Geometric Foundations of Universal Constants: The 600-Cell Theory of Fundamental Physics

We present a comprehensive framework in which the fundamental constants of 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, provides a discrete structure for spacetime at the Planck scale. We derive the fine structure constant α with 0.0001% precision, the strong coupling constant αₛ with 0.03% error, and the Higgs mass with 0.09% accuracy. The theory successfully maps the three generations of fermions onto the 96 vertices of the snub 24-cell substructure, explains the CKM and PMNS mixing angles through a ``phi-tower'' hierarchy, and achieves near-perfect gauge unification via the E₈ extension with only 2.4% discrepancy. The Lagrangian framework passes all validation tests including general relativity limits, trans-Planckian unitarity, and Weyl conformal invariance. We propose testable predictions for photon dispersion from gamma-ray bursts using the quadratic model ($n=2$).