Investigations on Geometric Unification of the Standard Model, Gravity and Flavor Physics via Sq³ Noncommutative Geometry

The unification of quantum mechanics and general relativity, as well as the first-principles interpretation of free parameters in the Standard Model (SM) of particle physics, remain two of the most fundamental unsolved problems in modern fundamental physics. In this work, we develop a comprehensive geometric framework for grand unification by grounding the construction in the noncommutative geometry of the quantum 3-sphere Sq³ and the associated real spectral triple on the total spacetime manifold Sq³. We rigorously derive the full SM gauge group SU (3) c SU (2) L U (1) Y and chiral fermion representations from the underlying geometric structure of spacetime, with no ad hoc introduction of symmetries or particle content. By establishing the Representation Splitting Theorem on the quantum flag manifold SUq (3) /T, we provide a first-principles geometric origin for the three generations of fermions, addressing a long-standing puzzle in the SM. Leveraging the topology of the quantum flag manifold and the Scale Self-Duality Axiom, we deduce the classical benchmark value of the fine-structure constant and perform a semi-quantitative estimation of higher-order corrections, yielding results in excellent agreement with experimental measurements. Via the Spectral Action Principle and Scale Self-Duality Axiom, we fix the universal geometric benchmark exponent for fermion masses, which matches the experimental value of the electron mass with high precision. We further prove the geometric necessity of vacuum alignment for flavor Higgs fields, fully derive the CKM quark mixing matrix and PMNS lepton mixing matrix, and interpret the intergenerational mass hierarchy and flavor mixing structure of fermions. Additionally, the Einstein-Hilbert action of general relativity is naturally derived from the spectral action, and the fine-tuning problem of the cosmological constant is resolved by the Scale Self-Duality Axiom. This work realizes a geometric unification of quantum mechanics, all four fundamental interactions, and the complete particle content of the SM from a single set of geometric axioms, and presents a series of quantitative, testable predictions for quantum gravity effects. The framework is self-consistent, mathematically rigorous, and experimentally falsifiable, offering a new perspective for the unification of fundamental physics.