Lean statement

Hyperbolic group with no geometric 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.HyperbolicObstruction.main

Lean statement file: HyperbolicObstruction.leanShow on the mapFamily 257. 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 : MainStatement.{u} := by sorry

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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.

structure OAI.HyperbolicObstruction.FiniteComplex line 26

structure FiniteComplex where  n : ℕ  complex : AbstractSimplicialComplex (Fin n)

abbreviation OAI.HyperbolicObstruction.FiniteComplex.Vertex line 32
```
abbrev Vertex (K : FiniteComplex) := Fin K.n
```

definition OAI.HyperbolicObstruction.FiniteComplex.Carrier line 34

def Carrier (K : FiniteComplex) :=  {x : K.Vertex → ℝ // (∀ i, 0 ≤ x i) ∧ (∑ i, x i) = 1 ∧    ∃ s ∈ K.complex.faces, ∀ i, i ∉ s → x i = 0}

definition OAI.HyperbolicObstruction.FiniteComplex.Edge line 41

def Edge (K : FiniteComplex) (a b : K.Vertex) : Prop :=  {a, b} ∈ K.complex.faces

definition OAI.HyperbolicObstruction.FiniteComplex.EdgePath line 44

def EdgePath (K : FiniteComplex) : List K.Vertex → Prop  | [] => False  | [_] => True  | a :: b :: t => K.Edge a b ∧ K.EdgePath (b :: t)

definition OAI.HyperbolicObstruction.FiniteComplex.Closed line 49

def Closed (K : FiniteComplex) (p : List K.Vertex) : Prop :=  ∃ a t, p = a :: (t ++ [a]) ∨ p = [a]

inductive type OAI.HyperbolicObstruction.FiniteComplex.DiskMove line 54

inductive DiskMove (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop  | stutter (l r) (a) :      DiskMove K (l ++ [a, a] ++ r) (l ++ [a] ++ r) 0  | spur (l r) (a b) (h : K.Edge a b) :      DiskMove K (l ++ [a, b, a] ++ r) (l ++ [a] ++ r) 0  | triangle (l r) (a b c) (h : {a, b, c} ∈ K.complex.faces)      (hne : ({a, b, c} : Finset K.Vertex).card = 3) :      DiskMove K (l ++ [a, b, c] ++ r) (l ++ [a, c] ++ r) 1

inductive type OAI.HyperbolicObstruction.FiniteComplex.DiskHomotopy line 65

inductive DiskHomotopy (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop  | refl (p) : DiskHomotopy K p p 0  | move {p q n} : DiskMove K p q n → DiskHomotopy K p q n  | symm {p q n} : DiskHomotopy K p q n → DiskHomotopy K q p n  | trans {p q r m n} : DiskHomotopy K p q m → DiskHomotopy K q r n →      DiskHomotopy K p r (m + n)

definition OAI.HyperbolicObstruction.FiniteComplex.Disk line 72

def Disk (K : FiniteComplex) (p : List K.Vertex) (faces : ℕ) : Prop :=  ∃ a, K.DiskHomotopy p [a] faces

definition OAI.HyperbolicObstruction.FiniteComplex.LinearDiskFilling line 75

def LinearDiskFilling (K : FiniteComplex) (C : ℝ) : Prop :=  ∀ p, K.EdgePath p → K.Closed p → (∃ a, K.Disk p a) →    ∃ a, K.Disk p a ∧ (a : ℝ) ≤ C * (p.length - 1 : ℕ)

definition OAI.HyperbolicObstruction.FiniteComplex.Aspherical line 79

def Aspherical (K : FiniteComplex) (x : K.Carrier) : Prop :=  ∀ n : ℕ, Subsingleton (HomotopyGroup.Pi (n + 2) K.Carrier x)

definition OAI.HyperbolicObstruction.IsSegment line 85

def IsSegment {X : Type*} [MetricSpace X] (γ : ℝ → X) (x y : 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.HyperbolicObstruction.IsGeodesic line 90

def IsGeodesic (X : Type*) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ, IsSegment γ x y

definition OAI.HyperbolicObstruction.CATZero line 94

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

definition OAI.HyperbolicObstruction.CATNegOne line 101

def CATNegOne (X : Type*) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ, IsSegment γ x y ∧    ∀ z : X, Real.cosh (dist z (γ (1 / 2))) ≤      (Real.cosh (dist z x) + Real.cosh (dist z y)) /        (2 * Real.cosh (dist x y / 2))

