Ionization Energies from Icosahedral Geometry: The Twin Prime Formula

We derive first ionization energies for alkali and alkaline earth metals using only icosahedral geometry and the vertex twin primes (11, 13). The effective nuclear charge is given by Z_eff = 13n/24 + 11/(30n) + (g−1)×11/24 + 11/(120n²), where n is the principal quantum number and g is the group number (1 or 2). In icosahedral form: Z_eff = (V+1)n/(2V) + (V−1)/(En) + (g−1)(V−1)/(2V) + (V−1)/(10Vn²), with V = 12 vertices and E = 30 edges. The formula yields exact Z_eff = 1 for hydrogen and achieves better than 5% accuracy for 8 of 11 elements tested. The quantum correction term 11/(120n²) emerges naturally from icosahedral geometry, where 120 = 10V is the order of the icosahedral rotation group. The formula uses only the vertex twin primes (11, 13), with 13 appearing once (matter branch) and 11 appearing three times (antimatter branch) — a 1:3 ratio that mirrors quark content in baryons. Systematic deviations identify real physics: sodium's −9.2% error reflects 3s orbital penetration, cesium's +6.6% error reflects relativistic contraction, and beryllium's +11.5% error reflects high electron density effects.