Lean statement

Gromov width of symmetric polar products

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.SymmetricPolar.symmetric_polar_main

Lean statement file: SymmetricPolar.leanShow on the mapFamily 087. 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 symmetric_polar_main {n : ℕ} (hn : 2 ≤ n)    (K : Set (Position n)) (hK : IsSymmetricConvexBody K) :    gromovWidth (polarProduct K) = 4 ∧      ∀ c : ℝ, 0 < c → c < 4 →        HasSymplecticEmbedding (capacityBall n c) (polarProduct K) := 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.SymmetricPolar.Position line 10

abbrev Position (n : ℕ) := EuclideanSpace ℝ (Fin n)

abbreviation OAI.SymmetricPolar.Phase line 11

abbrev Phase (n : ℕ) := Position n × Position n

definition OAI.SymmetricPolar.IsSymmetricConvexBody line 13

def IsSymmetricConvexBody {n : ℕ} (K : Set (Position n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty ∧    ∀ x, x ∈ K ↔ -x ∈ K

definition OAI.SymmetricPolar.polar line 17

def polar {n : ℕ} (K : Set (Position n)) : Set (Position n) :=  {p | ∀ q ∈ K, inner (𝕜 := ℝ) q p ≤ 1}

definition OAI.SymmetricPolar.polarProduct line 20

def polarProduct {n : ℕ} (K : Set (Position n)) : Set (Phase n) :=  interior K ×ˢ interior (polar K)

definition OAI.SymmetricPolar.capacityBall line 23

def capacityBall (n : ℕ) (c : ℝ) : Set (Phase n) :=  {z | Real.pi * (‖z.1‖ ^ 2 + ‖z.2‖ ^ 2) < c}

definition OAI.SymmetricPolar.omega0 line 26

noncomputable def omega0 {n : ℕ} (v w : Phase n) : ℝ :=  inner (𝕜 := ℝ) v.1 w.2 - inner (𝕜 := ℝ) w.1 v.2

definition OAI.SymmetricPolar.HasSymplecticEmbedding line 29

def HasSymplecticEmbedding {n : ℕ} (U V : Set (Phase n)) : Prop :=  ∃ e : Phase n → Phase n,    ContDiffOn ℝ ∞ e U ∧    Topology.IsEmbedding (fun z : U => e z) ∧    MapsTo e U V ∧    ∀ z ∈ U, ∀ v w : Phase n,      omega0 (fderiv ℝ e z v) (fderiv ℝ e z w) = omega0 v w

definition OAI.SymmetricPolar.gromovWidth line 37

noncomputable def gromovWidth {n : ℕ} (U : Set (Phase n)) : ℝ≥0∞ :=  sSup (ENNReal.ofReal '' {c : ℝ | 0 < c ∧ HasSymplecticEmbedding (capacityBall n c) U})

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: embedding, symplectic, symmetric, body, integer and convex. Found in the paper’s title, which its main theorem states: polar, products and balls. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: origin.

Statements in Symplectic Balls in Symmetric Polar Products come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Symplectic Balls in Symmetric Polar Products

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

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
$B^{2n}(c)\hookrightarrow U_K$.

Other possible paper matches, closest wording first

Theorem 2.1 of Symplectic Balls in Symmetric Polar Products. Words of the Lean statement, its names split into words, that also appear in the theorem: omega, compact, ball, pi, embedding, symplectic and one more.
Corollary 5.5 of Symplectic Balls in Symmetric Polar Products. Words of the Lean statement, its names split into words, that also appear in the theorem: balls, embeddings, symplectic, symmetric, body, integers and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: origin.
Proposition 4.1 of Symplectic Balls in Symmetric Polar Products. Words of the Lean statement, its names split into words, that also appear in the theorem: interior, embedding, symplectic and integer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times and circ.
Corollary 5.3 of Symplectic Balls in Symmetric Polar Products. Words of the Lean statement, its names split into words, that also appear in the theorem: symmetric, body, integer and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circ and origin.
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: product, symmetric, body and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circ and origin.
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, body, integer and convex.

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 202 s; peak memory 6.5 GiB).
Reported 11 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 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.

