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 mₖ, ₔ 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 φₘ = 1/5 imposed by T (2, 3) differs from the energetically optimal value φₘ ≈ 0. 218, producing a 22% effect on ΔE. The proton gauge field at the stasis sphere is κₚ = Kₚroton (r*) = 0. 94511, and Qbare × 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) = 23φₘ²/8. 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 Lₛcale) ; spatial observables require Branch I (confinement), currently open. The mass splitting ΔE = 1. 015 MeV is independent of this scale gap.
Adrien le Hardy de Beaulieu (Sat,) studied this question.