Lean statement

Superpolynomial lower bound for affine semidefinite lifts

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.affine_lift_lower_bound

Lean statement file: MatchingAffineLift.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.

theorem affine_lift_lower_bound :    ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧      ∀ n : ℕ, n₀ ≤ n → Even n → ∀ r : ℕ,        HasAffineLift n r → (n : ℝ) ^ C < (r : ℝ) := 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}

definition OAI.PerfectMatchingPSD.matchingPoint line 15

noncomputable def matchingPoint {n : ℕ} (M : PerfectMatching n) : Edge n → ℝ := by  classical  exact fun e => if e ∈ M.1 then 1 else 0

definition OAI.PerfectMatchingPSD.matchingPolytope line 19

noncomputable def matchingPolytope (n : ℕ) : Set (Edge n → ℝ) :=  convexHull ℝ (Set.range (matchingPoint (n := n)))

definition OAI.PerfectMatchingPSD.HasAffineLift line 22

def HasAffineLift (n r : ℕ) : Prop :=  ∃ (L : AffineSubspace ℝ (Matrix (Fin r) (Fin r) ℝ))    (T : Matrix (Fin r) (Fin r) ℝ →ᵃ[ℝ] (Edge n → ℝ)),    T '' {X | X ∈ L ∧ X.PosSemidef} = matchingPolytope 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: even, exact, lift, lifts, rank, semidefinite and size.

Corollary 1.2 of Exponential PSD rank of positively shifted matching matrices

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 $2^{c_0n}$.

Other possible paper matches, closest wording first

Corollary A.4 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: perfect, edge, matching, integers and even. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
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, real and integers. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: semidefinite and positive.
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: matching.
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: integer and even. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive.
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

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. 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

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, Classical.choice, Quot.sound

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

35 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: 10 to its own declarations, 16 to Mathlib’s documentation. 7 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 affine_lift_lower_bound 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 6 other declarations of the file; with the theorem itself they span 19 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
def matchingPointline 15library
def matchingPolytopeline 19library
def HasAffineLiftline 22in the statementlibrary
theorem affine_lift_lower_boundline 27selected 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}noncomputable def matchingPoint {n : ℕ} (M : PerfectMatching n) : Edge n → ℝ := by  classical  exact fun e => if e ∈ M.1 then 1 else 0noncomputable def matchingPolytope (n : ℕ) : Set (Edge n → ℝ) :=  convexHull ℝ (Set.range (matchingPoint (n := n)))def HasAffineLift (n r : ℕ) : Prop :=  ∃ (L : AffineSubspace ℝ (Matrix (Fin r) (Fin r) ℝ))    (T : Matrix (Fin r) (Fin r) ℝ →ᵃ[ℝ] (Edge n → ℝ)),    T '' {X | X ∈ L ∧ X.PosSemidef} = matchingPolytope ntheorem affine_lift_lower_bound :    ∀ C : ℝ, 0 < C → ∃ n₀ : ℕ, 4 ≤ n₀ ∧      ∀ n : ℕ, n₀ ≤ n → Even n → ∀ r : ℕ,        HasAffineLift n r → (n : ℝ) ^ C < (r : ℝ) := 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/MatchingAffineLift.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.AffineLift
Allowed axioms: propext, Classical.choice, Quot.soundThe 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: 6 local vocabulary declarations
Lines of Lean, without comments or blank lines: 19
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