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

Unifying quantum mechanics with general relativity and deriving the free parameters of the Standard Model (SM) of particle physics from first principles stand as the two most fundamental unsolved problems in theoretical physics. This work advances Einstein’s lifelong vision of geometric unification of all fundamental interactions—extending his pursuit of reducing physical laws to spacetime geometry—by elevating scale to an intrinsic, constitutive degree of freedom of spacetime, thus establishing the theoretical framework of Scale Geometry. We build a self-consistent grand unified framework based on the noncommutative geometry of the quantum 3-sphere Sq³, anchored by the core Scale Self-Duality Axiom (invariance of physical laws under ultraviolet-infrared scale duality) and the Spectral Action Principle. Rigorous proof of the Representation Splitting Theorem on the quantum flag manifold SUq (3) /T provides a first-principles geometric explanation for the three generations of SM fermions. Leveraging the manifold’s topology and scale self-duality, we derive the classical benchmark value of the fine-structure constant and semiquantitatively estimate higher-order corrections, achieving high consistency with experimental measurements. The spectral action yields a fermion mass power-law formula, with the geometric benchmark exponent ₀ 0. 63 locked to a value matching the experimental electron mass. From the quantum flag manifold’s Weyl group and line bundle topology, we naturally derive the S₃ quark flavor symmetry and A₄ lepton flavor symmetry, rigorously prove the geometric necessity of flavor Higgs field vacuum alignment, and systematically derive the CKM and PMNS mixing matrices consistent with observations—explaining intergenerational mass hierarchy and flavor mixing. Within this framework, the Einstein-Hilbert action of general relativity is derived naturally, the geometric origin of Newton’s gravitational constant is clarified, and a self-consistent solution to the cosmological constant fine-tuning problem is proposed (resolving the 120-order-of-magnitude discrepancy by tying the cosmological constant to the cosmic infrared scale). This theory unifies quantum mechanics, all four fundamental interactions, and the full SM particle content from a single geometric axiom, naturally yielding the SM gauge group SU (3) c SU (2) L U (1) Y and chiral fermion representations. We further establish nine falsifiable experimental predictions classified by verification time windows: (1) neutrino physics: inverted mass ordering, ₁₃ 8. 5^, and constraints on Majorana phases; (2) cosmology: closed S³ universe with ₖ < 0 (magnitude 10^-4) ; (3) quantum gravity: photon dispersion correction with t 10^-2 s for ultra-high-energy photons (O (1) ) ; (4) particle physics: proton decay dominated by p K^+ + (lifetime 10^35 years) ; (5) Higgs physics: Higgs self-coupling deviating from the SM by O (0. 1) ; (6) black hole physics: 10^-5 relative correction to the shadow radius of intermediate-mass black holes; (7) early universe physics: primordial gravitational wave spectral index deviating from slow-roll consistency by nₜ 0. 007; (8) gravitational physics: violation of the weak equivalence principle with parameter 10^-122; (9) cosmic topology: macroscopic topological signature of the closed S³ universe in CMB and all-sky surveys. Each prediction includes clear quantitative targets, corresponding experimental facilities, and explicit falsification criteria. This work breaks through the limitations of traditional noncommutative geometry, providing a self-consistent, rigorous, and experimentally testable scheme for realizing Einstein’s unfulfilled vision of geometric unification in fundamental physics.