PH-POS-002

Name: Maximum principle violation

Severity: error

Input modes: adapter+dump

PH-POS-002 enforces the maximum principle for harmonic functions: under a well-posed Dirichlet problem for \(-\Delta u = 0\), the minimum and maximum of \(u\) are attained on the boundary. The rule reads the boundary values supplied alongside the field and compares them against the interior min/max. The raw value is the interior overshoot — the magnitude by which the interior extremum exceeds the boundary extrema (zero when the maximum principle holds).

The rule fires (FAIL) when the overshoot exceeds a tight tolerance (1e-10). This is a hard check, not a noise-floor-banded one: any finite-precision-significant overshoot signals a spurious interior extremum. The rule emits SKIPPED when the configured PDE is not laplace, since the maximum principle does not extend to other operators without restatement.

PH-POS-002 is the rule whose firing is shown in the README hero screenshot: it catches the FNO baseline on a Dirichlet-homogeneous Laplace problem in the laplace-uq-bench dogfood corpus.