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

==== Spin-½ is best understood as a manifestation of the spinor representation of rotations: states transform under SU(2), the double cover of SO(3). Under a physical 2π spatial rotation a spinor picks up a minus sign; only after 4π does it return to itself. Geometrically: ====

• In ordinary SO(3) (vectors) the fundamental closed rotation corresponds to 2π2\pi2π.
• In SU(2) (spinors) the fundamental closed element corresponds to a 4π4\pi4π rotation to return to identity.

How does reff=r/2r_{\rm eff}=r/2reff=r/2 fit this? Think of the wavefunction phase accumulated when you traverse the spherical boundary once. If the physical boundary that carries the phase is shorter (half the geometric circumference), then one geometric 2π2\pi2π rotation corresponds to two traversals of the elementary boundary loop. In effect, the minimal closed phase loop of the system is half the geometric circumference — so the system needs two geometric rotations to return the elementary-phase loop to itself. This maps naturally to the SU(2) picture: the elementary loop (phase domain) has twice the periodicity of the geometric loop → spinor-like behavior.

So the group picture + half-radius boundary-layer yields the same qualitative conclusion: a fundamental domain that is half the geometric circle produces double-cover behavior and half-integer quantum labels.