The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 10: The Physical Mass Gap

Volume 10 of 14. Derives the physical mass gap from the arithmetic mass gap Δ = 13 and establishes the zero-parameter prediction for the lightest glueball mass. Proves Theorem 42 (Physical Mass Gap): m_gap/√σ_QCD = 2π√48/13 = 3.34855..., where 48 = 6 × 8 = (off-diagonal generators of su(3)) × (dim(su(3))) = N_off × φ(30). Comparison with lattice QCD (Morningstar-Peardon 1999): 3.3523 ± 0.0200. Agreement: 0.11%. Zero free parameters. Physical glueball mass: m_gap = 1.4671 GeV. Proves Theorem 43 (One-Loop Stability): the one-loop correction is 0.72%; the mass gap is stable under quantum corrections. Proves Theorem 44 (Physical Mass Gap Theorem): Δ = 13 > 0 (exact), spec(H_W) ∩ (0,m_gap) = ∅, m_gap = 1.4671 GeV. Proves Theorem 45 (Scale Invariance): Δ = 13 is independent of N — a renormalization group fixed point. The mass gap does not run. It is always already there in the arithmetic. Identifies the arithmetic of 48 = 6 × 8: the product of the number of off-diagonal generators of su(3) and the total dimension of su(3), both determined by the arithmetic of M = 30.