Geometric Foundations of Universal Constants: A 600-Cell Framework for Coupling Constants, Mass Hierarchies, and Spacetime Geometry

We present a framework in which the fundamental constants of particle physics emerge from the geometric and spectral properties of the 600-cell polytope. This four-dimensional regular polytope, identified as the universal energy minimizer in R⁴ by the Cohn-Kumar theorem, provides a discrete structure whose spectral properties encode the coupling constants with high precision: the fine structure constant (0. 0001\%), the strong coupling ₛ (0. 03\%), the Weinberg angle ²W (0. 19\%), and the Higgs mass (0. 09\%). The 24 non-fermionic vertices form the D₄ root system, from which the Standard Model gauge group SU (3) SU (2) U (1) emerges as a maximal-rank subalgebra. The remaining 96 vertices of the snub 24-cell encode fermions (96 = 16 3 2), with a geometric 4-to-3 selection mechanism via the temporal axis. The lepton mass hierarchy m_/mₑ ^11, m_/m_ ^6 is derived from holonomy on the Hopf fibration, where the exponents 5 (graph diameter) and 6 (decagons per vertex) are topological invariants of the 600-cell. Electroweak phase corrections reduce the lepton mass errors to 0. 25--0. 53\%. A unified mass formula mf = mₑ ^a d₄₅₅ + b k₄₅₅ with sector-dependent corrections predicts seven of nine fermion masses within 2\%. We identify Discrete Scale Invariance (DSI) as the physical mechanism: the 600-cell lattice generates a potential with minima at mₑ ⁿ, and the topological quantum numbers (a, b) count diameter traversals and decagonal loops on the graph. An exploratory gravitational model (Surface Tension Gravity) is presented in the appendix: it reproduces GR weak-field tests at 1PN order but predicts a black hole shadow half the GR size (7 tension with EHT), and should be regarded as a toy model. The topological quantum numbers (a, b) are uniquely determined by complexity minimization on the graph: (a, b) = |a|+|b| subject to 5a + 6b = n, where 5 and 6 are the intersection numbers of the 600-cell. The E₈ root system is constructed explicitly from the 600-cell via the icosian lattice: two copies S and T = ' S yield all 240 roots with exact inner products. Two algebraic constraints---Z norm restriction and a three-line rule in the (a, b) lattice---plus norm positivity on the b=1 line correctly classify all 31 mass levels from n=0 to n=30; the sole remaining pattern is the three-generation limit (b 2). The CKM exponents connect to H₄ Coxeter exponents and admit a cleaner formulation via H₄ H₄' cross-pairings from the E₈ branching. The Down quark prediction (5. 15\, MeV) lies within 1 of the PDG value (4. 67^+0. 48-₀. ₁₇\, MeV). While D₄ triality is preserved by the 600-cell geometry alone, the E₈ embedding breaks it: the three D₄ legs extend into E₈ with distinct tail lengths \0, 1, 3\ and Kac labels \3, 4, 5\, uniquely assigning each leg to SU (3) C, SU (2) L, or U (1) Y via the standard GUT breaking chain. The edge structure automatically forbids direct gauge--gauge couplings (A-A = A-B = B-B = 0), leaving exactly three interaction vertex types (ACC, BCC, CCC) in the ratio 1: 2: 2; the massless graph propagator changes sign at distance d=3, exhibiting confinement-like behavior. A detailed analysis of the local neighborhood reveals a natural 1 + 3 + 8 edge splitting that matches (U (1) SU (2) SU (3) ) = 12: each fermion has a geometrically unique ``special'' CC neighbor (degree~4, shared A-gauge vertex, zero shared B), and the effective gauge adjacency through fermion mediators reproduces the 24-cell exactly (A₄₅₅ = A²|₆₀ₔ₆₄/3). Among 1200 plaquettes, only SU (3) has pure plaquettes (288) ---zero for U (1) and SU (2) ---reproducing the non-Abelian self-interaction structure of QCD. A vertex-transitivity theorem proves that spectral projections are sector-blind, establishing that the Lagrangian form (from topology) and coupling values (from the spectrum) are two independent geometric properties of the 600-cell. We explicitly identify remaining open problems including the three-generation count, the CKM derivation mechanism, and the need for a complete Lagrangian formulation.