Universal Spectral Scaling and Hierarchical Organization in Atomic Spectra

A model-independent analysis of more than a century of empirical atomic spectral data across the periodic table reveals that atomic spectra exhibit a self-similar, scale-invariant multifractal-like organization, consistent with a hierarchical network of effectively coupled oscillatory modes. Across elements, core spectral observables, including line positions, spectral edges, bandwidths, fine-structure splittings, and spectral families, collapse onto deterministic global scaling relations governed by a single empirical spectral scale, the maximum observed frequency \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\). When additional atomic observables are indexed by this same spectral scale, including isotopic state structure and reported nuclear decay sequences and rates, they likewise collapse onto the same global relations. These observables follow simple, near-perfect power-law relations to \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) through systematic mode coupling, splitting, and hierarchical branching, without invoking particles, discrete energy transitions, or quantum-number formalisms as primitive organizing concepts. The spectral phylogeny (hierarchical spectral lineage) reconstructed from mode trajectories and splitting events shows that atomic spectra form a generational cascade of dynamically selected configurations within a single hierarchical spectral system, rather than a collection of independent objects. Atomic identity spectrally emerges from the discrete fractional spacing of stable \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) attractors, with spacing that is large at low \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\) and converges toward a universal minimum of approximately 2% of \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\), below which distinct stable spectral configurations are not observed. This empirically derived organization reproduces and refines major chemical groupings, indicating that regularities commonly attributed to shell structure and orbital occupation can arise directly from invariant spectral organization. Analysis of hydrogen demonstrates that the same \(\:{\nu\:}_{\text{m}\text{a}\text{x}}\)-governed hierarchy observed across the periodic table is already present internally within its spectrum, indicating a nested and recursively organized oscillatory structure. These results show that atomic spectra constitute a low-dimensional, self-similar oscillatory hierarchy governed by an empirically accessible spectral invariant, providing a model-independent basis for spectral regularities and elemental classification.