Lean statement

Unbounded nodal ratio on $S^4\times S^1$

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.Yau.Target.yau_nodal_set_upper_bound_counterexample

Lean statement file: YauCounterexample.leanShow on the mapFamily 350. 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 MainTarget : Prop :=  ∃ (g : SmoothMetric) (lam : ℕ → ℝ) (u : ℕ → Manifold5 → ℝ),    (∀ k, 0 < lam k) ∧    (∀ k, u k ≠ 0) ∧    (∀ k, ContMDiff modelWithCorners 𝓘(ℝ, ℝ) ∞ (u k)) ∧    (∀ k x, -chartLaplacian g (extChartAt modelWithCorners x) (u k) ((extChartAt modelWithCorners x) x) =      lam k * u k x) ∧    Tendsto lam atTop atTop ∧    Tendsto (fun k ↦ nodalMeasure g (u k) / ENNReal.ofReal (Real.sqrt (lam k)))      atTop (𝓝 ⊤)⋮theorem yau_nodal_set_upper_bound_counterexample : MainTarget := 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.Yau.Target.Base line 11

abbrev Base := Metric.sphere (0 : EuclideanSpace ℝ (Fin 5)) 1

abbreviation OAI.Yau.Target.Manifold5 line 13
```
abbrev Manifold5 := Base × Circle
```

abbreviation OAI.Yau.Target.Model line 15

abbrev Model := EuclideanSpace ℝ (Fin 4) × EuclideanSpace ℝ (Fin 1)

abbreviation OAI.Yau.Target.modelWithCorners line 17

abbrev modelWithCorners := (𝓡 4).prod (𝓡 1)

abbreviation OAI.Yau.Target.SmoothMetric line 19

abbrev SmoothMetric := ContMDiffRiemannianMetric modelWithCorners ∞ Model  (fun x : Manifold5 ↦ TangentSpace modelWithCorners x)

abbreviation OAI.Yau.Target.Index line 22

abbrev Index := Fin (Module.finrank ℝ Model)

definition OAI.Yau.Target.frame line 24

def frame : Module.Basis Index ℝ Model := Module.finBasis ℝ Model

definition OAI.Yau.Target.metricMatrix line 26

def metricMatrix (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (y : Model) : Matrix Index Index ℝ := fun i j ↦  g.inner (c.symm y)    (mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame i))    (mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame j))

definition OAI.Yau.Target.volumeFactor line 32

def volumeFactor (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (y : Model) : ℝ := Real.sqrt (metricMatrix g c y).det

definition OAI.Yau.Target.chartLaplacian line 35

def chartLaplacian (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (u : Manifold5 → ℝ) (y : Model) : ℝ :=  (volumeFactor g c y)⁻¹ * ∑ i, fderiv ℝ    (fun z ↦ volumeFactor g c z * ∑ j,      (metricMatrix g c z)⁻¹ i j * fderiv ℝ (u ∘ c.symm) z (frame j)) y (frame i)

definition OAI.Yau.Target.nodalMeasure line 41

def nodalMeasure (g : SmoothMetric) (u : Manifold5 → ℝ) : ℝ≥0∞ :=

Show the remaining 11 lines

  letI : RiemannianBundle (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=    ⟨g.toRiemannianMetric⟩  letI : IsContMDiffRiemannianBundle modelWithCorners ∞ Model      (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) := inferInstance  letI : MeasurableSpace Manifold5 := borel Manifold5  letI : BorelSpace Manifold5 := ⟨rfl⟩  letI : IsContinuousRiemannianBundle Model      (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=    ⟨g.inner, g.toContinuousRiemannianMetric.continuous, fun _ _ _ ↦ rfl⟩  letI : EMetricSpace Manifold5 := EMetricSpace.ofRiemannianMetric modelWithCorners Manifold5  MeasureTheory.Measure.hausdorffMeasure (4 : ℝ) {x | u x = 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: sqrt, riemannian, metric, real and smooth. Found in the paper’s title, which its main theorem states: counterexamples, yaus, upper, nodal and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: positive and nonzero.

Statements in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four come first because the Lean file and module names share words with that paper’s title.

