# Neumann eigenmode extrema validation

Family 369: The hot spots conjecture for simply connected planar domains. First source-informed review, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Spectral solver researchers can use the claimed first-positive-Neumann-eigenmode structure on smooth simply connected planar domains as a specialized consistency benchmark.

## Applicability and commercial boundary

The no-interior-critical-point claim concerns a nonzero first positive Neumann eigenfunction, including multiplicity. It does not locate maxima of arbitrary heat fields, mixed modes, sourced temperature distributions or multiply connected domains.

## Initial business decision

Research benchmark. Commercial demand and profitability remain hypotheses. Research usefulness is not evidence of a buyer budget.

## Next verification action

Select a smooth domain and verify eigenvalue ordering, discretization convergence and boundary treatment before using observed extrema as a diagnostic.

## Evidence scope

The catalog statement and main-paper abstract or introductory theorem passages were individually reviewed. Available family-level scope documents were inspected, with attention to version differences and exclusions. This is a first application triage, not a full proof audit or independent Lean verification. Family 376 has nine paper abstracts inspected; selected scopes are not assumed to cover all nine papers in full. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

## Pinned sources

- [Strict hot spots and absence of interior critical points on smooth simply connected planar domains](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026/main.pdf)
- [Selected formal scope](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/lean/docs/369.md)