Beyond Training Loss: Full-Space Certificates for Neural JKO and Wasserstein Evolution

A small neural training loss does not certify an inexact Jordan-Kinderlehrer-Otto (JKO) step. We develop a purely theoretical certification framework for entropy-potential-interaction energies in arbitrary finite dimension. A uniformly convex neural transport exposes an ambient residual, including the transported source score and the interaction force. We prove a supporting inequality against every finite-energy competitor, including nonsmooth Brenier maps; an audited residual norm consequently bounds both the unrestricted JKO gap and the one-step Wasserstein error. This is not a guarantee only within a network family. A second, explicitly differentiated transport-interpolation residual controls physical-time error without assuming that a general Wasserstein proximal map is contractive. We separate parameter-visible and missing directions, give a global integration procedure with Gaussian tail bounds, and prove an obstruction to certification from finitely many pointwise jets alone. Graph-consistent approximation yields conditional finite certification, while a separate regularity hypothesis is needed to exclude time-step accumulation. Exact noncommuting Gaussian and genuinely nonlinear manufactured models establish that the assumptions are nonempty. The results concern certifiability, not dimension-independent audit cost or convergence of arbitrary neural optimizers.