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

=== 1. The physical boundary is the 2-D spherical shell Sr2S^2_{r}Sr2 that surrounds the nucleus (the “electron sphere”). QAT treats the boundary as an active, dynamical 2-D manifold where phase/charge lives and photon–electron exchanges happen. ===
# The elementary microscopic phase domain is a boundary field living on that spherical surface. We model it as a spinor field Ψ(t,Ω)\Psi(t,\Omega)Ψ(t,Ω) (a section of the spin bundle over the sphere) because spinor fields naturally encode the double-cover behavior under rotations.
# Geometry assumption (QAT): the effective microscopic loop radius is half the geometric radius,

reff  =  r2,r_{\rm eff} \;=\; \tfrac{r}{2},reff=2r,
and the elementary phase circulation is associated with that half-loop. This will be the geometric origin of antiperiodicity.
# Keep the quantum of action visible as Squantum=h/2πS_{\rm quantum}=h/2\piSquantum=h/2π.