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…