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.Paper092.product_counterexample
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.
defproductWitness : Set (Euclidean 20) := {x | firstBlock x ∈ standardSimplex 10 ∧ secondBlock x ∈ standardSimplex 10}
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: dimension, proposed, thus, twenty and unit.
Statements in A product counterexample to the simplex maximum for projection-body volume come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1 of 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
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.
Other possible paper matches, closest wording first
Corollary 6 of A product counterexample to the simplex maximum for projection-body volume. Words of the Lean statement, its names split into words, that also appear in the theorem: constant, products, integer and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: simplices and dimensional.
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, first, nonempty, interior, volume, bodies and one more.
Proposition 5 of A product counterexample to the simplex maximum for projection-body volume. Words of the Lean statement, its names split into words, that also appear in the theorem: standard and simplex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimensional.
Proposition 3 of A product counterexample to the simplex maximum for projection-body volume. Words of the Lean statement, its names split into words, that also appear in the theorem: orthogonal, space, product and convex. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimensional.
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, normalized, space, volume, body and convex.
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: nonempty, interior, integers, body 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.
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 142 s; peak memory 6.3 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 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.
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: 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
Finds part of it stated, or a narrower or weaker statementTheir 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).
OpenAI’s own note on what this Lean statement covers
“The formalized counterexample is the product of two ten-dimensional simplices.”
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
61 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: 41 to its own declarations, 34 to Mathlib’s documentation. 12 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.
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 $T$.
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.
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.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.