Degenerate in Whose Frame? An Equivariance Condition for Degeneracy Detection in LiDAR Registration

Degeneracy detectors for LiDAR registration commonly return six per-axis binary labels. We ask whether these labels are properties of the scene. Under a body-frame change, the point-to-plane information matrix transforms by congruence, H' = Ad(T)^T H Ad(T), not similarity. Congruence preserves nullity and, through the adjoint reparameterization, identifies the same physical twist subspace; the per-axis footprint and a thresholded spectrum need not be invariant. In a noise-free circular tunnel, shifting the origin by one metre changes which degrees of freedom are flagged. A generalized criterion Hv = lambda Mv is universally frame-independent over positive-semidefinite information forms if and only if its metric rule is equivariant. No fixed metric qualifies, while a rig-adapted one exists only at zero screw pitch, met in one of nineteen surveyed calibrations. The equivariant point-displacement metric M = sum_i J_i^T J_i yields dimensionless, scene-scale-invariant generalized eigenvalues. They are invariant to body frame, consistent changes of length unit and scene scales; the threshold also transfers empirically across sequences. Across 365 frame pairs from four public sequences, labels rarely change at practical extrinsic magnitudes, yet a remapping estimator's correction differs between body-frame choices on 44.5-69.5% of pairs, with a median of 0.7-4.0 mm and a maximum of 0.87 m. The per-axis footprint changes even under the equivariant metric, placing the fundamental issue in the reported quantity.