Recovering 4f Electronic Shell Signatures from Collective Spectral Manifolds of Lanthanide Ions: A New Perspective for Rare-Earth Photonics

The electronic spectra of trivalent lanthanide ions constitute one of the most extensively characterised manifestations of partially filled 4f shells. Conventional rare-earth spectroscopy provides a detailed microscopic description based on assigned multiplets, angular-momentum structure, crystal-field effects and transition probabilities. Here, we examine a complementary question: whether information associated with the underlying 4f electronic architecture remains detectable after the spectroscopic labels normally used to identify it are deliberately removed. Using the multiplet-centre energies of Ln3+ ions in LaF3, each ion was represented as an energy-only spectral manifold, Mn={Ei}, obtained after removing term identity, angular-momentum labels and transition assignments. The resulting manifolds were analysed through a controlled information-reduction sequence involving spectral occupancy and four complementary descriptors: spectral entropy S, compactness C, mean manifold energy μE and spectral dispersion σE. The results show that the reduced manifolds do not collapse into featureless collections of energies. Instead, they retain structured organisation across the lanthanide series, with distinctive behaviour associated with the shell-edge configurations Ce3+ (4f1) and Yb3+ (4f13), and with the half-filled-shell configuration Gd3+ (4f7). A blind reconstruction test provides the strongest internal validation: when Gd3+ is excluded from the reference evolution and subsequently reintroduced, it emerges as the dominant positive deviation of the mean manifold-energy coordinate. This demonstrates that the 4f7 signature is not encoded only in explicit term labels or selected multiplets but leaves a measurable fingerprint in the collective energy distribution. The comparison of normalised descriptor responses further indicates that electronic shell information is distributed across state occupation, compactness, energetic displacement and spectral spreading. These results suggest that lanthanide spectra can be analysed not only as collections of assigned transitions, but also as collective spectral manifolds whose global organisation preserves physically meaningful information about shell filling. This approach does not replace conventional rare-earth spectroscopy; rather, it provides a complementary framework for testing which aspects of electronic structure remain accessible after microscopic spectroscopic identity has been removed.