CCR-k Arithmetic Refinement: p-Adic Resonance, Ramification, and Arithmetic Density Laws

Abstract

The paper develops the arithmetic refinement of our original CCR-k General Theory, which arose from the higher-order structure of chronological carry and exact pushdown normalization. Whereas the general theory establishes the higher-order transport equations and their Stirling-type structure, this work determines the arithmetic constraints governing their realizability and local splitting behavior. It establishes exact integer source lattices and obstruction formulas across all odd orders, analyzes exceptional p-adic phenomena at primes 73 and 419, and develops Galois-theoretic, Haar-measure, and adelic distribution results. A central distinction emerges between fixed integer sources, which admit only finitely many relevant splitting primes, and Haar-distributed profinite sources, which exhibit almost-sure infinite splitting phenomena. Its main novelty is the transformation of the CCR-k hierarchy from a formal chronological transport theory into an arithmetic framework with explicit source obstructions, local classification, and global distribution laws, thereby completing a substantial arithmetic layer of the original theory.