Lean statement

A common radial phase transition over a density interval

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.RadialTransition.densityInterval

Lean statement file: RadialDensityInterval.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 228 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 densityInterval :    ∃ (p : ℝ) (φ : Potential) (B C η βc : ℝ) (f : ℝ → ℝ → ℝ),      0 < p ∧ IsPackingDensity p ∧      IsBoundedPotential φ ∧ Continuous φ ∧      0 < B ∧ 0 < C ∧ StableWith φ B ∧ AlgebraicDecayWith φ C ∧      0 < η ∧ Ioo (5*p/3 - η) (5*p/3 + η) ⊆ Ioi 0 ∧      βc ∈ Icc (7/8:ℝ) (9/8) ∧      ∀ ρ ∈ Ioo (5*p/3 - η) (5*p/3 + η),        IsCanonicalFreeEnergy φ ρ (fun β => f β ρ) ∧        StrictTemperatureCorner (fun β => f β ρ) βc := 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.RadialTransition.Position line 12

abbrev Position := EuclideanSpace ℝ (Fin 3)

abbreviation OAI.RadialTransition.Potential line 13
```
abbrev Potential := ℝ≥0 → ℝ
```

definition OAI.RadialTransition.cube line 15

def cube (L : ℝ) : Set Position :=  {x | ∀ k : Fin 3, 0 ≤ x k ∧ x k ≤ L}

definition OAI.RadialTransition.configurationCube line 18

def configurationCube (L : ℝ) (N : ℕ) : Set (Fin N → Position) :=  {x | ∀ i : Fin N, x i ∈ cube L}

definition OAI.RadialTransition.energy line 21

def energy (φ : Potential) {N : ℕ} (x : Fin N → Position) : ℝ :=  ∑ i : Fin N, ∑ j : Fin N with i < j, φ (nndist (x i) (x j))

definition OAI.RadialTransition.partition line 24

def partition (φ : Potential) (L : ℝ) (N : ℕ) (β : ℝ) : ℝ :=  (N.factorial : ℝ)⁻¹ *    ∫ x in configurationCube L N, Real.exp (-β * energy φ x)

definition OAI.RadialTransition.packingNumber line 28

def packingNumber (L : ℝ) : ℕ :=  sSup {n : ℕ | ∃ X : Finset Position, X.card = n ∧    (∀ x ∈ X, x ∈ cube L) ∧    (∀ x ∈ X, ∀ y ∈ X, x ≠ y → 1 ≤ dist x y)}

definition OAI.RadialTransition.IsPackingDensity line 33

def IsPackingDensity (p : ℝ) : Prop :=  Tendsto (fun L : ℝ => (packingNumber L : ℝ) / L ^ 3) atTop (nhds p)

definition OAI.RadialTransition.IsBoundedPotential line 36

def IsBoundedPotential (φ : Potential) : Prop :=  ∃ M : ℝ, 0 ≤ M ∧ ∀ r : ℝ≥0, |φ r| ≤ M

definition OAI.RadialTransition.StableWith line 39

def StableWith (φ : Potential) (B : ℝ) : Prop :=  ∀ (N : ℕ) (x : Fin N → Position), -(B * N) ≤ energy φ x

definition OAI.RadialTransition.AlgebraicDecayWith line 42

def AlgebraicDecayWith (φ : Potential) (C : ℝ) : Prop :=  ∀ r : ℝ≥0, 1 ≤ r → |φ r| ≤ C * (r : ℝ) ^ (-(3 : ℝ) - 1 / 32)

definition OAI.RadialTransition.IsCanonicalFreeEnergy line 45

def IsCanonicalFreeEnergy (φ : Potential) (ρ : ℝ) (f : ℝ → ℝ) : Prop :=  ∀ β : ℝ, 0 < β →    Tendsto (fun L : ℝ =>      -(Real.log (partition φ L ⌊ρ * L ^ 3⌋₊ β) / (β * L ^ 3)))      atTop (nhds (f β))

definition OAI.RadialTransition.StrictTemperatureCorner line 51

def StrictTemperatureCorner (f : ℝ → ℝ) (βc : ℝ) : Prop :=  ∃ dLeft dRight : ℝ,    HasDerivWithinAt f dLeft (Iic βc) βc ∧    HasDerivWithinAt f dRight (Ici βc) βc ∧    dRight < dLeft ∧ ¬DifferentiableAt ℝ f βc

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: 5p/3, 7/8, 9/8, differentiable, 1/32, temperature, free, energy and five more. Found in the paper’s title, which its main theorem states: algebraic and decay. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: derivatives, inverse, phi, common, one and beta.

