The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 1: The Arithmetic Modulation Field

Volume 1 of 14. Establishes the arithmetic ground of the Prime Field Architecture: the modulus M = 30 = 2×3×5, the three fibers (P1, E2, O3), and the Prime Field Transform T(n) = (⌊n/30⌋, n mod 30). Proves Theorem 1 (Uniqueness of M = 30) via the Clausen-von Staudt theorem: M = 30 is the unique modulus satisfying five arithmetic properties simultaneously — palindromic gap symmetry, φ(M) = 8 = dim(su(3)), denom(B₄) = M, boundary primes {2,3,5}, and minimality. Proves Theorem 2 (Fiber Partition): every positive integer belongs to exactly one of the three fibers. Proves Theorem 3: P1 = (ℤ/30ℤ)* is closed under multiplication mod 30. Presents the abstract honoring Yang and Mills, the MSC codes, keywords, and two tension-building questions that the series answers: what if the mass was never a dynamical property but an arithmetic property always already there, and what does it mean that the arithmetic of a single number contains the complete structure of the forces?