Lean statement

Superpolynomial PSD rank for the perfect-matching slack matrix

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.PerfectMatchingPSD.main

Lean statement file: MatchingPSD.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 126 link it.

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.

def MainClaim : Prop :=  ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧    ∀ n : ℕ, n₀ ≤ n → Even n → (n : ℝ) ^ C < (psdRank n : ℝ)⋮theorem main : MainClaim := 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.PerfectMatchingPSD.Edge line 7

abbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}

definition OAI.PerfectMatchingPSD.IsPerfectMatching line 9

def IsPerfectMatching {n : ℕ} (M : Finset (Edge n)) : Prop :=  ∀ v : Fin n, (M.filter fun e => e.1.1 = v ∨ e.1.2 = v).card = 1

abbreviation OAI.PerfectMatchingPSD.PerfectMatching line 12

abbrev PerfectMatching (n : ℕ) :=  {M : Finset (Edge n) // IsPerfectMatching M}

abbreviation OAI.PerfectMatchingPSD.OddCut line 15

abbrev OddCut (n : ℕ) := {U : Finset (Fin n) // Odd U.card}

abbreviation OAI.PerfectMatchingPSD.Row line 17

abbrev Row (n : ℕ) := Edge n ⊕ OddCut n

definition OAI.PerfectMatchingPSD.Crosses line 19

def Crosses {n : ℕ} (U : Finset (Fin n)) (e : Edge n) : Prop :=  (e.1.1 ∈ U ∧ e.1.2 ∉ U) ∨ (e.1.1 ∉ U ∧ e.1.2 ∈ U)

definition OAI.PerfectMatchingPSD.crossingCount line 22

noncomputable def crossingCount {n : ℕ} (U : Finset (Fin n))    (M : PerfectMatching n) : ℕ := by  classical  exact (M.1.filter (Crosses U)).card

definition OAI.PerfectMatchingPSD.slack line 27

noncomputable def slack {n : ℕ} (i : Row n) (M : PerfectMatching n) : ℝ := by  classical  exact match i with  | Sum.inl e => if e ∈ M.1 then 1 else 0  | Sum.inr U => (crossingCount U.1 M : ℝ) - 1

definition OAI.PerfectMatchingPSD.HasFactorization line 34

def HasFactorization (n r : ℕ) : Prop :=  ∃ (A : Row n → Matrix (Fin r) (Fin r) ℝ)    (B : PerfectMatching n → Matrix (Fin r) (Fin r) ℝ),    (∀ i, (A i).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧    ∀ i M, slack i M = Matrix.trace (A i * B M)

definition OAI.PerfectMatchingPSD.psdRank line 41

noncomputable def psdRank (n : ℕ) : ℕ :=  sInf {r : ℕ | 0 < r ∧ HasFactorization n r}

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: perfect, edge, ordered, uniformly, counts, matching, one, positive and two more.

Corollary A.4 of Exponential PSD rank of positively shifted matching matrices

Corollary (Three-cell smoothing)

For every fixed K>0, there are an integer d\ge1 and N_0 with the following property. Let N\ge N_0 be even, and let a,b,c be positive integers summing to N. For a uniform ordered partition into cells of these sizes, prescribe x,y,z edges between the first and second, first and third, and second and third cells, respectively. Assume that x,y,z are nonnegative integers and that

a-x-y,\qquad b-x-z,\qquad c-y-z

are nonnegative even integers. Let T_{x,y,z} average functions on perfect matchings subject to these counts, and let \Pi_d project onto the edge-product span of degree at most d, using uniform measures as above. Then

\normop{T_{x,y,z}(I-\Pi_d)}\le N^{-K}

uniformly in all the sizes and admissible triples. For every fixed partition and matching function f, the values T_{x,y,z}\Pi_df are given by one polynomial of total degree at most d in (x,y,z) across all admissible triples.

Appendix, Uniform smoothing of matching averages Read it in the paper TeX source, line 324

Show the TeX

For every fixed $K>0$, there are an integer $d\ge1$ and $N_0$ with the following property. Let $N\ge N_0$ be even, and let $a,b,c$ be positive integers summing to $N$. For a uniform ordered partition into cells of these sizes, prescribe $x,y,z$ edges between the first and second, first and third, and second and third cells, respectively. Assume that $x,y,z$ are nonnegative integers and that
\[
 a-x-y,\qquad b-x-z,\qquad c-y-z
\]
are nonnegative even integers. Let $T_{x,y,z}$ average functions on perfect matchings subject to these counts, and let $\Pi_d$ project onto the edge-product span of degree at most $d$, using uniform measures as above. Then
\[
 \normop{T_{x,y,z}(I-\Pi_d)}\le N^{-K}
\]
uniformly in all the sizes and admissible triples. For every fixed partition and matching function $f$, the values $T_{x,y,z}\Pi_df$ are given by one polynomial of total degree at most $d$ in $(x,y,z)$ across all admissible triples.

Other possible paper matches, closest wording first

Corollary 1.2 of Exponential PSD rank of positively shifted matching matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: rank, psd, least, real and even.
Theorem 1.1 of Exponential PSD rank of positively shifted matching matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: even. Found in the paper’s title, which its main theorem states: psd, rank and matching.
Theorem 4.1 of Exponential PSD rank of positively shifted matching matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: matrix, one, least, real, positive and integers.
Theorem A.1 of Exponential PSD rank of positively shifted matching matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: counts, positive, integer and even.
Proposition 5.2 of Exponential PSD rank of positively shifted matching matrices. Words of the Lean statement, its names split into words, that also appear in the theorem: real and even.

