Lean statement

Nonuniqueness for every nonamenable Cayley graph

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.Percolation.arbitrary_cayley

Lean statement file: CayleyPercolation.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 214 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 arbitrary_cayley {Λ : Type u} [Group Λ] [Group.FG Λ]    (hna : ¬ AmenableGroup Λ) (S : Finset Λ)    (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set Λ) = ⊤)    (h1 : 1 ∉ S) :    OperatorBounded (cayleyGraph S) (pc (cayleyGraph S)) ∧    pc (cayleyGraph S) < ptwo (cayleyGraph S) ∧    ptwo (cayleyGraph S) ≤ pu (cayleyGraph S) ∧    (∃ p₁ p₂ : unitInterval, pc (cayleyGraph S) < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧      ∀ᵐ labels ∂(labelLaw : Measure (CayleyEdge S → unitInterval)),        ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters (cayleyGraph S) (openEdges labels p)).Infinite) ∧    (∀ p : unitInterval, pc (cayleyGraph S) < p → p < pu (cayleyGraph S) →      ∀ᵐ ω ∂law p, (infiniteClusters (cayleyGraph S) ω).Infinite) := 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.Percolation.BondGraph line 14

abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 V

definition OAI.Percolation.Conn line 16

def Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x y

definition OAI.Percolation.cluster line 19

def cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=  {y | Conn G ω x y}

definition OAI.Percolation.infiniteClusters line 22

def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}

definition OAI.Percolation.law line 25

noncomputable def law (p : unitInterval) : Measure (Set E) :=  setBernoulli Set.univ p

definition OAI.Percolation.pc line 28

noncomputable def pc (G : BondGraph V E) : unitInterval :=  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}

definition OAI.Percolation.pu line 31

noncomputable def pu (G : BondGraph V E) : unitInterval :=  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}

definition OAI.Percolation.twoPoint line 34

noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=  (law p {ω | Conn G ω x y}).toReal

abbreviation OAI.Percolation.Hilbert line 37

noncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2

definition OAI.Percolation.HasKernel line 39

noncomputable def HasKernel [DecidableEq V]    (G : BondGraph V E) (p : unitInterval)    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)

definition OAI.Percolation.OperatorBounded line 44

noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by  classical  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p A

definition OAI.Percolation.ptwo line 48

noncomputable def ptwo (G : BondGraph V E) : unitInterval :=  sSup {p | OperatorBounded G p}

definition OAI.Percolation.labelLaw line 51

noncomputable def labelLaw : Measure (E → unitInterval) :=  Measure.infinitePi (fun _ => (volume : Measure unitInterval))

definition OAI.Percolation.openEdges line 54

def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=  {e | labels e ≤ p}

definition OAI.Percolation.AmenableGroup line 57

noncomputable def AmenableGroup (Λ : Type u) [Group Λ] : Prop := by  classical  exact ∀ K : Finset Λ, ∀ ε : ℝ, 0 < ε → ∃ A : Finset Λ, A.Nonempty ∧    ∀ g ∈ K, (((A.image (fun x => x * g)) \ A).card : ℝ) < ε * A.card

definition OAI.Percolation.CayleyEdge line 62

def CayleyEdge {Λ : Type u} [Group Λ] (S : Finset Λ) :=  {e : Sym2 Λ // ∃ x : Λ, ∃ s ∈ S, e = s(x, x * s)}

definition OAI.Percolation.cayleyGraph line 65

def cayleyGraph {Λ : Type u} [Group Λ] (S : Finset Λ) :    BondGraph Λ (CayleyEdge S) := Subtype.val

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: group, measure, arbitrary, cayley, edges, interval, bernoulli, pu and eight more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: symmetric, generating, finitely, nonamenable, nonuniqueness and finite.

Corollary 1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs

Corollary (Arbitrary Cayley graphs)

Let \Lambda be a nonamenable finitely generated group, and let S be any finite symmetric generating set with the identity removed. For Bernoulli bond percolation on the simple undirected Cayley graph with edges \{x,xs\}, x\in\Lambda, s\in S, the critical connection operator is bounded on counting-measure \ell^2(\Lambda) and

\pc<\ptwo\le\pu.

