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

=== If you prefer the geometric / phase picture (useful to present alongside the field derivation): ===
* Let θ\thetaθ parametrize the elementary microscopic loop (the boundary elementary coordinate) with θ∈[0,2π)\theta\in[0,2\pi)θ∈[0,2π). The geometric azimuth φ\varphiφ on the physical sphere is related by φ=2θ\varphi = 2\thetaφ=2θ (one elementary loop advances the full geometry by half).
* A microscopic wave on the elementary loop obeys Ψ(θ+2π)=Ψ(θ)\Psi(\theta+2\pi)=\Psi(\theta)Ψ(θ+2π)=Ψ(θ). In φ\varphiφ coordinates this means Ψ(φ+4π)=Ψ(φ)\Psi(\varphi+4\pi)=\Psi(\varphi)Ψ(φ+4π)=Ψ(φ), so after a geometric 2π2\pi2π rotation the wave may change sign: Ψ(φ+2π)=±Ψ(φ)\Psi(\varphi+2\pi)=\pm\Psi(\varphi)Ψ(φ+2π)=±Ψ(φ). Antiperiodic choice −-− corresponds to the nontrivial spin structure.
* Quantization of angular momentum with the canonical action h/2πh/2\pih/2π then yields half-integer eigenvalues when the elementary phase loop is half the geometric circumference. (This is the Bohr–Sommerfeld style mapping you’ve used before, but now with boundary topology made explicit.)