Lean statement

Isoperimetric profile and all minimizers in the cubic flat three-torus

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.CubicTorus.unit_cubic_isoperimetric

Lean statement file: CubicTorus.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 354 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 unit_cubic_isoperimetric (V : ℝ) (hV : V ∈ Ioo 0 1) :    (∃ E : Set Torus, IsMinimizerAt V E) ∧    (∀ E : Set Torus, IsMinimizerAt V E →      perimeter E = ENNReal.ofReal (candidateProfile V)) ∧    (∀ E : Set Torus, IsMinimizerAt V E ↔ StandardRegionAt V E) := 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.CubicTorus.Cover line 11

abbrev Cover := EuclideanSpace ℝ (Fin 3)

abbreviation OAI.CubicTorus.Torus line 12

abbrev Torus := PiLp 2 (fun _ : Fin 3 => AddCircle (1 : ℝ))

definition OAI.CubicTorus.quotientMap line 15

def quotientMap (x : Cover) : Torus :=  WithLp.toLp 2 (fun i => (x i : AddCircle (1 : ℝ)))

definition OAI.CubicTorus.fundamentalCube line 19

def fundamentalCube : Set Cover := {x | ∀ i, 0 ≤ x i ∧ x i < 1}

definition OAI.CubicTorus.torusVolume line 22

def torusVolume : Measure Torus :=  Measure.map quotientMap (volume.restrict fundamentalCube)

definition OAI.CubicTorus.latticeVector line 26

def latticeVector (n : Fin 3 → ℤ) : Cover := WithLp.toLp 2 (fun i => (n i : ℝ))

definition OAI.CubicTorus.AdmissibleField line 31

def AdmissibleField (F : Cover → Cover) : Prop :=  ContDiff ℝ (⊤ : ℕ∞) F ∧  (∀ (n : Fin 3 → ℤ) (x : Cover), F (x + latticeVector n) = F x) ∧  ∀ x, ‖F x‖ ≤ 1

definition OAI.CubicTorus.divergence line 37

def divergence (F : Cover → Cover) (x : Cover) : ℝ :=  ∑ i : Fin 3, (fderiv ℝ F x (EuclideanSpace.single i 1)) i

definition OAI.CubicTorus.perimeter line 41

def perimeter (E : Set Torus) : ℝ≥0∞ :=  ⨆ (F : Cover → Cover) (_ : AdmissibleField F),    ENNReal.ofReal (∫ x in fundamentalCube ∩ quotientMap ⁻¹' E, divergence F x)

definition OAI.CubicTorus.FinitePerimeter line 46

def FinitePerimeter (E : Set Torus) : Prop :=  NullMeasurableSet E torusVolume ∧ perimeter E < ⊤

definition OAI.CubicTorus.IsMinimizerAt line 50

def IsMinimizerAt (V : ℝ) (E : Set Torus) : Prop :=  FinitePerimeter E ∧ torusVolume E = ENNReal.ofReal V ∧  ∀ F : Set Torus, FinitePerimeter F → torusVolume F = ENNReal.ofReal V →    perimeter E ≤ perimeter F

definition OAI.CubicTorus.candidateProfile line 56

def candidateProfile (V : ℝ) : ℝ :=  let v := min V (1 - V)  min ((36 * Real.pi) ^ (1 / 3 : ℝ) * v ^ (2 / 3 : ℝ))    (min (2 * Real.sqrt (Real.pi * v)) 2)

definition OAI.CubicTorus.canonicalBall line 62

def canonicalBall (V : ℝ) : Set Torus :=  Metric.ball 0 ((3 * V / (4 * Real.pi)) ^ (1 / 3 : ℝ))

definition OAI.CubicTorus.canonicalTube line 66

def canonicalTube (V : ℝ) : Set Torus :=  {x | ‖x 1‖ ^ 2 + ‖x 2‖ ^ 2 < (Real.sqrt (V / Real.pi)) ^ 2}

definition OAI.CubicTorus.canonicalSlab line 70

def canonicalSlab (V : ℝ) : Set Torus := {x | ‖x 0‖ < V / 2}

definition OAI.CubicTorus.CongruentAE line 73

def CongruentAE (E F : Set Torus) : Prop :=  ∃ φ : Torus ≃ᵢ Torus, E =ᵐ[torusVolume] (φ '' F)

