FOX Theory (Branch C): The Coupled Scalar-Gauge Soliton and the Neutron-Proton Mass Difference

We study the coupled scalar-gauge soliton of FOX Theory as a model of the nucleon. The coupled system consists of the FOX scalar field φ(r) and an SU(2) gauge field K(r), subject to two classes of boundary conditions at the stasis sphere r*: Neumann (K'(r*) = 0, proton) and Dirichlet (K(r*) = 1, neutron). Three principal results are established. (i) The BPS incompatibility theorem: no static BPS solution exists for the coupled system with f(φ) ≠ 1, confirming topological stabilization. (ii) The BVP is solved numerically, yielding K₂* = −1.7007, s* = 2.4649 (units mW,u r/ℏc), and r* = 0.885 fm (−1.67% from the experimental value). The coupling term f(φ)φ²K² carries 64.4% of the total energy. (iii) The mass splitting theorem: the two boundary conditions yield ΔE = 1.015 MeV, at 78.5% of the experimental neutron-proton mass difference Δm = 1.293 MeV, without any free parameter. The 21.5% gap is identified as a structural prediction: the topological constraint φ_m = 1/5 imposed by T(2,3) differs from the energetically optimal value φ_m ≈ 0.218, producing a 22% effect on ΔE. The proton gauge field at the stasis sphere is κ_p = K_proton(r*) = 0.94511, and Q_bare × det(T(2,3)) × π ≈ 1 at 0.62% (Column B).Three exact algebraic identities are derived: β = 200/π, R/r = 4, and V''(0)/(4πβ) = 23φ_m²/8. FOX Theory in its current form predicts the topological structure and mass splitting of the nucleon. The stasis radius r* = 4.43 fm exceeds the experimental charge radius by a factor 5.3 (Lacuna L_scale); spatial observables require Branch I (confinement), currently open. The mass splitting ΔE = 1.015 MeV is independent of this scale gap.