The Symmetry Stress-Test: Violation-Aware Benchmarking for Calibrated Equivariance

Approximate symmetry is by now uncontroversial: gravity, contacts, boundaries, and sensor pipelines routinely break the groups our models assume, and soft or relaxed equivariance can absorb the mismatch. What the field cannot yet do is measure when a symmetry prior helps, hurts, or is untested, because aggregate IID accuracy hides all three. We make it measurable with violation-aware benchmarking: sweep a controlled symmetry-breaking parameter, verify the parameter is a valid proxy for violation (monotone, stable, and predictive of the strict-versus-calibrated gap), and account for total cost. The three steps fit in a one-paragraph Symmetry Stress-Test Card that a reviewer can require. Across three settings, the same recipe exposes the same pattern. Our primary setting is N-body dynamics under an external field, which contracts O(3) to a horizontal subgroup, so the violation comes from the environment rather than the labels. There a strictly equivariant EGNN degrades from median MSE 0.006 to 1.56 as the field grows, while an otherwise identical calibrated model holds at 0.006: a 249× gap for 1.05× the cost. Rotation augmentation backfires, re-imposing the very symmetry the field removed, and the strict model catches up only when handed the field axis, which the calibrated model never needs. Two synthetic sweeps reproduce the pattern across group and output type (O(3) tensor, 134×; O(5) scalar,  1046×), and calibration keeps equivariance's data efficiency where the symmetry holds. The lesson is conditional: hard-code symmetry where the pipeline preserves it, calibrate where it does not, and report the card so the difference is visible rather than assumed.