Proves Seymour’s second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs. (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 proof of Seymour’s second-neighborhood 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.
Every nonempty finite oriented graph has a vertex v such that
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SeymourSecondNeighborhoodThe paper states
In A proof of Seymour’s second-neighborhood conjecture, 23 September 2026
Every nonempty finite oriented graph has a vertex v such that
The words they share: finite, graph, nonempty, oriented, vertex.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
SeymourSecondNeighborhood.lean, the release lists it as a main result.
theorem exists_goodVertex (r : V → V → Prop) (hr : IsOriented r) : ∃ v, GoodVertex r v := 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.
GoodVertex Model.lean, line 27MathVet 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: exists_goodVertex (r) (hr : IsOriented r) : ∃ v, GoodVertex r v over a nonempty Fintype V, with GoodVertex r v := (firstNeighbors r v).card ≤ (secondNeighbors r v).card and second neighbors {w | w ≠ v ∧ ¬ r v w ∧ ∃ u, r v u ∧ r u w} (directed distance exactly two); IsOriented = loopless + asymmetric. Exactly Seymour’s conjecture.
“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
Seymour’s second-neighborhood conjecture asserts that every nonempty finite oriented graph has a vertex with at least as many second out-neighbors as first out-neighbors. The formalized result proves this assertion, where the second neighborhood consists of vertices at directed distance exactly two. The initial vertex and first neighbors are excluded from that count, and sinks are included.
Seymour’s second-neighborhood conjecture SeymourSecondNeighborhood
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: every nonempty finite oriented graph has a vertex with at least as many second out-neighbours as first out-neighbours.
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SeymourSecondNeighborhoodThree 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 123 s; peak memory 2.3 GiB). | 11 October 2026 | Original report |
Other people’s accounts of running proof-verification software, with links to their reports.
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.
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Why this passage is shown
Theorem 1.1
In A proof of Seymour’s second-neighborhood conjecture, 23 September 2026
Independent assessment
MathVet
14c7f91Independent assessment
Carlos Toledo’s cert-machine
9814e4cProof-verification report
cert-machine, SeymourSecondNeighborhood
Proof-verification report
MathVet, SeymourSecondNeighborhood
Proof-verification report
Neal McBurnett, openai-math-lean-tooling, SeymourSecondNeighborhood
Lean statement
SeymourSecondNeighborhood