Lean statement

Coulomb Monge nonattainment with equality of infima

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.Problem356.coulomb_counterexample_and_equal_infima

Lean statement file: CoulombCounterexample.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 373 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 coulomb_counterexample_and_equal_infima :    ∃ rho : E3 → ℝ, HasFullCoulombConclusion rho := 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.Problem356.E3 line 12
```
abbrev E3 := EuclideanSpace ℝ (Fin 3)
```
abbreviation OAI.Problem356.Triple line 14
```
abbrev Triple := E3 × (E3 × E3)
```
definition OAI.Problem356.tripleFst line 16
```
def tripleFst (t : Triple) : E3 := t.1
```

definition OAI.Problem356.tripleSnd line 18

def tripleSnd (t : Triple) : E3 := t.2.1

definition OAI.Problem356.tripleThd line 20

def tripleThd (t : Triple) : E3 := t.2.2

definition OAI.Problem356.invDistance line 22

def invDistance (x y : E3) : ℝ≥0∞ :=  (ENNReal.ofReal ‖x - y‖)⁻¹

definition OAI.Problem356.coulombCost line 25

def coulombCost (t : Triple) : ℝ≥0∞ :=  invDistance (tripleFst t) (tripleSnd t) +    invDistance (tripleFst t) (tripleThd t) +    invDistance (tripleSnd t) (tripleThd t)

definition OAI.Problem356.IsThreeCoupling line 30

def IsThreeCoupling (mu : Measure E3) (pi : Measure Triple) : Prop :=  IsProbabilityMeasure pi ∧    Measure.map tripleFst pi = mu ∧    Measure.map tripleSnd pi = mu ∧    Measure.map tripleThd pi = mu

definition OAI.Problem356.densityMeasure line 36

def densityMeasure (rho : E3 → ℝ) : Measure E3 :=  Measure.withDensity (volume : Measure E3) (fun x => ENNReal.ofReal (rho x))

definition OAI.Problem356.Preserves line 39

def Preserves (mu : Measure E3) (T : E3 → E3) : Prop :=  Measurable T ∧ Measure.map T mu = mu

definition OAI.Problem356.graphCost line 42

def graphCost (mu : Measure E3) (T2 T3 : E3 → E3) : ℝ≥0∞ :=  ∫⁻ x, coulombCost (x, (T2 x, T3 x)) ∂mu

definition OAI.Problem356.kantorovichValue line 45

def kantorovichValue (mu : Measure E3) : ℝ≥0∞ :=  ⨅ (pi : Measure Triple), ⨅ (_h : IsThreeCoupling mu pi),    ∫⁻ t, coulombCost t ∂pi

definition OAI.Problem356.mongeValue line 49

def mongeValue (mu : Measure E3) : ℝ≥0∞ :=  ⨅ (T2 : E3 → E3), ⨅ (T3 : E3 → E3),    ⨅ (_h2 : Preserves mu T2), ⨅ (_h3 : Preserves mu T3),      graphCost mu T2 T3

definition OAI.Problem356.IsSmoothCompactProbabilityDensity line 54

def IsSmoothCompactProbabilityDensity (rho : E3 → ℝ) : Prop :=  (∀ x, 0 ≤ rho x) ∧    ContDiff ℝ (↑(⊤ : ENat) : WithTop ENat) rho ∧    HasCompactSupport rho ∧    ContDiff ℝ (↑(⊤ : ENat) : WithTop ENat) (fun x : E3 => Real.sqrt (rho x)) ∧    HasCompactSupport (fun x : E3 => Real.sqrt (rho x)) ∧    (∫ x : E3, rho x ∂(volume : Measure E3)) = 1

definition OAI.Problem356.KantorovichAttained line 62

def KantorovichAttained (mu : Measure E3) : Prop :=  ∃ pi : Measure Triple,    IsThreeCoupling mu pi ∧      (∫⁻ t, coulombCost t ∂pi) = kantorovichValue mu

definition OAI.Problem356.NoMongeOptimizer line 67

def NoMongeOptimizer (mu : Measure E3) : Prop :=  ∀ T2 T3 : E3 → E3,    Preserves mu T2 → Preserves mu T3 →      kantorovichValue mu < graphCost mu T2 T3

definition OAI.Problem356.FiniteMongeApproximation line 72

def FiniteMongeApproximation (mu : Measure E3) : Prop :=  ∀ epsilon : ℝ, 0 < epsilon →    ∃ T2 T3 : E3 → E3,      Preserves mu T2 ∧      Preserves mu T3 ∧      graphCost mu T2 T3 < ⊤ ∧      graphCost mu T2 T3 ≤        kantorovichValue mu + ENNReal.ofReal epsilon

definition OAI.Problem356.HasCoulombCounterexample line 81

def HasCoulombCounterexample (rho : E3 → ℝ) : Prop :=  IsSmoothCompactProbabilityDensity rho ∧    IsProbabilityMeasure (densityMeasure rho) ∧    kantorovichValue (densityMeasure rho) < ⊤ ∧    KantorovichAttained (densityMeasure rho) ∧    NoMongeOptimizer (densityMeasure rho)

definition OAI.Problem356.HasFullCoulombConclusion line 88

def HasFullCoulombConclusion (rho : E3 → ℝ) : Prop :=  HasCoulombCounterexample rho ∧    mongeValue (densityMeasure rho) =      kantorovichValue (densityMeasure rho) ∧    FiniteMongeApproximation (densityMeasure rho)