Theorem 1.1 of Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four

Theorem

Let g_* be the round metric on \Sph^3. Every neighborhood of g_* in the C^\infty topology contains a smooth Riemannian metric g_\infty for which there are nonzero smooth real functions u_j and positive numbers \lambda_j satisfying

-\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad \lambda_j\longrightarrow\infty,\qquad \frac{\HH^2_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}} \longrightarrow\infty.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

Let $g_*$ be the round metric on $\Sph^3$. Every neighborhood of $g_*$ in
the $C^\infty$ topology contains a smooth Riemannian metric $g_\infty$ for
which there are nonzero smooth real functions $u_j$ and positive numbers
$\lambda_j$ satisfying
\[
 -\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad
 \lambda_j\longrightarrow\infty,\qquad
 \frac{\HH^2_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}}
       \longrightarrow\infty.
\]

Other possible paper matches, closest wording first

Theorem 1.1 of Power-law violations of Yau’s nodal upper bound. Words of the Lean statement, its names split into words, that also appear in the theorem: riemannian, metric, real and smooth. Found in the paper’s title, which its main theorem states: yaus, upper, nodal and bound. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: throughout, sequence, times, eigenvalues, positive and nonzero.
Corollary 5.6 of Power-law violations of Yau’s nodal upper bound. Words of the Lean statement, its names split into words, that also appear in the theorem: manifold, nodal, measure, riemannian, integer and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: divided and dimensional.
Theorem 1.1 of Sharp nodal length on smooth surfaces. Words of the Lean statement, its names split into words, that also appear in the theorem: hausdorff, measure, sqrt, riemannian and real. Found in the paper’s title, which its main theorem states: nodal and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: eigenfunction, positive and nonzero.
Proposition 4.4 of Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four. Words of the Lean statement, its names split into words, that also appear in the theorem: continuous, upper, riemannian, integers, metric, real and one more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: fix, positive and nonzero.
Theorem 1.2 of Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four. Words of the Lean statement, its names split into words, that also appear in the theorem: sqrt, riemannian, metric, real and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: times, positive and nonzero.
Proposition 2.2 of Power-law violations of Yau’s nodal upper bound. Words of the Lean statement, its names split into words, that also appear in the theorem: base, chart, integer, real and smooth. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: eigenvalue, fix, positive and nonzero.

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 350’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: NodalLength.Main is the headline: on every compact connected smooth Riemannian surface there is C with H¹(nodal set) ≤ C√λ for every nonzero smooth eigenfunction with -Δu = λu, λ > 0; sphere_three (every C^∞ neighbourhood of the round S³ metric contains a metric with unbounded nodal ratio) and sphere_two_torus_two give dimensions 3 and 4. For dimension five, YauCounterexample (S⁴×S¹) only gives H⁴(nodal)/√λ → ∞, not the summary's power-law growth λ^(1/2+ε₀), so that claim is only stated in a weaker form.
Definitions the group asks readers to compare with the paper: Laplace-Beltrami and nodal measures are hand-written: laplaceBeltrami (det(g)^(-1/2) ∂_i(det(g)^(1/2) g^{ij} ∂_j u) in the centre chart of each point), nodalMeasure = d-dimensional Hausdorff measure for EMetricSpace.ofRiemannianMetric; NodalLength uses a MetricSpace M instance tied to the Riemannian structure only through [IsRiemannianManifold] ; IsSmoothNeighborhood g₀ N (finitely many compact-chart C^k seminorm tests with one ε) as the C^∞ topology on metrics; IsRound as the pullback of the Euclidean inner product
Carlos Toledo’s cert-machine
Finds part of it stated, or a narrower or weaker statement Their term: “NARROWER — UNDECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so. Their record
Their note: The dimension-five power-law violation (nodal measure ≥ λ^{1/2+ε₀} along a sequence, for a fixed ε₀ > 0) is not stated; only nodal/√λ → ∞ is.

