The Fine Structure Constant from the Hopf Fibration Tower

Title: The Fine Structure Constant from the Hopf Fibration Tower

Author: Alexander Novickis (alex.novickis@gmail.com)

We show that the fine structure constant satisfies $1/\alpha = \pi + \pi^2 + 4\pi^3 = 137.0363$ to 2.2 parts per million. The three terms correspond to the three Hopf fibrations ($S^1$, $S^3$, $S^7$). All three levels are derived: Levels 1 and 2 from exact sphere volume formulas ($\text{Vol}(S^1)/2 = \pi$ and $\text{Vol}(S^3)/2 = \pi^2$), and Level 3 from one-loop KK on $S^7$ via the Spin(7) branching $\mathbf{7} = (\mathbf{4}, \mathbf{1}) \oplus (\mathbf{1}, \mathbf{3})$ under SO(4)$\times$SO(3), where the coefficient 4 = dim($M_4$) counts spacetime directions that couple through the octonionic fiber. O'Neill's theorem ($A = 0$ for totally geodesic fibers) ensures exact factorization of the one-loop determinant on round $S^7$ (§III.4). The remaining refinement is verifying the factorization extends to deformed (non-round) $S^7$.

Keywords: fine structure constant, Hopf fibration, division algebras, pi, fundamental constants

DOI: 10.5281/zenodo.19432061

Series: Paper LXX in the Hopf Soliton Programme