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 $\alpha^{-1} = C + r/C - r\,T(n_c)/(T(r)\,C^2) + r\,T(n_c)/(T(r)\,T(n_c{+}1)\,C^3) - \cdots$ where $C = n \cdot N_g + r = 11 \times 12 + 5 = 137$ is the dimension of the interaction space, $n = 11$ is the number of independent degrees of freedom per lattice site, $N_g = 12$ is the number of gauge bosons, $r = 5$ is the number of independent Noether charges (four Cartan generators plus baryon number), $n_c = n - d = 7$ 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 $V = (\mathcal{C}_{11} \otimes \mathrm{Lie}(G)_{12}) \oplus Z(\mathcal{U}(\mathfrak{g}))_5$, 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 $T(n_c + k)$ in the denominator, producing a rapidly convergent series. The series converges by the twenty-fifth term (corrections fall below $10^{-100}$), yielding $\alpha^{-1} = 137.035\,999\,177\,301\,0\ldots$, which matches the CODATA 2022 value $137.035\,999\,177(21)$ to $0.01\sigma$. 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 $\alpha_s : \alpha_W : \alpha = 16 : 5 : 1$ 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.