Lean statement

Seymour’s second-neighborhood conjecture

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.

Seymour's second-neighborhood conjecture

Theorem selected for verification: OAI.SeymourSecondNeighborhood.exists_goodVertex

Lean statement file: SeymourSecondNeighborhood.leanShow on the mapFamily 173. OpenAI lists this as a main result in its Lean catalogue.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem exists_goodVertex (r : V → V → Prop) (hr : IsOriented r) :    ∃ v, GoodVertex r v := by  sorry

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.

structure OAI.SeymourSecondNeighborhood.IsOriented line 10

structure IsOriented (r : V → V → Prop) : Prop where  loopless : ∀ v, ¬ r v v  asymmetric : ∀ {u v}, r u v → ¬ r v u

definition OAI.SeymourSecondNeighborhood.firstNeighbors line 18

def firstNeighbors (r : V → V → Prop) (v : V) : Finset V := by  classical  exact Finset.univ.filter (r v)

definition OAI.SeymourSecondNeighborhood.secondNeighbors line 22

def secondNeighbors (r : V → V → Prop) (v : V) : Finset V := by  classical  exact Finset.univ.filter (fun w =>    w ≠ v ∧ ¬ r v w ∧ ∃ u, r v u ∧ r u w)

definition OAI.SeymourSecondNeighborhood.GoodVertex line 27

def GoodVertex (r : V → V → Prop) (v : V) : Prop :=  (firstNeighbors r v).card ≤ (secondNeighbors r v).card

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: finite, graph, nonempty, oriented and vertex.

Theorem 1.1 of A proof of Seymour’s second-neighborhood conjecture

Theorem

Every nonempty finite oriented graph has a vertex v such that

\abs{N_1^+(v)}\leq \abs{N_2^+(v)}.

Section 1, Introduction Read it in the paper TeX source, line 16

Show the TeX

Every nonempty finite oriented graph has a vertex \(v\) such that
\[
 \abs{N_1^+(v)}\leq \abs{N_2^+(v)}.
\]

Other possible paper matches, closest wording first

Corollary 6.1 of A proof of Seymour’s second-neighborhood conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: neighbors, vertex and oriented. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: directed, excluding, exactly, finite, nonempty and graph.
Corollary 5.1 of A proof of Seymour’s second-neighborhood conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: second, vertex and oriented. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: two, neighborhood, graph, nonempty and finite.
Proposition 4.1 of A proof of Seymour’s second-neighborhood conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: neighbor, vertex and oriented. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite, nonempty and graph.
Proposition 2.1 of A proof of Seymour’s second-neighborhood conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: vertex and oriented. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite, nonempty and graph.

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.

Carlos Toledo’s cert-machine

How they ran it, in their words

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…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

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.

Accepted Their term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Reported 8 October 2026, at the release version shown here. Original verification report

Neal McBurnett’s openai-math-lean-tooling

How they ran it, in their words

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).

Accepted Their term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 123 s; peak memory 2.3 GiB).
Reported 11 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

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.

Allowed axioms: propext, Quot.sound, Classical.choice

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 173’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.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
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.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement 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.”
Quoted from OpenAI’s Lean note for this family

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

40 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: 4 to its own declarations, 7 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves exists_goodVertex exactly as written here. To compare that statement with the paper, also read the definitions and lemmas listed below. Atlas found them by following names in the source text and lists each after the names it uses. For x.foo, Atlas includes every foo defined here for a type, even when the statement may use only one of them.

It reaches all 4 other declarations of the file; with the theorem itself they span 15 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.SeymourSecondNeighborhood are listed without it.

structure IsOrientedline 10in the statementlibrary
def firstNeighborsline 18library
def secondNeighborsline 22library
def GoodVertexline 27in the statementlibrary
theorem exists_goodVertexline 34selected theoremlibrary

import Mathlib.Data.Fintype.Cardimport Mathlib.Data.Finset.Unionnamespace OAInamespace SeymourSecondNeighborhoodvariable {V : Type*}structure IsOriented (r : V → V → Prop) : Prop where  loopless : ∀ v, ¬ r v v  asymmetric : ∀ {u v}, r u v → ¬ r v uvariable [Fintype V] [DecidableEq V]noncomputable sectiondef firstNeighbors (r : V → V → Prop) (v : V) : Finset V := by  classical  exact Finset.univ.filter (r v)def secondNeighbors (r : V → V → Prop) (v : V) : Finset V := by  classical  exact Finset.univ.filter (fun w =>    w ≠ v ∧ ¬ r v w ∧ ∃ u, r v u ∧ r u w)def GoodVertex (r : V → V → Prop) (v : V) : Prop :=  (firstNeighbors r v).card ≤ (secondNeighbors r v).cardendvariable [Nonempty V]theorem exists_goodVertex (r : V → V → Prop) (hr : IsOriented r) :    ∃ v, GoodVertex r v := by  sorryend SeymourSecondNeighborhoodend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

A proof of Seymour’s second-neighborhood conjecture

Theorem

Every nonempty finite oriented graph has a vertex v such that

\abs{N_1^+(v)}\leq \abs{N_2^+(v)}.

Section 1, Introduction Read it in the paper TeX source, line 16

Show the TeX

Every nonempty finite oriented graph has a vertex \(v\) such that
\[
 \abs{N_1^+(v)}\leq \abs{N_2^+(v)}.
\]

Configuration and file facts

File: lean/ComparatorChallenges/SeymourSecondNeighborhood.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/SecondNeighborhood/Main.lean
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.Data.Fintype.Card, Mathlib.Data.Finset.Union
Imports from outside Mathlib: None
Local vocabulary: 4 local vocabulary declarations
Lines of Lean, without comments or blank lines: 15
Second proof-verification program (nanoda): Off in OpenAI's configuration.

A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.

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