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

We prove that the stella octangula (two interpenetrating tetrahedra forming an 8-vertex compound) is the unique minimal three-dimensional polyhedral realization of the SU (3) weight structure, with the finite Weyl group W (SU (3) ) ≅ S₃ (order 6) embedded as a subgroup of the polyhedral symmetry group Oₕ (order 48) ---not claiming any isomorphism between the discrete polyhedron and the continuous 8-dimensional Lie group SU (3). Geometric foundations: (1) Under standard physics (GR + QM), spacetime dimension D=4 is uniquely compatible with stable bound-state observers. (2) SU (3) is topologically derived as the unique gauge group: the stella's intrinsic Z₃ rotational symmetry determines Z₃ ⊆ Z (G), and the rank constraint rank (G) ≤ 2 from D=4 uniquely forces G = SU (3). (3) The Killing form induces a Euclidean metric on 2D weight space, extending consistently to the 3D stella embedding. (4) Among all topological spaces satisfying the geometric realization conditions---including non-convex polyhedra, infinite structures, and fractals---the stella octangula is unique. Dynamical consequences: (5) Fermion masses follow m ∝ ω·η from phase-gradient coupling. (6) The mass hierarchy pattern mₙ ∝ λ^2n is derived from generation localization geometry; the Wolfenstein parameter λ = (1/φ³) sin 72° = 0. 2245 (0. 2σ from PDG). (7) The Strong CP problem is completely resolved: θbare = 0 from Z₃ superselection and arg det (Mq) = 0 from real overlap integrals. (8) Three fundamental asymmetries (weak chirality, time's arrow, matter dominance η ≈ 6×10⁻¹⁰) share unified topological origin from stella orientation. (9) Einstein's equations emerge as fixed-point conditions for metric iteration, with G = 1/ (8πf_χ²). (10) Neutrino reactor angle θ₁3 = 8. 54° derived via A₄ → Z₃ breaking formula. (11) Three fermion generations derived through four independent proofs. The framework reduces the Standard Model's 20 fermion-sector parameters to approximately 11. Machine-verified Lean 4 code and Python verification scripts are provided.