OpenAI: Yau’s nodal bound holds on surfaces, fails in higher dimensions
Awaiting review
Importance: historic (5 of 5)The takeaway
Three manuscripts in OpenAI’s 6 Oct 2026 math release (family 350, dated 23 Sept) claim to settle Yau’s 1982 conjecture on the size of nodal sets of Laplace eigenfunctions for smooth metrics. On every smooth closed surface the nodal length is at most C√λ, which completes the conjecture there. In dimensions 3 and 4, explicit smooth metrics, including some arbitrarily close to the round 3-sphere, violate the upper bound, and on S⁴×S¹ it fails by a power of λ. Lean proofs of all three statements passed an independent Comparator run. Whether the formal statements match the papers has not been checked by specialists.
Status
- Claim
Awaiting review
- Our reporting
- Medium confidence
- Verification
- Peer-reviewed
- Importance
- Historic (5 of 5)
- Last verified
- 29 October 2026
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Key facts
- Surfaces (abstract, verbatim): ‘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√λ. Together with the known lower bound, this proves Yau’s conjecture in this setting.’
- Dimensions 3–4 (verbatim): ‘We construct a smooth metric on the three-sphere, arbitrarily close to the round metric in the smooth topology, and a smooth metric on S²×𝕋² for which sequences of exact real Laplace eigenfunctions have unbounded nodal measure divided by the square root of the eigenvalue… Thus the upper-bound part of Yau’s nodal conjecture fails for smooth metrics in dimensions three and four.’
- Dimension 5 (verbatim): ‘We construct a smooth Riemannian metric on S⁴ × S¹ and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than λ^{1/2+ε₀} for one fixed ε₀ > 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.’
- Prior state (surface paper’s introduction): Donnelly–Fefferman proved both bounds for real-analytic metrics; Logunov proved the lower bound in every dimension; for smooth surfaces the upper bound was Cλ^{3/4} (Donnelly–Fefferman; Dong), improved to Cλ^{3/4−β} by Logunov and Malinnikova
- Lean (scope note lean/docs/350.md): the surface bound with a surface-dependent constant; ‘one fixed smooth metric’ near round S³ and one on S²×T² ‘with an eigenfunction sequence of unbounded nodal-measure ratio’; and one fixed metric on S⁴×S¹ with nodal measure over √λ tending to infinity. oai-math-revised’s Comparator run (25–26 Oct, standard axioms only) accepted all three; it checks the proofs against the Lean statements, not the statements against the papers
- Follow-up: Xie and Zhang (arXiv 2610.28391, 20 Oct) prove the sharp linear upper bound for interior nodal sets of Steklov eigenfunctions on smooth surfaces, listing OpenAI’s preprint in their references
What happened
When a drum vibrates at a high frequency, its “nodal set”, where the surface does not move, becomes a fine network of curves. Shing-Tung Yau conjectured in 1982 that its size grows like the square root of the eigenvalue λ, with upper and lower constants depending only on the manifold. Donnelly and Fefferman proved this for real-analytic metrics in 1988. For smooth metrics, Alexander Logunov proved the lower bound in 2018 together with a polynomial upper bound, but the sharp upper bound stayed open even on surfaces.
OpenAI’s release claims a split answer. On surfaces the upper bound holds for every smooth metric. In dimensions three and four it fails: there are smooth metrics, some as close as one likes to the round 3-sphere, with eigenfunctions whose nodal sets are too large for any constant C. A third paper builds a metric on S⁴×S¹ where the ratio grows like a power of λ, which also rules out the weaker form of the bound with a small power loss. All three statements come with Lean proofs, and an independent Comparator run by the oai-math-revised maintainers accepted them with only the standard axioms.
Why it matters
Yau’s nodal-set conjecture is one of the best-known problems in spectral geometry, and Logunov’s partial results were among its main recent advances. If the claims hold, the smooth conjecture is true only in dimension two. The Lean proofs make the result unusually checkable. The open question is whether the formal definitions (Hausdorff measure, eigenfunctions, the metrics) say what the papers say. That needs a specialist to read the statement files. We found no public reaction from spectral geometers as of 29 Oct.
What is disputed or not yet verified
| Verification | Lean 4 proofs of the three main statements accepted by an independent Comparator run; fidelity of the formal definitions to the papers not checked by specialists; manuscripts unrefereed |
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Sources
6 sources from 2 sites. Numbers match the chips in the text.
6 sources: 6 primary
Primary
- OpenAI: Sharp nodal length on smooth surfaces (PDF)github.com, paper
- OpenAI: Smooth counterexamples to Yau’s nodal upper bound in dimensions three and four (PDF)github.com, paper
- OpenAI: Power-law violations of Yau’s nodal upper bound (PDF)github.com, paper
- OpenAI Lean scope note 350github.com, code
- oai-math-revised: Comparator results (25–26 Oct)github.com, code
- Xie, Zhang: Sharp bounds for nodal sets of Steklov eigenfunctions on smooth surfaces (arXiv 2610.28391)arxiv.org, paper
Changes
- Filed (families 251–378; three abstracts, the surface paper’s introduction, Lean scope note, oai-math-revised Comparator results, arXiv 2610.28391)