Lean statement

No proper five-coloring

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.EuclideanFiveColor.no_proper_five_coloring

Lean statement file: EuclideanFiveColor.leanShow on the mapFamily 158. 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)

Problems reported

Problem reported A reader replaced the proper-colouring definition by false and Comparator accepted a trivial proof, because that definition is listed as a hole. Alex Meiburg, Several of the comparator challenges in the OpenAI release are broken and hackable, 2026-10-08 Source

The Lean statement

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

theorem no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ProperColoring 5 coloring := 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.EuclideanFiveColor.ProperColoring line 7

def ProperColoring (colorCount : ℕ) (coloring : ℂ → Fin colorCount) : Prop :=  ∀ point otherPoint : ℂ, ‖point - otherPoint‖ = 1 → coloring point ≠ coloring otherPoint

Definitions taken from OpenAI’s library 1

Comparator compares only the name and type of each definition listed here. When a theorem is configured, its proof uses the definition in OpenAI’s solution library. Read that definition as well as the one in this file when comparing the statement with the paper.

Comparator checks the name and type of ProperColoring; its definition is in OpenAI’s library, in the modules that OAI.Geometry.PlaneColoring.Five imports.

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: proper, five and coloring. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: arbitrary.

Theorem 1.1 of The Euclidean plane is not five-colorable

Theorem

The Euclidean plane has no proper five-coloring, even when arbitrary color classes are allowed. Consequently,

6\le \chi(\R^2)\le 7.

Section 1, Introduction Read it in the paper TeX source, line 62

Show the TeX

The Euclidean plane has no proper five-coloring, even when arbitrary
color classes are allowed. Consequently,
\[
  6\le \chi(\R^2)\le 7.
\]

Other possible paper matches, closest wording first

Proposition 4.2 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, proper and colorings. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable.
Theorem 1.3 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, proper and coloring. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable.
Theorem 1.4 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: five and coloring. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable.
Proposition 7.1 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: point, integer, five and coloring. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable.
Proposition 8.2 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: five.
Corollary 1.5 of The Euclidean plane is not five-colorable. Words of the Lean statement, its names split into words, that also appear in the theorem: five and coloring. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: measurable.

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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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

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.

Definitions compared by name and type only (1): ProperColoring. Their bodies come from OpenAI’s solution library.

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 158’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: no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ∀ p q, ‖p - q‖ = 1 → coloring p ≠ coloring q is the headline for arbitrary colorings (no measurability or color-class hypothesis), and properColoring_seven gives the classical seven-coloring of EuclideanSpace ℝ (Fin 2) (so only 6 or 7 remain). The two statements use different plane models (ℂ with its norm vs EuclideanSpace ℝ (Fin 2), namespace Problem160), both genuinely the Euclidean plane; no lemma identifies them, which does not matter for either claim.
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

“The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption; the upper bound includes every boundary point of the coloring regions.”
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

15 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: 1 to its own declarations, 2 to Mathlib’s documentation. 2 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 no_proper_five_coloring 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. For a definition marked “name and type only”, Comparator does not compare the definition’s body with the one here. When a theorem is configured, it verifies the proof using the definition in OpenAI’s library. Follow the library link to compare the two definitions.

It reaches the file’s only other declaration; with the theorem itself they span 4 lines. 1 of them is named in the statement itself.

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

def ProperColoringline 7in the statementname and type onlylibrary
theorem no_proper_five_coloringline 10selected theoremlibrary

import Mathlibnamespace OAInamespace EuclideanFiveColordef ProperColoring (colorCount : ℕ) (coloring : ℂ → Fin colorCount) : Prop :=  ∀ point otherPoint : ℂ, ‖point - otherPoint‖ = 1 → coloring point ≠ coloring otherPointtheorem no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ProperColoring 5 coloring := by  sorryend EuclideanFiveColorend 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.

