Sq³ Noncommutative Geometry Grand Unified Theory

The unification of quantum mechanics and general relativity, alongside the first-principles derivation of free parameters in the Standard Model (SM), remains two century-old cornerstones of fundamental physics. We establish a comprehensive grand unified framework rooted in Sq³ noncommutative geometry, starting from the intrinsic geometric structure of spacetime Sq³ R. Via the rigorously proven Representation Splitting Theorem on the SUq (3) quantum flag manifold Fq = SUq (3) /T, we provide the first geometric origin of three fermion generations from first principles. Leveraging the topology of Fq and the Scale Self-Duality Axiom, we derive the classical benchmark value of the fine-structure constant, with semi-quantitative higher-order corrections achieving exceptional consistency with the CODATA 2022 experimental value (^-1 136. 9137. 1). Through the Spectral Action Principle, we fix the geometric benchmark exponent of fermion masses (₀ 0. 63), perfectly matching the electron mass. We rigorously prove the geometric necessity of flavor Higgs vacuum alignment, fully derive the CKM and PMNS mixing matrices consistent with experimental data, and explain intergenerational mass hierarchies and flavor mixing. Additionally, the Einstein-Hilbert action of general relativity is naturally derived, and the cosmological constant fine-tuning problem is resolved. This work realizes, for the first time, a unified theoretical system encompassing quantum mechanics, all four fundamental interactions, and the full particle content of the SM 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 present testable predictions including cosmic spatial curvature (ₖ < 0) and photon dispersion corrections, offering a self-consistent, falsifiable scheme for fundamental physics unification.