# 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.