Exact p-adic prefix rigidity and the Archimedean obstruction in affine valuation dynamics

We develop a complete and exact structural framework for affine dynamics governed by valuation data and deterministic normalization by powers of a fixed prime. The dynamics is analyzed through valuation words, which encode the local normalization behavior step by step. For every finite valuation word, we construct an associated deterministic p-adic prefix system. We show that such a system is either inconsistent or selects a unique residue class modulo a power of the prime. This establishes a rigid and exact correspondence between finite valuation data and p-adic congruence classes, without any probabilistic or heuristic input. We also describe the inverse structure explicitly. All admissible preimages arise from finite inverse compositions and form affine families that can be written in closed form. The inverse dynamics is therefore completely classified and fully encoded by explicit arithmetic data. For infinite valuation words, we prove a fundamental rigidity result: the associated p-adic prefix system never blocks. Every infinite valuation pattern canonically determines a unique compatible p-adic integer. As a consequence, no obstruction to global finiteness or exclusion of infinite behaviors can arise from local, finite, or purely p-adic constraints alone. The exclusion of nontrivial infinite behaviors is therefore reduced to a single Archimedean realizability obstruction: the p-adic codes determined by infinite valuation patterns must fail to correspond to natural numbers. This isolates the precise structural bottleneck between p-adic coherence and Archimedean realizability. The framework is fully deterministic and non-probabilistic, and provides an exact separation between local p-adic structure and genuinely global arithmetic constraints.