We derive the Standard Model gauge structure and fermion quantum numbers for one generation from the geometry of S3 x S6. By identifying the spin connection with gauge fields (following Lawrence, arXiv: 2203. 09473), we obtain the gauge group SU (3) x SU (2) x U (1) ₁-₋ and the full electric charge formula geometrically, with all 32 spinor states matching one SM generation. The gauge coupling ratio g2²/g3² = 15/ (16 pi) at equal radii agrees with the SM value at MZ within 4. 0%. We prove that the Atiyah-Singer index of the twisted Dirac operator on S6 = G2/SU (3) equals exactly one, and conjecture Ngen = 3 from the G2 = Fix (Z3 subset Aut (O) ) triality structure permuting three SO (8) representations 8ᵥ, 8ₛ, 8c, each carrying index 1. A theorem (Bundle Obstruction, E-L3B) proves that the G2-equivariant path to channel independence is ruled out; distinguishing the three channels requires an additional Spin (8) fibre-symmetry input, which remains an explicitly stated open problem. This is a candidate mathematical mechanism, not an experimentally confirmed theory of nature. The companion verification suite (2484 tests, exact-arithmetic sympy derivations for every claim) is available at the linked GitHub repository. Status: preprint, submitted for arXiv endorsement (hep-th).
Sergey Boyko (Wed,) studied this question.