Lean statement

No infinite critical clusters on $\mathbb Z^3$

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.CriticalZ3.critical_no_infinite

Lean statement file: CriticalZ3.leanShow on the mapFamily 213. 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 critical_no_infinite :    (∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite) ∧    (∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite) := 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.CriticalZ3.Vertex line 9
```
abbrev Vertex := Fin 3 → ℤ
```
abbreviation OAI.CriticalZ3.Bond line 10
```
abbrev Bond := Vertex × Fin 3
```
abbreviation OAI.CriticalZ3.BondConfig line 11
```
abbrev BondConfig := Bond → Bool
```
abbreviation OAI.CriticalZ3.SiteConfig line 12
```
abbrev SiteConfig := Vertex → Bool
```

definition OAI.CriticalZ3.step line 14

def step (x : Vertex) (i : Fin 3) : Vertex := fun j => x j + if i = j then 1 else 0

definition OAI.CriticalZ3.NearestNeighbor line 16

def NearestNeighbor (x y : Vertex) : Prop :=  ∃ i : Fin 3, y = step x i ∨ x = step y i

definition OAI.CriticalZ3.BondOpen line 19

def BondOpen (ω : BondConfig) (x y : Vertex) : Prop :=  (∃ i : Fin 3, y = step x i ∧ ω (x, i) = true) ∨  (∃ i : Fin 3, x = step y i ∧ ω (y, i) = true)

definition OAI.CriticalZ3.SiteOpen line 23

def SiteOpen (ω : SiteConfig) (x y : Vertex) : Prop :=  NearestNeighbor x y ∧ ω x = true ∧ ω y = true

definition OAI.CriticalZ3.BondConnected line 26

def BondConnected (ω : BondConfig) (x y : Vertex) : Prop :=  Relation.ReflTransGen (BondOpen ω) x y

definition OAI.CriticalZ3.SiteConnected line 29

def SiteConnected (ω : SiteConfig) (x y : Vertex) : Prop :=  ω x = true ∧ Relation.ReflTransGen (SiteOpen ω) x y

definition OAI.CriticalZ3.bondCluster line 32

def bondCluster (ω : BondConfig) (x : Vertex) : Set Vertex :=  {y | BondConnected ω x y}

definition OAI.CriticalZ3.siteCluster line 35

def siteCluster (ω : SiteConfig) (x : Vertex) : Set Vertex :=  {y | SiteConnected ω x y}

definition OAI.CriticalZ3.bondInfiniteAt line 38

def bondInfiniteAt (x : Vertex) : Set BondConfig := {ω | (bondCluster ω x).Infinite}

definition OAI.CriticalZ3.siteInfiniteAt line 39

def siteInfiniteAt (x : Vertex) : Set SiteConfig := {ω | (siteCluster ω x).Infinite}

definition OAI.CriticalZ3.parameter line 41

noncomputable def parameter (p : ℝ) : Set.Icc (0 : ℝ) 1 :=  ⟨max 0 (min 1 p), le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩

definition OAI.CriticalZ3.bondLaw line 44

noncomputable def bondLaw (p : ℝ) : Measure BondConfig :=  Measure.infinitePi (fun _ : Bond => bernoulliMeasure true false (parameter p))

definition OAI.CriticalZ3.siteLaw line 47

noncomputable def siteLaw (p : ℝ) : Measure SiteConfig :=  Measure.infinitePi (fun _ : Vertex => bernoulliMeasure true false (parameter p))

definition OAI.CriticalZ3.bondCritical line 50

noncomputable def bondCritical : ℝ :=  sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < bondLaw p (bondInfiniteAt 0)}

definition OAI.CriticalZ3.siteCritical line 53

noncomputable def siteCritical : ℝ :=  sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < siteLaw p (siteInfiniteAt 0)}

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: nearest, neighbor, site, integer, parameter, critical, bernoulli, bond and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: surely, almost and percolation.

Statements in Critical bond and site percolation on the cubic lattice come first because the AI draft reading of OpenAI’s note shares 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

Corollary 10.1 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: nearest, neighbor, critical, bernoulli, bond, cluster and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: surely, almost, probability and percolation.
Corollary 2.3 of Critical bond and site percolation on the cubic lattice. Words of the Lean statement, its names split into words, that also appear in the theorem: site, vertex, bond and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: probabilities and percolation.
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: critical, bernoulli, connected, bond, cluster, infinite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: surely, almost and probability.
Theorem 5.1 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: critical, bernoulli, connected, bond, cluster, infinite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: surely, almost and percolation.
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: vertex, connected, cluster, infinite and finite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equivalently.
Proposition 9.3 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: step, parameters, bernoulli, bond, cluster and infinite. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: probability and percolation.

