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.NineStatement.nine_circle_configuration
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.
defRamsey {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
defliteralCircleConfig (t : Fin 9 → ℝ) (i : Fin 9) : Space 2 := WithLp.toLp 2 ![(1 - (t i)^2) / (1 + (t i)^2), (2 * t i) / (1 + (t i)^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: real, algebraically, nine, independent, points and ramsey. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: spherical.
Proposition 7.6 of A classification of finite Euclidean Ramsey configurations
Proposition
Let t_1,\ldots,t_9 be algebraically independent real numbers over \Q. The nine points
Let $t_1,\ldots,t_9$ be algebraically independent real numbers over
$\Q$. The nine points
\[
a_i=\left(\frac{1-t_i^2}{1+t_i^2},
\frac{2t_i}{1+t_i^2}\right),\qquad 1\le i\le9,
\]
form a spherical set that is not Ramsey.
Other possible paper matches, closest wording first
Proposition 7.7 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: unit, circle, point and ramsey. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: twelve.
Corollary 7.4 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: distinct, circle, points and ramsey.
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, integer, euclidean and ramsey. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: coloring.
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: independent and ramsey. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: spherical.
Theorem 1.1 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: points and ramsey. Found in the paper’s title, which its main theorem states: configurations and euclidean.
Corollary 7.1 of A classification of finite Euclidean Ramsey configurations. Words of the Lean statement, its names split into words, that also appear in the theorem: euclidean and ramsey. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: spherical.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 162 s; peak memory 6.2 GiB).
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.
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.
Finds part of it stated, or a narrower or weaker statementTheir 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.
Finds part of it stated, or a narrower or weaker statementTheir 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.
OpenAI’s own note on what this Lean statement covers
“The formalization also gives spherical non-Ramsey examples: the specified twelve-point set on the unit circle has a fifty-color obstruction in every positive dimension, and a nine-point circle configuration built from algebraically independent parameters is non-Ramsey.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
38 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: 11 to its own declarations, 17 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.
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
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.
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 $P$.
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.