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)) ≅ S_3 (order 6) embedded as a subgroup of the polyhedral symmetry group O_h (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 Z_3 rotational symmetry determines Z_3 ⊆ 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 m_n ∝ λ²ⁿ 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: θ_bare = 0 from Z_3 superselection and arg det(M_q) = 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 θ_13 = 8.54° derived via A_4 → Z_3 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.