Yau's nodal-set conjecture for smooth metrics: is nodal measure comparable to sqrt(lambda)?

Let $(M,g)$ be a smooth closed connected Riemannian manifold of dimension $n$ and $u$ a real eigenfunction, $-\Delta_gu=\lambda u$ with $\lambda>0$ . Yau (1982) conjectured that the nodal set $Z_u$ satisfies $c_g\sqrt\lambda\le\mathcal H^{n-1}(Z_u)\le C_g\sqrt\lambda$ with constants depending only on $(M,g)$ . Donnelly and Fefferman proved both bounds for real-analytic metrics. For smooth metrics Logunov proved the lower bound in every dimension and a polynomial upper bound; on smooth surfaces the best upper bound was $C\lambda^{3/4-\beta}$ (Logunov-Malinnikova). Does the two-sided bound $\mathcal H^{n-1}(Z_u)\asymp\sqrt\lambda$ hold for every smooth metric, in particular the upper bound?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Spectral geometry; nodal sets of Laplace eigenfunctions
Posed by: Shing-Tung Yau
Year posed: 1982
Years open: 44y
Solved: 2026-09-23
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 58 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Dimensions three and four (Theorems 1.1, 1.2 of the principal manuscript): a smooth metric arbitrarily close to the round metric on $S^3$ , and one on $S^2\times T^2$ , with exact eigenfunctions whose nodal measure divided by $\sqrt\lambda$ is unbounded. Dimension five and up (companion): a metric on $S^4\times S^1$ with nodal measure $\gg\lambda^{1/2+\epsilon_0}$ for a fixed $\epsilon_0>0$ , and products give every dimension above five. Surfaces (companion): $\mathcal H^1(Z_u)\le C(M,g)\sqrt\lambda$ , so with Bruning's lower bound Yau's conjecture holds in dimension two. So the upper bound is true for surfaces and false for smooth metrics in every dimension $\ge3$ ; the lower bound (Logunov) is untouched, and real-analytic metrics are unaffected. No power excess is claimed in dimensions 3 and 4.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The family has three manuscripts, all dated September 23, 2026: the principal one constructs counterexamples in dimensions three and four, a companion gives power-law violations in dimension five and above, and a second companion proves the upper bound on every smooth surface.

Verification

No independent mathematician has checked this yet. Checked here: the main theorems of all three manuscripts were read against Yau's conjecture as they state it (fixed smooth metric, all eigenfunctions). Proofs not refereed. lean/formalization.yaml lists SmoothYau (OAI.YauCounterexamples.sphere_three and sphere_two_torus_two) and YauCounterexample (OAI.Yau.Target.yau_nodal_set_upper_bound_counterexample). Their comparator statements were read: in every smooth neighborhood of the round metric on $S^3$ there is a metric with an eigenfunction sequence, $\lambda_j\to\infty$ , whose finite nodal measure over $\sqrt{\lambda_j}$ tends to infinity; likewise for one metric on $S^2\times T^2$ ; and for one metric on $S^4\times S^1$ the ratio of 4-dimensional nodal measure to $\sqrt\lambda$ tends to infinity. The last does not state the paper's power-law excess $\lambda^{1/2+\epsilon_0}$ . The surface bound has a challenge, NodalLength.json (solution OAI.Analysis.NodalLength.Main present), not in the catalogue; NodalLength.lean states $\mathcal H^1(Z_u)\le C\sqrt\lambda$ on every compact connected smooth Riemannian surface. Not rebuilt here.

Yau's nodal-set conjecture for smooth metrics: is nodal measure comparable to sqrt(lambda)?

What was actually shown

What the AI did

Verification

Sources

Discussion