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: pi, coupling, rho, value, sqrt, attained, density, mu and two more. Found in the paper’s title, which its main theorem states: three, monge, counterexample, coulomb and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: borel and pair.

Theorem 1.1 of A counterexample to the Monge ansatz for the three-marginal Coulomb cost

Theorem

There is a nonnegative density \rho on \R^3 such that

\int_{\R^3}\rho(x)\dd x=1,\qquad \rho\in C_c^\infty(\R^3),\qquad \sqrt{\rho}\in C_c^\infty(\R^3),

with the following properties. For \mu=\rho\dd x, the value \E(\mu) is finite and is attained by a coupling in \Pi(\mu,\mu,\mu), but every pair of Borel maps T_2,T_3:\R^3\to\R^3 satisfying (T_2)_\#\mu=(T_3)_\#\mu=\mu obeys

\int_{\R^3}c\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>\E(\mu).

The inequality permits the left-hand side to be infinite.

Section 1, The problem and the result Read it in the paper TeX source, line 29

Show the TeX

There is a nonnegative density \(\rho\) on \(\R^3\) such that
\[
 \int_{\R^3}\rho(x)\dd x=1,\qquad
 \rho\in C_c^\infty(\R^3),\qquad
 \sqrt{\rho}\in C_c^\infty(\R^3),
\]
with the following properties. For \(\mu=\rho\dd x\), the value
\(\E(\mu)\) is finite and is attained by a coupling in
\(\Pi(\mu,\mu,\mu)\), but every pair of Borel maps
\(T_2,T_3:\R^3\to\R^3\) satisfying
\((T_2)_\#\mu=(T_3)_\#\mu=\mu\) obeys
\[
 \int_{\R^3}c\bigl(x,T_2(x),T_3(x)\bigr)\dd\mu(x)>\E(\mu).
\]
The inequality permits the left-hand side to be infinite.

Other possible paper matches, closest wording first

Theorem 7.1 of A counterexample to the Monge ansatz for the three-marginal Coulomb cost. Words of the Lean statement, its names split into words, that also appear in the theorem: real, coupling, rho, value, sqrt, attained and five more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: borel and pairs.
Proposition 6.1 of A counterexample to the Monge ansatz for the three-marginal Coulomb cost. Words of the Lean statement, its names split into words, that also appear in the theorem: probability, measure, mu, finite and cost. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: borel and pairs.
Proposition 4.1 of A counterexample to the Monge ansatz for the three-marginal Coulomb cost. Words of the Lean statement, its names split into words, that also appear in the theorem: triples and maps. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: equality.

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 373’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: coulomb_counterexample_and_equal_infima: there is a smooth compactly supported probability density ρ on ℝ³ (with smooth compactly supported √ρ) whose three-marginal Kantorovich problem for the cost Σ 1/|x_i-x_j| with identical marginals has finite value attained by some coupling, no pair of measurable measure-preserving maps (T₂,T₃) attains it, yet the Monge infimum equals the Kantorovich value and finite-cost Monge maps approach it. The summary's extension to every inverse-power Riesz exponent in each dimension ≥ 2 is not stated (docs: 'outside this statement').
Definitions the group asks readers to compare with the paper: invDistance x y := (ofReal ‖x - y‖)⁻¹ in ℝ≥0∞ (value ⊤ on the diagonal), kantorovichValue/mongeValue as ⨅ over couplings / measure-preserving map pairs; all standard
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The inverse-power (Riesz s > 0) extensions in every dimension d ≥ 2.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“For the three-marginal Coulomb cost in $\mathbb R^3$, the formalization constructs a smooth compactly supported probability density, with smooth compactly supported square root, for which no pair of measure-preserving Borel maps attains the Kantorovich minimum.”
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

102 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: 91 to its own declarations, 41 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 coulomb_counterexample_and_equal_infima 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 66 lines. 2 of them are named in the statement itself.

