A Unied Geometric Scale of Fundamental Interactions through the Fine-Structure Constant in the Ontopole Field Model

Abstract The fine-structure constant α ≈ 1/137. 036 has resisted derivation from first principles for over a century. In the Ontopole field model, elementary particles are identified with topological vortex defects in a 12-dimensional field. We show that α arises from the dynamics of an unclosed vortex and can be expressed through a single geometric parameter — the effective number of vortex windings n — via the formula α⁻¹ (n) = 2πn − π/16 − 1. At n = 22 this yields α⁻¹ = 137. 034, deviating from the CODATA 2018 value by 0. 0017% without any fitted parameter. The phase deficit δφ = π/16, originating from four successive binary quantisations of the full phase 2π, is identified as the geometric origin of electric charge: a charged particle is a vortex whose phase permanently fails to close by π/16. All four fundamental interactions are unified on a single winding-number axis: strong (n ≈ 0. 350, αs ≈ 1), weak (n ≈ 4. 90, αw ≈ 1/30), electromagnetic (n = 22, α ≈ 1/137), and gravitational (n → ∞, αG ∼ 10⁻³⁹). Three structural results follow. First, the strong interaction lies exactly one quantum step 1/ (2π) above the field existence boundary nₘin = (1 + π/16) /2π ≈ 0. 190 — it is the first interaction the field can sustain. Second, the running of α with energy is discrete: each winding transition n → n−1 contributes exactly 2π to α⁻¹. Third, the prime number p = 137 occupies the same spectral orbit n = 22 as the electromagnetic vacuum coupling. The formula is validated against six independent physical constants (Rydberg constant, electron anomalous magnetic moment, von Klitzing resistance, hydrogen ground-state energy, Bohr radius, proton-to-electron mass ratio) ; all deviations are systematically 0. 001%–0. 003%, consistent with a single next-order lattice correction of order ε³ = 1/1728. Keywords fine-structure constant, fundamental interactions, Ontopole field model, vortex winding number, phase deficit, running coupling constant, geometric unification, field existence boundary, prime number spectrum, 12-dimensional field theory, discrete running, strong interaction, hierarchy problem Additional Zenodo metadata: Resource type: Preprint License: Creative Commons Attribution 4. 0 MSC 2020: 81V10, 11N05, 83E15 PACS: 06. 20. Jr, 12. 20. Ds, 11. 10. Kk Related publications: DOI 10. 5281/zenodo. 19411388 (K̃ operator paper)