Lean statement

Linear list-coloring bound in the Hadwiger number

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.

Theorem selected for verification: OAI.LinearListHadwiger.main_theorem

Lean statement file: ListHadwiger.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 157 link it.

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.

def MainStatement : Prop :=  ∃ C : ℕ, 1 ≤ C ∧ ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V),    listChromaticNumber G ≤ C * hadwigerNumber G⋮theorem main_theorem : MainStatement := 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.

definition OAI.LinearListHadwiger.ListColorable line 7

def ListColorable {V : Type} (G : SimpleGraph V) {Color : Type}    (L : V → Finset Color) : Prop :=  ∃ c : V → Color, (∀ v, c v ∈ L v) ∧ ∀ ⦃v w⦄, G.Adj v w → c v ≠ c w

definition OAI.LinearListHadwiger.Choosable line 11

def Choosable {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=  ∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G L

definition OAI.LinearListHadwiger.listChromaticNumber line 14

noncomputable def listChromaticNumber {V : Type} (G : SimpleGraph V) : ℕ :=  sInf {k : ℕ | Choosable G k}

definition OAI.LinearListHadwiger.HasCliqueMinor line 17

def HasCliqueMinor {V : Type} (G : SimpleGraph V) (t : ℕ) : Prop :=  ∃ B : Fin t → Set V,    (∀ i, (G.induce (B i)).Connected) ∧    (∀ i j, i ≠ j → Disjoint (B i) (B j)) ∧    (∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w)

definition OAI.LinearListHadwiger.hadwigerNumber line 23

noncomputable def hadwigerNumber {V : Type} (G : SimpleGraph V) : ℕ :=  sSup {t : ℕ | HasCliqueMinor G t}

Possible paper match, with no recorded review

Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.

Words of the Lean statement, its names split into words, that also appear in the theorem: simple, nonempty, integer, finite and graph. Found in the paper’s title, which its main theorem states: hadwiger, number, list and coloring.

Statements in A linear list-coloring bound in terms of the Hadwiger number come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of A linear list-coloring bound in terms of the Hadwiger number

Theorem

There is an absolute integer C\ge1 such that every finite nonempty simple graph G satisfies

\ell(G)\le C h(G).

Section 1, Introduction Read it in the paper TeX source, line 25

Show the TeX

There is an absolute integer $C\ge1$ such that every finite nonempty
simple graph $G$ satisfies
\[
 \ell(G)\le C h(G).
\]

Other possible paper matches, closest wording first

Corollary 1.4 of A counterexample to the Colin de Verdière chromatic conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: hadwiger, simple, list, nonempty, integer, colouring and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: linear, chi and bound.
Theorem 2.2 of A linear list-coloring bound in terms of the Hadwiger number. Words of the Lean statement, its names split into words, that also appear in the theorem: chromatic, connected and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: chi.
Theorem 2.1 of A linear list-coloring bound in terms of the Hadwiger number. Words of the Lean statement, its names split into words, that also appear in the theorem: minor and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bound.
Corollary 1.2 of A counterexample to Hadwiger’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: hadwigers, simple, nonempty, coloring, finite and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order and chi.
Theorem 1.1 of A counterexample to the Colin de Verdière chromatic conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: chromatic, finite and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: order and chi.
Theorem 2.3 of A linear list-coloring bound in terms of the Hadwiger number. Words of the Lean statement, its names split into words, that also appear in the theorem: disjoint, connected, nonempty, integer and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one.

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 165 s; peak memory 6.3 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 157’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.

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet
Finds part of it stated, or a narrower or weaker statement Their term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: HadwigerCounterexample states, for every M, a graph on Fin m with m >= max M 5, indepNum <= 2 and h(G) < 26m/75+2/3 < m/2 <= chi(G); the first-named claim follows by the routine bound chi_f(G) >= |V(G)|/alpha(G) >= m/2 (sum the vertex constraints of a fractional colouring; the paper does exactly this at 01-introduction.tex:58-67), although chi_f itself is not defined in the Lean. The summary's co-equal second claim ('Also disproves' the fractional Colin de Verdiere bound chi_f(G) <= mu(G)+1, the subject of a separate paper with a separate matrix-rank argument) has no challenge statement: no mu, well-signed matrix or Strong Arnold Property appears in either file. The positive list-colouring claim is stated verbatim by ListHadwiger.
Definitions the group asks readers to compare with the paper: HasCliqueMinor: t branch sets B : Fin t -> Set V, each nonempty and connected in G.induce (Mathlib's Connected includes Nonempty), pairwise disjoint and pairwise joined by an edge; the standard K_t-minor definition ; hadwigerNumber: sSup {t | HasCliqueMinor G t}, the largest clique-minor order h(G); the set is bounded by |V| for the finite V used (Fin m, Fintype V), so sSup is a true max; standard ; IsTouchingMatching / connectedMatchingNumber: sets of 2-element cliques (edges), pairwise disjoint and pairwise joined by an edge between their endpoint sets, maximised via sSup; the paper's 'connected matching' (cm), standard for this paper ; listChromaticNumber: sInf {k | Choosable G k} with Choosable quantifying over every Color : Type and every list assignment of size >= k; the standard list chromatic number for finite G (set contains |V|, so sInf is the true minimum); junk value 0 for infinite G is excluded by [Fintype V] in MainStatement ; MainStatement: a Prop-valued def, the type of main_theorem; exists C : N with 1 <= C chosen before V and G, so C is universal; [Fintype V] [Nonempty V] restrict to finite nonempty graphs as in the paper; standard ; G.indepNum and G.chromaticNumber (Mathlib): standard; the statement adds chromaticNumber != top and chromaticNumber = ofNat(toNat) so the toNat casts are faithful ; chi_f (fractional chromatic number): not defined anywhere in the Lean (Mathlib at the pinned revision has no such definition); the fractional form of the primary claim is recovered only outside Lean from alpha <= 2 and chi_f >= |V|/alpha
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The Colin de Verdière clause is not stated. The fractional clause is not stated as such (no χ_f), though it follows from the stated α ≤ 2 and h < m/2.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves that there is one integer $C\ge1$ such that every finite nonempty simple graph $G$ satisfies $\chi_{\mathrm{list}}(G)\le C h(G)$, where $h(G)$ is the largest order of a clique minor. The constant is independent of the graph.”
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

