Lean statement

Counterexample to Sidorenko’s conjecture

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Counterexample to Sidorenko's conjecture

Theorem selected for verification: OAI.SidorenkoCounterexample.main

Lean statement file: SidorenkoCounterexample.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 161 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 main : ∃ size : ℕ, 0 < size ∧ ∃ host : SimpleGraph (Fin size),    (∃ left right, host.Adj left right) ∧ homDensity H host < (edgeDensity host) ^ 66 := 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.SidorenkoCounterexample.faces line 13

def faces : Fin 22 → Finset (Fin 13) := ![  {0,1,3}, {0,1,9}, {0,2,3}, {0,2,9}, {1,3,10}, {1,7,10},  {1,7,12}, {1,9,12}, {2,3,4}, {2,4,9}, {3,4,11}, {3,10,11},  {4,8,9}, {4,8,11}, {5,6,7}, {5,6,11}, {5,7,10}, {5,10,11},  {6,7,12}, {6,8,11}, {6,8,12}, {8,9,12}]

abbreviation OAI.SidorenkoCounterexample.PatternVertex line 19
```
abbrev PatternVertex := Fin 13 ⊕ Fin 22
```

definition OAI.SidorenkoCounterexample.incidence line 21

def incidence : PatternVertex → PatternVertex → Prop  | .inl point, .inr face => point ∈ faces face  | .inr face, .inl point => point ∈ faces face  | _, _ => False

definition OAI.SidorenkoCounterexample.H line 26

def H : SimpleGraph PatternVertex where  Adj := incidence  symm.symm := by intro left right; cases left <;> cases right <;> simp [incidence]  loopless.irrefl := by intro vertex; cases vertex <;> simp [incidence]

definition OAI.SidorenkoCounterexample.homCount line 31

noncomputable def homCount {V W : Type*} (F : SimpleGraph V) (G : SimpleGraph W) : ℕ :=  Nat.card {hom : V → W // ∀ left right, F.Adj left right → G.Adj (hom left) (hom right)}

definition OAI.SidorenkoCounterexample.homDensity line 34

noncomputable def homDensity {V W : Type*} (F : SimpleGraph V) (G : SimpleGraph W) : ℝ :=  (homCount F G : ℝ) / (Nat.card W : ℝ) ^ Nat.card V

definition OAI.SidorenkoCounterexample.edgeDensity line 37

noncomputable def edgeDensity {W : Type*} (G : SimpleGraph W) : ℝ :=  2 * (Nat.card G.edgeSet : ℝ) / (Nat.card W : ℝ) ^ 2

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: edge, least, simple, points, incidence, one, finite and graph. Found in the paper’s title, which its main theorem states: sidorenkos. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: edges, vertices and bipartite.

Theorem 1.1 of A counterexample to Sidorenko’s conjecture

Theorem

Let H be the incidence graph of the 22 triples on 13 points in Table tab:complex. This bipartite graph has 35 vertices and 66 edges. There is a finite simple undirected graph G with at least one edge such that

t(H,G)<p(G)^{66}.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

Let \(H\) be the incidence graph of the \(22\) triples on \(13\) points
in Table~\ref{tab:complex}. This bipartite graph has \(35\) vertices
and \(66\) edges. There is a finite simple undirected graph \(G\)
with at least one edge such that
\[
t(H,G)<p(G)^{66}.
\]

Other possible paper matches, closest wording first

Corollary 1.3 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: density and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: forcing, conjecture and bipartite.
Proposition 2.1 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: faces, least, face, points, one and graph. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: separate and vertices.
Corollary 1.2 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: simple, finite and graph.
Proposition 4.7 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: patterns and incidence.
Proposition 6.3 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: integer, face, point, one, incidences and finite.
Proposition 3.5 of A counterexample to Sidorenko’s conjecture. Words of the Lean statement, its names split into words, that also appear in the theorem: finite.

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 349 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 161’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: main : ∃ size > 0, ∃ host : SimpleGraph (Fin size), (host has an edge) ∧ homDensity H host < (edgeDensity host)^66 for a fixed explicit H = incidence graph of 22 triples on 13 points (checked by script: 35 vertices, 66 edges, connected, bipartite by construction), with homomorphism counts including non-injective maps and edgeDensity = 2e/n² = t(K₂,G): a faithful Sidorenko counterexample. The second named claim (the same pattern disproves the Skokan-Thoma forcing conjecture) has no statement (docs: 'outside this statement'); connectivity and the counts 35/66 are properties of the defined H, not stated.
Definitions the group asks readers to compare with the paper: homDensity F G := homCount F G / (Nat.card W)^(Nat.card V) with homCount := Nat.card {hom : V → W // ∀ l r, F.Adj l r → G.Adj (hom l) (hom r)} and edgeDensity G := 2 * Nat.card G.edgeSet / (Nat.card W)^2: custom but the standard t(H,G) and t(K₂,G) (checked)
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: The forcing-conjecture sentence.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization disproves this for the paper's fixed bipartite graph with $35$ vertices and $66$ edges: it constructs a nonempty finite simple host graph with $t(H,G)<t(K_2,G)^{66}$. The separate forcing-conjecture consequence in the paper is outside this statement.”
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

47 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, 19 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 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 7 other declarations of the file; with the theorem itself they span 23 lines. 3 of them are named in the statement itself.