definition OAI.CubicTorus.SmallStandardRegion line 77

def SmallStandardRegion (V : ℝ) (E : Set Torus) : Prop :=  (V ≤ 4 * Real.pi / 81 ∧ CongruentAE E (canonicalBall V)) ∨  (4 * Real.pi / 81 ≤ V ∧ V ≤ 1 / Real.pi ∧ CongruentAE E (canonicalTube V)) ∨  (1 / Real.pi ≤ V ∧ CongruentAE E (canonicalSlab V))

definition OAI.CubicTorus.StandardRegionAt line 83

def StandardRegionAt (V : ℝ) (E : Set Torus) : Prop :=  if V ≤ 1 / 2 then SmallStandardRegion V E  else SmallStandardRegion (1 - V) Eᶜ

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: 1/3, 2/3, 1/2, radius, real, shortest, unit, geodesic and 26 more. Found in the paper’s title, which its main theorem states: cubic, torus, flat and isoperimetric. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circular.

Theorem 1.1 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus

Theorem

For V\in(0,1), put v=\min(V,1-V). Among finite-perimeter subsets of \mathbb R^3/\mathbb Z^3 of volume V, the minimum perimeter is

I_{\rm unit}(V)= \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.

For 0<V\le1/2, every minimizing region is, up to an ambient isometry and a null set, one of the following:

\begin{array}{c|c} \text{volume range}&\text{region}\\ \hline 0<V\le 4\pi/81 & \text{a ball of radius }(3V/(4\pi))^{1/3}\\[2pt] 4\pi/81\le V\le1/\pi &\text{a solid circular tube of radius }\sqrt{V/\pi}\\ &\text{about a shortest closed geodesic}\\[2pt] 1/\pi\le V\le1/2 &\text{a slab of width }V\text{ between coordinate tori}. \end{array}

Both listed types occur at each common endpoint of adjacent ranges, and there are no further equality cases. For V>1/2, the minimizing regions are the complements of those at volume 1-V. At zero or full volume the conclusion is trivial.

Section 1, Introduction Read it in the paper TeX source, line 56

Show the TeX

For \(V\in(0,1)\), put \(v=\min(V,1-V)\).
Among finite-perimeter subsets of \(\mathbb R^3/\mathbb Z^3\)
of volume \(V\), the minimum perimeter is
\begin{equation}
 I_{\rm unit}(V)=
 \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.
 
\end{equation}
For \(0<V\le1/2\), every minimizing region is, up to an ambient
isometry and a null set, one of the following:
\[
\begin{array}{c|c}
\text{volume range}&\text{region}\\ \hline
0<V\le 4\pi/81 & \text{a ball of radius }(3V/(4\pi))^{1/3}\\[2pt]
4\pi/81\le V\le1/\pi
 &\text{a solid circular tube of radius }\sqrt{V/\pi}\\
 &\text{about a shortest closed geodesic}\\[2pt]
1/\pi\le V\le1/2
 &\text{a slab of width }V\text{ between coordinate tori}.
\end{array}
\]
Both listed types occur at each common endpoint of adjacent ranges,
and there are no further equality cases.
For \(V>1/2\), the minimizing regions are the complements of those
at volume \(1-V\). At zero or full volume the conclusion is trivial.

Other possible paper matches, closest wording first

Theorem 2.1 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: quotient, boundary, parallel, flat, two, isoperimetric and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: circular and three.
Proposition 2.4 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: standard, two, one, volume, regions and minimizer.
Proposition 3.3 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: coordinates, volume, region and minimizer. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: three.
Proposition 5.3 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, minimizer and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: three.
Proposition 5.2 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: minimizer and pi. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: three.
Proposition 5.1 of The Isoperimetric Conjecture for the Cubic Flat Three-Torus. Words of the Lean statement, its names split into words, that also appear in the theorem: minimizer and pi.

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

One independent group reports that Comparator accepted a run for this Lean statement file: Carlos Toledo’s cert-machine. 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

