The Law of N: A Modular Framework for Prime Classification and Discovery

The Law of N (LN) defines a complete partition of the natural numbers greater than one into primes, prime powers, and composites, enabling modular classification of primes as residue-shadowed (PR) or structurally-isolated (PS). By rejecting composite-dense modular neighborhoods, LN supports deterministic prime prediction. We implement and validate an LN-based engine that discovers 4096-bit to 16,384-bit RSA-grade primes using modular exclusion alone. Results confirm LN as a mathematically sound and cryptographically applicable framework for deterministic prime generation.