Chiral Geometrogenesis: Deriving Gauge Structure, Quantum Mechanics, Mass, and Gravity from Geometric Foundations

We prove the stella octangula (two interpenetrating tetrahedra) is the unique minimal 3D polyhedral realization of the SU (3) weight structure, with the Weyl group W (SU (3) ) ≅ S₃ embedded in the polyhedral symmetry group Oₕ. Geometric foundations: (1) Spacetime dimension D=4 is uniquely compatible with stable observers. (2) SU (3) is topologically derived as the unique gauge group from the stella's Z₃ symmetry and rank constraints. (3) The Killing form induces a Euclidean metric on weight space, extending to the 3D embedding. (4) Among all topological spaces satisfying the realization conditions, the stella is unique. Genuine predictions: (5) Fermion masses follow m ∝ ω·η from phase-gradient coupling. (6) Mass hierarchy mₙ ∝ λ^2n derived geometrically; Wolfenstein λ = 0. 2245 (0. 2σ from PDG). (7) Strong CP problem resolved: θ = 0 geometrically required. (8) Fermions are topological solitons; weak chirality, time's arrow, and matter dominance share unified origin; Asymmetric Dark Matter candidate (~1. 7 TeV). (9) Einstein's equations emerge as fixed-point conditions; G = 6. 52×10⁻¹¹ m³/ (kg·s²) derived (2. 3% from CODATA). (10) Neutrino angle θ₁₃ = 8. 54° via A₄ → Z₃ breaking. (11) Three generations from four independent proofs. (12) Spectral index nₛ = 0. 9648 from stella topology (0. 02σ from Planck 2018). Self-consistency: Quantum mechanics emerges from chiral field dynamics; Lorentz invariance emerges from discrete symmetry coarse-graining. Complete scattering theory (Feynman rules, amplitudes, hadronization) follows from geometric structure. The framework reduces 20 Standard Model fermion-sector parameters to ~11. Machine-verified Lean 4 code and Python verification scripts provided.