Every word-hyperbolic group with boundary homeomorphic to S2 admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon’s conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold. (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
The paper states
In A Modulus Proof of Cannon’s Conjecture, 23 September 2026
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.
Let G be a hyperbolic group with \partial G homeomorphic to \Sphere. There is a homomorphism \rho:G\to\Isom(\HH^3) for which the action on \HH^3 is proper and cocompact. Its kernel is finite.
Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.
Lean is a language for mathematical statements and proofs that a computer can verify. The Lean statement below is shown beside its possible paper counterpart, with the reason they were linked.
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.
CannonGeometricActionThe paper states
In A Modulus Proof of Cannon’s Conjecture, 23 September 2026
Let G be a hyperbolic group with \partial G homeomorphic to \Sphere. There is a homomorphism \rho:G\to\Isom(\HH^3) for which the action on \HH^3 is proper and cocompact. Its kernel is finite.
The words they share: action, cannon, cocompact, conjecture, finite, group, homeomorphic, hyperbolic, kernel, proper.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
CannonGeometricAction.lean, the release lists it as a supporting statement.
theorem cannon (G : Type u) [Group G] [TopologicalSpace G] [DiscreteTopology G] (D : CayleyData G) (hthin : D.ThinTriangles) (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) : ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧ (ρ.ker : Set G).Finite := 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.
ThinTriangles Model.lean, line 19ProperAction Model.lean, line 81CocompactAction Model.lean, line 85MathVet 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: cannon (G) [Group G] [DiscreteTopology G] (D : CayleyData G) (hthin : D.ThinTriangles) (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) : ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧ (ρ.ker : Set G).Finite, with hyperbolicity as uniformly thin geodesic triangles in the Cayley graph, Boundary D the quotient of based geodesic rays by bounded synchronous distance (quotient of the pointwise topology), H³ the upper half-space with arcosh(1 + |x-y|²/(2 x₃ y₃)) and its full isometry group. The summary’s corollary (torsion-free ⇒ closed hyperbolic 3-manifold group) is not stated.
Definitions the group asks readers to compare with the paper: Custom hyperbolic-group boundary: BasedRay D := {r : ℕ → G // r 0 = 1 ∧ ∀ i j, dist (r i) (r j) = Nat.dist i j}, Boundary D := Quotient (raySetoid D) with raySetoid r s := ∃ C, ∀ n, dist (r n) (s n) ≤ C, topology inherited from ℕ → G (discrete G): the standard ray model of the Gromov boundary but hand-built rather than 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 or imply the whole advertised result (their term: “MATCHES”).
“MATCHES”, in their words: 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.
OpenAI’s own note on what its Lean covers
Cannon’s conjecture asks whether a hyperbolic group with boundary homeomorphic to the two-sphere acts geometrically on hyperbolic three-space. The formalization proves that such a group admits an isometric action on $\mathbb H^3$ that is proper and cocompact and has finite kernel. Hyperbolicity is expressed through uniformly thin geodesic triangles in a Cayley graph, and the boundary hypothesis is a homeomorphism with $S^2$.
Cannon’s geometric-action conclusion CannonGeometricAction
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (Cannon’s conjecture): a hyperbolic group with two-sphere boundary acts properly and cocompactly by isometries on H³ with finite kernel.
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.
CannonGeometricActionThree 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 218 s). | 10 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.
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
Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.
Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.
Offer to write an explanationCompare the Lean statement with the paperSuggest a source attributionSuggest a change
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 proof 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
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 (Cannon’s conjecture)
In A Modulus Proof of Cannon’s Conjecture, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, CannonGeometricAction
Proof-verification report
MathVet, CannonGeometricAction
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, CannonGeometricAction
Lean statement
CannonGeometricAction