Lean statement

Exponential correlation decay for classical $O(n)$ models

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.ClassicalON.main

Lean statement file: ClassicalON.leanShow on the mapFamily 215. 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.

def ExponentialDecay : Prop :=  ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →    ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),      (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,        0 ≤ correlation n G b x y ∧          correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)⋮theorem main : ExponentialDecay := 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.ClassicalON.Site line 11
```
abbrev Site := ℤ × ℤ
```

definition OAI.ClassicalON.PositiveNeighbor line 13

def PositiveNeighbor (x y : Site) : Prop :=  (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1)

structure OAI.ClassicalON.LatticeGraph line 16

structure LatticeGraph where  vertices : Finset Site  edges : Finset (vertices × vertices)  nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.val

abbreviation OAI.ClassicalON.Spin line 21

abbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)

abbreviation OAI.ClassicalON.Configuration line 22

abbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin n

definition OAI.ClassicalON.sphereProbability line 24

def sphereProbability (n : ℕ) : Measure (Spin n) :=  let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere  (surface Set.univ)⁻¹ • surface

definition OAI.ClassicalON.referenceLaw line 28

def referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) :=  Measure.pi (fun _ => sphereProbability n)

definition OAI.ClassicalON.interaction line 31

def interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)    (σ : Configuration n G) : ℝ :=  ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝ

definition OAI.ClassicalON.partition line 35

def partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ :=  ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n G

definition OAI.ClassicalON.correlation line 38

def correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)    (x y : G.vertices) : ℝ :=  (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ)    ∂referenceLaw n G) / partition n G b

definition OAI.ClassicalON.siteDistance line 43

def siteDistance (x y : Site) : ℝ :=  Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : 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: nearest, neighbor, exp, lattice and integer. Found in the paper’s title, which its main theorem states: decay and exponential. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subgraph, strength, square, finite, beta and constants.

Statements in Exponential decay in two-dimensional classical O(n) models come first because the release lists only that paper for this Lean statement.

Theorem 1.1 of Exponential decay in two-dimensional classical O(n) models

Theorem

For every integer n\ge3 and every finite \beta>0, there are constants A=A(n,\beta)<\infty and m=m(n,\beta)>0 such that, for every finite nearest-neighbor square-lattice subgraph G, every strength array b\in[0,\beta]^E, and all x,y\in V,

0\le \langle\sigma_x\cdot\sigma_y\rangle_{G,b}^{(n)} \le A\exp(-m\|x-y\|_2).

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

For every integer $n\ge3$ and every finite $\beta>0$, there are constants
$A=A(n,\beta)<\infty$ and $m=m(n,\beta)>0$ such that, for every finite
nearest-neighbor square-lattice subgraph $G$, every strength array
$b\in[0,\beta]^E$, and all $x,y\in V$,
\begin{equation}
 0\le \langle\sigma_x\cdot\sigma_y\rangle_{G,b}^{(n)}
 \le A\exp(-m\|x-y\|_2).
 
\end{equation}

Other possible paper matches, closest wording first

Proposition 4.5 of Exponential decay in two-dimensional classical O(n) models. Words of the Lean statement, its names split into words, that also appear in the theorem: probability, graph, spins and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strengths, nonnegative, free and finite.
Corollary 1.2 of Exponential decay in two-dimensional classical O(n) models. Words of the Lean statement, its names split into words, that also appear in the theorem: edges, nearest, neighbor and laws. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: boundary, strength, free, limits, two, bound and two more.
Proposition 4.6 of Exponential decay in two-dimensional classical O(n) models. Words of the Lean statement, its names split into words, that also appear in the theorem: probability, graph and lattice. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subgraph, strength, free, finite and beta.
Proposition 4.7 of Exponential decay in two-dimensional classical O(n) models. Words of the Lean statement, its names split into words, that also appear in the theorem: exp, lattice and law. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subgraph, strengths, square, free, finite, beta and one more.
Theorem 1.1 of The canonical massive continuum limit of the two-dimensional O(3) model. Words of the Lean statement, its names split into words, that also appear in the theorem: probability, real, nearest, neighbor, spin, law and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: model, limiting, two, finite, bounded, beta and one more.
Theorem 3.1 of Sharp mass bounds for the two-dimensional O(4) model. Words of the Lean statement, its names split into words, that also appear in the theorem: partition, distance, exp, lattice, sqrt, pi and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: subgraph, model, square, free, limit, bound and two 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 215’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 exponential decay of spin correlations is stated: ExponentialDecay : ∀ n ≥ 3, ∀ β > 0, ∃ A m, 0 < m ∧ ∀ finite lattice graph G, b ∈ [0,β] → 0 ≤ correlation ≤ A * exp(-m * siteDistance x y) (free boundary, finite volume, uniform in the graph; docs: 'The infinite-volume limit and the separate O(4) spectral-gap claim are not included'). The summary's headline results, the canonical O(3) continuum limit with positive mass gap and the exact O(4) asymptotic m_lat(β) ~ 32 e^{π/4-1/2} √β e^{-πβ}, are not stated.
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 O(3) canonical continuum limit (undeclared) and the O(4) exact mass asymptotic (declared)

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized result proves exponential decay of spin correlations for the two-dimensional classical $O(n)$ model at every temperature when $n\ge3$. For each interaction bound $\beta>0$, constants $A$ and $m>0$ work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most $\beta$: the correlation between sites $x,y$ is at most $Ae^{-m|x-y|}$. The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.”
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

60 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: 24 to its own declarations, 17 to Mathlib’s documentation. 13 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 12 other declarations of the file; with the theorem itself they span 33 lines. 1 of them is named in the statement itself.