The simultaneous nonuniqueness interval in thm:main exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

Let $\Lambda$ be a nonamenable finitely generated group, and let $S$
be any finite symmetric generating set with the identity removed.
For Bernoulli bond percolation on the simple undirected Cayley graph
with edges $\{x,xs\}$, $x\in\Lambda$, $s\in S$, the critical connection
operator is bounded on counting-measure $\ell^2(\Lambda)$ and
\[
 \pc<\ptwo\le\pu.
\]
The simultaneous nonuniqueness interval in \cref{thm:main} exists for
every such fixed generating set. At every fixed $p\in(\pc,\pu)$,
there are almost surely infinitely many infinite open clusters.

Other possible paper matches, closest wording first

Theorem 1.1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: edge, bernoulli, pu, bond, graph, clusters and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Proposition 9 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: edges, interval, edge, clusters, infinite, bounded and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Proposition 1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: pu, operator, ptwo, bounded and pc.
Corollary 2 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: laws, bond, bernoulli, graph, ptwo, infinite and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Theorem 2.1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: graph, infinite and pc. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: finite.
Proposition 7 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: operator.

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 170 s).
Reported 10 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 214’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: full_main (infinite, connected, locally finite, quasi-transitive multigraph with 0 < hV G) concludes MainConclusion G including pc G < ptwo G ∧ ptwo G ≤ pu G and a coupled interval [p₁,p₂] above pc with infinitely many infinite clusters a.s.; critical_laws gives the finite triangle diagram triangleDiagram G (pc G) x ≤ operatorNorm^3 ∧ < ⊤ plus mean-field laws; cayley_main/arbitrary_cayley handle every finite symmetric generating set of a non-Folner-amenable group for every p ∈ (pc, pu). Nonamenability is encoded as positive vertex-isoperimetric constant (graphs) or failure of the Folner condition (groups).
Definitions the group asks readers to compare with the paper: Custom 'amenable': def FolnerAmenable (Λ) : Prop := ∀ K : Finset Λ, ∀ ε > 0, ∃ A : Finset Λ, A.Nonempty ∧ ∀ k ∈ K, ((A.image (· * k)) \ A).card < ε * A.card (and the identical AmenableGroup in CayleyPercolation) is the standard right-Folner criterion, not Mathlib's; for general graphs nonamenability is 0 < hV G with hV G := sInf {|∂A| / |A| : A finite nonempty} (outer vertex boundary). Standard but hand-written.
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

“For Cayley graphs, the result holds for every finite symmetric generating set of a nonamenable finitely generated group and every $p\in(p_c,p_u)$.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a consequence of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

84 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, 53 to Mathlib’s documentation. 18 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 arbitrary_cayley 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 17 other declarations of the file; with the theorem itself they span 50 lines. 11 of them are named in the statement itself.

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

abbrev BondGraphline 14library
def Connline 16library
def clusterline 19library
def infiniteClustersline 22in the statementlibrary
def lawline 25in the statementlibrary
def pcline 28in the statementlibrary
def puline 31in the statementlibrary
def twoPointline 34library
abbrev Hilbertline 37library
def HasKernelline 39library
def OperatorBoundedline 44in the statementlibrary
def ptwoline 48in the statementlibrary
def labelLawline 51in the statementlibrary
def openEdgesline 54in the statementlibrary
def AmenableGroupline 57in the statementlibrary
def CayleyEdgeline 62in the statementlibrary
def cayleyGraphline 65in the statementlibrary
theorem arbitrary_cayleyline 68selected theoremlibrary

