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.SymmetricMahler.symmetric_mahler
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.
defcoordinatePolar (K : Set (I → ℝ)) : Set (I → ℝ) := {p | ∀ v ∈ K, (∑ i, p i*v i) ≤ 1}noncomputablesection
No confident match with a paper statement
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in The symmetric Mahler conjecture and its equality cases come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Theorem 1.1 of The symmetric Mahler conjecture and its equality cases. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, integer and convex. Found in the paper’s title, which its main theorem states: mahler. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hanner, equality, linear and body.
Corollary 6.2 of The symmetric Mahler conjecture and its equality cases. Words of the Lean statement, its names split into words, that also appear in the theorem: volume, symmetric and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: product, inequality and body.
Theorem 1.1 of The Mahler Conjecture for General Convex Bodies. Words of the Lean statement, its names split into words, that also appear in the theorem: real, mahler, integer and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equality and body.
Proposition 8.4 of The symmetric Mahler conjecture and its equality cases. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: hanner, equality, linear and body.
Theorem 1.1 of Symplectic Balls in Symmetric Polar Products. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, integer and convex. Found in the paper’s title, which its main theorem states: polar. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: body.
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.
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.
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 087’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 disagree on how much of the advertised result the Lean statements express.
Finds the whole advertised result stated or impliedTheir term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: Symmetric and general Mahler with all equality cases and the Gromov width 4 are stated; the sharp functional Mahler inequalities of the summary are not.
Definitions the group asks readers to compare with the paper: Custom but standard encodings: coordinatePolar (polar w.r.t. the dot product), IsHanner/hannerJoin/hannerProduct, volumeProduct := sInf (... '' interior K), HasSymplecticEmbedding (C^∞ on U, embedding, omega0 pullback condition) and gromovWidth := sSup {c | HasSymplecticEmbedding (capacityBall n c) U} with capacity π r^2
OpenAI’s own note on what this Lean statement covers
“The formalization establishes this for every $n\ge1$ and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins.”
An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
26 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, 11 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.
import Mathlibnamespace OAInamespace SymmetricMahlervariable {I : Type*} [Fintype I]defcoordinatePolar (K : Set (I → ℝ)) : Set (I → ℝ) := {p | ∀ v ∈ K, (∑ i, p i*v i) ≤ 1}noncomputablesectionopen Set MeasureTheorytheoremsymmetric_mahler {n : ℕ} (hn : 1 ≤ n) {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hconv : Convexℝ K) (hsym : ∀ x ∈ K, -x ∈ K) (hint : (interior K).Nonempty) : (4:ℝ)^n/(Nat.factorial n:ℝ) ≤ (volume K).toReal*(volume (coordinatePolar K)).toReal := bysorryendend SymmetricMahlerend 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.
For every integer $n\ge1$ and every origin-symmetric convex body
$K\subset\R^n$,
\begin{equation}
|K|\,|K^\circ|\ge\frac{4^n}{n!}.
\end{equation}
Equality holds if and only if $K$ is a linear Hanner body.
Corollary(Even functional Mahler inequality)
For every integer n\ge1 and every even proper lower-semicontinuous convex function \varphi:\R^n\to\R\cup\{+\infty\} such that 0<\int_{\R^n}e^{-\varphi(x)}\,dx<\infty,
For every integer $n\ge1$ and every even proper lower-semicontinuous
convex function $\varphi:\R^n\to\R\cup\{+\infty\}$ such that
$0<\int_{\R^n}e^{-\varphi(x)}\,dx<\infty$,
\[
\left(\int_{\R^n}e^{-\varphi(x)}\,dx\right)
\left(\int_{\R^n}e^{-\varphi^*(y)}\,dy\right)\ge4^n.
\]
The second integral is allowed to be infinite, in which case the
inequality is immediate.
Corollary(Symmetric entropy–transport inequality)
For every integer n\ge1, let \eta_i(dx)=e^{-V_i(x)}\,dx, i=1,2, be full-dimensional log-concave probability measures, both symmetric about the origin, with proper lower-semicontinuous convex potentials V_i. Suppose their densities are essentially continuous, meaning that e^{-V_i}=0 at \mathcal H^{n-1}-almost every point of \partial\operatorname{supp}\eta_i. Their moment measures \nu_i=(\nabla V_i)_\#\eta_i have finite first moments, H(\eta_i)=-\int V_i\,d\eta_i is finite, and
For every integer $n\ge1$, let $\eta_i(dx)=e^{-V_i(x)}\,dx$, $i=1,2$,
be full-dimensional log-concave probability measures, both symmetric
about the origin, with proper lower-semicontinuous convex potentials
$V_i$. Suppose their densities are essentially continuous, meaning that
$e^{-V_i}=0$ at $\mathcal H^{n-1}$-almost every point of
$\partial\operatorname{supp}\eta_i$. Their moment measures
$\nu_i=(\nabla V_i)_\#\eta_i$ have finite first moments,
$H(\eta_i)=-\int V_i\,d\eta_i$ is finite, and
\[
H(\eta_1)+H(\eta_2)\le -n\log(4e^2)+\mathcal T(\nu_1,\nu_2).
\]
For every integer $n\ge1$ and every convex body $K\subset\mathbb R^n$,
\[
P(K)\ge\frac{(n+1)^{n+1}}{(n!)^2}.
\]
Equality holds if and only if $K$ is a simplex.
Corollary(Functional Mahler inequality)
For every integer n\ge1 and every proper lower-semicontinuous convex function \varphi:\R^n\to\R\cup\{+\infty\} such that 0<\int_{\R^n}e^{-\varphi(x)}\,dx<\infty,
For every integer $n\ge1$ and every proper lower-semicontinuous convex
function $\varphi:\R^n\to\R\cup\{+\infty\}$ such that
$0<\int_{\R^n}e^{-\varphi(x)}\,dx<\infty$,
\[
\left(\int_{\R^n}e^{-\varphi(x)}\,dx\right)
\left(\int_{\R^n}e^{-\varphi^*(y)}\,dy\right)\ge e^n.
\]
The second integral is allowed to be infinite, in which case the
inequality is immediate.
For every integer $n\ge2$ and every origin-symmetric convex body
$K\subset\R^n$,
\[
c_G(U_K)=4.
\]
More precisely, for every $0<c<4$ there is a smooth symplectic embedding
$B^{2n}(c)\hookrightarrow U_K$.
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.
More about this statement
Follow-up result Winter shows the Mahler conjecture implies Kalai's flag conjecture, so OpenAI's claimed Mahler proof, if correct, would settle the flag conjecture. Martin Winter, A relation between Mahler volume and flag number for convex polytopes, 2026-10-07Source
Explanation A definition-by-definition statement check against Theorem 1.1 of the symmetric Mahler paper finds no added hypothesis or weakened conclusion; the challenge states the inequality for every n at least 1, and the equality classification is a separate challenge (SymmetricMahlerEquality). SUNNY99, Independent Comparator/Lean audit of two OpenAI mathematical results, 2026-10-07Source
Questions to resolve when comparing this statement with the paper
Two independent readers, MathVet and cert-machine, disagree on whether this family's challenges state its headline; a specialist reading would settle it.