# 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).