A Geometric Framework for Unifying the Standard Model and Gravity: Noncommutative Deformation of S^3 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 S_q^3 of the three-dimensional sphere S^3. Starting from the real spectral triple (A, H, D, J) with total spacetime S_q^3 × 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 SU_q(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 α ≈ 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.