Lean statement

No infinite cluster at the critical probability

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.CriticalPercolation.BondGraph.no_percolation_at_criticality

Lean statement file: CriticalPercolation.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 213 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 no_percolation_at_criticality [Infinite V]    (hconnected : G.fullGraph.Connected)    (hlocal : G.LocallyFinite)    (hquasi : G.QuasiTransitive)    (hpc : G.criticalProbability < 1) :    G.law G.criticalProbability G.percolates = 0 := 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.CriticalPercolation.BondGraph line 10

structure BondGraph (V E : Type*) where  ends : E → Sym2 V

definition OAI.CriticalPercolation.BondGraph.openGraph line 17

def openGraph (ω : Set E) : SimpleGraph V :=  SimpleGraph.fromEdgeSet (G.ends '' ω)

definition OAI.CriticalPercolation.BondGraph.fullGraph line 21
```
def fullGraph : SimpleGraph V := G.openGraph Set.univ
```

definition OAI.CriticalPercolation.BondGraph.LocallyFinite line 24

def LocallyFinite : Prop := ∀ v : V, {e : E | v ∈ G.ends e}.Finite

structure OAI.CriticalPercolation.BondGraph.Aut line 27

structure Aut where  vertex : V ≃ V  edge : E ≃ E  preserves : ∀ e, G.ends (edge e) = Sym2.map vertex (G.ends e)

definition OAI.CriticalPercolation.BondGraph.QuasiTransitive line 33

def QuasiTransitive : Prop :=  ∃ S : Set V, S.Finite ∧ ∀ v : V, ∃ s ∈ S, ∃ φ : G.Aut, φ.vertex s = v

definition OAI.CriticalPercolation.BondGraph.cluster line 37

def cluster (ω : Set E) (v : V) : Set V := {w | (G.openGraph ω).Reachable v w}

definition OAI.CriticalPercolation.BondGraph.percolates line 40

def percolates : Set (Set E) := {ω | ∃ v : V, (G.cluster ω v).Infinite}

definition OAI.CriticalPercolation.BondGraph.law line 43

noncomputable def law (_G : BondGraph V E) (p : unitInterval) : Measure (Set E) :=  ProbabilityTheory.setBernoulli Set.univ p

definition OAI.CriticalPercolation.BondGraph.criticalProbability line 47

noncomputable def criticalProbability : unitInterval :=  sInf {p : unitInterval | 0 < G.law p G.percolates}

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: neighbor, parameter, critical, bernoulli, bond, cluster, percolation and infinite.

Statements in Critical bond and site percolation on the cubic lattice come first because the Lean file and module names share words with that paper’s title.

Theorem 1.1 of Critical bond and site percolation on the cubic lattice

Theorem

For nearest-neighbor Bernoulli bond percolation on \mathbb Z^3, there is almost surely no infinite open cluster at the bond critical parameter. The same assertion holds for nearest-neighbor Bernoulli site percolation at the site critical parameter.

Section 1, Introduction Read it in the paper TeX source, line 17

Show the TeX

For nearest-neighbor Bernoulli bond percolation on $\mathbb Z^3$, there is
almost surely no infinite open cluster at the bond critical parameter.
The same assertion holds for nearest-neighbor Bernoulli site percolation
at the site critical parameter.

Other possible paper matches, closest wording first

Proposition 4.1 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: distinct, least, simple, positive, event, path and ten more.
Theorem 1.1 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: undirected, quasi, transitive, vertex, connected, locally and four more. Found in the paper’s title, which its main theorem states: criticality and percolation.
Theorem 7.1 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: automorphisms, criticality, orbits, group, finitely, many and nine more.
Proposition 7.2 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: aut, orbits, group, finitely, quasi, many and seven more.
Proposition 8.1 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: map, one, event, automorphisms, orbits, edge and twelve more.
Theorem 2.2 of No percolation at criticality on quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: quasi, transitive, critical, bernoulli, connected, locally and six more.

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, 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 213’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.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: no_percolation_at_criticality [Infinite V] (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) (hpc : G.criticalProbability < 1) : G.law G.criticalProbability G.percolates = 0 on BondGraph V E (multigraphs, law p = setBernoulli univ p, criticalProbability = sInf {p | 0 < P_p(∃ infinite cluster)}), plus CriticalZ3.critical_no_infinite: ∀ᵐ ω ∂bondLaw bondCritical, ∀ x, (bondCluster ω x).Finite and the same for site percolation, with bondCritical = sInf {p | 0 < P_p(origin cluster infinite)}. Both summary claims are stated.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves absence of an infinite cluster at criticality for Bernoulli bond percolation on every infinite connected locally finite quasi-transitive graph with critical probability $p_c<1$. At $p=p_c$, the probability that any infinite cluster exists is zero. The graph model permits bond multiplicities.”
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

63 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: 2 to its own declarations, 21 to Mathlib’s documentation. 11 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 no_percolation_at_criticality 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 10 other declarations of the file; with the theorem itself they span 25 lines. 6 of them are named in the statement itself.

