Derivation of the Fine Structure Constant from a Scalar Field Theory on a φ-Metric 7-Torus

We present a candidate derivation of three fundamental constants — the electromagnetic finestructure constant α, the strong coupling constant αs, and the proton-to-electron mass ratio mp/me—from a nonlinear scalar field equation on an 8-dimensional compact manifold, with no free parameters. The field is octonionic-valued, and the manifold is T7 ×S1 equipped with a metric whosecompact radii scale as successive inverse powers of the golden ratio φ = (1 + √5) /2. The equation□φΨ + (|Ψ|2 − φ) Ψ = 0 admits topological soliton solutions whose quantum fluctuations inducea non-perturbative vacuum displacement κ0. We identify specific geometric mechanisms — G2structure calibrations on the octonionic tangent bundle and Kaluza–Klein charge summation —that motivate the tree-level and one-loop contributions. The resulting formula, 1/α = (2π) ¹0/φ²8 + (49/32π²) · ln (φ²8/ln2) contains no free parameters and yields 1/α = 137. 088577, within 0. 038% of the CODATA 2022value 137. 035999177 (21). The same Fano triple (1, 2, 4) gives αs = φ7/ (2π) 3 = 0. 11705, within1. 05σ of the PDG value. A numerical study of the three-soliton bound state on the Fano tripleyields a classical energy ratio Rcl = 1237. 52; multiplied by the geometric factor φ4/5 = 1. 3708 fromthe compact directions of the 7-torus not captured by the radial ODE, this gives Rcorr = 1696. 42. The perturbation spectrum of the bound state yields a Sakharov confinement coefficient c = 0. 6762, producing the final prediction mp/me = Rcorr ecαs = 1836. 15, matching experiment to the precisionof the computation.