Add‑On A.5 extends the equilibrium‑mode electron framework into the domain of ionization thresholds and fine‑structure splitting. Building directly on the internal‑mode energy law (A.1), angular‑mode geometry (A.2), transition‑symmetry rules (A.3), and multi‑electron coupling structure (A.4), this add‑on derives threshold behavior and internal splitting as structural consequences of the electron’s mode architecture. Ionization is formulated as a stability‑loss transition in which radial‑mode expansion drives polarity‑boundary collapse and decoupling of the equilibrium configuration. Fine‑structure splitting emerges from angular‑mode asymmetry, curvature differences, phase‑twist compatibility, and coupling imbalance within the constrained angular operator—without invoking spin–orbit coupling, relativistic perturbations, or additional quantum numbers. A.5 provides the full mathematical and structural formulation of these mechanisms, including the radial‑expansion law, polarity‑boundary collapse condition, angular‑splitting operator, field‑perturbation operator, and multi‑electron shielding geometry. The resulting framework yields quantitative predictions for ionization energies, angular‑dependent threshold shifts, fine‑structure splitting ratios, high‑field distortion patterns, Rydberg‑state deviations, and multi‑electron threshold modifications. A complete falsifiability structure is included, spanning spectroscopy, Rydberg scaling, Stark/Zeeman distortions, and extreme‑field ionization. A.5 completes the atomic‑level stability layer of the Version 1.0 equilibrium‑mode architecture and establishes the structural foundation required for subsequent extensions into scattering (A.6), bonding (A.7), plasma behavior (A.8), and astrophysical spectra (A.9).
James Reeves (Tue,) studied this question.