Rational Interpreters for Discrete Dynamics: Existence, Exactness, and Decomposition over $p$-adic Fields

We address an inverse problem in non-Archimedean dynamics: given a finite discrete dynamical system (equivalently, a functional graph on $N$ states), construct a continuous $p$-adic dynamical system whose residue-level behavior reproduces the prescribed transitions. Using the cylinder partition of $\mathcal{O}_K$ (viewed as \emph{Witt cylinders} for unramified $K/\mathbb{Q}_p$), we encode states by pairwise disjoint closed balls and formalize an \textbf{interpreter} as a map sending each state ball into its target ball. Our main existence result constructs rational interpreters that are analytic (hence pole-free) on the prescribed state cylinders, combining rigid-analytic Runge approximation with finite interpolation constraints. Under a linear-dominance condition on each cylinder, ball images are explicit and locally affine, leading to a robust classification of discrete behavior into contractive, indifferent, and expansive regimes. Good reduction provides a selection principle for natural interpreters; effective degree and height bounds for general rational interpreters remain open. For composite alphabets we prove a \textbf{Dynamic Chinese Remainder Theorem} for congruence-preserving systems: the CRT isomorphism $\Theta:\mathbb{Z}/m\mathbb{Z}\xrightarrow{\sim}\prod_i\mathbb{Z}/p_i^{k_i}\mathbb{Z}$ (for $m=\prod p_i^{k_i}$) yields a factorization of the \emph{dynamics} (equivalently, the functional graph) on $\mathbb{Z}/m\mathbb{Z}$ into dynamics on the prime-power components, compatible with reduction. Finally, we discuss an inverse-limit (profinite) extension: compatible towers define a $1$-Lipschitz map on $\mathbb{Z}_p$, while selecting compatible analytic/rational interpreters across levels becomes a separate problem.