The State-Dependent Coefficient Matrix (SDCM) Architecture

Traditional numerical mechanics relies on continuous calculus and sequential differential solvers (such as Runge-Kutta or Crank-Nicolson) to model non-linear systems, inherently introducing artificial computational latencies, floating-point truncation traps, and the strict barrier of the Lyapunov horizon. This paper introduces the State-Dependent Coefficient Matrix (SDCM) architecture—a revolutionary, autonomous, discrete matrix framework that eliminates iterative step-size integration entirely. By mapping fully coupled systems of non-linear differential equations into a unified matrix populated by inline state-dependent lambda expressions, the architecture evaluates cross-couplings simultaneously rather than sequentially. Grounded in an ontological perspective of discrete physical reality rather than continuous approximations, the SDCM framework naturally manages floating-point round-off and bypasses mathematical singularities. Empirical validation across 17 diverse problem domains—ranging from classical chaotic attractors (Lorenz-63, Rössler) and non-linear oscillators to relativistic three-body celestial mechanics, fluid dynamics, and quantum wave models—demonstrates exceptional structural stability, universal scale variance resilience, and complete immunity to classical error explosion. Utilizing a State-Dependent Coefficient Matrix (SDCM) / SDC Conjugate Formulation, the model structurally couples spacetime curvature perturbations with quantum wavefunction dynamics via a mutual feedback parameter (). Crucially, we propose that this SDCM-based system provides a viable computational and analytical road toward a Grand Unified Theory (GUT) and Theory of Everything (TOE). Simulation results demonstrate that the framework successfully maintains bounded cyclical trajectories and stable energy exchange across multi-domain physical interactions without numerical instability, offering a robust foundation for future unified physical formulations.