Equivariance Transports, Closure Completes: Gauge Obstructions in Latent World Models

Equivariance does not make an exact latent-prediction optimum a predictive state. Two finite constructions prove this: a persistent covariant copy under autonomous self-prediction, and a hidden trajectory gauge despite canonical innovation, hard transport, a proper anchor, and a supplied pose–quotient reconstructor. In a free, observation-aligned Borel class, we prove that deterministic equivariant emission driven by the world's specified joint innovation exists if and only if the invariant code carries the point gauge. This specializes standard factorization and is distinct from full-future-law sufficiency with auxiliary noise. We separate the assumptions for quotient-future and full-future sufficiency. Controlled studies then test how supplied information enters learned dynamics. With supplied physical calibration and known association, calibrated routing with multi-step training attains mean H8 NMSE 0.127500 versus 0.675110 for persistence, winning all 20 frozen-encoder blocks. Readout calibration and one-step retraining fail their tested new-object criteria. A separate matched-update study reverses the advantage of root supervision, and sensor-mounting Pendulum favors system identification over learned dynamics. These contrasts locate calibration's benefits without claiming generic predictive superiority.