import Mathlibnamespace OAI/-! Nonuniqueness of Bernoulli bond percolation on arbitrary Cayley graphs. -/open MeasureTheory ProbabilityTheoryopen scoped ENNReal NNRealnamespace Percolationuniverse u vvariable {V : Type u} {E : Type v}abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 Vdef Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x ydef cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=  {y | Conn G ω x y}def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}noncomputable def law (p : unitInterval) : Measure (Set E) :=  setBernoulli Set.univ pnoncomputable def pc (G : BondGraph V E) : unitInterval :=  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}noncomputable def pu (G : BondGraph V E) : unitInterval :=  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=  (law p {ω | Conn G ω x y}).toRealnoncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2noncomputable def HasKernel [DecidableEq V]    (G : BondGraph V E) (p : unitInterval)    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by  classical  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p Anoncomputable def ptwo (G : BondGraph V E) : unitInterval :=  sSup {p | OperatorBounded G p}noncomputable def labelLaw : Measure (E → unitInterval) :=  Measure.infinitePi (fun _ => (volume : Measure unitInterval))def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=  {e | labels e ≤ p}noncomputable def AmenableGroup (Λ : Type u) [Group Λ] : Prop := by  classical  exact ∀ K : Finset Λ, ∀ ε : ℝ, 0 < ε → ∃ A : Finset Λ, A.Nonempty ∧    ∀ g ∈ K, (((A.image (fun x => x * g)) \ A).card : ℝ) < ε * A.carddef CayleyEdge {Λ : Type u} [Group Λ] (S : Finset Λ) :=  {e : Sym2 Λ // ∃ x : Λ, ∃ s ∈ S, e = s(x, x * s)}def cayleyGraph {Λ : Type u} [Group Λ] (S : Finset Λ) :    BondGraph Λ (CayleyEdge S) := Subtype.valtheorem arbitrary_cayley {Λ : Type u} [Group Λ] [Group.FG Λ]    (hna : ¬ AmenableGroup Λ) (S : Finset Λ)    (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (hgen : Subgroup.closure (S : Set Λ) = ⊤)    (h1 : 1 ∉ S) :    OperatorBounded (cayleyGraph S) (pc (cayleyGraph S)) ∧    pc (cayleyGraph S) < ptwo (cayleyGraph S) ∧    ptwo (cayleyGraph S) ≤ pu (cayleyGraph S) ∧    (∃ p₁ p₂ : unitInterval, pc (cayleyGraph S) < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧      ∀ᵐ labels ∂(labelLaw : Measure (CayleyEdge S → unitInterval)),        ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters (cayleyGraph S) (openEdges labels p)).Infinite) ∧    (∀ p : unitInterval, pc (cayleyGraph S) < p → p < pu (cayleyGraph S) →      ∀ᵐ ω ∂law p, (infiniteClusters (cayleyGraph S) ω).Infinite) := by  sorryend Percolationend 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.

Nonuniqueness of percolation on nonamenable quasi-transitive graphs

Theorem

Let G be an infinite connected locally finite quasi-transitive graph with h_V(G)>0. For Bernoulli bond percolation on G,

\sup_{0\le p<\pc}\norm{T_p}_{2\to2} =\norm{T_{\pc}}_{2\to2}<\infty, \qquad \pc(G)<\ptwo(G)\le\pu(G).

There exist \pc<p_1<p_2<1 such that, in the coupling obtained from independent uniform edge labels, almost surely there are infinitely many infinite open clusters simultaneously for every p\in[p_1,p_2].

Section 1, Introduction Read it in the paper TeX source, line 58

Show the TeX

Let $G$ be an infinite connected locally finite quasi-transitive graph
with $h_V(G)>0$. For Bernoulli bond percolation on $G$,
\[
 \sup_{0\le p<\pc}\norm{T_p}_{2\to2}
       =\norm{T_{\pc}}_{2\to2}<\infty,
 \qquad \pc(G)<\ptwo(G)\le\pu(G).
\]
There exist $\pc<p_1<p_2<1$ such that, in the coupling obtained from
independent uniform edge labels, almost surely there are infinitely
many infinite open clusters simultaneously for every $p\in[p_1,p_2]$.

