The Photon Coupling Formula: Icosahedral Geometry in QED and Mass Ratios

We show that the photon coupling coefficient c₂ = (E − χ²)/D² = 26/9, derived from the hydrogen spectrum, encodes how photons couple to matter: they travel on icosahedral edges (E = 30), cannot self-interact at tree level (subtract χ² = 4), and require area normalization (D² = 9) because the electromagnetic interaction occurs on surfaces. The same icosahedral geometry generates the proton-electron mass ratio through prime factorization: 1836 = 2² × 3³ × 17 = χ² × D³ × (V+d) = 108 × 17. The divisor sum σ(1836) = 5040 = 7! factors as σ(χ²) × σ(459) = 7 × 720, revealing that 7 = V − d emerges from the arithmetic of χ through σ(χ²) = 1 + χ + χ² = 7. This connects the mass ratio to Robin's theorem threshold for the Riemann Hypothesis, Plato's ideal number, and the boundary of balanced growth (7! is the last highly composite factorial). The primes dividing 5040 are exactly {2, 3, 5, 7} = {χ, D, d, V−d}—all icosahedral.