What question did this study set out to answer?

This analysis aims to extend the understanding of FOX scalar-gauge soliton behavior in the exterior domain.

March 27, 20260 citationsOpen Access

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

Key Points

This analysis aims to extend the understanding of FOX scalar-gauge soliton behavior in the exterior domain.
Extension of previous research from Branch C.
Identification of unique fixed points and matter vacuum states.
Use of oscillatory cancellations validated by Fast Fourier Transform (FFT) methods.
Topological reformulation of the neutron condition in terms of K(s*).
Established K(∞) = 0 as a unique fixed point of the soliton.
Demonstrated φ(∞) = φm, representing the matter vacuum.
Showed ΔEext approaches zero through oscillatory behavior.
Validated existence and uniqueness of K₂⁽ⁿ⁾ in the model.

Abstract

Branch D analyses the FOX scalar-gauge soliton in the exterior domain r*, ∞), extending Branch C. Five results are established: K (∞) = 0 (unique fixed point) ; φ (∞) = φm (matter vacuum) ; ΔEext → 0 by oscillatory cancellation (FFT-verified) ; K (s*) = 1 ⟺ NCS[0, s* = 0 (topological reformulation of the neutron condition) ; existence and uniqueness of K₂⁽ⁿ⁾. Three negative results close important directions. This note supplements Branch C (doi. org/10. 5281/zenodo. 19203591). (Version 2: NCS0, ∞=−1 corrected from −2; toroidal parameter p=2 replaces writhe Wr=2; charge signature phrasing clarified. )

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