`ComparatorChallenges/GrothendieckElementaryExpansion.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 120 curated AI-written annotations and 1273 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 9 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. Those 9 annotations were revised on 2026-10-14 to bind all of them to exact page-one quotations from 6 pinned OpenAI manuscripts, the matching pinned formalization notes, and named historical primary sources; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. The 3 manuscript-grounded matrix-multiplication annotations were composed from pinned OpenAI manuscripts and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Six additional target-specific annotations for Catalan's constant, the irrationality exponent of pi, dyadic Erdos similarity, Unique Games, quintic Lienard systems, and Yau nodal growth were composed from pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for Seymour's second-neighborhood conjecture, Cannon's conjecture, Koebe circle-domain uniformization, and circle-domain rigidity were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for simple Lebesgue spectrum on the three-torus, Rokhlin multiple mixing, the nine-dimensional Borsuk counterexample, and Tingley's sphere-isometry problem were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and named historical primary sources in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the Abhyankar-Sathaye counterexample, the circulant Hadamard conjecture, the Eilenberg-Ganea counterexample, and short Egyptian fractions were composed from 4 pinned OpenAI manuscripts, exact page-one quotations, exact Lean statements, and quotations from Sathaye, Turyn with Storer, Eilenberg with Ganea, and Erdos in a Codex session on 2026-10-14; no human review is claimed, and the exact runtime model identifier was not exposed to the artifact generator. Four additional target-specific annotations for the complete Crouzeix inequality, unrestricted Euclidean five-coloring, the general Mahler conjecture, and Thompson F nonamenability were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, the matching formalization notes, and linked historical sources by Crouzeix, de Grey, Mahler, Cannon, Floyd, and Parry; no human review is claimed. Four additional target-specific annotations for algorithmic strong thin trees, Barnette's Hamiltonian-cycle conjecture, the explicit Artin CAT(0) counterexample, and generalized amplitude-damping capacity were composed on 2026-10-14 from exact Lean statements, 4 pinned OpenAI manuscripts, matching formalization notes, and linked historical sources by Anari with Goddyn, Arts with Hopman and Timmermans, Charney, and Khatri with Sharma and Wilde; no human review is claimed. Fourteen solution-level annotations for Seymour's second-neighborhood theorem were added on 2026-10-14 from the pinned 15-page manuscript, covering the exact-neighborhood model, minimal-counterexample reduction, subset deficit, transitive blowup, pruning, extremal contradiction, and proof-bearing final theorem. The subset and blowup annotations also bind exact quotations from Tyler Seacrest (2019) and Brantner, Brockman, Kay, and Snively (2009); no human review is claimed. Thirty-seven declaration-level annotations were added on 2026-10-14 for the rest of ComparatorChallenges/AlgorithmicThinTrees.lean, covering its multigraph semantics, stack-machine cost model, explicit and binary-multiplicity encodings, output certificate, and complete algorithmic proposition. They are bound to the pinned OpenAI manuscript and linked earlier thin-tree sources by Luis Goddyn, Nima Anari, and Shayan Oveis Gharan; the annotation explicitly preserves that the concluding comparator theorem still has body by sorry, and no human review is claimed. Thirty-one supporting declarations in ComparatorChallenges/UniqueGamesTheorem.lean were upgraded or added on 2026-10-14, completing curated coverage of all thirty-two declarations in that file. The annotations cover the finite game model, Boolean formula and parser, binary-vector translations, explicit output serialization, finite-alphabet machine contract, and gap theorem; they distinguish manuscript-backed mathematics from Lean-specific canonical parsing and unary game output encoding, bind exact OpenAI manuscript quotations, and credit Subhash Khot through his 2010 survey. The comparator theorem remains explicitly labeled as having body by sorry; no human review is claimed. The source-derived generator was hardened on 2026-10-14 so a formalization note covering multiple comparator results no longer copies its combined Scope prose onto each declaration; those targets now use declaration-specific source comments when available and otherwise remain explicitly bounded to the exact Lean statement. The remaining source-derived annotations were generated mechanically from pinned formalization notes and source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the source prose.

