The Rydberg Energy and Icosahedral Geometry

We report a numerical relationship between the Rydberg energy and icosahedral geometry: E_R × (V+E)² = 2V keV, where V = 12 vertices and E = 30 edges. This gives E_R × 1764 = 24 keV, matching experiment to 18 ppm. Rearranged, this predicts the hydrogen ionization energy as E_R = 24000/1764 = 13.6054 eV, compared to the measured 13.6057 eV (99.998% agreement). The sum V + E = 42 is the center of twin primes (41, 43), and (V+E)² = 1764 = 41×43 + 1. The small correction factor (1.84 × 10⁻⁵) equals α/400 where α is the fine structure constant and 400 = F² (faces squared). This relationship connects atomic physics to the same icosahedral structure that appears in quark geometry and the fine structure constant. We present this as an empirical observation requiring theoretical explanation.