Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== - This derivation is a clean, widely used method: bulk action + surface action → matching conditions + conservation/continuity. It shows the concept you want: electron spheres as active membranes with JμAμJ^\mu A_\muJμAμ coupling that convert bulk EM quanta into changes of the boundary occupation (QAT picture of time as photon-by-photon updates). ===
* The antiperiodic sign is a direct geometrical consequence of spin-½ rotation by 2π2\pi2π (and can be shifted or combined with gauge holonomy).
* Things that call for extra care / independent checking: - Global spin-structure/topology on S2S^2S2 (mathematically subtle if you insist on single-valued global spinors). - Incorporating the Planck constant h/2πh/2\pih/2π explicitly into a semiclassical boundary quantization: quantum normalization of the surface fields and the relation between discrete photon absorption events and continuous boundary currents needs a quantum boundary treatment (e.g., second quantization of ψ\psiψ on the shell). - If you want a rigorous microscopic model of the surface in terms of atomic orbitals, you can attempt to derive the boundary action from integrating out degrees of freedom of the atom (effective action), which will produce specific forms and corrections. - If you want to claim that antiperiodic phases are set by geometry such that α (1/137) arises exactly, that’s a much stronger mathematical program: one must show how e2e^2e2, h/2πh/2\pih/2π, ccc, ε0\varepsilon_0ε0 all fall out from geometry or boundary quantization. The present derivation shows the mechanism (where charge/current live and how they exchange with bulk), not the full derivation of α from geometry.