PH-SYM-004

Name: Translation equivariance violation

Severity: warning

Input modes: adapter+dump

PH-SYM-004 is a structural stub in v1.0: once past its declared-symmetry + periodicity gates, it always emits SKIPPED. True translation equivariance is a model property (\(f(\mathrm{roll}(x)) = \mathrm{roll}(f(x))\) on a live callable) and requires adapter-mode plumbing deferred to v1.1.

An earlier v1 offline metric \(\|\mathrm{roll}(u) - u\| / \|\mathrm{roll}(u)\|\) was removed before release: np.roll preserves norm, so the triangle inequality caps the offline quantity at 2.0, and a PASS-if-\(<2.0\) threshold would rubber-stamp random noise, smooth ramps, and most structured fields. Rather than ship a false-pass metric, v1.0 ships the stub with SKIPPED reasons documenting the deferral.

The harness-level validation exercises the underlying mathematical primitive: shift_commutation_error on four controlled operators — identity, circular convolution, Fourier multiplier, and coordinate- dependent multiplication. The first three are translation-equivariant at float64 roundoff (max \(3.75 \times 10^{-16}\) over 100 random 2D trials); the fourth is non-equivariant by construction (error \(0.09\)–\(0.87\)) and serves as the negative-control. v1.1 will replace the stub with an adapter-mode implementation that compares \(f(\mathrm{roll}(x))\) against \(\mathrm{roll}(f(x))\) on a live callable, using the same shift_commutation_error primitive.

Per the v1.0 stability policy, the rule’s verdict surface is stable; only the implementation will change in a minor release.