structure OAI.HyperbolicObstruction.GeometricAction line 108

structure GeometricAction (G X : Type*) [Group G] [MetricSpace X] where  act : G →* (X ≃ᵢ X)  proper : ∀ C : Set X, IsCompact C →    {g : G | ∃ x ∈ C, act g x ∈ C}.Finite  cocompact : ∃ C : Set X, IsCompact C ∧ ∀ x : X, ∃ g : G, ∃ y ∈ C, act g y = x

definition OAI.HyperbolicObstruction.NoGeometricCATZeroAction line 116

def NoGeometricCATZeroAction (G : Type*) [Group G] : Prop :=  ∀ (X : Type u) (_ : MetricSpace X), Nonempty X → ProperSpace X → CompleteSpace X →    CATZero X → IsEmpty (GeometricAction G X)

definition OAI.HyperbolicObstruction.WordLength line 120

def WordLength {G : Type*} [Group G] (S : Finset G) (g : G) : ℕ :=  sInf {n | ∃ w : List G, w.length = n ∧    (∀ a ∈ w, a ∈ S ∨ a⁻¹ ∈ S) ∧ w.prod = g}

definition OAI.HyperbolicObstruction.WordDistance line 124

def WordDistance {G : Type*} [Group G] (S : Finset G) (g h : G) : ℝ :=  WordLength S (g⁻¹ * h)

definition OAI.HyperbolicObstruction.WordHyperbolic line 128

def WordHyperbolic (G : Type*) [Group G] : Prop :=  ∃ S : Finset G, Subgroup.closure (S : Set G) = ⊤ ∧    ∃ δ : ℝ, 0 ≤ δ ∧ ∀ a b c d : G,      WordDistance S a c + WordDistance S b d ≤        max (WordDistance S a b + WordDistance S c d)          (WordDistance S a d + WordDistance S b c) + 2 * δ

definition OAI.HyperbolicObstruction.CompatibleLocallyCATNegOne line 136

def CompatibleLocallyCATNegOne (K : FiniteComplex) : Prop :=  ∃ m : MetricSpace K.Carrier,    m.toUniformSpace.toTopologicalSpace = (inferInstance : TopologicalSpace K.Carrier) ∧    (letI := m     IsGeodesic K.Carrier ∧ ∀ x : K.Carrier, ∃ r : ℝ, 0 < r ∧       CATNegOne (Metric.ball x r))

definition OAI.HyperbolicObstruction.MainStatement line 145

def MainStatement : Prop :=  ∃ K : FiniteComplex, ∃ x : K.Carrier, ∃ C : ℝ,    ConnectedSpace K.Carrier ∧ K.Aspherical x ∧ 0 ≤ C ∧ K.LinearDiskFilling C ∧    NoGeometricCATZeroAction.{u} (FundamentalGroup K.Carrier x) ∧    WordHyperbolic (FundamentalGroup K.Carrier x) ∧    ∀ L : FiniteComplex, Nonempty (K.Carrier ≃ₕ L.Carrier) →      ¬ CompatibleLocallyCATNegOne L

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: topology, hyperbolic, homotopy, equivalent, compatible, connected, word, simplicial and 20 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounds, dimensional, model, admits and two.

Theorem 1.1 of A hyperbolic group with no geometric CAT(0) action

Theorem

There exist a finite connected two-dimensional aspherical simplicial complex K and a constant C<\infty with the following properties.

(i)
Every combinatorial edge loop of length \ell in the universal cover of K bounds a disk diagram with at most C\ell triangular faces.
(ii)
The group \pi_1K admits no geometric action on a proper complete \CAT(0) space.

In particular, \pi_1K is word-hyperbolic, and no finite simplicial complex homotopy equivalent to K admits a locally \CAT(0) geodesic length metric inducing its given topology. Consequently, no such model admits a compatible locally \CAT(-1) geodesic length metric.

Section 1, Introduction Read it in the paper TeX source, line 6

Show the TeX

There exist a finite connected two-dimensional aspherical simplicial complex $K$ and a constant $C<\infty$ with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every combinatorial edge loop of length $\ell$ in the universal cover of $K$ bounds a disk diagram with at most $C\ell$ triangular faces.
\item The group $\pi_1K$ admits no geometric action on a proper complete $\CAT(0)$ space.
\end{enumerate}
In particular, $\pi_1K$ is word-hyperbolic, and no finite simplicial complex homotopy equivalent to $K$ admits a locally $\CAT(0)$ geodesic length metric inducing its given topology. Consequently, no such model admits a compatible locally $\CAT(-1)$ geodesic length metric.

