Standard Model Gauge Structure from \(S^{3}\) Spectral Geometry: Deriving \(SU(2)_{1} ×U(1)_{Y} ×SU(3)\) from the Dirac Spectrum

This article constitutes the fourth paper within the Spectral Bifurcation Cosmology (SBC) series. Built upon the compact $S^3$ spectral geometric framework established in prior works, we demonstrate that the gauge group structure $SU(3)_C\times SU(2)_L\times U(1)_Y$ of the Standard Model can be intrinsically derived from topological spectral geometry. No extra gauge symmetries are manually introduced. This work connects cosmological spacetime geometry with particle physics, establishing a unified geometric origin for cosmic evolution and fundamental interactions. Further research on fermion representations and mass generation will be presented in follow-up papers.