Lean statement

Uniform sparsest-cut integrality gaps

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.UniformSparsestCut.mainGap

Lean statement file: UniformSparsestCut.leanShow on the mapFamily 117. 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 mainGap :    ∃ c : ℝ, 0 < c ∧      ∃ n : ℕ → ℕ, ∃ C : (j : ℕ) → Capacity (n j),        (∀ j, 2 ≤ n j) ∧        Tendsto n atTop atTop ∧        (∀ j, 0 < glValue (C j)) ∧        ∀ᶠ j in atTop,          c * Real.sqrt (Real.log (n j : ℝ)) / (Real.log (Real.log (n j : ℝ))) ^ 3 ≤            OPT (C j) / glValue (C j) := 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.

structure OAI.UniformSparsestCut.Capacity line 11

structure Capacity (n : ℕ) where  cap : Fin n → Fin n → ℝ  nonneg : ∀ i j, 0 ≤ cap i j  symm : ∀ i j, cap i j = cap j i  diagonal : ∀ i, cap i i = 0

definition OAI.UniformSparsestCut.pairSum line 17

def pairSum {n : ℕ} (f : Fin n → Fin n → ℝ) : ℝ :=  ∑ i, ∑ j, if i < j then f i j else 0

definition OAI.UniformSparsestCut.cutRatio line 20

noncomputable def cutRatio {n : ℕ} (C : Capacity n) (B : Finset (Fin n)) : ℝ :=  (∑ i ∈ B, ∑ j ∈ Bᶜ, C.cap i j) /    ((B.card : ℝ) * ((n : ℝ) - (B.card : ℝ)))

definition OAI.UniformSparsestCut.OPT line 24

noncomputable def OPT {n : ℕ} (C : Capacity n) : ℝ :=  sInf {r | ∃ B : Finset (Fin n), B.Nonempty ∧ B ≠ Finset.univ ∧ r = cutRatio C B}

definition OAI.UniformSparsestCut.NegativeType line 27

def NegativeType {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop :=  (∃ x : Fin n → EuclideanSpace ℝ (Fin n), ∀ i j, d i j = ‖x i - x j‖ ^ 2) ∧  ∀ i j k, d i k ≤ d i j + d j k

definition OAI.UniformSparsestCut.Feasible line 31

def Feasible {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop :=  NegativeType d ∧ pairSum d = 1

definition OAI.UniformSparsestCut.glValue line 34

noncomputable def glValue {n : ℕ} (C : Capacity n) : ℝ :=  sInf {r | ∃ d : Fin n → Fin n → ℝ, Feasible d ∧    r = pairSum (fun i j => C.cap i j * d i j)}

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: gl, opt, capacity, sqrt, log and pair. Found in the paper’s title, which its main theorem states: cut and gaps. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: one, absolute and demand.

Statements in Near-square-root logarithmic integrality gaps for uniform sparsest cut come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Near-square-root logarithmic integrality gaps for uniform sparsest cut

Theorem

There are an absolute constant c>0 and capacity instances C^{(j)} on n_j\to\infty vertices, with \GL(C^{(j)})>0, such that

\frac{\OPT(C^{(j)})}{\GL(C^{(j)})} \ge c\,\frac{\sqrt{\log n_j}}{(\log\log n_j)^3}

for all sufficiently large j. The demand is one between every pair of distinct vertices.

Section 1, Introduction Read it in the paper TeX source, line 152

Show the TeX

There are an absolute constant $c>0$ and capacity instances $C^{(j)}$
on $n_j\to\infty$ vertices, with $\GL(C^{(j)})>0$, such that
\[
\frac{\OPT(C^{(j)})}{\GL(C^{(j)})}
\ge c\,\frac{\sqrt{\log n_j}}{(\log\log n_j)^3}
\]
for all sufficiently large $j$. The demand is one between every pair
of distinct vertices.

Other possible paper matches, closest wording first

Proposition 1 of Near-square-root logarithmic integrality gaps for uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: negative, integer, sqrt and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: absolute and uniform.
Theorem 1.1 of Constant-factor hardness of uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: real, cut and capacities. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: unit, sparsest, positive, demands, size and uniform.
Proposition 2 of Near-square-root logarithmic integrality gaps for uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: absolute, bound and uniform.
Proposition 7.1 of Constant-factor hardness of uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: pair and gap. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, bound, demand and size.
Theorem 3.1 of Constant-factor hardness of uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: space and paired. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: least, one, positive, bound, size and uniform.
Theorem 6.1 of Constant-factor hardness of uniform sparsest cut. Words of the Lean statement, its names split into words, that also appear in the theorem: gap. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: bounded, demand and size.

