A p-adic non-invariance principle for affine normalization maps

We study a class of affine normalization dynamics generated by p-adicrenormalized maps of the form (a t + c) divided by the highest power of aprime p dividing (a t + c), where p is a prime, a is coprime with p, and c isa fixed integer. After introducing a natural index space, we show that the dynamics is governedentirely by exact p-adic valuation constraints. The local p-adic structure iscompletely rigid: for each valuation depth r, the condition that the p-adicvaluation is at least r selects a single arithmetic progression modulo p^r.These congruence classes form a strictly nested p-adic obstruction tower,independent of probabilistic or asymptotic considerations. Our main result establishes a p-adic congruence–dynamical non-invarianceprinciple. Beyond an explicit and computable depth threshold depending only onthe affine data, the induced dynamics on each deep p-adic congruence class isdeterministically contracting. As a consequence, no periodic orbit can besupported entirely within a sufficiently deep p-adic regime. This yields a global finiteness result: the p-adic valuation depths admissiblealong any periodic orbit are uniformly bounded by an explicit constant. Allarguments are elementary and exact, relying solely on linear congruences,divisibility, and direct comparison between affine growth and p-adiccontraction.