Statements in A radial continuum phase transition with algebraic decay come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of A radial continuum phase transition with algebraic decay

Theorem

There is a bounded continuous function \phi:[0,\infty)\to\R and finite constants B,C>0 such that

\sum_{i<j}\phi(\abs{x_i-x_j})\ge-BN, \qquad \abs{\phi(r)}\le C r^{-3-1/32}\quad(r\ge1).

There are an open interval I\subset(0,\infty) centered at 5p/3 and a common inverse temperature \beta_c\in[7/8,9/8] such that, for every \rho\in I, the canonical free energy

f(\beta,\rho)= -\lim_{L\to\infty}\frac1{\beta L^3} \log Z^\phi_{L,\lfloor\rho L^3\rfloor}(\beta)

exists and is finite for every \beta>0. Its finite one-sided derivatives at \beta_c satisfy

f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho).

In particular, f(\,\cdot\,,\rho) is not differentiable at \beta_c.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

There is a bounded continuous function
\(\phi:[0,\infty)\to\R\) and finite constants \(B,C>0\) such that
\[
 \sum_{i<j}\phi(\abs{x_i-x_j})\ge-BN,
 \qquad
 \abs{\phi(r)}\le C r^{-3-1/32}\quad(r\ge1).
\]
There are an open interval \(I\subset(0,\infty)\) centered at \(5p/3\)
and a common inverse temperature \(\beta_c\in[7/8,9/8]\) such that,
for every \(\rho\in I\), the canonical free energy
\[
 f(\beta,\rho)=
 -\lim_{L\to\infty}\frac1{\beta L^3}
       \log Z^\phi_{L,\lfloor\rho L^3\rfloor}(\beta)
\]
exists and is finite for every \(\beta>0\).
Its finite one-sided derivatives at \(\beta_c\) satisfy
\[
 f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho).
\]
In particular, \(f(\,\cdot\,,\rho)\) is not differentiable at \(\beta_c\).

Other possible paper matches, closest wording first

Theorem 1.1 of A continuum temperature singularity for a radial pair potential. Words of the Lean statement, its names split into words, that also appear in the theorem: strict, temperature, free, energy, interval, canonical and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: derivatives, radial, inverse, nonempty, phi, limit and three more.
Proposition 10 of A radial continuum phase transition with algebraic decay. Words of the Lean statement, its names split into words, that also appear in the theorem: 1/32, stable, algebraic, decay, continuous and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: cr and phi.
Proposition 2 of A radial continuum phase transition with algebraic decay. Words of the Lean statement, its names split into words, that also appear in the theorem: real, integer, partition, continuous, log and bounded. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: limits and beta.
Proposition 1 of A radial continuum phase transition with algebraic decay. Words of the Lean statement, its names split into words, that also appear in the theorem: configuration, stable and cube. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: common.
Corollary 2 of A radial continuum phase transition with algebraic decay. Words of the Lean statement, its names split into words, that also appear in the theorem: partition, number, cube, log and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta.
Proposition 3 of A radial continuum phase transition with algebraic decay. Words of the Lean statement, its names split into words, that also appear in the theorem: exp, energy, canonical and bounds. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta.

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 228’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: ContinuumTemperature.main_theorem : ∃ φ f I βc, Admissible φ ∧ φ = o(r^{-3}) ∧ HasCanonicalFreeEnergy φ f ∧ I open interval ⊆ (0,∞) ∧ βc ∈ (1/2, 3/2) ∧ ∀ ρ ∈ I, dplus < dminus (Admissible = stable, φ → +∞ at 0 (divergent core), integrable tail) and RadialTransition.densityInterval (bounded continuous φ, |φ r| ≤ C r^(-3-1/32), stable, a nonempty open density interval around 5p/3, common βc ∈ [7/8, 9/8], StrictTemperatureCorner); both potentials of the summary and the density-interval statement are covered.
Definitions the group asks readers to compare with the paper: Custom 'free energy': HasCanonicalFreeEnergy φ f := ∀ β ρ > 0, Tendsto (L^{-3} log Z(β, L, ⌊ρL^3⌋)) atTop (𝓝 (-β f β ρ)) / IsCanonicalFreeEnergy with explicit partition φ β L N := (N!)^{-1} ∫_{cube^N} exp(-β energy) and energy in EReal (⊤ at coincident points) for the first potential; standard thermodynamic-limit definitions, free boundary conditions.
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 constructs a bounded continuous stable radial pair potential in $\mathbb R^3$ satisfying $|\phi(r)|\le Cr^{-3-1/32}$ for $r\ge1$. There is a nonempty open interval of positive densities around $5p/3$, where $p$ is the unit-separated packing-density limit, on which the canonical free energy exists at every positive inverse temperature and has a strict downward derivative jump at one common $\beta_c\in[7/8,9/8]$.”
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

