Lean statement

Counterexample to the Eilenberg–Ganea conjecture

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorem selected for verification: OAI.EilenbergGanea.source_main

Lean statement file: EilenbergGanea.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 249 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem source_main :    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧    cohomologicalDimension SourceGroup = 2 ∧    HasClassifyingSpace.{0} SourceGroup 3 ∧    ¬ HasClassifyingSpace.{u} SourceGroup 2 := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

definition OAI.EilenbergGanea.seedWord line 12

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)]

definition OAI.EilenbergGanea.boundaryLength line 18

def boundaryLength (relation : Bool) : ℕ := 3 * (seedWord relation).length

abbreviation OAI.EilenbergGanea.RoseVertex line 20
```
abbrev RoseVertex := Option (Bool × Fin 2)
```
abbreviation OAI.EilenbergGanea.SeedVertex line 21
```
abbrev SeedVertex := RoseVertex ⊕ Bool
```
abbreviation OAI.EilenbergGanea.CircleEdge line 22
```
abbrev CircleEdge := Bool × Fin 3
```

abbreviation OAI.EilenbergGanea.Sector line 23

abbrev Sector := (relation : Bool) × Fin (boundaryLength relation)

abbreviation OAI.EilenbergGanea.SeedEdge line 24
```
abbrev SeedEdge := CircleEdge ⊕ Sector
```

abbreviation OAI.EilenbergGanea.SeedCell line 25

abbrev SeedCell := SeedVertex ⊕ (SeedEdge ⊕ Sector)

definition OAI.EilenbergGanea.circleStart line 27

def circleStart (edge : CircleEdge) : RoseVertex :=  ![none, some (edge.1, 0), some (edge.1, 1)] edge.2

definition OAI.EilenbergGanea.circleEnd line 30

def circleEnd (edge : CircleEdge) : RoseVertex :=  ![some (edge.1, 0), some (edge.1, 1), none] edge.2

definition OAI.EilenbergGanea.sectorLetter line 33

def sectorLetter (sector : Sector) : Bool × Bool :=  (seedWord sector.1).get ⟨sector.2.val / 3, by    have bound := sector.2.isLt    dsimp [boundaryLength] at bound    omega⟩

definition OAI.EilenbergGanea.sectorCircleEdge line 39

def sectorCircleEdge (sector : Sector) : CircleEdge :=  ((sectorLetter sector).1, if (sectorLetter sector).2 then    ⟨sector.2.val % 3, Nat.mod_lt _ (by decide)⟩ else    ⟨2 - sector.2.val % 3, by omega⟩)

definition OAI.EilenbergGanea.sectorStart line 44

def sectorStart (sector : Sector) : RoseVertex :=  if (sectorLetter sector).2 then circleStart (sectorCircleEdge sector)  else circleEnd (sectorCircleEdge sector)

definition OAI.EilenbergGanea.nextSector line 48

def nextSector (sector : Sector) : Sector :=  ⟨sector.1, ⟨(sector.2.val + 1) % boundaryLength sector.1,    Nat.mod_lt _ (by have := sector.2.isLt; omega)⟩⟩

definition OAI.EilenbergGanea.edgeVertices line 52

def edgeVertices : SeedEdge → Finset SeedVertex  | .inl edge => {.inl (circleStart edge), .inl (circleEnd edge)}  | .inr sector => {.inr sector.1, .inl (sectorStart sector)}

definition OAI.EilenbergGanea.sectorEdges line 56

def sectorEdges (sector : Sector) : Finset SeedEdge :=  {.inl (sectorCircleEdge sector), .inr sector, .inr (nextSector sector)}

definition OAI.EilenbergGanea.sectorVertices line 59

def sectorVertices (sector : Sector) : Finset SeedVertex :=  (sectorEdges sector).biUnion edgeVertices

definition OAI.EilenbergGanea.properFace line 62

def properFace : SeedCell → SeedCell → Prop  | .inl vertex, .inr (.inl edge) => vertex ∈ edgeVertices edge  | .inl vertex, .inr (.inr sector) => vertex ∈ sectorVertices sector  | .inr (.inl edge), .inr (.inr sector) => edge ∈ sectorEdges sector  | _, _ => False

