Lean statement

A nonflat one-homogeneous Bernoulli minimizer in dimension seven

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Theorem selected for verification: OAI.Bernoulli.nonflat_in_seven

Lean statement file: BernoulliNonflatInSeven.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 367 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 nonflat_in_seven : ∃ u : Space 7 → ℝ,    GlobalMinimizer u ∧ Nonzero u ∧ OneHomogeneous u ∧ ¬ Flat u := 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.Bernoulli.Space line 13

abbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d)

definition OAI.Bernoulli.Test line 16

def Test {d : ℕ} (D : Set (Space d)) (φ : Space d → ℝ) : Prop :=  ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧ tsupport φ ⊆ D

definition OAI.Bernoulli.WeakGradient line 20

def WeakGradient {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  ∀ φ, Test D φ → ∀ h : Space d,    (∫ x in D, u x * (fderiv ℝ φ x) h) =      -(∫ x in D, inner ℝ (G x) h * φ x)

definition OAI.Bernoulli.H1 line 27

def H1 {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  MemLp u 2 (volume.restrict D) ∧  MemLp G 2 (volume.restrict D) ∧ WeakGradient D u G

definition OAI.Bernoulli.H10 line 34

def H10 {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  H1 D u G ∧ ∀ ε : ℝ, 0 < ε → ∃ φ : Space d → ℝ,    Test D φ ∧    eLpNorm (fun x => u x - φ x) 2 (volume.restrict D) < ENNReal.ofReal ε ∧    eLpNorm (fun x => G x - gradient φ x) 2 (volume.restrict D) < ENNReal.ofReal ε

definition OAI.Bernoulli.H1Loc line 42

def H1Loc {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  ∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →    H1 (Metric.ball c r) u G

definition OAI.Bernoulli.energy line 48

def energy {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : ℝ :=  ∫ x in D, ‖G x‖ ^ 2 + if 0 < u x then (1 : ℝ) else 0

definition OAI.Bernoulli.LocalMinimizer line 54

def LocalMinimizer {d : ℕ} (D : Set (Space d)) (u : Space d → ℝ) : Prop :=  IsOpen D ∧ ∃ G : Space d → Space d,    H1Loc D u G ∧ (∀ᵐ x ∂volume.restrict D, 0 ≤ u x) ∧    ∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →      ∀ (v : Space d → ℝ) (V : Space d → Space d),        H1 (Metric.ball c r) v V →        (∀ᵐ x ∂volume.restrict (Metric.ball c r), 0 ≤ v x) →        H10 (Metric.ball c r) (fun x => v x - u x) (fun x => V x - G x) →        energy (Metric.ball c r) u G ≤ energy (Metric.ball c r) v V

definition OAI.Bernoulli.GlobalMinimizer line 65

def GlobalMinimizer {d : ℕ} (u : Space d → ℝ) : Prop :=  LocalMinimizer Set.univ u

definition OAI.Bernoulli.OneHomogeneous line 68

def OneHomogeneous {d : ℕ} (u : Space d → ℝ) : Prop :=  ∀ r : ℝ, 0 < r → ∀ᵐ x ∂volume, u (r • x) = r * u x

definition OAI.Bernoulli.Nonzero line 71

def Nonzero {d : ℕ} (u : Space d → ℝ) : Prop :=  ¬ (u =ᵐ[volume] fun _ => 0)

definition OAI.Bernoulli.Flat line 75

def Flat {d : ℕ} (u : Space d → ℝ) : Prop :=  ∃ e : Space d, ‖e‖ = 1 ∧ ∀ᵐ x ∂volume, u x = max (inner ℝ x e) 0

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: flat, energy, nonzero, nonflat, homogeneous, minimizer, one and global. Found in the paper’s title, which its main theorem states: bernoulli.

Theorem 1.1 of The critical dimension for one-phase Bernoulli minimizers

Theorem

Every nonzero one-homogeneous global minimizer of (eq:energy) in \R^d, 1\leq d\leq6, is flat. There is a nonflat one-homogeneous global minimizer in \R^7. Consequently,

d_*=7.

