Remark: Family 350. Yau's nodal conjecture: surfaces and counterexamples

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 $S^3$ arbitrarily close to round. In dimension five, nodal measure can grow faster than ${\lambda }^{1/2+{\epsilon }_0}$ for some fixed ${\epsilon }_0>0$, ruling out even arbitrarily small power losses.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.