Lean statement

Artin group with no geometric $\mathrm{CAT}(0)$ action

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.ArtinCAT0.main

Lean statement file: ArtinCAT0.leanShow on the mapFamily 254. OpenAI lists this as a main result in its Lean catalogue.

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 main :    (∀ s t : Letters, explicitMatrix s t = explicitMatrix t s) ∧    (∀ s : Letters, explicitMatrix s s = 1) ∧    (∀ s t : Letters, s ≠ t →      explicitMatrix s t = 2 ∨ explicitMatrix s t = 3 ∨ explicitMatrix s t = ⊤) ∧    (∀ (X : Type u) [MetricSpace X] [ProperSpace X] [Nonempty X]      [MulAction (ArtinGroup explicitMatrix) X],      ¬ GeometricAction (ArtinGroup explicitMatrix) X) := 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.

abbreviation OAI.ArtinCAT0.Letters line 9
```
abbrev Letters := Fin 116
```
abbreviation OAI.ArtinCAT0.Label line 10
```
abbrev Label := WithTop ℕ
```

definition OAI.ArtinCAT0.blockLetter line 12

def blockLetter (b : Fin 3) (i : Fin 39) : Letters :=  if h₀ : i.val = 0 then ⟨b.val, by omega⟩  else if h₁ : i.val = 1 then ⟨(b.val + 1) % 3, by omega⟩  else ⟨5 + 37 * b.val + (i.val - 2), by omega⟩

definition OAI.ArtinCAT0.Adjacent line 17

def Adjacent (s t : Letters) : Prop :=  ∃ (b : Fin 3) (i j : Fin 39), s = blockLetter b i ∧ t = blockLetter b j ∧    (i.val + 1 = j.val ∨ j.val + 1 = i.val)

definition OAI.ArtinCAT0.Distant line 21

def Distant (s t : Letters) : Prop :=  ∃ (b : Fin 3) (i j : Fin 39), s = blockLetter b i ∧ t = blockLetter b j ∧    (i.val + 2 ≤ j.val ∨ j.val + 2 ≤ i.val)

definition OAI.ArtinCAT0.Sentinel line 25

def Sentinel (s t : Letters) : Prop :=  ((s.val = 3 ∨ s.val = 4) ∧ t.val < 3) ∨  ((t.val = 3 ∨ t.val = 4) ∧ s.val < 3)

definition OAI.ArtinCAT0.explicitMatrix line 29

noncomputable def explicitMatrix (s t : Letters) : Label := by  classical  exact if s = t then 1 else if Adjacent s t then 3    else if Distant s t ∨ Sentinel s t then 2 else ⊤

definition OAI.ArtinCAT0.alternating line 34

def alternating {S : Type uS} (s t : S) : ℕ → FreeGroup S  | 0 => 1  | n + 1 => FreeGroup.of s * alternating t s n

definition OAI.ArtinCAT0.artinRelators line 38

def artinRelators {S : Type uS} (M : S → S → Label) : Set (FreeGroup S) :=  {w | ∃ (s t : S) (n : ℕ), M s t = (n : Label) ∧    w = alternating s t n * (alternating t s n)⁻¹}

abbreviation OAI.ArtinCAT0.ArtinGroup line 42

abbrev ArtinGroup {S : Type uS} (M : S → S → Label) := PresentedGroup (artinRelators M)

definition OAI.ArtinCAT0.Segment line 44

def Segment {X : Type uX} [MetricSpace X] (x y : X) (γ : ℝ → X) : Prop :=  γ 0 = x ∧ γ 1 = y ∧    ∀ s ∈ Set.Icc (0 : ℝ) 1, ∀ t ∈ Set.Icc (0 : ℝ) 1,      dist (γ s) (γ t) = |s - t| * dist x y

definition OAI.ArtinCAT0.CAT0 line 49

def CAT0 (X : Type uX) [MetricSpace X] : Prop :=  (∀ x y : X, ∃ γ : ℝ → X, Segment x y γ) ∧  ∀ (x y z : X) (γ η : ℝ → X), Segment x y γ → Segment x z η →    ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ u ∈ Set.Icc (0 : ℝ) 1,      dist (γ t) (η u) ^ 2 ≤        (t ^ 2 - t * u) * dist x y ^ 2 +        (u ^ 2 - t * u) * dist x z ^ 2 + t * u * dist y z ^ 2

definition OAI.ArtinCAT0.GeometricAction line 57

def GeometricAction (G : Type uG) (X : Type uX) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  CAT0 X ∧  (∀ g : G, Isometry (fun x : X => g • x)) ∧  (∀ K : Set X, IsCompact K → {g : G | ((fun x : X => g • x) '' K ∩ K).Nonempty}.Finite) ∧  ∃ K : Set X, IsCompact K ∧ ∀ x : X, ∃ g : G, ∃ k ∈ K, g • k = x

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: explicit, geometric, action, matrix and artin. Found in the paper’s title, which its main theorem states: cat and group. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: diagonal, off and generators.