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 142 s; peak memory 6.4 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 126’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: “weaker-statement”.
The challenge statements do not imply the headline claim: they state a nontrivially weaker statement. Their record
Their note: Superpolynomial (n^C for every C) lower bounds for exact affine SDP lifts and the unshifted PSD rank only; exponential size and the fixed-shift (0<rho<1) PSD-rank statements are not stated (superpolynomial still answers Rothvoss' polynomial-size question).
Definitions the group asks readers to compare with the paper: psdRank n := sInf {r | 0 < r ∧ HasFactorization n r} over real PSD factors of the (unshifted) matching slack matrix with slack defined on edge and odd-cut rows (Crosses, crossingCount); IsPerfectMatching by exactly one incident edge per vertex
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: Exponential (2^{Ω(n)}) lower bounds and the shifted matrices (0 < ρ < 1) are not stated; only superpolynomial bounds for the unshifted slack matrix and for affine lifts.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“These selected statements give superpolynomial growth. They do not state the paper's exponential bound for every fixed positive shift.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

55 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: 17 to its own declarations, 23 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.

Definitions and lemmas named by the statement

If Comparator accepts a proof, it proves 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 11 other declarations of the file; with the theorem itself they span 30 lines. 1 of them is named in the statement itself.

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

abbrev Edgeline 7library
def IsPerfectMatchingline 9library
abbrev PerfectMatchingline 12library
abbrev OddCutline 15library
abbrev Rowline 17library
def Crossesline 19library
def crossingCountline 22library
def slackline 27library
def HasFactorizationline 34library
def psdRankline 41library
def MainClaimline 46in the statementlibrary
theorem mainline 50selected theoremlibrary

import Mathlibnamespace OAInamespace PerfectMatchingPSDabbrev Edge (n : ℕ) := {p : Fin n × Fin n // p.1 < p.2}def IsPerfectMatching {n : ℕ} (M : Finset (Edge n)) : Prop :=  ∀ v : Fin n, (M.filter fun e => e.1.1 = v ∨ e.1.2 = v).card = 1abbrev PerfectMatching (n : ℕ) :=  {M : Finset (Edge n) // IsPerfectMatching M}abbrev OddCut (n : ℕ) := {U : Finset (Fin n) // Odd U.card}abbrev Row (n : ℕ) := Edge n ⊕ OddCut ndef Crosses {n : ℕ} (U : Finset (Fin n)) (e : Edge n) : Prop :=  (e.1.1 ∈ U ∧ e.1.2 ∉ U) ∨ (e.1.1 ∉ U ∧ e.1.2 ∈ U)noncomputable def crossingCount {n : ℕ} (U : Finset (Fin n))    (M : PerfectMatching n) : ℕ := by  classical  exact (M.1.filter (Crosses U)).cardnoncomputable def slack {n : ℕ} (i : Row n) (M : PerfectMatching n) : ℝ := by  classical  exact match i with  | Sum.inl e => if e ∈ M.1 then 1 else 0  | Sum.inr U => (crossingCount U.1 M : ℝ) - 1/-- Real PSD factors of order `r`, without equivariance or rank-one restrictions. -/def HasFactorization (n r : ℕ) : Prop :=  ∃ (A : Row n → Matrix (Fin r) (Fin r) ℝ)    (B : PerfectMatching n → Matrix (Fin r) (Fin r) ℝ),    (∀ i, (A i).PosSemidef) ∧ (∀ M, (B M).PosSemidef) ∧    ∀ i M, slack i M = Matrix.trace (A i * B M)/-- The least positive order of a real PSD factorization. -/noncomputable def psdRank (n : ℕ) : ℕ :=  sInf {r : ℕ | 0 < r ∧ HasFactorization n r}/-- Every positive real power is eventually smaller than the PSD rank,uniformly over all even vertex counts. The exponentiation is real `rpow`. -/def MainClaim : Prop :=  ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧    ∀ n : ℕ, n₀ ≤ n → Even n → (n : ℝ) ^ C < (psdRank n : ℝ)theorem main : MainClaim := by  sorryend PerfectMatchingPSDend 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.

Exponential PSD rank of positively shifted matching matrices

Theorem

There is a constant c>0 such that, for every fixed 0<\rho<1, there is n_0=n_0(\rho) for which

\rankpsd A_n(\rho)\ge2^{cn}

for every even n\ge n_0.

Section 1, Introduction Read it in the paper TeX source, line 37

Show the TeX

There is a constant $c>0$ such that, for every fixed $0<\rho<1$,
there is $n_0=n_0(\rho)$ for which
\[
 \rankpsd A_n(\rho)\ge2^{cn}
\]
for every even $n\ge n_0$.

Corollary (Unshifted rank and exact lifts)

There are constants c_0>0 and n_1 such that, for every even n\ge n_1, both A_n(0) and T_n(0) have real PSD rank at least 2^{c_0n}, and every exact semidefinite lift of P_n has size at least 2^{c_0n}.

Section 1, Introduction Read it in the paper TeX source, line 74

Show the TeX

There are constants $c_0>0$ and $n_1$ such that, for every even
$n\ge n_1$, both $A_n(0)$ and $T_n(0)$ have real PSD rank at least
$2^{c_0n}$, and every exact semidefinite lift of $P_n$ has size at
least $2^{c_0n}$.

Configuration and file facts

File: lean/ComparatorChallenges/MatchingPSD.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 126 link it.
Solution module: OAI.Combinatorics.MatchingPSD.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: 11 local vocabulary declarations
Lines of Lean, without comments or blank lines: 30
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