An Analytical Floor Does Not Imply a Confined Correction: Learning Inside Equivariant Observers

We study which guarantees of a symmetry-structured estimator survive when a learned component is placed inside it, within an instance where the structure is known: an equiv- ariant observer for inertial navigation aided only by a kinematic constraint imposed as a pseudo-measurement. Decomposing the update into measurement model, gain and correc- tion yields three positions at which a network may enter; of each we ask whether it retains an analytical floor (a setting reproducing the constructed observer) and whether it confines the correction to the admitted subspace. The two properties are independent: we instan- tiate every combination, including one that keeps the floor while the correction leaves the plane. This configuration arises whenever a residual connection modifies a model-based gain, and we identify it as a symmetry breaking of the learned module. We also prove that for any equivariant observer whose output Jacobian annihilates an estimate-independent unobservable subspace, no gain, learned or not, adds information along it under a Joseph- form covariance recursion. A per-epoch diagnostic confirms the predicted confinement to machine precision. For the unconstrained gain, the correction instead concentrates in state blocks that the measurement Jacobian annihilates. On held-out KITTI drives (Geiger et al., 2013), from inertial measurements alone, the positions behave as predicted: upstream re- duces mean trajectory error by 48%, midstream improves on every metric, downstream diverges.