Other possible paper matches, closest wording first

Proposition 5.9 of A hyperbolic group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/2, triangle, midpoints, distance, zero, cocompact and 14 more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: whose and two.
Proposition 7.2 of A hyperbolic group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: group, geometric, proper, complete, action, cat and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: admits.
Theorem 3.1 of A hyperbolic group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: linear, disk, aspherical and one.
Proposition 4.5 of A hyperbolic group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty, uniform, finite, metric, one, action and one more.
Proposition 6.5 of A hyperbolic group with no geometric CAT(0) action. Words of the Lean statement, its names split into words, that also appear in the theorem: complete, cat and space.

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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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

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 257’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: MainStatement : ∃ K : FiniteComplex, ∃ x C, ConnectedSpace K.Carrier ∧ K.Aspherical x ∧ 0 ≤ C ∧ K.LinearDiskFilling C ∧ NoGeometricCATZeroAction (FundamentalGroup K.Carrier x) ∧ WordHyperbolic (FundamentalGroup K.Carrier x) ∧ ∀ L, K ≃ₕ L → ¬ CompatibleLocallyCATNegOne L: a finite aspherical complex (hence a finite classifying space) with word-hyperbolic π₁ (four-point condition) admitting no geometric action on any nonempty proper complete CAT(0) space (NoGeometricCATZeroAction, CN-inequality CAT(0)). Matches the summary.
Definitions the group asks readers to compare with the paper: Hand-written geometry: CATZero X := ∀ x y, ∃ γ, IsSegment γ x y ∧ ∀ z, ∀ t ∈ [0,1], dist z (γ t)^2 ≤ (1-t) dist z x^2 + t dist z y^2 - t(1-t) dist x y^2 (CN inequality), WordHyperbolic as the four-point condition with WordDistance from sInf-defined word length, FiniteComplex as barycentric realizations of AbstractSimplicialComplex (Fin n); standard formulations but not Mathlib notions.
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 constructs a finite connected aspherical complex with a linear disk-filling bound whose fundamental group is word-hyperbolic but admits no geometric action on a nonempty proper complete $\mathrm{CAT}(0)$ space. It also excludes locally $\mathrm{CAT}(-1)$ geodesic metrics on every finite complex homotopy equivalent to the witness. The stronger claims that the witness is two-dimensional and that those models admit no locally $\mathrm{CAT}(0)$ metric are not included.”
Quoted from OpenAI’s Lean note for this family

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

Supporting material

The complete Lean statement file

163 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: 43 to its own declarations, 70 to Mathlib’s documentation. 23 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. 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 22 of the file’s 23 other declarations; with the theorem itself they span 85 lines. 1 of them is named in the statement itself.

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

structure FiniteComplexline 26library
abbrev FiniteComplex.Vertexline 32library
def FiniteComplex.Carrierline 34library
def FiniteComplex.Edgeline 41library
def FiniteComplex.EdgePathline 44library
def FiniteComplex.Closedline 49library
inductive FiniteComplex.DiskMoveline 54library
inductive FiniteComplex.DiskHomotopyline 65library
def FiniteComplex.Diskline 72library
def FiniteComplex.LinearDiskFillingline 75library
def FiniteComplex.Asphericalline 79library
def IsSegmentline 85library
def IsGeodesicline 90library
def CATZeroline 94library
def CATNegOneline 101library
structure GeometricActionline 108library
def NoGeometricCATZeroActionline 116library
def WordLengthline 120library
def WordDistanceline 124library
def WordHyperbolicline 128library
def CompatibleLocallyCATNegOneline 136library
def MainStatementline 145in the statementlibrary
theorem mainline 154selected theoremlibrary

Other Lean rules that may affect the statement 1

Lean can use instances, notation and attributes without naming them in the statement. These are rules for interpreting or finding mathematical objects. Atlas’s scan of names cannot follow all such uses; read the listed source as well.

