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/φ^28 + (49/32π²) · ln(φ^28/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.