definition OAI.EilenbergGanea.seedGraph line 69

def seedGraph : SimpleGraph SeedCell where  Adj first second := properFace first second ∨ properFace second first  symm := ⟨fun _ _ adjacent => adjacent.elim Or.inr Or.inl⟩  loopless := ⟨by    intro cell    rcases cell with vertex | cell    · simp [properFace]    · rcases cell with edge | sector <;> simp [properFace]⟩

definition OAI.EilenbergGanea.artinRelations line 78

def artinRelations {Vertex : Type*} (graph : SimpleGraph Vertex) : Set (FreeGroup Vertex) :=  {word | ∃ first second, graph.Adj first second ∧    word = FreeGroup.of first * FreeGroup.of second *      (FreeGroup.of first)⁻¹ * (FreeGroup.of second)⁻¹}

abbreviation OAI.EilenbergGanea.ArtinGroup line 83

abbrev ArtinGroup {Vertex : Type*} (graph : SimpleGraph Vertex) :=  PresentedGroup (artinRelations graph)

definition OAI.EilenbergGanea.height line 86

def height {Vertex : Type*} (graph : SimpleGraph Vertex) :    ArtinGroup graph →* Multiplicative ℤ :=  PresentedGroup.toGroup (f := fun _ : Vertex => Multiplicative.ofAdd (1 : ℤ)) (by    rintro word ⟨first, second, _, rfl⟩    simp)

abbreviation OAI.EilenbergGanea.SourceGroup line 92
```
abbrev SourceGroup := (height seedGraph).ker
```

definition OAI.EilenbergGanea.cohomologicalDimension line 95

def cohomologicalDimension (GroupType : Type*) [Group GroupType] : WithBot ℕ∞ :=  CategoryTheory.projectiveDimension    (ModuleCat.of (MonoidAlgebra ℤ GroupType)      (Representation.trivial ℤ GroupType ℤ).asModule)

1 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: cells, classifying, dimensional, finite, finitely, generated, group, residually and one more.

Theorem 1.1 of A finitely generated counterexample to the Eilenberg–Ganea conjecture

Theorem

There is a finitely generated residually finite group G such that

\cdim G=2,\qquad \gdim G=3.

One may take G to be the height kernel in the right-angled Artin group associated to a finite acyclic flag triangulation of the presentation complex

\langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle.

No two-dimensional classifying space for this G exists, regardless of the number of its cells.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

There is a finitely generated residually finite group $G$ such that
\[
 \cdim G=2,\qquad \gdim G=3.
\]
One may take $G$ to be the height kernel in the right-angled Artin group associated to a finite acyclic flag triangulation of the presentation complex
\begin{equation}

 \langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle.
\end{equation}
No two-dimensional classifying space for this $G$ exists, regardless of the number of its cells.

Other possible paper matches, closest wording first

Proposition 4.5 of A finitely generated counterexample to the Eilenberg–Ganea conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, boundary, vertex, representation, ker, edges and eleven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finitely, generating, one, dimensional and two.
Theorem 3.1 of A finitely generated counterexample to the Eilenberg–Ganea conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: map, dimension, ker, free, list, trivial and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two.
Proposition 5.1 of A finitely generated counterexample to the Eilenberg–Ganea conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: path, edges, cell and edge. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 1996 s; peak memory 6.2 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 249’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: source_main : Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧ cohomologicalDimension SourceGroup = 2 ∧ HasClassifyingSpace.{0} SourceGroup 3 ∧ ¬ HasClassifyingSpace.{u} SourceGroup 2, with SourceGroup the kernel of the height map of an explicit Artin group, cohomologicalDimension := projectiveDimension (trivial ℤ[G]-module) and HasClassifyingSpace Γ d = aspherical T2 path-connected CW complex with cells only up to dimension d (arbitrary cardinality of cells, any universe). Matches the summary including 'no two-dimensional classifying space, even with infinitely many cells'.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization constructs a finitely generated residually finite group of cohomological dimension two that has a three-dimensional classifying space but no two-dimensional one. Thus its geometric dimension is three, contradicting the conjecture.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

124 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 64 to its own declarations, 98 to Mathlib’s documentation. 26 declared names also link to the same declaration in OpenAI’s library. 2 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. 2 of them could name more than one declaration. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves source_main exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 25 other declarations of the file; with the theorem itself they span 82 lines. 3 of them are named in the statement itself.

