Three Generations from the Geometry of S3 x S6

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)B-L and the full electric charge formula geometrically, with all 32 spinor states matching one SM generation. The gauge coupling ratio g2^2/g3^2 = 15/(16 pi) at equal radii agrees with the SM value at M_Z 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 N_gen = 3 from the G2 = Fix(Z3 subset Aut(O)) triality structure permuting three SO(8) representations 8_v, 8_s, 8_c, 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).