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.MatchingEntropyBounds.Refined.pointwise_entropy
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremRefined.pointwise_entropy (G : LooplessGraph V E) (m : ℕ) (hm : 0 < m) (hV : Fintype.card V = 2 * m) (hPM : Nonempty G.Matching) (x : E → ℝ) (hx : x ∈ G.polytope) : entropy x - (2 - 2 / (m : ℝ)) * complementEntropy x ≤ maxMatchingEntropy G x ∧ maxMatchingEntropy G x ≤ entropy x := bysorry
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.
defcomplementEntropy (x : E → ℝ) : ℝ := entropy (fun e => 1 - x e)
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: bounds, entropy, least, loopless, matching, multigraph, perfect, pointwise and one more.
Statements in Entropy and Face Dimension of the Perfect-Matching Polytope come first because the release’s name for this Lean statement shares words with that paper’s title.
Theorem 1.1 of Entropy and Face Dimension of the Perfect-Matching Polytope
Theorem(Pointwise entropy bounds)
Let G be a loopless multigraph on 2m\ge2 vertices with at least one perfect matching. For every x\in P(G),
Let $G$ be a loopless multigraph on $2m\ge2$ vertices with at least one
perfect matching. For every $x\in P(G)$,
\begin{equation}
F(x)-\left(2-\frac2m\right)B(x)
\le H(x)\le F(x).
\end{equation}
Other possible paper matches, closest wording first
Proposition 5.2 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: mean, integer, law, entropy, graph, perfect and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: marginal and point.
Theorem A.4 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: natural, nonempty, finite, entropy, loopless, perfect and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounds and multigraph.
Theorem 2.2 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: polytope, real, graph, perfect and matching.
Corollary 6.2 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: weighted, real, log, finite, loopless, perfect and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: multigraph and vertices.
Theorem 4.3 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, loopless, perfect and matchings. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: edge, multigraph and vertices.
Theorem 1.2 of Entropy and Face Dimension of the Perfect-Matching Polytope. Words of the Lean statement, its names split into words, that also appear in the theorem: integer and graph. Found in the paper’s title, which its main theorem states: polytope, entropy, perfect and matching. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: includes and vertices.
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 429 s; peak memory 6.4 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 113’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 agree 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: None for the FPRAS and the pointwise entropy bounds; the sharp face-dimension bound is not stated (TriangleFace is only the triangle-expansion face correspondence).
Definitions the group asks readers to compare with the paper: Randomized machine is hand-rolled: RandomMachine (single finite transition table on Turing.Tape (Fin 8), one fair coin per tick, tick = one write/move via Turing.TM0.Stmt), input encoded sparsely with binary encodeNat; maxMatchingEntropy := sSup (entropy '' feasibleLaws y) (nonempty and bounded on the matching polytope)
OpenAI’s own note on what this Lean statement covers
“For feasible edge marginals $x$ of perfect matchings in a loopless multigraph on $2m$ vertices, let $F(x)=-\sum_e x_e\log x_e$, $B(x)=-\sum_e(1-x_e)\log(1-x_e)$, and let $H(x)$ be the maximum entropy of a matching law with those marginals. The formalization proves $F(x)-(2-2/m)B(x)\le H(x)\le F(x)$ for $m\ge1$, including boundary points of the polytope.”
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
84 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: 19 to its own declarations, 25 to Mathlib’s documentation. 15 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
There is a uniform classical randomized algorithm which, given G and rational parameters 0<\eps<1 and 0<\delta<1/2, returns a nonnegative rational number \widehat Z such that
If Z(G)=0, the output is zero with certainty. The worst-case bit running time is polynomial in the input encoding length, \eps^{-1}, and \log\delta^{-1}.
There is a uniform classical randomized algorithm which, given $G$ and
rational parameters $0<\eps<1$ and $0<\delta<1/2$, returns a nonnegative
rational number $\widehat Z$ such that
\[
\Prb\bigl[(1-\eps)Z(G)\le \widehat Z\le(1+\eps)Z(G)\bigr]
\ge 1-\delta.
\]
If $Z(G)=0$, the output is zero with certainty. The worst-case bit running
time is polynomial in the input encoding length, $\eps^{-1}$, and
$\log\delta^{-1}$.
Let $G$ be a loopless multigraph on $2m\ge2$ vertices with at least one
perfect matching. For every $x\in P(G)$,
\begin{equation}
F(x)-\left(2-\frac2m\right)B(x)
\le H(x)\le F(x).
\end{equation}
Theorem(Deterministic approximate counting)
There is one uniform deterministic algorithm which, on a graph on n vertices with nonnegative binary integer pair multiplicities, returns an integer A\ge0 satisfying
N(G)/512^n\le A\le N(G).
Its running time and output length are polynomial in the input bit length. In particular, A=0 exactly when N(G)=0. The guarantee includes a compact input consisting of binary n and a list of positive pair records.
There is one uniform deterministic algorithm which, on a graph on $n$
vertices with nonnegative binary integer pair multiplicities, returns an
integer $A\ge0$ satisfying
\[
N(G)/512^n\le A\le N(G).
\]
Its running time and output length are polynomial in the input bit length.
In particular, $A=0$ exactly when $N(G)=0$. The guarantee includes a compact
input consisting of binary $n$ and a list of positive pair records.
OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 113 link it.
Solution module
OAI.Combinatorics.MatchingEntropy.Main
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
14 local vocabulary declarations
Lines of Lean, without comments or blank lines
35
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Refined pointwise perfect-matching entropy bounds
Family 113: Approximate counting and entropy of perfect matchings
Candidate pairing
Theorem (Pointwise entropy bounds)
Let G be a loopless multigraph on 2m\ge2 vertices with at least one perfect matching. For every x\in P(G),
F(x)-\left(2-\frac2m\right)B(x)
\le H(x)\le F(x).
Entropy and Face Dimension of the Perfect-Matching Polytope
TeX label ent:main, build/sections/01-introduction.tex, line 33
The Lean statement says
theoremRefined.pointwise_entropy (G : LooplessGraph V E) (m : ℕ) (hm : 0 < m) (hV : Fintype.card V = 2 * m) (hPM : Nonempty G.Matching) (x : E → ℝ) (hx : x ∈ G.polytope) : entropy x - (2 - 2 / (m : ℝ)) * complementEntropy x ≤ maxMatchingEntropy G x ∧ maxMatchingEntropy G x ≤ entropy x := bysorry
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.