Nodal counterexamples in dimensions three and four
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Theorems selected for verification:OAI.YauCounterexamples.sphere_threeOAI.YauCounterexamples.sphere_two_torus_two
The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.
theoremsphere_three (gRound : SmoothMetric (Euclidean 3) (Sphere 3)) (hRound : IsRound gRound) : ∀ N : Set (SmoothMetric (Euclidean 3) (Sphere 3)), IsSmoothNeighborhood gRound N → ∃ g ∈ N, HasUnboundedNodalRatio g 3 := bysorry⋮theoremsphere_two_torus_two : ∃ g : SmoothMetric FourModel FourManifold, HasUnboundedNodalRatio g 4 := bysorry
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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.
defmetricCoefficients (g : SmoothMetric E M) (p : M) (y : E) : Matrix (CoordIndex E) (CoordIndex E) ℝ :=fun i j => g.inner ((chartAt E p).symm y) (coordinateVector p y i) (coordinateVector p y j)
deflaplaceBeltrami (g : SmoothMetric E M) (u : M → ℝ) (x : M) : ℝ :=let c := chartAt E xlet a := metricCoefficients g xlet density := fun y => Real.sqrt (Matrix.det (a y))let u' := u ∘ c.symm (density (c x))⁻¹ * ∑ i, fderiv ℝ (fun y => density y * ∑ j, (a y)⁻¹ i j * fderiv ℝ u' y (Module.finBasis ℝ E j)) (c x) (Module.finBasis ℝ E i)
structureCoordinateTest (E : Type*) [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] (M : Type*) [TopologicalSpace M] [ChartedSpace E M] where center : M compactSet : Set E isCompact : IsCompact compactSet inTarget : compactSet ⊆ (chartAt E center).target order : ℕ row : CoordIndex E column : CoordIndex E
defIsSmoothNeighborhood (g₀ : SmoothMetric E M) (N : Set (SmoothMetric E M)) : Prop := ∃ tests : List (CoordinateTest E M), ∃ ε : ℝ, 0 < ε ∧ ∀ g : SmoothMetric E M, (∀ t ∈ tests, ∀ y ∈ t.compactSet, ‖iteratedFDeriv ℝ t.order (fun z => metricCoefficients g t.center z t.row t.column - metricCoefficients g₀ t.center z t.row t.column) y‖ < ε) → g ∈ N
defnodalMeasure [T3Space M] (g : SmoothMetric E M) (d : ℝ) (u : M → ℝ) : ℝ≥0∞ :=letI : RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, E) x) := ⟨g.toRiemannianMetric⟩letI : IsContinuousRiemannianBundle E (fun x : M => TangentSpace 𝓘(ℝ, E) x) := ⟨⟨g.inner, g.contMDiff.continuous, fun _ _ _ => rfl⟩⟩letI : EMetricSpace M := EMetricSpace.ofRiemannianMetric 𝓘(ℝ, E) MletI : MeasurableSpace M := borel MletI : BorelSpace M := ⟨rfl⟩ Measure.hausdorffMeasure d {x | u x = 0}
defIsRound (g : SmoothMetric (Euclidean 3) (Sphere 3)) : Prop := ∀ (x : Sphere 3) (v w : TangentSpace 𝓘(ℝ, Euclidean 3) x), g.inner x v w = inner ℝ (mfderiv 𝓘(ℝ, Euclidean 3) 𝓘(ℝ, Euclidean 4) (fun y : Sphere 3 => (y : Euclidean 4)) x v) (mfderiv 𝓘(ℝ, Euclidean 3) 𝓘(ℝ, Euclidean 4) (fun y : Sphere 3 => (y : Euclidean 4)) x w)
No confident match with a paper statement
No paper theorem is paired with this Lean statement.
Why none: no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four come first because the release’s name for this Lean statement shares words with that paper’s title.
Other possible paper matches, closest wording first
Theorem 1.1 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: topology, round, sqrt, riemannian, neighborhood, positive and three more. Found in the paper’s title, which its main theorem states: three, four and nodal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: contains and lambda.
Proposition 2.6 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: two, ambient, points, exactly, distance, coordinate and seven more.
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: laplace, hausdorff, distance, measure, sqrt, riemannian and three more. 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 and lambda.
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: unit, exactly, continuous, round, 1/2, riemannian and six more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: upper, property and lambda.
Proposition 2.2 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: three, manifold, dimensional, compact, coordinate, one and five more.
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.
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…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
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AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
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.
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?
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Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir 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
Finds part of it stated, or a narrower or weaker statementTheir 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.
OpenAI’s own note on what this Lean statement covers
“The formalized results contradict Yau's proposed $O(\sqrt\lambda)$ upper bound for nodal measure in dimensions three and four. On $S^3$, every prescribed smooth neighborhood of the round metric contains one fixed smooth metric with an eigenfunction sequence of unbounded nodal-measure ratio. A second fixed metric on $S^2\times T^2$ has the same property. In both cases the eigenvalues tend to infinity and the intrinsic nodal measures are finite.”
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
149 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: 71 to its own declarations, 97 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.
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.
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.
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
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
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
\[
-\Delta_{g_\infty}u_j=\lambda_j u_j,\qquad
\frac{\cH^3_{g_\infty}(Z_{u_j})}{\sqrt{\lambda_j}}
\longrightarrow\infty.
\]
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
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.
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
88
Second proof-verification program (nanoda)
Off in OpenAI's configuration.
Statement comparison
A shareable summary of this page: the proposed paper match, what the Lean statement expresses, and which reviews are recorded.
Plectis Atlas
Nodal counterexamples in dimensions three and four
Family 350: Yau’s nodal bounds: surfaces and higher dimensions
No paper statement selected
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
The Lean statement says
theoremsphere_three (gRound : SmoothMetric (Euclidean 3) (Sphere 3)) (hRound : IsRound gRound) : ∀ N : Set (SmoothMetric (Euclidean 3) (Sphere 3)), IsSmoothNeighborhood gRound N → ∃ g ∈ N, HasUnboundedNodalRatio g 3 := bysorry⋮theoremsphere_two_torus_two : ∃ g : SmoothMetric FourModel FourManifold, HasUnboundedNodalRatio g 4 := bysorry
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.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
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