The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 3: The Prime Arithmetic Theorem and the Conservation Law

Volume 3 of 14. Establishes the governing identity of the Prime Field Architecture: the Prime Arithmetic Theorem Π(N) + C(N) = 3 + S(N), the Conservation Law of the Arithmetic Modulation Field. Proves Theorem 7 (The Prime Arithmetic Theorem): for all N ≥ 7, Π(N) + C(N) = 3 + S(N), where S(N) = ⌊N/30⌋ × 8 + ρ(N mod 30) is the exact count of P1 Fiber positions. This identity is exact at every positive integer, with zero error, verified computationally at 13 independent scales from N = 10³ to N = 10¹⁵ by two independent engines (PFAE Sieve and PFA_PiN) with zero divergence. Establishes the Five Prime-Square Laws: Law I (Residue Collapse: p² mod 30 ∈ {1,19}), Law II (p-Fold Magnification: S(p²)/S(p) = p), Law III (Inverse Prime Reaction: F(p)/F(q) = q/p, verified to 7 decimal places), Law IV (√N Field Amplification), Law V (Action-Reaction Balance = the Conservation Law). Presents the trecade structural signature S(3×10^k) = 8×10^{k-1} exactly, verified at T1 through T11 by the SJQBL v8 engine. Establishes the isomorphism between the Conservation Law and Yang-Mills gauge charge conservation d_μ J^μ_a = 0.