Recoverability of Atomic Quantum Structure from Emission Spectra

Atomic emission spectra encode physical structure through observable photon energies, yet
the inverse mapping from spectra to atomic configuration is inherently information-entropic.
In this work, we empirically assess which aspects of atomic quantum structure are recover-
able from spectral observables alone and which are irretrievably compressed under radiative
projection by quantifying information retention and loss. Using the NIST Atomic Spectra
Database, we first demonstrate near-perfect statistical recovery of the Planck–Einstein
relation, establishing a baseline for invertible physical structure. We then show that global
reconstruction of Moseley–Rydberg scaling fails decisively for neutral multi-electron atoms, re-
flecting genuine degeneracy rather than model inadequacy. Constrained neural network models
are subsequently used as diagnostic probes of information content, revealing a clear hierarchy
of recoverability: initial-state quantum numbers exhibit statistically significant recoverabil-
ity in restricted atomic regimes, while final-state quantum numbers are generally unstable
and non-invertible, collapsing toward chance under mixed-element conditions. These results
demonstrate that radiative emission preserves only partial and regime-dependent information
about atomic structure, with systematic loss increasing under electronic screening and config-
uration mixing.