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 92 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 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 formalized result proves that, at their respective critical probabilities, nearest-neighbor bond and site percolation on $\mathbb Z^3$ almost surely have no infinite cluster. Equivalently, almost surely every vertex belongs to a finite cluster in each model.”
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: 53 to its own declarations, 39 to Mathlib’s documentation. 20 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 critical_no_infinite 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 19 other declarations of the file; with the theorem itself they span 36 lines. 7 of them are named in the statement itself.

Names in the namespace OAI.CriticalZ3 are listed without it.

abbrev Vertexline 9in the statementlibrary
abbrev Bondline 10library
abbrev BondConfigline 11library
abbrev SiteConfigline 12library
def stepline 14library
def NearestNeighborline 16library
def BondOpenline 19library
def SiteOpenline 23library
def BondConnectedline 26library
def SiteConnectedline 29library
def bondClusterline 32in the statementlibrary
def siteClusterline 35in the statementlibrary
def bondInfiniteAtline 38library
def siteInfiniteAtline 39library
def parameterline 41library
def bondLawline 44in the statementlibrary
def siteLawline 47in the statementlibrary
def bondCriticalline 50in the statementlibrary
def siteCriticalline 53in the statementlibrary
theorem critical_no_infiniteline 56selected theoremlibrary

import Mathlibnamespace OAIopen MeasureTheory ProbabilityTheorynamespace CriticalZ3abbrev Vertex := Fin 3 → ℤabbrev Bond := Vertex × Fin 3abbrev BondConfig := Bond → Boolabbrev SiteConfig := Vertex → Booldef step (x : Vertex) (i : Fin 3) : Vertex := fun j => x j + if i = j then 1 else 0def NearestNeighbor (x y : Vertex) : Prop :=  ∃ i : Fin 3, y = step x i ∨ x = step y idef BondOpen (ω : BondConfig) (x y : Vertex) : Prop :=  (∃ i : Fin 3, y = step x i ∧ ω (x, i) = true) ∨  (∃ i : Fin 3, x = step y i ∧ ω (y, i) = true)def SiteOpen (ω : SiteConfig) (x y : Vertex) : Prop :=  NearestNeighbor x y ∧ ω x = true ∧ ω y = truedef BondConnected (ω : BondConfig) (x y : Vertex) : Prop :=  Relation.ReflTransGen (BondOpen ω) x ydef SiteConnected (ω : SiteConfig) (x y : Vertex) : Prop :=  ω x = true ∧ Relation.ReflTransGen (SiteOpen ω) x ydef bondCluster (ω : BondConfig) (x : Vertex) : Set Vertex :=  {y | BondConnected ω x y}def siteCluster (ω : SiteConfig) (x : Vertex) : Set Vertex :=  {y | SiteConnected ω x y}def bondInfiniteAt (x : Vertex) : Set BondConfig := {ω | (bondCluster ω x).Infinite}def siteInfiniteAt (x : Vertex) : Set SiteConfig := {ω | (siteCluster ω x).Infinite}noncomputable def parameter (p : ℝ) : Set.Icc (0 : ℝ) 1 :=  ⟨max 0 (min 1 p), le_max_left _ _, max_le (by norm_num) (min_le_left _ _)⟩noncomputable def bondLaw (p : ℝ) : Measure BondConfig :=  Measure.infinitePi (fun _ : Bond => bernoulliMeasure true false (parameter p))noncomputable def siteLaw (p : ℝ) : Measure SiteConfig :=  Measure.infinitePi (fun _ : Vertex => bernoulliMeasure true false (parameter p))noncomputable def bondCritical : ℝ :=  sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < bondLaw p (bondInfiniteAt 0)}noncomputable def siteCritical : ℝ :=  sInf {p : ℝ | p ∈ Set.Icc (0 : ℝ) 1 ∧ 0 < siteLaw p (siteInfiniteAt 0)}theorem critical_no_infinite :    (∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite) ∧    (∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite) := by  sorryend CriticalZ3end 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/CriticalZ3.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/CriticalZ3/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: 19 local vocabulary declarations
Lines of Lean, without comments or blank lines: 36
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