Lean statement

Petty’s projection-volume inequality

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.

Petty's projection-volume inequality

Theorem selected for verification: OAI.PettyProjection.petty_projection_volume

Lean statement file: PettyProjectionVolume.leanShow on the mapFamily 088. 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 petty_projection_volume (n : ℕ) (hn : 4 ≤ n)    (K : Set (Space n)) (hK : IsConvexBody K) :    pettyConstant n ≤ projectionRatio K ∧      (projectionRatio K = pettyConstant n ↔ IsEllipsoid 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.PettyProjection.Space line 11

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

definition OAI.PettyProjection.IsConvexBody line 13

def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty

definition OAI.PettyProjection.unitBall line 16

def unitBall (n : ℕ) : Set (Space n) := Metric.closedBall 0 1

definition OAI.PettyProjection.kappa line 18

def kappa (n : ℕ) : ℝ := volume.real (unitBall n)

definition OAI.PettyProjection.perpendicular line 20

def perpendicular {n : ℕ} (u : Space n) : Submodule ℝ (Space n) :=  (Submodule.span ℝ {u})ᗮ

definition OAI.PettyProjection.shadowVolume line 24

def shadowVolume {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ :=  (volume : Measure (perpendicular u)).real    ((perpendicular u).orthogonalProjectionOnto '' K)

definition OAI.PettyProjection.projectionBody line 30

def projectionBody {n : ℕ} (K : Set (Space n)) : Set (Space n) :=  {x | ∀ u : Space n, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u}

definition OAI.PettyProjection.projectionRatio line 33

def projectionRatio {n : ℕ} (K : Set (Space n)) : ℝ :=  volume.real (projectionBody K) / volume.real K ^ (n - 1)

definition OAI.PettyProjection.pettyConstant line 36

def pettyConstant (n : ℕ) : ℝ :=  kappa (n - 1) ^ n * kappa n ^ (2 - (n : ℤ))

definition OAI.PettyProjection.IsEllipsoid line 39

def IsEllipsoid {n : ℕ} (K : Set (Space n)) : Prop :=  ∃ (a : Space n) (T : Space n ≃ₗ[ℝ] Space n),    K = (fun x => a + T x) '' unitBall n

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: body, convex and equality.

Statements in Petty’s projection-volume conjecture in dimensions at least four come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of Petty’s projection-volume conjecture in dimensions at least four

Theorem

For every integer n\ge4 and every convex body K\subset\R^n,

\frac{|\Pi K|}{|K|^{n-1}} \ge \kappa_{n-1}^{\,n}\kappa_n^{\,2-n}.

Equality holds if and only if K=a+TB_2^n for some a\in\R^n and some invertible linear map T.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

For every integer $n\ge4$ and every convex body $K\subset\R^n$,
\begin{equation}
 \frac{|\Pi K|}{|K|^{n-1}}
 \ge \kappa_{n-1}^{\,n}\kappa_n^{\,2-n}.
\end{equation}
Equality holds if and only if $K=a+TB_2^n$ for some $a\in\R^n$ and some
invertible linear map $T$.

Other possible paper matches, closest wording first

Corollary 7.3 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: real, ball, space, ellipsoid, unit, volume and four more.
Proposition 2.5 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: measure, ball, support, unit, projection, functions and four more.
Corollary 7.1 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: petty, nonempty, interior, integers, inequalities, body and two more.
Corollary 7.2 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: constants, ellipsoid, functions, inequality, integer, equality and one more.
Theorem 2.2 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: inequality, integer, volume, bodies, equality and convex.
Corollary 2.3 of Petty’s projection-volume conjecture in dimensions at least four. Words of the Lean statement, its names split into words, that also appear in the theorem: compact, nonempty, interior, volume, inequality, equality and three more.

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 494 s; peak memory 6.6 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 088’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: Petty (n >= 4, with ellipsoid equality case) fully stated; the product-of-simplices counterexample is stated only in dimension 20 (ratio 1.015...), not the exponential factor in every sufficiently large dimension; the full Lutwak-Petty inequality family is not stated.
Definitions the group asks readers to compare with the paper: Projection body defined by hand: projectionBody K := {x | ∀ u, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u} with shadowVolume the volume of the orthogonal projection on the subtype (span {u})ᗮ (three files use slightly different encodings: volume on the subtype vs Measure.euclideanHausdorffMeasure (d - 1)), and simplexConstant d := (d+1) * d^d / d! in ProjectionVolume is an asserted value for the simplex, not computed in the statement
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 Lutwak–Petty projection inequalities, and the exponential-factor counterexample in every sufficiently large dimension (only n = 20 is stated).

