This paper presents the complete generating function for the fine structure constant, with derived coefficients Cₖ = 2^(k²). The coefficient at rung k counts the directed coupling configurations among k self-referential modes at maximum entropy equilibrium — a combinatorial derivation, not a fitting. The series shares the asymptotic character of QED's own perturbation expansion (Dyson 1952) and has optimal truncation near k = 6. A pre-registered fifth term (magnitude 1.3×10⁻⁸) awaits future measurement at 10⁻¹⁰ precision. The fine structure constant is a theorem of self-referential geometry.