Corollary (Arbitrary Cayley graphs)

\pc<\ptwo\le\pu.

The simultaneous nonuniqueness interval in thm:main exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

Let $\Lambda$ be a nonamenable finitely generated group, and let $S$
be any finite symmetric generating set with the identity removed.
For Bernoulli bond percolation on the simple undirected Cayley graph
with edges $\{x,xs\}$, $x\in\Lambda$, $s\in S$, the critical connection
operator is bounded on counting-measure $\ell^2(\Lambda)$ and
\[
 \pc<\ptwo\le\pu.
\]
The simultaneous nonuniqueness interval in \cref{thm:main} exists for
every such fixed generating set. At every fixed $p\in(\pc,\pu)$,
there are almost surely infinitely many infinite open clusters.

Corollary (Critical laws and exponential connectivity)

Let G be an infinite connected locally finite quasi-transitive graph with h_V(G)>0. For Bernoulli bond percolation on G,

\nabla_{\pc}(v)\le\norm{T_{\pc}}_{2\to2}^{3}<\infty \qquad\text{for every }v\in V.

For every v\in V,

\begin{aligned}\chi_v(p)&\asymp(\pc-p)^{-1} &&(p\uparrow\pc), \\ \theta_v(p)&\asymp p-\pc &&(p\downarrow\pc), \\ \Prob_{\pc}(|C_v|\ge n)&\asymp n^{-1/2} &&(n\to\infty), \\ \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{int}}(C_v)\ge n\bigr) &\asymp n^{-1} &&(n\to\infty), \\ \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{ext}}(C_v)\ge n\bigr) &\asymp n^{-1} &&(n\to\infty).\end{aligned}

Here n tends through positive integers. The comparison constants and the one-sided neighborhoods of \pc may depend on G and v. Moreover,

\pc(G)<\ptwo(G)\le\pexp(G).

For each fixed p\in[0,\ptwo(G)), there are constants 0<A_{G,p}<\infty and a_{G,p}>0 such that

T_p(x,y)\le A_{G,p}e^{-a_{G,p}\dist_G(x,y)} \qquad\text{for every }x,y\in V.

Section 1, Introduction Read it in the paper TeX source, line 257

Show the TeX

Let $G$ be an infinite connected locally finite quasi-transitive graph
with $h_V(G)>0$. For Bernoulli bond percolation on $G$,
\begin{equation}
 \nabla_{\pc}(v)\le\norm{T_{\pc}}_{2\to2}^{3}<\infty
 \qquad\text{for every }v\in V.
\end{equation}
For every $v\in V$,
\begin{align}
 \chi_v(p)&\asymp(\pc-p)^{-1} &&(p\uparrow\pc), \\
 \theta_v(p)&\asymp p-\pc &&(p\downarrow\pc), \\
 \Prob_{\pc}(|C_v|\ge n)&\asymp n^{-1/2} &&(n\to\infty), \\
 \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{int}}(C_v)\ge n\bigr)
    &\asymp n^{-1} &&(n\to\infty), \\
 \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{ext}}(C_v)\ge n\bigr)
    &\asymp n^{-1} &&(n\to\infty). 
\end{align}
Here $n$ tends through positive integers. The comparison constants
and the one-sided neighborhoods of $\pc$ may depend on $G$ and $v$.
Moreover,
\begin{equation}
 \pc(G)<\ptwo(G)\le\pexp(G).
\end{equation}
For each fixed $p\in[0,\ptwo(G))$, there are constants
$0<A_{G,p}<\infty$ and $a_{G,p}>0$ such that
\begin{equation}
 T_p(x,y)\le A_{G,p}e^{-a_{G,p}\dist_G(x,y)}
 \qquad\text{for every }x,y\in V.
\end{equation}

Configuration and file facts

File: lean/ComparatorChallenges/CayleyPercolation.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 214 link it.
Solution module: OAI.Probability.UnimodularPercolation.CayleyMain
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: 17 local vocabulary declarations
Lines of Lean, without comments or blank lines: 50
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