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.
Cannon's geometric-action conclusion
Theorem selected for verification:OAI.CannonRelease.cannon
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.
defCayleyData.ThinTriangles {G : Type u} [Group G] (D : CayleyData G) : Prop := ∃ δ : ℕ, ∀ x y z : G, ∀ p : D.graph.Walk x y, ∀ q : D.graph.Walk y z, ∀ r : D.graph.Walk z x, p.length = D.graph.dist x y → q.length = D.graph.dist y z → r.length = D.graph.dist z x → ∀ a ∈ p.support, ∃ b ∈ q.support ++ r.support, D.graph.dist a b ≤ δ
defraySetoid {G : Type u} [Group G] (D : CayleyData G) : Setoid (BasedRay D) where
Show the remaining 11 lines
r r s := ∃ C : ℕ, ∀ n, D.graph.dist (r.val n) (s.val n) ≤ C iseqv := by constructor · intro r exact ⟨0, fun n => by simp⟩ · rintro r s ⟨C, hC⟩ exact ⟨C, fun n => by rw [SimpleGraph.dist_comm]; exact hC n⟩ · rintro r s t ⟨C, hC⟩ ⟨E, hE⟩ refine ⟨C + E, fun n => ?_⟩ exact le_trans ((D.connected (r.val n) (s.val n)).dist_triangle_left (t.val n)) (Nat.add_le_add (hC n) (hE n))
defhyperbolicIsometries : Subgroup (HyperbolicThree ≃ₜ HyperbolicThree) where
Show the remaining 11 lines
carrier := {f | ∀ x y, hyperbolicDist (f x) (f y) = hyperbolicDist x y} one_mem' := by change ∀ x y, hyperbolicDist x y = hyperbolicDist x y intros rfl mul_mem' := by intro f g hf hg x y exact (hf (g x) (g y)).trans (hg x y) inv_mem' := by intro f hf x y simpa using (hf (f.symm x) (f.symm y)).symm
defCocompactAction {G : Type u} [Group G] (ρ : G →* H3Isom) : Prop := ∃ K : Set HyperbolicThree, IsCompact K ∧ ∀ x : HyperbolicThree, ∃ g : G, ∃ y ∈ K, (ρ g).val y = x
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: action, cannon, cocompact, conjecture, finite, group, homeomorphic, hyperbolic and two more.
Theorem 1.1 of A Modulus Proof of Cannon’s Conjecture
Theorem(Cannon’s conjecture)
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.
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.
Other possible paper matches, closest wording first
Proposition 1 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: boundary, space, action, cocompact, proper, hyperbolic and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: isometric, admits, kernel and homeomorphic.
Corollary 1 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: three, group, hyperbolic and sphere. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homeomorphic.
Corollary 2 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: subgroup, connected, orientable, compact, group, hyperbolic and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homeomorphic.
Proposition 4 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: nonempty, distance and finite.
Proposition 2 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: one, compact and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: uniformly.
Theorem 2.1 of A Modulus Proof of Cannon’s Conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: sphere. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: homeomorphic.
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 218 s).
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 246’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.
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: 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.
OpenAI’s own note on what this Lean statement 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 $S^2$.”
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
99 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: 32 to its own declarations, 49 to Mathlib’s documentation. 14 declared names also link to the same declaration in OpenAI’s library. 3 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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.
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.
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.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 246 link it.
Solution module
OAI.Geometry.Cannon.Main
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
16 local vocabulary declarations
Lines of Lean, without comments or blank lines
64
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
Cannon's geometric-action conclusion
Family 246: Cannon’s conjecture
Candidate pairing
Theorem (Cannon’s conjecture)
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.
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.