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, Oₕ 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.