Credit: OpenAI's pinned repository says an unreleased internal model produced the manuscripts and supporting proof artifacts. The collection builds on the mathematics community's accumulated problems, results, language, and formal libraries; each manuscript's BibTeX and bibliography are the authority for individual scholarly credit.

Project sources: Pinned openai/math documentation, Pinned formalization catalogue

77 declarations, 1 contain sorry, 0 declared as axioms

structure OAI.Grothendieck.GlobularSet line 9
Also declared: OAI/CategoryTheory/Globular/Model.lean:9 (same declaration text).
structure OAI.Grothendieck.GlobularSet.Hom line 19
Also declared: OAI/CategoryTheory/Globular/Model.lean:19 (same declaration text).
def OAI.Grothendieck.GlobularSet.Parallel line 24
Also declared: OAI/CategoryTheory/Globular/Model.lean:24 (same declaration text).
def OAI.Grothendieck.GlobularSet.Joined line 28
Also declared: OAI/CategoryTheory/Globular/Model.lean:28 (same declaration text).
def OAI.Grothendieck.GlobularSet.Boundary line 31
Also declared: OAI/CategoryTheory/Globular/Model.lean:43 (same declaration text).
def OAI.Grothendieck.GlobularSet.boundary line 35
Also declared: OAI/CategoryTheory/Globular/Model.lean:47 (same declaration text).
def OAI.Grothendieck.GlobularSet.Hom.boundaryMap line 42
Also declared: OAI/CategoryTheory/Globular/Model.lean:64 (same declaration text).
structure OAI.Grothendieck.Globe line 56
Also declared: OAI/CategoryTheory/Globular/Model.lean:78 (same declaration text).
inductive OAI.Grothendieck.Globe.Face line 61
Also declared: OAI/CategoryTheory/Globular/Model.lean:83 (same declaration text).
def OAI.Grothendieck.Globe.Face.comp line 67
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:89 (different declaration text). Qualify references as OAI.Grothendieck.Globe.Face.comp@path.
instance OAI.Grothendieck.Globe.(instance at line 74) line 74
def OAI.Grothendieck.Globe.face line 84
Also declared: OAI/CategoryTheory/Globular/Model.lean:116 (same declaration text).
def OAI.Grothendieck.Globe.source line 87
Also declared: OAI/CategoryTheory/Globular/Model.lean:119 (same declaration text).
def OAI.Grothendieck.Globe.target line 89
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:121 (same statement text; different proof or body). Qualify references as OAI.Grothendieck.Globe.target@path.
abbrev OAI.Grothendieck.GlobularPresheaf line 93
Also declared: OAI/CategoryTheory/Globular/Model.lean:131 (same declaration text).
def OAI.Grothendieck.GlobularPresheaf.underlying line 97
Also declared: OAI/CategoryTheory/Globular/Model.lean:135 (same declaration text).
def OAI.Grothendieck.GlobularPresheaf.underlyingMap line 108
Also declared: OAI/CategoryTheory/Globular/Model.lean:146 (same declaration text).
def OAI.Grothendieck.GlobularPresheaf.disk line 115
Also declared: OAI/CategoryTheory/Globular/Model.lean:153 (same declaration text).
def OAI.Grothendieck.GlobularPresheaf.face line 117
Also declared: OAI/CategoryTheory/Globular/Model.lean:155 (same declaration text).
inductive OAI.Grothendieck.Table line 122
Also declared: OAI/CategoryTheory/Globular/Model.lean:160 (same declaration text).
def OAI.Grothendieck.Table.height line 128
Also declared: OAI/CategoryTheory/Globular/Model.lean:166 (same declaration text).
def OAI.Grothendieck.Table.realizeWithLast line 132
Also declared: OAI/CategoryTheory/Globular/Model.lean:170 (same declaration text).
def OAI.Grothendieck.Table.realize line 141
Also declared: OAI/CategoryTheory/Globular/Model.lean:179 (same declaration text).
def OAI.Grothendieck.Table.lastInclusion line 143
Also declared: OAI/CategoryTheory/Globular/Model.lean:181 (same declaration text).
abbrev OAI.Grothendieck.Shape line 148
Also declared: OAI/CategoryTheory/Globular/Model.lean:186 (same declaration text).
