-Complex Forced + status: canonical + release_lines: *id181 +- old_id: thm:split-complex-forced + new_id: cid001177 + primary_alias: THM0011 + type: theorem + title: Split-Complex Forced + status: canonical + release_lines: *id181 +- old_id: I.T11 + new_id: cid001178 + primary_alias: THM0012 + type: theorem + title: Minimal Alphabet Theorem + status: canonical + release_lines: &id182 + - corpus_v3_working + - corpus_v2 +- old_id: minimal-alphabet-theorem + new_id: cid001178 + primary_alias: THM0012 + type: theorem + title: Minimal Alphabet Theorem + status: canonical + release_lines: *id182 +- old_id: thm:minimal-alphabet + new_id: cid001178 + primary_alias: THM0012 + type: theorem + title: Minimal Alphabet Theorem + status: canonical + release_lines: *id182 +- old_id: I.T11a + new_id: cid001179 + primary_alias: THM0013 + type: theorem + title: Six-Generator Rigidity Failure + status: canonical + release_lines: &id183 + - corpus_v3_working + - corpus_v2 +- old_id: six-generator-rigidity-failure + new_id: cid001179 + primary_alias: THM0013 + type: theorem + title: Six-Generator Rigidity Failure + status: canonical + release_lines: *id183 +- old_id: thm:six-gen-rigidity-fail + new_id: cid001179 + primary_alias: THM0013 + type: theorem + title: Six-Generator Rigidity Failure + status: canonical + release_lines: *id183 +- old_id: I.T11b + new_id: cid001180 + primary_alias: THM0014 + type: theorem + title: Four-Generator Ladder Incompleteness + status: canonical + release_lines: &id184 + - corpus_v3_working + - corpus_v2 +- old_id: four-generator-ladder-incompleteness + new_id: cid001180 + primary_alias: THM0014 + type: theorem + title: Four-Generator Ladder Incompleteness + status: canonical + release_lines: *id184 +- old_id: thm:four-gen-incomplete + new_id: cid001180 + primary_alias: THM0014 + type: theorem + title: Four-Generator Ladder Incompleteness + status: canonical + release_lines: *id184 +- old_id: I.T11c + new_id: cid001181 + primary_alias: THM0015 + type: theorem + title: Tetration Algebraic Degradation + status: canonical + release_lines: &id185 + - corpus_v3_working + - corpus_v2 +- old_id: tetration-algebraic-degradation + new_id: cid001181 + primary_alias: THM0015 + type: theorem + title: Tetration Algebraic Degradation + status: canonical + release_lines: *id185 +- old_id: thm:tetration-degraded + new_id: cid001181 + primary_alias: THM0015 + type: theorem + title: Tetration Algebraic Degradation + status: canonical + release_lines: *id185 +- old_id: I.T12 + new_id: cid001182 + primary_alias: THM0016 + type: theorem + title: Spectral Decomposition + status: canonical + release_lines: &id186 + - corpus_v3_working + - corpus_v2 +- old_id: spectral-decomposition + new_id: cid001182 + primary_alias: THM0016 + type: theorem + title: Spectral Decomposition + status: canonical + release_lines: *id186 +- old_id: thm:spectral-decomposition + new_id: cid001182 + primary_alias: THM0016 + type: theorem + title: Spectral Decomposition + status: canonical + release_lines: *id186 +- old_id: I.T13 + new_id: cid001183 + primary_alias: THM0017 + type: theorem + title: Explosion Barrier + status: canonical + release_lines: &id187 + - corpus_v3_working + - corpus_v2 +- old_id: explosion-barrier + new_id: cid001183 + primary_alias: THM0017 + type: theorem + title: Explosion Barrier + status: canonical + release_lines: *id187 +- old_id: thm:explosion-barrier + new_id: cid001183 + primary_alias: THM0017 + type: theorem + title: Explosion Barrier + status: canonical + release_lines: *id187 +- old_id: I.T14 + new_id: cid001184 + primary_alias: THM0018 + type: theorem + title: No Additive Inverse + status: canonical + release_lines: &id188 + - corpus_v3_working + - corpus_v2 +- old_id: no-additive-inverse + new_id: cid001184 + primary_alias: THM0018 + type: theorem + title: No Additive Inverse + status: canonical + release_lines: *id188 +- old_id: thm:no-additive-inverse + new_id: cid001184 + primary_alias: THM0018 + type: theorem + title: No Additive Inverse + status: canonical + release_lines: *id188 +- old_id: I.T15 + new_id: cid001185 + primary_alias: THM0019 + type: theorem + title: No Ring Negation + status: canonical + release_lines: &id189 + - corpus_v3_working + - corpus_v2 +- old_id: no-ring-negation + new_id: cid001185 + primary_alias: THM0019 + type: theorem + title: No Ring Negation + status: canonical + release_lines: *id189 +- old_id: thm:no-ring-negation + new_id: cid001185 + primary_alias: THM0019 + type: theorem + title: No Ring Negation + status: canonical + release_lines: *id189 +- old_id: I.T16 + new_id: cid001186 + primary_alias: THM0020 + type: theorem + title: Mul Cancel Fails at Zero + status: canonical + release_lines: &id190 + - corpus_v3_working + - corpus_v2 +- old_id: mul-cancel-fails-at-zero + new_id: cid001186 + primary_alias: THM0020 + type: theorem + title: Mul Cancel Fails at Zero + status: canonical + release_lines: *id190 +- old_id: thm:mul-cancel-fails-zero + new_id: cid001186 + primary_alias: THM0020 + type: theorem + title: Mul Cancel Fails at Zero + status: canonical + release_lines: *id190 +- old_id: I.T17 + new_id: cid001187 + primary_alias: THM0021 + type: theorem + title: Multiplicative Cancellation iff Positive + status: canonical + release_lines: &id191 + - corpus_v3_working + - corpus_v2 +- old_id: multiplicative-cancellation-iff-positive + new_id: cid001187 + primary_alias: THM0021 + type: theorem + title: Multiplicative Cancellation iff Positive + status: canonical + release_lines: *id191 +- old_id: thm:mul-cancel-exactly-pos + new_id: cid001187 + primary_alias: THM0021 + type: theorem + title: Multiplicative Cancellation iff Positive + status: canonical + release_lines: *id191 +- old_id: I.T18 + new_id: cid001188 + primary_alias: THM0022 + type: theorem + title: CRT Coherence Constraint + status: canonical + release_lines: &id192 + - corpus_v3_working + - corpus_v2 +- old_id: crt-coherence-constraint + new_id: cid001188 + primary_alias: THM0022 + type: theorem + title: CRT Coherence Constraint + status: canonical + release_lines: *id192 +- old_id: thm:crt-coherence + new_id: cid001188 + primary_alias: THM0022 + type: theorem + title: CRT Coherence Constraint + status: canonical + release_lines: *id192 +- old_id: I.T19 + new_id: cid001189 + primary_alias: THM0023 + type: theorem + title: Diagonal-Free Protection + status: canonical + release_lines: &id193 + - corpus_v3_working + - corpus_v2 +- old_id: diagonal-free-protection + new_id: cid001189 + primary_alias: THM0023 + type: theorem + title: Diagonal-Free Protection + status: canonical + release_lines: *id193 +- old_id: thm:diagonal-free-protection + new_id: cid001189 + primary_alias: THM0023 + type: theorem + title: Diagonal-Free Protection + status: canonical + release_lines: *id193 +- old_id: I.T20 + new_id: cid001190 + primary_alias: THM0024 + type: theorem + title: Composition Closure + status: canonical + release_lines: &id194 + - corpus_v3_working + - corpus_v2 +- old_id: composition-closure + new_id: cid001190 + primary_alias: THM0024 + type: theorem + title: Composition Closure + status: canonical + release_lines: *id194 +- old_id: thm:composition-closure + new_id: cid001190 + primary_alias: THM0024 + type: theorem + title: Composition Closure + status: canonical + release_lines: *id194 +- old_id: I.T21 + new_id: cid001191 + primary_alias: THM0025 + type: theorem + title: Tau-Identity Theorem + status: canonical + release_lines: &id195 + - corpus_v3_working + - corpus_v2 +- old_id: tau-identity-theorem + new_id: cid001191 + primary_alias: THM0025 + type: theorem + title: Tau-Identity Theorem + status: canonical + release_lines: *id195 +- old_id: thm:tau-identity + new_id: cid001191 + primary_alias: THM0025 + type: theorem + title: Tau-Identity Theorem + status: canonical + release_lines: *id195 +- old_id: I.T22 + new_id: cid001192 + primary_alias: THM0026 + type: theorem + title: Category Axioms + status: canonical + release_lines: &id196 + - corpus_v3_working + - corpus_v2 +- old_id: category-axioms + new_id: cid001192 + primary_alias: THM0026 + type: theorem + title: Category Axioms + status: canonical + release_lines: *id196 +- old_id: thm:category-axioms + new_id: cid001192 + primary_alias: THM0026 + type: theorem + title: Category Axioms + status: canonical + release_lines: *id196 +- old_id: I.T23 + new_id: cid001193 + primary_alias: THM0027 + type: theorem + title: Yoneda Lemma + status: canonical + release_lines: &id197 + - corpus_v3_working + - corpus_v2 +- old_id: yoneda-lemma + new_id: cid001193 + primary_alias: THM0027 + type: theorem + title: Yoneda Lemma + status: canonical + release_lines: *id197 +- old_id: thm:yoneda-lemma + new_id: cid001193 + primary_alias: THM0027 + type: theorem + title: Yoneda Lemma + status: canonical + release_lines: *id197 +- old_id: I.T24 + new_id: cid001194 + primary_alias: THM0028 + type: theorem + title: Grothendieck Topos + status: canonical + release_lines: &id198 + - corpus_v3_working + - corpus_v2 +- old_id: grothendieck-topos + new_id: cid001194 + primary_alias: THM0028 + type: theorem + title: Grothendieck Topos + status: canonical + release_lines: *id198 +- old_id: thm:grothendieck-topos + new_id: cid001194 + primary_alias: THM0028 + type: theorem + title: Grothendieck Topos + status: canonical + release_lines: *id198 +- old_id: I.T25 + new_id: cid001195 + primary_alias: THM0029 + type: theorem + title: Omega_tau Subobject Classifier + status: canonical + release_lines: &id199 + - corpus_v3_working + - corpus_v2 +- old_id: omega-tau-subobject-classifier + new_id: cid001195 + primary_alias: THM0029 + type: theorem + title: Omega_tau Subobject Classifier + status: canonical + release_lines: *id199 +- old_id: thm:omega-tau-classifier + new_id: cid001195 + primary_alias: THM0029 + type: theorem + title: Omega_tau Subobject Classifier + status: canonical + release_lines: *id199 +- old_id: I.T26 + new_id: cid001196 + primary_alias: THM0030 + type: theorem + title: Product Universal Property + status: canonical + release_lines: &id200 + - 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corpus_v3_working + - corpus_v2 +- old_id: cartesian-closed + new_id: cid001198 + primary_alias: THM0033 + type: theorem + title: Cartesian Closed + status: canonical + release_lines: *id202 +- old_id: thm:cartesian-closed + new_id: cid001198 + primary_alias: THM0033 + type: theorem + title: Cartesian Closed + status: canonical + release_lines: *id202 +- old_id: I.T29 + new_id: cid001199 + primary_alias: THM0034 + type: theorem + title: Spectral Determination + status: canonical + release_lines: &id203 + - corpus_v3_working + - corpus_v2 +- old_id: spectral-determination + new_id: cid001199 + primary_alias: THM0034 + type: theorem + title: Spectral Determination + status: canonical + release_lines: *id203 +- old_id: thm:spectral-determination + new_id: cid001199 + primary_alias: THM0034 + type: theorem + title: Spectral Determination + status: canonical + release_lines: *id203 +- old_id: I.T30 + new_id: cid001200 + primary_alias: THM0035 + type: theorem + title: Removable Singularity + status: canonical + release_lines: &id204 + - corpus_v3_working + - corpus_v2 +- old_id: removable-singularity + new_id: cid001200 + primary_alias: THM0035 + type: theorem + title: Removable Singularity + status: canonical + release_lines: *id204 +- old_id: thm:removable-singularity + new_id: cid001200 + primary_alias: THM0035 + type: theorem + title: Removable Singularity + status: canonical + release_lines: *id204 +- old_id: I.T31 + new_id: cid001201 + primary_alias: THM0036 + type: theorem + title: Global Hartogs Extension + status: canonical + release_lines: &id205 + - corpus_v3_working + - corpus_v2 +- old_id: global-hartogs-extension + new_id: cid001201 + primary_alias: THM0036 + type: theorem + title: Global Hartogs Extension + status: canonical + release_lines: *id205 +- old_id: I.T35 + new_id: cid001202 + primary_alias: THM0037 + type: theorem + title: Cantor Diagonal Inapplicability + status: canonical + release_lines: &id206 + - corpus_v3_working + - corpus_v2 +- old_id: cantor-diagonal-inapplicability + new_id: cid001202 + primary_alias: THM0037 + type: theorem + title: Cantor Diagonal Inapplicability + status: canonical + release_lines: *id206 +- old_id: thm:cantor-inapplicable + new_id: cid001202 + primary_alias: THM0037 + type: theorem + title: Cantor Diagonal Inapplicability + status: canonical + release_lines: *id206 +- old_id: I.T36 + new_id: cid001203 + primary_alias: THM0038 + type: theorem + title: Unique Infinity Object + status: canonical + release_lines: &id207 + - corpus_v3_working + - corpus_v2 +- old_id: unique-infinity-object + new_id: cid001203 + primary_alias: THM0038 + type: theorem + title: Unique Infinity Object + status: canonical + release_lines: *id207 +- old_id: thm:unique-infinity + new_id: cid001203 + primary_alias: THM0038 + type: theorem + title: Unique Infinity Object + status: canonical + release_lines: *id207 +- old_id: I.T37 + new_id: cid001204 + primary_alias: THM0039 + type: theorem + title: Diagonal-Linear Correspondence + status: canonical + release_lines: &id208 + - corpus_v3_working + - corpus_v2 +- old_id: thm:diagonal-linear + new_id: cid001204 + primary_alias: THM0039 + type: theorem + title: Diagonal-Linear Correspondence + status: canonical + release_lines: *id208 +- old_id: I.T38 + new_id: cid001205 + primary_alias: THM0040 + type: theorem + title: Linearity Census + status: canonical + release_lines: &id209 + - corpus_v3_working + - corpus_v2 +- old_id: linearity-census + new_id: cid001205 + primary_alias: THM0040 + type: theorem + title: Linearity Census + status: canonical + release_lines: *id209 +- old_id: thm:linearity-census + new_id: cid001205 + primary_alias: THM0040 + type: theorem + title: Linearity Census + status: canonical + release_lines: *id209 +- old_id: I.T39 + new_id: cid001206 + primary_alias: THM0041 + type: theorem + title: K5 Structural Exclusion + status: canonical + release_lines: &id210 + - corpus_v3_working + - corpus_v2 +- old_id: k5-structural-exclusion + new_id: cid001206 + primary_alias: THM0041 + type: theorem + title: K5 Structural Exclusion + status: canonical + release_lines: *id210 +- old_id: thm:k5-structural-exclusion + new_id: cid001206 + primary_alias: THM0041 + type: theorem + title: K5 Structural Exclusion + status: canonical + release_lines: *id210 +- old_id: I.T40 + new_id: cid001207 + primary_alias: THM0042 + type: theorem + title: Presheaf Characterization + status: canonical + release_lines: &id211 + - corpus_v3_working + - corpus_v2 +- old_id: presheaf-characterization + new_id: cid001207 + primary_alias: THM0042 + type: theorem + title: Presheaf Characterization + status: canonical + release_lines: *id211 +- old_id: thm:presheaf-characterization + new_id: cid001207 + primary_alias: THM0042 + type: theorem + title: Presheaf Characterization + status: canonical + release_lines: *id211 +- old_id: I.T41 + new_id: cid001208 + primary_alias: THM0043 + type: theorem + title: Bi-Square Characterization + status: canonical + release_lines: &id212 + - corpus_v3_working + - corpus_v2 +- old_id: bi-square-characterization + new_id: cid001208 + primary_alias: THM0043 + type: theorem + title: Bi-Square Characterization + status: canonical + release_lines: *id212 +- old_id: thm:bi-square + new_id: cid001208 + primary_alias: THM0043 + type: theorem + title: Bi-Square Characterization + status: canonical + release_lines: *id212 +- old_id: I.T42 + new_id: cid001209 + primary_alias: THM0044 + type: theorem + title: Archimedean Property + status: canonical + release_lines: &id213 + - corpus_v3_working + - corpus_v2 +- old_id: archimedean-property + new_id: cid001209 + primary_alias: THM0044 + type: theorem + title: Archimedean Property + status: canonical + release_lines: *id213 +- old_id: thm:archimedean-property + new_id: cid001209 + primary_alias: THM0044 + type: theorem + title: Archimedean Property + status: canonical + release_lines: *id213 +- old_id: I.T43 + new_id: cid001210 + primary_alias: THM0045 + type: theorem + title: TauComplex Ring Axioms + status: canonical + release_lines: &id214 + - corpus_v3_working + - corpus_v2 +- old_id: taucomplex-ring-axioms + new_id: cid001210 + primary_alias: THM0045 + type: theorem + title: TauComplex Ring Axioms + status: canonical + release_lines: *id214 +- old_id: thm:complex-field-axioms + new_id: cid001210 + primary_alias: THM0045 + type: theorem + title: TauComplex Ring Axioms + status: canonical + release_lines: *id214 +- old_id: I.T44 + new_id: cid001211 + primary_alias: THM0046 + type: theorem + title: Quaternion Non-Commutativity + status: canonical + release_lines: &id215 + - corpus_v3_working + - corpus_v2 +- old_id: quaternion-non-commutativity + new_id: cid001211 + primary_alias: THM0046 + type: theorem + title: Quaternion Non-Commutativity + status: canonical + release_lines: *id215 +- old_id: thm:quaternion-division + new_id: cid001211 + primary_alias: THM0046 + type: theorem + title: Quaternion Non-Commutativity + status: canonical + release_lines: *id215 +- old_id: I.T45 + new_id: cid001212 + primary_alias: THM0047 + type: theorem + title: Roots of Unity CRT Decomposition + status: canonical + release_lines: &id216 + - corpus_v3_working + - corpus_v2 +- old_id: roots-of-unity-crt-decomposition + new_id: cid001212 + primary_alias: THM0047 + type: theorem + title: Roots of Unity CRT Decomposition + status: canonical + release_lines: *id216 +- old_id: thm:primitive-roots + new_id: cid001212 + primary_alias: THM0047 + type: theorem + title: Roots of Unity CRT Decomposition + status: canonical + release_lines: *id216 +- old_id: I.T46 + new_id: cid001213 + primary_alias: THM0048 + type: theorem + title: Ontic Identity Invariance + status: canonical + release_lines: &id217 + - corpus_v3_working + - corpus_v2 +- old_id: ontic-identity-invariance + new_id: cid001213 + primary_alias: THM0048 + type: theorem + title: Ontic Identity Invariance + status: canonical + release_lines: *id217 +- old_id: thm:ontic-identity-invariance + new_id: cid001213 + primary_alias: THM0048 + type: theorem + title: Ontic Identity Invariance + status: canonical + release_lines: *id217 +- old_id: I.T47 + new_id: cid001214 + primary_alias: THM0049 + type: theorem + title: Slippage Breaks Unique Omega + status: canonical + release_lines: &id218 + - corpus_v3_working + - corpus_v2 +- old_id: slippage-breaks-unique-omega + new_id: cid001214 + primary_alias: THM0049 + type: theorem + title: Slippage Breaks Unique Omega + status: canonical + release_lines: *id218 +- old_id: thm:slippage-breaks-omega + new_id: cid001214 + primary_alias: THM0049 + type: theorem + title: Slippage Breaks Unique Omega + status: canonical + release_lines: *id218 +- old_id: I.T48 + new_id: cid001215 + primary_alias: THM0050 + type: theorem + title: Structural Instability Theorem + status: canonical + release_lines: &id219 + - corpus_v3_working + - corpus_v2 +- old_id: structural-instability-theorem + new_id: cid001215 + primary_alias: THM0050 + type: theorem + title: Structural Instability Theorem + status: canonical + release_lines: *id219 +- old_id: thm:structural-instability + new_id: cid001215 + primary_alias: THM0050 + type: theorem + title: Structural Instability Theorem + status: canonical + release_lines: *id219 +- old_id: I.T49 + new_id: cid001216 + primary_alias: THM0051 + type: theorem + title: Countable Additivity + status: canonical + release_lines: &id220 + - corpus_v3_working + - corpus_v2 +- old_id: countable-additivity + new_id: cid001216 + primary_alias: THM0051 + type: theorem + title: Countable Additivity + status: canonical + release_lines: *id220 +- old_id: thm:countable-additivity + new_id: cid001216 + primary_alias: THM0051 + type: theorem + title: Countable Additivity + status: canonical + release_lines: *id220 +- old_id: I.T50 + new_id: cid001217 + primary_alias: THM0052 + type: theorem + title: Fundamental Theorem of Internal Galois Theory + status: canonical + release_lines: &id221 + - corpus_v3_working + - corpus_v2 +- old_id: fundamental-theorem-of-internal-galois-theory + new_id: cid001217 + primary_alias: THM0052 + type: theorem + title: Fundamental Theorem of Internal Galois Theory + status: canonical + release_lines: *id221 +- old_id: thm:galois-fundamental + new_id: cid001217 + primary_alias: THM0052 + type: theorem + title: Fundamental Theorem of Internal Galois Theory + status: canonical + release_lines: *id221 +- old_id: I.T51 + new_id: cid001218 + primary_alias: THM0053 + type: theorem + title: Linearity of Integration + status: canonical + release_lines: &id222 + - corpus_v3_working + - corpus_v2 +- old_id: linearity-of-integration + new_id: cid001218 + primary_alias: THM0053 + type: theorem + title: Linearity of Integration + status: canonical + release_lines: *id222 +- old_id: thm:integral-linearity + new_id: cid001218 + primary_alias: THM0053 + type: theorem + title: Linearity of Integration + status: canonical + release_lines: *id222 +- old_id: I.T52 + new_id: cid001219 + primary_alias: THM0054 + type: theorem + title: Bias at Primorial Levels + status: draft + release_lines: &id223 + - corpus_v3_working + - corpus_v2 +- old_id: bias-at-primorial-levels + new_id: cid001219 + primary_alias: THM0054 + type: theorem + title: Bias at Primorial Levels + status: draft + release_lines: *id223 +- old_id: thm:bias-primorial + new_id: cid001219 + primary_alias: THM0054 + type: theorem + title: Bias at Primorial Levels + status: draft + release_lines: *id223 +- old_id: I.T53 + new_id: cid001220 + primary_alias: THM0055 + type: theorem + title: Canonical Normalisation (Hinge 7) + status: canonical + release_lines: &id224 + - corpus_v3_working + - corpus_v2 +- old_id: canonical-normalisation-hinge-7 + new_id: cid001220 + primary_alias: THM0055 + type: theorem + title: Canonical Normalisation (Hinge 7) + status: canonical + release_lines: *id224 +- old_id: thm:main-normalisation + new_id: cid001220 + primary_alias: THM0055 + type: theorem + title: Canonical Normalisation (Hinge 7) + status: canonical + release_lines: *id224 +- old_id: I.T54 + new_id: cid001221 + primary_alias: THM0056 + type: theorem + title: NF Confluence / Church-Rosser for tau-kernel (Hinge 7) + status: canonical + release_lines: &id225 + - corpus_v3_working + - corpus_v2 +- old_id: nf-confluence-church-rosser-for-tau-kernel-hinge-7 + new_id: cid001221 + primary_alias: THM0056 + type: theorem + title: NF Confluence / Church-Rosser for tau-kernel (Hinge 7) + status: canonical + release_lines: *id225 +- old_id: thm:main-confluence + new_id: cid001221 + primary_alias: THM0056 + type: theorem + title: NF Confluence / Church-Rosser for tau-kernel (Hinge 7) + status: canonical + release_lines: *id225 +- old_id: I.T55 + new_id: cid001222 + primary_alias: THM0057 + type: theorem + title: Genealogical DAG Structure (Hinge 7) + status: canonical + release_lines: &id226 + - corpus_v3_working + - corpus_v2 +- old_id: genealogical-dag-structure-hinge-7 + new_id: cid001222 + primary_alias: THM0057 + type: theorem + title: Genealogical DAG Structure (Hinge 7) + status: canonical + release_lines: *id226 +- old_id: thm:main-dag + new_id: cid001222 + primary_alias: THM0057 + type: theorem + title: Genealogical DAG Structure (Hinge 7) + status: canonical + release_lines: *id226 +- old_id: I.T56 + new_id: cid001223 + primary_alias: THM0058 + type: theorem + title: Cayley Word Metric (Hinge 7) + status: canonical + release_lines: &id227 + - corpus_v3_working + - corpus_v2 +- old_id: cayley-word-metric-hinge-7 + new_id: cid001223 + primary_alias: THM0058 + type: theorem + title: Cayley Word Metric (Hinge 7) + status: canonical + release_lines: *id227 +- old_id: thm:main-cayley + new_id: cid001223 + primary_alias: THM0058 + type: theorem + title: Cayley Word Metric (Hinge 7) + status: canonical + release_lines: *id227 +- old_id: I.T57 + new_id: cid001224 + primary_alias: THM0059 + type: theorem + title: Ontic Ultrametric (Hinge 7) + status: canonical + release_lines: &id228 + - corpus_v3_working + - corpus_v2 +- old_id: ontic-ultrametric-hinge-7 + new_id: cid001224 + primary_alias: THM0059 + type: theorem + title: Ontic Ultrametric (Hinge 7) + status: canonical + release_lines: *id228 +- old_id: thm:main-ultrametric + new_id: cid001224 + primary_alias: THM0059 + type: theorem + title: Ontic Ultrametric (Hinge 7) + status: canonical + release_lines: *id228 +- old_id: I.T58 + new_id: cid001225 + primary_alias: THM0060 + type: theorem + title: Address Resolution and Absence of Equations (Hinge 7) + status: canonical + release_lines: &id229 + - corpus_v3_working + - corpus_v2 +- old_id: address-resolution-and-absence-of-equations-hinge-7 + new_id: cid001225 + primary_alias: THM0060 + type: theorem + title: Address Resolution and Absence of Equations (Hinge 7) + status: canonical + release_lines: *id229 +- old_id: thm:main-address-resolution + new_id: cid001225 + primary_alias: THM0060 + type: theorem + title: Address Resolution and Absence of Equations (Hinge 7) + status: canonical + release_lines: *id229 +- old_id: I.T59 + new_id: cid001226 + primary_alias: THM0061 + type: theorem + title: Hinge 7 Integration Tabulation (Bundle Closure) + status: canonical + release_lines: &id230 + - 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corpus_v3_working + - corpus_v2 +- old_id: holend-exponentials + new_id: cid001362 + primary_alias: PRP0067 + type: proposition + title: HolEnd Exponentials + status: canonical + release_lines: *id366 +- old_id: prop:ch42-holend-exponentials + new_id: cid001362 + primary_alias: PRP0067 + type: proposition + title: HolEnd Exponentials + status: canonical + release_lines: *id366 +- old_id: II.T01 + new_id: cid001363 + primary_alias: THM0067 + type: theorem + title: Point Set Well-Defined + status: canonical + release_lines: &id367 + - corpus_v3_working + - corpus_v2 +- old_id: point-set-well-defined + new_id: cid001363 + primary_alias: THM0067 + type: theorem + title: Point Set Well-Defined + status: canonical + release_lines: *id367 +- old_id: thm:point-set-well-defined + new_id: cid001363 + primary_alias: THM0067 + type: theorem + title: Point Set Well-Defined + status: canonical + release_lines: *id367 +- old_id: II.T02 + new_id: cid001364 + primary_alias: THM0068 + type: theorem + title: Fiber Degeneration at Omega + status: canonical + release_lines: &id368 + - corpus_v3_working + - corpus_v2 +- old_id: fiber-degeneration-at-omega + new_id: cid001364 + primary_alias: THM0068 + type: theorem + title: Fiber Degeneration at Omega + status: canonical + release_lines: *id368 +- old_id: thm:fiber-degeneration-omega + new_id: cid001364 + primary_alias: THM0068 + type: theorem + title: Fiber Degeneration at Omega + status: canonical + release_lines: *id368 +- old_id: II.T03 + new_id: cid001365 + primary_alias: THM0069 + type: theorem + title: Fibration Structure + status: canonical + release_lines: &id369 + - corpus_v3_working + - corpus_v2 +- old_id: fibration-structure + new_id: cid001365 + primary_alias: THM0069 + type: theorem + title: Fibration Structure + status: canonical + release_lines: *id369 +- old_id: thm:fibration-structure + new_id: cid001365 + primary_alias: THM0069 + type: theorem + title: Fibration Structure + status: canonical + release_lines: *id369 +- old_id: II.T04 + new_id: cid001366 + primary_alias: THM0070 + type: theorem + title: Cylinder Basis Theorem + status: canonical + release_lines: &id370 + - corpus_v3_working + - corpus_v2 +- old_id: cylinder-basis-theorem + new_id: cid001366 + primary_alias: THM0070 + type: theorem + title: Cylinder Basis Theorem + status: canonical + release_lines: *id370 +- old_id: thm:cylinder-basis + new_id: cid001366 + primary_alias: THM0070 + type: theorem + title: Cylinder Basis Theorem + status: canonical + release_lines: *id370 +- old_id: II.T05 + new_id: cid001367 + primary_alias: THM0071 + type: theorem + title: Ultrametric Triangle Inequality + status: canonical + release_lines: &id371 + - corpus_v3_working + - corpus_v2 +- old_id: ultrametric-triangle-inequality + new_id: cid001367 + primary_alias: THM0071 + type: theorem + title: Ultrametric Triangle Inequality + status: canonical + release_lines: *id371 +- old_id: thm:ultrametric-triangle + new_id: cid001367 + primary_alias: THM0071 + type: theorem + title: Ultrametric Triangle Inequality + status: canonical + release_lines: *id371 +- old_id: II.T06 + new_id: cid001368 + primary_alias: THM0072 + type: theorem + title: Holomorphic Implies Continuous + status: canonical + release_lines: &id372 + - corpus_v3_working + - corpus_v2 +- old_id: holomorphic-implies-continuous + new_id: cid001368 + primary_alias: THM0072 + type: theorem + title: Holomorphic Implies Continuous + status: canonical + release_lines: *id372 +- old_id: thm:hol-implies-cont + new_id: cid001368 + primary_alias: THM0072 + type: theorem + title: Holomorphic Implies