Names in the namespace OAI.Problem356 are listed without it.

abbrev E3line 12in the statementlibrary
abbrev Tripleline 14library
def tripleFstline 16library
def tripleSndline 18library
def tripleThdline 20library
def invDistanceline 22library
def coulombCostline 25library
def IsThreeCouplingline 30library
def densityMeasureline 36library
def Preservesline 39library
def graphCostline 42library
def kantorovichValueline 45library
def mongeValueline 49library
def IsSmoothCompactProbabilityDensityline 54library
def KantorovichAttainedline 62library
def NoMongeOptimizerline 67library
def FiniteMongeApproximationline 72library
def HasCoulombCounterexampleline 81library
def HasFullCoulombConclusionline 88in the statementlibrary
theorem coulomb_counterexample_and_equal_infimaline 94selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheoryopen scoped ENNRealnamespace Problem356abbrev E3 := EuclideanSpace ℝ (Fin 3)abbrev Triple := E3 × (E3 × E3)def tripleFst (t : Triple) : E3 := t.1def tripleSnd (t : Triple) : E3 := t.2.1def tripleThd (t : Triple) : E3 := t.2.2def invDistance (x y : E3) : ℝ≥0∞ :=  (ENNReal.ofReal ‖x - y‖)⁻¹def coulombCost (t : Triple) : ℝ≥0∞ :=  invDistance (tripleFst t) (tripleSnd t) +    invDistance (tripleFst t) (tripleThd t) +    invDistance (tripleSnd t) (tripleThd t)def IsThreeCoupling (mu : Measure E3) (pi : Measure Triple) : Prop :=  IsProbabilityMeasure pi ∧    Measure.map tripleFst pi = mu ∧    Measure.map tripleSnd pi = mu ∧    Measure.map tripleThd pi = mudef densityMeasure (rho : E3 → ℝ) : Measure E3 :=  Measure.withDensity (volume : Measure E3) (fun x => ENNReal.ofReal (rho x))def Preserves (mu : Measure E3) (T : E3 → E3) : Prop :=  Measurable T ∧ Measure.map T mu = mudef graphCost (mu : Measure E3) (T2 T3 : E3 → E3) : ℝ≥0∞ :=  ∫⁻ x, coulombCost (x, (T2 x, T3 x)) ∂mudef kantorovichValue (mu : Measure E3) : ℝ≥0∞ :=  ⨅ (pi : Measure Triple), ⨅ (_h : IsThreeCoupling mu pi),    ∫⁻ t, coulombCost t ∂pidef mongeValue (mu : Measure E3) : ℝ≥0∞ :=  ⨅ (T2 : E3 → E3), ⨅ (T3 : E3 → E3),    ⨅ (_h2 : Preserves mu T2), ⨅ (_h3 : Preserves mu T3),      graphCost mu T2 T3def IsSmoothCompactProbabilityDensity (rho : E3 → ℝ) : Prop :=  (∀ x, 0 ≤ rho x) ∧    ContDiff ℝ (↑(⊤ : ENat) : WithTop ENat) rho ∧    HasCompactSupport rho ∧    ContDiff ℝ (↑(⊤ : ENat) : WithTop ENat) (fun x : E3 => Real.sqrt (rho x)) ∧    HasCompactSupport (fun x : E3 => Real.sqrt (rho x)) ∧    (∫ x : E3, rho x ∂(volume : Measure E3)) = 1def KantorovichAttained (mu : Measure E3) : Prop :=  ∃ pi : Measure Triple,    IsThreeCoupling mu pi ∧      (∫⁻ t, coulombCost t ∂pi) = kantorovichValue mudef NoMongeOptimizer (mu : Measure E3) : Prop :=  ∀ T2 T3 : E3 → E3,    Preserves mu T2 → Preserves mu T3 →      kantorovichValue mu < graphCost mu T2 T3def FiniteMongeApproximation (mu : Measure E3) : Prop :=  ∀ epsilon : ℝ, 0 < epsilon →    ∃ T2 T3 : E3 → E3,      Preserves mu T2 ∧      Preserves mu T3 ∧      graphCost mu T2 T3 < ⊤ ∧      graphCost mu T2 T3 ≤        kantorovichValue mu + ENNReal.ofReal epsilondef HasCoulombCounterexample (rho : E3 → ℝ) : Prop :=  IsSmoothCompactProbabilityDensity rho ∧    IsProbabilityMeasure (densityMeasure rho) ∧    kantorovichValue (densityMeasure rho) < ⊤ ∧    KantorovichAttained (densityMeasure rho) ∧    NoMongeOptimizer (densityMeasure rho)def HasFullCoulombConclusion (rho : E3 → ℝ) : Prop :=  HasCoulombCounterexample rho ∧    mongeValue (densityMeasure rho) =      kantorovichValue (densityMeasure rho) ∧    FiniteMongeApproximation (densityMeasure rho)theorem coulomb_counterexample_and_equal_infima :    ∃ rho : E3 → ℝ, HasFullCoulombConclusion rho := by  sorryend Problem356endend 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 counterexample to the Monge ansatz for the three-marginal Coulomb cost