The Wave System for Mapping Prime Numbers: A Deterministic Phase-Shift Theory in Composite Disruption

This research introduces a novel, rigorous framework for parsing the distribution of prime numbers by analyzing the deterministic mappings of composite fields. Instead of tracking stochastic prime behavior, we model composite number generation within the linear arithmetic progression system 2n+1 as a set of periodic interference patterns, termed "harmonic wave systems." We demonstrate that every composite value within this domain is strictly generated via a sequence of waves (W_k) governed by the initial anchor point 3k+1 and a linear progression rule (3k+1) + (2k+1)m. Prime numbers are consequently defined as the absolute structural residues escaping these wave disruptions. The proposed system offers absolute algebraic certainty, bypassing probabilistic zero-counting methods, and provides a hardware-friendly topological foundation for discrete computational mathematics.