Arithmetic Classification and Multiplicative Monoid Structure of Natural Numbers: A Seven-Class Partition, Prime Coordinate Representation, and Deterministic Addition Laws

Natural numbers are conventionally treated as ordered magnitudes. This paper develops a complementary framework: every natural number n encodes intrinsic structural information through its prime factorization — independent of magnitude and computable in deterministic, finite time. Five contributions (C1–C5):- C1: Seven-class partition of ℕ with a finite commutative monoid (S,×) and complete 49-entry Cayley table- C2: Power Dynamics universality theorem — every n > 1 reaches the absorbing class S₆ in at most 3 steps- C3: Seven deterministic addition laws, one non-trivial: Solar + Lunar → S₆ (verified on 90,000 pairs)- C4: SpectralAddress: K-dimensional p-adic valuation vector with four O(K) isomorphisms- C5: Resonance relation R(i,j) = Ω(gcd(i,j)) with ≈61% non-resonant pairs (6/π²) All results verified in Python (100,000 pairs), C++20 (≥475M Cayley lookups/s), and Verilog RTL (14/14 test vectors).