# PH-SYM-003

**Name:** SO(2) Lie derivative equivariance violation

**Severity:** warning

**Input modes:** adapter

PH-SYM-003 validates infinitesimal Lie-derivative equivariance for
scalar models under SO(2). The rule is **adapter-mode-only**: dump-mode
inputs emit `SKIPPED` because the diagnostic requires forward-mode
automatic differentiation on a live callable, which a frozen tensor
cannot supply. In adapter mode, the rule computes
$(L_A f)(x) = \frac{d}{d\theta}\big|_{\theta=0} f(R_\theta x)$ via
`torch.autograd.functional.jvp` and reports the per-point $L^2$ norm
against a calibrated roundoff floor.

The rule applies under five gates:

1. The adapter declares `SO(2)` symmetry on the domain spec.
2. The input is an adapter-mode `CallableField` (not a dumped tensor).
3. The grid is 2D.
4. The grid is centered on the origin.
5. The domain is square (so the rotation $R_\theta$ stays inside the
   domain).

Failing any gate emits `SKIPPED`. When all gates pass, the rule emits
`PASS` (per-point $L^2$ norm at machine roundoff) or `FAIL` (per-point
$L^2$ norm above the calibrated floor; the closed-form magnitude is
reported on common toy fixtures).

**Scope — infinitesimal, not global finite.** PH-SYM-003 validates the
rule's **single-generator, pointwise-$L^2$, scalar-invariant** subset
of the finite identity. It does **not** prove global finite
equivariance for arbitrary models. The finite-vs-infinitesimal
direction is subtle: finite ⇒ infinitesimal is trivial by
differentiation, but infinitesimal ⇒ finite requires smoothness +
connected group + generator coverage + exact constraint. The rule's
empirical grid-sampled diagnostic does not establish all four
conditions; it does, however, detect the most common failure mode
(a model that breaks SO(2) at first order).

Higher-dimensional Lie groups (SO(3), SE(3)), disconnected groups
(O(2), E(2)), and vector-output equivariance are out of v1.0 scope.