A Geometric Framework for Unifying the Standard Model and Gravity: Noncommutative Deformation of S³ Spacetime

We propose a consistent geometric framework for unifying the Standard Model (SM) of particle physics and general relativity, based on the noncommutative quantum deformation Sq³ of the three-dimensional sphere S³. Starting from the real spectral triple (A, H, D, J) with total spacetime Sq³ R, we systematically derive the core structures of the SM, including the gauge group SU (3) c SU (2) L U (1) Y, chiral fermion representations, and three generations of fermions (via the Representation Splitting Theorem on the quantum flag manifold SUq (3) /T). Through the spectral action principle and Scale Self-Duality Axiom, we explore the geometric origin of fundamental physical quantities: the fine-structure constant 1/137. 036 is locked by the topology of the quantum flag manifold and cosmic scale ratio, the Einstein-Hilbert action of gravity is naturally derived, and the fermion mass hierarchy, CKM/PMNS mixing matrices are explained by combining geometric constraints with discrete flavor symmetries. This framework avoids artificial free parameters, resolves the fine-tuning problem of the cosmological constant, and provides testable predictions (e. g. , spatial curvature of closed universe, photon dispersion corrections). The work contributes to the exploration of first-principles unification of fundamental interactions from a geometric perspective.