`ComparatorChallenges/EilenbergGanea.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.

CW: closure-finite weak topology
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

26 declarations, 1 contain sorry, 0 declared as axioms

def OAI.EilenbergGanea.seedWord The signed words encode the relators x²y⁻⁵ and x²(yx⁻¹)³ line 12
Source documentation
The signed words encode the relators x²y⁻⁵ and x²(yx⁻¹)³.
L4DEL explanation
The signed words encode the relators x²y⁻⁵ and x²(yx⁻¹)³.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:14 (same declaration text).
```
def seedWord : Bool → List (Bool × Bool)
  | false => [(false, true), (false, true), (true, false), (true, false),
      (true, false), (true, false), (true, false)]
  | true => [(false, true), (false, true), (true, true), (false, false),
      (true, true), (false, false), (true, true), (false, false)]
```
def OAI.EilenbergGanea.boundaryLength line 18
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:21 (different declaration text). Qualify references as OAI.EilenbergGanea.boundaryLength@path.
abbrev OAI.EilenbergGanea.RoseVertex line 20
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:23 (same declaration text).
abbrev OAI.EilenbergGanea.SeedVertex line 21
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:24 (same declaration text).
abbrev OAI.EilenbergGanea.CircleEdge line 22
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:25 (same declaration text).
abbrev OAI.EilenbergGanea.Sector line 23
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:26 (same statement text; different proof or body). Qualify references as OAI.EilenbergGanea.Sector@path.
abbrev OAI.EilenbergGanea.SeedEdge line 24
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:27 (same declaration text).
abbrev OAI.EilenbergGanea.SeedCell line 25
Also declared: OAI/Topology/EilenbergGanea/Seed.lean:28 (same declaration text).
def OAI.EilenbergGanea.circleStart line 27
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:30 (different declaration text). Qualify references as OAI.EilenbergGanea.circleStart@path.
def OAI.EilenbergGanea.circleEnd line 30
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:33 (different declaration text). Qualify references as OAI.EilenbergGanea.circleEnd@path.
def OAI.EilenbergGanea.sectorLetter line 33
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:36 (different declaration text). Qualify references as OAI.EilenbergGanea.sectorLetter@path.
def OAI.EilenbergGanea.sectorCircleEdge line 39
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:42 (different declaration text). Qualify references as OAI.EilenbergGanea.sectorCircleEdge@path.
def OAI.EilenbergGanea.sectorStart line 44
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:47 (different declaration text). Qualify references as OAI.EilenbergGanea.sectorStart@path.
def OAI.EilenbergGanea.nextSector line 48
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:51 (different declaration text). Qualify references as OAI.EilenbergGanea.nextSector@path.
def OAI.EilenbergGanea.edgeVertices line 52
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:55 (different declaration text). Qualify references as OAI.EilenbergGanea.edgeVertices@path.
def OAI.EilenbergGanea.sectorEdges line 56
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:59 (different declaration text). Qualify references as OAI.EilenbergGanea.sectorEdges@path.
def OAI.EilenbergGanea.sectorVertices line 59
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:62 (different declaration text). Qualify references as OAI.EilenbergGanea.sectorVertices@path.
def OAI.EilenbergGanea.properFace line 62
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:66 (different declaration text). Qualify references as OAI.EilenbergGanea.properFace@path.
def OAI.EilenbergGanea.seedGraph Comparability graph of the seven vertices, 51 edges and 45 sectors line 69
Source documentation
Comparability graph of the seven vertices, 51 edges and 45 sectors.
L4DEL explanation
Comparability graph of the seven vertices, 51 edges and 45 sectors.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Seed.lean:72 (same statement text; different proof or body). Qualify references as OAI.EilenbergGanea.seedGraph@path.
```
def seedGraph : SimpleGraph SeedCell where
…
```
Mentions: OAI.EilenbergGanea.SeedCell
def OAI.EilenbergGanea.artinRelations line 78