72 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: 25 to its own declarations, 33 to Mathlib’s documentation. 14 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 densityInterval 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 13 other declarations of the file; with the theorem itself they span 44 lines. 7 of them are named in the statement itself.

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

abbrev Positionline 12library
abbrev Potentialline 13in the statementlibrary
def cubeline 15library
def configurationCubeline 18library
def energyline 21library
def partitionline 24library
def packingNumberline 28library
def IsPackingDensityline 33in the statementlibrary
def IsBoundedPotentialline 36in the statementlibrary
def StableWithline 39in the statementlibrary
def AlgebraicDecayWithline 42in the statementlibrary
def IsCanonicalFreeEnergyline 45in the statementlibrary
def StrictTemperatureCornerline 51in the statementlibrary
theorem densityIntervalline 57selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen scoped BigOperators NNReal ContDiff ENNRealopen Filter MeasureTheory Setnamespace RadialTransitionabbrev Position := EuclideanSpace ℝ (Fin 3)abbrev Potential := ℝ≥0 → ℝdef cube (L : ℝ) : Set Position :=  {x | ∀ k : Fin 3, 0 ≤ x k ∧ x k ≤ L}def configurationCube (L : ℝ) (N : ℕ) : Set (Fin N → Position) :=  {x | ∀ i : Fin N, x i ∈ cube L}def energy (φ : Potential) {N : ℕ} (x : Fin N → Position) : ℝ :=  ∑ i : Fin N, ∑ j : Fin N with i < j, φ (nndist (x i) (x j))def partition (φ : Potential) (L : ℝ) (N : ℕ) (β : ℝ) : ℝ :=  (N.factorial : ℝ)⁻¹ *    ∫ x in configurationCube L N, Real.exp (-β * energy φ x)def packingNumber (L : ℝ) : ℕ :=  sSup {n : ℕ | ∃ X : Finset Position, X.card = n ∧    (∀ x ∈ X, x ∈ cube L) ∧    (∀ x ∈ X, ∀ y ∈ X, x ≠ y → 1 ≤ dist x y)}def IsPackingDensity (p : ℝ) : Prop :=  Tendsto (fun L : ℝ => (packingNumber L : ℝ) / L ^ 3) atTop (nhds p)def IsBoundedPotential (φ : Potential) : Prop :=  ∃ M : ℝ, 0 ≤ M ∧ ∀ r : ℝ≥0, |φ r| ≤ Mdef StableWith (φ : Potential) (B : ℝ) : Prop :=  ∀ (N : ℕ) (x : Fin N → Position), -(B * N) ≤ energy φ xdef AlgebraicDecayWith (φ : Potential) (C : ℝ) : Prop :=  ∀ r : ℝ≥0, 1 ≤ r → |φ r| ≤ C * (r : ℝ) ^ (-(3 : ℝ) - 1 / 32)def IsCanonicalFreeEnergy (φ : Potential) (ρ : ℝ) (f : ℝ → ℝ) : Prop :=  ∀ β : ℝ, 0 < β →    Tendsto (fun L : ℝ =>      -(Real.log (partition φ L ⌊ρ * L ^ 3⌋₊ β) / (β * L ^ 3)))      atTop (nhds (f β))def StrictTemperatureCorner (f : ℝ → ℝ) (βc : ℝ) : Prop :=  ∃ dLeft dRight : ℝ,    HasDerivWithinAt f dLeft (Iic βc) βc ∧    HasDerivWithinAt f dRight (Ici βc) βc ∧    dRight < dLeft ∧ ¬DifferentiableAt ℝ f βctheorem densityInterval :    ∃ (p : ℝ) (φ : Potential) (B C η βc : ℝ) (f : ℝ → ℝ → ℝ),      0 < p ∧ IsPackingDensity p ∧      IsBoundedPotential φ ∧ Continuous φ ∧      0 < B ∧ 0 < C ∧ StableWith φ B ∧ AlgebraicDecayWith φ C ∧      0 < η ∧ Ioo (5*p/3 - η) (5*p/3 + η) ⊆ Ioi 0 ∧      βc ∈ Icc (7/8:ℝ) (9/8) ∧      ∀ ρ ∈ Ioo (5*p/3 - η) (5*p/3 + η),        IsCanonicalFreeEnergy φ ρ (fun β => f β ρ) ∧        StrictTemperatureCorner (fun β => f β ρ) βc := by  sorryend RadialTransitionendend 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 continuum temperature singularity for a radial pair potential

