PHI-OMEGA / M: Theoretical Foundations of Open-System Nonlinear Dynamics, Reservoir Energy Balance, and RK4 Numerical Stability Validation

This study formalizes the theoretical architecture and numerical validation of the PHI-OMEGA / M physics framework, focusing on open-system dynamics, nonlinear energy operator evolution, and thermodynamic reservoir balance. We derive the explicit temporal energy dissipation rate dE/dt under a non-linearly coupled Hamiltonian structure, incorporating the anti-commutator loss dynamics {H, Gamma} to guarantee exact local loss conservation across operator modes. Furthermore, we establish the non-negative damping criteria over the phase space, demonstrating system boundedness under positive heat capacity (Cv > 0) reservoir exchange conditions. The dynamic stability of the frozen candidate model is numerically verified via a fourth-order Runge-Kutta (RK4) integration scheme across a 100-step temporal horizon. The quantitative simulation confirms monotonic energy attenuation from E(0) = 2.5000 to E(20.0) = 0.3452, validating long-term stability and thermodynamic consistency without divergent mode oscillations.