Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette’s conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle. (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
The paper states
In Paired states and Hamiltonian cycles in cubic bipartite planar graphs, 24 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.
Every finite simple undirected graph that is cubic, bipartite, planar, and 3-vertex-connected has a Hamiltonian cycle.
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.
BarnetteHamiltonianThe paper states
In Paired states and Hamiltonian cycles in cubic bipartite planar graphs, 24 September 2026
Every finite simple undirected graph that is cubic, bipartite, planar, and 3-vertex-connected has a Hamiltonian cycle.
The words they share: planar, connected, finite, bipartite, simple, cubic, vertex, graph, hamiltonian, cycle.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
BarnetteHamiltonian.lean, the release lists it as a supporting statement.
theorem main : MainStatement.{u} := 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.
Planar Model.lean, line 29ThreeVertexConnected Model.lean, line 33HasHamiltonianCycle Model.lean, line 38MainStatement Model.lean, line 42MathVet 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 — UNDECLARED”).
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: Barnette.main : MainStatement = ∀ V [Fintype V] G, G.IsRegularOfDegree 3 → G.IsBipartite → Planar G → ThreeVertexConnected G → HasHamiltonianCycle G, with Planar an explicit crossing-free topological embedding (PlaneEmbedding with Path arcs in ℝ×ℝ), ThreeVertexConnected = card >= 4 and connected after deleting any <= 2 vertices, and Mathlib IsHamiltonianCycle. The summary’s ‘equivalently... Pfaffian graph’ reformulation is not stated.
“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 — UNDECLARED”).
The gap they name: the Pfaffian three-edge-path formulation
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say 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
Barnette’s conjecture states that every finite simple cubic bipartite planar graph that is 3-vertex-connected has a Hamiltonian cycle. The formalization proves this statement for every such graph. The selected result is the existence of a cycle visiting every vertex exactly once.
Barnette’s Hamiltonian-cycle conjecture BarnetteHamiltonian
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every cubic bipartite planar 3-connected graph has a Hamiltonian cycle.
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.
BarnetteHamiltonianThree 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 349 s; peak memory 6.3 GiB). | 11 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
Cameron Nottingham, cjnott1999/barnette-conjecture: AI-assisted programme and verification of OpenAI’s proof, 9 October 2026. Source
Lean build, axiom check and statement comparison pass; written proof re-derived with no error found.
Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The review reports that Comparator accepted OpenAI’s solution for this challenge, run end to end on Lean 4.34.1 with the default kernel and only the three standard axioms.
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
Hacker News readers, Sharing AI progress in mathematics, 6 October 2026. Source
Several commenters report that the Barnette’s conjecture manuscript looks approachable at first glance (one had failed on it with frontier models months earlier).
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 Paired states and Hamiltonian cycles in cubic bipartite planar graphs, 24 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, BarnetteHamiltonian
Proof-verification report
MathVet, BarnetteHamiltonian
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, BarnetteHamiltonian
Lean statement
BarnetteHamiltonian