Primality as Cyclotomic Saturation: Primes as Finite Fields, Composites as Cartesian Products, and the Structural Blindness of Peano Arithmetic

This paper proposes a shift from a computational to a structural vision of the integers. The central contribution is the Cyclotomic Saturation Theorem: an integer N is prime if and only if its multiplicative structure is algebraically saturated with respect to cyclotomic polynomials. The bidirectional proof uses Bézout's identity in Zx and does not rely on the assumption of cyclic group structure. On this basis we develop a spectroscopic metaphor: primes as quantized atoms whose energy levels are completely saturated — the algebraic analogue of noble gases — and composites as molecules arising from the Cartesian product of prime atoms via the Chinese Remainder Theorem. Bond electrons, solutions of x²≡1 mod N, deterministically factorize N. Rather than asking how primes are distributed among the naturals, the framework reverses the question: naturals grow from primes, generated progressively by prime generators through successive powers mod P. The framework reveals that Peano Arithmetic is inadequate to capture the constitutive internal structure of integers: its linear additive logic treats all naturals as successors of zero. It is precisely this inability — PA's blindness to the internal structure of numbers derived from syntactic encoding — that allows Gödel to prove incompleteness through the construction of the undecidable proposition.