OpenAI · 350-03 · 2026-09-23 · Differential geometry
Power-law violations of Yau's nodal upper bound
preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026
Family 350. Yau's nodal conjecture: surfaces and counterexamples.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the sharp 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 arbitrarily close to round. In dimension five, nodal measure can grow faster than for some fixed , ruling out even arbitrarily small power losses.
We construct a smooth Riemannian metric on and a sequence of real Laplace eigenfunctions whose nodal four-volume grows faster than for one fixed . This disproves the smooth upper-bound assertion in Yau's nodal-set conjecture and the proposed bound with an arbitrarily small positive power loss.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 350: Yau’s nodal bounds: surfaces and higher dimensions. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.