What question did this study set out to answer?

The aim is to derive the fine-structure constant from the spectral zeta function of the Laplacian on S³.

March 29, 20260 citationsOpen Access

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

LMLuqman Omar MahmoodUniversity of Mosul

Key Points

The aim is to derive the fine-structure constant from the spectral zeta function of the Laplacian on S³.
Derived the formula α⁻¹ = N + ζ_u(2)/c + ζ_u(3)·c/N² using spectral zeta function.
Utilized exact values for ζ_u(2) and ζ_u(3), integrating framework constants.
Calculated the fine-structure constant and compared with existing values, confirming accuracy.
Obtained α⁻¹ = 137.035999081, closely matching the PDG value 137.035999084(21).
Achieved accuracy of 10.6 significant figures with zero fitted parameters.
Identified all integers in the formula as framework constants, linking to theoretical physics.

Abstract

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).

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