Exact Derivation of the Fine-Structure Constant from the S³ Spectral Zeta Function

The fine-structure constant reciprocal is derived from the spectral zeta function of the Laplacian on the three-sphere S³: α⁻¹ = N + ζ_u(2)/c + ζ_u(3)·c/N² where N = 137 (the Koide integer), ζ_u(2) = π²/12 − 11/16 and ζ_u(3) = −π²/16 + 21/32 are proven closed-form spectral zeta values, and c = dim(SU(4))/N_PS = 15/4. The result is 137.035999081, matching the PDG value 137.035999084(21) at 0.15σ — 10.6 significant figures with zero fitted parameters. The one-loop and two-loop coefficients (4/15 and 15/4) are exact reciprocals. Every integer in the formula is a framework constant: 12 = N_c×N_PS, 11 = b₀, 16 = N_PS², 21 = N_c×b_QCD, 15 = dim(SU(4)). This result is derived within the Vacuum Mechanics framework (DOI: 10.5281/zenodo.19240140).