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

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.

Exact FrontierDelta

Prior state unknown→disproved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

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

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

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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.