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

def facesline 13library
abbrev PatternVertexline 19library
def incidenceline 21library
def Hline 26in the statementlibrary
def homCountline 31library
def homDensityline 34in the statementlibrary
def edgeDensityline 37in the statementlibrary
theorem mainline 41selected theoremlibrary

import Mathlibnamespace OAI/-! # A finite counterexample to Sidorenko's conjectureThe pattern is the incidence graph of the 22 triples in Table 1 of the source.Homomorphism density counts all edge-preserving maps, including noninjective maps.-/namespace SidorenkoCounterexampledef faces : Fin 22 → Finset (Fin 13) := ![  {0,1,3}, {0,1,9}, {0,2,3}, {0,2,9}, {1,3,10}, {1,7,10},  {1,7,12}, {1,9,12}, {2,3,4}, {2,4,9}, {3,4,11}, {3,10,11},  {4,8,9}, {4,8,11}, {5,6,7}, {5,6,11}, {5,7,10}, {5,10,11},  {6,7,12}, {6,8,11}, {6,8,12}, {8,9,12}]abbrev PatternVertex := Fin 13 ⊕ Fin 22def incidence : PatternVertex → PatternVertex → Prop  | .inl point, .inr face => point ∈ faces face  | .inr face, .inl point => point ∈ faces face  | _, _ => Falsedef H : SimpleGraph PatternVertex where  Adj := incidence  symm.symm := by intro left right; cases left <;> cases right <;> simp [incidence]  loopless.irrefl := by intro vertex; cases vertex <;> simp [incidence]noncomputable def homCount {V W : Type*} (F : SimpleGraph V) (G : SimpleGraph W) : ℕ :=  Nat.card {hom : V → W // ∀ left right, F.Adj left right → G.Adj (hom left) (hom right)}noncomputable def homDensity {V W : Type*} (F : SimpleGraph V) (G : SimpleGraph W) : ℝ :=  (homCount F G : ℝ) / (Nat.card W : ℝ) ^ Nat.card Vnoncomputable def edgeDensity {W : Type*} (G : SimpleGraph W) : ℝ :=  2 * (Nat.card G.edgeSet : ℝ) / (Nat.card W : ℝ) ^ 2/-- A finite simple host with at least one edge violates Sidorenko's inequality. -/theorem main : ∃ size : ℕ, 0 < size ∧ ∃ host : SimpleGraph (Fin size),    (∃ left right, host.Adj left right) ∧ homDensity H host < (edgeDensity host) ^ 66 := by  sorryend SidorenkoCounterexampleend 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 counterexample to Sidorenko’s conjecture

Theorem

t(H,G)<p(G)^{66}.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

Let \(H\) be the incidence graph of the \(22\) triples on \(13\) points
in Table~\ref{tab:complex}. This bipartite graph has \(35\) vertices
and \(66\) edges. There is a finite simple undirected graph \(G\)
with at least one edge such that
\[
t(H,G)<p(G)^{66}.
\]

Corollary (Amplification and injective supersaturation)

For the same graph H as in Theorem thm:main, the following hold.

(i)
For every c>0, there is a finite simple undirected graph F with p(F)>0 such that
t(H,F)<c\,p(F)^{66}.
(ii)
There are p_*\in(0,1), \delta_*>0, and finite simple undirected graphs F_s of orders n_s\to\infty for which d_s=|E(F_s)|/\binom{n_s}{2}\to p_* and, for all sufficiently large s,
\operatorname{Inj}(H,F_s) \le(1-\delta_*)(n_s)_{35}d_s^{66}.

Section 1, Introduction Read it in the paper TeX source, line 279

Show the TeX

For the same graph \(H\) as in Theorem~\ref{thm:main}, the following
hold.
\begin{enumerate}[label=(\roman*)]
\item For every \(c>0\), there is a finite simple undirected graph
\(F\) with \(p(F)>0\) such that
\[
t(H,F)<c\,p(F)^{66}.
\]
\item There are \(p_*\in(0,1)\), \(\delta_*>0\), and finite simple
undirected graphs \(F_s\) of orders \(n_s\to\infty\) for which
\(d_s=|E(F_s)|/\binom{n_s}{2}\to p_*\) and, for all sufficiently
large \(s\),
\[
\operatorname{Inj}(H,F_s)
\le(1-\delta_*)(n_s)_{35}d_s^{66}.
\]
\end{enumerate}

Corollary (Failure of the forcing conjecture)

For the same connected bipartite graph H of Theorem thm:main, there are p\in(0,1) and a nonconstant graphon W such that

t(K_2,W)=p,\qquad t(H,W)=p^{66}.

Consequently, H is not p-forcing at this density and hence is not forcing, despite being bipartite and containing a cycle.

Section 1, Introduction Read it in the paper TeX source, line 335

Show the TeX

For the same connected bipartite graph \(H\) of
Theorem~\ref{thm:main}, there are \(p\in(0,1)\) and a nonconstant
graphon \(W\) such that
\[
t(K_2,W)=p,\qquad t(H,W)=p^{66}.
\]
Consequently, \(H\) is not \(p\)-forcing at this density and hence is
not forcing, despite being bipartite and containing a cycle.

Configuration and file facts

File: lean/ComparatorChallenges/SidorenkoCounterexample.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 161 link it.
Solution module: OAI.Combinatorics.Sidorenko.RandomHost
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: 23
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.

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