The Rydberg Energy and Icosahedral Geometry

We derive the Rydberg energy from icosahedral geometry with zero free parameters. The complete formula E_R = (4V² × 5^D) / ((V+E) × 126) eV, where V = 12 vertices, E = 30 edges, and D = 3 dimensions, yields 13.6054 eV compared to the measured 13.6057 eV (18 ppm agreement). The apparent "1000" in E_R × (V+E)² = 2V keV is not a unit conversion but encodes geometry: 1000 = 1008 × 125/126 = (V+E) × 2V × 5^D/(5^D+1). The formula separates into three components: proton structure (2V/(V+E)²), proton-electron bond ((V+E) × 2V), and electron cap correction (5^D/(5^D+1)). The denominator 126 = D² × (V+χ) also appears in the α⁶ coefficient of the hydrogen harmonic expansion, providing independent geometric confirmation. The 18 ppm residual matches QED corrections at orders α⁶ and α⁷, derived separately from icosahedral harmonics.