Certified Frozen-Midpoint Neural Integrators: Original-Energy Dissipation and Second-Order Physical Time

Neural gradient-flow solvers can decrease the correct energy while evolving on an incorrect physical clock. With state-dependent mobility, even an exactly solved, energy-dissipative step can be only first order. We develop a certified frozen-midpoint interface that separates three obligations: mobility consistency, full-state nonlinear accuracy, and signed energy work. A predicted midpoint freezes the mobility before optimization, preserving a globally strongly convex average-vector-field subproblem whenever its stated curvature and existence conditions hold. Deterministic oracle envelopes give computable objective-gap and endpoint-radius certificates, without assuming global neural training convergence. We prove variable-step second-order accuracy, finite correction for nonstationary states under explicit solver assumptions, and sharp worst-case tolerance scalings. Applications include coupled positive quantile diffusions and full-covariance Gaussian Wasserstein flows with noncommuting matrices. For smooth quantile reconstructions, a separate posterior theorem certifies error against the continuum evolution, including an explicit spatial defect. We also characterize validation under adaptive sampling and the obstruction at exact equilibrium. This is a purely theoretical study: it establishes conditional guarantees and computability constructions, not empirical speedups, unrestricted-generator convergence, or priority for discrete-gradient energy preservation itself.