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

==== Mainstream fact: spin-½ is described by spinors (SU(2) double cover). Physically experiments show electrons have half-integer angular momentum and fermionic statistics. Wikipedia<ref>{{cite web|title=Wikipedia|url=https://en.wikipedia.org/wiki/Spin_%28physics%29|publisher=en.wikipedia.org|access-date=2025-12-16}}</ref> ====

QAT picture (geometric intuition):
* Picture the electron sphere (the 2-D probability shell around the nucleus) as a boundary where an internal phase degree of freedom lives — a phase that registers how the EM tangential field has been absorbed and re-emitted by that shell.
* If that boundary carries an antiperiodic condition (phase flips sign after a 360° rotation of the local frame), then a 360° rotation is not the identity: you need 720°. That directly reproduces the spinor behaviour (Dirac belt trick / two-turn rule). This antiperiodicity is the same structure that in QFT produces spin-½ fields. Harvard Scholar<ref>{{cite web|title=Harvard Scholar|url=https://scholar.harvard.edu/files/noahmiller/files/dirac_belt_trick_faq.pdf|publisher=Harvard Scholar|access-date=2025-12-16}}</ref>PhilArchive<ref>{{cite web|title=PhilArchive|url=https://philarchive.org/archive/NIETTUv146|publisher=philarchive.org|access-date=2025-12-16}}</ref>
* Geometric origin in QAT: you suggested the “half-radius” (r_eff = r/2) and a central nucleus as a privileged center. In a boundary Lagrangian picture, a field restricted to that shell can naturally have antiperiodic boundary conditions (a Möbius-like phase twist), giving the half-spin outcome. The boundary becomes a spinor bundle rather than an ordinary vector bundle.

Takeaway: spin-½ can be described as a topological phase living on the 2-D shell: an antiperiodic boundary condition ↔ the observed ½-spin transformation law. This is conceptually consistent with mainstream spinor topology (SU(2)/double cover). Rochester School of Arts & Sciences<ref>{{cite web|title=Rochester School of Arts & Sciences|url=https://www.sas.rochester.edu/pas/assets/pdf/undergraduate/su-2s_double_covering_of_so-3.pdf|publisher=Rochester School of Arts & Sciences|access-date=2025-12-16}}</ref>PhilArchive<ref>{{cite web|title=PhilArchive|url=https://philarchive.org/archive/NIETTUv146|publisher=philarchive.org|access-date=2025-12-16}}</ref>