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

This preface to the fourteen-volume Prime Field Architecture Yang-Mills proof series establishes the arithmetic foundation from which the complete proof grows. It presents the Prime Arithmetic Theorem — the exact identity Π(N) + C(N) = 3 + S(N), proved by Skerl (2026) — as the Conservation Law of the Arithmetic Modulation Field of modulus M = 30 = 2×3×5, and demonstrates that this exact identity is the arithmetic statement of Yang-Mills gauge charge conservation.