Names in the namespace OAI.EilenbergGanea are listed without it.

def seedWordline 12library
def boundaryLengthline 18library
abbrev RoseVertexline 20library
abbrev SeedVertexline 21library
abbrev CircleEdgeline 22library
abbrev Sectorline 23library
abbrev SeedEdgeline 24library
abbrev SeedCellline 25library
def circleStartline 27library
def circleEndline 30library
def sectorLetterline 33library
def sectorCircleEdgeline 39library
def sectorStartline 44library
def nextSectorline 48library
def edgeVerticesline 52library
def sectorEdgesline 56library
def sectorVerticesline 59library
def properFaceline 62library
def seedGraphline 69library
def artinRelationsline 78library
abbrev ArtinGroupline 83library
def heightline 86library
abbrev SourceGroupline 92in the statementlibrary
def cohomologicalDimensionline 95in the statementlibrary
def HasClassifyingSpaceline 103in the statementlibrary
theorem source_mainline 113selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Classical Set Topologynamespace EilenbergGanea/-- The signed words encode the relators `x²y⁻⁵` and `x²(yx⁻¹)³`. -/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 boundaryLength (relation : Bool) : ℕ := 3 * (seedWord relation).lengthabbrev RoseVertex := Option (Bool × Fin 2)abbrev SeedVertex := RoseVertex ⊕ Boolabbrev CircleEdge := Bool × Fin 3abbrev Sector := (relation : Bool) × Fin (boundaryLength relation)abbrev SeedEdge := CircleEdge ⊕ Sectorabbrev SeedCell := SeedVertex ⊕ (SeedEdge ⊕ Sector)def circleStart (edge : CircleEdge) : RoseVertex :=  ![none, some (edge.1, 0), some (edge.1, 1)] edge.2def circleEnd (edge : CircleEdge) : RoseVertex :=  ![some (edge.1, 0), some (edge.1, 1), none] edge.2def sectorLetter (sector : Sector) : Bool × Bool :=  (seedWord sector.1).get ⟨sector.2.val / 3, by    have bound := sector.2.isLt    dsimp [boundaryLength] at bound    omega⟩def sectorCircleEdge (sector : Sector) : CircleEdge :=  ((sectorLetter sector).1, if (sectorLetter sector).2 then    ⟨sector.2.val % 3, Nat.mod_lt _ (by decide)⟩ else    ⟨2 - sector.2.val % 3, by omega⟩)def sectorStart (sector : Sector) : RoseVertex :=  if (sectorLetter sector).2 then circleStart (sectorCircleEdge sector)  else circleEnd (sectorCircleEdge sector)def nextSector (sector : Sector) : Sector :=  ⟨sector.1, ⟨(sector.2.val + 1) % boundaryLength sector.1,    Nat.mod_lt _ (by have := sector.2.isLt; omega)⟩⟩def edgeVertices : SeedEdge → Finset SeedVertex  | .inl edge => {.inl (circleStart edge), .inl (circleEnd edge)}  | .inr sector => {.inr sector.1, .inl (sectorStart sector)}def sectorEdges (sector : Sector) : Finset SeedEdge :=  {.inl (sectorCircleEdge sector), .inr sector, .inr (nextSector sector)}def sectorVertices (sector : Sector) : Finset SeedVertex :=  (sectorEdges sector).biUnion edgeVerticesdef properFace : SeedCell → SeedCell → Prop  | .inl vertex, .inr (.inl edge) => vertex ∈ edgeVertices edge  | .inl vertex, .inr (.inr sector) => vertex ∈ sectorVertices sector  | .inr (.inl edge), .inr (.inr sector) => edge ∈ sectorEdges sector  | _, _ => False/-- Comparability graph of the seven vertices, 51 edges and 45 sectors. -/def seedGraph : SimpleGraph SeedCell where  Adj first second := properFace first second ∨ properFace second first  symm := ⟨fun _ _ adjacent => adjacent.elim Or.inr Or.inl⟩  loopless := ⟨by    intro cell    rcases cell with vertex | cell    · simp [properFace]    · rcases cell with edge | sector <;> simp [properFace]⟩def artinRelations {Vertex : Type*} (graph : SimpleGraph Vertex) : Set (FreeGroup Vertex) :=  {word | ∃ first second, graph.Adj first second ∧    word = FreeGroup.of first * FreeGroup.of second *      (FreeGroup.of first)⁻¹ * (FreeGroup.of second)⁻¹}abbrev ArtinGroup {Vertex : Type*} (graph : SimpleGraph Vertex) :=  PresentedGroup (artinRelations graph)def height {Vertex : Type*} (graph : SimpleGraph Vertex) :    ArtinGroup graph →* Multiplicative ℤ :=  PresentedGroup.toGroup (f := fun _ : Vertex => Multiplicative.ofAdd (1 : ℤ)) (by    rintro word ⟨first, second, _, rfl⟩    simp)abbrev SourceGroup := (height seedGraph).ker/-- Projective dimension of the trivial integral left group-ring module. -/def cohomologicalDimension (GroupType : Type*) [Group GroupType] : WithBot ℕ∞ :=  CategoryTheory.projectiveDimension    (ModuleCat.of (MonoidAlgebra ℤ GroupType)      (Representation.trivial ℤ GroupType ℤ).asModule)universe u/-- Ordinary classifying CW spaces, with no bound on the cardinality of their cells. -/def HasClassifyingSpace (GroupType : Type*) [Group GroupType] (dimension : ℕ) : Prop :=  ∃ (Space : Type u) (_ : TopologicalSpace Space) (_ : T2Space Space)    (_ : Topology.CWComplex (Set.univ : Set Space)),    (∀ degree : ℕ, dimension < degree →      IsEmpty (Topology.CWComplex.cell (Set.univ : Set Space) degree)) ∧    PathConnectedSpace Space ∧    ∃ basepoint : Space, Nonempty (FundamentalGroup Space basepoint ≃* GroupType) ∧    ∃ (Cover : Type u) (_ : TopologicalSpace Cover), ContractibleSpace Cover ∧    ∃ projection : Cover → Space, IsCoveringMap projection ∧ Function.Surjective projectiontheorem source_main :    Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧    cohomologicalDimension SourceGroup = 2 ∧    HasClassifyingSpace.{0} SourceGroup 3 ∧    ¬ HasClassifyingSpace.{u} SourceGroup 2 := by  sorryend EilenbergGaneaendend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

