The Fine Structure Constant from Lattice Gauge Theory: A Combinatorial Derivation with Zero Free Parameters

We show that the inverse fine structure constant admits a convergent structural expansion α⁻¹ = C + r/C - r, T(nc)/T(r), C² + r, T(nc)/T(r), T(nc+1), C³ - ⋯ where C = n · Ng + r = 11 × 12 + 5 = 137 is the dimension of the interaction space, $n = 11$ is the number of independent degrees of freedom per lattice site, Ng = 12 is the number of gauge bosons, $r = 5$ is the number of independent Noether charges (four Cartan generators plus baryon number), nc = n - d = 7 is the number of compact dimensions ($d = 4$ being spacetime), and $T(k) = k(k+1)/2$ denotes the k-th triangular number. This decomposition arises from a vertex tensor product factorization: the interaction space at each lattice site decomposes as V = (C11 ⊗ Lie(G)12) ⊕ Z(U(g))₅, yielding $132 + 5 = 137$ independent interaction sectors. The electromagnetic coupling is the reciprocal of this count. Each higher-order correction introduces the next triangular number T(nc + k) in the denominator, producing a rapidly convergent series. The series converges by the twenty-fifth term (corrections fall below 10⁻¹⁰⁰), yielding α⁻¹ = 137.035,999,177,301,0…, which matches the CODATA 2022 value $137.035,999,177(21)$ to 0.014σ. We tabulate eighty significant figures of the converged value, of which only twelve are currently measured; the remaining digits are forward predictions for next-generation experiments. The same framework predicts the ratios αₛ : αW : α = 16 : 5 : 1 from the generator-charge structure of each gauge factor, in agreement with measured values to within a few percent. The integer $137$ is not a free parameter but a combinatorial invariant of the Standard Model placed on a four-dimensional lattice, determined entirely by the gauge group structure, the spacetime dimension, and the vertex factorization principle.