# Spectral nodal-set scope and mesh benchmark

Family 350: Yau’s nodal bounds: surfaces and higher dimensions. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Spectral solver researchers can distinguish the sharp surface nodal-length law from its failure in higher dimensions.

## Applicability and commercial boundary

The surface constant depends on the fixed metric, and numeric zero contours require resolution control. The selected dimension-five scope proves an unbounded square-root ratio, not automatically the broader power-law headline.

## Initial business decision

Research or conditional engineering only. A practical implementation, recurring buyer problem and willingness to pay have not been established. Mathematical usefulness alone is insufficient evidence for a standalone lucrative product.

## Next verification action

Build a fixed-surface eigenmode fixture and keep higher-dimensional exclusions separate.

## Evidence scope

The catalog statement was individually reviewed. Main-paper abstracts and available selected formal scope notes were also inspected; family 354’s explicit main profile formula was read. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Sharp nodal length on smooth surfaces](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026/paper.pdf).