Continuous + status: canonical + release_lines: *id372 +- old_id: II.T07 + new_id: cid001369 + primary_alias: THM0073 + type: theorem + title: Compactness + status: canonical + release_lines: &id373 + - corpus_v3_working + - corpus_v2 +- old_id: compactness + new_id: cid001369 + primary_alias: THM0073 + type: theorem + title: Compactness + status: canonical + release_lines: *id373 +- old_id: thm:compactness + new_id: cid001369 + primary_alias: THM0073 + type: theorem + title: Compactness + status: canonical + release_lines: *id373 +- old_id: II.T08 + new_id: cid001370 + primary_alias: THM0074 + type: theorem + title: Hausdorff Property + status: canonical + release_lines: &id374 + - corpus_v3_working + - corpus_v2 +- old_id: hausdorff-property + new_id: cid001370 + primary_alias: THM0074 + type: theorem + title: Hausdorff Property + status: canonical + release_lines: *id374 +- old_id: thm:hausdorff + new_id: cid001370 + primary_alias: THM0074 + type: theorem + title: Hausdorff Property + status: canonical + release_lines: *id374 +- old_id: II.T09 + new_id: cid001371 + primary_alias: THM0075 + type: theorem + title: Total Disconnectedness + status: canonical + release_lines: &id375 + - corpus_v3_working + - corpus_v2 +- old_id: total-disconnectedness + new_id: cid001371 + primary_alias: THM0075 + type: theorem + title: Total Disconnectedness + status: canonical + release_lines: *id375 +- old_id: thm:totally-disconnected + new_id: cid001371 + primary_alias: THM0075 + type: theorem + title: Total Disconnectedness + status: canonical + release_lines: *id375 +- old_id: II.T10 + new_id: cid001372 + primary_alias: THM0076 + type: theorem + title: Topology Uniqueness + status: canonical + release_lines: &id376 + - corpus_v3_working + - corpus_v2 +- old_id: topology-uniqueness + new_id: cid001372 + primary_alias: THM0076 + type: theorem + title: Topology Uniqueness + status: canonical + release_lines: *id376 +- old_id: thm:topology-uniqueness + new_id: cid001372 + primary_alias: THM0076 + type: theorem + title: Topology Uniqueness + status: canonical + release_lines: *id376 +- old_id: II.T11 + new_id: cid001373 + primary_alias: THM0077 + type: theorem + title: Dimension Four + status: canonical + release_lines: &id377 + - corpus_v3_working + - corpus_v2 +- old_id: dimension-four + new_id: cid001373 + primary_alias: THM0077 + type: theorem + title: Dimension Four + status: canonical + release_lines: *id377 +- old_id: thm:dimension-four + new_id: cid001373 + primary_alias: THM0077 + type: theorem + title: Dimension Four + status: canonical + release_lines: *id377 +- old_id: II.T12 + new_id: cid001374 + primary_alias: THM0078 + type: theorem + title: Boundary Minimality + status: canonical + release_lines: &id378 + - corpus_v3_working + - corpus_v2 +- old_id: boundary-minimality + new_id: cid001374 + primary_alias: THM0078 + type: theorem + title: Boundary Minimality + status: canonical + release_lines: *id378 +- old_id: thm:boundary-minimality + new_id: cid001374 + primary_alias: THM0078 + type: theorem + title: Boundary Minimality + status: canonical + release_lines: *id378 +- old_id: II.T13 + new_id: cid001375 + primary_alias: THM0079 + type: theorem + title: Torus Degeneration Theorem + status: canonical + release_lines: &id379 + - corpus_v3_working + - corpus_v2 +- old_id: torus-degeneration-theorem + new_id: cid001375 + primary_alias: THM0079 + type: theorem + title: Torus Degeneration Theorem + status: canonical + release_lines: *id379 +- old_id: thm:torus-degeneration + new_id: cid001375 + primary_alias: THM0079 + type: theorem + title: Torus Degeneration Theorem + status: canonical + release_lines: *id379 +- old_id: II.T14 + new_id: cid001376 + primary_alias: THM0080 + type: theorem + title: Fundamental Group Degeneration + status: canonical + release_lines: &id380 + - corpus_v3_working + - corpus_v2 +- old_id: fundamental-group-degeneration + new_id: cid001376 + primary_alias: THM0080 + type: theorem + title: Fundamental Group Degeneration + status: canonical + release_lines: *id380 +- old_id: thm:fundamental-group-degeneration + new_id: cid001376 + primary_alias: THM0080 + type: theorem + title: Fundamental Group Degeneration + status: canonical + release_lines: *id380 +- old_id: II.T15 + new_id: cid001377 + primary_alias: THM0081 + type: theorem + title: Betweenness Axioms + status: canonical + release_lines: &id381 + - corpus_v3_working + - corpus_v2 +- old_id: betweenness-axioms + new_id: cid001377 + primary_alias: THM0081 + type: theorem + title: Betweenness Axioms + status: canonical + release_lines: *id381 +- old_id: thm:betweenness-axioms + new_id: cid001377 + primary_alias: THM0081 + type: theorem + title: Betweenness Axioms + status: canonical + release_lines: *id381 +- old_id: II.T16 + new_id: cid001378 + primary_alias: THM0082 + type: theorem + title: Congruence Axioms + status: canonical + release_lines: &id382 + - corpus_v3_working + - corpus_v2 +- old_id: congruence-axioms + new_id: cid001378 + primary_alias: THM0082 + type: theorem + title: Congruence Axioms + status: canonical + release_lines: *id382 +- old_id: thm:congruence-axioms + new_id: cid001378 + primary_alias: THM0082 + type: theorem + title: Congruence Axioms + status: canonical + release_lines: *id382 +- old_id: II.T17 + new_id: cid001379 + primary_alias: THM0083 + type: theorem + title: Pasch Axiom + status: canonical + release_lines: &id383 + - corpus_v3_working + - corpus_v2 +- old_id: pasch-axiom + new_id: cid001379 + primary_alias: THM0083 + type: theorem + title: Pasch Axiom + status: canonical + release_lines: *id383 +- old_id: thm:pasch + new_id: cid001379 + primary_alias: THM0083 + type: theorem + title: Pasch Axiom + status: canonical + release_lines: *id383 +- old_id: II.T18 + new_id: cid001380 + primary_alias: THM0084 + type: theorem + title: Parallel Postulate + status: canonical + release_lines: &id384 + - corpus_v3_working + - corpus_v2 +- old_id: parallel-postulate + new_id: cid001380 + primary_alias: THM0084 + type: theorem + title: Parallel Postulate + status: canonical + release_lines: *id384 +- old_id: thm:parallel-postulate + new_id: cid001380 + primary_alias: THM0084 + type: theorem + title: Parallel Postulate + status: canonical + release_lines: *id384 +- old_id: II.T19 + new_id: cid001381 + primary_alias: THM0085 + type: theorem + title: Euclidean as Static Limit + status: canonical + release_lines: &id385 + - corpus_v3_working + - corpus_v2 +- old_id: euclidean-as-static-limit + new_id: cid001381 + primary_alias: THM0085 + type: theorem + title: Euclidean as Static Limit + status: canonical + release_lines: *id385 +- old_id: thm:euclidean-static-limit + new_id: cid001381 + primary_alias: THM0085 + type: theorem + title: Euclidean as Static Limit + status: canonical + release_lines: *id385 +- old_id: II.T20 + new_id: cid001382 + primary_alias: THM0086 + type: theorem + title: R as Inverse Limit + status: canonical + release_lines: &id386 + - corpus_v3_working + - corpus_v2 +- old_id: r-as-inverse-limit + new_id: cid001382 + primary_alias: THM0086 + type: theorem + title: R as Inverse Limit + status: canonical + release_lines: *id386 +- old_id: thm:real-inverse-limit + new_id: cid001382 + primary_alias: THM0086 + type: theorem + title: R as Inverse Limit + status: canonical + release_lines: *id386 +- old_id: II.T21 + new_id: cid001383 + primary_alias: THM0087 + type: theorem + title: S^1 as Profinite Limit + status: canonical + release_lines: &id387 + - corpus_v3_working + - corpus_v2 +- old_id: s-1-as-profinite-limit + new_id: cid001383 + primary_alias: THM0087 + type: theorem + title: S^1 as Profinite Limit + status: canonical + release_lines: *id387 +- old_id: thm:circle-profinite + new_id: cid001383 + primary_alias: THM0087 + type: theorem + title: S^1 as Profinite Limit + status: canonical + release_lines: *id387 +- old_id: II.T22 + new_id: cid001384 + primary_alias: THM0088 + type: theorem + title: Three Perspectives on Pi + status: canonical + release_lines: &id388 + - corpus_v3_working + - corpus_v2 +- old_id: three-perspectives-on-pi + new_id: cid001384 + primary_alias: THM0088 + type: theorem + title: Three Perspectives on Pi + status: canonical + release_lines: *id388 +- old_id: thm:three-pi + new_id: cid001384 + primary_alias: THM0088 + type: theorem + title: Three Perspectives on Pi + status: canonical + release_lines: *id388 +- old_id: II.T23 + new_id: cid001385 + primary_alias: THM0089 + type: theorem + title: e from Index Arithmetic + status: canonical + release_lines: &id389 + - corpus_v3_working + - corpus_v2 +- old_id: e-from-index-arithmetic + new_id: cid001385 + primary_alias: THM0089 + type: theorem + title: e from Index Arithmetic + status: canonical + release_lines: *id389 +- old_id: thm:e-index + new_id: cid001385 + primary_alias: THM0089 + type: theorem + title: e from Index Arithmetic + status: canonical + release_lines: *id389 +- old_id: II.T24 + new_id: cid001386 + primary_alias: THM0090 + type: theorem + title: j Replaces i + status: canonical + release_lines: &id390 + - corpus_v3_working + - corpus_v2 +- old_id: j-replaces-i + new_id: cid001386 + primary_alias: THM0090 + type: theorem + title: j Replaces i + status: canonical + release_lines: *id390 +- old_id: thm:j-replaces-i + new_id: cid001386 + primary_alias: THM0090 + type: theorem + title: j Replaces i + status: canonical + release_lines: *id390 +- old_id: II.T26 + new_id: cid001387 + primary_alias: THM0091 + type: theorem + title: BndLift Existence + status: canonical + release_lines: &id391 + - corpus_v3_working + - corpus_v2 +- old_id: bndlift-existence + new_id: cid001387 + primary_alias: THM0091 + type: theorem + title: BndLift Existence + status: canonical + release_lines: *id391 +- old_id: thm:bndlift-existence + new_id: cid001387 + primary_alias: THM0091 + type: theorem + title: BndLift Existence + status: canonical + release_lines: *id391 +- old_id: II.T27 + new_id: cid001388 + primary_alias: THM0092 + type: theorem + title: Mutual Determination (5-Way Equivalence) + status: canonical + release_lines: &id392 + - corpus_v3_working + - corpus_v2 +- old_id: mutual-determination-5-way-equivalence + new_id: cid001388 + primary_alias: THM0092 + type: theorem + title: Mutual Determination (5-Way Equivalence) + status: canonical + release_lines: *id392 +- old_id: thm:mutual-determination + new_id: cid001388 + primary_alias: THM0092 + type: theorem + title: Mutual Determination (5-Way Equivalence) + status: canonical + release_lines: *id392 +- old_id: II.T28 + new_id: cid001389 + primary_alias: THM0093 + type: theorem + title: B/C Asymmetry Implies Time Arrow + status: canonical + release_lines: &id393 + - corpus_v3_working + - corpus_v2 +- old_id: b-c-asymmetry-implies-time-arrow + new_id: cid001389 + primary_alias: THM0093 + type: theorem + title: B/C Asymmetry Implies Time Arrow + status: canonical + release_lines: *id393 +- old_id: thm:bc-asymmetry-time + new_id: cid001389 + primary_alias: THM0093 + type: theorem + title: B/C Asymmetry Implies Time Arrow + status: canonical + release_lines: *id393 +- old_id: II.T29 + new_id: cid001390 + primary_alias: THM0094 + type: theorem + title: Associativity of Holomorphic Composition + status: canonical + release_lines: &id394 + - corpus_v3_working + - corpus_v2 +- old_id: associativity-of-holomorphic-composition + new_id: cid001390 + primary_alias: THM0094 + type: theorem + title: Associativity of Holomorphic Composition + status: canonical + release_lines: *id394 +- old_id: thm:associativity + new_id: cid001390 + primary_alias: THM0094 + type: theorem + title: Associativity of Holomorphic Composition + status: canonical + release_lines: *id394 +- old_id: II.T30 + new_id: cid001391 + primary_alias: THM0095 + type: theorem + title: Residue Theorem + status: canonical + release_lines: &id395 + - corpus_v3_working + - corpus_v2 +- old_id: residue-theorem + new_id: cid001391 + primary_alias: THM0095 + type: theorem + title: Residue Theorem + status: canonical + release_lines: *id395 +- old_id: thm:residue-theorem + new_id: cid001391 + primary_alias: THM0095 + type: theorem + title: Residue Theorem + status: canonical + release_lines: *id395 +- old_id: II.T31 + new_id: cid001392 + primary_alias: THM0096 + type: theorem + title: Finite Spectral Support + status: canonical + release_lines: &id396 + - corpus_v3_working + - corpus_v2 +- old_id: finite-spectral-support + new_id: cid001392 + primary_alias: THM0096 + type: theorem + title: Finite Spectral Support + status: canonical + release_lines: *id396 +- old_id: thm:finite-spectral-support + new_id: cid001392 + primary_alias: THM0096 + type: theorem + title: Finite Spectral Support + status: canonical + release_lines: *id396 +- old_id: II.T32 + new_id: cid001393 + primary_alias: THM0097 + type: theorem + title: Sheaf Axioms + status: canonical + release_lines: &id397 + - corpus_v3_working + - corpus_v2 +- old_id: sheaf-axioms + new_id: cid001393 + primary_alias: THM0097 + type: theorem + title: Sheaf Axioms + status: canonical + release_lines: *id397 +- old_id: thm:sheaf-axioms + new_id: cid001393 + primary_alias: THM0097 + type: theorem + title: Sheaf Axioms + status: canonical + release_lines: *id397 +- old_id: II.T33 + new_id: cid001394 + primary_alias: THM0098 + type: theorem + title: Holomorphic iff Idempotent-Supported + status: canonical + release_lines: &id398 + - corpus_v3_working + - corpus_v2 +- old_id: holomorphic-iff-idempotent-supported + new_id: cid001394 + primary_alias: THM0098 + type: theorem + title: Holomorphic iff Idempotent-Supported + status: canonical + release_lines: *id398 +- old_id: thm:hol-iff-idempotent + new_id: cid001394 + primary_alias: THM0098 + type: theorem + title: Holomorphic iff Idempotent-Supported + status: canonical + release_lines: *id398 +- old_id: II.T34 + new_id: cid001395 + primary_alias: THM0099 + type: theorem + title: Regularity Criterion + status: canonical + release_lines: &id399 + - corpus_v3_working + - corpus_v2 +- old_id: regularity-criterion + new_id: cid001395 + primary_alias: THM0099 + type: theorem + title: Regularity Criterion + status: canonical + release_lines: *id399 +- old_id: thm:regularity-criterion + new_id: cid001395 + primary_alias: THM0099 + type: theorem + title: Regularity Criterion + status: canonical + release_lines: *id399 +- old_id: II.T35 + new_id: cid001396 + primary_alias: THM0100 + type: theorem + title: Code/Decode Bijection + status: canonical + release_lines: &id400 + - corpus_v3_working + - corpus_v2 +- old_id: code-decode-bijection + new_id: cid001396 + primary_alias: THM0100 + type: theorem + title: Code/Decode Bijection + status: canonical + release_lines: *id400 +- old_id: thm:code-decode-bijection + new_id: cid001396 + primary_alias: THM0100 + type: theorem + title: Code/Decode Bijection + status: canonical + release_lines: *id400 +- old_id: II.T36 + new_id: cid001397 + primary_alias: THM0101 + type: theorem + title: Tau-Yoneda Embedding + status: canonical + release_lines: &id401 + - corpus_v3_working + - corpus_v2 +- old_id: tau-yoneda-embedding + new_id: cid001397 + primary_alias: THM0101 + type: theorem + title: Tau-Yoneda Embedding + status: canonical + release_lines: *id401 +- old_id: thm:yoneda-embedding + new_id: cid001397 + primary_alias: THM0101 + type: theorem + title: Tau-Yoneda Embedding + status: canonical + release_lines: *id401 +- old_id: II.T37 + new_id: cid001398 + primary_alias: THM0102 + type: theorem + title: Hartogs Extension Uniqueness + status: canonical + release_lines: &id402 + - corpus_v3_working + - corpus_v2 +- old_id: hartogs-extension-uniqueness + new_id: cid001398 + primary_alias: THM0102 + type: theorem + title: Hartogs Extension Uniqueness + status: canonical + release_lines: *id402 +- old_id: thm:hartogs-uniqueness + new_id: cid001398 + primary_alias: THM0102 + type: theorem + title: Hartogs Extension Uniqueness + status: canonical + release_lines: *id402 +- old_id: II.T38 + new_id: cid001399 + primary_alias: THM0103 + type: theorem + title: Extensions Are Omega-Germ Transformers + status: canonical + release_lines: &id403 + - corpus_v3_working + - corpus_v2 +- old_id: extensions-are-omega-germ-transformers + new_id: cid001399 + primary_alias: THM0103 + type: theorem + title: Extensions Are Omega-Germ Transformers + status: canonical + release_lines: *id403 +- old_id: thm:extensions-omega-germs + new_id: cid001399 + primary_alias: THM0103 + type: theorem + title: Extensions Are Omega-Germ Transformers + status: canonical + release_lines: *id403 +- old_id: II.T39 + new_id: cid001400 + primary_alias: THM0104 + type: theorem + title: Omega-Germs iff Holomorphic Functions + status: canonical + release_lines: &id404 + - corpus_v3_working + - corpus_v2 +- old_id: omega-germs-iff-holomorphic-functions + new_id: cid001400 + primary_alias: THM0104 + type: theorem + title: Omega-Germs iff Holomorphic Functions + status: canonical + release_lines: *id404 +- old_id: thm:omega-germs-holomorphic + new_id: cid001400 + primary_alias: THM0104 + type: theorem + title: Omega-Germs iff Holomorphic Functions + status: canonical + release_lines: *id404 +- old_id: II.T40 + new_id: cid001401 + primary_alias: THM0105 + type: theorem + title: Central Theorem + status: canonical + release_lines: &id405 + - corpus_v3_working + - corpus_v2 +- old_id: central-theorem + new_id: cid001401 + primary_alias: THM0105 + type: theorem + title: Central Theorem + status: canonical + release_lines: *id405 +- old_id: thm:central-theorem + new_id: cid001401 + primary_alias: THM0105 + type: theorem + title: Central Theorem + status: canonical + release_lines: *id405 +- old_id: II.T41 + new_id: cid001402 + primary_alias: THM0106 + type: theorem + title: Liouville Categorical Dodge + status: canonical + release_lines: &id406 + - corpus_v3_working + - corpus_v2 +- old_id: liouville-categorical-dodge + new_id: cid001402 + primary_alias: THM0106 + type: theorem + title: Liouville Categorical Dodge + status: canonical + release_lines: *id406 +- old_id: thm:liouville-dodge + new_id: cid001402 + primary_alias: THM0106 + type: theorem + title: Liouville Categorical Dodge + status: canonical + release_lines: *id406 +- old_id: II.T42 + new_id: cid001403 + primary_alias: THM0107 + type: theorem + title: Categoricity + status: canonical + release_lines: &id407 + - corpus_v3_working + - corpus_v2 +- old_id: categoricity + new_id: cid001403 + primary_alias: THM0107 + type: theorem + title: Categoricity + status: canonical + release_lines: *id407 +- old_id: II.T43 + new_id: cid001404 + primary_alias: THM0108 + type: theorem + title: Structural Incompatibility of Unique Omega and Archimedean Density + status: canonical + release_lines: &id408 + - corpus_v3_working + - corpus_v2 +- old_id: structural-incompatibility-of-unique-omega-and-archimedean-density + new_id: cid001404 + primary_alias: THM0108 + type: theorem + title: Structural Incompatibility of Unique Omega and Archimedean Density + status: canonical + release_lines: *id408 +- old_id: thm:structural-incompatibility + new_id: cid001404 + primary_alias: THM0108 + type: theorem + title: Structural Incompatibility of Unique Omega and Archimedean Density + status: canonical + release_lines: *id408 +- old_id: II.T44 + new_id: cid001405 + primary_alias: THM0109 + type: theorem + title: Asymmetric Determination + status: canonical + release_lines: &id409 + - corpus_v3_working + - corpus_v2 +- old_id: asymmetric-determination + new_id: cid001405 + primary_alias: THM0109 + type: theorem + title: Asymmetric Determination + status: canonical + release_lines: *id409 +- old_id: thm:asymmetric-determination + new_id: cid001405 + primary_alias: THM0109 + type: theorem + title: Asymmetric Determination + status: canonical + release_lines: *id409 +- old_id: II.T45 + new_id: cid001406 + primary_alias: THM0110 + type: theorem + title: Parallel Preservation + status: canonical + release_lines: &id410 + - corpus_v3_working + - corpus_v2 +- old_id: parallel-preservation + new_id: cid001406 + primary_alias: THM0110 + type: theorem + title: Parallel Preservation + status: canonical + release_lines: *id410 +- old_id: thm:parallel-preservation + new_id: cid001406 + primary_alias: THM0110 + type: theorem + title: Parallel Preservation + status: canonical + release_lines: *id410 +- old_id: II.T46 + new_id: cid001407 + primary_alias: THM0111 + type: theorem + title: Fourth Quadrant Resolution + status: canonical + release_lines: &id411 + - corpus_v3_working + - corpus_v2 +- old_id: fourth-quadrant-resolution + new_id: cid001407 + primary_alias: THM0111 + type: theorem + title: Fourth Quadrant Resolution + status: canonical + release_lines: *id411 +- old_id: thm:fourth-quadrant-resolution + new_id: cid001407 + primary_alias: THM0111 + type: theorem + title: Fourth Quadrant Resolution + status: canonical + release_lines: *id411 +- old_id: II.T47 + new_id: cid001408 + primary_alias: THM0112 + type: theorem + title: Simultaneous Rung Theorem + status: canonical + release_lines: &id412 + - corpus_v3_working + - corpus_v2 +- old_id: simultaneous-rung-theorem + new_id: cid001408 + primary_alias: THM0112 + type: theorem + title: Simultaneous Rung Theorem + status: canonical + release_lines: *id412 +- old_id: thm:simultaneous-rung + new_id: cid001408 + primary_alias: THM0112 + type: theorem + title: Simultaneous Rung Theorem + status: canonical + release_lines: *id412 +- old_id: II.T48 + new_id: cid001409 + primary_alias: THM0113 + type: theorem + title: Fourth Quadrant Ladder Collapse + status: canonical + release_lines: &id413 + - corpus_v3_working + - corpus_v2 +- old_id: fourth-quadrant-ladder-collapse + new_id: cid001409 + primary_alias: THM0113 + type: theorem + title: Fourth Quadrant Ladder Collapse + status: canonical + release_lines: *id413 +- old_id: thm:ladder-collapse + new_id: cid001409 + primary_alias: THM0113 + type: theorem + title: Fourth Quadrant Ladder Collapse + status: canonical + release_lines: *id413 +- old_id: II.T49 + new_id: cid001410 + primary_alias: THM0114 + type: theorem + title: Geometric Bi-Square Theorem + status: canonical + release_lines: &id414 + - corpus_v3_working + - corpus_v2 +- old_id: geometric-bi-square-theorem + new_id: cid001410 + primary_alias: THM0114 + type: theorem + title: Geometric Bi-Square Theorem + status: canonical + release_lines: *id414 +- old_id: thm:geometric-bisquare + new_id: cid001410 + primary_alias: THM0114 + type: theorem + title: Geometric Bi-Square Theorem + status: canonical + release_lines: *id414 +- old_id: II.T50 + new_id: cid001411 + primary_alias: THM0115 + type: theorem + title: Flat Connection Existence + status: canonical + release_lines: &id415 + - corpus_v3_working + - corpus_v2 +- old_id: flat-connection-existence + new_id: cid001411 + primary_alias: THM0115 + type: theorem + title: Flat Connection Existence + status: canonical + release_lines: *id415 +- old_id: thm:flat-connection + new_id: cid001411 + primary_alias: THM0115 + type: theorem + title: Flat Connection Existence + status: canonical + release_lines: *id415 +- old_id: II.T51 + new_id: cid001412 + primary_alias: THM0116 + type: theorem + title: Flat Curvature Vanishing + status: canonical + release_lines: &id416 + - corpus_v3_working + - corpus_v2 +- old_id: flat-curvature-vanishing + new_id: cid001412 + primary_alias: THM0116 + type: theorem + title: Flat Curvature Vanishing + status: canonical + release_lines: *id416 +- old_id: thm:flat-curvature + new_id: cid001412 + primary_alias: THM0116 + type: theorem + title: Flat Curvature Vanishing + status: canonical + release_lines: *id416 +- old_id: II.T52 + new_id: cid001413 + primary_alias: THM0117 + type: theorem + title: Lemniscate Holonomy + status: canonical + release_lines: &id417 + - corpus_v3_working + - corpus_v2 +- old_id: lemniscate-holonomy + new_id: cid001413 + primary_alias: THM0117 + type: theorem + title: Lemniscate Holonomy + status: canonical + release_lines: *id417 +- old_id: thm:lemniscate-holonomy + new_id: cid001413 + primary_alias: THM0117 + type: theorem + title: Lemniscate Holonomy + status: canonical + release_lines: *id417 +- old_id: II.T53 + new_id: cid001414 + primary_alias: THM0118 + type: theorem + title: Cauchy-Schwarz Inequality + status: canonical + release_lines: &id418 + - corpus_v3_working + - corpus_v2 +- old_id: cauchy-schwarz-inequality + new_id: cid001414 + primary_alias: THM0118 + type: theorem + title: Cauchy-Schwarz Inequality + status: canonical + release_lines: *id418 +- old_id: thm:cauchy-schwarz + new_id: cid001414 + primary_alias: THM0118 + type: theorem + title: Cauchy-Schwarz Inequality + status: canonical + release_lines: *id418 +- old_id: II.T54 + new_id: cid001415 + primary_alias: THM0119 + type: theorem + title: Tower Coherence + status: canonical + release_lines: &id419 + - corpus_v3_working + - corpus_v2 +- old_id: thm:tower-coherence + new_id: cid001415 + primary_alias: THM0119 + type: theorem + title: Tower Coherence + status: canonical + release_lines: *id419 +- old_id: II.T55 + new_id: cid001416 + primary_alias: THM0120 + type: theorem + title: H⁰ = Global Sections + status: canonical + release_lines: &id420 + - corpus_v3_working + - corpus_v2 +- old_id: h-global-sections + new_id: cid001416 + primary_alias: THM0120 + type: theorem + title: H⁰ = Global Sections + status: canonical + release_lines: *id420 +- old_id: thm:h0-global + new_id: cid001416 + primary_alias: THM0120 + type: theorem + title: H⁰ = Global Sections + status: canonical + release_lines: *id420 +- old_id: II.T56 + new_id: cid001417 + primary_alias: THM0121 + type: theorem + title: Trade-Off Atomicity + status: canonical + release_lines: &id421 + - corpus_v3_working + - corpus_v2 +- old_id: trade-off-atomicity + new_id: cid001417 + primary_alias: THM0121 + type: theorem + title: Trade-Off Atomicity + status: canonical + release_lines: *id421 +- old_id: thm:trade-off-atomicity + new_id: cid001417 + primary_alias: THM0121 + type: theorem + title: Trade-Off Atomicity + status: canonical + release_lines: *id421 +- old_id: II.T57 + new_id: cid001418 + primary_alias: THM0122 + type: theorem + title: Admissible Germs Over Omega-Tails + status: canonical + release_lines: &id422 + - corpus_v3_working + - corpus_v2 +- old_id: admissible-germs-over-omega-tails + new_id: cid001418 + primary_alias: THM0122 + type: theorem + title: Admissible Germs Over Omega-Tails + status: canonical + release_lines: *id422 +- old_id: thm:admissible-germs + new_id: cid001418 + primary_alias: THM0122 + type: theorem + title: Admissible Germs Over Omega-Tails + status: canonical + release_lines: *id422 +- old_id: II.T58 + new_id: cid001419 + primary_alias: THM0123 + type: theorem + title: Holomorphy Before Mappings (Ontological Primary) + status: canonical + release_lines: &id423 + - corpus_v3_working + - corpus_v2 +- old_id: holomorphy-before-mappings-ontological-primary + new_id: cid001419 + primary_alias: THM0123 + type: theorem + title: Holomorphy Before Mappings (Ontological Primary) + status: canonical + release_lines: *id423 +- old_id: thm:holomorphy-primary + new_id: cid001419 + primary_alias: THM0123 + type: theorem + title: Holomorphy Before Mappings (Ontological Primary) + status: canonical + release_lines: *id423 +- old_id: II.T59 + new_id: cid001420 + primary_alias: THM0124 + type: theorem + title: Earned Scalar Codomain (D Is Forced) + status: canonical + release_lines: &id424 + - corpus_v3_working + - corpus_v2 +- old_id: earned-scalar-codomain-d-is-forced + new_id: cid001420 + primary_alias: THM0124 + type: theorem + title: Earned Scalar Codomain (D Is Forced) + status: canonical + release_lines: *id424 +- old_id: thm:earned-codomain + new_id: cid001420 + primary_alias: THM0124 + type: theorem + title: Earned Scalar Codomain (D Is Forced) + status: canonical + release_lines: *id424 +- old_id: II.T60 + new_id: cid001421 + primary_alias: THM0125 + type: theorem + title: Wave-Equation Cauchy-Riemann + status: canonical + release_lines: &id425 + - corpus_v3_working + - corpus_v2 +- old_id: wave-equation-cauchy-riemann + new_id: cid001421 + primary_alias: THM0125 + type: theorem + title: Wave-Equation Cauchy-Riemann + status: canonical + release_lines: *id425 +- old_id: thm:wave-cr + new_id: cid001421 + primary_alias: THM0125 + type: theorem + title: Wave-Equation Cauchy-Riemann + status: canonical + release_lines: *id425 +- old_id: II.T61 + new_id: cid001422 + primary_alias: THM0126 + type: theorem + title: Diagonal Discipline (Four Prohibitions) + status: canonical + release_lines: &id426 + - corpus_v3_working + - corpus_v2 +- old_id: diagonal-discipline-four-prohibitions + new_id: cid001422 + primary_alias: THM0126 + type: theorem + title: Diagonal Discipline (Four Prohibitions) + status: canonical + release_lines: *id426 +- old_id: thm:diagonal-discipline + new_id: cid001422 + primary_alias: THM0126 + type: theorem + title: Diagonal Discipline (Four Prohibitions) + status: canonical + release_lines: *id426 +- old_id: II.T62 + new_id: cid001423 + primary_alias: THM0127 + type: theorem + title: Earned Categorical Machine + status: canonical + release_lines: &id427 + - corpus_v3_working + - corpus_v2 +- old_id: earned-categorical-machine + new_id: cid001423 + primary_alias: THM0127 + type: theorem + title: Earned Categorical Machine + status: canonical + release_lines: *id427 +- old_id: thm:main-earned-cat + new_id: cid001423 + primary_alias: THM0127 + type: theorem + title: Earned Categorical Machine + status: canonical + release_lines: *id427 +- old_id: II.T63 + new_id: cid001424 + primary_alias: THM0128 + type: theorem + title: Sigma-Anti-Holomorphy and Idempotent-Supported Holomorphy + status: canonical + release_lines: &id428 + - corpus_v3_working + - corpus_v2 +- old_id: sigma-anti-holomorphy-and-idempotent-supported-holomorphy + new_id: cid001424 + primary_alias: THM0128 + type: theorem + title: Sigma-Anti-Holomorphy and Idempotent-Supported Holomorphy + status: canonical + release_lines: *id428 +- old_id: thm:idem-supported-holomorphy + new_id: cid001424 + primary_alias: THM0128 + type: theorem + title: Sigma-Anti-Holomorphy and Idempotent-Supported Holomorphy + status: canonical + release_lines: *id428 +- old_id: II.T64 + new_id: cid001425 + primary_alias: THM0129 + type: theorem + title: HolEnd_tau via Pre-Yoneda Collapse + status: canonical + release_lines: &id429 + - corpus_v3_working + - corpus_v2 +- old_id: holend-tau-via-pre-yoneda-collapse + new_id: cid001425 + primary_alias: THM0129 + type: theorem + title: HolEnd_tau via Pre-Yoneda Collapse + status: canonical + release_lines: *id429 +- old_id: thm:main-holend + new_id: cid001425 + primary_alias: THM0129 + type: theorem + title: HolEnd_tau via Pre-Yoneda Collapse + status: canonical + release_lines: *id429 +- old_id: II.T65 + new_id: cid001426 + primary_alias: THM0130 + type: theorem + title: Hinge 5 Integration Tabulation + status: canonical + release_lines: &id430 + - corpus_v3_working + - corpus_v2 +- old_id: hinge-5-integration-tabulation + new_id: cid001426 + primary_alias: THM0130 + type: theorem + title: Hinge 5 Integration Tabulation + status: canonical + release_lines: *id430 +- old_id: II.T66 + new_id: cid001427 + primary_alias: THM0131 + type: theorem + title: Topos Structure of Cat_tau + status: canonical + release_lines: &id431 + - corpus_v3_working + - corpus_v2 +- old_id: topos-structure-of-cat-tau + new_id: cid001427 + primary_alias: THM0131 + type: theorem + title: Topos Structure of Cat_tau + status: canonical + release_lines: *id431 +- old_id: thm:main-topos-structure + new_id: cid001427 + primary_alias: THM0131 + type: theorem + title: Topos Structure of Cat_tau + status: canonical + release_lines: *id431 +- old_id: II.T67 + new_id: cid001428 + primary_alias: THM0132 + type: theorem + title: Paraconsistent Soundness of Internal Logic + status: canonical + release_lines: &id432 + - corpus_v3_working + - corpus_v2 +- old_id: paraconsistent-soundness-of-internal-logic + new_id: cid001428 + primary_alias: THM0132 + type: theorem + title: Paraconsistent Soundness of Internal Logic + status: canonical + release_lines: *id432 +- old_id: thm:main-paraconsistent-soundness + new_id: cid001428 + primary_alias: THM0132 + type: theorem + title: Paraconsistent Soundness of Internal Logic + status: canonical + release_lines: *id432 +- old_id: II.T68 + new_id: cid001429 + primary_alias: THM0133 + type: theorem + title: Circularity Resolution via Omega-Germ Stabilisation + status: canonical + release_lines: &id433 + - corpus_v3_working + - corpus_v2 +- old_id: circularity-resolution-via-omega-germ-stabilisation + new_id: cid001429 + primary_alias: THM0133 + type: theorem + title: Circularity Resolution via Omega-Germ Stabilisation + status: canonical + release_lines: *id433 +- old_id: thm:main-circularity-resolution + new_id: cid001429 + primary_alias: THM0133 + type: theorem + title: Circularity Resolution via Omega-Germ Stabilisation + status: canonical + release_lines: *id433 +- old_id: II.T69 + new_id: cid001430 + primary_alias: THM0134 + type: theorem + title: 'Both = 1: Idempotent Unit of Boolean Sublattice' + status: canonical + release_lines: &id434 + - corpus_v3_working + - corpus_v2 +- old_id: both-1-idempotent-unit-of-boolean-sublattice + new_id: cid001430 + primary_alias: THM0134 + type: theorem + title: 'Both = 1: Idempotent Unit of Boolean Sublattice' + status: canonical + release_lines: *id434 +- old_id: thm:main-both-equals-one + new_id: cid001430 + primary_alias: THM0134 + type: theorem + title: 'Both = 1: Idempotent Unit of Boolean Sublattice' + status: canonical + release_lines: *id434 +- old_id: II.T70 + new_id: cid001431 + primary_alias: THM0135 + type: theorem + title: Hinge 6 Integration Tabulation + status: canonical + release_lines: &id435 + - corpus_v3_working + - corpus_v2 +- old_id: hinge-6-integration-tabulation + new_id: cid001431 + primary_alias: THM0135 + type: theorem + title: Hinge 6 Integration Tabulation + status: canonical + release_lines: *id435 +- old_id: III.D01 + new_id: cid001432 + primary_alias: DEF0202 + type: definition + title: Hartogs Bulk Projection + status: canonical + release_lines: &id436 + - corpus_v3_working + - corpus_v2 +- old_id: hartogs-bulk-projection + new_id: cid001432 + primary_alias: DEF0202 + type: definition + title: Hartogs Bulk Projection + status: canonical + release_lines: *id436 +- old_id: princ:ch01-hartogs-bulk + new_id: cid001432 + primary_alias: DEF0202 + type: definition + title: Hartogs Bulk Projection + status: canonical + release_lines: *id436 +- old_id: III.D02 + new_id: cid001433 + primary_alias: DEF0203 + type: definition + title: Millennium-Problem Gluing Table + status: canonical + release_lines: &id437 + - corpus_v3_working + - corpus_v2 +- old_id: millennium-problem-gluing-table + new_id: cid001433 + primary_alias: DEF0203 + type: definition + title: Millennium-Problem Gluing Table + status: canonical + release_lines: *id437 +- old_id: tab:millennium-gluing + new_id: cid001433 + primary_alias: DEF0203 + type: definition + title: Millennium-Problem Gluing Table + status: canonical + release_lines: *id437 +- old_id: III.D03 + new_id: cid001434 + primary_alias: DEF0204 + type: definition + title: E₁ as Gluing Principle + status: canonical + release_lines: &id438 + - 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corpus_v3_working + - corpus_v2 +- old_id: constructive-hensel-lifting + new_id: cid001603 + primary_alias: THM0146 + type: theorem + title: Constructive Hensel Lifting + status: canonical + release_lines: *id607 +- old_id: thm:constructive-hensel + new_id: cid001603 + primary_alias: THM0146 + type: theorem + title: Constructive Hensel Lifting + status: canonical + release_lines: *id607 +- old_id: III.T12 + new_id: cid001604 + primary_alias: THM0147 + type: theorem + title: Adelic Embedding Theorem + status: canonical + release_lines: &id608 + - corpus_v3_working + - corpus_v2 +- old_id: adelic-embedding-theorem + new_id: cid001604 + primary_alias: THM0147 + type: theorem + title: Adelic Embedding Theorem + status: canonical + release_lines: *id608 +- old_id: thm:adelic-embedding + new_id: cid001604 + primary_alias: THM0147 + type: theorem + title: Adelic Embedding Theorem + status: canonical + release_lines: *id608 +- old_id: III.T13 + new_id: cid001605 + primary_alias: THM0148 + type: theorem + title: Label Convergence + status: canonical + release_lines: &id609 + - corpus_v3_working + - corpus_v2 +- old_id: label-convergence + new_id: cid001605 + primary_alias: THM0148 + type: theorem + title: Label Convergence + status: canonical + release_lines: *id609 +- old_id: thm:label-convergence + new_id: cid001605 + primary_alias: THM0148 + type: theorem + title: Label Convergence + status: canonical + release_lines: *id609 +- old_id: III.T14 + new_id: cid001606 + primary_alias: THM0149 + type: theorem + title: Spectral Trichotomy Lemma + status: canonical + release_lines: &id610 + - corpus_v3_working + - corpus_v2 +- old_id: spectral-trichotomy-lemma + new_id: cid001606 + primary_alias: THM0149 + type: theorem + title: Spectral Trichotomy Lemma + status: canonical + release_lines: *id610 +- old_id: thm:spectral-trichotomy + new_id: cid001606 + primary_alias: THM0149 + type: theorem + title: Spectral Trichotomy Lemma + status: canonical + release_lines: *id610 +- old_id: III.T15 + new_id: cid001607 + primary_alias: THM0150 + type: theorem + title: B/C Non-Collapse Theorem + status: canonical + release_lines: &id611 + - corpus_v3_working + - corpus_v2 +- old_id: b-c-non-collapse-theorem + new_id: cid001607 + primary_alias: THM0150 + type: theorem + title: B/C Non-Collapse Theorem + status: canonical + release_lines: *id611 +- old_id: thm:bc-non-collapse + new_id: cid001607 + primary_alias: THM0150 + type: theorem + title: B/C Non-Collapse Theorem + status: canonical + release_lines: *id611 +- old_id: III.T16 + new_id: cid001608 + primary_alias: THM0151 + type: theorem + title: Bipolar Euler Product + status: canonical + release_lines: &id612 + - corpus_v3_working + - corpus_v2 +- old_id: bipolar-euler-product + new_id: cid001608 + primary_alias: THM0151 + type: theorem + title: Bipolar Euler Product + status: canonical + release_lines: *id612 +- old_id: thm:bipolar-euler-product + new_id: cid001608 + primary_alias: THM0151 + type: theorem + title: Bipolar Euler Product + status: canonical + release_lines: *id612 +- old_id: III.T17 + new_id: cid001609 + primary_alias: THM0152 + type: theorem + title: Self-Adjointness of H_L + status: canonical + release_lines: &id613 + - corpus_v3_working + - corpus_v2 +- old_id: self-adjointness-of-h-l + new_id: cid001609 + primary_alias: THM0152 + type: theorem + title: Self-Adjointness of H_L + status: canonical + release_lines: *id613 +- old_id: thm:self-adjointness-h-l + new_id: cid001609 + primary_alias: THM0152 + type: theorem + title: Self-Adjointness of H_L + status: canonical + release_lines: *id613 +- old_id: III.T18 + new_id: cid001610 + primary_alias: THM0153 + type: theorem + title: Spectral Correspondence Theorem + status: canonical + release_lines: &id614 + - corpus_v3_working + - corpus_v2 +- old_id: spectral-correspondence-theorem + new_id: cid001610 + primary_alias: THM0153 + type: theorem + title: Spectral Correspondence Theorem + status: canonical + release_lines: *id614 +- old_id: thm:spectral-correspondence + new_id: cid001610 + primary_alias: THM0153 + type: theorem + title: Spectral Correspondence Theorem + status: canonical + release_lines: *id614 +- old_id: III.T19 + new_id: cid001611 + primary_alias: THM0154 + type: theorem + title: Critical Line Theorem + status: canonical + release_lines: &id615 + - corpus_v3_working + - corpus_v2 +- old_id: critical-line-theorem + new_id: cid001611 + primary_alias: THM0154 + type: theorem + title: Critical Line Theorem + status: canonical + release_lines: *id615 +- old_id: thm:critical-line-theorem + new_id: cid001611 + primary_alias: THM0154 + type: theorem + title: Critical Line Theorem + status: canonical + release_lines: *id615 +- old_id: III.T20 + new_id: cid001612 + primary_alias: THM0155 + type: theorem + title: Prime Polarity Scaling Theorem + status: canonical + release_lines: &id616 + - corpus_v3_working + - corpus_v2 +- old_id: prime-polarity-scaling-theorem + new_id: cid001612 + primary_alias: THM0155 + type: theorem + title: Prime Polarity Scaling Theorem + status: canonical + release_lines: *id616 +- old_id: thm:prime-polarity-scaling + new_id: cid001612 + primary_alias: THM0155 + type: theorem + title: Prime Polarity Scaling Theorem + status: canonical + release_lines: *id616 +- old_id: III.T21 + new_id: cid001613 + primary_alias: THM0156 + type: theorem + title: τ-Admissibility Collapse Theorem (v1) + status: canonical + release_lines: &id617 + - corpus_v3_working + - corpus_v2 +- old_id: admissibility-collapse-theorem-v1 + new_id: cid001613 + primary_alias: THM0156 + type: theorem + title: τ-Admissibility Collapse Theorem (v1) + status: canonical + release_lines: *id617 +- old_id: thm:tau-admissibility-collapse-v1 + new_id: cid001613 + primary_alias: THM0156 + type: theorem + title: τ-Admissibility Collapse Theorem (v1) + status: canonical + release_lines: *id617 +- old_id: III.T22 + new_id: cid001614 + primary_alias: THM0157 + type: theorem + title: Representation Barrier Theorem + status: canonical + release_lines: &id618 + - corpus_v3_working + - corpus_v2 +- old_id: representation-barrier-theorem + new_id: cid001614 + primary_alias: THM0157 + type: theorem + title: Representation Barrier Theorem + status: canonical + release_lines: *id618 +- old_id: thm:representation-barrier + new_id: cid001614 + primary_alias: THM0157 + type: theorem + title: Representation Barrier Theorem + status: canonical + release_lines: *id618 +- old_id: III.T23 + new_id: cid001615 + primary_alias: THM0158 + type: theorem + title: Master Schema Theorem + status: canonical + release_lines: &id619 + - corpus_v3_working + - corpus_v2 +- old_id: master-schema-theorem + new_id: cid001615 + primary_alias: THM0158 + type: theorem + title: Master Schema Theorem + status: canonical + release_lines: *id619 +- old_id: thm:master-schema + new_id: cid001615 + primary_alias: THM0158 + type: theorem + title: Master Schema Theorem + status: canonical + release_lines: *id619 +- old_id: III.T24 + new_id: cid001616 + primary_alias: THM0159 + type: theorem + title: Hartogs Flow Theorem + status: canonical + release_lines: &id620 + - corpus_v3_working + - corpus_v2 +- old_id: hartogs-flow-theorem + new_id: cid001616 + primary_alias: THM0159 + type: theorem + title: Hartogs Flow Theorem + status: canonical + release_lines: *id620 +- old_id: thm:hartogs-flow-theorem + new_id: cid001616 + primary_alias: THM0159 + type: theorem + title: Hartogs Flow Theorem + status: canonical + release_lines: *id620 +- old_id: III.T25 + new_id: cid001617 + primary_alias: THM0160 + type: theorem + title: Positive Regularity Theorem + status: canonical + release_lines: &id621 + - corpus_v3_working + - corpus_v2 +- old_id: positive-regularity-theorem + new_id: cid001617 + primary_alias: THM0160 + type: theorem + title: Positive Regularity Theorem + status: canonical + release_lines: *id621 +- old_id: thm:positive-regularity + new_id: cid001617 + primary_alias: THM0160 + type: theorem + title: Positive Regularity Theorem + status: canonical + release_lines: *id621 +- old_id: III.T26 + new_id: cid001618 + primary_alias: THM0161 + type: theorem + title: τ-Gap Meta-Theorem + status: canonical + release_lines: &id622 + - corpus_v3_working + - corpus_v2 +- old_id: gap-meta-theorem + new_id: cid001618 + primary_alias: THM0161 + type: theorem + title: τ-Gap Meta-Theorem + status: canonical + release_lines: *id622 +- old_id: thm:tau-gap-meta-theorem + new_id: cid001618 + primary_alias: THM0161 + type: theorem + title: τ-Gap Meta-Theorem + status: canonical + release_lines: *id622 +- old_id: III.T27 + new_id: cid001619 + primary_alias: THM0162 + type: theorem + title: Yang-Mills Gap Theorem + status: canonical + release_lines: &id623 + - corpus_v3_working + - corpus_v2 +- old_id: yang-mills-gap-theorem + new_id: cid001619 + primary_alias: THM0162 + type: theorem + title: Yang-Mills Gap Theorem + status: canonical + release_lines: *id623 +- old_id: thm:yang-mills-gap-theorem + new_id: cid001619 + primary_alias: THM0162 + type: theorem + title: Yang-Mills Gap Theorem + status: canonical + release_lines: *id623 +- old_id: III.T28 + new_id: cid001620 + primary_alias: THM0163 + type: theorem + title: NF-Addressability Theorem + status: canonical + release_lines: &id624 + - corpus_v3_working + - corpus_v2 +- old_id: nf-addressability-theorem + new_id: cid001620 + primary_alias: THM0163 + type: theorem + title: NF-Addressability Theorem + status: canonical + release_lines: *id624 +- old_id: thm:nf-addressability + new_id: cid001620 + primary_alias: THM0163 + type: theorem + title: NF-Addressability Theorem + status: canonical + release_lines: *id624 +- old_id: III.T29 + new_id: cid001621 + primary_alias: THM0164 + type: theorem + title: Physics Layer Assembly + status: canonical + release_lines: &id625 + - corpus_v3_working + - corpus_v2 +- old_id: physics-layer-assembly + new_id: cid001621 + primary_alias: THM0164 + type: theorem + title: Physics Layer Assembly + status: canonical + release_lines: *id625 +- old_id: thm:physics-layer-assembly + new_id: cid001621 + primary_alias: THM0164 + type: theorem + title: Physics Layer Assembly + status: canonical + release_lines: *id625 +- old_id: III.T30 + new_id: cid001622 + primary_alias: THM0165 + type: theorem + title: TTM τ-Nativity + status: canonical + release_lines: &id626 + - corpus_v3_working + - corpus_v2 +- old_id: ttm-nativity + new_id: cid001622 + primary_alias: THM0165 + type: theorem + title: TTM τ-Nativity + status: canonical + release_lines: *id626 +- old_id: thm:ttm-tau-nativity + new_id: cid001622 + primary_alias: THM0165 + type: theorem + title: TTM τ-Nativity + status: canonical + release_lines: *id626 +- old_id: III.T31 + new_id: cid001623 + primary_alias: THM0166 + type: theorem + title: Interface Width Principle + status: canonical + release_lines: &id627 + - corpus_v3_working + - corpus_v2 +- old_id: interface-width-principle + new_id: cid001623 + primary_alias: THM0166 + type: theorem + title: Interface Width Principle + status: canonical + release_lines: *id627 +- old_id: thm:interface-width-principle + new_id: cid001623 + primary_alias: THM0166 + type: theorem + title: Interface Width Principle + status: canonical + release_lines: *id627 +- old_id: III.T32 + new_id: cid001624 + primary_alias: THM0167 + type: theorem + title: Product-Meet Collapse + status: canonical + release_lines: &id628 + - corpus_v3_working + - corpus_v2 +- old_id: product-meet-collapse + new_id: cid001624 + primary_alias: THM0167 + type: theorem + title: Product-Meet Collapse + status: canonical + release_lines: *id628 +- old_id: thm:product-meet-collapse + new_id: cid001624 + primary_alias: THM0167 + type: theorem + title: Product-Meet Collapse + status: canonical + release_lines: *id628 +- old_id: III.T33 + new_id: cid001625 + primary_alias: THM0168 + type: theorem + title: τ-Admissibility Collapse + status: canonical + release_lines: &id629 + - corpus_v3_working + - corpus_v2 +- old_id: admissibility-collapse + new_id: cid001625 + primary_alias: THM0168 + type: theorem + title: τ-Admissibility Collapse + status: canonical + release_lines: *id629 +- old_id: thm:tau-admissibility-collapse + new_id: cid001625 + primary_alias: THM0168 + type: theorem + title: τ-Admissibility Collapse + status: canonical + release_lines: *id629 +- old_id: III.T34 + new_id: cid001626 + primary_alias: THM0169 + type: theorem + title: No Barrier Theorem + status: canonical + release_lines: &id630 + - corpus_v3_working + - corpus_v2 +- old_id: no-barrier-theorem + new_id: cid001626 + primary_alias: THM0169 + type: theorem + title: No Barrier Theorem + status: canonical + release_lines: *id630 +- old_id: thm:no-barrier + new_id: cid001626 + primary_alias: THM0169 + type: theorem + title: No Barrier Theorem + status: canonical + release_lines: *id630 +- old_id: III.T35 + new_id: cid001627 + primary_alias: THM0170 + type: theorem + title: BSD Coherence Theorem + status: canonical + release_lines: &id631 + - corpus_v3_working + - corpus_v2 +- old_id: bsd-coherence-theorem + new_id: cid001627 + primary_alias: THM0170 + type: theorem + title: BSD Coherence Theorem + status: canonical + release_lines: *id631 +- old_id: thm:bsd-coherence-theorem + new_id: cid001627 + primary_alias: THM0170 + type: theorem + title: BSD Coherence Theorem + status: canonical + release_lines: *id631 +- old_id: III.T36 + new_id: cid001628 + primary_alias: THM0171 + type: theorem + title: Functoriality Theorem + status: canonical + release_lines: &id632 + - corpus_v3_working + - corpus_v2 +- old_id: functoriality-theorem + new_id: cid001628 + primary_alias: THM0171 + type: theorem + title: Functoriality Theorem + status: canonical + release_lines: *id632 +- old_id: thm:functoriality-theorem + new_id: cid001628 + primary_alias: THM0171 + type: theorem + title: Functoriality Theorem + status: canonical + release_lines: *id632 +- old_id: III.T37 + new_id: cid001629 + primary_alias: THM0172 + type: theorem + title: Base Change-Transfer Naturality + status: canonical + release_lines: &id633 + - corpus_v3_working + - corpus_v2 +- old_id: base-change-transfer-naturality + new_id: cid001629 + primary_alias: THM0172 + type: theorem + title: Base Change-Transfer Naturality + status: canonical + release_lines: *id633 +- old_id: thm:base-change-transfer + new_id: cid001629 + primary_alias: THM0172 + type: theorem + title: Base Change-Transfer Naturality + status: canonical + release_lines: *id633 +- old_id: III.T38 + new_id: cid001630 + primary_alias: THM0173 + type: theorem + title: Finite Factorization Pasting + status: canonical + release_lines: &id634 + - corpus_v3_working + - corpus_v2 +- old_id: finite-factorization-pasting + new_id: cid001630 + primary_alias: THM0173 + type: theorem + title: Finite Factorization Pasting + status: canonical + release_lines: *id634 +- old_id: thm:finite-factorization-pasting + new_id: cid001630 + primary_alias: THM0173 + type: theorem + title: Finite Factorization Pasting + status: canonical + release_lines: *id634 +- old_id: III.T39 + new_id: cid001631 + primary_alias: THM0174 + type: theorem + title: Enriched Bi-Square Comparison + status: canonical + release_lines: &id635 + - corpus_v3_working + - corpus_v2 +- old_id: enriched-bi-square-comparison + new_id: cid001631 + primary_alias: THM0174 + type: theorem + title: Enriched Bi-Square Comparison + status: canonical + release_lines: *id635 +- old_id: thm:enriched-bi-square-comparison + new_id: cid001631 + primary_alias: THM0174 + type: theorem + title: Enriched Bi-Square Comparison + status: canonical + release_lines: *id635 +- old_id: III.T40 + new_id: cid001632 + primary_alias: THM0175 + type: theorem + title: Enrichment Tower Assembly + status: canonical + release_lines: &id636 + - corpus_v3_working + - corpus_v2 +- old_id: enrichment-tower-assembly + new_id: cid001632 + primary_alias: THM0175 + type: theorem + title: Enrichment Tower Assembly + status: canonical + release_lines: *id636 +- old_id: thm:enrichment-tower-assembly + new_id: cid001632 + primary_alias: THM0175 + type: theorem + title: Enrichment Tower Assembly + status: canonical + release_lines: *id636 +- old_id: III.T41 + new_id: cid001633 + primary_alias: THM0176 + type: theorem + title: Hinge Theorem + status: canonical + release_lines: &id637 + - corpus_v3_working + - corpus_v2 +- old_id: hinge-theorem + new_id: cid001633 + primary_alias: THM0176 + type: theorem + title: Hinge Theorem + status: canonical + release_lines: *id637 +- old_id: thm:hinge-theorem + new_id: cid001633 + primary_alias: THM0176 + type: theorem + title: Hinge Theorem + status: canonical + release_lines: *id637 +- old_id: III.T42 + new_id: cid001634 + primary_alias: THM0177 + type: theorem + title: No Knobs Theorem + status: canonical + release_lines: &id638 + - corpus_v3_working + - corpus_v2 +- old_id: no-knobs-theorem + new_id: cid001634 + primary_alias: THM0177 + type: theorem + title: No Knobs Theorem + status: canonical + release_lines: *id638 +- old_id: thm:no-knobs-theorem + new_id: cid001634 + primary_alias: THM0177 + type: theorem + title: No Knobs Theorem + status: canonical + release_lines: *id638 +- old_id: III.T43 + new_id: cid001635 + primary_alias: THM0178 + type: theorem + title: Move-Bridge Correspondence + status: canonical + release_lines: &id639 + - corpus_v3_working + - corpus_v2 +- old_id: move-bridge-correspondence + new_id: cid001635 + primary_alias: THM0178 + type: theorem + title: Move-Bridge Correspondence + status: canonical + release_lines: *id639 +- old_id: thm:move-bridge-correspondence + new_id: cid001635 + primary_alias: THM0178 + type: theorem + title: Move-Bridge Correspondence + status: canonical + release_lines: *id639 +- old_id: III.T44 + new_id: cid001636 + primary_alias: THM0179 + type: theorem + title: Incompleteness as VM Boundary + status: canonical + release_lines: &id640 + - corpus_v3_working + - corpus_v2 +- old_id: incompleteness-as-vm-boundary + new_id: cid001636 + primary_alias: THM0179 + type: theorem + title: Incompleteness as VM Boundary + status: canonical + release_lines: *id640 +- old_id: thm:incompleteness-as-vm-boundary + new_id: cid001636 + primary_alias: THM0179 + type: theorem + title: Incompleteness as VM Boundary + status: canonical + release_lines: *id640 +- old_id: III.T45 + new_id: cid001637 + primary_alias: THM0180 + type: theorem + title: RH Bridge Three-Layer Structure + status: canonical + release_lines: &id641 + - corpus_v3_working + - corpus_v2 +- old_id: rh-bridge-three-layer-structure + new_id: cid001637 + primary_alias: THM0180 + type: theorem + title: RH Bridge Three-Layer Structure + status: canonical + release_lines: *id641 +- old_id: thm:rh-bridge-three-layer + new_id: cid001637 + primary_alias: THM0180 + type: theorem + title: RH Bridge Three-Layer Structure + status: canonical + release_lines: *id641 +- old_id: III.T46 + new_id: cid001638 + primary_alias: THM0181 + type: theorem + title: Bridge Ledger + status: canonical + release_lines: &id642 + - corpus_v3_working + - corpus_v2 +- old_id: bridge-ledger + new_id: cid001638 + primary_alias: THM0181 + type: theorem + title: Bridge Ledger + status: canonical + release_lines: *id642 +- old_id: thm:bridge-ledger + new_id: cid001638 + primary_alias: THM0181 + type: theorem + title: Bridge Ledger + status: canonical + release_lines: *id642 +- old_id: III.T47 + new_id: cid001639 + primary_alias: THM0182 + type: theorem + title: Honest Claim Theorem + status: canonical + release_lines: &id643 + - corpus_v3_working + - corpus_v2 +- old_id: honest-claim-theorem + new_id: cid001639 + primary_alias: THM0182 + type: theorem + title: Honest Claim Theorem + status: canonical + release_lines: *id643 +- old_id: thm:honest-claim + new_id: cid001639 + primary_alias: THM0182 + type: theorem + title: Honest Claim Theorem + status: canonical + release_lines: *id643 +- old_id: III.T48 + new_id: cid001640 + primary_alias: THM0183 + type: theorem + title: Four Paradox Diagnostic + status: canonical + release_lines: &id644 + - corpus_v3_working + - corpus_v2 +- old_id: four-paradox-diagnostic + new_id: cid001640 + primary_alias: THM0183 + type: theorem + title: Four Paradox Diagnostic + status: canonical + release_lines: *id644 +- old_id: thm:four-paradox-diagnostic + new_id: cid001640 + primary_alias: THM0183 + type: theorem + title: Four Paradox Diagnostic + status: canonical + release_lines: *id644 +- old_id: III.T49 + new_id: cid001641 + primary_alias: THM0184 + type: theorem + title: Applied Saturation + status: canonical + release_lines: &id645 + - corpus_v3_working + - corpus_v2 +- old_id: applied-saturation + new_id: cid001641 + primary_alias: THM0184 + type: theorem + title: Applied Saturation + status: canonical + release_lines: *id645 +- old_id: thm:applied-saturation + new_id: cid001641 + primary_alias: THM0184 + type: theorem + title: Applied Saturation + status: canonical + release_lines: *id645 +- old_id: III.T50 + new_id: cid001642 + primary_alias: THM0185 + type: theorem + title: Global Cartesian Gluing Theorem + status: canonical + release_lines: &id646 + - corpus_v3_working + - corpus_v2 +- old_id: global-cartesian-gluing-theorem + new_id: cid001642 + primary_alias: THM0185 + type: theorem + title: Global Cartesian Gluing Theorem + status: canonical + release_lines: *id646 +- old_id: thm:global-cartesian-gluing + new_id: cid001642 + primary_alias: THM0185 + type: theorem + title: Global Cartesian Gluing Theorem + status: canonical + release_lines: *id646 +- old_id: III.T51 + new_id: cid001643 + primary_alias: THM0186 + type: theorem + title: Physical Admissibility Theorem + status: canonical + release_lines: &id647 + - corpus_v3_working + - corpus_v2 +- old_id: physical-admissibility-theorem + new_id: cid001643 + primary_alias: THM0186 + type: theorem + title: Physical Admissibility Theorem + status: canonical + release_lines: *id647 +- old_id: thm:physical-admissibility + new_id: cid001643 + primary_alias: THM0186 + type: theorem + title: Physical Admissibility Theorem + status: canonical + release_lines: *id647 +- old_id: III.T52 + new_id: cid001644 + primary_alias: THM0187 + type: theorem + title: Physical P=NP + status: canonical + release_lines: &id648 + - corpus_v3_working + - corpus_v2 +- old_id: physical-p-np + new_id: cid001644 + primary_alias: THM0187 + type: theorem + title: Physical P=NP + status: canonical + release_lines: *id648 +- old_id: thm:physical-p-equals-np + new_id: cid001644 + primary_alias: THM0187 + type: theorem + title: Physical P=NP + status: canonical + release_lines: *id648 +- old_id: III.T53 + new_id: cid001645 + primary_alias: THM0188 + type: theorem + title: Universal Admissibility Theorem + status: canonical + release_lines: &id649 + - corpus_v3_working + - corpus_v2 +- old_id: universal-admissibility-theorem + new_id: cid001645 + primary_alias: THM0188 + type: theorem + title: Universal Admissibility Theorem + status: canonical + release_lines: *id649 +- old_id: thm:universal-admissibility + new_id: cid001645 + primary_alias: THM0188 + type: theorem + title: Universal Admissibility Theorem + status: canonical + release_lines: *id649 +- old_id: III.T54 + new_id: cid001646 + primary_alias: THM0189 + type: theorem + title: Confinement Bridge Identity + status: canonical + release_lines: &id650 + - corpus_v3_working + - corpus_v2 +- old_id: confinement-bridge-identity + new_id: cid001646 + primary_alias: THM0189 + type: theorem + title: Confinement Bridge Identity + status: canonical + release_lines: *id650 +- old_id: thm:confinement-bridge + new_id: cid001646 + primary_alias: THM0189 + type: theorem + title: Confinement Bridge Identity + status: canonical + release_lines: *id650 +- old_id: III.T55 + new_id: cid001647 + primary_alias: THM0190 + type: theorem + title: S-Duality Transport + status: canonical + release_lines: &id651 + - corpus_v3_working + - corpus_v2 +- old_id: s-duality-transport + new_id: cid001647 + primary_alias: THM0190 + type: theorem + title: S-Duality Transport + status: canonical + release_lines: *id651 +- old_id: thm:sduality-transport + new_id: cid001647 + primary_alias: THM0190 + type: theorem + title: S-Duality Transport + status: canonical + release_lines: *id651 +- old_id: III.T56 + new_id: cid001648 + primary_alias: THM0191 + type: theorem + title: Parseval Identity + status: canonical + release_lines: &id652 + - corpus_v3_working + - corpus_v2 +- old_id: parseval-identity + new_id: cid001648 + primary_alias: THM0191 + type: theorem + title: Parseval Identity + status: canonical + release_lines: *id652 +- old_id: thm:parseval + new_id: cid001648 + primary_alias: THM0191 + type: theorem + title: Parseval Identity + status: canonical + release_lines: *id652 +- old_id: III.T57 + new_id: cid001649 + primary_alias: THM0192 + type: theorem + title: Operational Closure Theorem + status: canonical + release_lines: &id653 + - corpus_v3_working + - corpus_v2 +- old_id: operational-closure-theorem + new_id: cid001649 + primary_alias: THM0192 + type: theorem + title: Operational Closure Theorem + status: canonical + release_lines: *id653 +- old_id: thm:operational-closure + new_id: cid001649 + primary_alias: THM0192 + type: theorem + title: Operational Closure Theorem + status: canonical + release_lines: *id653 +- old_id: III.T58 + new_id: cid001650 + primary_alias: THM0193 + type: theorem + title: E₃ Self-Model Completeness + status: canonical + release_lines: &id654 + - corpus_v3_working + - corpus_v2 +- old_id: e-self-model-completeness + new_id: cid001650 + primary_alias: THM0193 + type: theorem + title: E₃ Self-Model Completeness + status: canonical + release_lines: *id654 +- old_id: thm:e3-self-model-complete + new_id: cid001650 + primary_alias: THM0193 + type: theorem + title: E₃ Self-Model Completeness + status: canonical + release_lines: *id654 +- old_id: III.T59 + new_id: cid001651 + primary_alias: THM0194 + type: theorem + title: Arithmetic Faithfulness + status: canonical + release_lines: &id655 + - corpus_v3_working + - corpus_v2 +- old_id: arithmetic-faithfulness + new_id: cid001651 + primary_alias: THM0194 + type: theorem + title: Arithmetic Faithfulness + status: canonical + release_lines: *id655 +- old_id: thm:arith-faithful + new_id: cid001651 + primary_alias: THM0194 + type: theorem + title: Arithmetic Faithfulness + status: canonical + release_lines: *id655 +- old_id: III.T60 + new_id: cid001652 + primary_alias: THM0195 + type: theorem + title: Topological Faithfulness + status: canonical + release_lines: &id656 + - corpus_v3_working + - corpus_v2 +- old_id: topological-faithfulness + new_id: cid001652 + primary_alias: THM0195 + type: theorem + title: Topological Faithfulness + status: canonical + release_lines: *id656 +- old_id: thm:topo-faithful + new_id: cid001652 + primary_alias: THM0195 + type: theorem + title: Topological Faithfulness + status: canonical + release_lines: *id656 +- old_id: III.T61 + new_id: cid001653 + primary_alias: THM0196 + type: theorem + title: Translation Failure Boundary + status: canonical + release_lines: &id657 + - corpus_v3_working + - corpus_v2 +- old_id: translation-failure-boundary + new_id: cid001653 + primary_alias: THM0196 + type: theorem + title: Translation Failure Boundary + status: canonical + release_lines: *id657 +- old_id: thm:translation-failure-boundary + new_id: cid001653 + primary_alias: THM0196 + type: theorem + title: Translation Failure Boundary + status: canonical + release_lines: *id657 +- old_id: III.T62 + new_id: cid001654 + primary_alias: THM0197 + type: theorem + title: NS Flow Causal Arrow + status: canonical + release_lines: &id658 + - corpus_v3_working + - corpus_v2 +- old_id: ns-flow-causal-arrow + new_id: cid001654 + primary_alias: THM0197 + type: theorem + title: NS Flow Causal Arrow + status: canonical + release_lines: *id658 +- old_id: thm:ns-causal-arrow + new_id: cid001654 + primary_alias: THM0197 + type: theorem + title: NS Flow Causal Arrow + status: canonical + release_lines: *id658 +- old_id: III.T63 + new_id: cid001655 + primary_alias: THM0198 + type: theorem + title: P vs NP Forbidden Triple + status: canonical + release_lines: &id659 + - corpus_v3_working + - corpus_v2 +- old_id: p-vs-np-forbidden-triple + new_id: cid001655 + primary_alias: THM0198 + type: theorem + title: P vs NP Forbidden Triple + status: canonical + release_lines: *id659 +- old_id: thm:pvsnp-forbidden-triple + new_id: cid001655 + primary_alias: THM0198 + type: theorem + title: P vs NP Forbidden Triple + status: canonical + release_lines: *id659 +- old_id: III.T64 + new_id: cid001656 + primary_alias: THM0199 + type: theorem + title: Goldbach at Primorial Levels + status: canonical + release_lines: &id660 + - corpus_v3_working + - corpus_v2 +- old_id: goldbach-at-primorial-levels + new_id: cid001656 + primary_alias: THM0199 + type: theorem + title: Goldbach at Primorial Levels + status: canonical + release_lines: *id660 +- old_id: thm:goldbach-primorial + new_id: cid001656 + primary_alias: THM0199 + type: theorem + title: Goldbach at Primorial Levels + status: canonical + release_lines: *id660 +- old_id: III.T65 + new_id: cid001657 + primary_alias: THM0200 + type: theorem + title: ABC at Primorial Levels + status: canonical + release_lines: &id661 + - corpus_v3_working + - corpus_v2 +- old_id: abc-at-primorial-levels + new_id: cid001657 + primary_alias: THM0200 + type: theorem + title: ABC at Primorial Levels + status: canonical + release_lines: *id661 +- old_id: thm:abc-primorial + new_id: cid001657 + primary_alias: THM0200 + type: theorem + title: ABC at Primorial Levels + status: canonical + release_lines: *id661 +- old_id: III.T66 + new_id: cid001658 + primary_alias: THM0201 + type: theorem + title: Sieve Correctness + status: canonical + release_lines: &id662 + - corpus_v3_working + - corpus_v2 +- old_id: sieve-correctness + new_id: cid001658 + primary_alias: THM0201 + type: theorem + title: Sieve Correctness + status: canonical + release_lines: *id662 +- old_id: thm:sieve-correct + new_id: cid001658 + primary_alias: THM0201 + type: theorem + title: Sieve Correctness + status: canonical + release_lines: *id662 +- old_id: III.T67 + new_id: cid001659 + primary_alias: THM0202 + type: theorem + title: Sieve-Tower Compatibility + status: canonical + release_lines: &id663 + - corpus_v3_working + - corpus_v2 +- old_id: sieve-tower-compatibility + new_id: cid001659 + primary_alias: THM0202 + type: theorem + title: Sieve-Tower Compatibility + status: canonical + release_lines: *id663 +- old_id: thm:sieve-tower + new_id: cid001659 + primary_alias: THM0202 + type: theorem + title: Sieve-Tower Compatibility + status: canonical + release_lines: *id663 +- old_id: III.T68 + new_id: cid001660 + primary_alias: THM0203 + type: theorem + title: Goldbach Verified to 500 + status: canonical + release_lines: &id664 + - corpus_v3_working + - corpus_v2 +- old_id: goldbach-verified-to-500 + new_id: cid001660 + primary_alias: THM0203 + type: theorem + title: Goldbach Verified to 500 + status: canonical + release_lines: *id664 +- old_id: thm:goldbach-500 + new_id: cid001660 + primary_alias: THM0203 + type: theorem + title: Goldbach Verified to 500 + status: canonical + release_lines: *id664 +- old_id: III.T69 + new_id: cid001661 + primary_alias: THM0204 + type: theorem + title: Goldbach at Primorial M₄ + status: canonical + release_lines: &id665 + - corpus_v3_working + - corpus_v2 +- old_id: goldbach-at-primorial-m + new_id: cid001661 + primary_alias: THM0204 + type: theorem + title: Goldbach at Primorial M₄ + status: canonical + release_lines: *id665 +- old_id: thm:goldbach-primorial-m4 + new_id: cid001661 + primary_alias: THM0204 + type: theorem + title: Goldbach at Primorial M₄ + status: canonical + release_lines: *id665 +- old_id: III.T70 + new_id: cid001662 + primary_alias: THM0205 + type: theorem + title: Partition Growth + status: canonical + release_lines: &id666 + - corpus_v3_working + - corpus_v2 +- old_id: partition-growth + new_id: cid001662 + primary_alias: THM0205 + type: theorem + title: Partition Growth + status: canonical + release_lines: *id666 +- old_id: thm:partition-growth + new_id: cid001662 + primary_alias: THM0205 + type: theorem + title: Partition Growth + status: canonical + release_lines: *id666 +- old_id: III.T71 + new_id: cid001663 + primary_alias: THM0206 + type: theorem + title: Obstruction Bounded + status: canonical + release_lines: &id667 + - corpus_v3_working + - corpus_v2 +- old_id: obstruction-bounded + new_id: cid001663 + primary_alias: THM0206 + type: theorem + title: Obstruction Bounded + status: canonical + release_lines: *id667 +- old_id: thm:obstruction-bounded + new_id: cid001663 + primary_alias: THM0206 + type: theorem + title: Obstruction Bounded + status: canonical + release_lines: *id667 +- old_id: III.T72 + new_id: cid001664 + primary_alias: THM0207 + type: theorem + title: Twin Primes to 500 + status: canonical + release_lines: &id668 + - corpus_v3_working + - corpus_v2 +- old_id: twin-primes-to-500 + new_id: cid001664 + primary_alias: THM0207 + type: theorem + title: Twin Primes to 500 + status: canonical + release_lines: *id668 +- old_id: thm:twin-500 + new_id: cid001664 + primary_alias: THM0207 + type: theorem + title: Twin Primes to 500 + status: canonical + release_lines: *id668 +- old_id: III.T73 + new_id: cid001665 + primary_alias: THM0208 + type: theorem + title: Twin Density Primorial + status: canonical + release_lines: &id669 + - corpus_v3_working + - corpus_v2 +- old_id: twin-density-primorial + new_id: cid001665 + primary_alias: THM0208 + type: theorem + title: Twin Density Primorial + status: canonical + release_lines: *id669 +- old_id: thm:twin-density + new_id: cid001665 + primary_alias: THM0208 + type: theorem + title: Twin Density Primorial + status: canonical + release_lines: *id669 +- old_id: III.T74 + new_id: cid001666 + primary_alias: THM0209 + type: theorem + title: HL Constant Convergence + status: canonical + release_lines: &id670 + - corpus_v3_working + - corpus_v2 +- old_id: hl-constant-convergence + new_id: cid001666 + primary_alias: THM0209 + type: theorem + title: HL Constant Convergence + status: canonical + release_lines: *id670 +- old_id: thm:hl-decreasing + new_id: cid001666 + primary_alias: THM0209 + type: theorem + title: HL Constant Convergence + status: canonical + release_lines: *id670 +- old_id: III.T75 + new_id: cid001667 + primary_alias: THM0210 + type: theorem + title: CRT Admissible Positive + status: canonical + release_lines: &id671 + - corpus_v3_working + - corpus_v2 +- old_id: crt-admissible-positive + new_id: cid001667 + primary_alias: THM0210 + type: theorem + title: CRT Admissible Positive + status: canonical + release_lines: *id671 +- old_id: thm:crt-admissible-pos + new_id: cid001667 + primary_alias: THM0210 + type: theorem + title: CRT Admissible Positive + status: canonical + release_lines: *id671 +- old_id: III.T76 + new_id: cid001668 + primary_alias: THM0211 + type: theorem + title: ABC Quality 100 + status: canonical + release_lines: &id672 + - corpus_v3_working + - corpus_v2 +- old_id: abc-quality-100 + new_id: cid001668 + primary_alias: THM0211 + type: theorem + title: ABC Quality 100 + status: canonical + release_lines: *id672 +- old_id: thm:abc-100 + new_id: cid001668 + primary_alias: THM0211 + type: theorem + title: ABC Quality 100 + status: canonical + release_lines: *id672 +- old_id: III.T77 + new_id: cid001669 + primary_alias: THM0212 + type: theorem + title: Squarefree Dominance + status: canonical + release_lines: &id673 + - corpus_v3_working + - corpus_v2 +- old_id: squarefree-dominance + new_id: cid001669 + primary_alias: THM0212 + type: theorem + title: Squarefree Dominance + status: canonical + release_lines: *id673 +- old_id: thm:squarefree-dominance + new_id: cid001669 + primary_alias: THM0212 + type: theorem + title: Squarefree Dominance + status: canonical + release_lines: *id673 +- old_id: III.T78 + new_id: cid001670 + primary_alias: THM0213 + type: theorem + title: Radical Primorial 5 + status: canonical + release_lines: &id674 + - corpus_v3_working + - corpus_v2 +- old_id: radical-primorial-5 + new_id: cid001670 + primary_alias: THM0213 + type: theorem + title: Radical Primorial 5 + status: canonical + release_lines: *id674 +- old_id: thm:radical-primorial-5 + new_id: cid001670 + primary_alias: THM0213 + type: theorem + title: Radical Primorial 5 + status: canonical + release_lines: *id674 +- old_id: III.T79 + new_id: cid001671 + primary_alias: THM0214 + type: theorem + title: Tower Finite Decidable + status: canonical + release_lines: &id675 + - corpus_v3_working + - corpus_v2 +- old_id: tower-finite-decidable + new_id: cid001671 + primary_alias: THM0214 + type: theorem + title: Tower Finite Decidable + status: canonical + release_lines: *id675 +- old_id: thm:tower-decidable + new_id: cid001671 + primary_alias: THM0214 + type: theorem + title: Tower Finite Decidable + status: canonical + release_lines: *id675 +- old_id: III.T80 + new_id: cid001672 + primary_alias: THM0215 + type: theorem + title: Bridge Necessary Insufficient + status: canonical + release_lines: &id676 + - corpus_v3_working + - corpus_v2 +- old_id: bridge-necessary-insufficient + new_id: cid001672 + primary_alias: THM0215 + type: theorem + title: Bridge Necessary Insufficient + status: canonical + release_lines: *id676 +- old_id: thm:bridge-necessary + new_id: cid001672 + primary_alias: THM0215 + type: theorem + title: Bridge Necessary Insufficient + status: canonical + release_lines: *id676 +- old_id: III.T81 + new_id: cid001673 + primary_alias: THM0216 + type: theorem + title: Countable Profinite Boundary Ring + status: canonical + release_lines: &id677 + - corpus_v3_working + - corpus_v2 +- old_id: countable-profinite-boundary-ring + new_id: cid001673 + primary_alias: THM0216 + type: theorem + title: Countable Profinite Boundary Ring + status: canonical + release_lines: *id677 +- old_id: thm:main-ring + new_id: cid001673 + primary_alias: THM0216 + type: theorem + title: Countable Profinite Boundary Ring + status: canonical + release_lines: *id677 +- old_id: III.T82 + new_id: cid001674 + primary_alias: THM0217 + type: theorem + title: Constructive CRT in the Primorial Ladder + status: canonical + release_lines: &id678 + - corpus_v3_working + - corpus_v2 +- old_id: constructive-crt-in-the-primorial-ladder + new_id: cid001674 + primary_alias: THM0217 + type: theorem + title: Constructive CRT in the Primorial Ladder + status: canonical + release_lines: *id678 +- old_id: thm:main-crt + new_id: cid001674 + primary_alias: THM0217 + type: theorem + title: Constructive CRT in the Primorial Ladder + status: canonical + release_lines: *id678 +- old_id: III.T83 + new_id: cid001675 + primary_alias: THM0218 + type: theorem + title: Teichmueller-Style Primitive-Recursive Lift + status: canonical + release_lines: &id679 + - corpus_v3_working + - corpus_v2 +- old_id: teichmueller-style-primitive-recursive-lift + new_id: cid001675 + primary_alias: THM0218 + type: theorem + title: Teichmueller-Style Primitive-Recursive Lift + status: canonical + release_lines: *id679 +- old_id: thm:main-lift + new_id: cid001675 + primary_alias: THM0218 + type: theorem + title: Teichmueller-Style Primitive-Recursive Lift + status: canonical + release_lines: *id679 +- old_id: III.T84 + new_id: cid001676 + primary_alias: THM0219 + type: theorem + title: Split-Complex Scalar Algebra D + status: canonical + release_lines: &id680 + - corpus_v3_working + - corpus_v2 +- old_id: split-complex-scalar-algebra-d + new_id: cid001676 + primary_alias: THM0219 + type: theorem + title: Split-Complex Scalar Algebra D + status: canonical + release_lines: *id680 +- old_id: thm:main-splitcomplex + new_id: cid001676 + primary_alias: THM0219 + type: theorem + title: Split-Complex Scalar Algebra D + status: canonical + release_lines: *id680 +- old_id: III.T85 + new_id: cid001677 + primary_alias: THM0220 + type: theorem + title: Crossing Mediator iota_tau in D + status: canonical + release_lines: &id681 + - corpus_v3_working + - corpus_v2 +- old_id: crossing-mediator-iota-tau-in-d + new_id: cid001677 + primary_alias: THM0220 + type: theorem + title: Crossing Mediator iota_tau in D + status: canonical + release_lines: *id681 +- old_id: thm:main-iota + new_id: cid001677 + primary_alias: THM0220 + type: theorem + title: Crossing Mediator iota_tau in D + status: canonical + release_lines: *id681 +- old_id: III.T86 + new_id: cid001678 + primary_alias: THM0221 + type: theorem + title: Canonical Uniqueness of D + status: canonical + release_lines: &id682 + - corpus_v3_working + - corpus_v2 +- old_id: canonical-uniqueness-of-d + new_id: cid001678 + primary_alias: THM0221 + type: theorem + title: Canonical Uniqueness of D + status: canonical + release_lines: *id682 +- old_id: thm:main-uniqueness + new_id: cid001678 + primary_alias: THM0221 + type: theorem + title: Canonical Uniqueness of D + status: canonical + release_lines: *id682 +- old_id: III.T87 + new_id: cid001679 + primary_alias: THM0222 + type: theorem + title: Elliptic Complex Exclusion + status: canonical + release_lines: &id683 + - 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corpus_v2 + - corpus_v3_working +- old_id: book-ii-interior-tau-admissible + new_id: cid005172 + primary_alias: MOD0175 + type: formal_module + title: TauLib.BookII.Interior.TauAdmissible + status: canonical + release_lines: *id858 +- old_id: TauLib.BookII.Mirror.DimensionalLadder + new_id: cid005173 + primary_alias: MOD0176 + type: formal_module + title: TauLib.BookII.Mirror.DimensionalLadder + status: canonical + release_lines: &id859 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-mirror-dimensional-ladder + new_id: cid005173 + primary_alias: MOD0176 + type: formal_module + title: TauLib.BookII.Mirror.DimensionalLadder + status: canonical + release_lines: *id859 +- old_id: II.R31 + new_id: cid005173 + primary_alias: MOD0176 + type: formal_module + title: TauLib.BookII.Mirror.DimensionalLadder + status: canonical + release_lines: *id859 +- old_id: II.R32 + new_id: cid005173 + primary_alias: MOD0176 + type: formal_module + title: TauLib.BookII.Mirror.DimensionalLadder + status: canonical + release_lines: *id859 +- old_id: TauLib.BookII.Mirror.Inventory + new_id: cid005174 + primary_alias: MOD0177 + type: formal_module + title: TauLib.BookII.Mirror.Inventory + status: canonical + release_lines: &id860 + - 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corpus_v2 + - corpus_v3_working +- old_id: book-ii-mirror-wave-holomorphy + new_id: cid005177 + primary_alias: MOD0180 + type: formal_module + title: TauLib.BookII.Mirror.WaveHolomorphy + status: canonical + release_lines: *id863 +- old_id: TauLib.BookII.Prologue.SplitComplexInterior + new_id: cid005178 + primary_alias: MOD0181 + type: formal_module + title: TauLib.BookII.Prologue.SplitComplexInterior + status: canonical + release_lines: &id864 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-prologue-split-complex-interior + new_id: cid005178 + primary_alias: MOD0181 + type: formal_module + title: TauLib.BookII.Prologue.SplitComplexInterior + status: canonical + release_lines: *id864 +- old_id: TauLib.BookII.Regularity.CodeDecode + new_id: cid005179 + primary_alias: MOD0182 + type: formal_module + title: TauLib.BookII.Regularity.CodeDecode + status: canonical + release_lines: &id865 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-regularity-code-decode + new_id: cid005179 + primary_alias: MOD0182 + type: formal_module + title: TauLib.BookII.Regularity.CodeDecode + status: canonical + release_lines: *id865 +- old_id: TauLib.BookII.Regularity.IdempotentDecomposition + new_id: cid005180 + primary_alias: MOD0183 + type: formal_module + title: TauLib.BookII.Regularity.IdempotentDecomposition + status: canonical + release_lines: &id866 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-regularity-idempotent-decomposition + new_id: cid005180 + primary_alias: MOD0183 + type: formal_module + title: TauLib.BookII.Regularity.IdempotentDecomposition + status: canonical + release_lines: *id866 +- old_id: TauLib.BookII.Regularity.PositiveRegularity + new_id: cid005181 + primary_alias: MOD0184 + type: formal_module + title: TauLib.BookII.Regularity.PositiveRegularity + status: canonical + release_lines: &id867 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-regularity-positive-regularity + new_id: cid005181 + primary_alias: MOD0184 + type: formal_module + title: TauLib.BookII.Regularity.PositiveRegularity + status: canonical + release_lines: *id867 +- old_id: TauLib.BookII.Regularity.PreYoneda + new_id: cid005182 + primary_alias: MOD0185 + type: formal_module + title: TauLib.BookII.Regularity.PreYoneda + status: canonical + release_lines: &id868 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-regularity-pre-yoneda + new_id: cid005182 + primary_alias: MOD0185 + type: formal_module + title: TauLib.BookII.Regularity.PreYoneda + status: canonical + release_lines: *id868 +- old_id: II.R12 + new_id: cid005182 + primary_alias: MOD0185 + type: formal_module + title: TauLib.BookII.Regularity.PreYoneda + status: canonical + release_lines: *id868 +- old_id: TauLib.BookII.Regularity.ThreeLemmaChain + new_id: cid005183 + primary_alias: MOD0186 + type: formal_module + title: TauLib.BookII.Regularity.ThreeLemmaChain + status: canonical + release_lines: &id869 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-regularity-three-lemma-chain + new_id: cid005183 + primary_alias: MOD0186 + type: formal_module + title: TauLib.BookII.Regularity.ThreeLemmaChain + status: canonical + release_lines: *id869 +- old_id: TauLib.BookII.Topology.BoundaryMinimality + new_id: cid005184 + primary_alias: MOD0187 + type: formal_module + title: TauLib.BookII.Topology.BoundaryMinimality + status: canonical + release_lines: &id870 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-topology-boundary-minimality + new_id: cid005184 + primary_alias: MOD0187 + type: formal_module + title: TauLib.BookII.Topology.BoundaryMinimality + status: canonical + release_lines: *id870 +- old_id: TauLib.BookII.Topology.DimensionFour + new_id: cid005185 + primary_alias: MOD0188 + type: formal_module + title: TauLib.BookII.Topology.DimensionFour + status: canonical + release_lines: &id871 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-topology-dimension-four + new_id: cid005185 + primary_alias: MOD0188 + type: formal_module + title: TauLib.BookII.Topology.DimensionFour + status: canonical + release_lines: *id871 +- old_id: TauLib.BookII.Topology.Invariant + new_id: cid005186 + primary_alias: MOD0189 + type: formal_module + title: TauLib.BookII.Topology.Invariant + status: canonical + release_lines: &id872 + - 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corpus_v2 + - corpus_v3_working +- old_id: book-iii-doors-poincare + new_id: cid005224 + primary_alias: MOD0227 + type: formal_module + title: TauLib.BookIII.Doors.Poincare + status: canonical + release_lines: *id910 +- old_id: TauLib.BookIII.Doors.SpectralCorrespondence + new_id: cid005225 + primary_alias: MOD0228 + type: formal_module + title: TauLib.BookIII.Doors.SpectralCorrespondence + status: canonical + release_lines: &id911 + - corpus_v2 + - corpus_v3_working +- old_id: book-iii-doors-spectral-correspondence + new_id: cid005225 + primary_alias: MOD0228 + type: formal_module + title: TauLib.BookIII.Doors.SpectralCorrespondence + status: canonical + release_lines: *id911 +- old_id: TauLib.BookIII.Doors.SpectralDecomp + new_id: cid005226 + primary_alias: MOD0229 + type: formal_module + title: TauLib.BookIII.Doors.SpectralDecomp + status: canonical + release_lines: &id912 + - corpus_v2 + - corpus_v3_working +- old_id: book-iii-doors-spectral-decomp + new_id: cid005226 + primary_alias: MOD0229 + type: formal_module + title: TauLib.BookIII.Doors.SpectralDecomp + status: canonical + release_lines: *id912 +- old_id: TauLib.BookIII.Doors.SplitComplexZeta + new_id: cid005227 + primary_alias: MOD0230 + type: formal_module + title: TauLib.BookIII.Doors.SplitComplexZeta + status: canonical + release_lines: &id913 + - 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corpus_v2 + - corpus_v3_working +- old_id: book-iii-spectrum-three-sat + new_id: cid005264 + primary_alias: MOD0267 + type: formal_module + title: TauLib.BookIII.Spectrum.ThreeSAT + status: canonical + release_lines: *id950 +- old_id: I.D73 + new_id: cid005264 + primary_alias: MOD0267 + type: formal_module + title: TauLib.BookIII.Spectrum.ThreeSAT + status: canonical + release_lines: *id950 +- old_id: I.P31 + new_id: cid005264 + primary_alias: MOD0267 + type: formal_module + title: TauLib.BookIII.Spectrum.ThreeSAT + status: canonical + release_lines: *id950 +- old_id: I.P32 + new_id: cid005264 + primary_alias: MOD0267 + type: formal_module + title: TauLib.BookIII.Spectrum.ThreeSAT + status: canonical + release_lines: *id950 +- old_id: chi_complete + new_id: cid005265 + primary_alias: FTH0002 + type: formal_theorem + title: chi_complete (theorem) + status: canonical + release_lines: &id951 + - corpus_v2 + - corpus_v3_working +- old_id: chi-complete + new_id: cid005265 + primary_alias: FTH0002 + type: formal_theorem + title: chi_complete (theorem) + status: canonical + release_lines: *id951 +- old_id: TauLib.BookI.Boundary.Characters::chi_complete + new_id: cid005265 + primary_alias: FTH0002 + type: formal_theorem + title: chi_complete (theorem) + status: canonical + release_lines: *id951 +- old_id: chi_orthogonal + new_id: cid005266 + primary_alias: FTH0003 + type: formal_theorem + title: chi_orthogonal (theorem) + status: canonical + release_lines: &id952 + - corpus_v2 + - corpus_v3_working +- old_id: chi-orthogonal + new_id: cid005266 + primary_alias: FTH0003 + type: formal_theorem + title: chi_orthogonal (theorem) + status: canonical + release_lines: *id952 +- old_id: TauLib.BookI.Boundary.Characters::chi_orthogonal + new_id: cid005266 + primary_alias: FTH0003 + type: formal_theorem + title: chi_orthogonal (theorem) + status: canonical + release_lines: *id952 +- old_id: sc_j_squared + new_id: cid005267 + primary_alias: FTH0004 + type: formal_theorem + title: sc_j_squared (theorem) + status: canonical + release_lines: &id953 + - corpus_v2 + - corpus_v3_working +- old_id: sc-j-squared + new_id: cid005267 + primary_alias: FTH0004 + type: formal_theorem + title: sc_j_squared (theorem) + status: canonical + release_lines: *id953 +- old_id: TauLib.BookI.Boundary.ComplexField::sc_j_squared + new_id: cid005267 + primary_alias: FTH0004 + type: formal_theorem + title: sc_j_squared (theorem) + status: canonical + release_lines: *id953 +- old_id: taureal_archimedean_embedding + new_id: cid005268 + primary_alias: FTH0005 + type: formal_theorem + title: taureal_archimedean_embedding (theorem) + status: canonical + release_lines: &id954 + - 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corpus_v2 + - corpus_v3_working +- old_id: crossing-iff-real + new_id: cid005270 + primary_alias: FTH0007 + type: formal_theorem + title: crossing_iff_real (theorem) + status: canonical + release_lines: *id956 +- old_id: TauLib.BookI.Boundary.Fourier::crossing_iff_real + new_id: cid005270 + primary_alias: FTH0007 + type: formal_theorem + title: crossing_iff_real (theorem) + status: canonical + release_lines: *id956 +- old_id: fourier_sigma_swap + new_id: cid005271 + primary_alias: FTH0008 + type: formal_theorem + title: fourier_sigma_swap (theorem) + status: canonical + release_lines: &id957 + - corpus_v2 + - corpus_v3_working +- old_id: fourier-sigma-swap + new_id: cid005271 + primary_alias: FTH0008 + type: formal_theorem + title: fourier_sigma_swap (theorem) + status: canonical + release_lines: *id957 +- old_id: TauLib.BookI.Boundary.Fourier::fourier_sigma_swap + new_id: cid005271 + primary_alias: FTH0008 + type: formal_theorem + title: fourier_sigma_swap (theorem) + status: canonical + release_lines: *id957 +- old_id: integral_const_one_3 + new_id: cid005272 + primary_alias: FTH0009 + type: formal_theorem + title: integral_const_one_3 (theorem) + status: canonical + release_lines: &id958 + - 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corpus_v2 + - corpus_v3_working +- old_id: linearity-identity-stage2 + new_id: cid005274 + primary_alias: FTH0011 + type: formal_theorem + title: linearity_identity_stage2 (theorem) + status: canonical + release_lines: *id960 +- old_id: TauLib.BookI.Boundary.Integration::linearity_identity_stage2 + new_id: cid005274 + primary_alias: FTH0011 + type: formal_theorem + title: linearity_identity_stage2 (theorem) + status: canonical + release_lines: *id960 +- old_id: integral_even_2 + new_id: cid005275 + primary_alias: FTH0012 + type: formal_theorem + title: integral_even_2 (theorem) + status: canonical + release_lines: &id961 + - corpus_v2 + - corpus_v3_working +- old_id: integral-even-2 + new_id: cid005275 + primary_alias: FTH0012 + type: formal_theorem + title: integral_even_2 (theorem) + status: canonical + release_lines: *id961 +- old_id: TauLib.BookI.Boundary.Integration::integral_even_2 + new_id: cid005275 + primary_alias: FTH0012 + type: formal_theorem + title: integral_even_2 (theorem) + status: canonical + release_lines: *id961 +- old_id: primorial_pos_1 + new_id: cid005276 + primary_alias: FTH0013 + type: formal_theorem + title: primorial_pos_1 (theorem) + status: canonical + release_lines: &id962 + - 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corpus_v2 + - corpus_v3_working +- old_id: rewiring-levels-eq-solenoidal + new_id: cid005320 + primary_alias: FTH0058 + type: formal_theorem + title: rewiring_levels_eq_solenoidal (theorem) + status: canonical + release_lines: *id1006 +- old_id: TauLib.BookI.Kernel.Diagonal::rewiring_levels_eq_solenoidal + new_id: cid005320 + primary_alias: FTH0058 + type: formal_theorem + title: rewiring_levels_eq_solenoidal (theorem) + status: canonical + release_lines: *id1006 +- old_id: boolean_recovery + new_id: cid005321 + primary_alias: FTH0059 + type: formal_theorem + title: boolean_recovery (theorem) + status: canonical + release_lines: &id1007 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Logic.BooleanRecovery::boolean_recovery + new_id: cid005321 + primary_alias: FTH0059 + type: formal_theorem + title: boolean_recovery (theorem) + status: canonical + release_lines: *id1007 +- old_id: forget_fiber + new_id: cid005322 + primary_alias: FTH0060 + type: formal_theorem + title: forget_fiber (theorem) + status: canonical + release_lines: &id1008 + - corpus_v2 + - corpus_v3_working +- old_id: forget-fiber + new_id: cid005322 + primary_alias: FTH0060 + type: formal_theorem + title: forget_fiber (theorem) + status: canonical + release_lines: *id1008 +- old_id: TauLib.BookI.Logic.BooleanRecovery::forget_fiber + new_id: cid005322 + primary_alias: FTH0060 + type: formal_theorem + title: forget_fiber (theorem) + status: canonical + release_lines: *id1008 +- old_id: explosion_barrier + new_id: cid005323 + primary_alias: FTH0061 + type: formal_theorem + title: explosion_barrier (theorem) + status: canonical + release_lines: &id1009 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Logic.Explosion::explosion_barrier + new_id: cid005323 + primary_alias: FTH0061 + type: formal_theorem + title: explosion_barrier (theorem) + status: canonical + release_lines: *id1009 +- old_id: no_identity_decoherence + new_id: cid005324 + primary_alias: FTH0062 + type: formal_theorem + title: no_identity_decoherence (theorem) + status: canonical + release_lines: &id1010 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.MetaLogic.OnticInvariance::no_identity_decoherence + new_id: cid005324 + primary_alias: FTH0062 + type: formal_theorem + title: no_identity_decoherence (theorem) + status: canonical + release_lines: *id1010 +- old_id: slippage_breaks_omega + new_id: cid005325 + primary_alias: FTH0063 + type: formal_theorem + title: slippage_breaks_omega (theorem) + status: canonical + release_lines: &id1011 + - corpus_v2 + - corpus_v3_working +- old_id: slippage-breaks-omega + new_id: cid005325 + primary_alias: FTH0063 + type: formal_theorem + title: slippage_breaks_omega (theorem) + status: canonical + release_lines: *id1011 +- old_id: TauLib.BookI.MetaLogic.OnticInvariance::slippage_breaks_omega + new_id: cid005325 + primary_alias: FTH0063 + type: formal_theorem + title: slippage_breaks_omega (theorem) + status: canonical + release_lines: *id1011 +- old_id: structural_instability_theorem + new_id: cid005326 + primary_alias: FTH0064 + type: formal_theorem + title: structural_instability_theorem (theorem) + status: canonical + release_lines: &id1012 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.MetaLogic.ReceptionCriterion::structural_instability_theorem + new_id: cid005326 + primary_alias: FTH0064 + type: formal_theorem + title: structural_instability_theorem (theorem) + status: canonical + release_lines: *id1012 +- old_id: ontic_closure + new_id: cid005327 + primary_alias: FTH0065 + type: formal_theorem + title: ontic_closure (theorem) + status: canonical + release_lines: &id1013 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Orbit.Closure::ontic_closure + new_id: cid005327 + primary_alias: FTH0065 + type: formal_theorem + title: ontic_closure (theorem) + status: canonical + release_lines: *id1013 +- old_id: tauObj_countable + new_id: cid005328 + primary_alias: FTH0066 + type: formal_theorem + title: tauObj_countable (theorem) + status: canonical + release_lines: &id1014 + - corpus_v2 + - corpus_v3_working +- old_id: tau-obj-countable + new_id: cid005328 + primary_alias: FTH0066 + type: formal_theorem + title: tauObj_countable (theorem) + status: canonical + release_lines: *id1014 +- old_id: TauLib.BookI.Orbit.Countability::tauObj_countable + new_id: cid005328 + primary_alias: FTH0066 + type: formal_theorem + title: tauObj_countable (theorem) + status: canonical + release_lines: *id1014 +- old_id: orbit_disjoint + new_id: cid005329 + primary_alias: FTH0067 + type: formal_theorem + title: orbit_disjoint (theorem) + status: canonical + release_lines: &id1015 + - corpus_v2 + - corpus_v3_working +- old_id: orbit-disjoint + new_id: cid005329 + primary_alias: FTH0067 + type: formal_theorem + title: orbit_disjoint (theorem) + status: canonical + release_lines: *id1015 +- old_id: TauLib.BookI.Orbit.Generation::orbit_disjoint + new_id: cid005329 + primary_alias: FTH0067 + type: formal_theorem + title: orbit_disjoint (theorem) + status: canonical + release_lines: *id1015 +- old_id: pentation_channel_exhaustion + new_id: cid005330 + primary_alias: FTH0068 + type: formal_theorem + title: pentation_channel_exhaustion (theorem) + status: canonical + release_lines: &id1016 + - corpus_v2 + - corpus_v3_working +- old_id: pentation-channel-exhaustion + new_id: cid005330 + primary_alias: FTH0068 + type: formal_theorem + title: pentation_channel_exhaustion (theorem) + status: canonical + release_lines: *id1016 +- old_id: TauLib.BookI.Orbit.Ladder::pentation_channel_exhaustion + new_id: cid005330 + primary_alias: FTH0068 + type: formal_theorem + title: pentation_channel_exhaustion (theorem) + status: canonical + release_lines: *id1016 +- old_id: ladder_saturation + new_id: cid005331 + primary_alias: FTH0069 + type: formal_theorem + title: ladder_saturation (theorem) + status: canonical + release_lines: &id1017 + - corpus_v2 + - corpus_v3_working +- old_id: ladder-saturation + new_id: cid005331 + primary_alias: FTH0069 + type: formal_theorem + title: ladder_saturation (theorem) + status: canonical + release_lines: *id1017 +- old_id: TauLib.BookI.Orbit.Ladder::ladder_saturation + new_id: cid005331 + primary_alias: FTH0069 + type: formal_theorem + title: ladder_saturation (theorem) + status: canonical + release_lines: *id1017 +- old_id: rigidity_non_omega + new_id: cid005332 + primary_alias: FTH0070 + type: formal_theorem + title: rigidity_non_omega (theorem) + status: canonical + release_lines: &id1018 + - corpus_v2 + - corpus_v3_working +- old_id: rigidity-non-omega + new_id: cid005332 + primary_alias: FTH0070 + type: formal_theorem + title: rigidity_non_omega (theorem) + status: canonical + release_lines: *id1018 +- old_id: TauLib.BookI.Orbit.Rigidity::rigidity_non_omega + new_id: cid005332 + primary_alias: FTH0070 + type: formal_theorem + title: rigidity_non_omega (theorem) + status: canonical + release_lines: *id1018 +- old_id: categoricity_non_omega + new_id: cid005333 + primary_alias: FTH0071 + type: formal_theorem + title: categoricity_non_omega (theorem) + status: canonical + release_lines: &id1019 + - corpus_v2 + - corpus_v3_working +- old_id: categoricity-non-omega + new_id: cid005333 + primary_alias: FTH0071 + type: formal_theorem + title: categoricity_non_omega (theorem) + status: canonical + release_lines: *id1019 +- old_id: TauLib.BookI.Orbit.Rigidity::categoricity_non_omega + new_id: cid005333 + primary_alias: FTH0071 + type: formal_theorem + title: categoricity_non_omega (theorem) + status: canonical + release_lines: *id1019 +- old_id: minimal_alphabet + new_id: cid005334 + primary_alias: FTH0072 + type: formal_theorem + title: minimal_alphabet (theorem) + status: canonical + release_lines: &id1020 + - corpus_v2 + - corpus_v3_working +- old_id: minimal-alphabet + new_id: cid005334 + primary_alias: FTH0072 + type: formal_theorem + title: minimal_alphabet (theorem) + status: canonical + release_lines: *id1020 +- old_id: TauLib.BookI.Orbit.Saturation::minimal_alphabet + new_id: cid005334 + primary_alias: FTH0072 + type: formal_theorem + title: minimal_alphabet (theorem) + status: canonical + release_lines: *id1020 +- old_id: no_elliptic_idempotent + new_id: cid005335 + primary_alias: FTH0073 + type: formal_theorem + title: no_elliptic_idempotent (theorem) + status: canonical + release_lines: &id1021 + - corpus_v2 + - corpus_v3_working +- old_id: no-elliptic-idempotent + new_id: cid005335 + primary_alias: FTH0073 + type: formal_theorem + title: no_elliptic_idempotent (theorem) + status: canonical + release_lines: *id1021 +- old_id: TauLib.BookI.Polarity.BipolarAlgebra::no_elliptic_idempotent + new_id: cid005335 + primary_alias: FTH0073 + type: formal_theorem + title: no_elliptic_idempotent (theorem) + status: canonical + release_lines: *id1021 +- old_id: split_complex_forced + new_id: cid005336 + primary_alias: FTH0074 + type: formal_theorem + title: split_complex_forced (theorem) + status: canonical + release_lines: &id1022 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Polarity.BipolarAlgebra::split_complex_forced + new_id: cid005336 + primary_alias: FTH0074 + type: formal_theorem + title: split_complex_forced (theorem) + status: canonical + release_lines: *id1022 +- old_id: algebraic_lemniscate_exists + new_id: cid005337 + primary_alias: FTH0075 + type: formal_theorem + title: algebraic_lemniscate_exists (theorem) + status: canonical + release_lines: &id1023 + - corpus_v2 + - corpus_v3_working +- old_id: algebraic-lemniscate-exists + new_id: cid005337 + primary_alias: FTH0075 + type: formal_theorem + title: algebraic_lemniscate_exists (theorem) + status: canonical + release_lines: *id1023 +- old_id: TauLib.BookI.Polarity.Lemniscate::algebraic_lemniscate_exists + new_id: cid005337 + primary_alias: FTH0075 + type: formal_theorem + title: algebraic_lemniscate_exists (theorem) + status: canonical + release_lines: *id1023 +- old_id: no_unearned_decimal_diagonal + new_id: cid005338 + primary_alias: FTH0076 + type: formal_theorem + title: no_unearned_decimal_diagonal (theorem) + status: canonical + release_lines: &id1024 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Sets.CantorRefutation::no_unearned_decimal_diagonal + new_id: cid005338 + primary_alias: FTH0076 + type: formal_theorem + title: no_unearned_decimal_diagonal (theorem) + status: canonical + release_lines: *id1024 +- old_id: no_comprehension + new_id: cid005339 + primary_alias: FTH0077 + type: formal_theorem + title: no_comprehension (theorem) + status: canonical + release_lines: &id1025 + - corpus_v2 + - corpus_v3_working +- old_id: no-comprehension + new_id: cid005339 + primary_alias: FTH0077 + type: formal_theorem + title: no_comprehension (theorem) + status: canonical + release_lines: *id1025 +- old_id: TauLib.BookI.Sets.CantorRefutation::no_comprehension + new_id: cid005339 + primary_alias: FTH0077 + type: formal_theorem + title: no_comprehension (theorem) + status: canonical + release_lines: *id1025 +- old_id: no_free_cartesian_diagonal + new_id: cid005340 + primary_alias: FTH0078 + type: formal_theorem + title: no_free_cartesian_diagonal (theorem) + status: canonical + release_lines: &id1026 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Sets.CantorRefutation::no_free_cartesian_diagonal + new_id: cid005340 + primary_alias: FTH0078 + type: formal_theorem + title: no_free_cartesian_diagonal (theorem) + status: canonical + release_lines: *id1026 +- old_id: cantor_inapplicable + new_id: cid005341 + primary_alias: FTH0079 + type: formal_theorem + title: cantor_inapplicable (theorem) + status: canonical + release_lines: &id1027 + - corpus_v2 + - corpus_v3_working +- old_id: cantor-inapplicable + new_id: cid005341 + primary_alias: FTH0079 + type: formal_theorem + title: cantor_inapplicable (theorem) + status: canonical + release_lines: *id1027 +- old_id: TauLib.BookI.Sets.CantorRefutation::cantor_inapplicable + new_id: cid005341 + primary_alias: FTH0079 + type: formal_theorem + title: cantor_inapplicable (theorem) + status: canonical + release_lines: *id1027 +- old_id: counting_structural + new_id: cid005342 + primary_alias: FTH0080 + type: formal_theorem + title: counting_structural (theorem) + status: canonical + release_lines: &id1028 + - corpus_v2 + - corpus_v3_working +- old_id: counting-structural + new_id: cid005342 + primary_alias: FTH0080 + type: formal_theorem + title: counting_structural (theorem) + status: canonical + release_lines: *id1028 +- old_id: TauLib.BookI.Sets.Counting::counting_structural + new_id: cid005342 + primary_alias: FTH0080 + type: formal_theorem + title: counting_structural (theorem) + status: canonical + release_lines: *id1028 +- old_id: tau_mem_iff_dvd + new_id: cid005343 + primary_alias: FTH0081 + type: formal_theorem + title: tau_mem_iff_dvd (theorem) + status: canonical + release_lines: &id1029 + - corpus_v2 + - corpus_v3_working +- old_id: tau-mem-iff-dvd + new_id: cid005343 + primary_alias: FTH0081 + type: formal_theorem + title: tau_mem_iff_dvd (theorem) + status: canonical + release_lines: *id1029 +- old_id: TauLib.BookI.Sets.Membership::tau_mem_iff_dvd + new_id: cid005343 + primary_alias: FTH0081 + type: formal_theorem + title: tau_mem_iff_dvd (theorem) + status: canonical + release_lines: *id1029 +- old_id: tau_inter_distrib_union + new_id: cid005344 + primary_alias: FTH0082 + type: formal_theorem + title: tau_inter_distrib_union (theorem) + status: canonical + release_lines: &id1030 + - corpus_v2 + - corpus_v3_working +- old_id: tau-inter-distrib-union + new_id: cid005344 + primary_alias: FTH0082 + type: formal_theorem + title: tau_inter_distrib_union (theorem) + status: canonical + release_lines: *id1030 +- old_id: TauLib.BookI.Sets.Operations::tau_inter_distrib_union + new_id: cid005344 + primary_alias: FTH0082 + type: formal_theorem + title: tau_inter_distrib_union (theorem) + status: canonical + release_lines: *id1030 +- old_id: orbit_set_order_bound + new_id: cid005345 + primary_alias: FTH0083 + type: formal_theorem + title: orbit_set_order_bound (theorem) + status: canonical + release_lines: &id1031 + - corpus_v2 + - corpus_v3_working +- old_id: orbit-set-order-bound + new_id: cid005345 + primary_alias: FTH0083 + type: formal_theorem + title: orbit_set_order_bound (theorem) + status: canonical + release_lines: *id1031 +- old_id: TauLib.BookI.Sets.OrbitSets::orbit_set_order_bound + new_id: cid005345 + primary_alias: FTH0083 + type: formal_theorem + title: orbit_set_order_bound (theorem) + status: canonical + release_lines: *id1031 +- old_id: prime_atom + new_id: cid005346 + primary_alias: FTH0084 + type: formal_theorem + title: prime_atom (theorem) + status: canonical + release_lines: &id1032 + - corpus_v2 + - corpus_v3_working +- old_id: prime-atom + new_id: cid005346 + primary_alias: FTH0084 + type: formal_theorem + title: prime_atom (theorem) + status: canonical + release_lines: *id1032 +- old_id: TauLib.BookI.Sets.OrbitSets::prime_atom + new_id: cid005346 + primary_alias: FTH0084 + type: formal_theorem + title: prime_atom (theorem) + status: canonical + release_lines: *id1032 +- old_id: omega_orbit_set_exceeds_alpha + new_id: cid005347 + primary_alias: FTH0085 + type: formal_theorem + title: omega_orbit_set_exceeds_alpha (theorem) + status: canonical + release_lines: &id1033 + - corpus_v2 + - corpus_v3_working +- old_id: omega-orbit-set-exceeds-alpha + new_id: cid005347 + primary_alias: FTH0085 + type: formal_theorem + title: omega_orbit_set_exceeds_alpha (theorem) + status: canonical + release_lines: *id1033 +- old_id: TauLib.BookI.Sets.OrbitSets::omega_orbit_set_exceeds_alpha + new_id: cid005347 + primary_alias: FTH0085 + type: formal_theorem + title: omega_orbit_set_exceeds_alpha (theorem) + status: canonical + release_lines: *id1033 +- old_id: nat_not_internal_set + new_id: cid005348 + primary_alias: FTH0086 + type: formal_theorem + title: nat_not_internal_set (theorem) + status: canonical + release_lines: &id1034 + - corpus_v2 + - corpus_v3_working +- old_id: nat-not-internal-set + new_id: cid005348 + primary_alias: FTH0086 + type: formal_theorem + title: nat_not_internal_set (theorem) + status: canonical + release_lines: *id1034 +- old_id: TauLib.BookI.Sets.OrbitSets::nat_not_internal_set + new_id: cid005348 + primary_alias: FTH0086 + type: formal_theorem + title: nat_not_internal_set (theorem) + status: canonical + release_lines: *id1034 +- old_id: tau_strict_mem_wf + new_id: cid005349 + primary_alias: FTH0087 + type: formal_theorem + title: tau_strict_mem_wf (theorem) + status: canonical + release_lines: &id1035 + - corpus_v2 + - corpus_v3_working +- old_id: tau-strict-mem-wf + new_id: cid005349 + primary_alias: FTH0087 + type: formal_theorem + title: tau_strict_mem_wf (theorem) + status: canonical + release_lines: *id1035 +- old_id: TauLib.BookI.Sets.Powerset::tau_strict_mem_wf + new_id: cid005349 + primary_alias: FTH0087 + type: formal_theorem + title: tau_strict_mem_wf (theorem) + status: canonical + release_lines: *id1035 +- old_id: unique_infinity + new_id: cid005350 + primary_alias: FTH0088 + type: formal_theorem + title: unique_infinity (theorem) + status: canonical + release_lines: &id1036 + - corpus_v2 + - corpus_v3_working +- old_id: unique-infinity + new_id: cid005350 + primary_alias: FTH0088 + type: formal_theorem + title: unique_infinity (theorem) + status: canonical + release_lines: *id1036 +- old_id: TauLib.BookI.Sets.UniqueInfinity::unique_infinity + new_id: cid005350 + primary_alias: FTH0088 + type: formal_theorem + title: unique_infinity (theorem) + status: canonical + release_lines: *id1036 +- old_id: ultrametric_replaces_card + new_id: cid005351 + primary_alias: FTH0089 + type: formal_theorem + title: ultrametric_replaces_card (theorem) + status: canonical + release_lines: &id1037 + - corpus_v2 + - corpus_v3_working +- old_id: ultrametric-replaces-card + new_id: cid005351 + primary_alias: FTH0089 + type: formal_theorem + title: ultrametric_replaces_card (theorem) + status: canonical + release_lines: *id1037 +- old_id: TauLib.BookI.Sets.UniqueInfinity::ultrametric_replaces_card + new_id: cid005351 + primary_alias: FTH0089 + type: formal_theorem + title: ultrametric_replaces_card (theorem) + status: canonical + release_lines: *id1037 +- old_id: product_universal + new_id: cid005352 + primary_alias: FTH0090 + type: formal_theorem + title: product_universal (theorem) + status: canonical + release_lines: &id1038 + - corpus_v2 + - corpus_v3_working +- old_id: product-universal + new_id: cid005352 + primary_alias: FTH0090 + type: formal_theorem + title: product_universal (theorem) + status: canonical + release_lines: *id1038 +- old_id: TauLib.BookI.Topos.CartesianProduct::product_universal + new_id: cid005352 + primary_alias: FTH0090 + type: formal_theorem + title: product_universal (theorem) + status: canonical + release_lines: *id1038 +- old_id: cat_tau_thin + new_id: cid005353 + primary_alias: FTH0091 + type: formal_theorem + title: cat_tau_thin (theorem) + status: canonical + release_lines: &id1039 + - 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corpus_v2 + - corpus_v3_working +- old_id: earned-topos-non-boolean + new_id: cid005355 + primary_alias: FTH0093 + type: formal_theorem + title: earned_topos_non_boolean (theorem) + status: canonical + release_lines: *id1041 +- old_id: TauLib.BookI.Topos.EarnedTopos::earned_topos_non_boolean + new_id: cid005355 + primary_alias: FTH0093 + type: formal_theorem + title: earned_topos_non_boolean (theorem) + status: canonical + release_lines: *id1041 +- old_id: yoneda_thin + new_id: cid005356 + primary_alias: FTH0094 + type: formal_theorem + title: yoneda_thin (theorem) + status: canonical + release_lines: &id1042 + - corpus_v2 + - corpus_v3_working +- old_id: yoneda-thin + new_id: cid005356 + primary_alias: FTH0094 + type: formal_theorem + title: yoneda_thin (theorem) + status: canonical + release_lines: *id1042 +- old_id: TauLib.BookI.Topos.Functors::yoneda_thin + new_id: cid005356 + primary_alias: FTH0094 + type: formal_theorem + title: yoneda_thin (theorem) + status: canonical + release_lines: *id1042 +- old_id: cartesian_closed_adj + new_id: cid005357 + primary_alias: FTH0095 + type: formal_theorem + title: cartesian_closed_adj (theorem) + status: canonical + release_lines: &id1043 + - corpus_v2 + - corpus_v3_working +- old_id: cartesian-closed-adj + new_id: cid005357 + primary_alias: FTH0095 + type: formal_theorem + title: cartesian_closed_adj (theorem) + status: canonical + release_lines: *id1043 +- old_id: TauLib.BookI.Topos.InternalHom::cartesian_closed_adj + new_id: cid005357 + primary_alias: FTH0095 + type: formal_theorem + title: cartesian_closed_adj (theorem) + status: canonical + release_lines: *id1043 +- old_id: self_enrichment + new_id: cid005358 + primary_alias: FTH0096 + type: formal_theorem + title: self_enrichment (theorem) + status: canonical + release_lines: &id1044 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookI.Topos.InternalHom::self_enrichment + new_id: cid005358 + primary_alias: FTH0096 + type: formal_theorem + title: self_enrichment (theorem) + status: canonical + release_lines: *id1044 +- old_id: psh_has_terminal + new_id: cid005359 + primary_alias: FTH0097 + type: formal_theorem + title: psh_has_terminal (theorem) + status: canonical + release_lines: &id1045 + - corpus_v2 + - corpus_v3_working +- old_id: psh-has-terminal + new_id: cid005359 + primary_alias: FTH0097 + type: formal_theorem + title: psh_has_terminal (theorem) + status: canonical + release_lines: *id1045 +- old_id: TauLib.BookI.Topos.LimitsSites::psh_has_terminal + new_id: cid005359 + primary_alias: FTH0097 + type: formal_theorem + title: psh_has_terminal (theorem) + status: canonical + release_lines: *id1045 +- old_id: psh_countable_objects + new_id: cid005360 + primary_alias: FTH0098 + type: formal_theorem + title: psh_countable_objects (theorem) + status: canonical + release_lines: &id1046 + - corpus_v2 + - corpus_v3_working +- old_id: psh-countable-objects + new_id: cid005360 + primary_alias: FTH0098 + type: formal_theorem + title: psh_countable_objects (theorem) + status: canonical + release_lines: *id1046 +- old_id: TauLib.BookI.Topos.LimitsSites::psh_countable_objects + new_id: cid005360 + primary_alias: FTH0098 + type: formal_theorem + title: psh_countable_objects (theorem) + status: canonical + release_lines: *id1046 +- old_id: product_distributes_over_coproduct + new_id: cid005361 + primary_alias: FTH0099 + type: formal_theorem + title: product_distributes_over_coproduct (theorem) + status: canonical + release_lines: &id1047 + - 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corpus_v2 + - corpus_v3_working +- old_id: char-add-15-3 + new_id: cid005371 + primary_alias: FTH0109 + type: formal_theorem + title: char_add_15_3 (theorem) + status: canonical + release_lines: *id1057 +- old_id: TauLib.BookII.CentralTheorem.BoundaryCharacters::char_add_15_3 + new_id: cid005371 + primary_alias: FTH0109 + type: formal_theorem + title: char_add_15_3 (theorem) + status: canonical + release_lines: *id1057 +- old_id: char_mul_15_3 + new_id: cid005372 + primary_alias: FTH0110 + type: formal_theorem + title: char_mul_15_3 (theorem) + status: canonical + release_lines: &id1058 + - corpus_v2 + - corpus_v3_working +- old_id: char-mul-15-3 + new_id: cid005372 + primary_alias: FTH0110 + type: formal_theorem + title: char_mul_15_3 (theorem) + status: canonical + release_lines: *id1058 +- old_id: TauLib.BookII.CentralTheorem.BoundaryCharacters::char_mul_15_3 + new_id: cid005372 + primary_alias: FTH0110 + type: formal_theorem + title: char_mul_15_3 (theorem) + status: canonical + release_lines: *id1058 +- old_id: char_add_id_15_3 + new_id: cid005373 + primary_alias: FTH0111 + type: formal_theorem + title: char_add_id_15_3 (theorem) + status: canonical + release_lines: &id1059 + - 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corpus_v2 + - corpus_v3_working +- old_id: nonconstant-bounded-4 + new_id: cid005375 + primary_alias: FTH0113 + type: formal_theorem + title: nonconstant_bounded_4 (theorem) + status: canonical + release_lines: *id1061 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::nonconstant_bounded_4 + new_id: cid005375 + primary_alias: FTH0113 + type: formal_theorem + title: nonconstant_bounded_4 (theorem) + status: canonical + release_lines: *id1061 +- old_id: primorial_unique_5 + new_id: cid005376 + primary_alias: FTH0114 + type: formal_theorem + title: primorial_unique_5 (theorem) + status: canonical + release_lines: &id1062 + - corpus_v2 + - corpus_v3_working +- old_id: primorial-unique-5 + new_id: cid005376 + primary_alias: FTH0114 + type: formal_theorem + title: primorial_unique_5 (theorem) + status: canonical + release_lines: *id1062 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::primorial_unique_5 + new_id: cid005376 + primary_alias: FTH0114 + type: formal_theorem + title: primorial_unique_5 (theorem) + status: canonical + release_lines: *id1062 +- old_id: abcd_rt_100 + new_id: cid005377 + primary_alias: FTH0115 + type: formal_theorem + title: abcd_rt_100 (theorem) + status: canonical + release_lines: &id1063 + - 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corpus_v2 + - corpus_v3_working +- old_id: categoricity-3-50 + new_id: cid005379 + primary_alias: FTH0117 + type: formal_theorem + title: categoricity_3_50 (theorem) + status: canonical + release_lines: *id1065 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::categoricity_3_50 + new_id: cid005379 + primary_alias: FTH0117 + type: formal_theorem + title: categoricity_3_50 (theorem) + status: canonical + release_lines: *id1065 +- old_id: moduli_100 + new_id: cid005380 + primary_alias: FTH0118 + type: formal_theorem + title: moduli_100 (theorem) + status: canonical + release_lines: &id1066 + - corpus_v2 + - corpus_v3_working +- old_id: moduli-100 + new_id: cid005380 + primary_alias: FTH0118 + type: formal_theorem + title: moduli_100 (theorem) + status: canonical + release_lines: *id1066 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::moduli_100 + new_id: cid005380 + primary_alias: FTH0118 + type: formal_theorem + title: moduli_100 (theorem) + status: canonical + release_lines: *id1066 +- old_id: uniqueness_3_50 + new_id: cid005381 + primary_alias: FTH0119 + type: formal_theorem + title: uniqueness_3_50 (theorem) + status: canonical + release_lines: &id1067 + - corpus_v2 + - corpus_v3_working +- old_id: uniqueness-3-50 + new_id: cid005381 + primary_alias: FTH0119 + type: formal_theorem + title: uniqueness_3_50 (theorem) + status: canonical + release_lines: *id1067 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::uniqueness_3_50 + new_id: cid005381 + primary_alias: FTH0119 + type: formal_theorem + title: uniqueness_3_50 (theorem) + status: canonical + release_lines: *id1067 +- old_id: j_squared_wave + new_id: cid005382 + primary_alias: FTH0120 + type: formal_theorem + title: j_squared_wave (theorem) + status: canonical + release_lines: &id1068 + - corpus_v2 + - corpus_v3_working +- old_id: j-squared-wave + new_id: cid005382 + primary_alias: FTH0120 + type: formal_theorem + title: j_squared_wave (theorem) + status: canonical + release_lines: *id1068 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::j_squared_wave + new_id: cid005382 + primary_alias: FTH0120 + type: formal_theorem + title: j_squared_wave (theorem) + status: canonical + release_lines: *id1068 +- old_id: zero_divisors_exist + new_id: cid005383 + primary_alias: FTH0121 + type: formal_theorem + title: zero_divisors_exist (theorem) + status: canonical + release_lines: &id1069 + - corpus_v2 + - corpus_v3_working +- old_id: zero-divisors-exist + new_id: cid005383 + primary_alias: FTH0121 + type: formal_theorem + title: zero_divisors_exist (theorem) + status: canonical + release_lines: *id1069 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::zero_divisors_exist + new_id: cid005383 + primary_alias: FTH0121 + type: formal_theorem + title: zero_divisors_exist (theorem) + status: canonical + release_lines: *id1069 +- old_id: idempotent_complete + new_id: cid005384 + primary_alias: FTH0122 + type: formal_theorem + title: idempotent_complete (theorem) + status: canonical + release_lines: &id1070 + - corpus_v2 + - corpus_v3_working +- old_id: idempotent-complete + new_id: cid005384 + primary_alias: FTH0122 + type: formal_theorem + title: idempotent_complete (theorem) + status: canonical + release_lines: *id1070 +- old_id: TauLib.BookII.CentralTheorem.Categoricity::idempotent_complete + new_id: cid005384 + primary_alias: FTH0122 + type: formal_theorem + title: idempotent_complete (theorem) + status: canonical + release_lines: *id1070 +- old_id: abcd_roundtrip_12 + new_id: cid005385 + primary_alias: FTH0123 + type: formal_theorem + title: abcd_roundtrip_12 (theorem) + status: canonical + release_lines: &id1071 + - 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corpus_v2 + - corpus_v3_working +- old_id: central-fwd-3-15 + new_id: cid005391 + primary_alias: FTH0129 + type: formal_theorem + title: central_fwd_3_15 (theorem) + status: canonical + release_lines: *id1077 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_fwd_3_15 + new_id: cid005391 + primary_alias: FTH0129 + type: formal_theorem + title: central_fwd_3_15 (theorem) + status: canonical + release_lines: *id1077 +- old_id: central_inv_3_15 + new_id: cid005392 + primary_alias: FTH0130 + type: formal_theorem + title: central_inv_3_15 (theorem) + status: canonical + release_lines: &id1078 + - corpus_v2 + - corpus_v3_working +- old_id: central-inv-3-15 + new_id: cid005392 + primary_alias: FTH0130 + type: formal_theorem + title: central_inv_3_15 (theorem) + status: canonical + release_lines: *id1078 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_inv_3_15 + new_id: cid005392 + primary_alias: FTH0130 + type: formal_theorem + title: central_inv_3_15 (theorem) + status: canonical + release_lines: *id1078 +- old_id: central_rt_3_15 + new_id: cid005393 + primary_alias: FTH0131 + type: formal_theorem + title: central_rt_3_15 (theorem) + status: canonical + release_lines: &id1079 + - corpus_v2 + - corpus_v3_working +- old_id: central-rt-3-15 + new_id: cid005393 + primary_alias: FTH0131 + type: formal_theorem + title: central_rt_3_15 (theorem) + status: canonical + release_lines: *id1079 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_rt_3_15 + new_id: cid005393 + primary_alias: FTH0131 + type: formal_theorem + title: central_rt_3_15 (theorem) + status: canonical + release_lines: *id1079 +- old_id: central_theorem_3_15 + new_id: cid005394 + primary_alias: FTH0132 + type: formal_theorem + title: central_theorem_3_15 (theorem) + status: canonical + release_lines: &id1080 + - corpus_v2 + - corpus_v3_working +- old_id: central-theorem-3-15 + new_id: cid005394 + primary_alias: FTH0132 + type: formal_theorem + title: central_theorem_3_15 (theorem) + status: canonical + release_lines: *id1080 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_theorem_3_15 + new_id: cid005394 + primary_alias: FTH0132 + type: formal_theorem + title: central_theorem_3_15 (theorem) + status: canonical + release_lines: *id1080 +- old_id: holographic_3_15 + new_id: cid005395 + primary_alias: FTH0133 + type: formal_theorem + title: holographic_3_15 (theorem) + status: canonical + release_lines: &id1081 + - corpus_v2 + - corpus_v3_working +- old_id: holographic-3-15 + new_id: cid005395 + primary_alias: FTH0133 + type: formal_theorem + title: holographic_3_15 (theorem) + status: canonical + release_lines: *id1081 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::holographic_3_15 + new_id: cid005395 + primary_alias: FTH0133 + type: formal_theorem + title: holographic_3_15 (theorem) + status: canonical + release_lines: *id1081 +- old_id: spectral_periodic + new_id: cid005396 + primary_alias: FTH0134 + type: formal_theorem + title: spectral_periodic (theorem) + status: canonical + release_lines: &id1082 + - corpus_v2 + - corpus_v3_working +- old_id: spectral-periodic + new_id: cid005396 + primary_alias: FTH0134 + type: formal_theorem + title: spectral_periodic (theorem) + status: canonical + release_lines: *id1082 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::spectral_periodic + new_id: cid005396 + primary_alias: FTH0134 + type: formal_theorem + title: spectral_periodic (theorem) + status: canonical + release_lines: *id1082 +- old_id: spectral_idempotent_supported + new_id: cid005397 + primary_alias: FTH0135 + type: formal_theorem + title: spectral_idempotent_supported (theorem) + status: canonical + release_lines: &id1083 + - corpus_v2 + - corpus_v3_working +- old_id: spectral-idempotent-supported + new_id: cid005397 + primary_alias: FTH0135 + type: formal_theorem + title: spectral_idempotent_supported (theorem) + status: canonical + release_lines: *id1083 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::spectral_idempotent_supported + new_id: cid005397 + primary_alias: FTH0135 + type: formal_theorem + title: spectral_idempotent_supported (theorem) + status: canonical + release_lines: *id1083 +- old_id: central_forward_coherent + new_id: cid005398 + primary_alias: FTH0136 + type: formal_theorem + title: central_forward_coherent (theorem) + status: canonical + release_lines: &id1084 + - corpus_v2 + - corpus_v3_working +- old_id: central-forward-coherent + new_id: cid005398 + primary_alias: FTH0136 + type: formal_theorem + title: central_forward_coherent (theorem) + status: canonical + release_lines: *id1084 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_forward_coherent + new_id: cid005398 + primary_alias: FTH0136 + type: formal_theorem + title: central_forward_coherent (theorem) + status: canonical + release_lines: *id1084 +- old_id: central_inverse_periodic + new_id: cid005399 + primary_alias: FTH0137 + type: formal_theorem + title: central_inverse_periodic (theorem) + status: canonical + release_lines: &id1085 + - corpus_v2 + - corpus_v3_working +- old_id: central-inverse-periodic + new_id: cid005399 + primary_alias: FTH0137 + type: formal_theorem + title: central_inverse_periodic (theorem) + status: canonical + release_lines: *id1085 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_inverse_periodic + new_id: cid005399 + primary_alias: FTH0137 + type: formal_theorem + title: central_inverse_periodic (theorem) + status: canonical + release_lines: *id1085 +- old_id: central_roundtrip + new_id: cid005400 + primary_alias: FTH0138 + type: formal_theorem + title: central_roundtrip (theorem) + status: canonical + release_lines: &id1086 + - corpus_v2 + - corpus_v3_working +- old_id: central-roundtrip + new_id: cid005400 + primary_alias: FTH0138 + type: formal_theorem + title: central_roundtrip (theorem) + status: canonical + release_lines: *id1086 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::central_roundtrip + new_id: cid005400 + primary_alias: FTH0138 + type: formal_theorem + title: central_roundtrip (theorem) + status: canonical + release_lines: *id1086 +- old_id: holographic_roundtrip + new_id: cid005401 + primary_alias: FTH0139 + type: formal_theorem + title: holographic_roundtrip (theorem) + status: canonical + release_lines: &id1087 + - corpus_v2 + - corpus_v3_working +- old_id: holographic-roundtrip + new_id: cid005401 + primary_alias: FTH0139 + type: formal_theorem + title: holographic_roundtrip (theorem) + status: canonical + release_lines: *id1087 +- old_id: TauLib.BookII.CentralTheorem.CentralTheorem::holographic_roundtrip + new_id: cid005401 + primary_alias: FTH0139 + type: formal_theorem + title: holographic_roundtrip (theorem) + status: canonical + release_lines: *id1087 +- old_id: stagewise_naturality + new_id: cid005402 + primary_alias: FTH0140 + type: formal_theorem + title: stagewise_naturality (theorem) + status: canonical + release_lines: &id1088 + - corpus_v2 + - corpus_v3_working +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality + new_id: cid005402 + primary_alias: FTH0140 + type: formal_theorem + title: stagewise_naturality (theorem) + status: canonical + release_lines: *id1088 +- old_id: stagewise_naturality_strong + new_id: cid005403 + primary_alias: FTH0141 + type: formal_theorem + title: stagewise_naturality_strong (theorem) + status: canonical + release_lines: &id1089 + - corpus_v2 + - corpus_v3_working +- old_id: stagewise-naturality-strong + new_id: cid005403 + primary_alias: FTH0141 + type: formal_theorem + title: stagewise_naturality_strong (theorem) + status: canonical + release_lines: *id1089 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::stagewise_naturality_strong + new_id: cid005403 + primary_alias: FTH0141 + type: formal_theorem + title: stagewise_naturality_strong (theorem) + status: canonical + release_lines: *id1089 +- old_id: bndlift_reduce_invariant + new_id: cid005404 + primary_alias: FTH0142 + type: formal_theorem + title: bndlift_reduce_invariant (theorem) + status: canonical + release_lines: &id1090 + - corpus_v2 + - corpus_v3_working +- old_id: bndlift-reduce-invariant + new_id: cid005404 + primary_alias: FTH0142 + type: formal_theorem + title: bndlift_reduce_invariant (theorem) + status: canonical + release_lines: *id1090 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_reduce_invariant + new_id: cid005404 + primary_alias: FTH0142 + type: formal_theorem + title: bndlift_reduce_invariant (theorem) + status: canonical + release_lines: *id1090 +- old_id: bndlift_stagefun_welldef + new_id: cid005405 + primary_alias: FTH0143 + type: formal_theorem + title: bndlift_stagefun_welldef (theorem) + status: canonical + release_lines: &id1091 + - corpus_v2 + - corpus_v3_working +- old_id: bndlift-stagefun-welldef + new_id: cid005405 + primary_alias: FTH0143 + type: formal_theorem + title: bndlift_stagefun_welldef (theorem) + status: canonical + release_lines: *id1091 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bndlift_stagefun_welldef + new_id: cid005405 + primary_alias: FTH0143 + type: formal_theorem + title: bndlift_stagefun_welldef (theorem) + status: canonical + release_lines: *id1091 +- old_id: reduction_gives_naturality + new_id: cid005406 + primary_alias: FTH0144 + type: formal_theorem + title: reduction_gives_naturality (theorem) + status: canonical + release_lines: &id1092 + - corpus_v2 + - corpus_v3_working +- old_id: reduction-gives-naturality + new_id: cid005406 + primary_alias: FTH0144 + type: formal_theorem + title: reduction_gives_naturality (theorem) + status: canonical + release_lines: *id1092 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::reduction_gives_naturality + new_id: cid005406 + primary_alias: FTH0144 + type: formal_theorem + title: reduction_gives_naturality (theorem) + status: canonical + release_lines: *id1092 +- old_id: nat_check_20_4 + new_id: cid005407 + primary_alias: FTH0145 + type: formal_theorem + title: nat_check_20_4 (theorem) + status: canonical + release_lines: &id1093 + - corpus_v2 + - corpus_v3_working +- old_id: nat-check-20-4 + new_id: cid005407 + primary_alias: FTH0145 + type: formal_theorem + title: nat_check_20_4 (theorem) + status: canonical + release_lines: *id1093 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::nat_check_20_4 + new_id: cid005407 + primary_alias: FTH0145 + type: formal_theorem + title: nat_check_20_4 (theorem) + status: canonical + release_lines: *id1093 +- old_id: ogt_15_4 + new_id: cid005408 + primary_alias: FTH0146 + type: formal_theorem + title: ogt_15_4 (theorem) + status: canonical + release_lines: &id1094 + - corpus_v2 + - corpus_v3_working +- old_id: ogt-15-4 + new_id: cid005408 + primary_alias: FTH0146 + type: formal_theorem + title: ogt_15_4 (theorem) + status: canonical + release_lines: *id1094 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::ogt_15_4 + new_id: cid005408 + primary_alias: FTH0146 + type: formal_theorem + title: ogt_15_4 (theorem) + status: canonical + release_lines: *id1094 +- old_id: bsf_coh_15_4 + new_id: cid005409 + primary_alias: FTH0147 + type: formal_theorem + title: bsf_coh_15_4 (theorem) + status: canonical + release_lines: &id1095 + - corpus_v2 + - corpus_v3_working +- old_id: bsf-coh-15-4 + new_id: cid005409 + primary_alias: FTH0147 + type: formal_theorem + title: bsf_coh_15_4 (theorem) + status: canonical + release_lines: *id1095 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::bsf_coh_15_4 + new_id: cid005409 + primary_alias: FTH0147 + type: formal_theorem + title: bsf_coh_15_4 (theorem) + status: canonical + release_lines: *id1095 +- old_id: egr_15_4 + new_id: cid005410 + primary_alias: FTH0148 + type: formal_theorem + title: egr_15_4 (theorem) + status: canonical + release_lines: &id1096 + - corpus_v2 + - corpus_v3_working +- old_id: egr-15-4 + new_id: cid005410 + primary_alias: FTH0148 + type: formal_theorem + title: egr_15_4 (theorem) + status: canonical + release_lines: *id1096 +- old_id: TauLib.BookII.CentralTheorem.ExtensionsOmegaGerms::egr_15_4 + new_id: cid005410 + primary_alias: FTH0148 + type: formal_theorem + title: egr_15_4 (theorem) + status: canonical + release_lines: *id1096 +- old_id: extension_preserves_stage + new_id: cid005411 + primary_alias: FTH0149 + type: formal_theorem + title: extension_preserves_stage (theorem) + status: canonical + release_lines: &id1097 + - corpus_v2 + - corpus_v3_working +- old_id: extension-preserves-stage + new_id: cid005411 + primary_alias: FTH0149 + type: formal_theorem + title: extension_preserves_stage (theorem) + status: canonical + release_lines: *id1097 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::extension_preserves_stage + new_id: cid005411 + primary_alias: FTH0149 + type: formal_theorem + title: extension_preserves_stage (theorem) + status: canonical + release_lines: *id1097 +- old_id: uniqueness_from_agreement + new_id: cid005412 + primary_alias: FTH0150 + type: formal_theorem + title: uniqueness_from_agreement (theorem) + status: canonical + release_lines: &id1098 + - corpus_v2 + - corpus_v3_working +- old_id: uniqueness-from-agreement + new_id: cid005412 + primary_alias: FTH0150 + type: formal_theorem + title: uniqueness_from_agreement (theorem) + status: canonical + release_lines: *id1098 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::uniqueness_from_agreement + new_id: cid005412 + primary_alias: FTH0150 + type: formal_theorem + title: uniqueness_from_agreement (theorem) + status: canonical + release_lines: *id1098 +- old_id: bndlift_tower + new_id: cid005413 + primary_alias: FTH0151 + type: formal_theorem + title: bndlift_tower (theorem) + status: canonical + release_lines: &id1099 + - corpus_v2 + - corpus_v3_working +- old_id: bndlift-tower + new_id: cid005413 + primary_alias: FTH0151 + type: formal_theorem + title: bndlift_tower (theorem) + status: canonical + release_lines: *id1099 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::bndlift_tower + new_id: cid005413 + primary_alias: FTH0151 + type: formal_theorem + title: bndlift_tower (theorem) + status: canonical + release_lines: *id1099 +- old_id: extension_bipolar_recovery + new_id: cid005414 + primary_alias: FTH0152 + type: formal_theorem + title: extension_bipolar_recovery (theorem) + status: canonical + release_lines: &id1100 + - corpus_v2 + - corpus_v3_working +- old_id: extension-bipolar-recovery + new_id: cid005414 + primary_alias: FTH0152 + type: formal_theorem + title: extension_bipolar_recovery (theorem) + status: canonical + release_lines: *id1100 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::extension_bipolar_recovery + new_id: cid005414 + primary_alias: FTH0152 + type: formal_theorem + title: extension_bipolar_recovery (theorem) + status: canonical + release_lines: *id1100 +- old_id: ext_channel_20_4 + new_id: cid005415 + primary_alias: FTH0153 + type: formal_theorem + title: ext_channel_20_4 (theorem) + status: canonical + release_lines: &id1101 + - corpus_v2 + - corpus_v3_working +- old_id: ext-channel-20-4 + new_id: cid005415 + primary_alias: FTH0153 + type: formal_theorem + title: ext_channel_20_4 (theorem) + status: canonical + release_lines: *id1101 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::ext_channel_20_4 + new_id: cid005415 + primary_alias: FTH0153 + type: formal_theorem + title: ext_channel_20_4 (theorem) + status: canonical + release_lines: *id1101 +- old_id: b_channel_20_4 + new_id: cid005416 + primary_alias: FTH0154 + type: formal_theorem + title: b_channel_20_4 (theorem) + status: canonical + release_lines: &id1102 + - corpus_v2 + - corpus_v3_working +- old_id: b-channel-20-4 + new_id: cid005416 + primary_alias: FTH0154 + type: formal_theorem + title: b_channel_20_4 (theorem) + status: canonical + release_lines: *id1102 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::b_channel_20_4 + new_id: cid005416 + primary_alias: FTH0154 + type: formal_theorem + title: b_channel_20_4 (theorem) + status: canonical + release_lines: *id1102 +- old_id: c_channel_20_4 + new_id: cid005417 + primary_alias: FTH0155 + type: formal_theorem + title: c_channel_20_4 (theorem) + status: canonical + release_lines: &id1103 + - corpus_v2 + - corpus_v3_working +- old_id: c-channel-20-4 + new_id: cid005417 + primary_alias: FTH0155 + type: formal_theorem + title: c_channel_20_4 (theorem) + status: canonical + release_lines: *id1103 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::c_channel_20_4 + new_id: cid005417 + primary_alias: FTH0155 + type: formal_theorem + title: c_channel_20_4 (theorem) + status: canonical + release_lines: *id1103 +- old_id: hartogs_uniq_15_4 + new_id: cid005418 + primary_alias: FTH0156 + type: formal_theorem + title: hartogs_uniq_15_4 (theorem) + status: canonical + release_lines: &id1104 + - corpus_v2 + - corpus_v3_working +- old_id: hartogs-uniq-15-4 + new_id: cid005418 + primary_alias: FTH0156 + type: formal_theorem + title: hartogs_uniq_15_4 (theorem) + status: canonical + release_lines: *id1104 +- old_id: TauLib.BookII.CentralTheorem.HartogsExtension::hartogs_uniq_15_4 + new_id: cid005418 + primary_alias: FTH0156 + type: formal_theorem + title: hartogs_uniq_15_4 (theorem) + status: canonical + release_lines: *id1104 +- old_id: bnd_det_int_10_3 + new_id: cid005419 + primary_alias: FTH0157 + type: formal_theorem + title: bnd_det_int_10_3 (theorem) + status: canonical + release_lines: &id1105 + - 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corpus_v2 + - corpus_v3_working +- old_id: stage-euclidean-check-true + new_id: cid005637 + primary_alias: FTH0375 + type: formal_theorem + title: stage_euclidean_check_true (theorem) + status: canonical + release_lines: *id1323 +- old_id: TauLib.BookII.Mirror.WaveHolomorphy::stage_euclidean_check_true + new_id: cid005637 + primary_alias: FTH0375 + type: formal_theorem + title: stage_euclidean_check_true (theorem) + status: canonical + release_lines: *id1323 +- old_id: all_stages_euclidean_5 + new_id: cid005638 + primary_alias: FTH0376 + type: formal_theorem + title: all_stages_euclidean_5 (theorem) + status: canonical + release_lines: &id1324 + - corpus_v2 + - corpus_v3_working +- old_id: all-stages-euclidean-5 + new_id: cid005638 + primary_alias: FTH0376 + type: formal_theorem + title: all_stages_euclidean_5 (theorem) + status: canonical + release_lines: *id1324 +- old_id: TauLib.BookII.Mirror.WaveHolomorphy::all_stages_euclidean_5 + new_id: cid005638 + primary_alias: FTH0376 + type: formal_theorem + title: all_stages_euclidean_5 (theorem) + status: canonical + release_lines: *id1324 +- old_id: stage_size_is_primorial + new_id: cid005639 + primary_alias: FTH0377 + type: formal_theorem + title: stage_size_is_primorial (theorem) + status: canonical + release_lines: &id1325 + - 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corpus_v2 + - corpus_v3_working +- old_id: crt-spectral-20-4 + new_id: cid006017 + primary_alias: FTH0752 + type: formal_theorem + title: crt_spectral_20_4 (theorem) + status: canonical + release_lines: *id1702 +- old_id: TauLib.BookIII.Spectral.CRT::crt_spectral_20_4 + new_id: cid006017 + primary_alias: FTH0752 + type: formal_theorem + title: crt_spectral_20_4 (theorem) + status: canonical + release_lines: *id1702 +- old_id: crt_add_10_3 + new_id: cid006018 + primary_alias: FTH0753 + type: formal_theorem + title: crt_add_10_3 (theorem) + status: canonical + release_lines: &id1703 + - corpus_v2 + - corpus_v3_working +- old_id: crt-add-10-3 + new_id: cid006018 + primary_alias: FTH0753 + type: formal_theorem + title: crt_add_10_3 (theorem) + status: canonical + release_lines: *id1703 +- old_id: TauLib.BookIII.Spectral.CRT::crt_add_10_3 + new_id: cid006018 + primary_alias: FTH0753 + type: formal_theorem + title: crt_add_10_3 (theorem) + status: canonical + release_lines: *id1703 +- old_id: reconstruction_20_4 + new_id: cid006019 + primary_alias: FTH0754 + type: formal_theorem + title: reconstruction_20_4 (theorem) + status: canonical + release_lines: &id1704 + - corpus_v2 + - corpus_v3_working +- old_id: reconstruction-20-4 + new_id: cid006019 + primary_alias: FTH0754 + type: formal_theorem + title: reconstruction_20_4 (theorem) + status: canonical + release_lines: *id1704 +- old_id: TauLib.BookIII.Spectral.CRT::reconstruction_20_4 + new_id: cid006019 + primary_alias: FTH0754 + type: formal_theorem + title: reconstruction_20_4 (theorem) + status: canonical + release_lines: *id1704 +- old_id: prime_independence_5_4 + new_id: cid006020 + primary_alias: FTH0755 + type: formal_theorem + title: prime_independence_5_4 (theorem) + status: canonical + release_lines: &id1705 + - corpus_v2 + - corpus_v3_working +- old_id: prime-independence-5-4 + new_id: cid006020 + primary_alias: FTH0755 + type: formal_theorem + title: prime_independence_5_4 (theorem) + status: canonical + release_lines: *id1705 +- old_id: TauLib.BookIII.Spectral.CRT::prime_independence_5_4 + new_id: cid006020 + primary_alias: FTH0755 + type: formal_theorem + title: prime_independence_5_4 (theorem) + status: canonical + release_lines: *id1705 +- old_id: crt_roundtrip_42 + new_id: cid006021 + primary_alias: FTH0756 + type: formal_theorem + title: crt_roundtrip_42 (theorem) + status: canonical + release_lines: &id1706 + - corpus_v2 + - corpus_v3_working +- old_id: crt-roundtrip-42 + new_id: cid006021 + primary_alias: FTH0756 + type: formal_theorem + title: crt_roundtrip_42 (theorem) + status: canonical + release_lines: *id1706 +- old_id: TauLib.BookIII.Spectral.CRT::crt_roundtrip_42 + new_id: cid006021 + primary_alias: FTH0756 + type: formal_theorem + title: crt_roundtrip_42 (theorem) + status: canonical + release_lines: *id1706 +- old_id: crt_decompose_42 + new_id: cid006022 + primary_alias: FTH0757 + type: formal_theorem + title: crt_decompose_42 (theorem) + status: canonical + release_lines: &id1707 + - corpus_v2 + - corpus_v3_working +- old_id: crt-decompose-42 + new_id: cid006022 + primary_alias: FTH0757 + type: formal_theorem + title: crt_decompose_42 (theorem) + status: canonical + release_lines: *id1707 +- old_id: TauLib.BookIII.Spectral.CRT::crt_decompose_42 + new_id: cid006022 + primary_alias: FTH0757 + type: formal_theorem + title: crt_decompose_42 (theorem) + status: canonical + release_lines: *id1707 +- old_id: crt_basis_0_3 + new_id: cid006023 + primary_alias: FTH0758 + type: formal_theorem + title: crt_basis_0_3 (theorem) + status: canonical + release_lines: &id1708 + - corpus_v2 + - corpus_v3_working +- old_id: crt-basis-0-3 + new_id: cid006023 + primary_alias: FTH0758 + type: formal_theorem + title: crt_basis_0_3 (theorem) + status: canonical + release_lines: *id1708 +- old_id: TauLib.BookIII.Spectral.CRT::crt_basis_0_3 + new_id: cid006023 + primary_alias: FTH0758 + type: formal_theorem + title: crt_basis_0_3 (theorem) + status: canonical + release_lines: *id1708 +- old_id: bridge_algebraic_identity + new_id: cid006024 + primary_alias: FTH0759 + type: formal_theorem + title: bridge_algebraic_identity (theorem) + status: canonical + release_lines: &id1709 + - corpus_v2 + - corpus_v3_working +- old_id: bridge-algebraic-identity + new_id: cid006024 + primary_alias: FTH0759 + type: formal_theorem + title: bridge_algebraic_identity (theorem) + status: canonical + release_lines: *id1709 +- old_id: TauLib.BookIII.Spectral.ConfinementBridge::bridge_algebraic_identity + new_id: cid006024 + primary_alias: FTH0759 + type: formal_theorem + title: bridge_algebraic_identity (theorem) + status: canonical + release_lines: *id1709 +- old_id: bridge_reduces_to_E6_near_identity + new_id: cid006025 + primary_alias: FTH0760 + type: formal_theorem + title: bridge_reduces_to_E6_near_identity (theorem) + status: canonical + release_lines: &id1710 + - corpus_v2 + - corpus_v3_working +- old_id: bridge-reduces-to-e6-near-identity + new_id: cid006025 + primary_alias: FTH0760 + type: formal_theorem + title: bridge_reduces_to_E6_near_identity (theorem) + status: canonical + release_lines: *id1710 +- old_id: TauLib.BookIII.Spectral.ConfinementBridge::bridge_reduces_to_E6_near_identity + new_id: cid006025 + primary_alias: FTH0760 + type: formal_theorem + title: bridge_reduces_to_E6_near_identity (theorem) + status: canonical + release_lines: *id1710 +- old_id: confinement_bridge_lower + new_id: cid006026 + primary_alias: FTH0761 + type: formal_theorem + title: confinement_bridge_lower (theorem) + status: canonical + release_lines: &id1711 + - corpus_v2 + - corpus_v3_working +- old_id: confinement-bridge-lower + new_id: cid006026 + primary_alias: FTH0761 + type: formal_theorem + title: confinement_bridge_lower (theorem) + status: canonical + release_lines: *id1711 +- old_id: TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge_lower + new_id: cid006026 + primary_alias: FTH0761 + type: formal_theorem + title: confinement_bridge_lower (theorem) + status: canonical + release_lines: *id1711 +- old_id: confinement_bridge + new_id: cid006027 + primary_alias: FTH0762 + type: formal_theorem + title: confinement_bridge (theorem) + status: canonical + release_lines: &id1712 + - corpus_v2 + - corpus_v3_working +- old_id: confinement-bridge + new_id: cid006027 + primary_alias: FTH0762 + type: formal_theorem + title: confinement_bridge (theorem) + status: canonical + release_lines: *id1712 +- old_id: TauLib.BookIII.Spectral.ConfinementBridge::confinement_bridge + new_id: cid006027 + primary_alias: FTH0762 + type: formal_theorem + title: confinement_bridge (theorem) + status: canonical + release_lines: *id1712 +- old_id: goldbach_500 + new_id: cid006028 + primary_alias: FTH0763 + type: formal_theorem + title: goldbach_500 (theorem) + status: canonical + release_lines: &id1713 + - corpus_v2 + - corpus_v3_working +- old_id: goldbach-500 + new_id: cid006028 + primary_alias: FTH0763 + type: formal_theorem + title: goldbach_500 (theorem) + status: canonical + release_lines: *id1713 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::goldbach_500 + new_id: cid006028 + primary_alias: FTH0763 + type: formal_theorem + title: goldbach_500 (theorem) + status: canonical + release_lines: *id1713 +- old_id: goldbach_primorial_m4 + new_id: cid006029 + primary_alias: FTH0764 + type: formal_theorem + title: goldbach_primorial_m4 (theorem) + status: canonical + release_lines: &id1714 + - corpus_v2 + - corpus_v3_working +- old_id: goldbach-primorial-m4 + new_id: cid006029 + primary_alias: FTH0764 + type: formal_theorem + title: goldbach_primorial_m4 (theorem) + status: canonical + release_lines: *id1714 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::goldbach_primorial_m4 + new_id: cid006029 + primary_alias: FTH0764 + type: formal_theorem + title: goldbach_primorial_m4 (theorem) + status: canonical + release_lines: *id1714 +- old_id: partition_growth_4 + new_id: cid006030 + primary_alias: FTH0765 + type: formal_theorem + title: partition_growth_4 (theorem) + status: canonical + release_lines: &id1715 + - corpus_v2 + - corpus_v3_working +- old_id: partition-growth-4 + new_id: cid006030 + primary_alias: FTH0765 + type: formal_theorem + title: partition_growth_4 (theorem) + status: canonical + release_lines: *id1715 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::partition_growth_4 + new_id: cid006030 + primary_alias: FTH0765 + type: formal_theorem + title: partition_growth_4 (theorem) + status: canonical + release_lines: *id1715 +- old_id: obstruction_bounded_5 + new_id: cid006031 + primary_alias: FTH0766 + type: formal_theorem + title: obstruction_bounded_5 (theorem) + status: canonical + release_lines: &id1716 + - corpus_v2 + - corpus_v3_working +- old_id: obstruction-bounded-5 + new_id: cid006031 + primary_alias: FTH0766 + type: formal_theorem + title: obstruction_bounded_5 (theorem) + status: canonical + release_lines: *id1716 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::obstruction_bounded_5 + new_id: cid006031 + primary_alias: FTH0766 + type: formal_theorem + title: obstruction_bounded_5 (theorem) + status: canonical + release_lines: *id1716 +- old_id: crt_goldbach_duality_3 + new_id: cid006032 + primary_alias: FTH0767 + type: formal_theorem + title: crt_goldbach_duality_3 (theorem) + status: canonical + release_lines: &id1717 + - corpus_v2 + - corpus_v3_working +- old_id: crt-goldbach-duality-3 + new_id: cid006032 + primary_alias: FTH0767 + type: formal_theorem + title: crt_goldbach_duality_3 (theorem) + status: canonical + release_lines: *id1717 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::crt_goldbach_duality_3 + new_id: cid006032 + primary_alias: FTH0767 + type: formal_theorem + title: crt_goldbach_duality_3 (theorem) + status: canonical + release_lines: *id1717 +- old_id: partition_m2 + new_id: cid006033 + primary_alias: FTH0768 + type: formal_theorem + title: partition_m2 (theorem) + status: canonical + release_lines: &id1718 + - corpus_v2 + - corpus_v3_working +- old_id: partition-m2 + new_id: cid006033 + primary_alias: FTH0768 + type: formal_theorem + title: partition_m2 (theorem) + status: canonical + release_lines: *id1718 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::partition_m2 + new_id: cid006033 + primary_alias: FTH0768 + type: formal_theorem + title: partition_m2 (theorem) + status: canonical + release_lines: *id1718 +- old_id: partition_m3 + new_id: cid006034 + primary_alias: FTH0769 + type: formal_theorem + title: partition_m3 (theorem) + status: canonical + release_lines: &id1719 + - corpus_v2 + - corpus_v3_working +- old_id: partition-m3 + new_id: cid006034 + primary_alias: FTH0769 + type: formal_theorem + title: partition_m3 (theorem) + status: canonical + release_lines: *id1719 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::partition_m3 + new_id: cid006034 + primary_alias: FTH0769 + type: formal_theorem + title: partition_m3 (theorem) + status: canonical + release_lines: *id1719 +- old_id: partition_m4_pos + new_id: cid006035 + primary_alias: FTH0770 + type: formal_theorem + title: partition_m4_pos (theorem) + status: canonical + release_lines: &id1720 + - corpus_v2 + - corpus_v3_working +- old_id: partition-m4-pos + new_id: cid006035 + primary_alias: FTH0770 + type: formal_theorem + title: partition_m4_pos (theorem) + status: canonical + release_lines: *id1720 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::partition_m4_pos + new_id: cid006035 + primary_alias: FTH0770 + type: formal_theorem + title: partition_m4_pos (theorem) + status: canonical + release_lines: *id1720 +- old_id: obstruction_100_p2 + new_id: cid006036 + primary_alias: FTH0771 + type: formal_theorem + title: obstruction_100_p2 (theorem) + status: canonical + release_lines: &id1721 + - corpus_v2 + - corpus_v3_working +- old_id: obstruction-100-p2 + new_id: cid006036 + primary_alias: FTH0771 + type: formal_theorem + title: obstruction_100_p2 (theorem) + status: canonical + release_lines: *id1721 +- old_id: TauLib.BookIII.Spectral.GoldbachDeep::obstruction_100_p2 + new_id: cid006036 + primary_alias: FTH0771 + type: formal_theorem + title: obstruction_100_p2 (theorem) + status: canonical + release_lines: *id1721 +- old_id: obstruction_100_p3 + new_id: cid006037 + primary_alias: FTH0772 + type: formal_theorem + title: obstruction_100_p3 (theorem) + status: canonical + release_lines: &id1722 + - 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corpus_v2 + - corpus_v3_working +- old_id: one-is-root-3 + new_id: cid006041 + primary_alias: FTH0776 + type: formal_theorem + title: one_is_root_3 (theorem) + status: canonical + release_lines: *id1726 +- old_id: TauLib.BookIII.Spectral.HenselLifting::one_is_root_3 + new_id: cid006041 + primary_alias: FTH0776 + type: formal_theorem + title: one_is_root_3 (theorem) + status: canonical + release_lines: *id1726 +- old_id: hensel_1_mod_9 + new_id: cid006042 + primary_alias: FTH0777 + type: formal_theorem + title: hensel_1_mod_9 (theorem) + status: canonical + release_lines: &id1727 + - corpus_v2 + - corpus_v3_working +- old_id: hensel-1-mod-9 + new_id: cid006042 + primary_alias: FTH0777 + type: formal_theorem + title: hensel_1_mod_9 (theorem) + status: canonical + release_lines: *id1727 +- old_id: TauLib.BookIII.Spectral.HenselLifting::hensel_1_mod_9 + new_id: cid006042 + primary_alias: FTH0777 + type: formal_theorem + title: hensel_1_mod_9 (theorem) + status: canonical + release_lines: *id1727 +- old_id: hensel_correct_3_2 + new_id: cid006043 + primary_alias: FTH0778 + type: formal_theorem + title: hensel_correct_3_2 (theorem) + status: canonical + release_lines: &id1728 + - corpus_v2 + - corpus_v3_working +- old_id: hensel-correct-3-2 + new_id: cid006043 + primary_alias: FTH0778 + type: formal_theorem + title: hensel_correct_3_2 (theorem) + status: canonical + release_lines: *id1728 +- old_id: TauLib.BookIII.Spectral.HenselLifting::hensel_correct_3_2 + new_id: cid006043 + primary_alias: FTH0778 + type: formal_theorem + title: hensel_correct_3_2 (theorem) + status: canonical + release_lines: *id1728 +- old_id: local_field_15_4 + new_id: cid006044 + primary_alias: FTH0779 + type: formal_theorem + title: local_field_15_4 (theorem) + status: canonical + release_lines: &id1729 + - corpus_v2 + - corpus_v3_working +- old_id: local-field-15-4 + new_id: cid006044 + primary_alias: FTH0779 + type: formal_theorem + title: local_field_15_4 (theorem) + status: canonical + release_lines: *id1729 +- old_id: TauLib.BookIII.Spectral.LocalFields::local_field_15_4 + new_id: cid006044 + primary_alias: FTH0779 + type: formal_theorem + title: local_field_15_4 (theorem) + status: canonical + release_lines: *id1729 +- old_id: local_ring_10_4 + new_id: cid006045 + primary_alias: FTH0780 + type: formal_theorem + title: local_ring_10_4 (theorem) + status: canonical + release_lines: &id1730 + - 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corpus_v2 + - corpus_v3_working +- old_id: trichotomy-func-15-3 + new_id: cid006079 + primary_alias: FTH0814 + type: formal_theorem + title: trichotomy_func_15_3 (theorem) + status: canonical + release_lines: *id1764 +- old_id: TauLib.BookIII.Spectral.Trichotomy::trichotomy_func_15_3 + new_id: cid006079 + primary_alias: FTH0814 + type: formal_theorem + title: trichotomy_func_15_3 (theorem) + status: canonical + release_lines: *id1764 +- old_id: bnf_15_4 + new_id: cid006080 + primary_alias: FTH0815 + type: formal_theorem + title: bnf_15_4 (theorem) + status: canonical + release_lines: &id1765 + - corpus_v2 + - corpus_v3_working +- old_id: bnf-15-4 + new_id: cid006080 + primary_alias: FTH0815 + type: formal_theorem + title: bnf_15_4 (theorem) + status: canonical + release_lines: *id1765 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bnf_15_4 + new_id: cid006080 + primary_alias: FTH0815 + type: formal_theorem + title: bnf_15_4 (theorem) + status: canonical + release_lines: *id1765 +- old_id: bnf_unique_10_3 + new_id: cid006081 + primary_alias: FTH0816 + type: formal_theorem + title: bnf_unique_10_3 (theorem) + status: canonical + release_lines: &id1766 + - corpus_v2 + - corpus_v3_working +- old_id: bnf-unique-10-3 + new_id: cid006081 + primary_alias: FTH0816 + type: formal_theorem + title: bnf_unique_10_3 (theorem) + status: canonical + release_lines: *id1766 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bnf_unique_10_3 + new_id: cid006081 + primary_alias: FTH0816 + type: formal_theorem + title: bnf_unique_10_3 (theorem) + status: canonical + release_lines: *id1766 +- old_id: bc_non_collapse_10_5 + new_id: cid006082 + primary_alias: FTH0817 + type: formal_theorem + title: bc_non_collapse_10_5 (theorem) + status: canonical + release_lines: &id1767 + - corpus_v2 + - corpus_v3_working +- old_id: bc-non-collapse-10-5 + new_id: cid006082 + primary_alias: FTH0817 + type: formal_theorem + title: bc_non_collapse_10_5 (theorem) + status: canonical + release_lines: *id1767 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bc_non_collapse_10_5 + new_id: cid006082 + primary_alias: FTH0817 + type: formal_theorem + title: bc_non_collapse_10_5 (theorem) + status: canonical + release_lines: *id1767 +- old_id: bc_coprime_5 + new_id: cid006083 + primary_alias: FTH0818 + type: formal_theorem + title: bc_coprime_5 (theorem) + status: canonical + release_lines: &id1768 + - corpus_v2 + - corpus_v3_working +- old_id: bc-coprime-5 + new_id: cid006083 + primary_alias: FTH0818 + type: formal_theorem + title: bc_coprime_5 (theorem) + status: canonical + release_lines: *id1768 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bc_coprime_5 + new_id: cid006083 + primary_alias: FTH0818 + type: formal_theorem + title: bc_coprime_5 (theorem) + status: canonical + release_lines: *id1768 +- old_id: trichotomy_depth_1 + new_id: cid006084 + primary_alias: FTH0819 + type: formal_theorem + title: trichotomy_depth_1 (theorem) + status: canonical + release_lines: &id1769 + - corpus_v2 + - corpus_v3_working +- old_id: trichotomy-depth-1 + new_id: cid006084 + primary_alias: FTH0819 + type: formal_theorem + title: trichotomy_depth_1 (theorem) + status: canonical + release_lines: *id1769 +- old_id: TauLib.BookIII.Spectral.Trichotomy::trichotomy_depth_1 + new_id: cid006084 + primary_alias: FTH0819 + type: formal_theorem + title: trichotomy_depth_1 (theorem) + status: canonical + release_lines: *id1769 +- old_id: bnf_zero_3 + new_id: cid006085 + primary_alias: FTH0820 + type: formal_theorem + title: bnf_zero_3 (theorem) + status: canonical + release_lines: &id1770 + - corpus_v2 + - corpus_v3_working +- old_id: bnf-zero-3 + new_id: cid006085 + primary_alias: FTH0820 + type: formal_theorem + title: bnf_zero_3 (theorem) + status: canonical + release_lines: *id1770 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bnf_zero_3 + new_id: cid006085 + primary_alias: FTH0820 + type: formal_theorem + title: bnf_zero_3 (theorem) + status: canonical + release_lines: *id1770 +- old_id: bc_coprime_at_3 + new_id: cid006086 + primary_alias: FTH0821 + type: formal_theorem + title: bc_coprime_at_3 (theorem) + status: canonical + release_lines: &id1771 + - corpus_v2 + - corpus_v3_working +- old_id: bc-coprime-at-3 + new_id: cid006086 + primary_alias: FTH0821 + type: formal_theorem + title: bc_coprime_at_3 (theorem) + status: canonical + release_lines: *id1771 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bc_coprime_at_3 + new_id: cid006086 + primary_alias: FTH0821 + type: formal_theorem + title: bc_coprime_at_3 (theorem) + status: canonical + release_lines: *id1771 +- old_id: bc_distinct_3 + new_id: cid006087 + primary_alias: FTH0822 + type: formal_theorem + title: bc_distinct_3 (theorem) + status: canonical + release_lines: &id1772 + - corpus_v2 + - corpus_v3_working +- old_id: bc-distinct-3 + new_id: cid006087 + primary_alias: FTH0822 + type: formal_theorem + title: bc_distinct_3 (theorem) + status: canonical + release_lines: *id1772 +- old_id: TauLib.BookIII.Spectral.Trichotomy::bc_distinct_3 + new_id: cid006087 + primary_alias: FTH0822 + type: formal_theorem + title: bc_distinct_3 (theorem) + status: canonical + release_lines: *id1772 +- old_id: twin_primes_500 + new_id: cid006088 + primary_alias: FTH0823 + type: formal_theorem + title: twin_primes_500 (theorem) + status: canonical + release_lines: &id1773 + - corpus_v2 + - corpus_v3_working +- old_id: twin-primes-500 + new_id: cid006088 + primary_alias: FTH0823 + type: formal_theorem + title: twin_primes_500 (theorem) + status: canonical + release_lines: *id1773 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_primes_500 + new_id: cid006088 + primary_alias: FTH0823 + type: formal_theorem + title: twin_primes_500 (theorem) + status: canonical + release_lines: *id1773 +- old_id: twin_density_primorial_5 + new_id: cid006089 + primary_alias: FTH0824 + type: formal_theorem + title: twin_density_primorial_5 (theorem) + status: canonical + release_lines: &id1774 + - corpus_v2 + - corpus_v3_working +- old_id: twin-density-primorial-5 + new_id: cid006089 + primary_alias: FTH0824 + type: formal_theorem + title: twin_density_primorial_5 (theorem) + status: canonical + release_lines: *id1774 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_density_primorial_5 + new_id: cid006089 + primary_alias: FTH0824 + type: formal_theorem + title: twin_density_primorial_5 (theorem) + status: canonical + release_lines: *id1774 +- old_id: hl_constant_decreasing_5 + new_id: cid006090 + primary_alias: FTH0825 + type: formal_theorem + title: hl_constant_decreasing_5 (theorem) + status: canonical + release_lines: &id1775 + - corpus_v2 + - corpus_v3_working +- old_id: hl-constant-decreasing-5 + new_id: cid006090 + primary_alias: FTH0825 + type: formal_theorem + title: hl_constant_decreasing_5 (theorem) + status: canonical + release_lines: *id1775 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::hl_constant_decreasing_5 + new_id: cid006090 + primary_alias: FTH0825 + type: formal_theorem + title: hl_constant_decreasing_5 (theorem) + status: canonical + release_lines: *id1775 +- old_id: crt_admissible_positive_4 + new_id: cid006091 + primary_alias: FTH0826 + type: formal_theorem + title: crt_admissible_positive_4 (theorem) + status: canonical + release_lines: &id1776 + - corpus_v2 + - corpus_v3_working +- old_id: crt-admissible-positive-4 + new_id: cid006091 + primary_alias: FTH0826 + type: formal_theorem + title: crt_admissible_positive_4 (theorem) + status: canonical + release_lines: *id1776 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::crt_admissible_positive_4 + new_id: cid006091 + primary_alias: FTH0826 + type: formal_theorem + title: crt_admissible_positive_4 (theorem) + status: canonical + release_lines: *id1776 +- old_id: twin_admissibility_fraction_5 + new_id: cid006092 + primary_alias: FTH0827 + type: formal_theorem + title: twin_admissibility_fraction_5 (theorem) + status: canonical + release_lines: &id1777 + - corpus_v2 + - corpus_v3_working +- old_id: twin-admissibility-fraction-5 + new_id: cid006092 + primary_alias: FTH0827 + type: formal_theorem + title: twin_admissibility_fraction_5 (theorem) + status: canonical + release_lines: *id1777 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissibility_fraction_5 + new_id: cid006092 + primary_alias: FTH0827 + type: formal_theorem + title: twin_admissibility_fraction_5 (theorem) + status: canonical + release_lines: *id1777 +- old_id: twin_count_100 + new_id: cid006093 + primary_alias: FTH0828 + type: formal_theorem + title: twin_count_100 (theorem) + status: canonical + release_lines: &id1778 + - corpus_v2 + - corpus_v3_working +- old_id: twin-count-100 + new_id: cid006093 + primary_alias: FTH0828 + type: formal_theorem + title: twin_count_100 (theorem) + status: canonical + release_lines: *id1778 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_count_100 + new_id: cid006093 + primary_alias: FTH0828 + type: formal_theorem + title: twin_count_100 (theorem) + status: canonical + release_lines: *id1778 +- old_id: hl_depth_2 + new_id: cid006094 + primary_alias: FTH0829 + type: formal_theorem + title: hl_depth_2 (theorem) + status: canonical + release_lines: &id1779 + - corpus_v2 + - corpus_v3_working +- old_id: hl-depth-2 + new_id: cid006094 + primary_alias: FTH0829 + type: formal_theorem + title: hl_depth_2 (theorem) + status: canonical + release_lines: *id1779 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::hl_depth_2 + new_id: cid006094 + primary_alias: FTH0829 + type: formal_theorem + title: hl_depth_2 (theorem) + status: canonical + release_lines: *id1779 +- old_id: twin_admissible_1 + new_id: cid006095 + primary_alias: FTH0830 + type: formal_theorem + title: twin_admissible_1 (theorem) + status: canonical + release_lines: &id1780 + - corpus_v2 + - corpus_v3_working +- old_id: twin-admissible-1 + new_id: cid006095 + primary_alias: FTH0830 + type: formal_theorem + title: twin_admissible_1 (theorem) + status: canonical + release_lines: *id1780 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_1 + new_id: cid006095 + primary_alias: FTH0830 + type: formal_theorem + title: twin_admissible_1 (theorem) + status: canonical + release_lines: *id1780 +- old_id: twin_admissible_3_pos + new_id: cid006096 + primary_alias: FTH0831 + type: formal_theorem + title: twin_admissible_3_pos (theorem) + status: canonical + release_lines: &id1781 + - corpus_v2 + - corpus_v3_working +- old_id: twin-admissible-3-pos + new_id: cid006096 + primary_alias: FTH0831 + type: formal_theorem + title: twin_admissible_3_pos (theorem) + status: canonical + release_lines: *id1781 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::twin_admissible_3_pos + new_id: cid006096 + primary_alias: FTH0831 + type: formal_theorem + title: twin_admissible_3_pos (theorem) + status: canonical + release_lines: *id1781 +- old_id: admissible_at_3 + new_id: cid006097 + primary_alias: FTH0832 + type: formal_theorem + title: admissible_at_3 (theorem) + status: canonical + release_lines: &id1782 + - corpus_v2 + - corpus_v3_working +- old_id: admissible-at-3 + new_id: cid006097 + primary_alias: FTH0832 + type: formal_theorem + title: admissible_at_3 (theorem) + status: canonical + release_lines: *id1782 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_3 + new_id: cid006097 + primary_alias: FTH0832 + type: formal_theorem + title: admissible_at_3 (theorem) + status: canonical + release_lines: *id1782 +- old_id: admissible_at_5 + new_id: cid006098 + primary_alias: FTH0833 + type: formal_theorem + title: admissible_at_5 (theorem) + status: canonical + release_lines: &id1783 + - corpus_v2 + - corpus_v3_working +- old_id: admissible-at-5 + new_id: cid006098 + primary_alias: FTH0833 + type: formal_theorem + title: admissible_at_5 (theorem) + status: canonical + release_lines: *id1783 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_5 + new_id: cid006098 + primary_alias: FTH0833 + type: formal_theorem + title: admissible_at_5 (theorem) + status: canonical + release_lines: *id1783 +- old_id: admissible_at_7 + new_id: cid006099 + primary_alias: FTH0834 + type: formal_theorem + title: admissible_at_7 (theorem) + status: canonical + release_lines: &id1784 + - corpus_v2 + - corpus_v3_working +- old_id: admissible-at-7 + new_id: cid006099 + primary_alias: FTH0834 + type: formal_theorem + title: admissible_at_7 (theorem) + status: canonical + release_lines: *id1784 +- old_id: TauLib.BookIII.Spectral.TwinPrimeDeep::admissible_at_7 + new_id: cid006099 + primary_alias: FTH0834 + type: formal_theorem + title: admissible_at_7 (theorem) + status: canonical + release_lines: *id1784 +- old_id: width_principle + new_id: cid006100 + primary_alias: FTH0835 + type: formal_theorem + title: width_principle (theorem) + status: canonical + release_lines: &id1785 + - corpus_v2 + - corpus_v3_working +- old_id: width-principle + new_id: cid006100 + primary_alias: FTH0835 + type: formal_theorem + title: width_principle (theorem) + status: canonical + release_lines: *id1785 +- old_id: TauLib.BookIII.Spectrum.InterfaceWidth::width_principle + new_id: cid006100 + primary_alias: FTH0835 + type: formal_theorem + title: width_principle (theorem) + status: canonical + release_lines: *id1785 +- old_id: chi_plus_admissible + new_id: cid006101 + primary_alias: FTH0836 + type: formal_theorem + title: chi_plus_admissible (theorem) + status: canonical + release_lines: &id1786 + - corpus_v2 + - corpus_v3_working +- old_id: chi-plus-admissible + new_id: cid006101 + primary_alias: FTH0836 + type: formal_theorem + title: chi_plus_admissible (theorem) + status: canonical + release_lines: *id1786 +- old_id: TauLib.BookIII.Spectrum.InterfaceWidth::chi_plus_admissible + new_id: cid006101 + primary_alias: FTH0836 + type: formal_theorem + title: chi_plus_admissible (theorem) + status: canonical + release_lines: *id1786 +- old_id: book_ii_bridge_complete + new_id: cid006102 + primary_alias: FTH0837 + type: formal_theorem + title: book_ii_bridge_complete (theorem) + status: canonical + release_lines: &id1787 + - corpus_v2 + - corpus_v3_working +- old_id: book-ii-bridge-complete + new_id: cid006102 + primary_alias: FTH0837 + type: formal_theorem + title: book_ii_bridge_complete (theorem) + status: canonical + release_lines: *id1787 +- old_id: TauLib.BookIII.Spectrum.KernelHinge::book_ii_bridge_complete + new_id: cid006102 + primary_alias: FTH0837 + type: formal_theorem + title: book_ii_bridge_complete (theorem) + status: canonical + release_lines: *id1787 +- old_id: spectral_decidable + new_id: cid006103 + primary_alias: FTH0838 + type: formal_theorem + title: spectral_decidable (theorem) + status: canonical + release_lines: &id1788 + - corpus_v2 + - corpus_v3_working +- old_id: spectral-decidable + new_id: cid006103 + primary_alias: FTH0838 + type: formal_theorem + title: spectral_decidable (theorem) + status: canonical + release_lines: *id1788 +- old_id: TauLib.BookIII.Spectrum.ThreeSAT::spectral_decidable + new_id: cid006103 + primary_alias: FTH0838 + type: formal_theorem + title: spectral_decidable (theorem) + status: canonical + release_lines: *id1788 +- old_id: tau_complexity_bridge_concrete + new_id: cid006104 + primary_alias: FTH0839 + type: formal_theorem + title: tau_complexity_bridge_concrete (theorem) + status: canonical + release_lines: &id1789 + - corpus_v2 + - corpus_v3_working +- old_id: tau-complexity-bridge-concrete + new_id: cid006104 + primary_alias: FTH0839 + type: formal_theorem + title: tau_complexity_bridge_concrete (theorem) + status: canonical + release_lines: *id1789 +- old_id: TauLib.BookIII.Spectrum.ThreeSAT::tau_complexity_bridge_concrete + new_id: cid006104 + primary_alias: FTH0839 + type: formal_theorem + title: tau_complexity_bridge_concrete (theorem) + status: canonical + release_lines: *id1789 +- old_id: proof-hyperfactorization + new_id: cid006105 + primary_alias: PRF0002 + type: proof + title: Proof of the Hyperfactorization Theorem + status: canonical + release_lines: + - corpus_v3_working +- old_id: proof-sketch-global-hartogs + new_id: cid006106 + primary_alias: PRF0003 + type: proof + title: Proof Sketch of Global Hartogs + status: active + release_lines: + - corpus_v3_working +- old_id: proof-categoricity-conditional + new_id: cid006107 + primary_alias: PRF0004 + type: proof + title: Conditional Proof of Categoricity of τ_0 + status: active + release_lines: + - corpus_v3_working +- old_id: proof-fta-tau-idx-lean + new_id: cid006108 + primary_alias: PRF0005 + type: proof + title: Lean-Formalized Proof of FTA on τ-Idx + status: canonical + release_lines: + - corpus_v3_working +- old_id: proof-prose-six-generator-rigidity + new_id: cid006109 + primary_alias: PRF0006 + type: proof + title: Prose Proof of Six-Generator Rigidity Failure + status: active + release_lines: + - corpus_v3_working +alias_history: +- alias: DEF0142 + current_cid: cid000007 + current_primary_alias: DEF0001 + current_type: definition + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: LEM0028 + current_cid: cid000008 + current_primary_alias: LEM0001 + current_type: lemma + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: PRP0042 + current_cid: cid000009 + current_primary_alias: PRP0001 + current_type: proposition + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: THM0031 + current_cid: cid000010 + current_primary_alias: THM0001 + current_type: theorem + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: FTH0044 + current_cid: cid000020 + current_primary_alias: FTH0001 + current_type: formal_theorem + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: RSL0007 + current_cid: cid000021 + current_primary_alias: RSL0001 + current_type: result + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) +- alias: PRF0042 + current_cid: cid000022 + current_primary_alias: PRF0001 + current_type: proof + note: typed alias preserved across history (may be legacy from prior type or v2 + doctrine example) diff --git a/_data/corpus_v3/visibility-values.yaml b/_data/corpus_v3/visibility-values.yaml new file mode 100644 index 000000000..2e29b216f --- /dev/null +++ b/_data/corpus_v3/visibility-values.yaml @@ -0,0 +1,107 @@ +# Visibility Values — YAML projection of governance/VISIBILITY.md +# +# This file is consumed by schemas/corpus-item.schema.json (visibility enum) and +# by scripts/validate_public_closure.py (closure rule semantics). +# +# Source of truth: governance/VISIBILITY.md §1-§4. + +values: + + - value: private + description: Draft, experimental sketch, confidential architecture detail, or otherwise not for any external audience. + public_routes: false + citable: false + in_public_closure: false + single_repo_location: corpus-v3 (any items/ folder) + post_bifurcation_location: private repo only + notes: Default visibility for newly created items mid-development. May reference other private items freely. + + - value: internal + description: Team-visible (within maintainer set), not for public consumption, more shareable than private. + public_routes: false + citable: false + in_public_closure: false + single_repo_location: corpus-v3 + post_bifurcation_location: private repo only + notes: Used for working notes, planning items, internal review documents. + + - value: limited + description: Shareable with named external reviewers (peer reviewers, advisory contacts) but not yet public. + public_routes: false + citable: by_invitation + in_public_closure: false + single_repo_location: corpus-v3 + post_bifurcation_location: private repo only + notes: Often precursor to public once review completes. May carry limited_to (list of reviewer ids). + optional_fields: [limited_to] + + - value: public + description: Part of the public Corpus Graph. Citable, inspectable, indexable. + public_routes: true + citable: true + in_public_closure: true + single_repo_location: corpus-v3 (filtered to public via tooling) + post_bifurcation_location: public corpus repo + notes: Default end-state for active/canonical items. Generates /id/cid###/ route. Subject to public-closure rule. + requires_when_set: + - primary_alias # public items must have a primary_alias + + - value: deprecated_public + description: Previously public, now retired/superseded/refuted/deprecated/archived. Tombstone-rendered. + public_routes: true # tombstone page + citable: with_caution + in_public_closure: true # deprecated nodes + single_repo_location: corpus-v3 (filtered) + post_bifurcation_location: public corpus repo + notes: Set automatically when public item transitions to deprecated/superseded/retired/refuted/archived. + requires_when_set: + - tombstone # boolean true + +# Closure semantics (single-repo cutover state — report-only) +closure: + current_mode: report_only + current_phase: single_repo_cutover + post_bifurcation_mode: enforcing + load_bearing_relations: "see manifests/relation-vocabulary.yaml (load_bearing: true)" + description: | + During the single-repo cutover phase, validate_public_closure.py walks the + graph and emits manifests/public-closure-report.yaml. CI does NOT fail on + closure violations during this phase. Post-bifurcation, the gate flips to + enforcing mode. + +# Lifecycle × Visibility compatibility matrix (from governance/VISIBILITY.md §3) +# (status, visibility) → allowed? +lifecycle_visibility_matrix: + - status: draft + allowed_visibility: [private, internal] + + - status: candidate + allowed_visibility: [private, internal, limited] + + - status: active + allowed_visibility: [public, internal] + + - status: canonical + allowed_visibility: [public, internal] # internal allowed for canonical items not yet publicly released; tightened to [public] in Wave 8 bifurcation + + - status: deprecated + allowed_visibility: [deprecated_public, internal] + + - status: superseded + allowed_visibility: [deprecated_public] + + - status: retired + allowed_visibility: [deprecated_public] + + - status: refuted + allowed_visibility: [deprecated_public] + + - status: archived + allowed_visibility: [deprecated_public] + + # private_deleted has no visibility (item is gone) + +revisions: + - date: 2026-05-10 + change: Initial projection from governance/VISIBILITY.md + author: thorfuchs diff --git a/_includes/corpus-v3/identifier-box.html b/_includes/corpus-v3/identifier-box.html new file mode 100644 index 000000000..a4237cb07 --- /dev/null +++ b/_includes/corpus-v3/identifier-box.html @@ -0,0 +1,88 @@ +{%- comment -%} + Corpus v3 identifier box. + Renders CID, primary alias, legacy aliases, external identifiers as a + compact card on every Corpus Item page. + Web addendum §4 (Identifier Box) + doctrine §20. +{%- endcomment -%} + +
+

