Lean statement

Logarithmic independence bound for clique-free graphs

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.CliqueFreeLog.logarithmic_independence_bound

Lean statement file: CliqueFreeLog.leanShow on the mapFamily 184. 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 logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) :    ∃ c : ℝ, 0 < c ∧      ∀ {V : Type} [Fintype V] (G : SimpleGraph V),        G.CliqueFree r → 2 ≤ averageDegree G →        c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤          (G.indepNum : 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.

definition OAI.CliqueFreeLog.averageDegree line 7

noncomputable def averageDegree {V : Type*} [Fintype V] (G : SimpleGraph V) : ℝ := by  classical  exact 2 * (G.edgeFinset.card : ℝ) / (Fintype.card V : 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: average, simple, integer, degree, free, finite, log and graph. Found in the paper’s title, which its main theorem states: logarithmic, independence, clique and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vertices and constant.

Statements in A logarithmic independence bound for clique-free graphs come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of A logarithmic independence bound for clique-free graphs

Theorem

For every integer r\ge4, there is a constant c_r>0 such that every finite simple K_r-free graph G with n vertices and average degree d\ge2 satisfies

\alpha(G)\ge c_r\frac{n\log d}{d}.

Section 1, Introduction Read it in the paper TeX source, line 14

Show the TeX

For every integer $r\ge4$, there is a constant $c_r>0$ such that every
finite simple $K_r$-free graph $G$ with $n$ vertices and average degree $d\ge2$
satisfies
\[
 \alpha(G)\ge c_r\frac{n\log d}{d}.
\]

Other possible paper matches, closest wording first

Proposition 6.1 of A logarithmic independence bound for clique-free graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: real, integer, degree, free, finite, log and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: vertices.
Theorem 1.1 of Correspondence coloring graphs with a forbidden clique. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, degree, free, log and graph. Found in the paper’s title, which its main theorem states: clique. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: depend and constants.
Theorem 3.1 of A logarithmic independence bound for clique-free graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: simple, finite and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Corollary 1.2 of Correspondence coloring graphs with a forbidden clique. Words of the Lean statement, its names split into words, that also appear in the theorem: degree, free, finite, log and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constants.
Proposition 5.4 of A logarithmic independence bound for clique-free graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: finite, log and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: constant.
Proposition 2.5 of Correspondence coloring graphs with a forbidden clique. Words of the Lean statement, its names split into words, that also appear in the theorem: bound and graphs. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: depends and constant.

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 358 s; peak memory 6.3 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 184’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: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Only the independence half is stated: logarithmic_independence_bound (r) (hr : 4 ≤ r) : ∃ c > 0, ∀ G, G.CliqueFree r → 2 ≤ averageDegree G → c * n * log d / d ≤ G.indepNum (averageDegree = 2|E|/|V|). The summary's first and primary claim, the Alon-Krivelevich-Sudakov coloring conjecture in correspondence-coloring form (O_F(Δ/log Δ) colors for any fixed forbidden subgraph F), has no Lean statement, and the docs scope also only mentions the independence bound.
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNLINKED”.
The challenge statements state less than the headline; the missing part comes from a paper the Lean note does not link. Their record
Their note: The correspondence-coloring theorem (χ_DP = O_F(Δ/log Δ) for F-free graphs of large maximum degree), the headline's first clause, is not stated.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For every fixed integer $r\ge4$, there is $c_r>0$ such that every finite $K_r$-free simple graph on $n$ vertices with average degree $d\ge2$ has an independent set of size at least $c_r n\log d/d$. The constant depends only on $r$.”
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

21 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: 3 to its own declarations, 7 to Mathlib’s documentation. 2 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 logarithmic_independence_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 the file’s only other declaration; with the theorem itself they span 10 lines. 1 of them is named in the statement itself.

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

def averageDegreeline 7in the statementlibrary
theorem logarithmic_independence_boundline 11selected theoremlibrary

import Mathlibnamespace OAInamespace CliqueFreeLognoncomputable def averageDegree {V : Type*} [Fintype V] (G : SimpleGraph V) : ℝ := by  classical  exact 2 * (G.edgeFinset.card : ℝ) / (Fintype.card V : ℝ)theorem logarithmic_independence_bound (r : ℕ) (hr : 4 ≤ r) :    ∃ c : ℝ, 0 < c ∧      ∀ {V : Type} [Fintype V] (G : SimpleGraph V),        G.CliqueFree r → 2 ≤ averageDegree G →        c * (Fintype.card V : ℝ) * Real.log (averageDegree G) / averageDegree G ≤          (G.indepNum : ℝ) := by  sorryend CliqueFreeLogend 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.

A logarithmic independence bound for clique-free graphs

Theorem

For every integer r\ge4, there is a constant c_r>0 such that every finite simple K_r-free graph G with n vertices and average degree d\ge2 satisfies

\alpha(G)\ge c_r\frac{n\log d}{d}.

Section 1, Introduction Read it in the paper TeX source, line 14

Show the TeX

For every integer $r\ge4$, there is a constant $c_r>0$ such that every
finite simple $K_r$-free graph $G$ with $n$ vertices and average degree $d\ge2$
satisfies
\[
 \alpha(G)\ge c_r\frac{n\log d}{d}.
\]

Configuration and file facts

File: lean/ComparatorChallenges/CliqueFreeLog.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Combinatorics/CliqueFree/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: 1 local vocabulary declaration
Lines of Lean, without comments or blank lines: 10
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