Distinct definitions of this name: OAI/GroupTheory/RightAngledArtin/Basic.lean:14 (different declaration text). Qualify references as OAI.EilenbergGanea.artinRelations@path.
abbrev OAI.EilenbergGanea.ArtinGroup line 83
Distinct definitions of this name: OAI/GroupTheory/RightAngledArtin/Basic.lean:18 (different declaration text). Qualify references as OAI.EilenbergGanea.ArtinGroup@path.
def OAI.EilenbergGanea.height line 86
Distinct definitions of this name: OAI/GroupTheory/RightAngledArtin/Basic.lean:21 (different declaration text). Qualify references as OAI.EilenbergGanea.height@path.
abbrev OAI.EilenbergGanea.SourceGroup line 92
Also declared: OAI/Topology/EilenbergGanea/Model.lean:17 (same declaration text).
def OAI.EilenbergGanea.cohomologicalDimension Projective dimension of the trivial integral left group-ring module line 95
Source documentation
Projective dimension of the trivial integral left group-ring module.
L4DEL explanation
Projective dimension of the trivial integral left group-ring module.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Model.lean:20 (different declaration text). Qualify references as OAI.EilenbergGanea.cohomologicalDimension@path.
```
def cohomologicalDimension (GroupType : Type*) [Group GroupType] : WithBot ℕ∞ :=
…
```
def OAI.EilenbergGanea.HasClassifyingSpace Ordinary classifying CW spaces, with no bound on the cardinality of their cells line 103
Source documentation
Ordinary classifying CW spaces, with no bound on the cardinality of their cells.
L4DEL explanation
Ordinary classifying CW spaces, with no bound on the cardinality of their cells.
1. This explanation is a condensed rendering of the declaration's own doc comment at the pinned source commit; consult the exact Lean statement below for authority.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f)
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Model.lean:28 (different declaration text). Qualify references as OAI.EilenbergGanea.HasClassifyingSpace@path.
```
def HasClassifyingSpace (GroupType : Type*) [Group GroupType] (dimension : ℕ) : Prop :=
…
```
theorem OAI.EilenbergGanea.source_main Challenge statement: A finitely generated Eilenberg-Ganea counterexample sorry line 113
L4DEL explanation
a finitely generated residually finite group of integral cohomological dimension two and geometric dimension three
— OpenAI Eilenberg-Ganea counterexample manuscript (abstract)
The least dimension of such a complex KΠ is called the geometric dimension of Π
— Samuel Eilenberg and Tudor Ganea (1957), original paper
One explicit height-kernel group is finitely generated and residually finite, has integral cohomological dimension two, admits a three-dimensional classifying space, and admits no two-dimensional classifying space.
1. SourceGroup is the kernel of the homomorphism sending every generator of a right-angled Artin group to 1; the Artin group is built from the finite flag-complex data encoded by seedGraph.
2. The first two conjuncts assert finite generation and residual finiteness. The third identifies the projective dimension of the trivial integral group-ring module as exactly two.
3. The final conjuncts give a three-dimensional connected CW classifying space with contractible cover and rule out every two-dimensional one. Together, these say geometric dimension three while cohomological dimension is two.
4. This is an existence result for the specifically defined SourceGroup, not a statement about all groups of cohomological dimension two. The comparator body is sorry, paired with OAI.Topology.EilenbergGanea.Main.
Sources and attribution: Pinned Lean source (fd4aeeb2ee4f), OpenAI Eilenberg-Ganea counterexample manuscript (abstract), Pinned OpenAI formalization note 249, Samuel Eilenberg and Tudor Ganea (1957), original paper
Distinct definitions of this name: OAI/Topology/EilenbergGanea/Main.lean:30 (same statement text; different proof or body). Qualify references as OAI.EilenbergGanea.source_main@path.

cohomological dimension: the projective dimension of the trivial integral module over the group's integral group ring
geometric dimension: the least dimension of a classifying CW space for the group
```
theorem source_main :
    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧
    cohomologicalDimension SourceGroup = 2 ∧
    HasClassifyingSpace.{0} SourceGroup 3 ∧
    ¬ HasClassifyingSpace.{u} SourceGroup 2 := by
…
```
Mentions: OAI.EilenbergGanea.SourceGroup