A Preliminary Study on Geometric Unification of Fundamental Physics via Sq³ Noncommutative Geometry

The unification of quantum mechanics and general relativity, as well as the first-principles explanation of the free parameters in the Standard Model of particle physics, have long been core unsolved problems in fundamental physics. This paper attempts to construct a grand unified theoretical framework based on the noncommutative geometry of Sq³, the quantum deformation of the three-dimensional sphere, and explore the geometric unification path of the Standard Model and gravity starting from the underlying structure of spacetime geometry. In this study, we derive the complete gauge group and chiral fermion representations of the Standard Model based on the noncommutative spectral triple of Sq³×R. By virtue of the representation splitting theorem on the quantum flag manifold, we attempt to interpret the geometric origin of the three generations of fermions. Through the spectral action principle and the scale self-duality axiom, we lock the geometric benchmark exponent of fermion masses, and combined with the S₃ and A₄ discrete flavor symmetries naturally derived from geometry, we deduce the CKM matrix of quark mixing and the PMNS matrix of lepton mixing, trying to explain the intergenerational mass differences and flavor mixing structure. Meanwhile, the Einstein-Hilbert action of general relativity is naturally derived from the noncommutative geometric framework in this paper, providing a possible solution to the fine-tuning problem of the cosmological constant. This study makes a preliminary exploration on unifying quantum mechanics, the four fundamental interactions and all particle content of the Standard Model from a single geometric axiom, and presents several quantitative predictions testable by future experiments, aiming to provide a self-consistent preliminary scheme for the research on the grand unification of fundamental physics. This research is still in the preliminary exploration stage, and some conclusions need to be further verified and improved.