Theorem

There exist an admissible radial difference-only pair potential \Phi(x)=\phi(|x|) on \R^3, a nonempty open interval I\subset(0,\infty), and one \beta_c\in(1/2,3/2) with the following properties. The potential satisfies \phi(r)=o(r^{-3}) as r\to\infty. The free-cube limit (eq:free-energy-definition) exists and is finite for every \beta>0 and every density \rho>0. For every \rho\in I, the finite one-sided derivatives at the same critical point satisfy

D_\beta^- f_\Phi(\beta_c,\rho) >D_\beta^+ f_\Phi(\beta_c,\rho).

Thus one potential and one critical inverse temperature give a strict canonical temperature singularity throughout I.

Section 1, The continuum existence problem Read it in the paper TeX source, line 44

Show the TeX

There exist an admissible radial difference-only pair potential
$\Phi(x)=\phi(|x|)$ on $\R^3$, a nonempty open interval
$I\subset(0,\infty)$, and one $\beta_c\in(1/2,3/2)$ with the following
properties. The potential satisfies $\phi(r)=o(r^{-3})$ as $r\to\infty$.
The free-cube limit \eqref{eq:free-energy-definition} exists and is finite
for every $\beta>0$ and every density $\rho>0$. For every $\rho\in I$,
the finite one-sided derivatives at the same critical point satisfy
\[
 D_\beta^- f_\Phi(\beta_c,\rho)
 >D_\beta^+ f_\Phi(\beta_c,\rho).
\]
Thus one potential and one critical inverse temperature give a strict
canonical temperature singularity throughout $I$.

A radial continuum phase transition with algebraic decay

Theorem

There is a bounded continuous function \phi:[0,\infty)\to\R and finite constants B,C>0 such that

\sum_{i<j}\phi(\abs{x_i-x_j})\ge-BN, \qquad \abs{\phi(r)}\le C r^{-3-1/32}\quad(r\ge1).

There are an open interval I\subset(0,\infty) centered at 5p/3 and a common inverse temperature \beta_c\in[7/8,9/8] such that, for every \rho\in I, the canonical free energy

f(\beta,\rho)= -\lim_{L\to\infty}\frac1{\beta L^3} \log Z^\phi_{L,\lfloor\rho L^3\rfloor}(\beta)

exists and is finite for every \beta>0. Its finite one-sided derivatives at \beta_c satisfy

f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho).

In particular, f(\,\cdot\,,\rho) is not differentiable at \beta_c.

Section 1, Introduction Read it in the paper TeX source, line 32

Show the TeX

There is a bounded continuous function
\(\phi:[0,\infty)\to\R\) and finite constants \(B,C>0\) such that
\[
 \sum_{i<j}\phi(\abs{x_i-x_j})\ge-BN,
 \qquad
 \abs{\phi(r)}\le C r^{-3-1/32}\quad(r\ge1).
\]
There are an open interval \(I\subset(0,\infty)\) centered at \(5p/3\)
and a common inverse temperature \(\beta_c\in[7/8,9/8]\) such that,
for every \(\rho\in I\), the canonical free energy
\[
 f(\beta,\rho)=
 -\lim_{L\to\infty}\frac1{\beta L^3}
       \log Z^\phi_{L,\lfloor\rho L^3\rfloor}(\beta)
\]
exists and is finite for every \(\beta>0\).
Its finite one-sided derivatives at \(\beta_c\) satisfy
\[
 f'_-(\beta_c,\rho)>f'_+(\beta_c,\rho).
\]
In particular, \(f(\,\cdot\,,\rho)\) is not differentiable at \(\beta_c\).

Configuration and file facts

File: lean/ComparatorChallenges/RadialDensityInterval.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 228 link it.
Solution module: OAI.Probability.RadialTransition.DensityInterval
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: 13 local vocabulary declarations
Lines of Lean, without comments or blank lines: 44
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