Universal Shell Structures Across Physical Domains Radial Eigenmode Hierarchies in Physical Systems. Universal Shell Structure Catalog

This paper advances a universal shell framework in which shell-like organization is treated not as a local quantum accident but as a general consequence of three-dimensional rotational symmetry. We argue that whenever a system combines approximate spherical symmetry with a confining radial operator, a stability boundary, or an effective stratifying constraint, its admissible modes separate into radial and angular parts governed by SO(3), producing hierarchies indexed by radial order and angular degree. On this basis, shell structure is shown to recur across atomic orbitals, nuclear shells, planetary and stellar layering, normal-mode families, turbulence in spectral space, and magnetospheric plasma confinement. We further argue that three-dimensional space occupies a privileged stability regime: two dimensions under-resolve angular closure, while four and higher dimensions overpopulate angular sectors and generate spectral crowding. Only in three dimensions does the linear 2l+1 multiplicity law provide the balanced regime required for persistent hierarchical shells. Finally, the paper situates this result within a broader symmetryclosure ladder - 2, 3, 8, and 24 - interpreted as thresholds of binary, rotational, exceptional algebraic, and lattice-modular organization. The central claim is therefore structural: shell hierarchies are a recurrent law of organization in a three-dimensional rotational universe. Keywords: SO(3), shell hierarchy, spherical harmonics, degeneracy, radial eigenmodes, stability optimum, closure projection, symmetry ladder