FOX Theory (Branch D): The Exterior Soliton: Asymptotic Structure, Topological Classification, and the Neutron Condition (Note to Branch C)

Branch D analyses the FOX scalar-gauge soliton in the exterior domain [r*, ∞), extending Branch C beyond the stasis sphere. Five Column A results are established. (i) K(∞) = 0 is the unique fixed point of the gauge equation of motion; K = ±1 are not equilibria. (ii) φ(∞) = φ_m = 0.2, the matter vacuum of Branch A, not φ = 1. (iii) ΔE_ext → 0 by oscillatory cancellation: φ_n and φ_p oscillate at identical asymptotic frequencies (k_n/k_p = 1.000000, FFT-verified), confirming ΔE_int = 1.015 MeV from Branch C. (iv) The exact algebraic identity K(s*) = 1 ⟺ N_CS[0, s*] = 0, where N_CS is the partial Chern-Simons number. The neutron is the configuration whose interior core is topologically trivial. (v) Existence and uniqueness of K₂⁽ⁿ⁾ such that K(s*) = 1, established by the intermediate value theorem and strict monotonicity.Three Column A negative results close important directions: the formula Q_em = 1 − K(∞)² is invalid (identical for proton and neutron); the conjecture K(r*)² = 1 − 1/(det(T) × π) is analytically refuted;H6 is not the variational minimum of E_total. Version 3 adds: the degenerate critical point ∂N_CS/∂K|K=1 = 0 (LICQ fails, ruling out a Lagrange derivation of H6); the H5/H6 duality (two structurally independent stationarity mechanisms on N_CS); and Postulate P6 formalised as a working hypothesis.