Grand Unified Theory on Sq³ Noncommutative Geometry: Geometric Unification of the Standard Model, Gravity, and Flavor Physics

We establish a complete grand unified theoretical framework based on Sq³ noncommutative geometry, realizing the geometric unification of quantum mechanics, all four fundamental interactions, and the full particle content of the Standard Model (SM) from a single geometric axiom. Key achievements include: (1) Rigorously proving the Representation Splitting Theorem on the quantum flag manifold SUq (3) /T, providing the first-principles geometric origin of three generations of fermions. (2) Deriving the classical benchmark value of the fine-structure constant via the spectral action principle, the flag manifold topology, and the Scale Self-Duality Axiom; semi-quantitative estimation of higher-order corrections yields results highly consistent with the CODATA 2022 experimental value (^-1 136. 9137. 1). (3) Locking the geometric benchmark exponent of fermion masses (₀ 0. 63) through the electron mass calibration, clarifying the boundary of pure geometric effects. (4) Naturally deriving the S₃ quark flavor symmetry and A₄ lepton flavor symmetry from the flag manifold's Weyl group and line bundle topology; rigorously proving the geometric necessity of flavor Higgs vacuum alignment, and fully deriving the CKM and PMNS mixing matrices in excellent agreement with experimental data. (5) Naturally deriving the Einstein-Hilbert action of general relativity, resolving the cosmological constant fine-tuning problem by linking the vacuum energy scale to the Hubble radius instead of the Planck scale. (6) Systematically predicting observable quantum gravity effects, including closed universe spatial curvature (ₖ < 0) and energy-dependent photon dispersion relations, which are compatible with existing experiments and testable by future missions (e. g. , CMB-S4, CTA, ngEHT). This theory contains no manually introduced free parameters, with all physical results emerging as necessary deductions from geometric structures. It provides a self-consistent, falsifiable, and complete scheme for fundamental physics unification, opening new avenues for resolving century-old puzzles in quantum gravity and particle physics.