On identifiability limits of phase-space information in Schr\"odinger Bridges for post shell-crossing gravitational structure formation

Abstract

Many nonlinear dynamical systems remain deterministic and invertible in their full state space while becoming non-identifiable after projection onto reduced observables. Gravitational shell crossing provides a canonical example: the collisionless phase-space flow remains Hamiltonian, whereas the configuration-space map folds and several velocities coexist at the same Eulerian position. We formulate a fixed-reference Schr\"odinger-bridge framework to diagnose the resulting loss of closure at the density--current level. The reference law is obtained by freezing the drift along a Jordan--Kinderlehrer--Otto diffusion derived from an isothermal overdamped closure. An independently specified adhesion--Kardar--Parisi--Zhang stochastic lift, or a diffusion constructed directly from a measured density--current history, provides the candidate law. A finite drift-only path-space Kullback--Leibler divergence requires common state variables, a common time coordinate, matched path covariance, feasible finite-entropy endpoints, and integrable drift control. Thus, in the scalar model, equality of the diffusivities is a covariance-matching condition and does not imply dynamical equivalence of the two closures. We formulate the endpoint-constrained bridge, but the numerical experiments evaluate a current-matching Markov proxy relative to a fixed unforced Burgers reference rather than solve the bridge optimization. The central result is a structural identifiability limit. At fixed smoothing, reference drift, and diffusivity, the proxy is determined entirely by the resolved density--current history. Distinct phase-space histories with the same resolved history therefore generate the same proxy law and cost even when their velocity covariance, higher cumulants, and branch populations differ. An explicit homogeneous counter-stream family establishes this no-go result. Periodic one-dimensional kinematic and self-gravitating Vlasov--Poisson sheet tests illustrate the distinction: the proxy cost and velocity dispersion are both enhanced after shell crossing, but exhibit different local time dependence. The proxy consequently diagnoses failure of a specified reduced closure, but neither reconstructs the full multistream kinetic law nor measures physical entropy production. Faithful extensions require kinetic or branch-state augmentation, temporal memory, or both.