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalized counterexample fixes one smooth metric on $S^4\times S^1$ and a sequence of nonzero smooth eigenfunctions whose positive eigenvalues tend to infinity, while the intrinsic four-dimensional nodal measures divided by the square roots of the eigenvalues tend to infinity. The metric is fixed throughout the sequence.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a weaker form of a paper’s theorem”. 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: 61 to its own declarations, 48 to Mathlib’s documentation. 13 declared names also link to the same declaration in OpenAI’s library. 3 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names 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 yau_nodal_set_upper_bound_counterexample 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 44 lines. 1 of them is named in the statement itself.

Names in the namespace OAI.Yau.Target are listed without it.

abbrev Baseline 11library
abbrev Manifold5line 13library
abbrev Modelline 15library
abbrev modelWithCornersline 17library
abbrev SmoothMetricline 19library
abbrev Indexline 22library
def frameline 24library
def metricMatrixline 26library
def volumeFactorline 32library
def chartLaplacianline 35library
def nodalMeasureline 41library
def MainTargetline 54in the statementlibrary
theorem yau_nodal_set_upper_bound_counterexampleline 65selected theoremlibrary

import Mathlibnamespace OAInamespace Yaunamespace Targetopen Bundle Manifold Matrix Filteropen scoped ContDiff Topology ENNRealnoncomputable sectionabbrev Base := Metric.sphere (0 : EuclideanSpace ℝ (Fin 5)) 1abbrev Manifold5 := Base × Circleabbrev Model := EuclideanSpace ℝ (Fin 4) × EuclideanSpace ℝ (Fin 1)abbrev modelWithCorners := (𝓡 4).prod (𝓡 1)abbrev SmoothMetric := ContMDiffRiemannianMetric modelWithCorners ∞ Model  (fun x : Manifold5 ↦ TangentSpace modelWithCorners x)abbrev Index := Fin (Module.finrank ℝ Model)def frame : Module.Basis Index ℝ Model := Module.finBasis ℝ Modeldef metricMatrix (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (y : Model) : Matrix Index Index ℝ := fun i j ↦  g.inner (c.symm y)    (mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame i))    (mfderiv 𝓘(ℝ, Model) modelWithCorners c.symm y (frame j))def volumeFactor (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (y : Model) : ℝ := Real.sqrt (metricMatrix g c y).detdef chartLaplacian (g : SmoothMetric) (c : PartialEquiv Manifold5 Model)    (u : Manifold5 → ℝ) (y : Model) : ℝ :=  (volumeFactor g c y)⁻¹ * ∑ i, fderiv ℝ    (fun z ↦ volumeFactor g c z * ∑ j,      (metricMatrix g c z)⁻¹ i j * fderiv ℝ (u ∘ c.symm) z (frame j)) y (frame i)def nodalMeasure (g : SmoothMetric) (u : Manifold5 → ℝ) : ℝ≥0∞ :=  letI : RiemannianBundle (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=    ⟨g.toRiemannianMetric⟩  letI : IsContMDiffRiemannianBundle modelWithCorners ∞ Model      (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) := inferInstance  letI : MeasurableSpace Manifold5 := borel Manifold5  letI : BorelSpace Manifold5 := ⟨rfl⟩  letI : IsContinuousRiemannianBundle Model      (fun x : Manifold5 ↦ TangentSpace modelWithCorners x) :=    ⟨g.inner, g.toContinuousRiemannianMetric.continuous, fun _ _ _ ↦ rfl⟩  letI : EMetricSpace Manifold5 := EMetricSpace.ofRiemannianMetric modelWithCorners Manifold5  MeasureTheory.Measure.hausdorffMeasure (4 : ℝ) {x | u x = 0}def MainTarget : Prop :=  ∃ (g : SmoothMetric) (lam : ℕ → ℝ) (u : ℕ → Manifold5 → ℝ),    (∀ k, 0 < lam k) ∧    (∀ k, u k ≠ 0) ∧    (∀ k, ContMDiff modelWithCorners 𝓘(ℝ, ℝ) ∞ (u k)) ∧    (∀ k x, -chartLaplacian g (extChartAt modelWithCorners x) (u k) ((extChartAt modelWithCorners x) x) =      lam k * u k x) ∧    Tendsto lam atTop atTop ∧    Tendsto (fun k ↦ nodalMeasure g (u k) / ENNReal.ofReal (Real.sqrt (lam k)))      atTop (𝓝 ⊤)theorem yau_nodal_set_upper_bound_counterexample : MainTarget := by  sorryendend Targetend Yauend 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.