A finitely generated counterexample to the Eilenberg–Ganea conjecture

Theorem

There is a finitely generated residually finite group G such that

\cdim G=2,\qquad \gdim G=3.

One may take G to be the height kernel in the right-angled Artin group associated to a finite acyclic flag triangulation of the presentation complex

\langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle.

No two-dimensional classifying space for this G exists, regardless of the number of its cells.

Section 1, Introduction Read it in the paper TeX source, line 15

Show the TeX

There is a finitely generated residually finite group $G$ such that
\[
 \cdim G=2,\qquad \gdim G=3.
\]
One may take $G$ to be the height kernel in the right-angled Artin group associated to a finite acyclic flag triangulation of the presentation complex
\begin{equation}

 \langle x,y\mid x^2=y^5,\ x^2=(xy^{-1})^3\rangle.
\end{equation}
No two-dimensional classifying space for this $G$ exists, regardless of the number of its cells.

Configuration and file facts

File: lean/ComparatorChallenges/EilenbergGanea.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 249 link it.
Solution module: OAI.Topology.EilenbergGanea.Main
Allowed axioms: propext, Quot.sound, Classical.choiceThe axioms this configuration allows a proof to use. A replay shows which of them a proof uses.
Imports: Mathlib
Imports from outside Mathlib: None
Local vocabulary: 25 local vocabulary declarations
Lines of Lean, without comments or blank lines: 82
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

Plectis Atlas