Lean statement

Temperature singularity 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.ContinuumTemperature.main_theorem

Lean statement file: ContinuumTransition.leanShow on the mapFamily 228. 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 MainStatement : Prop :=  ∃ (φ : ℝ → ℝ) (f : ℝ → ℝ → ℝ) (I : Set ℝ) (βc : ℝ),    Admissible φ ∧    Asymptotics.IsLittleO atTop φ (fun r : ℝ => r ^ (-3 : ℤ)) ∧    HasCanonicalFreeEnergy φ f ∧    IsOpen I ∧ OrdConnected I ∧ I.Nonempty ∧ I ⊆ Ioi 0 ∧    βc ∈ Ioo (1 / 2 : ℝ) (3 / 2 : ℝ) ∧    ∀ ρ ∈ I, ∃ dminus dplus : ℝ,      HasDerivWithinAt (fun β => f β ρ) dminus (Iic βc) βc ∧      HasDerivWithinAt (fun β => f β ρ) dplus (Ici βc) βc ∧      dplus < dminus⋮theorem main_theorem : MainStatement := 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.ContinuumTemperature.Space line 10
```
abbrev Space := EuclideanSpace ℝ (Fin 3)
```

definition OAI.ContinuumTemperature.radialPotential line 12

noncomputable def radialPotential (φ : ℝ → ℝ) (x : Space) : EReal :=  if x = 0 then ⊤ else (φ ‖x‖ : EReal)

definition OAI.ContinuumTemperature.energy line 15

noncomputable def energy (φ : ℝ → ℝ) {N : ℕ} (x : Fin N → Space) : EReal :=  ∑ i : Fin N, ∑ j ∈ Finset.univ.filter (fun j : Fin N => i < j),    radialPotential φ (x i - x j)

definition OAI.ContinuumTemperature.Admissible line 19

def Admissible (φ : ℝ → ℝ) : Prop :=  Measurable (radialPotential φ) ∧  (∀ a b : ℝ, 0 < a → a ≤ b →    BddAbove (φ '' Icc a b) ∧ BddBelow (φ '' Icc a b)) ∧  Tendsto φ (𝓝[>] (0 : ℝ)) atTop ∧  (∃ r₁ r₂ a : ℝ, 0 < r₁ ∧ r₁ < r₂ ∧ 0 < a ∧    ∀ r ∈ Icc r₁ r₂, φ r ≤ -a) ∧  (∃ R : ℝ, 0 < R ∧ IntegrableOn (fun r : ℝ => r ^ 2 * |φ r|) (Ioi R)) ∧  (∃ B : ℝ, ∀ (N : ℕ) (x : Fin N → Space),    ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x)

definition OAI.ContinuumTemperature.cube line 30

def cube (L : ℝ) : Set Space :=  {x | ∀ i, x i ∈ Icc (-L / 2) (L / 2)}

definition OAI.ContinuumTemperature.boltzmann line 33

noncomputable def boltzmann (φ : ℝ → ℝ) (β : ℝ) {N : ℕ}    (x : Fin N → Space) : ℝ :=  if energy φ x = ⊤ then 0 else Real.exp (-β * (energy φ x).toReal)

definition OAI.ContinuumTemperature.partition line 37

noncomputable def partition (φ : ℝ → ℝ) (β L : ℝ) (N : ℕ) : ℝ :=  if N = 0 then 1 else    (1 / (N.factorial : ℝ)) *      ∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β x

definition OAI.ContinuumTemperature.HasCanonicalFreeEnergy line 42

def HasCanonicalFreeEnergy (φ : ℝ → ℝ) (f : ℝ → ℝ → ℝ) : Prop :=  ∀ β ρ : ℝ, 0 < β → 0 < ρ →    Tendsto (fun L : ℝ => L⁻¹ ^ 3 * Real.log (partition φ β L ⌊ρ * L ^ 3⌋₊))      atTop (𝓝 (-β * f β ρ))

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: 1/2, 3/2, radial, free, energy, admissible, nonempty, canonical and two more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: strict, singularity, derivatives, temperature, inverse, sided, interval, pair and four more.

Statements in A continuum temperature singularity for a radial pair potential come first because the release’s name for this Lean statement shares words with that paper’s title.

Theorem 1.1 of 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 $I$.

Other possible paper matches, closest wording first

Theorem 1.1 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: free, energy, canonical and log. Found in the paper’s title, which its main theorem states: radial. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: derivatives, temperature, inverse, sided, interval, one and two more.
Proposition 2.2 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: partition and potential. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive, core and finite.
Proposition 3.1 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: potentials and cube. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: pair, finite and beta.
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, admissible, partition and log. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: sides, finite and 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, measure, energy and canonical. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: beta.
Proposition 2.4 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: potentials. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: density and finite.

