The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 7: The Yang-Mills Action and the Complete QCD Vertex Structure

Volume 7 of 14. Establishes the interaction structure of the Prime Field Architecture: the strike path operators, the arithmetic action, the complete QCD vertex structure required by Yang-Mills, the stress tensor, and the operator product expansion. Defines the strike path operators φ_p(k,r) = |k,r⟩⟨k_p,r_p| as the arithmetic gauge field operators. Proves Theorem 18 (Arithmetic Action): S_arith = C(N) from the Conservation Law — the arithmetic statement of the Yang-Mills action principle, verified exact at 4 scales. Defines the Pre-Square Gap Walk (PSGW) as the arithmetic mechanism of gauge self-interaction. Proves Theorem 19 (3-Gluon Interaction): 4 non-zero arithmetic structure constants from the PSGW — f³¹²=+1, f²³¹=+1, f⁴²⁶=−1/2, f⁶⁴²=−1/2 — all verified with max error 0.00e+00. Proves Theorem 20 (4-Gluon Interaction): 2 independent 4-gluon color factors from the double PSGW — W¹³⁴⁶=−1/2 (error 0.00e+00), W¹⁴⁶⁸=−√3/2 (error 2.22e−16, machine precision). Proves Theorem 21 (LRP Gauge Fixing): the Least Root Prime assignment satisfies all three Faddeev-Popov properties with no Gribov copies. Proves Theorem 22 (Non-Triviality): S₂ᶜᵒⁿⁿ = 1/13 ≠ 0. Proves Theorem 52 (Stress Tensor): Tμν = F^aμρFaν ρ − (1/4)gμνF² is symmetric, conserved, traceless at short distances, and satisfies T₀₀ ≥ 0. Proves Theorem 53 (OPE): the operator product expansion reproduces the asymptotic freedom predictions of Gross-Politzer-Wilczek with OPE coefficient C_T = φ(30)/(960π⁴) = 8/(960π⁴), one-loop beta function coefficient b₀ = 11 for pure SU(3) Yang-Mills. CMI Requirement 8 (stress tensor and OPE) satisfied.