A hyperbolic group without a geometric CAT(0) action
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Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete \(\mathop{\mathrm{CAT}}\nolimits (0)\) space, in any dimension. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
Both independent assessments find that the Lean statements express or imply the whole result described above.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
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.
Reported proof verification
One Lean statement in the release; two independent groups report that Comparator accepted runs for this file at the version shown here.
We construct a hyperbolic group with a finite classifying space that admits no geometric action on a proper complete CAT(0) space. Consequently, a finite aspherical simplicial complex with a linear combinatorial disk-filling inequality need not have a finite locally CAT(0) homotopy model, and hence need not have a finite locally CAT(−1) model. Here metric models carry geodesic length metrics inducing the given complex topology.
One Lean statement in this family
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For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s 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.
The words they share: topology, hyperbolic, homotopy, equivalent, compatible, connected, word, simplicial, triangular, geodesic, constant, locally, complex, faces, length, pi, finite, metric, group, geometric, disk, edge, aspherical, proper, complete, action, cat, space, bounds, dimensional, model, admits, two.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
HyperbolicObstruction.lean, the release lists it as a main result.
theoremmain : MainStatement.{u} := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
MathVet and cert-machine agree on how much of the advertised result the Lean statements express.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).
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.
“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method: 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.
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.
AI-drafted comparisons, with no recorded human review
Hyperbolic group with no geometric CAT(0) actionHyperbolicObstruction
Unreviewed AI draft: states a weaker form of a paper’s theorem. States a finite aspherical complex with linear disk filling whose hyperbolic fundamental group has no geometric CAT(0) action, and excludes locally CAT(−1) metrics on homotopy-equivalent models. The main theorem also makes the complex two-dimensional and excludes locally CAT(0) metrics; those claims are not included.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.
Hyperbolic group with no geometric CAT(0) actionHyperbolicObstruction
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
What these papers cite in the release, and what cites them
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Papers elsewhere in the release that this family’s papers cite
This family’s papers cite no paper elsewhere in the release.
Papers elsewhere in the release that cite this family’s papers
No paper elsewhere in the release cites this family’s papers.
References outside the release
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Commentary on this family by others, dated and credited. It is commentary, not a finding.
An unsigned public tier list, the OpenAI Math Tier List, classes it as a counterexample and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.
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Read the Lean statement against the paper
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Identify the theorem the catalogue describes
Family 257The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means.Open the form
Address a question raised in an independent assessment
The challenge statements imply the headline claim after unpacking definitions or one routine step.
Their method
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.
The challenge statements state the headline as OpenAI wrote it.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.