The Representation Splitting Theorem and Geometric Unification on Quantum Flag Manifolds

Unifying quantum mechanics with general relativity and explaining the free parameters of the Standard Model (SM) from first principles remain the most profound challenges in fundamental physics. This work establishes a complete grand unified theory (GUT) based on the quantum-deformed 3-sphere S_q^3 and quantum flag manifold SU_q(3)/T within the framework of noncommutative geometry (NCG). A key innovation is the Representation Splitting Theorem, which rigorously demonstrates that the fundamental representation of SU_q(3) decomposes into three inequivalent 1-dimensional representations on the quantum flag manifold—providing a geometric origin for the three generations of fermions, a long-standing puzzle in the SM. By incorporating the Scale Self-Duality Axiom, which links the Planck scale (l_p) to the cosmic Hubble radius (R_H), we derive a power-law formula for fermion masses with a geometric benchmark exponent α_0 ≈ 0.63 (calibrated via the electron mass). The flavor symmetries S_3 (quarks) and A_4 (leptons) naturally emerge from the Weyl group and line bundle topology of the quantum flag manifold, and we rigorously prove the geometric necessity of flavor Higgs vacuum alignment—leading to exact derivations of the CKM and PMNS mixing matrices that match experimental data. Additionally, the Einstein-Hilbert action of general relativity is derived from the spectral action principle, and the cosmological constant problem is resolved by showing the effective cosmological constant is determined by the infrared cosmic scale (not the ultraviolet Planck scale), eliminating the 10¹²⁰ fine-tuning discrepancy. The theory is parameter-free (all physical quantities depend only on l_p/R_H) and falsifiable, with key predictions including a closed universe (Ω_k < 0), energy-dependent photon dispersion (Lorentz invariance violation), and quantum-gravitational corrections to black hole shadows—all testable by upcoming experiments (CMB-S4, CTA, LHAASO, ngEHT) within 3–10 years. This work realizes a geometric unification of the four fundamental interactions, the SM particle content, and flavor physics from a single set of axioms, offering a self-consistent, predictive alternative to string theory and loop quantum gravity.