The Prime Field Architecture of the Arithmetic Modulation Field: Yang-Mills Existence and Mass Gap on R⁴ — Volume 4: The Gauge Group Identification

Volume 4 of 14. Establishes the gauge group of the Prime Field Architecture: G = SU(3), demanded by the gap arithmetic of M = 30. Proves Theorem 9 (g_PFA ≅ su(3), G = SU(3)) via a nine-step non-circular derivation: gap arithmetic {2,4,6} → inner product ⟨α₁,α₂⟩ = −1 → Cartan matrix [[2,−1],[−1,2]] → Serre construction → unique Lie algebra g_PFA → Cartan-Killing classification → g_PFA ≅ su(3). At no step is su(3) assumed. All 15 Serre relations verified with max error 0.00e+00. Proves Theorem 10 (Explicit su(3) generators): all 64 commutator pairs [T^a,T^b] verified with max error < 10⁻¹⁶. Proves Theorem 11 (G = SU(2) from count-3 sector). Establishes three independent identifications of G = SU(3): algebraic (φ(30) = 8 = dim(su(3))), geometric (A2 root system from count-6 sector), and combinatorial (8 simultaneous positions at gap 30). Presents the complete gauge sector identification table mapping gap counts to gauge groups: SU(2) from gap-4 pairs, SU(3) from gap-6 pairs, and the Standard Model gauge group U(1) × SU(2) × SU(3) from the three fibers.