PH-RES-003

Name: Spectral-vs-FD residual discrepancy on periodic grid

Severity: warning

Input modes: adapter+dump

PH-RES-003 compares the spectral-backend Laplacian against a 4th-order central FD Laplacian on periodic grids. The emitted quantity is the maximum relative discrepancy on an analytic periodic MMS fixture; under refinement, the spectral term collapses exponentially while the FD term decays polynomially at \(O(h^4)\). The rule passes when the residual is within the calibrated 0.01 relative threshold and fires when the residual exceeds the band, signalling a spectral-path bug or a non-analytic fixture. PH-RES-003 is the FFT-based companion to PH-RES-002 and requires no torch dependency (pure numpy FFT).