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: 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
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: the sharp functional Mahler inequalities

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For an origin-symmetric convex body $K\subset\mathbb R^n$, $n\ge2$, the formalized result determines the symplectic ball capacity of $\mathrm{int}\,K\times\mathrm{int}\,K^\circ$: its Gromov width is $4$, and every ball of capacity $0<c<4$ embeds symplectically into it. The normalization assigns capacity $\pi r^2$ to a ball of radius $r$.”
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

49 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: 26 to its own declarations, 27 to Mathlib’s documentation. 10 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 symmetric_polar_main 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 9 other declarations of the file; with the theorem itself they span 28 lines. 6 of them are named in the statement itself.

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

abbrev Positionline 10in the statementlibrary
abbrev Phaseline 11library
def IsSymmetricConvexBodyline 13in the statementlibrary
def polarline 17library
def polarProductline 20in the statementlibrary
def capacityBallline 23in the statementlibrary
def omega0line 26library
def HasSymplecticEmbeddingline 29in the statementlibrary
def gromovWidthline 37in the statementlibrary
theorem symmetric_polar_mainline 40selected theoremlibrary

import Mathlibnamespace OAIopen Setopen scoped ContDiff ENNRealnamespace SymmetricPolarabbrev Position (n : ℕ) := EuclideanSpace ℝ (Fin n)abbrev Phase (n : ℕ) := Position n × Position ndef IsSymmetricConvexBody {n : ℕ} (K : Set (Position n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty ∧    ∀ x, x ∈ K ↔ -x ∈ Kdef polar {n : ℕ} (K : Set (Position n)) : Set (Position n) :=  {p | ∀ q ∈ K, inner (𝕜 := ℝ) q p ≤ 1}def polarProduct {n : ℕ} (K : Set (Position n)) : Set (Phase n) :=  interior K ×ˢ interior (polar K)def capacityBall (n : ℕ) (c : ℝ) : Set (Phase n) :=  {z | Real.pi * (‖z.1‖ ^ 2 + ‖z.2‖ ^ 2) < c}noncomputable def omega0 {n : ℕ} (v w : Phase n) : ℝ :=  inner (𝕜 := ℝ) v.1 w.2 - inner (𝕜 := ℝ) w.1 v.2def HasSymplecticEmbedding {n : ℕ} (U V : Set (Phase n)) : Prop :=  ∃ e : Phase n → Phase n,    ContDiffOn ℝ ∞ e U ∧    Topology.IsEmbedding (fun z : U => e z) ∧    MapsTo e U V ∧    ∀ z ∈ U, ∀ v w : Phase n,      omega0 (fderiv ℝ e z v) (fderiv ℝ e z w) = omega0 v wnoncomputable def gromovWidth {n : ℕ} (U : Set (Phase n)) : ℝ≥0∞ :=  sSup (ENNReal.ofReal '' {c : ℝ | 0 < c ∧ HasSymplecticEmbedding (capacityBall n c) U})theorem symmetric_polar_main {n : ℕ} (hn : 2 ≤ n)    (K : Set (Position n)) (hK : IsSymmetricConvexBody K) :    gromovWidth (polarProduct K) = 4 ∧      ∀ c : ℝ, 0 < c → c < 4 →        HasSymplecticEmbedding (capacityBall n c) (polarProduct K) := by  sorryend SymmetricPolarend 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 symmetric Mahler conjecture and its equality cases

Theorem

For every integer n\ge1 and every origin-symmetric convex body K\subset\R^n,

|K|\,|K^\circ|\ge\frac{4^n}{n!}.

Equality holds if and only if K is a linear Hanner body.

Section 1, Introduction Read it in the paper TeX source, line 34

Show the TeX

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,

\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.

Section 1, Introduction Read it in the paper TeX source, line 210

Show the TeX

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

H(\eta_1)+H(\eta_2)\le -n\log(4e^2)+\mathcal T(\nu_1,\nu_2).

Section 1, Introduction Read it in the paper TeX source, line 270

Show the TeX

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
\[
 H(\eta_1)+H(\eta_2)\le -n\log(4e^2)+\mathcal T(\nu_1,\nu_2).
\]

The Mahler Conjecture for General Convex Bodies

Theorem (General Mahler conjecture)

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.

Section 1, Introduction Read it in the paper TeX source, line 26

Show the TeX

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,

\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.

Section 1, Introduction Read it in the paper TeX source, line 171

Show the TeX

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.

Symplectic Balls in Symmetric Polar Products

Theorem

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.

Section 1, Introduction Read it in the paper TeX source, line 22

Show the TeX

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
$B^{2n}(c)\hookrightarrow U_K$.

Configuration and file facts

File: lean/ComparatorChallenges/SymmetricPolar.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/PolarProducts/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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 28
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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