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

==== - Mathematically: if the microscopic boundary on which the wavefunction is defined has half the geometric circumference (or equivalently the physical single‑valuedness condition is that the wave is single only after a 4π4\pi4π geometric rotation), then the allowed eigenstates of L^z\hat L_zL^z on the full circle are half‑integers: Lz=(k+12)ℏL_z=(k+\tfrac12)\hbarLz=(k+21)ℏ. This follows directly from imposing antiperiodic boundary conditions Ψ(φ+2π)=−Ψ(φ)\Psi(\varphi+2\pi)=-\Psi(\varphi)Ψ(φ+2π)=−Ψ(φ). ====
* Physically: the step “reff=r/2r_{\rm eff}=r/2reff=r/2 ⇒ antiperiodic BC” must be justified by a microphysical boundary mechanism (why is the elementary phase domain half the geometric circumference?). If QAT provides that geometric/physical reason (boundary-layer, concave/convex charge partition, photon‑electron dipole geometry), then the above is a direct, rigorous one‑dimensional quantum demonstration of half‑integer quantization.
* A caution: this is not a replacement for the full QFT explanation of spin (which arises from representation theory of the Lorentz group and field‑theory construction). What we have derived is a clear quantum‑mechanical mechanism that produces the same discrete eigenvalue pattern (half‑integers) from a geometric boundary condition — exactly the behavior you want to attribute geometrically in QAT.

If you like, I’ll now:
• produce a short, slide‑ready block of LaTeX you can paste into your Core Equation document, or
• write the boundary‑action derivation (Lagrangian on the 2‑D sphere boundary leading to identical antiperiodic condition), which is the next step to make this fully rigorous.

Which would you prefer next?