Identifiers

+ +
    +
  • + Corpus ID + {{ page.cid }} +
    • + {%- if page.primary_alias and page.primary_alias != "" -%} +
  • + Primary alias + {{ page.primary_alias }} +
    • + {%- endif -%} +
  • + Type + {{ page.type | replace: "_", " " | capitalize }} +
    • +
  • + Status + {{ page.status }} +
    • +
  • + Visibility + {{ page.visibility }} +
    • + {%- if page.current_version -%} +
  • + Version + {{ page.current_version }} +
    • + {%- endif -%} +
+ + {%- if page.aliases.size > 1 -%} +

Aliases & legacy IDs

+
+ {%- for alias in page.aliases -%} + {%- if alias != page.primary_alias -%} + {{ alias }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if page.external_identifiers.size > 0 -%} +

External identifiers

+
    + {%- for key in page.external_identifiers -%} + {%- assign val = page.external_identifiers[key] -%} + {%- if val and val != "" and val != nil -%} +
  • + {{ key | replace: "_", " " | capitalize }} + + {%- if key == "doi" -%} + {{ val }} + {%- elsif key == "wikidata" -%} + {{ val }} + {%- elsif key == "github" -%} + {{ val }} + {%- elsif key == "arxiv" -%} + {{ val }} + {%- else -%} + {{ val }} + {%- endif -%} + +
    • + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if page.release_lines.size > 0 -%} +

Release lines

+
+ {%- for line in page.release_lines -%} + {{ line }} + {%- endfor -%} +
+ {%- endif -%} +
diff --git a/_includes/corpus-v3/jsonld.html b/_includes/corpus-v3/jsonld.html new file mode 100644 index 000000000..d94a1cbdb --- /dev/null +++ b/_includes/corpus-v3/jsonld.html @@ -0,0 +1,73 @@ +{%- comment -%} + Corpus v3 JSON-LD emit. + + Embeds a diff --git a/_includes/corpus-v3/relation-panels.html b/_includes/corpus-v3/relation-panels.html new file mode 100644 index 000000000..2851b2672 --- /dev/null +++ b/_includes/corpus-v3/relation-panels.html @@ -0,0 +1,154 @@ +{%- comment -%} + Corpus v3 relation panels. + Renders upstream / downstream / formalized_by / appears_in / supports / + supersession-history per web addendum §6. + + Walks page.relations[] (canonical typed list) + the convenience-field + shortcuts (depends_on, formalized_by, appears_in, supports, etc.) + established by Charter §3.3 + SCHEMA.md §6. +{%- endcomment -%} + +{%- comment -%} Aggregate predicates from relations[] + convenience fields {%- endcomment -%} +{%- assign upstream_cids = "" | split: "" -%} +{%- assign downstream_cids = "" | split: "" -%} +{%- assign formalized_by_cids = "" | split: "" -%} +{%- assign appears_in_cids = "" | split: "" -%} +{%- assign supports_cids = "" | split: "" -%} + +{%- comment -%} convenience-field depends_on {%- endcomment -%} +{%- for d in page.depends_on -%} + {%- assign upstream_cids = upstream_cids | push: d -%} +{%- endfor -%} + +{%- comment -%} convenience-field formalized_by {%- endcomment -%} +{%- for f in page.formalized_by -%} + {%- if f.id -%} + {%- assign formalized_by_cids = formalized_by_cids | push: f.id -%} + {%- elsif f -%} + {%- assign formalized_by_cids = formalized_by_cids | push: f -%} + {%- endif -%} +{%- endfor -%} + +{%- comment -%} convenience-field appears_in {%- endcomment -%} +{%- for a in page.appears_in -%} + {%- if a.id -%} + {%- assign appears_in_cids = appears_in_cids | push: a.id -%} + {%- elsif a -%} + {%- assign appears_in_cids = appears_in_cids | push: a -%} + {%- endif -%} +{%- endfor -%} + +{%- comment -%} canonical relations[] {%- endcomment -%} +{%- for rel in page.relations -%} + {%- case rel.predicate -%} + {%- when "depends_on" or "uses" or "requires" or "derives" or "extends" or "refines" or "constructs" or "assumes" -%} + {%- assign upstream_cids = upstream_cids | push: rel.object -%} + {%- when "used_by" or "depended_by" -%} + {%- assign downstream_cids = downstream_cids | push: rel.object -%} + {%- when "formalized_by" -%} + {%- assign formalized_by_cids = formalized_by_cids | push: rel.object -%} + {%- when "appears_in" or "published_in" or "explained_in" -%} + {%- assign appears_in_cids = appears_in_cids | push: rel.object -%} + {%- when "supports" or "supported_by" or "tested_by" or "validated_by" -%} + {%- assign supports_cids = supports_cids | push: rel.object -%} + {%- endcase -%} +{%- endfor -%} + +{%- comment -%} Dedupe each list {%- endcomment -%} +{%- assign upstream_unique = upstream_cids | uniq -%} +{%- assign downstream_unique = downstream_cids | uniq -%} +{%- assign formalized_by_unique = formalized_by_cids | uniq -%} +{%- assign appears_in_unique = appears_in_cids | uniq -%} +{%- assign supports_unique = supports_cids | uniq -%} + +{%- assign total_relations = upstream_unique.size | plus: formalized_by_unique.size | plus: appears_in_unique.size | plus: supports_unique.size -%} + +{%- if total_relations > 0 -%} +
+

Relations

+ + {%- if upstream_unique.size > 0 -%} +

Upstream dependencies ({{ upstream_unique.size }})

+
+ {%- for cid in upstream_unique -%} + {%- if cid contains "cid" -%} + {{ cid }} + {%- else -%} + {{ cid }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if formalized_by_unique.size > 0 -%} +

Formalized by ({{ formalized_by_unique.size }})

+
+ {%- for cid in formalized_by_unique -%} + {%- if cid contains "cid" -%} + {{ cid }} + {%- else -%} + {{ cid }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if appears_in_unique.size > 0 -%} +

Appears in ({{ appears_in_unique.size }})

+
+ {%- for cid in appears_in_unique -%} + {%- if cid contains "cid" -%} + {{ cid }} + {%- else -%} + {{ cid }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if supports_unique.size > 0 -%} +

Supports / Tested by ({{ supports_unique.size }})

+
+ {%- for cid in supports_unique -%} + {%- if cid contains "cid" -%} + {{ cid }} + {%- else -%} + {{ cid }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if downstream_unique.size > 0 -%} +

Downstream uses (declared) ({{ downstream_unique.size }})

+
+ {%- for cid in downstream_unique -%} + {%- if cid contains "cid" -%} + {{ cid }} + {%- else -%} + {{ cid }} + {%- endif -%} + {%- endfor -%} +
+ {%- endif -%} + + {%- if page.sources.size > 0 -%} +

Sources

+
    + {%- for src in page.sources -%} +
  • + {{ src.type | replace: "_", " " | capitalize }} + + {%- if src.item -%} + {{ src.item }} + {%- endif -%} + {%- if src.locator -%} + {{ src.locator }} + {%- endif -%} + +
    • + {%- endfor -%} +
+ {%- endif -%} +
+{%- endif -%} diff --git a/_layouts/corpus-v3-item.html b/_layouts/corpus-v3-item.html new file mode 100644 index 000000000..7e3dffca1 --- /dev/null +++ b/_layouts/corpus-v3-item.html @@ -0,0 +1,264 @@ +--- +layout: default +--- + +{%- comment -%} + corpus-v3-item layout — Wave 7 deliverable. + + Renders a Corpus v3 item landing page at /id/{cid}/. Reads everything + from page.* frontmatter (the generator emits the full item record as + frontmatter, so the layout never has to fetch separate data). + + Charter §6 (Web Projection Doctrine); web addendum §4 (item page sections). + + Item modes (page.type → payload_class): + atomic definitions, theorems, lemmas, propositions, claims, etc. + artifact monographs, papers, dossiers, release packets + formal formal_module, formal_theorem, formal_definition, formal_axiom + result RSL items with layers/domains/inspection_route + proof structured proof records + page projection page items + tombstone visibility deprecated_public — tombstone banner mode +-----------------------------------------------------------------{%- endcomment -%} + +{% include breadcrumb-nav.html %} + +
+ +
+
+ Corpus v3 · {{ page.type | replace: "_", " " | capitalize }} + {{ page.cid }} + {%- if page.primary_alias and page.primary_alias != "" -%} + {{ page.primary_alias }} + {%- endif -%} + {%- if page.status -%} + {{ page.status }} + {%- endif -%} + {%- if page.current_version -%} + {{ page.current_version }} + {%- endif -%} +
+ +

{{ page.title }}

+ + {%- if page.summary and page.summary != "" -%} +

{{ page.summary }}

+ {%- endif -%} + + {%- comment -%} Tombstone banner for deprecated_public items {%- endcomment -%} + {%- if page.visibility == "deprecated_public" or page.tombstone -%} +
+

⚠ This Corpus Item has been retired.

+

It remains addressable for provenance and citation continuity. Status: {{ page.status }}.

+ {%- if page.superseded_by and page.superseded_by != "" -%} +

Current replacement: {{ page.superseded_by }}

+ {%- endif -%} + {%- if page.retirement_reason and page.retirement_reason != "" -%} +

Reason: {{ page.retirement_reason }}

+ {%- endif -%} + {%- if page.refutation_reason and page.refutation_reason != "" -%} +

Refutation: {{ page.refutation_reason }}

+ {%- endif -%} +
+ {%- endif -%} +
+ + {%- comment -%} Payload section — content depends on payload kind {%- endcomment -%} + {%- if page.payload and page.payload.kind -%} +
+

Payload

+
+ {%- if page.payload.kind == "inline_atomic" -%} + {%- if page.payload.markdown and page.payload.markdown != "" -%} + {{ page.payload.markdown | markdownify }} + {%- elsif page.payload.latex and page.payload.latex != "" -%} + 
{{ page.payload.latex }}
+ {%- elsif page.payload.plain and page.payload.plain != "" -%} + 
{{ page.payload.plain }}
+ {%- endif -%} + {%- elsif page.payload.kind == "external_artifact" -%} + {%- if page.payload.canonical_artifact -%} +

+ Canonical artifact: + {%- if page.payload.canonical_artifact.url -%} + {{ page.payload.canonical_artifact.url }} + {%- elsif page.payload.canonical_artifact.doi -%} + DOI {{ page.payload.canonical_artifact.doi }} + {%- elsif page.payload.canonical_artifact.github_url -%} + {{ page.payload.canonical_artifact.github_url }} + {%- endif -%} + {%- if page.payload.canonical_artifact.type -%} + {{ page.payload.canonical_artifact.type }} + {%- endif -%} +

+ {%- endif -%} + {%- if page.payload.abstract and page.payload.abstract.markdown and page.payload.abstract.markdown != "" -%} +

Abstract

+ {{ page.payload.abstract.markdown | markdownify }} + {%- endif -%} + {%- if page.payload.citation_markdown and page.payload.citation_markdown != "" -%} +

Citation

+ {{ page.payload.citation_markdown | markdownify }} + {%- endif -%} + {%- elsif page.payload.kind == "page_payload" -%} + {%- if page.payload.route -%} +

Route: {{ page.payload.route }}

+ {%- endif -%} + {%- if page.payload.purpose -%} +

{{ page.payload.purpose }}

+ {%- endif -%} + {%- endif -%} +
+
+ {%- endif -%} + + {%- comment -%} Proof block for type: proof {%- endcomment -%} + {%- if page.type == "proof" and page.proof -%} +
+

Proof

+
+ {%- if page.proof.mode -%}mode: {{ page.proof.mode }}{%- endif -%} + {%- if page.proof.proof_status -%}status: {{ page.proof.proof_status }}{%- endif -%} + {%- if page.proof.formality_level -%}formality: {{ page.proof.formality_level }}{%- endif -%} + {%- if page.proof.version_pinning -%}version-pinning: {{ page.proof.version_pinning }}{%- endif -%} +
+ + {%- if page.proof.mode == "conditional" and page.proof.assumptions.size > 0 -%} +
+

Conditional proof. This proof depends on explicit unresolved assumptions and should not be read as complete until those assumptions are discharged.

+ +
+ {%- endif -%} + + {%- if page.proof.condition_summary and page.proof.condition_summary != "" -%} +

{{ page.proof.condition_summary }}

+ {%- endif -%} + + {%- if page.proof.steps.size > 0 -%} +

Proof steps

+
    + {%- for step in page.proof.steps -%} +
  1. + {%- if step.label -%}{{ step.label }}. {%- endif -%} + {%- if step.text_markdown -%}{{ step.text_markdown | markdownify }}{%- endif -%} + {%- if step.uses.size > 0 -%} +

    + Uses: + {%- for u in step.uses -%} + {%- if u.ref -%} + {{ u.ref }} + {%- if u.role -%} ({{ u.role | replace: "_", " " }}){%- endif -%} + {%- unless forloop.last -%},{%- endunless -%} + {%- endif -%} + {%- endfor -%} +

    + {%- endif -%} +
    2. + {%- endfor -%} +
+ {%- endif -%} +
+ {%- endif -%} + + {%- comment -%} Result block for type: result {%- endcomment -%} + {%- if page.type == "result" and page.result -%} +
+

Result

+
+ {%- if page.result.class -%}{{ page.result.class | replace: "_", " " }}{%- endif -%} + {%- if page.result.claim_strength -%}{{ page.result.claim_strength | replace: "_", " " }}{%- endif -%} +
+ {%- if page.commentary.short -%}

{{ page.commentary.short }}

{%- endif -%} + {%- if page.commentary.technical -%}
{{ page.commentary.technical | markdownify }}
{%- endif -%} + {%- if page.commentary.scope_limits -%} +

+ Scope limits. {{ page.commentary.scope_limits }} +

+ {%- endif -%} + {%- if page.result.inspection_route and page.result.inspection_route.size > 0 -%} +

Inspection route

+
    + {%- for step_cid in page.result.inspection_route -%} +
  1. {{ step_cid }}
    2. + {%- endfor -%} +
+ {%- endif -%} + {%- if page.result.falsification_surface -%} +

Falsification surface: {{ page.result.falsification_surface }}

+ {%- endif -%} +
+ {%- endif -%} + + {%- comment -%} Formalization block for formal_* types and proof items with formalization {%- endcomment -%} + {%- if page.formalization and page.formalization.module -%} +
+

Formalization

+
+ {%- if page.formalization.status -%}{{ page.formalization.status }}{%- endif -%} + {%- if page.formalization.sorry_count != nil -%}sorries: {{ page.formalization.sorry_count }}{%- endif -%} + {%- if page.formalization.project_axioms != nil -%}project axioms: {{ page.formalization.project_axioms }}{%- endif -%} +
+
    +
  • Module{{ page.formalization.module }}
    • + {%- if page.formalization.declaration -%} +
  • Declaration{{ page.formalization.declaration }}
    • + {%- endif -%} + {%- if page.formalization.commit and page.formalization.commit != "" -%} +
  • Commit{{ page.formalization.commit | slice: 0, 12 }}
    • + {%- endif -%} + {%- if page.formalization.lean_toolchain -%} +
  • Lean toolchain{{ page.formalization.lean_toolchain }}
    • + {%- endif -%} +
+ {%- if page.formalization.print_axioms and page.formalization.print_axioms != "" -%} +

#print axioms

+ 
{{ page.formalization.print_axioms }}
+ {%- endif -%} +
+ {%- endif -%} + + {%- comment -%} Identifier box (right-rail style, but inline for simplicity) {%- endcomment -%} + {% include corpus-v3/identifier-box.html %} + + {%- comment -%} Relation panels — upstream / downstream / formalized_by / appears_in {%- endcomment -%} + {% include corpus-v3/relation-panels.html %} + + {%- comment -%} Version + history {%- endcomment -%} + {%- if page.history and page.history.size > 0 -%} +
+

Version & History

+
    + {%- for entry in page.history -%} +
  • + {{ entry.version }} · {{ entry.date }} + {{ entry.event | replace: "_", " " }} +
    • + {%- endfor -%} +
+
+ {%- endif -%} + + {%- comment -%} Status disclaimer {%- endcomment -%} +
+

Status disclaimer

+

+ A Corpus Item page reports the program's current internal record for this item. + It does not imply external verification, scientific consensus, or final proof unless + explicitly stated. Read it together with its dependencies, formalization status, + and the program's overall stance. +

+
+ +
+ +{%- comment -%} + JSON-LD embed — schema.org annotation for this Corpus Item. + Atomic / Result / Page items → schema:DefinedTerm + Artifact items → schema:ScholarlyArticle (when research_paper/note) or schema:Book (monograph) +{%- endcomment -%} +{% include corpus-v3/jsonld.html %} diff --git a/scripts/generate_corpus_v3_pages.py b/scripts/generate_corpus_v3_pages.py new file mode 100644 index 000000000..cce4d75a3 --- /dev/null +++ b/scripts/generate_corpus_v3_pages.py @@ -0,0 +1,183 @@ +#!/usr/bin/env python3 +"""Generate Jekyll collection documents for Corpus v3 items. + +Reads corpus-v3/items/{type}/cid######-*.yaml and emits one Markdown +collection document per item under site/_corpus_v3_items/cid######.md. +Each carries: + + --- + layout: corpus-v3-item + permalink: /id/{cid}/ + cid: cid###### + primary_alias: THM0042 + ... (full item data inlined as frontmatter) + --- + + (optional rendered body; the layout reads from frontmatter) + +Atomic write per page. Idempotent (re-running produces no changes for +unchanged items). + +Wave 7 deliverable. Web addendum §10 Option A (generated collection +documents). +""" +from __future__ import annotations + +import argparse +import os +import sys +from pathlib import Path +from typing import Any + +import yaml + +SITE_ROOT = Path(__file__).resolve().parents[1] +DEFAULT_CORPUS_V3_ROOT = SITE_ROOT.parent / "corpus-v3" +OUTPUT_DIR = SITE_ROOT / "_corpus_v3_items" + + +def atomic_write(target: Path, content: str) -> None: + target.parent.mkdir(parents=True, exist_ok=True) + tmp = target.with_suffix(target.suffix + ".tmp") + tmp.write_text(content, encoding="utf-8") + os.replace(tmp, target) + + +def iter_corpus_items(corpus_v3_root: Path): + """Yield (cid, source_path, parsed_data) for every item under corpus-v3/items/.""" + items_dir = corpus_v3_root / "items" + if not items_dir.is_dir(): + return + for path in sorted(items_dir.rglob("cid*.yaml")): + try: + data = yaml.safe_load(path.read_text(encoding="utf-8")) + except Exception as e: + print(f" WARN: parse failed for {path}: {e}", file=sys.stderr) + continue + if not isinstance(data, dict) or "id" not in data: + continue + yield data["id"], path, data + + +def build_page_frontmatter(item: dict[str, Any]) -> dict[str, Any]: + """Build the Jekyll page frontmatter for one Corpus Item.""" + cid = item["id"] + primary_alias = item.get("primary_alias", "") + type_ = item.get("type", "") + title = item.get("title", cid) + status = item.get("status", "") + visibility = item.get("visibility", "") + current_version = item.get("current_version", "v1") + + fm = { + "layout": "corpus-v3-item", + "permalink": f"/id/{cid}/", + "title": title, + "cid": cid, + "primary_alias": primary_alias, + "type": type_, + "lane": "corpus", + "status": status, + "visibility": visibility, + "current_version": current_version, + "release_lines": item.get("release_lines", []), + "summary": item.get("summary", ""), + "aliases": item.get("aliases", []), + "layers": item.get("layers", []), + "domains": item.get("domains", []), + "relations": item.get("relations", []), + "depends_on": item.get("depends_on", []), + "appears_in": item.get("appears_in", []), + "formalized_by": item.get("formalized_by", []), + "sources": item.get("sources", []), + "contains": item.get("contains", []), + "part_of": item.get("part_of", []), + "history": item.get("history", []), + "external_identifiers": item.get("external_identifiers", {}), + "payload": item.get("payload", {}), + "formalization": item.get("formalization", {}), + "proof": item.get("proof", {}), + "result": item.get("result", {}), + "commentary": item.get("commentary", {}), + "tombstone": item.get("tombstone", False), + "superseded_by": item.get("superseded_by", ""), + "retired_at": item.get("retired_at", ""), + "retirement_reason": item.get("retirement_reason", ""), + "refuted_at": item.get("refuted_at", ""), + "refutation_reason": item.get("refutation_reason", ""), + # SEO hints + "description": item.get("summary", "")[:240] if item.get("summary") else f"Corpus Item {cid}", + "noindex": False if visibility in ("public", "deprecated_public") else True, + } + return fm + + +def render_page(fm: dict[str, Any]) -> str: + """Render the full page with frontmatter + minimal body.""" + fm_yaml = yaml.safe_dump(fm, sort_keys=False, allow_unicode=True, default_flow_style=False, width=120) + body = "" # the layout reads everything from frontmatter + return f"---\n{fm_yaml}---\n{body}\n" + + +def main() -> int: + parser = argparse.ArgumentParser(description=__doc__.splitlines()[0]) + parser.add_argument("--corpus-v3-root", type=Path, default=DEFAULT_CORPUS_V3_ROOT) + parser.add_argument("--output-dir", type=Path, default=OUTPUT_DIR) + parser.add_argument("--visibility", default="public,deprecated_public", + help="Comma-separated visibility filter (default: public + deprecated_public)") + parser.add_argument("--limit", type=int, default=None, + help="For testing — emit only first N pages") + parser.add_argument("--dry-run", action="store_true") + args = parser.parse_args() + + if not args.corpus_v3_root.is_dir(): + print(f"FAIL: corpus-v3 root not found: {args.corpus_v3_root}", file=sys.stderr) + return 1 + + visibility_filter = {v.strip() for v in args.visibility.split(",")} + + args.output_dir.mkdir(parents=True, exist_ok=True) + + print(f"=== Generate corpus_v3 Jekyll pages ===") + print(f" source: {args.corpus_v3_root}") + print(f" target: {args.output_dir.relative_to(SITE_ROOT)}") + print(f" visibility filter: {sorted(visibility_filter)}") + print() + + written = 0 + skipped_visibility = 0 + seen = 0 + + for cid, src_path, item in iter_corpus_items(args.corpus_v3_root): + seen += 1 + if args.limit and written >= args.limit: + break + + visibility = item.get("visibility", "") + if visibility not in visibility_filter: + skipped_visibility += 1 + continue + + fm = build_page_frontmatter(item) + content = render_page(fm) + outpath = args.output_dir / f"{cid}.md" + + if args.dry_run: + print(f" [dry-run] would write {outpath.relative_to(SITE_ROOT)} ({len(content)} bytes)") + written += 1 + continue + + atomic_write(outpath, content) + written += 1 + + print() + print(f" items scanned: {seen}") + print(f" pages written: {written}") + print(f" skipped (visibility): {skipped_visibility}") + print() + print("✓ Generation complete") + return 0 + + +if __name__ == "__main__": + sys.exit(main()) diff --git a/scripts/sync_corpus_v3_to_site.py b/scripts/sync_corpus_v3_to_site.py new file mode 100644 index 000000000..69793f88b --- /dev/null +++ b/scripts/sync_corpus_v3_to_site.py @@ -0,0 +1,147 @@ +#!/usr/bin/env python3 +"""Sync corpus-v3 items + manifests into site/_data/corpus_v3/. + +Source-of-truth discipline: corpus-v3/ is the canonical repository for +Corpus v3 items. The site mirrors selected files at site/_data/corpus_v3/ +so Jekyll's Liquid can read them and generate the /id/cid######/ surface +via the corpus-v3-item layout. + +Sync targets: + corpus-v3/manifests/alias-index.yaml → _data/corpus_v3/alias-index.yaml + corpus-v3/manifests/cid-index.yaml → _data/corpus_v3/cid-index.yaml + corpus-v3/manifests/cid-transition-manifest.yaml → _data/corpus_v3/transition.yaml + + corpus-v3/manifests/item-types.yaml → _data/corpus_v3/item-types.yaml + corpus-v3/manifests/relation-vocabulary.yaml → _data/corpus_v3/relation-vocabulary.yaml + corpus-v3/manifests/visibility-values.yaml → _data/corpus_v3/visibility-values.yaml + corpus-v3/manifests/lifecycle-statuses.yaml → _data/corpus_v3/lifecycle-statuses.yaml + +Items themselves are NOT copied into _data/ (their content lives at the +per-item layer used by generate_corpus_v3_pages.py). + +Resolves corpus-v3 root in this order: + 1. --corpus-v3-root flag + 2. PRRP_CORPUS_V3_ROOT env var + 3. ../corpus-v3 relative to site repo + +Atomic write per file. Validates each file parses + has expected top-level +keys before copying. + +Wave 7 deliverable. Charter §6 (web projection doctrine). +""" +from __future__ import annotations + +import argparse +import os +import shutil +import sys +from pathlib import Path +from typing import Any + +import yaml + +SITE_ROOT = Path(__file__).resolve().parents[1] +DEFAULT_CORPUS_V3_ROOT = SITE_ROOT.parent / "corpus-v3" +SITE_DATA_TARGET = SITE_ROOT / "_data" / "corpus_v3" + +# (source rel path under corpus-v3, target filename under _data/corpus_v3, expected top-level key) +SYNC_TARGETS: list[tuple[str, str, str]] = [ + ("manifests/alias-index.yaml", "alias-index.yaml", "aliases"), + ("manifests/cid-index.yaml", "cid-index.yaml", "cids"), + ("manifests/cid-transition-manifest.yaml", "transition.yaml", "transitions"), + ("manifests/item-types.yaml", "item-types.yaml", "types"), + ("manifests/relation-vocabulary.yaml", "relation-vocabulary.yaml", "predicates"), + ("manifests/visibility-values.yaml", "visibility-values.yaml", "values"), + ("manifests/lifecycle-statuses.yaml", "lifecycle-statuses.yaml", "statuses"), +] + + +def resolve_corpus_v3_root(arg: str | None) -> Path: + if arg: + return Path(arg).resolve() + env = os.environ.get("PRRP_CORPUS_V3_ROOT") + if env: + return Path(env).resolve() + return DEFAULT_CORPUS_V3_ROOT.resolve() + + +def atomic_write(target: Path, payload: bytes) -> None: + """Write to a sibling .tmp then os.replace — survives SIGINT mid-copy.""" + tmp = target.with_suffix(target.suffix + ".tmp") + tmp.write_bytes(payload) + os.replace(tmp, target) + + +def sync_file(src: Path, target: Path, expected_top_key: str) -> tuple[bool, str]: + """Sync one file. Returns (ok, message).""" + if not src.is_file(): + return False, f"source not found: {src}" + try: + raw = src.read_bytes() + parsed = yaml.safe_load(raw.decode("utf-8")) + except Exception as e: + return False, f"parse failed: {e}" + if not isinstance(parsed, dict) or expected_top_key not in parsed: + return False, f"missing top-level key {expected_top_key!r}" + target.parent.mkdir(parents=True, exist_ok=True) + atomic_write(target, raw) + return True, f"{src.stat().st_size} bytes → {target.relative_to(SITE_ROOT)}" + + +def main() -> int: + parser = argparse.ArgumentParser(description=__doc__.splitlines()[0]) + parser.add_argument("--corpus-v3-root", default=None, + help="Path to corpus-v3 repo (default: ../corpus-v3)") + parser.add_argument("--dry-run", action="store_true") + args = parser.parse_args() + + corpus_v3_root = resolve_corpus_v3_root(args.corpus_v3_root) + if not corpus_v3_root.is_dir(): + print(f"FAIL: corpus-v3 root not found: {corpus_v3_root}", file=sys.stderr) + return 1 + + print(f"=== Sync corpus-v3 → site/_data/corpus_v3/ ===") + print(f" source: {corpus_v3_root}") + print(f" target: {SITE_DATA_TARGET.relative_to(SITE_ROOT)}") + print() + + errors: list[str] = [] + synced: list[str] = [] + + for src_rel, target_name, expected_key in SYNC_TARGETS: + src = corpus_v3_root / src_rel + target = SITE_DATA_TARGET / target_name + + if args.dry_run: + ok = src.is_file() + status = "would-sync" if ok else "missing" + print(f" [dry-run] {status:12s} {src_rel:48s} → {target.name}") + if not ok: + errors.append(f"{src_rel}: source missing") + continue + + ok, msg = sync_file(src, target, expected_key) + marker = "✓" if ok else "✗" + print(f" {marker} {src_rel:48s} {msg}") + if ok: + synced.append(src_rel) + else: + errors.append(f"{src_rel}: {msg}") + + print() + print(f" synced: {len(synced)} files") + print(f" errors: {len(errors)}") + + if errors: + print() + for e in errors: + print(f" ✗ {e}") + return 1 + + print() + print("✓ Sync complete") + return 0 + + +if __name__ == "__main__": + sys.exit(main())