The Prime Remainder Theorem: A Structural and Algebraic Framework for Prime Classification

This work introduces a constructive and deterministic method for identifying prime numbers by removing all classifiable non-prime structures from the natural numbers. By filtering out composites, prime powers, and near-composite numbers, we isolate the primes as the only remaining elements. The model is validated computationally up to ten billion and extended algebraically to offer new insights into prime gaps and cryptographic key generation. The framework opens a new perspective on prime theory rooted in structure rather than indivisibility.