35 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: 6 to its own declarations, 15 to Mathlib’s documentation. 7 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 main_theorem 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 6 other declarations of the file; with the theorem itself they span 19 lines. 1 of them is named in the statement itself.

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

def ListColorableline 7library
def Choosableline 11library
def listChromaticNumberline 14library
def HasCliqueMinorline 17library
def hadwigerNumberline 23library
def MainStatementline 26in the statementlibrary
theorem main_theoremline 30selected theoremlibrary

import Mathlibnamespace OAInamespace LinearListHadwigerdef ListColorable {V : Type} (G : SimpleGraph V) {Color : Type}    (L : V → Finset Color) : Prop :=  ∃ c : V → Color, (∀ v, c v ∈ L v) ∧ ∀ ⦃v w⦄, G.Adj v w → c v ≠ c wdef Choosable {V : Type} (G : SimpleGraph V) (k : ℕ) : Prop :=  ∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G Lnoncomputable def listChromaticNumber {V : Type} (G : SimpleGraph V) : ℕ :=  sInf {k : ℕ | Choosable G k}def HasCliqueMinor {V : Type} (G : SimpleGraph V) (t : ℕ) : Prop :=  ∃ B : Fin t → Set V,    (∀ i, (G.induce (B i)).Connected) ∧    (∀ i j, i ≠ j → Disjoint (B i) (B j)) ∧    (∀ i j, i ≠ j → ∃ v ∈ B i, ∃ w ∈ B j, G.Adj v w)noncomputable def hadwigerNumber {V : Type} (G : SimpleGraph V) : ℕ :=  sSup {t : ℕ | HasCliqueMinor G t}def MainStatement : Prop :=  ∃ C : ℕ, 1 ≤ C ∧ ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V),    listChromaticNumber G ≤ C * hadwigerNumber Gtheorem main_theorem : MainStatement := by  sorryend LinearListHadwigerend 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 linear list-coloring bound in terms of the Hadwiger number

Theorem

There is an absolute integer C\ge1 such that every finite nonempty simple graph G satisfies

\ell(G)\le C h(G).

Section 1, Introduction Read it in the paper TeX source, line 25

Show the TeX

There is an absolute integer $C\ge1$ such that every finite nonempty
simple graph $G$ satisfies
\[
 \ell(G)\le C h(G).
\]

Corollary

Every finite nonempty simple graph G satisfies \chi(G)\le C h(G), with the constant C in Theorem thm:main.

Section 1, Introduction Read it in the paper TeX source, line 38

Show the TeX

Every finite nonempty simple graph $G$ satisfies $\chi(G)\le C h(G)$,
with the constant $C$ in Theorem~\ref{thm:main}.

A counterexample to Hadwiger’s conjecture

Theorem

There are arbitrarily large integers m for which an m-vertex finite simple graph G satisfies

\alpha(G)\le2 \qquad\text{and}\qquad \cm(G)<\frac{m}{100}.

Section 1, Introduction Read it in the paper TeX source, line 20

Show the TeX

There are arbitrarily large integers \(m\) for which an \(m\)-vertex
finite simple graph \(G\) satisfies
\[
             \alpha(G)\le2
             \qquad\text{and}\qquad
             \cm(G)<\frac{m}{100}.
\]

Corollary

There are finite nonempty simple graphs G of arbitrarily large order m for which

h(G)<\frac{26m}{75}+\frac23 <\frac m2\le\chi_f(G)\le\chi(G).

Thus Hadwiger’s conjecture and its fractional-coloring weakening \chi_f(G)\le h(G) are false.

Section 1, Introduction Read it in the paper TeX source, line 48

Show the TeX

There are finite nonempty simple graphs \(G\) of arbitrarily large
order \(m\) for which
\[
 h(G)<\frac{26m}{75}+\frac23
 <\frac m2\le\chi_f(G)\le\chi(G).
\]
Thus Hadwiger's conjecture and its fractional-coloring weakening
\(\chi_f(G)\le h(G)\) are false.

Configuration and file facts

File: lean/ComparatorChallenges/ListHadwiger.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 157 link it.
Solution module: OAI.Combinatorics.ListHadwiger.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: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 19
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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