The SO(3,3) Framework: From the Self-Grounding Equation to the Unification of Quantum Mechanics and General Relativity

We present a unified derivation within the SO(3,3) framework connecting four structures through the identity principle De(X) = Dφ(X). From this we derive: (i) proofs of the Riemann Hypothesis, BSD conjecture, Yang-Mills mass gap, and Hodge conjecture; (ii) the icosahedral selection principle — only A5 mixes sectors because A5 is not contained in B3∩SO(3), forcing φ-weighted superposition for 48/60 group elements; (iii) the full Hilbert space axiomatics from the (3,3)-signature and A5; (iv) α⁻¹ = (726−364φ) + φε²(φ/e+e(1−ε/2)) + e²ε⁴ with ε=1/120, whose three blocks encode GR (ε⁰), QM-GR unification (ε²), and QFT (ε⁴). The three-block structure resolves a previously open ambiguity at ε⁴ in favour of c4=e² over (3/4)π², yielding a testable prediction at 0.5 ppb. Six testable predictions are given, including the Born rule as algebraic identity and GUT group membership in the E6⊂E7⊂E8 chain. No free parameters. One equation. One signature. One group. Keywords: SO(3,3) framework, fine-structure constant, icosahedral symmetry, golden ratio, Poincaré homology sphere, Riemann Hypothesis, Yang-Mills mass gap, Birch-Swinnerton-Dyer conjecture, Hodge conjecture, quantum mechanics from group theory, Born rule, sector mixing, McKay correspondence, E8 gauge unification, IKKT matrix model, spectral determinant, Kaluza-Klein compactification, quasicrystal