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 + ζᵤ (2) /c + ζᵤ (3) ·c/N² where N = 137 (the Koide integer), ζᵤ (2) = π²/12 − 11/16 and ζᵤ (3) = −π²/16 + 21/32 are proven closed-form spectral zeta values, and c = dim (SU (4) ) /NPS = 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 = Nc×NPS, 11 = b₀, 16 = NPS², 21 = Nc×bQCD, 15 = dim (SU (4) ). This result is derived within the Vacuum Mechanics framework (DOI: 10. 5281/zenodo. 19240140).