In the foundational proposal of the Concentric Shell Theory (CST), elementary particles were modeled as extended spatial structures generated by a nonlinear complex scalar field. Previous numerical explorations relied on a heuristic, neutralized damped-oscillatory profile to test the viability of emergent long-range forces. In this paper, the radial Euler-Lagrange equation derived from the original CST Lagrangian is analyzed. It is demonstrated analytically that as the field amplitude decays, the effective linear mass term vanishes, reducing the far-field equation to the free spherical wave equation. Consequently, the field admits an asymptotic oscillatory solution whose effective shell density exhibits an asymptotic 1/r² envelope. Numerical integration of the complete radial ODE confirms that a localized core transitions into an extended oscillatory shell structure with a 1/r² envelope. While this does not yet constitute a full derivation of the two-body gravitational dynamics, it provides analytical and numerical support for the geometric scaling assumption utilized in previous CST studies.
Ernesto De Luca (2026) studied this question.