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 354’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: unit_cubic_isoperimetric for every V ∈ (0,1): a minimizer exists; every minimizer has perimeter candidateProfile V = min((36π)^(1/3)v^(2/3), 2√(πv), 2) with v = min(V,1-V); and minimizers are exactly (up to isometry and null sets, with complements for V > 1/2) balls, tubes about a shortest geodesic, or slabs on the ranges V ≤ 4π/81, 4π/81 ≤ V ≤ 1/π, V ≥ 1/π, with both adjacent types allowed at the transition volumes.
Definitions the group asks readers to compare with the paper: perimeter E := ⨆ F admissible periodic C^∞ field with ‖F‖ ≤ 1, ofReal ∫_{cube ∩ E} div F (De Giorgi perimeter by hand on the cover) and torusVolume := map quotientMap (volume.restrict fundamentalCube): standard
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 determines the isoperimetric profile and all minimizers in the unit cubic flat three-torus. At every volume $0<V<1$, a minimizer exists, and the minimizers, up to null-set changes, are exactly balls, circular tubes about shortest closed geodesics, coordinate slabs, and their complements in the appropriate volume ranges.”
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

98 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: 58 to its own declarations, 51 to Mathlib’s documentation. 19 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 unit_cubic_isoperimetric 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 18 other declarations of the file; with the theorem itself they span 47 lines. 5 of them are named in the statement itself.

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

abbrev Coverline 11library
abbrev Torusline 12in the statementlibrary
def quotientMapline 15library
def fundamentalCubeline 19library
def torusVolumeline 22library
def latticeVectorline 26library
def AdmissibleFieldline 31library
def divergenceline 37library
def perimeterline 41in the statementlibrary
def FinitePerimeterline 46library
def IsMinimizerAtline 50in the statementlibrary
def candidateProfileline 56in the statementlibrary
def canonicalBallline 62library
def canonicalTubeline 66library
def canonicalSlabline 70library
def CongruentAEline 73library
def SmallStandardRegionline 77library
def StandardRegionAtline 83in the statementlibrary
theorem unit_cubic_isoperimetricline 88selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory Setopen scoped ENNRealnamespace CubicTorusabbrev Cover := EuclideanSpace ℝ (Fin 3)abbrev Torus := PiLp 2 (fun _ : Fin 3 => AddCircle (1 : ℝ))/-- The quotient R³ → R³/Z³ in orthonormal coordinates. -/def quotientMap (x : Cover) : Torus :=  WithLp.toLp 2 (fun i => (x i : AddCircle (1 : ℝ)))/-- One unit-volume fundamental domain; its boundary is immaterial to integration. -/def fundamentalCube : Set Cover := {x | ∀ i, 0 ≤ x i ∧ x i < 1}/-- The Riemannian volume measure of the unit flat torus. -/def torusVolume : Measure Torus :=  Measure.map quotientMap (volume.restrict fundamentalCube)/-- An integer lattice vector in the Euclidean cover. -/def latticeVector (n : Fin 3 → ℤ) : Cover := WithLp.toLp 2 (fun i => (n i : ℝ))/-- The smooth vector fields on the flat torus with pointwise length at most one,expressed as Z³-periodic fields on its Euclidean cover. The explicit ℕ∞ indexis C^∞; top of WithTop ℕ∞ would instead mean analytic in this Mathlib. -/def AdmissibleField (F : Cover → Cover) : Prop :=  ContDiff ℝ (⊤ : ℕ∞) F ∧  (∀ (n : Fin 3 → ℤ) (x : Cover), F (x + latticeVector n) = F x) ∧  ∀ x, ‖F x‖ ≤ 1/-- Euclidean divergence in the orthonormal coordinate frame. -/def divergence (F : Cover → Cover) (x : Cover) : ℝ :=  ∑ i : Fin 3, (fderiv ℝ F x (EuclideanSpace.single i 1)) i/-- De Giorgi perimeter, allowed to be infinite; no regularity of E is imposed. -/def perimeter (E : Set Torus) : ℝ≥0∞ :=  ⨆ (F : Cover → Cover) (_ : AdmissibleField F),    ENNReal.ofReal (∫ x in fundamentalCube ∩ quotientMap ⁻¹' E, divergence F x)/-- Finite-perimeter measurable regions, including all null-set modifications. -/def FinitePerimeter (E : Set Torus) : Prop :=  NullMeasurableSet E torusVolume ∧ perimeter E < ⊤/-- A global perimeter minimizer under the prescribed volume constraint. -/def IsMinimizerAt (V : ℝ) (E : Set Torus) : Prop :=  FinitePerimeter E ∧ torusVolume E = ENNReal.ofReal V ∧  ∀ F : Set Torus, FinitePerimeter F → torusVolume F = ENNReal.ofReal V →    perimeter E ≤ perimeter F/-- The candidate unit-torus isoperimetric profile. -/def candidateProfile (V : ℝ) : ℝ :=  let v := min V (1 - V)  min ((36 * Real.pi) ^ (1 / 3 : ℝ) * v ^ (2 / 3 : ℝ))    (min (2 * Real.sqrt (Real.pi * v)) 2)/-- Canonical centered ball. Its radius on the listed branch is below 1/2. -/def canonicalBall (V : ℝ) : Set Torus :=  Metric.ball 0 ((3 * V / (4 * Real.pi)) ^ (1 / 3 : ℝ))/-- Canonical tube around the shortest closed geodesic in coordinate direction 0. -/def canonicalTube (V : ℝ) : Set Torus :=  {x | ‖x 1‖ ^ 2 + ‖x 2‖ ^ 2 < (Real.sqrt (V / Real.pi)) ^ 2}/-- Canonical slab of width V between two parallel coordinate tori. -/def canonicalSlab (V : ℝ) : Set Torus := {x | ‖x 0‖ < V / 2}/-- Equality of regions up to ambient flat isometry and a null set. -/def CongruentAE (E F : Set Torus) : Prop :=  ∃ φ : Torus ≃ᵢ Torus, E =ᵐ[torusVolume] (φ '' F)/-- Exactly the closed phase ranges, so adjacent types are BOTH included at ties. -/def SmallStandardRegion (V : ℝ) (E : Set Torus) : Prop :=  (V ≤ 4 * Real.pi / 81 ∧ CongruentAE E (canonicalBall V)) ∨  (4 * Real.pi / 81 ≤ V ∧ V ≤ 1 / Real.pi ∧ CongruentAE E (canonicalTube V)) ∨  (1 / Real.pi ≤ V ∧ CongruentAE E (canonicalSlab V))/-- The equality class at every interior volume, including complementation. -/def StandardRegionAt (V : ℝ) (E : Set Torus) : Prop :=  if V ≤ 1 / 2 then SmallStandardRegion V E  else SmallStandardRegion (1 - V) Eᶜ/-- Attainment, the sharp profile, and all equality cases at interior volumes. -/theorem unit_cubic_isoperimetric (V : ℝ) (hV : V ∈ Ioo 0 1) :    (∃ E : Set Torus, IsMinimizerAt V E) ∧    (∀ E : Set Torus, IsMinimizerAt V E →      perimeter E = ENNReal.ofReal (candidateProfile V)) ∧    (∀ E : Set Torus, IsMinimizerAt V E ↔ StandardRegionAt V E) := by  sorryend CubicTorusendend 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.

