Add‑On A.4 extends the equilibrium‑mode interpretation of the electron into the multi‑electron domain, demonstrating that shell structure, orbital ordering, shielding, periodicity, and multi‑electron transition anomalies arise directly from the internal geometry of coupled equilibrium modes. Building on the global internal‑mode energy law of Add‑On A.1, the angular geometry of Add‑On A.2, and the transition‑symmetry framework of Add‑On A.3, this work shows that multi‑electron behavior does not require new ontological assumptions or imposed quantum numbers. Instead, the structural features that govern single‑electron equilibrium modes—polarity inversion, two‑axis oscillation, standing‑wave angular nodes, phase‑twist symmetry, and stability‑preserving coupling—extend naturally to systems of multiple electrons sharing a nuclear anchoring field. The paper develops a complete structural and mathematical framework for multi‑electron atoms, including polarity allocation, angular‑mode compatibility, radial‑mode spacing, mode‑mode interference, and stability‑preserving coupling. These mechanisms generate the observed periodic table row lengths, shielding constants, orbital ordering patterns, and weakly allowed transition behavior. The model yields quantitative predictions for deviations in shielding constants, orbital‑ordering shifts under extreme fields, weak‑line intensity scaling, periodic anomalies, multi‑electron transition suppression, exotic‑configuration stability thresholds, and high‑field distortion behavior. A full falsifiability structure is provided, offering experimentally testable criteria across spectroscopy, scattering, Rydberg‑state measurements, and high‑field atomic physics. Add‑On A.4 completes the structural foundation for multi‑electron atomic behavior and establishes the basis for Add‑On A.5, which will extend the framework into ionization structure and fine‑structure splitting.
James Reeves (Tue,) studied this question.