Reimagining Dynamical Systems

Traditional computational modeling of dynamical systems relies heavily on continuous calculus, sequential time-stepping, and floating-point arithmetic, which inevitably introduces compounding truncation errors, temporal lag, and numerical drift. To mitigate these artifacts, models are frequently burdened with complex stabilization heuristics and artificial boundary-forcingfunctions. This paper introduces the Integral State-Dependent Coefficient Matrix (ISDCM) architecture, a self-contained computational paradigm that replaces traditional differential equations with a unified matrix operator. By mapping simultaneous algebraic and differential relationships into a state-dependent coefficient matrix and operating entirely within Diophantine space via Global Least Common Multiple (LCM) scaling, ISDCM eliminates floating-pointrounding errors at the root. The architecture achieves absolute step-wise zero truncation error(ϵstep = 0), autonomous closure, and extreme structural stability, yielding bounded phase-spacetopologies such as discrete cubic attractors and quasi-periodic tori without external damping ornumerical decay