S³ Noncommutative Geometry and Fundamental Physics Unification

This work presents a complete grand unified theory based on the noncommutative geometry of the three-sphere S³q. Starting from first principles of quantum spacetime geometry, we unify general relativity and the Standard Model within a single mathematical framework. The theory geometrically derives the gauge group, chiral fermion representations, and the origin of three fermion generations via the Representation Splitting Theorem on quantum flag manifolds. Fermion mass hierarchies, flavor mixing matrices (CKM/PMNS), and neutrino masses via the seesaw mechanism are naturally realized without free parameters. The spectral action principle yields the Einstein–Hilbert action and resolves the cosmological constant fine-tuning problem through scale self-duality. Testable predictions include cosmic spatial curvature, quantum gravity corrections to photon dispersion, and black hole shadow modifications. The theory is self-consistent, falsifiable, and provides a unified geometric foundation for fundamental physics.