Statements in An Artin group with no geometric CAT(0) action come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of An Artin group with no geometric CAT(0) action

Theorem

There is an Artin matrix M on 116 generators, with every off-diagonal label in \{2,3,\infty\}, such that A_M admits no geometric action. One explicit matrix is given in Definition def:matrix.

Section 1, Introduction Read it in the paper TeX source, line 61

Show the TeX

There is an Artin matrix \(M\) on 116 generators, with every
off-diagonal label in \(\{2,3,\infty\}\), such that \(A_M\) admits no
geometric action.  One explicit matrix is given in
Definition~\ref{def:matrix}.

Other possible paper matches, closest wording first

Proposition 3.1 of An Artin group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: cat, space and group.
Proposition 4.1 of An Artin group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: geometric and action.
Corollary 10.2 of Harmonic heights and the Artin K(pi,1) conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, integer, matrix, group, artin and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: entries, off and diagonal.
Corollary 1.4 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: proper, group and artin.
Corollary 1.3 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix, group, artin and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: generators.
Theorem 1.1 of Parabolic intersections in Artin groups. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix and finite. Found in the paper’s title, which its main theorem states: groups and artin.

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 72 s; peak memory 6.2 GiB).
Reported 10 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 254’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: All three claims are stated: HarmonicArtin.salvetti_cover_contractible [Finite S] (M : CoxeterMatrix S) : ContractibleSpace (SalvettiCover M) (K(π,1): the cover is the realization of the nerve of the lifted spherical-cell poset), ArtinCAT0.main (the explicit 116-generator matrix has entries in {1,2,3,∞} and ¬ GeometricAction (ArtinGroup explicitMatrix) X for every nonempty proper metric space X with a MulAction, CAT0 via the CN inequality), and ArtinParabolicIntersections (unconditional_arbitrary_intersections: sInf F of any family of parabolics is parabolic and equals sInf of at most Nat.card S members; unique parabolic closure; acylindrical hyperbolicity and weak malnormality for irreducible infinite-label groups). Config notes: ArtinParabolicIntersections.json has enable_nanoda: true (an extra independent-kernel check, not a weakening); tmp/import_cones.json flags external OAI for 254 only because it mis-parses the module MulBlockDiagonal'Entry (apostrophe), so the true cone is Mathlib-only and ~32 lines larger.
Definitions the group asks readers to compare with the paper: SalvettiCover M := SSet.toTop.obj (nerve (LiftedCell M)) with LiftedCell M := Artin M × SphericalType M and the face order generated by LiftedFace (minimal-length Coxeter coset representatives lifted to the Artin group): a hand-built model of the universal cover of the Salvetti complex (nerve of its face poset). Nothing in the challenge statement ties this space to a Mathlib notion of Salvetti complex or to the action of the Artin group, so 'asphericity of the Salvetti complex' is only as faithful as this model (the docs call it the equivalent statement that the universal cover is contractible).
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 formalized result gives a counterexample: an Artin group defined by a symmetric matrix on $116$ generators, with diagonal entries $1$ and off-diagonal entries in $\{2,3,\infty\}$.”
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

77 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: 38 to its own declarations, 37 to Mathlib’s documentation. 14 declared names also link to the same declaration in OpenAI’s library. 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 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 13 other declarations of the file; with the theorem itself they span 51 lines. 4 of them are named in the statement itself.

