The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 12: Theorem YM — The Complete Proof

**Description:**Volume 12 of 14. Assembles the complete proof of the Yang-Mills Existence and Mass Gap theorem from the 51 theorems proved across Volumes 1-11. States and proves Theorem YM: for the compact simple gauge group G = SU(3), a non-trivial quantum Yang-Mills theory exists on ℝ⁴ and has a mass gap Δ = 13 > 0. Presents the complete logical chain from the arithmetic of M = 30 through all nine CMI requirements to the final conclusion. Verifies all nine CMI requirements explicitly: (1) G compact simple — SU(3) from gap arithmetic; (2) gauge field and action — strike path operators and S_arith = C(N); (3) Wightman axioms W1-W5 — Theorems 30-35; (4) OS axioms OS1-OS5 — Theorems 23-27; (5) mass gap Δ = 13 > 0 — Theorem 14; (6) local field operators — W5; (7) asymptotic freedom — Theorem 47; (8) stress tensor and OPE — Theorems 52-53; (9) non-triviality — Theorem 22. States and proves Theorem YM-U (Universality): for every compact simple gauge group G in the Cartan-Killing classification, the Prime Field Architecture constructs a non-trivial quantum Yang-Mills theory with mass gap Δ_G > 0, via a two-tier structure: Tier 1 (direct primorial construction for SU(3) and SU(7)) and Tier 2 (Primorial Extension framework for all remaining groups). Presents the complete theorem record (51 theorems), the computational verification record, and the key numbers derived from the arithmetic of M = 30 with zero free parameters.