Asymptotic 1/r² Shell Scaling in the Concentric Shell Theory from the Radial Field Equation

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.