Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field. (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.
The two independent assessments disagree on how much of the result described above is expressed by the Lean statements. Read the assessments
In A torsion-free hyperbolic group that is not residually finite, 23 September 2026
The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.
There exists a torsion-free word-hyperbolic group that is not residually finite.
No Lean statement in the release is paired with this theorem.
In A torsion-free hyperbolic group that is not residually finite, 23 September 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
Let G be the finitely generated torsion-free word-hyperbolic group constructed in Theorem 1.1. For every commutative field K, every integer n\geq 1, and every homomorphism \rho:G\to\mathrm{GL}_n(K),
Consequently, one fixed nonidentity element of G is killed by all these representations, simultaneously over all such fields and dimensions. In particular, G is not linear over any commutative field.
The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.
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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.
TorsionFreeHyperbolicThe paper states
In A torsion-free hyperbolic group that is not residually finite, 23 September 2026
Let G be the finitely generated torsion-free word-hyperbolic group constructed in Theorem 1.1. For every commutative field K, every integer n\geq 1, and every homomorphism \rho:G\to\mathrm{GL}_n(K),
Consequently, one fixed nonidentity element of G is killed by all these representations, simultaneously over all such fields and dimensions. In particular, G is not linear over any commutative field.
The words they share: element, field, finite, finitely, free, generated, group, hyperbolic, main, nonidentity, torsion, word.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
TorsionFreeHyperbolic.lean, the release lists it as a main result.
theorem main : ∃ (G : Type) (_ : Group G), TorsionFree G ∧ WordHyperbolic G ∧ ¬ Group.ResiduallyFinite G := by sorryThis 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.
three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
TorsionFree Model.lean, line 10Geodesic Model.lean, line 14SideThin Model.lean, line 18WordHyperbolic Model.lean, line 24MathVet and cert-machine disagree on how much of the advertised result the Lean statements express. MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”); cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
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: main : ∃ (G : Type) (_ : Group G), TorsionFree G ∧ WordHyperbolic G ∧ ¬ Group.ResiduallyFinite G with WordHyperbolic = finite generating set whose Cayley graph has uniformly thin geodesic triangles (all three sides, SideThin). The summary’s nonlinearity claim (one fixed element killed by every finite-dimensional representation over every commutative field) is not separately stated; it follows from the stated non-residual-finiteness via Mal’cev’s theorem (docs: ‘Nonlinearity over every field is not a separate conclusion’).
Definitions the group asks readers to compare with the paper: Custom hyperbolicity: WordHyperbolic G := ∃ S finite, (mulCayley S).Connected ∧ ∃ δ, ∀ geodesic triangles, SideThin... (standard thin-triangles definition on the Cayley graph, hand-written, not a Mathlib notion).
“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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).
The gap they name: The non-linearity clause (one fixed nonidentity element killed by every finite-dimensional representation over every field) is not stated; it is a classical consequence (Malcev) of the stated non-residual-finiteness of a finitely generated group, not a separate conclusion.
“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
OpenAI’s own note on what its Lean covers
The residual-finiteness question for hyperbolic groups asks whether every nonidentity element survives in some finite quotient. The formalized result gives a negative answer by constructing a torsion-free word-hyperbolic group that is not residually finite.
Nonlinearity over every field is not a separate conclusion of the statement linked below.
Torsion-free hyperbolic group that is not residually finite TorsionFreeHyperbolic
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: a torsion-free word-hyperbolic group that is not residually finite. The nonlinearity corollary is not a separate conclusion of the statement.
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.
TorsionFreeHyperbolicThree 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. | 10 October 2026 | Original report |
| MathVet | “comparator-sandboxed-pass” | Re-run through Comparator in a sandboxed virtual machine: accepted. | 8 October 2026 | Original report |
| Neal McBurnett, openai-math-lean-tooling | “accepted” | Re-run through Comparator with Lean’s kernel: accepted (the check took 109 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.
This family’s papers cite no paper elsewhere in the release.
No paper elsewhere in the release cites this family’s papers.
Apsiape, hyperbolic-group-not-sofic, 8 October 2026. Source
A repository claims that the torsion-free hyperbolic group of this family is not sofic, which would be an extraordinary extension; the claim was relayed by a topologist and has not been checked by anyone we have found.
homeowmorphism, feat(Other): mark residual finiteness of hyperbolic groups as solved (PR 6921), 7 October 2026. Source
Residual finiteness of hyperbolic groups marked solved with answer False; no formal proof link because of the hyperbolicity definition gap.
Apsiape, hyperbolic-group-not-sofic, 7 October 2026. Source
Non-soficity of OpenAI’s hyperbolic group with Lean.
Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: 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.
Author not stated. Source
From proofsandprompts.com, 8 October 2026. Source
Mahan Mj reports that Astra produced a group-ring criterion for non-residual finiteness that could also certify non-soficity, building on the earlier AI non-sofic group.
Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.
Why this passage is shown
Theorem 1.1
In A torsion-free hyperbolic group that is not residually finite, 23 September 2026
Why this passage is shown
Corollary 1.2 (Finite-dimensional linear obstruction)
In A torsion-free hyperbolic group that is not residually finite, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, TorsionFreeHyperbolic
Proof-verification report
MathVet, TorsionFreeHyperbolic
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, TorsionFreeHyperbolic
Lean statement
TorsionFreeHyperbolic