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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.