Emergent Core--Shell Atomic Structure from a Universal Variational Principle

This work extends the atomic analysis developed in Emergent Core–Shell Atomic Structure from a Universal Variational Principle (Livolsi, 2026), where the hydrogen atom was derived as a unique coherent configuration of a closed quartic functional without external assumptions. Building on that result, the present study examines the full second variation of the universal Ψ–Γ functional around the emergent hydrogenic solution and extracts the discrete spectrum of admissible excitations selected by the global decisional principle Ω. All known spectral transitions of hydrogen arise directly from the Hessian eigenvalue structure, with no potentials, no fitting, and no model-dependent postulates. Crucially, the analysis reveals an additional coherent and stable excitation mode not corresponding to any catalogued hydrogen line. This constitutes a parameter-free prediction of a new hydrogen spectral line, emerging solely from the internal geometry of the universal quartic variational framework. The result demonstrates that spectroscopic structure is not an input but an unavoidable consequence of the universal principle, further reinforcing the generative completeness of the Ψ–Γ framework.