Abstract
For centuries, the pursuit of non-linear dynamics has been constrained by a fundamental methodological trap: the assumption of a continuous cosmos modeled through differential
equations, which are subsequently forced into artificial discretization via iterative numerical solvers. This continuous paradigm has accumulated centuries of intractable bottlenecks, most famously exemplified by the Newtonian three-body problem—whose chaotic
complexities prompted King Oscar II of Sweden's legendary 1887 prize competition—and the Navier-Stokes fluid dynamics governing the chaotic weather forecasting problem. Traditional numerical integration inevitably hits the Lyapunov horizon, resulting in exponential error
divergence and computational gridlock.
In this work, we present a radical paradigm shift: the State-Dependent Coefficient Matrix (SDCM) architecture. Operating on the premise of a natively discrete cosmos, SDCM
factorizes complex non-linear systems into linear parameter-varying structures evaluated simultaneously via map-fold operations, entirely bypassing step-size integration and
differential solvers. We demonstrate that the SDCM framework resolves centuries-old classical intractabilities in celestial mechanics and meteorology with absolute stability. Furthermore, because the architecture is domain-agnostic, it uniquely accommodates conjugate models, successfully bridging the micro-scale linear operators of Quantum
Mechanics (QM) with the macro-scale non-linear geometry of General Relativity (GR) into a shared state-space vector. By enabling these disparate domains to evolve together
simultaneously, the SDCM architecture provides a rigorous, solver-free computational pathway toward a Grand Unified Theory (GUT) and a Theory of Everything (TOE).