Lean statement

Tensor-field classification of Ramsey configurations

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.EuclideanRamsey.classification

Lean statement file: EuclideanRamsey.leanShow on the mapFamily 172. 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 classification {s d : ℕ} (a : Fin s → Space d)    (hs : 2 ≤ s) (hd : 1 ≤ d) (ha : Function.Injective a)    (hspan : affineSpan ℝ (Set.range a) = ⊤) :    Ramsey a ↔ FieldCriterion a := 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.EuclideanRamsey.Space line 9

abbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d)

definition OAI.EuclideanRamsey.Congruent line 11

def Congruent {s d D : ℕ} (a : Fin s → Space d) (b : Fin s → Space D) : Prop :=  ∀ i j, dist (b i) (b j) = dist (a i) (a j)

definition OAI.EuclideanRamsey.Ramsey line 14

def Ramsey {s d : ℕ} (a : Fin s → Space d) : Prop :=  ∀ r : ℕ, 2 ≤ r → ∃ D : ℕ, 1 ≤ D ∧    ∀ c : Space D → Fin r, ∃ b : Fin s → Space D,      Congruent a b ∧ ∃ k : Fin r, ∀ i, c (b i) = k

definition OAI.EuclideanRamsey.coordinateField line 19

def coordinateField {s d : ℕ} (a : Fin s → Space d) : IntermediateField ℚ ℝ :=  IntermediateField.adjoin ℚ (Set.range (fun ij : Fin s × Fin d => a ij.1 ij.2))

abbreviation OAI.EuclideanRamsey.Coeff line 22

abbrev Coeff {s d : ℕ} (a : Fin s → Space d) := ↥(coordinateField a)

abbreviation OAI.EuclideanRamsey.TensorRing line 23

abbrev TensorRing {s d : ℕ} (a : Fin s → Space d) := Coeff a ⊗[ℚ] Coeff a

definition OAI.EuclideanRamsey.coordinate line 25

def coordinate {s d : ℕ} (a : Fin s → Space d) (i : Fin s) (j : Fin d) : Coeff a :=  ⟨a i j, IntermediateField.subset_adjoin ℚ _ (Set.mem_range_self (i, j))⟩

definition OAI.EuclideanRamsey.augmented line 28

def augmented {s d : ℕ} (a : Fin s → Space d) (i : Fin s) : Option (Fin d) → Coeff a  | none => 1  | some j => coordinate a i j

definition OAI.EuclideanRamsey.multiply line 32

def multiply {s d : ℕ} (a : Fin s → Space d) : TensorRing a →ₐ[ℚ] Coeff a :=  Algebra.TensorProduct.lmul' ℚ

definition OAI.EuclideanRamsey.FieldCriterion line 35

def FieldCriterion {s d : ℕ} (a : Fin s → Space d) : Prop :=  ∃ P : Matrix (Option (Fin d)) (Option (Fin d)) (TensorRing a),    (∀ i : Fin s, ∑ α, ∑ β,      ((augmented a i α) ⊗ₜ[ℚ] (1 : Coeff a)) * P α β *      ((1 : Coeff a) ⊗ₜ[ℚ] (augmented a i β)) = 0) ∧    (∀ α β : Fin d, multiply a (P (some α) (some β)) = if α = β then 1 else 0)

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: affine, classification, finite, points, ramsey and span.

Theorem 1.1 of A classification of finite Euclidean Ramsey configurations

Theorem (Classification)

Let A=\{a_1,\ldots,a_s\}\subset\R^d be a finite set of at least two points with affine span \R^d, and define p_i,F,B,m_F as above. Then A is Ramsey if and only if there is a matrix P\in\operatorname{Mat}_{d+1}(B) such that

\begin{aligned}(p_i\otimes1)^T P(1\otimes p_i)&=0 &&(1\le i\le s),\\ m_F(P_{\alpha\beta})&=\delta_{\alpha\beta} &&(1\le\alpha,\beta\le d).\end{aligned}

Here the entries of p_i\otimes1 and 1\otimes p_i are formed componentwise. There are no separate conditions on the multiplied constant row or constant column of P.

Section 1, The classification problem Read it in the paper TeX source, line 74

Show the TeX

Let $A=\{a_1,\ldots,a_s\}\subset\R^d$ be a finite set of at least two
points with affine span $\R^d$, and define $p_i,F,B,m_F$ as above.
Then $A$ is Ramsey if and only if there is a matrix
$P\in\operatorname{Mat}_{d+1}(B)$ such that
\begin{align}
 (p_i\otimes1)^T P(1\otimes p_i)&=0
       &&(1\le i\le s),\\
 m_F(P_{\alpha\beta})&=\delta_{\alpha\beta}
       &&(1\le\alpha,\beta\le d).
\end{align}
Here the entries of $p_i\otimes1$ and $1\otimes p_i$ are formed
componentwise. There are no separate conditions on the multiplied
constant row or constant column of $P$.

Other possible paper matches, closest wording first

Proposition 2.4 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: congruent, tensor, integer, classification, matrix, euclidean and one more.
Proposition 7.3 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: field, span, coordinate and ramsey.
Proposition 5.1 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: criterion, integer, euclidean and ramsey.
Proposition 2.1 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: tensor, classification and matrix.
Proposition 7.6 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: real and ramsey.
Proposition 6.4 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: coordinate.