Names in the namespace OAI.ArtinCAT0 are listed without it.

abbrev Lettersline 9in the statementlibrary
abbrev Labelline 10library
def blockLetterline 12library
def Adjacentline 17library
def Distantline 21library
def Sentinelline 25library
def explicitMatrixline 29in the statementlibrary
def alternatingline 34library
def artinRelatorsline 38library
abbrev ArtinGroupline 42in the statementlibrary
def Segmentline 44library
def CAT0line 49library
def GeometricActionline 57in the statementlibrary
theorem mainline 65selected theoremlibrary

import Mathlibnamespace OAInamespace ArtinCAT0universe uS uG uXabbrev Letters := Fin 116abbrev Label := WithTop ℕdef blockLetter (b : Fin 3) (i : Fin 39) : Letters :=  if h₀ : i.val = 0 then ⟨b.val, by omega⟩  else if h₁ : i.val = 1 then ⟨(b.val + 1) % 3, by omega⟩  else ⟨5 + 37 * b.val + (i.val - 2), by omega⟩def Adjacent (s t : Letters) : Prop :=  ∃ (b : Fin 3) (i j : Fin 39), s = blockLetter b i ∧ t = blockLetter b j ∧    (i.val + 1 = j.val ∨ j.val + 1 = i.val)def Distant (s t : Letters) : Prop :=  ∃ (b : Fin 3) (i j : Fin 39), s = blockLetter b i ∧ t = blockLetter b j ∧    (i.val + 2 ≤ j.val ∨ j.val + 2 ≤ i.val)def Sentinel (s t : Letters) : Prop :=  ((s.val = 3 ∨ s.val = 4) ∧ t.val < 3) ∨  ((t.val = 3 ∨ t.val = 4) ∧ s.val < 3)noncomputable def explicitMatrix (s t : Letters) : Label := by  classical  exact if s = t then 1 else if Adjacent s t then 3    else if Distant s t ∨ Sentinel s t then 2 else ⊤def alternating {S : Type uS} (s t : S) : ℕ → FreeGroup S  | 0 => 1  | n + 1 => FreeGroup.of s * alternating t s ndef artinRelators {S : Type uS} (M : S → S → Label) : Set (FreeGroup S) :=  {w | ∃ (s t : S) (n : ℕ), M s t = (n : Label) ∧    w = alternating s t n * (alternating t s n)⁻¹}abbrev ArtinGroup {S : Type uS} (M : S → S → Label) := PresentedGroup (artinRelators M)def Segment {X : Type uX} [MetricSpace X] (x y : X) (γ : ℝ → X) : Prop :=  γ 0 = x ∧ γ 1 = y ∧    ∀ s ∈ Set.Icc (0 : ℝ) 1, ∀ t ∈ Set.Icc (0 : ℝ) 1,      dist (γ s) (γ t) = |s - t| * dist x ydef CAT0 (X : Type uX) [MetricSpace X] : Prop :=  (∀ x y : X, ∃ γ : ℝ → X, Segment x y γ) ∧  ∀ (x y z : X) (γ η : ℝ → X), Segment x y γ → Segment x z η →    ∀ t ∈ Set.Icc (0 : ℝ) 1, ∀ u ∈ Set.Icc (0 : ℝ) 1,      dist (γ t) (η u) ^ 2 ≤        (t ^ 2 - t * u) * dist x y ^ 2 +        (u ^ 2 - t * u) * dist x z ^ 2 + t * u * dist y z ^ 2def GeometricAction (G : Type uG) (X : Type uX) [Group G] [MetricSpace X] [MulAction G X] : Prop :=  CAT0 X ∧  (∀ g : G, Isometry (fun x : X => g • x)) ∧  (∀ K : Set X, IsCompact K → {g : G | ((fun x : X => g • x) '' K ∩ K).Nonempty}.Finite) ∧  ∃ K : Set X, IsCompact K ∧ ∀ x : X, ∃ g : G, ∃ k ∈ K, g • k = xuniverse utheorem main :    (∀ s t : Letters, explicitMatrix s t = explicitMatrix t s) ∧    (∀ s : Letters, explicitMatrix s s = 1) ∧    (∀ s t : Letters, s ≠ t →      explicitMatrix s t = 2 ∨ explicitMatrix s t = 3 ∨ explicitMatrix s t = ⊤) ∧    (∀ (X : Type u) [MetricSpace X] [ProperSpace X] [Nonempty X]      [MulAction (ArtinGroup explicitMatrix) X],      ¬ GeometricAction (ArtinGroup explicitMatrix) X) := by  sorryend ArtinCAT0end 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.

Harmonic heights and the Artin K(pi,1) conjecture

Theorem

For every Coxeter matrix on a finite set S, the universal cover of X(W,S) is contractible. Equivalently,

\pi_n\bigl(X(W,S)\bigr)=0\qquad(n\ge2).

