Grand Unified Theory on Sₐ^3 Noncommutative Geometry: Geometric Unification of the Standard Model, Gravity, and Flavor Physics

The unification of quantum mechanics and general relativity, as well as the first-principles explanation of free parameters in the Standard Model (SM), are long-standing fundamental problems in physics. This work establishes a self-consistent grand unified theoretical framework based on Sₐ^3 noncommutative geometry, starting from the geometric structure of spacetime. Via the Representation Splitting Theorem on the quantum flag manifold, we provide a first-principles geometric interpretation of the three generations of fermions. Using the topology of the quantum flag manifold and the Scale Self-Duality Axiom, we derive the classical benchmark value of the fine-structure constant and complete a semi-quantitative estimation of higher-order corrections, which is highly consistent with experimental data. Through the Spectral Action Principle, we fix the geometric benchmark exponent of fermion masses (consistent with the electron mass experiment) and rigorously prove the geometric necessity of vacuum alignment for flavor Higgs fields, systematically deriving the CKM and PMNS mixing matrices that match experimental observations. Meanwhile, the Einstein-Hilbert action of general relativity is naturally derived, and a self-consistent solution to the cosmological constant fine-tuning problem is proposed. This theory unifies quantum mechanics, all four fundamental interactions, and the full particle content of the SM from a single geometric axiom, naturally deriving the SM gauge group SU (3) c × SU (2) L × U (1) Y and chiral fermion representations. Notably, we establish a complete, falsifiable system of 9 testable experimental predictions classified by verification time windows: 1. Neutrino Physics (3-10 years): Strict normal mass ordering, ₁₃≈8. 5^, and constraints on Majorana phases (tested by JUNO, DUNE, Hyper-K) ; 2. Cosmology (5-10 years): Closed universe with ₖ<0 (magnitude ~10^-3 to 10^-4) (tested by CMB-S4, Euclid, LSST) ; 3. Quantum Gravity (3-10 years): Photon dispersion correction with O (1) and ultra-high-energy photon time delay ~10^-2 s (tested by LHAASO, CTA) ; 4. Particle Physics (10-20 years): Proton lifetime ₚ 10^34 years with dominant decay channel p K^+ (tested by Hyper-K, DUNE) ; 5. Higgs Physics (10-20 years): Higgs self-coupling constant deviating from the SM by O (0. 1) (tested by HL-LHC, CEPC, FCC-ee) ; 6. Black Hole Physics (10-20 years): ~10^-5 order correction to the shadow size of intermediate-mass black holes (tested by ngEHT) ; 7. Early Universe Physics (over 20 years): Primordial gravitational wave spectral index deviation nₜ≈0. 007 from the slow-roll consistency relation (tested by next-generation CMB polarization satellites, LISA, DECIGO) ; 8. Gravitational Physics (over 20 years): Violation of the weak equivalence principle with Eötvös parameter 10^-122 (tested by next-generation space equivalence principle experiments) ; 9. Cosmic Topology (over 20 years): Macroscopic multiple imaging effect and CMB topological signature of the S³ closed universe (tested by JWST, CSST). This work provides a self-consistent, falsifiable, and systematic scheme for the unification of fundamental physics, with clear quantitative targets, corresponding experimental facilities, and falsification criteria for all predictions.