Structural Compatibility Enables Exact Solvability in Wasserstein Proximal Dynamics

Finite-dimensional Wasserstein updates replace an optimization over probability measures by one over a structured family. We ask when this restriction is mathematically lossless rather than merely convenient. The key property is structural compatibility: on every statistic fiber, the same canonical representative must minimize the energy and saturate the Wasserstein transport lower bound. We show that this compatibility is equivalent to all-step fiberwise exactness. For mean–covariance fibers, this yields full-space finite-step Gaussian and matched $q$-Gaussian closure for Shannon and power entropies under quadratic confinement. Our main converse result is a rigidity theorem: within regular fixed-generator radial location–scatter families, all-step exactness forces the internal energy into the Shannon/positive-power class and fixes the radial generator. When exactness fails, the normal Wasserstein gradient gives the first-order distance to the family in one-dimensional quantile geometry; a strongly convex counterexample shows that vanishing first-order defect still does not imply finite-step closure. The associated $d$-dimensional radial defect is directly estimable. Across 375 frozen Monte Carlo configurations with $d\le 100$, every mismatch has positive estimated defect, matched $p=1$ cases are numerically zero, the mean relative error at $d=100,n=10^5$ is $0.17\%$, and all six preregistered bootstrap intervals contain the analytic target. Together, the results separate exact finite-dimensional solvability, infinitesimal compatibility, and structurally misspecified approximation.