Names in the namespace OAI.CriticalPercolation.BondGraph are listed without it.

structure OAI.CriticalPercolation.BondGraphline 10library
def openGraphline 17library
def fullGraphline 21in the statementlibrary
def LocallyFiniteline 24in the statementlibrary
structure Autline 27library
def QuasiTransitiveline 33in the statementlibrary
def clusterline 37library
def percolatesline 40in the statementlibrary
def lawline 43in the statementlibrary
def criticalProbabilityline 47in the statementlibrary
theorem no_percolation_at_criticalityline 52selected theoremlibrary

import Mathlibnamespace OAIopen Set MeasureTheory ProbabilityTheorynamespace CriticalPercolation/-- An undirected graph with labelled bonds, including loops and parallel bonds. -/structure BondGraph (V E : Type*) where  ends : E → Sym2 Vnamespace BondGraphvariable {V E : Type*} (G : BondGraph V E)/-- The graph of retained bonds. `fromEdgeSet` discards loops. -/def openGraph (ω : Set E) : SimpleGraph V :=  SimpleGraph.fromEdgeSet (G.ends '' ω)/-- The underlying connectivity graph with every bond retained. -/def fullGraph : SimpleGraph V := G.openGraph Set.univ/-- Local finiteness counts bonds, not just distinct neighbors. -/def LocallyFinite : Prop := ∀ v : V, {e : E | v ∈ G.ends e}.Finite/-- Automorphisms preserve the vertex and bond structure, including multiplicities. -/structure Aut where  vertex : V ≃ V  edge : E ≃ E  preserves : ∀ e, G.ends (edge e) = Sym2.map vertex (G.ends e)/-- Finitely many vertex orbits under the full automorphism group. -/def QuasiTransitive : Prop :=  ∃ S : Set V, S.Finite ∧ ∀ v : V, ∃ s ∈ S, ∃ φ : G.Aut, φ.vertex s = v/-- Finite open paths, including the path of length zero, define the open cluster. -/def cluster (ω : Set E) (v : V) : Set V := {w | (G.openGraph ω).Reachable v w}/-- The event that at least one open cluster is infinite. -/def percolates : Set (Set E) := {ω | ∃ v : V, (G.cluster ω v).Infinite}/-- Independent retention with parameter `p ∈ [0,1]` of each labelled bond. -/noncomputable def law (_G : BondGraph V E) (p : unitInterval) : Measure (Set E) :=  ProbabilityTheory.setBernoulli Set.univ p/-- The infimum of parameters with positive probability of an infinite cluster. -/noncomputable def criticalProbability : unitInterval :=  sInf {p : unitInterval | 0 < G.law p G.percolates}/-- Critical Bernoulli bond percolation has no infinite cluster when the criticalprobability is below one. -/theorem no_percolation_at_criticality [Infinite V]    (hconnected : G.fullGraph.Connected)    (hlocal : G.LocallyFinite)    (hquasi : G.QuasiTransitive)    (hpc : G.criticalProbability < 1) :    G.law G.criticalProbability G.percolates = 0 := by  sorryend BondGraphend CriticalPercolationend 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.

Critical bond and site percolation on the cubic lattice

Theorem

Section 1, Introduction Read it in the paper TeX source, line 17

Show the TeX

For nearest-neighbor Bernoulli bond percolation on $\mathbb Z^3$, there is
almost surely no infinite open cluster at the bond critical parameter.
The same assertion holds for nearest-neighbor Bernoulli site percolation
at the site critical parameter.

No percolation at criticality on quasi-transitive graphs

Theorem

Let G be an infinite connected locally finite quasi-transitive undirected graph. If \pc(G)<1, then

\PP_{\pc(G)}(\text{there is an infinite open cluster})=0.

Equivalently, \PP_{\pc(G)}(|C_x|=\infty)=0 for every vertex x.

Section 1, Introduction Read it in the paper TeX source, line 26

Show the TeX

Let $G$ be an infinite connected locally finite quasi-transitive
undirected graph. If $\pc(G)<1$, then
\[
 \PP_{\pc(G)}(\text{there is an infinite open cluster})=0.
\]
Equivalently, $\PP_{\pc(G)}(|C_x|=\infty)=0$ for every vertex $x$.

Configuration and file facts

File: lean/ComparatorChallenges/CriticalPercolation.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 213 link it.
Solution module: OAI.Probability.CriticalPercolation.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: 10 local vocabulary declarations
Lines of Lean, without comments or blank lines: 25
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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