Structural Diagnosis and Finite-Sample Replacement Certification for Factorized Priors in RSSMs

Observation-wise product posteriors can become dependent after observations are marginalized out. Under an access model in which full posterior probabilities are available, the representation is frozen, and scoring is independent, we cer-tify whether a given mixture-of-products candidate outperforms the entire product family in normalized Hamming Wasserstein distance. Under an exact fractional cover, the approximation loss of the local compression lower bound is at most twice the compression distortion plus a structural remainder, without requiring in-dependence across groups. On the candidate side, we use direct component trans-port as a baseline, combine preffx aggregation and state lifting into a TV hierarchy with nonincreasing costs, and enlarge the set of valid couplings through a stopping tree that retains parent nodes. Once the reference, candidate, and metric are ffxed, a common empirical event converts valid coupling costs into ffnite-sample upper endpoints, yielding the structural replacement criterion U < L. Validity holds for any ffxed ffnite number of components. In synthetic experiments, 60 of 80 audits using codebooks frozen after learning pass numerically; a further 180 perturba-tion comparisons and 400 reference-structure comparisons verify reductions in coupling costs and upper endpoints. The additional certiffcation gains from com-bining the new couplings with the codebook lower bound, applicability to trained RSSMs, and certiffcation with rigorous rounding remain unveriffed.