Lean statement

Nine-dimensional Borsuk counterexample

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.BorsukNine.main_theorem

Lean statement file: BorsukNine.leanShow on the mapFamily 156. 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 main_theorem :    IsCompact projectorSet ∧    projectorSet ⊆ traceOneSymmetric ∧    Metric.diam projectorSet = Real.sqrt 2 ∧    ¬ HasTenSmallCover := 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.

abbreviation OAI.BorsukNine.Vector4 line 7

abbrev Vector4 := EuclideanSpace ℝ (Fin 4)

abbreviation OAI.BorsukNine.Matrix4 line 8

abbrev Matrix4 := EuclideanSpace ℝ (Fin 4 × Fin 4)

definition OAI.BorsukNine.projector line 10

def projector (u : Vector4) : Matrix4 :=  WithLp.toLp 2 (fun ij => u ij.1 * u ij.2)

definition OAI.BorsukNine.projectorSet line 13

def projectorSet : Set Matrix4 :=  projector '' {u : Vector4 | ‖u‖ = 1}

definition OAI.BorsukNine.traceOneSymmetric line 16

def traceOneSymmetric : Set Matrix4 :=  {A | (∀ i j : Fin 4, A (i, j) = A (j, i)) ∧ ∑ i : Fin 4, A (i, i) = 1}

definition OAI.BorsukNine.HasTenSmallCover line 19

def HasTenSmallCover : Prop :=  ∃ C : Fin 10 → Set Matrix4,    (∀ i, C i ⊆ projectorSet) ∧    projectorSet ⊆ ⋃ i, C i ∧    ∀ i, Metric.diam (C i) < Real.sqrt 2

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: ten, metric, compact, covered, sqrt and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: frobenius, cannot and diameter.

Theorem 1.1 of A nine-dimensional counterexample to Borsuk’s covering assertion

Theorem

The compact set

X=\{uu^{\mathsf T}:u\in\mathbb R^4,\ \|u\|=1\} \subset\{A\in\operatorname{Sym}_4(\mathbb R):\operatorname{tr}A=1\},

with the Frobenius metric, has diameter \sqrt2 and cannot be covered by ten subsets of diameter strictly less than \sqrt2.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

The compact set
\[
 X=\{uu^{\mathsf T}:u\in\mathbb R^4,\ \|u\|=1\}
 \subset\{A\in\operatorname{Sym}_4(\mathbb R):\operatorname{tr}A=1\},
\]
with the Frobenius metric, has diameter $\sqrt2$ and cannot be covered
by ten subsets of diameter strictly less than $\sqrt2$.

Other possible paper matches, closest wording first

Corollary 7.1 of A nine-dimensional counterexample to Borsuk’s covering assertion. Words of the Lean statement, its names split into words, that also appear in the theorem: compact, covered, sqrt and real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: smaller, cannot and diameter.
Proposition 4.1 of A nine-dimensional counterexample to Borsuk’s covering assertion. Words of the Lean statement, its names split into words, that also appear in the theorem: one and real.
Theorem 4.3 of A nine-dimensional counterexample to Borsuk’s covering assertion. Words of the Lean statement, its names split into words, that also appear in the theorem: real. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: lies and dimensional.
Theorem 6.5 of A nine-dimensional counterexample to Borsuk’s covering assertion. Words of the Lean statement, its names split into words, that also appear in the theorem: real.

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.

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 156’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: main_theorem: the set {uu^T : ‖u‖ = 1} ⊆ EuclideanSpace ℝ (Fin 4 × Fin 4) (Frobenius metric) is compact, lies in the trace-one symmetric matrices, has Metric.diam = √2, and admits no cover by ten sets of diameter < √2 (pieces taken inside the set, w.l.o.g.). The routine step not stated in Lean is that the trace-one symmetric 4×4 matrices form a 9-dimensional affine subspace (10 - 1), isometric to ℝ⁹.
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 counterexample is the compact set of rank-one orthogonal projectors onto lines in $\mathbb R^4$, with the Frobenius metric. It lies in the nine-dimensional affine space of trace-one symmetric matrices, has diameter $\sqrt2$, and cannot be covered by ten arbitrary sets of smaller diameter.”
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

34 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: 14 to its own declarations, 17 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. 3 of them are named in the statement itself.

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

abbrev Vector4line 7library
abbrev Matrix4line 8library
def projectorline 10library
def projectorSetline 13in the statementlibrary
def traceOneSymmetricline 16in the statementlibrary
def HasTenSmallCoverline 19in the statementlibrary
theorem main_theoremline 25selected theoremlibrary

import Mathlibnamespace OAInamespace BorsukNineabbrev Vector4 := EuclideanSpace ℝ (Fin 4)abbrev Matrix4 := EuclideanSpace ℝ (Fin 4 × Fin 4)def projector (u : Vector4) : Matrix4 :=  WithLp.toLp 2 (fun ij => u ij.1 * u ij.2)def projectorSet : Set Matrix4 :=  projector '' {u : Vector4 | ‖u‖ = 1}def traceOneSymmetric : Set Matrix4 :=  {A | (∀ i j : Fin 4, A (i, j) = A (j, i)) ∧ ∑ i : Fin 4, A (i, i) = 1}def HasTenSmallCover : Prop :=  ∃ C : Fin 10 → Set Matrix4,    (∀ i, C i ⊆ projectorSet) ∧    projectorSet ⊆ ⋃ i, C i ∧    ∀ i, Metric.diam (C i) < Real.sqrt 2theorem main_theorem :    IsCompact projectorSet ∧    projectorSet ⊆ traceOneSymmetric ∧    Metric.diam projectorSet = Real.sqrt 2 ∧    ¬ HasTenSmallCover := by  sorryend BorsukNineend 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 nine-dimensional counterexample to Borsuk’s covering assertion

Theorem

The compact set

X=\{uu^{\mathsf T}:u\in\mathbb R^4,\ \|u\|=1\} \subset\{A\in\operatorname{Sym}_4(\mathbb R):\operatorname{tr}A=1\},

with the Frobenius metric, has diameter \sqrt2 and cannot be covered by ten subsets of diameter strictly less than \sqrt2.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

The compact set
\[
 X=\{uu^{\mathsf T}:u\in\mathbb R^4,\ \|u\|=1\}
 \subset\{A\in\operatorname{Sym}_4(\mathbb R):\operatorname{tr}A=1\},
\]
with the Frobenius metric, has diameter $\sqrt2$ and cannot be covered
by ten subsets of diameter strictly less than $\sqrt2$.

Configuration and file facts

File: lean/ComparatorChallenges/BorsukNine.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/Borsuk/Counterexample.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: 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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