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 (∞) = φₘ = 0. 2, the matter vacuum of Branch A, not φ = 1. (iii) ΔEₑxt → 0 by oscillatory cancellation: φₙ and φₚ oscillate at identical asymptotic frequencies (kₙ/kₚ = 1. 000000, FFT-verified), confirming ΔEᵢnt = 1. 015 MeV from Branch C. (iv) The exact algebraic identity K (s*) = 1 ⟺ NCS[0, s* = 0, where NCS 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 Qₑm = 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 Eₜotal. Version 3 adds: the degenerate critical point ∂NCS/∂K|₊=₁ = 0 (LICQ fails, ruling out a Lagrange derivation of H6) ; the H5/H6 duality (two structurally independent stationarity mechanisms on NCS) ; and Postulate P6 formalised as a working hypothesis.