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 one stable radial pair potential in three-dimensional continuum space with a divergent repulsive core and an integrable tail that is $o(r^{-3})$. Its canonical free energy exists for every positive inverse temperature and density. At one inverse temperature $\beta_c\in(1/2,3/2)$, the free energy has a strict finite jump between its one-sided derivatives with respect to inverse temperature, for every density in one nonempty open interval.”
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

64 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: 17 to its own declarations, 40 to Mathlib’s documentation. 10 declared names also link to the same declaration in OpenAI’s library. 1 identifier stays plain because it resolves to no single declaration: a variable Lean binds automatically, notation, or a name the scan of the pinned sources did not find. 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_theorem 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 9 other declarations of the file; with the theorem itself they span 42 lines. 1 of them is named in the statement itself.

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

abbrev Spaceline 10library
def radialPotentialline 12library
def energyline 15library
def Admissibleline 19library
def cubeline 30library
def boltzmannline 33library
def partitionline 37library
def HasCanonicalFreeEnergyline 42library
def MainStatementline 47in the statementlibrary
theorem main_theoremline 59selected theoremlibrary

import Mathlibnamespace OAIopen scoped Topology BigOperators ENNRealopen Filter MeasureTheory Setnamespace ContinuumTemperatureabbrev Space := EuclideanSpace ℝ (Fin 3)noncomputable def radialPotential (φ : ℝ → ℝ) (x : Space) : EReal :=  if x = 0 then ⊤ else (φ ‖x‖ : EReal)noncomputable def energy (φ : ℝ → ℝ) {N : ℕ} (x : Fin N → Space) : EReal :=  ∑ i : Fin N, ∑ j ∈ Finset.univ.filter (fun j : Fin N => i < j),    radialPotential φ (x i - x j)def Admissible (φ : ℝ → ℝ) : Prop :=  Measurable (radialPotential φ) ∧  (∀ a b : ℝ, 0 < a → a ≤ b →    BddAbove (φ '' Icc a b) ∧ BddBelow (φ '' Icc a b)) ∧  Tendsto φ (𝓝[>] (0 : ℝ)) atTop ∧  (∃ r₁ r₂ a : ℝ, 0 < r₁ ∧ r₁ < r₂ ∧ 0 < a ∧    ∀ r ∈ Icc r₁ r₂, φ r ≤ -a) ∧  (∃ R : ℝ, 0 < R ∧ IntegrableOn (fun r : ℝ => r ^ 2 * |φ r|) (Ioi R)) ∧  (∃ B : ℝ, ∀ (N : ℕ) (x : Fin N → Space),    ((-B * (N : ℝ) : ℝ) : EReal) ≤ energy φ x)def cube (L : ℝ) : Set Space :=  {x | ∀ i, x i ∈ Icc (-L / 2) (L / 2)}noncomputable def boltzmann (φ : ℝ → ℝ) (β : ℝ) {N : ℕ}    (x : Fin N → Space) : ℝ :=  if energy φ x = ⊤ then 0 else Real.exp (-β * (energy φ x).toReal)noncomputable def partition (φ : ℝ → ℝ) (β L : ℝ) (N : ℕ) : ℝ :=  if N = 0 then 1 else    (1 / (N.factorial : ℝ)) *      ∫ x in {x : Fin N → Space | ∀ i, x i ∈ cube L}, boltzmann φ β xdef HasCanonicalFreeEnergy (φ : ℝ → ℝ) (f : ℝ → ℝ → ℝ) : Prop :=  ∀ β ρ : ℝ, 0 < β → 0 < ρ →    Tendsto (fun L : ℝ => L⁻¹ ^ 3 * Real.log (partition φ β L ⌊ρ * L ^ 3⌋₊))      atTop (𝓝 (-β * f β ρ))def MainStatement : Prop :=  ∃ (φ : ℝ → ℝ) (f : ℝ → ℝ → ℝ) (I : Set ℝ) (βc : ℝ),    Admissible φ ∧    Asymptotics.IsLittleO atTop φ (fun r : ℝ => r ^ (-3 : ℤ)) ∧    HasCanonicalFreeEnergy φ f ∧    IsOpen I ∧ OrdConnected I ∧ I.Nonempty ∧ I ⊆ Ioi 0 ∧    βc ∈ Ioo (1 / 2 : ℝ) (3 / 2 : ℝ) ∧    ∀ ρ ∈ I, ∃ dminus dplus : ℝ,      HasDerivWithinAt (fun β => f β ρ) dminus (Iic βc) βc ∧      HasDerivWithinAt (fun β => f β ρ) dplus (Ici βc) βc ∧      dplus < dminustheorem main_theorem : MainStatement := by  sorryend ContinuumTemperatureend 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

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/ContinuumTransition.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Probability/ContinuumTransition/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: 9 local vocabulary declarations
Lines of Lean, without comments or blank lines: 42
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