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 m_H=v/2=123 GeV (1.8% error); and (8) a G2 compact-dimension scale ratio R /R =50=val(N). The string scale candidate M_s=M_GUT/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.