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

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 n_min = (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)