Section 1, Introduction Read it in the paper TeX source, line 73

Show the TeX

Every nonzero one-homogeneous global minimizer of \eqref{eq:energy}
in \(\R^d\), \(1\leq d\leq6\), is flat. There is a nonflat
one-homogeneous global minimizer in \(\R^7\). Consequently,
\[
  d_*=7.
\]

Other possible paper matches, closest wording first

Proposition 2.1 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: representative, positive, nonnegative, euclidean, loc, volume and nine more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: side.
Corollary 1.2 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: bernoulli, loc, nonnegative, energy, local, minimizer and three more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension.
Corollary 3.6 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: arbitrary, real, support, compact, closure, function and two more.
Proposition 3.5 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: euclidean, real, support, compact, closure, function and three more.
Theorem 7.3 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: homogeneous, nonflat, minimizer, one and global. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: dimension.
Proposition 3.4 of The critical dimension for one-phase Bernoulli minimizers. Words of the Lean statement, its names split into words, that also appear in the theorem: inside, metric, volume, nonzero and local. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: side.

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 367’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: nonflat_in_seven: there is u : ℝ⁷ → ℝ that is a nonnegative global minimizer of ∫ |∇u|² + 1_{u>0} (ballwise comparison with H¹ competitors agreeing on the boundary, all balls), nonzero, 1-homogeneous and not a half-space solution max(x·e,0): the existence half of 'seven is the first dimension'. The other half (flatness of such minimizers in dimensions ≤ 6) and the consequences (smooth free boundaries through dimension six, singular set dimension ≤ n-7) are not stated (docs: 'outside this selected statement').
Definitions the group asks readers to compare with the paper: LocalMinimizer/GlobalMinimizer written with hand-built H¹, H¹₀ (L²-approximation by compactly supported smooth functions) and the exact energy ∫ ‖G‖² + if 0 < u x then 1 else 0 with an explicit weak-gradient representative G; flatness, homogeneity and nonvanishing are a.e. statements
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: Flatness of one-homogeneous global minimizers in dimensions ≤ 6 (the 'first' in the headline) and the regularity and singular-set consequences.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The linked formalization proves the existence side: in dimension seven there is a nonzero one-homogeneous global minimizer that is not a half-space solution.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states part of a paper’s theorem”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

86 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: 54 to its own declarations, 42 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 nonflat_in_seven 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 45 lines. 5 of them are named in the statement itself.

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

abbrev Spaceline 13in the statementlibrary
def Testline 16library
def WeakGradientline 20library
def H1line 27library
def H10line 34library
def H1Locline 42library
def energyline 48library
def LocalMinimizerline 54library
def GlobalMinimizerline 65in the statementlibrary
def OneHomogeneousline 68in the statementlibrary
def Nonzeroline 71in the statementlibrary
def Flatline 75in the statementlibrary
theorem nonflat_in_sevenline 78selected theoremlibrary

