Certifiable Contraction Dynamics: Lyapunov-Stable Neural ODEs with Wasserstein Sensitivity Bounds

I introduce Invariant Neural Dynamical Systems (INDS), treating the Lyapunov dissipation inequality as an explicit optimization constraint for continuous-time neural dynamics. I deliberately separate three claims: (i) Lyapunov dissipation gives exponential error contraction; (ii) contraction gives a Wasserstein-1 OOD sensitivity bound; and (iii) statistical complexity requires a separate metric-entropy assumption. INDS explicitly trades predictive performance for certifiable dynamical constraint enforcement. By enforcing this explicit dynamical constraint, INDS achieves a substantial reduction in the intrinsic empirical Lyapunov constraint-violation rate from 98.02% to 6.69% (a 93.2% relative reduction, corresponding to a 14.7-fold lower violation rate). Across five paired seeds, the small sample size inherently limits the strength of strict statistical inference (p = 0.0625, the minimum attainable two-sided exact value for N = 5). Under the stated entropy assumption, the autonomous formulation yields a finite conditional Rademacher upper bound of Rn ≤ 0.485 at α = 0.5 via a learned anisotropic metric (κ(P ) = 1.418).