The conclusion allows arbitrary finite labels, infinite labels, and connected or disconnected diagrams.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

For every Coxeter matrix on a finite set \(S\), the universal cover
of \(X(W,S)\) is contractible.  Equivalently,
\[
  \pi_n\bigl(X(W,S)\bigr)=0\qquad(n\ge2).
\]
The conclusion allows arbitrary finite labels, infinite labels,
and connected or disconnected diagrams.

An Artin group with no geometric CAT(0) action

Theorem

There is an Artin matrix M on 116 generators, with every off-diagonal label in \{2,3,\infty\}, such that A_M admits no geometric action. One explicit matrix is given in Definition def:matrix.

Section 1, Introduction Read it in the paper TeX source, line 61

Show the TeX

There is an Artin matrix \(M\) on 116 generators, with every
off-diagonal label in \(\{2,3,\infty\}\), such that \(A_M\) admits no
geometric action.  One explicit matrix is given in
Definition~\ref{def:matrix}.

Parabolic intersections in Artin groups

Theorem (Parabolic intersections)

For every Coxeter matrix M on a finite set S, subsets X,Y\subseteq S, and g,h\in A_M, there are Z\subseteq S and k\in A_M such that

gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}.

Section 1, Introduction Read it in the paper TeX source, line 21

Show the TeX

For every Coxeter matrix $M$ on a finite set $S$, subsets $X,Y\subseteq S$, and
$g,h\in A_M$, there are $Z\subseteq S$ and $k\in A_M$ such that
\[
 gA_Xg^{-1}\cap hA_Yh^{-1}=kA_Zk^{-1}.
\]

Corollary (Arbitrary intersections and parabolic closure)

For every family \mathcal F of parabolic subgroups of A, there is a subfamily \mathcal F_0\subseteq\mathcal F with |\mathcal F_0|\le |S| such that

\bigcap_{P\in\mathcal F}P=\bigcap_{P\in\mathcal F_0}P.

Here the intersection of an empty family is A. In particular, arbitrary intersections of parabolic subgroups are parabolic, and every subset E\subseteq A is contained in a unique smallest parabolic subgroup.

Section 1, Introduction Read it in the paper TeX source, line 37

Show the TeX

For every family $\mathcal F$ of parabolic subgroups of $A$, there is a
subfamily $\mathcal F_0\subseteq\mathcal F$ with $|\mathcal F_0|\le |S|$ such
that
\[
 \bigcap_{P\in\mathcal F}P=\bigcap_{P\in\mathcal F_0}P.
\]
Here the intersection of an empty family is $A$. In particular, arbitrary
intersections of parabolic subgroups are parabolic, and every subset
$E\subseteq A$ is contained in a unique smallest parabolic subgroup.

Corollary (Word and parabolic membership problems)

There is an algorithm which, given a finite Coxeter matrix M, a subset X\subseteq S, and words u,v in the standard generators of A_M and their inverses, decides whether the element represented by v belongs to uA_Xu^{-1}. In particular, the word problem is decidable for every finite-rank Artin group.

Section 1, Introduction Read it in the paper TeX source, line 59

Show the TeX

There is an algorithm which, given a finite Coxeter matrix $M$, a subset
$X\subseteq S$, and words $u,v$ in the standard generators of $A_M$ and their inverses, decides
whether the element represented by $v$ belongs to $uA_Xu^{-1}$.
In particular, the word problem is decidable for every finite-rank
Artin group.

Corollary (Non-clique Artin-group dynamics)

Suppose M is irreducible, meaning that there is no nontrivial partition S=S_1\sqcup S_2 with m_{st}=2 for every s\in S_1, t\in S_2, and suppose m_{st}=\infty for some pair. Then A is acylindrically hyperbolic. Every proper parabolic subgroup P has a conjugate meeting it trivially: there is g\in A such that P\cap gPg^{-1}=\{1\}.

Section 1, Introduction Read it in the paper TeX source, line 77

Show the TeX

Suppose $M$ is irreducible, meaning that there is no nontrivial partition
$S=S_1\sqcup S_2$ with $m_{st}=2$ for every $s\in S_1$, $t\in S_2$, and
suppose $m_{st}=\infty$ for some pair. Then $A$ is acylindrically hyperbolic.
Every proper parabolic subgroup $P$ has a conjugate meeting it trivially:
there is $g\in A$ such that $P\cap gPg^{-1}=\{1\}$.

Configuration and file facts

File: lean/ComparatorChallenges/ArtinCAT0.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/GroupTheory/ArtinCAT0/Main.lean
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: 13 local vocabulary declarations
Lines of Lean, without comments or blank lines: 51
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