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

abbrev Siteline 11library
def PositiveNeighborline 13library
structure LatticeGraphline 16library
abbrev Spinline 21library
abbrev Configurationline 22library
def sphereProbabilityline 24library
def referenceLawline 28library
def interactionline 31library
def partitionline 35library
def correlationline 38library
def siteDistanceline 43library
def ExponentialDecayline 46in the statementlibrary
theorem mainline 53selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheoryopen scoped BigOperators InnerProductSpacenamespace ClassicalONabbrev Site := ℤ × ℤdef PositiveNeighbor (x y : Site) : Prop :=  (y.1 = x.1 + 1 ∧ y.2 = x.2) ∨ (y.1 = x.1 ∧ y.2 = x.2 + 1)structure LatticeGraph where  vertices : Finset Site  edges : Finset (vertices × vertices)  nearest : ∀ e ∈ edges, PositiveNeighbor e.1.val e.2.valabbrev Spin (n : ℕ) := ↥(Metric.sphere (0 : EuclideanSpace ℝ (Fin n)) 1)abbrev Configuration (n : ℕ) (G : LatticeGraph) := G.vertices → Spin ndef sphereProbability (n : ℕ) : Measure (Spin n) :=  let surface := (volume : Measure (EuclideanSpace ℝ (Fin n))).toSphere  (surface Set.univ)⁻¹ • surfacedef referenceLaw (n : ℕ) (G : LatticeGraph) : Measure (Configuration n G) :=  Measure.pi (fun _ => sphereProbability n)def interaction (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)    (σ : Configuration n G) : ℝ :=  ∑ e : G.edges, b e * ⟪(σ e.val.1).val, (σ e.val.2).val⟫_ℝdef partition (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ) : ℝ :=  ∫ σ, Real.exp (interaction n G b σ) ∂referenceLaw n Gdef correlation (n : ℕ) (G : LatticeGraph) (b : G.edges → ℝ)    (x y : G.vertices) : ℝ :=  (∫ σ, ⟪(σ x).val, (σ y).val⟫_ℝ * Real.exp (interaction n G b σ)    ∂referenceLaw n G) / partition n G bdef siteDistance (x y : Site) : ℝ :=  Real.sqrt (((x.1 : ℝ) - (y.1 : ℝ)) ^ 2 + ((x.2 : ℝ) - (y.2 : ℝ)) ^ 2)def ExponentialDecay : Prop :=  ∀ (n : ℕ), 3 ≤ n → ∀ (β : ℝ), 0 < β →    ∃ A m : ℝ, 0 < m ∧ ∀ (G : LatticeGraph) (b : G.edges → ℝ),      (∀ e, 0 ≤ b e ∧ b e ≤ β) → ∀ x y : G.vertices,        0 ≤ correlation n G b x y ∧          correlation n G b x y ≤ A * Real.exp (-m * siteDistance x.val y.val)theorem main : ExponentialDecay := by  sorryend ClassicalONendend 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.

Exponential decay in two-dimensional classical O(n) models

Theorem

0\le \langle\sigma_x\cdot\sigma_y\rangle_{G,b}^{(n)} \le A\exp(-m\|x-y\|_2).

Section 1, Introduction Read it in the paper TeX source, line 76

Show the TeX

For every integer $n\ge3$ and every finite $\beta>0$, there are constants
$A=A(n,\beta)<\infty$ and $m=m(n,\beta)>0$ such that, for every finite
nearest-neighbor square-lattice subgraph $G$, every strength array
$b\in[0,\beta]^E$, and all $x,y\in V$,
\begin{equation}
 0\le \langle\sigma_x\cdot\sigma_y\rangle_{G,b}^{(n)}
 \le A\exp(-m\|x-y\|_2).
 
\end{equation}

Corollary (Free-boundary limits)

Let \Lambda_L=[-L,L]^2\cap\Z^2, with all its nearest-neighbor edges, free boundary conditions, and constant strength \beta>0. For every n\ge3 and x\in\Z^2,

\limsup_{L\to\infty} \left|\langle\sigma_0\cdot\sigma_x\rangle_{\Lambda_L,\beta}^{(n)}\right| \le A e^{-m\|x\|_2},

with the constants of Theorem thm:main. Every subsequential local weak limit of these laws satisfies the same two-point bound.

Section 1, Introduction Read it in the paper TeX source, line 89

Show the TeX

Let $\Lambda_L=[-L,L]^2\cap\Z^2$, with all its nearest-neighbor edges,
free boundary conditions, and constant strength $\beta>0$. For every
$n\ge3$ and $x\in\Z^2$,
\begin{equation}
 \limsup_{L\to\infty}
 \left|\langle\sigma_0\cdot\sigma_x\rangle_{\Lambda_L,\beta}^{(n)}\right|
 \le A e^{-m\|x\|_2},
 
\end{equation}
with the constants of Theorem~\ref{thm:main}. Every subsequential local
weak limit of these laws satisfies the same two-point bound.

Configuration and file facts

File: lean/ComparatorChallenges/ClassicalON.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/ClassicalON/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: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 33
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