Geometric Foundations of SU(3): Deriving Gauge Structure, Spacetime Dimension, and Spatial Extension from a Single Polyhedron

We present a self-contained derivation of the geometric foundations underlying Chiral Geometrogenesis (CG), a framework in which gauge symmetry, spacetime dimension, and spatial structure emerge from a single polyhedral object. Starting from three irreducible inputs — observer existence, the requirement that the pre-geometric substrate have finite information content, and the restriction to compact simple gauge groups — we establish a chain of theorems: (1) Spacetime dimension D = 4 is uniquely compatible with stable bound-state observers, via five independent physical constraints. (2) The gauge group SU(3) is derived (not selected) via two complementary paths — topological (Z₃ center from stella geometry plus rank constraint) and information-theoretic (Fisher metric non-degeneracy eliminates N ≤ 2; a parsimony criterion selects N = 3 as the smallest stable prime) — and verified by categorical consistency (Tannaka reconstruction from polyhedral data). (3) The stella octangula — the compound of two interpenetrating tetrahedra — is the unique minimal geometric realization of SU(3) among all topological spaces satisfying weight correspondence, Weyl symmetry, and charge conjugation conditions. No other polyhedron works. (4) The Euclidean metric on R³ emerges from the SU(3) Killing form rather than being assumed. (5) Extended 3D space is derived as the unique FCC lattice from SU(3) representation theory, with four constraints (12-regularity, triangle prohibition, 4-squares-per-edge, O_h symmetry) selecting the FCC tiling uniquely. (6) The Standard Model gauge group SU(3)_C × SU(2)_L × U(1)_Y is the minimal phenomenologically viable completion of the D₄ root system geometrically encoded by the stella's polytope embedding chain via D₄ → so(10) → su(5). (7) Color fields exist as a derived consequence of distinguishability on the stella boundary, with phases (0, 2π/3, 4π/3) uniquely fixed by Z₃ symmetry and color neutrality, bridging to dynamics. The framework rests on 8 independent inputs (3 irreducible, 5 supporting/redundant) and is subject to 5 explicit falsification conditions. Lorentz violation is bounded at 10⁻³² with a distinctive ℓ = 4 angular pattern. This paper establishes the kinematic geometric foundations; dynamical consequences (mass generation, confinement, gravity emergence) are deferred to the companion paper. Machine-verified Lean 4 proofs and Python verification scripts are provided in the supplementary repository.