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization establishes this for every convex body $K\subset\mathbb R^n$ and every $n\ge4$.”
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

54 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: 29 to its own declarations, 21 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 petty_projection_volume 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 24 lines. 5 of them are named in the statement itself.

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

abbrev Spaceline 11in the statementlibrary
def IsConvexBodyline 13in the statementlibrary
def unitBallline 16library
def kappaline 18library
def perpendicularline 20library
def shadowVolumeline 24library
def projectionBodyline 30library
def projectionRatioline 33in the statementlibrary
def pettyConstantline 36in the statementlibrary
def IsEllipsoidline 39in the statementlibrary
theorem petty_projection_volumeline 45selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen Set MeasureTheoryopen scoped RealInnerProductSpacenamespace PettyProjectionabbrev Space (n : ℕ) := EuclideanSpace ℝ (Fin n)def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonemptydef unitBall (n : ℕ) : Set (Space n) := Metric.closedBall 0 1def kappa (n : ℕ) : ℝ := volume.real (unitBall n)def perpendicular {n : ℕ} (u : Space n) : Submodule ℝ (Space n) :=  (Submodule.span ℝ {u})ᗮ/-- Volume in the induced Euclidean structure on the orthogonal hyperplane. -/def shadowVolume {n : ℕ} (K : Set (Space n)) (u : Space n) : ℝ :=  (volume : Measure (perpendicular u)).real    ((perpendicular u).orthogonalProjectionOnto '' K)/-- The body with support function equal to the volumes of orthogonal shadows,expressed as its defining intersection of supporting half-spaces. -/def projectionBody {n : ℕ} (K : Set (Space n)) : Set (Space n) :=  {x | ∀ u : Space n, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u}def projectionRatio {n : ℕ} (K : Set (Space n)) : ℝ :=  volume.real (projectionBody K) / volume.real K ^ (n - 1)def pettyConstant (n : ℕ) : ℝ :=  kappa (n - 1) ^ n * kappa n ^ (2 - (n : ℤ))def IsEllipsoid {n : ℕ} (K : Set (Space n)) : Prop :=  ∃ (a : Space n) (T : Space n ≃ₗ[ℝ] Space n),    K = (fun x => a + T x) '' unitBall n/-- The exact unrestricted inequality and its global equality characterization.-/theorem petty_projection_volume (n : ℕ) (hn : 4 ≤ n)    (K : Set (Space n)) (hK : IsConvexBody K) :    pettyConstant n ≤ projectionRatio K ∧      (projectionRatio K = pettyConstant n ↔ IsEllipsoid K) := by  sorryend PettyProjectionendend 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.

Petty’s projection-volume conjecture in dimensions at least four

Theorem

For every integer n\ge4 and every convex body K\subset\R^n,

\frac{|\Pi K|}{|K|^{n-1}} \ge \kappa_{n-1}^{\,n}\kappa_n^{\,2-n}.

Equality holds if and only if K=a+TB_2^n for some a\in\R^n and some invertible linear map T.

Section 1, Introduction Read it in the paper TeX source, line 36

Show the TeX

For every integer $n\ge4$ and every convex body $K\subset\R^n$,
\begin{equation}
 \frac{|\Pi K|}{|K|^{n-1}}
 \ge \kappa_{n-1}^{\,n}\kappa_n^{\,2-n}.
\end{equation}
Equality holds if and only if $K=a+TB_2^n$ for some $a\in\R^n$ and some
invertible linear map $T$.

A product counterexample to the simplex maximum for projection-body volume

Theorem

Let T_{10}=\conv(0,e_1,\ldots,e_{10})\subset\R^{10}, where the e_i are the coordinate unit vectors, and let K=T_{10}\times T_{10}\subset\R^{20}. Then

\frac{R_{20}(K)}{c_{20}} =\frac{121\binom{20}{10}}{21\cdot2^{20}} =\frac{22\,355\,476}{22\,020\,096}>1.

Thus K violates (eq:proposed-bound) in dimension twenty.

Section 1, Introduction Read it in the paper TeX source, line 90

Show the TeX

Let \(T_{10}=\conv(0,e_1,\ldots,e_{10})\subset\R^{10}\), where the
\(e_i\) are the coordinate unit vectors, and let
\(K=T_{10}\times T_{10}\subset\R^{20}\). Then
\[
 \frac{R_{20}(K)}{c_{20}}
 =\frac{121\binom{20}{10}}{21\cdot2^{20}}
 =\frac{22\,355\,476}{22\,020\,096}>1.
\]
Thus \(K\) violates \eqref{eq:proposed-bound} in dimension twenty.

Configuration and file facts

File: lean/ComparatorChallenges/PettyProjectionVolume.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/ProjectionBodies/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: 24
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.

Plectis Atlas