The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 2: The Gapping Mechanism and the Gap Spectrum

Volume 2 of 14. Establishes the arrangement of the Prime Field: the Gapping Mechanism Δ_P₁^{dynamic} = (6,4,2,4,2,4,6,2), the Gap Spectrum Γ(P1) = {2,4,6}, and the Sarpa~Jala Universal Spacing Law. Proves Theorem 4 (Sarpa~Jala Universal Spacing Law): the gap sequence Δ_P₁ is palindromic, has sum 30 = M, and contains exclusively multiples of 2. Proves Theorem 5 (Rotation Invariance): the gap spectrum is invariant under row translation — the arithmetic expression of gauge invariance. Proves Theorem 6: the Gap Spectrum IS the mass spectrum of the Yang-Mills theory. Establishes the Mod-60 extension (Form iv of the Gapping Mechanism), the bilateral reflection R₃₀: r → 30−r as the arithmetic Cartan involution, and the Seam S = 15 as the arithmetic center of the P1 Fiber. Demonstrates that the gap vocabulary {2,4,6} is the same vocabulary that encodes the A2 root system of su(3) — the first explicit connection between the Gapping Mechanism and the Yang-Mills gauge group.