UV--IR Consistency and Core--Shell Structure of Deformation Modes in an Electron Core

This work presents a detailed numerical and geometric analysis of the deformation sector of a finite-energy electron core. Building on previous spectral and scaling studies, the spatial organization of the low-lying deformation modes is investigated using a unified set of diagnostics, including radial profiles, cumulative measures, subspace comparisons, and Fourier-space analysis. The results show that the deformation modes form a well-defined outer shell rather than being localized at the core center. A stable dimensionless ratio between the shell scale and the intrinsic core scale is identified, indicating an intrinsic geometric relation between the two structures. Domain-size analysis demonstrates that the deformation sector persists as a finite-dimensional branch while its spatial embedding shifts outward, corresponding to an infrared continuation rather than the emergence of a distinct shell. Subspace-based diagnostics confirm that the deformation sector remains stable across domains, providing a representation-independent validation of its structure. Independent Fourier-space analysis further supports the existence of a consistent intrinsic length scale, in agreement with real-space diagnostics. Taken together, these results establish a coherent geometric picture in which a compact electron core and a larger deformation shell are linked through stable, resolution-independent relations, providing a concrete realization of UV–IR consistency in the structure of the electron core.