instance (K : FiniteComplex) : TopologicalSpace K.Carrier line 38

import Mathlibnamespace OAIsectionsection/-!Finite complexes are standard barycentric realizations of finite abstractsimplicial complexes. Area-counted elementary edge homotopies encode theusual triangular disk diagrams, allowing spurs and collapsed edges. Null edgeloops (equivalently closed lifts to the universal cover) are those admittingsuch a disk, without any area bound. Asphericity is vanishing of all higherhomotopy groups of this finite polyhedron. The CAT conditions use the standardsquared-distance / hyperbolic-cosh comparison formulations.-/noncomputable sectionopen scoped Topology ContinuousMapnamespace HyperbolicObstructionstructure FiniteComplex where  n : ℕ  complex : AbstractSimplicialComplex (Fin n)namespace FiniteComplexabbrev Vertex (K : FiniteComplex) := Fin K.ndef Carrier (K : FiniteComplex) :=  {x : K.Vertex → ℝ // (∀ i, 0 ≤ x i) ∧ (∑ i, x i) = 1 ∧    ∃ s ∈ K.complex.faces, ∀ i, i ∉ s → x i = 0}instance (K : FiniteComplex) : TopologicalSpace K.Carrier :=  inferInstanceAs (TopologicalSpace (Subtype _))def Edge (K : FiniteComplex) (a b : K.Vertex) : Prop :=  {a, b} ∈ K.complex.facesdef EdgePath (K : FiniteComplex) : List K.Vertex → Prop  | [] => False  | [_] => True  | a :: b :: t => K.Edge a b ∧ K.EdgePath (b :: t)def Closed (K : FiniteComplex) (p : List K.Vertex) : Prop :=  ∃ a t, p = a :: (t ++ [a]) ∨ p = [a]/-- An elementary disk move. A noncollapsed triangular face costs one;    a spur or a collapsed edge costs zero. Contexts are unchanged. -/inductive DiskMove (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop  | stutter (l r) (a) :      DiskMove K (l ++ [a, a] ++ r) (l ++ [a] ++ r) 0  | spur (l r) (a b) (h : K.Edge a b) :      DiskMove K (l ++ [a, b, a] ++ r) (l ++ [a] ++ r) 0  | triangle (l r) (a b c) (h : {a, b, c} ∈ K.complex.faces)      (hne : ({a, b, c} : Finset K.Vertex).card = 3) :      DiskMove K (l ++ [a, b, c] ++ r) (l ++ [a, c] ++ r) 1/-- A disk certificate obtained by elementary triangle attachments/removals,    with spurs/collapses allowed. The index is the number of triangular faces. -/inductive DiskHomotopy (K : FiniteComplex) : List K.Vertex → List K.Vertex → ℕ → Prop  | refl (p) : DiskHomotopy K p p 0  | move {p q n} : DiskMove K p q n → DiskHomotopy K p q n  | symm {p q n} : DiskHomotopy K p q n → DiskHomotopy K q p n  | trans {p q r m n} : DiskHomotopy K p q m → DiskHomotopy K q r n →      DiskHomotopy K p r (m + n)def Disk (K : FiniteComplex) (p : List K.Vertex) (faces : ℕ) : Prop :=  ∃ a, K.DiskHomotopy p [a] facesdef LinearDiskFilling (K : FiniteComplex) (C : ℝ) : Prop :=  ∀ p, K.EdgePath p → K.Closed p → (∃ a, K.Disk p a) →    ∃ a, K.Disk p a ∧ (a : ℝ) ≤ C * (p.length - 1 : ℕ)def Aspherical (K : FiniteComplex) (x : K.Carrier) : Prop :=  ∀ n : ℕ, Subsingleton (HomotopyGroup.Pi (n + 2) K.Carrier x)end FiniteComplex/-- A constant-speed metric segment, parametrized on [0,1]. -/def IsSegment {X : Type*} [MetricSpace X] (γ : ℝ → X) (x y : 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 IsGeodesic (X : Type*) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ, IsSegment γ x y/-- The CN characterization of CAT(0), with whole metric segments. -/def CATZero (X : Type*) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ, IsSegment γ x y ∧    ∀ z : X, ∀ t ∈ Set.Icc (0 : ℝ) 1,      dist z (γ t) ^ 2 ≤ (1 - t) * dist z x ^ 2 + t * dist z y ^ 2 -        t * (1 - t) * dist x y ^ 2/-- The hyperbolic-cosh midpoint characterization of CAT(-1). -/def CATNegOne (X : Type*) [MetricSpace X] : Prop :=  ∀ x y : X, ∃ γ, IsSegment γ x y ∧    ∀ z : X, Real.cosh (dist z (γ (1 / 2))) ≤      (Real.cosh (dist z x) + Real.cosh (dist z y)) /        (2 * Real.cosh (dist x y / 2))/-- A proper, cocompact isometric action of a discrete group. -/structure GeometricAction (G X : Type*) [Group G] [MetricSpace X] where  act : G →* (X ≃ᵢ X)  proper : ∀ C : Set X, IsCompact C →    {g : G | ∃ x ∈ C, act g x ∈ C}.Finite  cocompact : ∃ C : Set X, IsCompact C ∧ ∀ x : X, ∃ g : G, ∃ y ∈ C, act g y = xuniverse udef NoGeometricCATZeroAction (G : Type*) [Group G] : Prop :=  ∀ (X : Type u) (_ : MetricSpace X), Nonempty X → ProperSpace X → CompleteSpace X →    CATZero X → IsEmpty (GeometricAction G X)def WordLength {G : Type*} [Group G] (S : Finset G) (g : G) : ℕ :=  sInf {n | ∃ w : List G, w.length = n ∧    (∀ a ∈ w, a ∈ S ∨ a⁻¹ ∈ S) ∧ w.prod = g}def WordDistance {G : Type*} [Group G] (S : Finset G) (g h : G) : ℝ :=  WordLength S (g⁻¹ * h)/-- Four-point word hyperbolicity, for some finite generating set. -/def WordHyperbolic (G : Type*) [Group G] : Prop :=  ∃ S : Finset G, Subgroup.closure (S : Set G) = ⊤ ∧    ∃ δ : ℝ, 0 ≤ δ ∧ ∀ a b c d : G,      WordDistance S a c + WordDistance S b d ≤        max (WordDistance S a b + WordDistance S c d)          (WordDistance S a d + WordDistance S b c) + 2 * δ/-- A metric on the given finite polyhedron, not on a different topology. -/def CompatibleLocallyCATNegOne (K : FiniteComplex) : Prop :=  ∃ m : MetricSpace K.Carrier,    m.toUniformSpace.toTopologicalSpace = (inferInstance : TopologicalSpace K.Carrier) ∧    (letI := m     IsGeodesic K.Carrier ∧ ∀ x : K.Carrier, ∃ r : ℝ, 0 < r ∧       CATNegOne (Metric.ball x r))/-- A finite aspherical complex with hyperbolic fundamental group, no geometric CAT(0)action, and no homotopy equivalent finite complex admitting a locally CAT(-1) metric. -/def MainStatement : Prop :=  ∃ K : FiniteComplex, ∃ x : K.Carrier, ∃ C : ℝ,    ConnectedSpace K.Carrier ∧ K.Aspherical x ∧ 0 ≤ C ∧ K.LinearDiskFilling C ∧    NoGeometricCATZeroAction.{u} (FundamentalGroup K.Carrier x) ∧    WordHyperbolic (FundamentalGroup K.Carrier x) ∧    ∀ L : FiniteComplex, Nonempty (K.Carrier ≃ₕ L.Carrier) →      ¬ CompatibleLocallyCATNegOne Ltheorem main : MainStatement.{u} := by sorryend HyperbolicObstructionendendendend 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 hyperbolic group with no geometric CAT(0) action