The Euclidean plane is not five-colorable

Theorem

The Euclidean plane has no proper five-coloring, even when arbitrary color classes are allowed. Consequently,

6\le \chi(\R^2)\le 7.

Section 1, Introduction Read it in the paper TeX source, line 62

Show the TeX

The Euclidean plane has no proper five-coloring, even when arbitrary
color classes are allowed. Consequently,
\[
  6\le \chi(\R^2)\le 7.
\]

Theorem (Transfer of colorability)

For every positive integer k, in ZFC,

\begin{gathered} \text{a proper $k$-coloring of $\R^2$ exists} \\ \Longleftrightarrow \\ \text{a weak measurable $k$-coloring of $\R^2$ exists}. \end{gathered}

Section 1, Introduction Read it in the paper TeX source, line 104

Show the TeX

For every positive integer $k$, in ZFC,
\begin{equation}
  \begin{gathered}
    \text{a proper $k$-coloring of $\R^2$ exists}
    \\ \Longleftrightarrow \\
    \text{a weak measurable $k$-coloring of $\R^2$ exists}.
  \end{gathered}
\end{equation}

Theorem

There is no weak measurable five-coloring of the Euclidean plane.

Section 1, Introduction Read it in the paper TeX source, line 115

Show the TeX

There is no weak measurable five-coloring of the Euclidean plane.

Corollary (Positive five-color badness)

There is a constant \delta>0 with the following properties.

(i)
For every coloring c:\R^2\to\{1,\ldots,5\} and every normalized finitely additive left-invariant measure \mu on all subsets of the plane-isometry group E(2), the invariant-mean badness satisfies
p_5^\mu(c):=\mu\bigl(\{T\in E(2):c(T^{-1}0)=c(T^{-1}1)\}\bigr) \ge\delta,
where \R^2 is identified with \C. • [(ii)] If c is a Lebesgue-measurable five-coloring with two linearly independent periods, then \mathbb P(c(A)=c(A+U))\ge\delta, where A is uniform in a fundamental parallelogram and U is an independent uniform unit direction. For any Lebesgue-measurable five-coloring, if the conditional square-table frequencies
\mathbb P\bigl(c(A_R)=c(A_R+U)\mid A_R+U\in[-R,R]^2\bigr) \qquad(R>1)
converge as R\to\infty, their limit is at least \delta; here A_R is uniform in [-R,R]^2 and independent of U.

Section 1, Introduction Read it in the paper TeX source, line 282

Show the TeX

There is a constant $\delta>0$ with the following properties.
\begin{enumerate}
\item[(i)] For every coloring $c:\R^2\to\{1,\ldots,5\}$ and every
normalized finitely additive left-invariant measure $\mu$ on all subsets
of the plane-isometry group $E(2)$, the invariant-mean badness satisfies
\[
 p_5^\mu(c):=\mu\bigl(\{T\in E(2):c(T^{-1}0)=c(T^{-1}1)\}\bigr)
 \ge\delta,
\]
where $\R^2$ is identified with $\C$.
\item[(ii)] If $c$ is a Lebesgue-measurable five-coloring with two
linearly independent periods, then
$\mathbb P(c(A)=c(A+U))\ge\delta$, where $A$ is uniform in a
fundamental parallelogram and $U$ is an independent uniform unit
direction. For any Lebesgue-measurable five-coloring, if the conditional
square-table frequencies
\[
 \mathbb P\bigl(c(A_R)=c(A_R+U)\mid A_R+U\in[-R,R]^2\bigr)
 \qquad(R>1)
\]
converge as $R\to\infty$, their limit is at least $\delta$; here
$A_R$ is uniform in $[-R,R]^2$ and independent of $U$.
\end{enumerate}

Configuration and file facts

File: lean/ComparatorChallenges/EuclideanFiveColor.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/PlaneColoring/Five.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
Imports from outside Mathlib: None
Local vocabulary: 1 local vocabulary declaration
Lines of Lean, without comments or blank lines: 4
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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