import Mathlibnamespace OAInoncomputable sectionopen MeasureTheory Set Filteropen scoped ENNReal Topology ContDiffnamespace Bernoulliopen scoped _root_.Bernoulliabbrev Space (d : ℕ) := EuclideanSpace ℝ (Fin d)/-- Compactly supported smooth real tests in the open set. -/def Test {d : ℕ} (D : Set (Space d)) (φ : Space d → ℝ) : Prop :=  ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧ tsupport φ ⊆ D/-- Distributional gradient, tested in every constant direction. -/def WeakGradient {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  ∀ φ, Test D φ → ∀ h : Space d,    (∫ x in D, u x * (fderiv ℝ φ x) h) =      -(∫ x in D, inner ℝ (G x) h * φ x)/-- H¹(D), with its weak gradient. -/def H1 {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  MemLp u 2 (volume.restrict D) ∧  MemLp G 2 (volume.restrict D) ∧ WeakGradient D u G/-- H¹₀(D) is H¹-closure of C∞ compactly supported functions.Both L² errors tend to zero. This does not restrict competitors to smooth ones. -/def H10 {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  H1 D u G ∧ ∀ ε : ℝ, 0 < ε → ∃ φ : Space d → ℝ,    Test D φ ∧    eLpNorm (fun x => u x - φ x) 2 (volume.restrict D) < ENNReal.ofReal ε ∧    eLpNorm (fun x => G x - gradient φ x) 2 (volume.restrict D) < ENNReal.ofReal ε/-- Ballwise H¹_loc, on an open set (balls have compact closure inside D). -/def H1Loc {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : Prop :=  ∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →    H1 (Metric.ball c r) u G/-- The exact volume-one Bernoulli energy, not just Dirichlet energy. -/def energy {d : ℕ} (D : Set (Space d))    (u : Space d → ℝ) (G : Space d → Space d) : ℝ :=  ∫ x in D, ‖G x‖ ^ 2 + if 0 < u x then (1 : ℝ) else 0/-- Full ballwise local comparison. Competitor and gradient are arbitrarySobolev representatives on the ball, with zero-trace difference. -/def LocalMinimizer {d : ℕ} (D : Set (Space d)) (u : Space d → ℝ) : Prop :=  IsOpen D ∧ ∃ G : Space d → Space d,    H1Loc D u G ∧ (∀ᵐ x ∂volume.restrict D, 0 ≤ u x) ∧    ∀ c : Space d, ∀ r : ℝ, 0 < r → Metric.closedBall c r ⊆ D →      ∀ (v : Space d → ℝ) (V : Space d → Space d),        H1 (Metric.ball c r) v V →        (∀ᵐ x ∂volume.restrict (Metric.ball c r), 0 ≤ v x) →        H10 (Metric.ball c r) (fun x => v x - u x) (fun x => V x - G x) →        energy (Metric.ball c r) u G ≤ energy (Metric.ball c r) v V/-- Nonnegative global minimizer, with all centers and all positive radii. -/def GlobalMinimizer {d : ℕ} (u : Space d → ℝ) : Prop :=  LocalMinimizer Set.univ udef OneHomogeneous {d : ℕ} (u : Space d → ℝ) : Prop :=  ∀ r : ℝ, 0 < r → ∀ᵐ x ∂volume, u (r • x) = r * u xdef Nonzero {d : ℕ} (u : Space d → ℝ) : Prop :=  ¬ (u =ᵐ[volume] fun _ => 0)/-- Flatness is a.e. equality of H¹ representatives. -/def Flat {d : ℕ} (u : Space d → ℝ) : Prop :=  ∃ e : Space d, ‖e‖ = 1 ∧ ∀ᵐ x ∂volume, u x = max (inner ℝ x e) 0theorem nonflat_in_seven : ∃ u : Space 7 → ℝ,    GlobalMinimizer u ∧ Nonzero u ∧ OneHomogeneous u ∧ ¬ Flat u := by  sorryend Bernoulliendend 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 critical dimension for one-phase Bernoulli minimizers

Theorem

Every nonzero one-homogeneous global minimizer of (eq:energy) in \R^d, 1\leq d\leq6, is flat. There is a nonflat one-homogeneous global minimizer in \R^7. Consequently,

d_*=7.

Section 1, Introduction Read it in the paper TeX source, line 73

Show the TeX

Every nonzero one-homogeneous global minimizer of \eqref{eq:energy}
in \(\R^d\), \(1\leq d\leq6\), is flat. There is a nonflat
one-homogeneous global minimizer in \(\R^7\). Consequently,
\[
  d_*=7.
\]

Corollary (Sharp singular-set bound)

Let n\ge2, let D\subset\R^n be open, and let u\in H^1_{\mathrm{loc}}(D) be a nonnegative local minimizer of the one-phase Bernoulli energy in D. Its interior free boundary is smooth outside \operatorname{Sing}(u). This set is empty when n\le6, is locally finite when n=7, and satisfies

\dim_{\mathrm H}\operatorname{Sing}(u)\le n-7\qquad(n\ge7).

For every n\ge7, a global minimizer attains this dimension bound.

Section 1, Introduction Read it in the paper TeX source, line 90

Show the TeX

Let $n\ge2$, let $D\subset\R^n$ be open, and let $u\in H^1_{\mathrm{loc}}(D)$ be a nonnegative local
minimizer of the one-phase Bernoulli energy in $D$.
Its interior free boundary is smooth outside $\operatorname{Sing}(u)$.
This set is empty when $n\le6$, is locally finite when $n=7$, and satisfies
\[
 \dim_{\mathrm H}\operatorname{Sing}(u)\le n-7\qquad(n\ge7).
\]
For every $n\ge7$, a global minimizer attains this dimension bound.

Configuration and file facts

File: lean/ComparatorChallenges/BernoulliNonflatInSeven.lean
OpenAI’s Lean catalogue: OpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 367 link it.
Solution module: OAI.Analysis.BernoulliCone.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: 12 local vocabulary declarations
Lines of Lean, without comments or blank lines: 45
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.

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