def OAI.Grothendieck.Shape.realize line 152
Also declared: OAI/CategoryTheory/Globular/Model.lean:190 (same declaration text).
def OAI.Grothendieck.Shape.height line 154
Also declared: OAI/CategoryTheory/Globular/Model.lean:192 (same declaration text).
def OAI.Grothendieck.Shape.globe line 156
Also declared: OAI/CategoryTheory/Globular/Model.lean:194 (same declaration text).
instance OAI.Grothendieck.Shape.(instance at line 158) line 158
def OAI.Grothendieck.Shape.face line 166
Also declared: OAI/CategoryTheory/Globular/Model.lean:205 (same declaration text).
def OAI.Grothendieck.Shape.source line 169
Also declared: OAI/CategoryTheory/Globular/Model.lean:208 (same declaration text).
def OAI.Grothendieck.Shape.target line 171
Also declared: OAI/CategoryTheory/Globular/Model.lean:210 (same declaration text).
def OAI.Grothendieck.Shape.attach line 173
Also declared: OAI/CategoryTheory/Globular/Model.lean:212 (same declaration text).
def OAI.Grothendieck.Shape.seam line 176
Also declared: OAI/CategoryTheory/Globular/Model.lean:215 (same declaration text).
def OAI.Grothendieck.Shape.inl line 180
Also declared: OAI/CategoryTheory/Globular/Model.lean:219 (same declaration text).
def OAI.Grothendieck.Shape.inr line 185
Also declared: OAI/CategoryTheory/Globular/Model.lean:224 (same declaration text).
structure OAI.Grothendieck.TheoryData line 192
Also declared: OAI/CategoryTheory/Globular/Model.lean:231 (same declaration text).
structure OAI.Grothendieck.TheoryData.Obj line 206
Also declared: OAI/CategoryTheory/Globular/Model.lean:245 (same declaration text).
instance OAI.Grothendieck.TheoryData.(instance at line 209) line 209
def OAI.Grothendieck.TheoryData.arity line 217
Also declared: OAI/CategoryTheory/Globular/Model.lean:256 (same declaration text).
def OAI.Grothendieck.TheoryData.PreservesGlobularSums line 223
Also declared: OAI/CategoryTheory/Globular/Model.lean:262 (same declaration text).
structure OAI.Grothendieck.GlobularTheory line 230
Also declared: OAI/CategoryTheory/Globular/Model.lean:269 (same declaration text).
abbrev OAI.Grothendieck.GlobularTheory.Obj line 235
Also declared: OAI/CategoryTheory/Globular/Model.lean:274 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.Morphism line 238
Also declared: OAI/CategoryTheory/Globular/Model.lean:277 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Morphism.id line 247
Also declared: OAI/CategoryTheory/Globular/Model.lean:286 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Morphism.comp line 253
Also declared: OAI/CategoryTheory/Globular/Model.lean:292 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Parallel line 261
Also declared: OAI/CategoryTheory/Globular/Model.lean:300 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.AdmissiblePair line 268
Also declared: OAI/CategoryTheory/Globular/Model.lean:307 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Fills line 276
Also declared: OAI/CategoryTheory/Globular/Model.lean:315 (same declaration text).
def OAI.Grothendieck.GlobularTheory.IsFreeExtension line 281
Also declared: OAI/CategoryTheory/Globular/Model.lean:320 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.CellularPresentation line 290
Also declared: OAI/CategoryTheory/Globular/Model.lean:329 (same declaration text).
def OAI.Grothendieck.GlobularTheory.IsCoherator line 304
Also declared: OAI/CategoryTheory/Globular/Model.lean:343 (same declaration text).
def OAI.Grothendieck.GlobularTheory.shapeGlobes line 308
Also declared: OAI/CategoryTheory/Globular/Model.lean:348 (same declaration text).
def OAI.Grothendieck.GlobularTheory.globes line 314
Also declared: OAI/CategoryTheory/Globular/Model.lean:354 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.Model line 317
Also declared: OAI/CategoryTheory/Globular/Model.lean:357 (same declaration text).