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 490 s; peak memory 6.4 GiB).
Reported 10 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 172’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: Classification is only for full-affine-span configurations: classification (hs : 2 ≤ s) (hd : 1 ≤ d) (ha : Injective a) (hspan : affineSpan ℝ (range a) = ⊤) : Ramsey a ↔ FieldCriterion a (docs: 'for full-affine-span configurations'; the isometric re-embedding reduction to this case is not a challenge statement). The summary's disproof of the Leader-Russell-Walters conjecture has no statement: only the Ramsey half is present (at_most_five_circle_points_ramsey, nine_circle_configuration, GrahamSpherical.full_main = a 12-point non-Ramsey spherical set), nothing says some Ramsey configuration is not contained in a finite transitive set.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: The existence of a Ramsey configuration that is not subtransitive — the actual counterexample to the LRW conjecture. Its non-subtransitivity half is a cited prior theorem in the paper, not proved there and not in Lean.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization gives the tensor-field classification for full-affine-span configurations, covers singleton and affine-span reductions, and proves that every Ramsey configuration is cospherical.”
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

52 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: 30 to its own declarations, 44 to Mathlib’s documentation. 11 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 classification 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 10 other declarations of the file; with the theorem itself they span 29 lines. 3 of them are named in the statement itself.

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

abbrev Spaceline 9in the statementlibrary
def Congruentline 11library
def Ramseyline 14in the statementlibrary
def coordinateFieldline 19library
abbrev Coeffline 22library
abbrev TensorRingline 23library
def coordinateline 25library
def augmentedline 28library
def multiplyline 32library
def FieldCriterionline 35in the statementlibrary
theorem classificationline 42selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped TensorProductnamespace EuclideanRamseyabbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d)def Congruent {s d D : ℕ} (a : Fin s → Space d) (b : Fin s → Space D) : Prop :=  ∀ i j, dist (b i) (b j) = dist (a i) (a j)def Ramsey {s d : ℕ} (a : Fin s → Space d) : Prop :=  ∀ r : ℕ, 2 ≤ r → ∃ D : ℕ, 1 ≤ D ∧    ∀ c : Space D → Fin r, ∃ b : Fin s → Space D,      Congruent a b ∧ ∃ k : Fin r, ∀ i, c (b i) = kdef coordinateField {s d : ℕ} (a : Fin s → Space d) : IntermediateField ℚ ℝ :=  IntermediateField.adjoin ℚ (Set.range (fun ij : Fin s × Fin d => a ij.1 ij.2))abbrev Coeff {s d : ℕ} (a : Fin s → Space d) := ↥(coordinateField a)abbrev TensorRing {s d : ℕ} (a : Fin s → Space d) := Coeff a ⊗[ℚ] Coeff adef coordinate {s d : ℕ} (a : Fin s → Space d) (i : Fin s) (j : Fin d) : Coeff a :=  ⟨a i j, IntermediateField.subset_adjoin ℚ _ (Set.mem_range_self (i, j))⟩def augmented {s d : ℕ} (a : Fin s → Space d) (i : Fin s) : Option (Fin d) → Coeff a  | none => 1  | some j => coordinate a i jdef multiply {s d : ℕ} (a : Fin s → Space d) : TensorRing a →ₐ[ℚ] Coeff a :=  Algebra.TensorProduct.lmul' ℚdef FieldCriterion {s d : ℕ} (a : Fin s → Space d) : Prop :=  ∃ P : Matrix (Option (Fin d)) (Option (Fin d)) (TensorRing a),    (∀ i : Fin s, ∑ α, ∑ β,      ((augmented a i α) ⊗ₜ[ℚ] (1 : Coeff a)) * P α β *      ((1 : Coeff a) ⊗ₜ[ℚ] (augmented a i β)) = 0) ∧    (∀ α β : Fin d, multiply a (P (some α) (some β)) = if α = β then 1 else 0)theorem classification {s d : ℕ} (a : Fin s → Space d)    (hs : 2 ≤ s) (hd : 1 ≤ d) (ha : Function.Injective a)    (hspan : affineSpan ℝ (Set.range a) = ⊤) :    Ramsey a ↔ FieldCriterion a := by  sorryend EuclideanRamseyendend 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 classification of finite Euclidean Ramsey configurations

Theorem (Classification)

\begin{aligned}(p_i\otimes1)^T P(1\otimes p_i)&=0 &&(1\le i\le s),\\ m_F(P_{\alpha\beta})&=\delta_{\alpha\beta} &&(1\le\alpha,\beta\le d).\end{aligned}

Here the entries of p_i\otimes1 and 1\otimes p_i are formed componentwise. There are no separate conditions on the multiplied constant row or constant column of P.

Section 1, The classification problem Read it in the paper TeX source, line 74

Show the TeX

Let $A=\{a_1,\ldots,a_s\}\subset\R^d$ be a finite set of at least two
points with affine span $\R^d$, and define $p_i,F,B,m_F$ as above.
Then $A$ is Ramsey if and only if there is a matrix
$P\in\operatorname{Mat}_{d+1}(B)$ such that
\begin{align}
 (p_i\otimes1)^T P(1\otimes p_i)&=0
       &&(1\le i\le s),\\
 m_F(P_{\alpha\beta})&=\delta_{\alpha\beta}
       &&(1\le\alpha,\beta\le d).
\end{align}
Here the entries of $p_i\otimes1$ and $1\otimes p_i$ are formed
componentwise. There are no separate conditions on the multiplied
constant row or constant column of $P$.

Configuration and file facts

File: lean/ComparatorChallenges/EuclideanRamsey.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/EuclideanRamsey/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
Imports from outside Mathlib: None
Local vocabulary: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 29
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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