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.

Move First, Commit Later: Selective LiDAR-to-BIM Global Initialization via Sequential Consensus with Symmetry-Aware Abstention

Global LiDAR-to-BIM initialization must place a robot within an as-designed building model without a prior pose. In repetitive interiors, the principal failure mode is not low-confidence registration but confident aliasing: a submap can match several BIM regions with comparable scores, producing a high-scoring pose displaced by symmetry. We present Move First, Commit Later, a selective layer that treats a registration front-end as an evidence source and decides whether to commit. Candidates from multiple submaps are mapped to a common $\mathrm{SE}(2)$ anchor; a top-1 consensus $B_m$, invariant to non-champion multiplicity, aggregates cross-submap evidence; and topology serves only as a binary feasibility gate. The decision is typed--COMMIT, DEFER, or AMBIGUOUS($τ$), reporting the detected symmetry period--and reversible: symmetry-breaking motion upgrades AMBIGUOUS to COMMIT. On a real multi-room building and a controlled symmetric simulation, the layer commits correctly in every tested trajectory-scale configuration and otherwise abstains with a typed state, whereas forced-choice policies on the same front-end select wrong rooms in most cases. Committed anchors are within 0.02--0.36 m of an independent laser-tracker position reference and within $1.9^\circ$ of a BIM-registration orientation proxy. The evaluation covers one building and one front-end; the layer is designed to be front-end modular.