Theorem

There exist a finite connected two-dimensional aspherical simplicial complex K and a constant C<\infty with the following properties.

(i)
Every combinatorial edge loop of length \ell in the universal cover of K bounds a disk diagram with at most C\ell triangular faces.
(ii)
The group \pi_1K admits no geometric action on a proper complete \CAT(0) space.

Section 1, Introduction Read it in the paper TeX source, line 6

Show the TeX

There exist a finite connected two-dimensional aspherical simplicial complex $K$ and a constant $C<\infty$ with the following properties.
\begin{enumerate}[label=\textup{(\roman*)}]
\item Every combinatorial edge loop of length $\ell$ in the universal cover of $K$ bounds a disk diagram with at most $C\ell$ triangular faces.
\item The group $\pi_1K$ admits no geometric action on a proper complete $\CAT(0)$ space.
\end{enumerate}
In particular, $\pi_1K$ is word-hyperbolic, and no finite simplicial complex homotopy equivalent to $K$ admits a locally $\CAT(0)$ geodesic length metric inducing its given topology. Consequently, no such model admits a compatible locally $\CAT(-1)$ geodesic length metric.

Configuration and file facts

File: lean/ComparatorChallenges/HyperbolicObstruction.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/HyperbolicGroups/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: 23 local vocabulary declarations
Lines of Lean, without comments or blank lines: 87
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