The Parallel Mathematical Domains Solver: A State-Variable Control Extension for Arithmetic and Symbolic Sieves

We present a unified mathematical framework that treats Number Theory not as a static field of discrete values, but as a dynamic Multiple-Input Multiple-Output (MIMO) control system. This paper synthesizes two distinct analytic operators: the \textit{Symbolic Trigonometric Sieve} (STS), which transforms discrete digit-exclusion rules into analytic singularities via $\Omega \cdot \tan(\pi n)$ [1], and the \textit{Arithmetic Resistance} ($f_r$), which linearizes the Prime Number Theorem through fractional root differences. By defining the integer $n$ as a coupled state vector $\mathbf{x}(n)$ evolving through arithmetic, symbolic, and $p$-adic domains simultaneously, we establish a ``Parallel Solver.'' This state-space formulation allows for the application of control theory concepts—such as observability, transfer functions, and Lyapunov stability—to classical problems in Diophantine approximation and prime distribution. We demonstrate that the Riemann Hypothesis can be reinterpreted as a feedback error correction signal within this critically damped system.