Weinberg Angle, GUT Coupling, and Three Matter Generations from a G2 Algebraic Framework

Building on Papers I and II, this paper presents algebraically derived results connecting the G2 framework to the Standard Model of particle physics. The central results are: (1) G2 branching rules showing 7→1+3+3* giving the quark-antiquark decomposition of QCD, with dim (G2) −dim (SM) =2=rank (G2) exactly; (2) the Weinberg angle sin² (W) |GUT=3/8 matching the SU (5) value exactly; (3) three matter θ generations from SO (8) triality; (4) the Koide formula 2/3 derived from the Z triality structure (0. 005%) ; ₃ (5) the EM coupling boundary condition EM (GUT) =1/64 from (1/24) × (3/8) ; (6) the GUT coupling α α α GUT=1/42=1/ (6×7) giving the chain 1/64 + running ≈ 137; (7) the Higgs self-coupling λ=1/8=1/SO (8) ₛpinor giving mH=v/2=123 GeV (1. 8% error) ; and (8) a G2 compact-dimension scale ratio R /R =50=val (N). The string scale candidate Mₛ=MGUT/50 and the string coupling gap (1/647 vs 1/42) ₂ ₁ are identified as open problems. All structural integers cited in the body are drawn from the corpus database of Paper I; their sources are tabulated in the Appendix for independent verification. Keywords: G2 algebra, Weinberg angle, GUT coupling, grand unification, Koide formula, SO (8) triality, three generations, Higgs mass, Froggatt-Nielsen, exceptional Lie algebras