Torsion-free hyperbolic group that is not residually finite
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Theorem selected for verification:OAI.Release075.main
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremmain : ∃ (G : Type) (_ : Group G), TorsionFree G ∧ WordHyperbolic G ∧ ¬ Group.ResiduallyFinite G := bysorry
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.
defWordHyperbolic (G : Type u) [Group G] : Prop :=
Show the remaining 11 lines
∃ S : Set G, S.Finite ∧ (SimpleGraph.mulCayley S).Connected ∧ ∃ δ : ℕ, ∀ (x y z : G) (p : (SimpleGraph.mulCayley S).Walk x y) (q : (SimpleGraph.mulCayley S).Walk y z) (r : (SimpleGraph.mulCayley S).Walk z x), Geodesic (SimpleGraph.mulCayley S) p → Geodesic (SimpleGraph.mulCayley S) q → Geodesic (SimpleGraph.mulCayley S) r → SideThin (SimpleGraph.mulCayley S) δ p q r ∧ SideThin (SimpleGraph.mulCayley S) δ q r p ∧ SideThin (SimpleGraph.mulCayley S) δ r p q
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: element, field, finite, finitely, free, generated, group, hyperbolic and four more.
Corollary 1.2 of A torsion-free hyperbolic group that is not residually finite
Corollary(Finite-dimensional linear obstruction)
Let G be the finitely generated torsion-free word-hyperbolic group constructed in Theorem thm:main. For every commutative field K, every integer n\geq 1, and every homomorphism \rho:G\to\mathrm{GL}_n(K),
R_f(G)\subseteq\ker\rho.
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.
Let $G$ be the finitely generated torsion-free word-hyperbolic group constructed in Theorem~\ref{thm:main}. For every commutative field $K$, every integer $n\geq 1$, and every homomorphism $\rho:G\to\mathrm{GL}_n(K)$,
\[
R_f(G)\subseteq\ker\rho.
\]
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.
Other possible paper matches, closest wording first
Theorem 1.1 of A torsion-free hyperbolic group that is not residually finite. Words of the Lean statement, its names split into words, that also appear in the theorem: residually, finite, group, word, hyperbolic, torsion and one more.
Theorem 2.1 of A torsion-free hyperbolic group that is not residually finite. Words of the Lean statement, its names split into words, that also appear in the theorem: length, vertex, triangle, connected, walk, path and six more.
Proposition 4.3 of A torsion-free hyperbolic group that is not residually finite. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, cayley, group, word and hyperbolic.
Proposition 3.3 of A torsion-free hyperbolic group that is not residually finite. Words of the Lean statement, its names split into words, that also appear in the theorem: whose, path and edge.
Proposition 3.4 of A torsion-free hyperbolic group that is not residually finite. Words of the Lean statement, its names split into words, that also appear in the theorem: group, torsion and free.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 109 s; peak memory 6.3 GiB).
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.
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 252’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 disagree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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).
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: 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.
OpenAI’s own note on what this Lean statement covers
“The formalized result gives a negative answer by constructing a torsion-free word-hyperbolic group that is not residually finite.”
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
43 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: 8 to its own declarations, 17 to Mathlib’s documentation. 5 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.
There exists a torsion-free word-hyperbolic group that is not residually finite.
Corollary(Finite-dimensional linear obstruction)
Let G be the finitely generated torsion-free word-hyperbolic group constructed in Theorem thm:main. For every commutative field K, every integer n\geq 1, and every homomorphism \rho:G\to\mathrm{GL}_n(K),
R_f(G)\subseteq\ker\rho.
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.
Let $G$ be the finitely generated torsion-free word-hyperbolic group constructed in Theorem~\ref{thm:main}. For every commutative field $K$, every integer $n\geq 1$, and every homomorphism $\rho:G\to\mathrm{GL}_n(K)$,
\[
R_f(G)\subseteq\ker\rho.
\]
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.
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
4 local vocabulary declarations
Lines of Lean, without comments or blank lines
22
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Torsion-free hyperbolic group that is not residually finite
Family 252: A torsion-free hyperbolic group that is neither residually finite nor linear over any field
Candidate pairing
Corollary (Finite-dimensional linear obstruction)
Let G be the finitely generated torsion-free word-hyperbolic group constructed in Theorem thm:main. For every commutative field K, every integer n\geq 1, and every homomorphism \rho:G\to\mathrm{GL}_n(K),
R_f(G)\subseteq\ker\rho.
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.
A torsion-free hyperbolic group that is not residually finite
TeX label cor:nonlinear, build/paper.tex, line 66
The Lean statement says
theoremmain : ∃ (G : Type) (_ : Group G), TorsionFree G ∧ WordHyperbolic G ∧ ¬ Group.ResiduallyFinite G := bysorry
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.