Adaptive Spectral Regularization Closed-Loop Stabilization of Nonlinear Dynamics in High-Dimensional State Spaces

This manuscript and accompanying computational framework introduce a closed-loop, adaptive regularization mechanism for the stabilization of high-frequency variance cascades in continuous-time nonlinear dynamical systems. Standard simulations of high-dimensional energy transfer, fluid dynamics, and continuous-depth neural networks (such as Neural ODEs) often exhibit pathological spectral bias. This leads to ultraviolet (UV) divergence and finite-time numerical breakdown during trajectory rollout. We demonstrate that an adaptive Ω-Σ governor can autonomously detect and suppress these instabilities. By dynamically scaling effective system dissipation via the temporal derivative of the spectral variance, our JAX-accelerated 4th-Order Runge-Kutta integration achieves a 10¹2 magnitude suppression of high-frequency concentration. The system converges to a bounded, active equilibrium, establishing that nonlinear transport can be globally stabilized without relying on artificial low-pass truncation or static Lipschitz constraints. Key Contributions: Formulation of the state-aware Ω-Σ dynamic regularizer for continuous-time vector fields. Empirical demonstration of a 12-order-of-magnitude suppression of spectral divergence using JAX Just-In-Time (JIT) compilation. Establishment of a regulated Lyapunov functional that ensures global trajectory boundedness while preserving the expressivity of underlying nonlinear transport. This work serves as the computational realization of the theoretical mechanisms established in our foundational paper, "Coherence Under Constraint"