The Adaptive Memory-Driven Coupling Framework (AMDCF): A Unified Kernel for Stabilizing Nonlinear Dynamics in Quantum and Classical Systems

Since we always learn from our mistakes... here we are. This preprint introduces a unified computational kernel for the Adaptive Memory-Driven Coupling Framework (AMDCF) to prevent numerical instabilities in highly nonlinear systems. We demonstrate its cross-disciplinary applicability by applying the same core architecture to two distinct physical domains: (1) preserving quantum entanglement in non-Markovian environments, and (2) stabilizing the 1D Burgers' equation for turbulent fluid flow. By utilizing a historical memory integral and a smooth, sigmoid-based dynamic safety valve, the framework prevents numerical blow-ups. In the quantum case, it preserves up to 0.5418 of concurrence compared to complete decay (0.0005) in Markovian models. In fluid dynamics, it maintains strict stability (Final Energy: 0.1279), dissipating kinetic energy monotonically without the typical infinity blow-up. This work establishes AMDCF as a robust algorithmic tool for nonlinear control and numerical stabilization