Sharp nodal length on smooth surfaces

Theorem

Let (M,g) be a closed connected C^\infty Riemannian surface. There is a finite constant C=C(M,g) such that every nonzero real Laplace eigenfunction satisfying

-\Delta_g u=\lambda u,\qquad \lambda>0,

obeys

\mathcal H_g^1(Z_u)\le C\sqrt\lambda.

Here \mathcal H_g^1 is Hausdorff measure for the Riemannian distance. The same constant applies to every eigenfunction in every positive eigenspace.

Section 1, Introduction and main result Read it in the paper TeX source, line 7

Show the TeX

Let $(M,g)$ be a closed connected $C^\infty$ Riemannian surface.
There is a finite constant $C=C(M,g)$ such that every nonzero real
Laplace eigenfunction satisfying
\[
 -\Delta_g u=\lambda u,\qquad \lambda>0,
\]
obeys
\[
 \mathcal H_g^1(Z_u)\le C\sqrt\lambda.
\]
Here $\mathcal H_g^1$ is Hausdorff measure for the Riemannian distance.
The same constant applies to every eigenfunction in every positive
eigenspace.

Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four

Theorem

-\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad \lambda_j\longrightarrow\infty,\qquad \frac{\HH^2_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}} \longrightarrow\infty.

Section 1, Introduction Read it in the paper TeX source, line 18

Show the TeX

Let $g_*$ be the round metric on $\Sph^3$. Every neighborhood of $g_*$ in
the $C^\infty$ topology contains a smooth Riemannian metric $g_\infty$ for
which there are nonzero smooth real functions $u_j$ and positive numbers
$\lambda_j$ satisfying
\[
 -\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad
 \lambda_j\longrightarrow\infty,\qquad
 \frac{\HH^2_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}}
       \longrightarrow\infty.
\]

Theorem

There are a smooth Riemannian metric g_\infty on S^2\times\T^2, nonzero smooth real functions u_j, and positive numbers \lambda_j\to\infty such that

-\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad \frac{\cH^3_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}} \longrightarrow\infty.

Section 1, Introduction Read it in the paper TeX source, line 31

Show the TeX

There are a smooth Riemannian metric $g_\infty$ on $S^2\times\T^2$,
nonzero smooth real functions $u_j$, and positive numbers
$\lambda_j\to\infty$ such that
\[
 -\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad
 \frac{\cH^3_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}}
       \longrightarrow\infty.
\]

Power-law violations of Yau’s nodal upper bound

Theorem

There exist a smooth Riemannian metric g on S^4\times S^1, a number \epsilon_0>0, and nonzero real smooth functions u_j with positive eigenvalues \lambda_j such that

-\Delta_g u_j=\lambda_j u_j,\qquad \lambda_j\longrightarrow\infty,\qquad \frac{\Hh_g^4(\Z(u_j))} {\lambda_j^{1/2+\epsilon_0}}\longrightarrow\infty.

The metric and \epsilon_0 are fixed throughout the sequence.

Section 1, Introduction Read it in the paper TeX source, line 28

Show the TeX

There exist a smooth Riemannian metric $g$ on $S^4\times S^1$, a number
$\epsilon_0>0$, and nonzero real smooth functions $u_j$ with positive
eigenvalues $\lambda_j$ such that
\[
 -\Delta_g u_j=\lambda_j u_j,\qquad
 \lambda_j\longrightarrow\infty,\qquad
 \frac{\Hh_g^4(\Z(u_j))}
      {\lambda_j^{1/2+\epsilon_0}}\longrightarrow\infty.
\]
The metric and $\epsilon_0$ are fixed throughout the sequence.

Configuration and file facts

File: lean/ComparatorChallenges/YauCounterexample.lean
OpenAI’s Lean catalogue: OpenAI lists this as a main result in its Lean catalogue.
Solution module: OAI/Geometry/NodalSets/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: 12 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