Yau’s nodal bounds: surfaces and higher dimensions
OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:
Proves the sharp \(C\sqrt\lambda\) upper bound for nodal length on every fixed smooth closed surface, completing Yau’s conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S3 arbitrarily close to round. In dimension five, nodal measure can grow faster than \(\lambda^{1/2+\varepsilon_0}\) for some fixed \(\varepsilon_0\gt 0\), ruling out even arbitrarily small power losses. (OpenAI’s Lean note)
Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.
The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement.Read the assessments
Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.
Three other possible matches are listed below the card, among them Theorem 1.1 in Power-law violations of Yau’s nodal upper bound and Theorem 1.2 in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four.
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.
Three more possible matches
Theorem 1.1in Power-law violations of Yau’s nodal upper bound, 23 September 2026.
Theorem 1.2in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Theorem 1.1in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Reported proof verification
Three Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
We prove that a nonzero real Laplace eigenfunction with eigenvalue λ > 0 on a fixed smooth closed connected Riemannian surface has nodal length at most \(C\sqrt\lambda\). Together with the known lower bound, this proves Yau’s conjecture in this setting.
We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on \(S^2\times\mathbb T^2\) for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue. Each sequence belongs to one fixed metric. Thus the upper-bound part of Yau’s nodal conjecture fails for smooth metrics in dimensions three and four.
We construct a smooth Riemannian metric on \(S^4\times S^1\) and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than \(\lambda^{1/2+\epsilon_0}\) for one fixed \(\epsilon_0\gt 0\). This disproves the smooth upper-bound assertion in Yau’s nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
Three Lean statements in this family
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SmoothYau
The paper states
No paper theorem is paired with this Lean statement.
no paper statement matches the Lean text clearly enough to pair, so the closest candidates are listed below.
Statements it shares words with:
Theorem 1.1in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Proposition 2.6(Spherical profiles)in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Theorem 1.1in Sharp nodal length on smooth surfaces, 23 September 2026.
Proposition 4.4(Exactification with an unchanged exterior)in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Proposition 2.2(Amplification of the mean gradient)in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Proposition 8.1(Finite-order packets)in Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four, 23 September 2026.
Proposition 4.3(Packets with arbitrary finite accuracy)in Power-law violations of Yau’s nodal upper bound, 23 September 2026.
Corollary 5.6in Power-law violations of Yau’s nodal upper bound, 23 September 2026.
SmoothYau.lean, the release lists it as a main result.
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
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
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.
The words they share: laplace, hausdorff, connected, measure, sqrt, riemannian, real, length, nodal, surface, closed, eigenfunction, nonzero, lambda, constant.
Possible Lean counterpart
Selected from shared wording. No review of whether the statements mean the same thing is recorded.
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
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
YauCounterexample.lean, the release lists it as a main result.
defMainTarget : 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 ⊤)⋮theoremyau_nodal_set_upper_bound_counterexample : MainTarget := bysorry
This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.
Other mathematical statements in OpenAI’s Lean library
Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.
Independent comparisons and proof-verification reports
Does the Lean express the advertised result?
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.
MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
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
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method: 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.
cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name: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.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.
Yau’s nodal-set conjecture predicts nodal size of order $\sqrt\lambda$ for Laplace eigenfunctions of eigenvalue $\lambda$. The formalization proves the upper bound on every fixed smooth closed connected Riemannian surface: there is a surface-dependent constant $C$ such that the one-dimensional Hausdorff measure of the zero set of every nonzero real eigenfunction with $\lambda>0$ is at most $C\sqrt\lambda$. The known lower bound is not part of this selected statement.
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.
Yau’s nodal upper-bound conjecture predicts that the nodal measure of an eigenfunction is bounded by a constant times the square root of its eigenvalue. 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.
AI-drafted comparisons, with no recorded human review
Nodal counterexamples in dimensions three and fourSmoothYau
Unreviewed AI draft: states a paper’s theorem as the paper does. States both main theorems of the counterexample paper: metrics on S³ near the round metric and on S² × T² with eigenfunction sequences whose nodal-measure ratio to √λ is unbounded.
Sharp upper bound for nodal length on smooth surfacesNodalLength
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: on every closed smooth surface, nodal length of an eigenfunction is at most C√λ.
Unbounded nodal ratio on $S^4\times S^1$YauCounterexample
Unreviewed AI draft: states a weaker form of a paper’s theorem. States a fixed metric on S⁴ × S¹ with nodal measure divided by √λ tending to infinity, whereas the main theorem gives a power-law violation, with nodal measure divided by λ^{1/2+ε₀} tending to infinity for a fixed ε₀ > 0.
Who ran the proof-verification program?
These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.
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Nodal counterexamples in dimensions three and fourSmoothYau
Two reports, all at the release’s recorded version.
Reported by
Their word
Their account
Date
Original report
cert-machine
“CERTIFIED”
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
The program that collects Atlas’s source records, Statement Map, could not obtain or interpret some source material. These notes describe those limits; they do not identify mathematical errors.
SmoothYau: Tracked source lean/OAI/Geometry/SmoothYau/Main.lean unavailable: bytes were not captured at fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
What these papers cite in the release, and what cites them
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Robert Joseph George, Do the Lean statements in openai/math match their papers?, 7 October 2026. Source
The Lean statement drops the fixed power-saving exponent the paper proves, so it states the ratio tends to infinity without that epsilon.
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Independent assessment
MathVet
Their term
“partial”
What their term means
Only a special case is stated, or a co-equal headline claim has no challenge statement.
Their method
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.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method
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…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
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.