The Isoperimetric Conjecture for the Cubic Flat Three-Torus

Theorem

For V\in(0,1), put v=\min(V,1-V). Among finite-perimeter subsets of \mathbb R^3/\mathbb Z^3 of volume V, the minimum perimeter is

I_{\rm unit}(V)= \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.

For 0<V\le1/2, every minimizing region is, up to an ambient isometry and a null set, one of the following:

Section 1, Introduction Read it in the paper TeX source, line 56

Show the TeX

For \(V\in(0,1)\), put \(v=\min(V,1-V)\).
Among finite-perimeter subsets of \(\mathbb R^3/\mathbb Z^3\)
of volume \(V\), the minimum perimeter is
\begin{equation}
 I_{\rm unit}(V)=
 \min\left\{(36\pi)^{1/3}v^{2/3},\ 2\sqrt{\pi v},\ 2\right\}.
 
\end{equation}
For \(0<V\le1/2\), every minimizing region is, up to an ambient
isometry and a null set, one of the following:
\[
\begin{array}{c|c}
\text{volume range}&\text{region}\\ \hline
0<V\le 4\pi/81 & \text{a ball of radius }(3V/(4\pi))^{1/3}\\[2pt]
4\pi/81\le V\le1/\pi
 &\text{a solid circular tube of radius }\sqrt{V/\pi}\\
 &\text{about a shortest closed geodesic}\\[2pt]
1/\pi\le V\le1/2
 &\text{a slab of width }V\text{ between coordinate tori}.
\end{array}
\]
Both listed types occur at each common endpoint of adjacent ranges,
and there are no further equality cases.
For \(V>1/2\), the minimizing regions are the complements of those
at volume \(1-V\). At zero or full volume the conclusion is trivial.

Configuration and file facts

File: lean/ComparatorChallenges/CubicTorus.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 354 link it.
Solution module: OAI.Geometry.CubicTorus.Main
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: 18 local vocabulary declarations
Lines of Lean, without comments or blank lines: 47
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