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

==== QAT hypothesis: the elementary phase domain for the microscopic boundary is a loop of radius reff=r/2r_{\rm eff}=r/2reff=r/2. Equivalently, the phase unit is accumulated along a path whose geometric angular advance corresponds to half the full geometric circle. ====

Mathematically implement this by introducing an elementary angular coordinate θ\thetaθ that parameterizes the elementary loop and runs 0≤θ<2π0\le\theta<2\pi0≤θ<2π. The geometric angle φ\varphiφ (full circle) is related to θ\thetaθ by

φ=2θ.\varphi = 2\theta.φ=2θ.
Thus advancing θ\thetaθ by 2π2\pi2π advances φ\varphiφ by 4π4\pi4π. In words: one elementary loop equals half the geometric circumference; two elementary loops equal one full geometric 2π2\pi2π rotation.

Because the physical microscopic wave is defined on the elementary loop, impose single‑valuedness on θ\thetaθ:

Ψ(θ+2π)=Ψ(θ).\Psi(\theta+2\pi)=\Psi(\theta).Ψ(θ+2π)=Ψ(θ).
Rewrite that condition in terms of φ\varphiφ: since θ=φ/2\theta=\varphi/2θ=φ/2,

Ψ ⁣(φ+4π2)=Ψ ⁣(φ2)⟹Ψ(φ+4π)=Ψ(φ).\Psi\!\big(\tfrac{\varphi+4\pi}{2}\big)=\Psi\!\big(\tfrac{\varphi}{2}\big)
\quad\Longrightarrow\quad
\Psi(\varphi+4\pi)=\Psi(\varphi).Ψ(2φ+4π)=Ψ(2φ)⟹Ψ(φ+4π)=Ψ(φ).
So the wavefunction is 4π4\pi4π-periodic in the geometric angle φ\varphiφ. That allows two possibilities for the behavior under a geometric 2π2\pi2π rotation:

Ψ(φ+2π)=± Ψ(φ).\Psi(\varphi+2\pi)=\pm\,\Psi(\varphi).Ψ(φ+2π)=±Ψ(φ).
The physically interesting case giving half‑integers is the antiperiodic choice

Ψ(φ+2π)=−Ψ(φ),\Psi(\varphi+2\pi)=-\Psi(\varphi),Ψ(φ+2π)=−Ψ(φ),
because this encodes the fact that one geometric 2π2\pi2π rotation equals two elementary loops (so the wave picks up a minus sign after one geometric 2π2\pi2π but returns after 4π4\pi4π) — exactly the spinor double‑cover behaviour.