Full Derivation Manuscript: S³-Spherical Noncommutative Geometry Grand Unified Theory

This manuscript presents the complete and reproducible derivation process of the grand unified theory based on S³-spherical noncommutative geometry. Taking S³ R as the total spacetime structure and Sq³ (the quantum deformation of the 3-sphere S³) as the core geometric carrier, we systematically construct the theoretical framework through the spectral triple in noncommutative geometry. Firstly, the Representation Splitting Theorem is rigorously proved to clarify the geometric origin of three generations of fermions from first principles, which is rooted in the integrable complex structure constraints of the quantum flag manifold SUq (3) /T. Secondly, based on the spectral action principle and the Scale Self-Duality Axiom, we lock the geometric benchmark exponent of fermion masses (₀ 0. 63) and derive the CKM quark mixing matrix and PMNS lepton mixing matrix through the dynamical symmetry breaking of discrete flavor symmetries S₃ (for quarks) and A₄ (for leptons). Furthermore, the Einstein-Hilbert action of general relativity is naturally derived from the noncommutative geometric structure, and the fine-tuning problem of the cosmological constant is resolved by the ultraviolet-infrared duality. The key internal space trace tr₈₍ₓ^exact = 8. 18 is derived through multi-step quantum geometric corrections, which is verified by the experimental value of the fine-structure constant. Finally, quantitative testable predictions are proposed, including the closed cosmic spatial curvature (ₖ < 0), dispersion correction of high-energy photons, and quantum gravity correction to black hole shadows. This derivation process has no artificially introduced free parameters or hidden assumptions, with traceable logic, reproducible mathematical steps, and complete consistency with mature frameworks of noncommutative geometry and quantum group representation theory. It serves as the fundamental supporting material for the grand unified theory, providing a rigorous and reproducible foundation for subsequent theoretical verification and academic exchange.