instance OAI.Grothendieck.GlobularTheory.Model.(instance at line 328) line 328
def OAI.Grothendieck.GlobularTheory.Model.underlying line 336
Also declared: OAI/CategoryTheory/Globular/Model.lean:376 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.underlyingMap line 339
Also declared: OAI/CategoryTheory/Globular/Model.lean:379 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.freeShape line 343
Also declared: OAI/CategoryTheory/Globular/Model.lean:383 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.disk line 347
Also declared: OAI/CategoryTheory/Globular/Model.lean:387 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.J line 350
Also declared: OAI/CategoryTheory/Globular/Model.lean:390 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.Model.BoundaryAttachment line 354
Also declared: OAI/CategoryTheory/Globular/Model.lean:394 (same declaration text).
structure OAI.Grothendieck.GlobularTheory.Model.CellularPresentation line 366
Also declared: OAI/CategoryTheory/Globular/Model.lean:406 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.IsCellular line 380
Also declared: OAI/CategoryTheory/Globular/Model.lean:420 (same declaration text).
def OAI.Grothendieck.GlobularTheory.unitPair line 384
Also declared: OAI/CategoryTheory/Globular/Model.lean:424 (same declaration text).
def OAI.Grothendieck.GlobularTheory.unit line 392
Also declared: OAI/CategoryTheory/Globular/Model.lean:432 (same declaration text).
def OAI.Grothendieck.GlobularTheory.Model.eval line 395
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:439 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.eval@path.
def OAI.Grothendieck.GlobularTheory.Model.unitCell line 399
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:443 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.unitCell@path.
def OAI.Grothendieck.GlobularTheory.Model.baseCell line 403
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:447 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.baseCell@path.
def OAI.Grothendieck.GlobularTheory.Model.Loops line 408
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:468 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.Loops@path.
def OAI.Grothendieck.GlobularTheory.Model.Components line 413
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:473 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.Components@path.
def OAI.Grothendieck.GlobularTheory.Model.HomotopyGroup line 416
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:476 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.HomotopyGroup@path.
def OAI.Grothendieck.GlobularTheory.Model.loopsMap line 420
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:480 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.loopsMap@path.
def OAI.Grothendieck.GlobularTheory.Model.componentsMap line 439
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:499 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.componentsMap@path.
def OAI.Grothendieck.GlobularTheory.Model.homotopyMap line 447
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:507 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.homotopyMap@path.
def OAI.Grothendieck.GlobularTheory.Model.WeakEquivalence line 456
Distinct definitions of this name: OAI/CategoryTheory/Globular/Model.lean:516 (different declaration text). Qualify references as OAI.Grothendieck.GlobularTheory.Model.WeakEquivalence@path.
theorem OAI.Grothendieck.elementary_expansion Challenge statement: The Grothendieck homotopy hypothesis sorry line 465
L4DEL explanation
The Grothendieck homotopy hypothesis asks whether algebraic $\infty$-groupoids recover the homotopy theory of spaces. The formalized result is the elementary-expansion theorem used in this approach: for every Grothendieck coherator in the Ara–Henry convention and every boundary-cellular model, attaching an $(n+1)$-disk along its source $n$-face induces a weak equivalence. The later semi-model structure and full comparison with the homotopy theory of spaces are not included.
1. This is an exact comparator target linked by the pinned repository's formalization note for this result.
2. The repository's comparator configuration selects OAI.CategoryTheory.Globular.Main as the released solution module checked against this declaration.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/CategoryTheory/Globular/ElementaryExpansion.lean:34 (different declaration text). Qualify references as OAI.Grothendieck.elementary_expansion@path.
```
theorem elementary_expansion (C : GlobularTheory.{v}) (hC : C.IsCoherator)
    (X : C.Model) (hX : X.IsCellular) (n : ℕ)
    (a : GlobularTheory.Model.disk C n ⟶ X)
    (Y : C.Model) (i : X ⟶ Y)
    (b : GlobularTheory.Model.disk C (n + 1) ⟶ Y)
    (square : IsPushout a (GlobularTheory.Model.J C n) i b) :
    GlobularTheory.Model.WeakEquivalence hC i := by
…
```
Mentions: OAI.Grothendieck.GlobularTheory.Model.disk, OAI.Grothendieck.GlobularTheory.Model.J