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 399 s; peak memory 6.5 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 117’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: Only the Goemans-Linial integrality gap (real capacities, along a sequence of n); the NP-hardness of constant-factor approximation (the summary's first claim) is not stated.
Definitions the group asks readers to compare with the paper: NegativeType d is ∃ x : Fin n → EuclideanSpace ℝ (Fin n), d i j = ‖x i - x j‖^2 plus triangle inequalities (n dimensions suffice, standard); OPT/glValue are sInfs over sets that must be nonempty and bounded below (true)
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: constant-factor NP-hardness of Uniform Sparsest Cut

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“Along a sequence of instance sizes $n\to\infty$, the ratio of the integral optimum to the positive relaxation optimum is at least $c\sqrt{\log n}/(\log\log n)^3$ for one absolute $c>0$. Every unordered pair has unit demand, and the relaxation uses squared Euclidean distances satisfying the triangle inequalities. The result is an existential sequence, not a bound for every size or a computational hardness theorem.”
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

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: 12 to its own declarations, 28 to Mathlib’s documentation. 8 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 mainGap 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 7 other declarations of the file; with the theorem itself they span 30 lines. 3 of them are named in the statement itself.

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

structure Capacityline 11in the statementlibrary
def pairSumline 17library
def cutRatioline 20library
def OPTline 24in the statementlibrary
def NegativeTypeline 27library
def Feasibleline 31library
def glValueline 34in the statementlibrary
theorem mainGapline 40selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperatorsnamespace UniformSparsestCutstructure Capacity (n : ℕ) where  cap : Fin n → Fin n → ℝ  nonneg : ∀ i j, 0 ≤ cap i j  symm : ∀ i j, cap i j = cap j i  diagonal : ∀ i, cap i i = 0def pairSum {n : ℕ} (f : Fin n → Fin n → ℝ) : ℝ :=  ∑ i, ∑ j, if i < j then f i j else 0noncomputable def cutRatio {n : ℕ} (C : Capacity n) (B : Finset (Fin n)) : ℝ :=  (∑ i ∈ B, ∑ j ∈ Bᶜ, C.cap i j) /    ((B.card : ℝ) * ((n : ℝ) - (B.card : ℝ)))noncomputable def OPT {n : ℕ} (C : Capacity n) : ℝ :=  sInf {r | ∃ B : Finset (Fin n), B.Nonempty ∧ B ≠ Finset.univ ∧ r = cutRatio C B}def NegativeType {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop :=  (∃ x : Fin n → EuclideanSpace ℝ (Fin n), ∀ i j, d i j = ‖x i - x j‖ ^ 2) ∧  ∀ i j k, d i k ≤ d i j + d j kdef Feasible {n : ℕ} (d : Fin n → Fin n → ℝ) : Prop :=  NegativeType d ∧ pairSum d = 1noncomputable def glValue {n : ℕ} (C : Capacity n) : ℝ :=  sInf {r | ∃ d : Fin n → Fin n → ℝ, Feasible d ∧    r = pairSum (fun i j => C.cap i j * d i j)}open Filtertheorem mainGap :    ∃ c : ℝ, 0 < c ∧      ∃ n : ℕ → ℕ, ∃ C : (j : ℕ) → Capacity (n j),        (∀ j, 2 ≤ n j) ∧        Tendsto n atTop atTop ∧        (∀ j, 0 < glValue (C j)) ∧        ∀ᶠ j in atTop,          c * Real.sqrt (Real.log (n j : ℝ)) / (Real.log (Real.log (n j : ℝ))) ^ 3 ≤            OPT (C j) / glValue (C j) := by  sorryend UniformSparsestCutendend 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.

Near-square-root logarithmic integrality gaps for uniform sparsest cut

Theorem

There are an absolute constant c>0 and capacity instances C^{(j)} on n_j\to\infty vertices, with \GL(C^{(j)})>0, such that

\frac{\OPT(C^{(j)})}{\GL(C^{(j)})} \ge c\,\frac{\sqrt{\log n_j}}{(\log\log n_j)^3}

for all sufficiently large j. The demand is one between every pair of distinct vertices.

Section 1, Introduction Read it in the paper TeX source, line 152

Show the TeX

There are an absolute constant $c>0$ and capacity instances $C^{(j)}$
on $n_j\to\infty$ vertices, with $\GL(C^{(j)})>0$, such that
\[
\frac{\OPT(C^{(j)})}{\GL(C^{(j)})}
\ge c\,\frac{\sqrt{\log n_j}}{(\log\log n_j)^3}
\]
for all sufficiently large $j$. The demand is one between every pair
of distinct vertices.

Configuration and file facts

File: lean/ComparatorChallenges/UniformSparsestCut.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/SparsestCut/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: 7 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