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

==== What I’ve just given is a coherent heuristic (two complementary ways) but not a full proof. To make it rigorous would require one of two next technical steps (I’ll happily do either): ====
# Lagrangian/wave derivation (recommended path): - Write a microscopic field/wavefunction living on the spherical boundary (2-D manifold). Include a boundary action term with the boundary radius rrr and an explicit reff=r/2r_{\rm eff}=r/2reff=r/2 factor that modifies the kinetic/phase term. - Derive stationary-phase (path integral) condition or Schrödinger equation on the boundary and enforce single-valuedness. Show eigenvalues of angular momentum operator are half-integers because the fundamental domain of the boundary wavefunction is the half-circle. - This yields a proper operator derivation of L=sℏL = s\hbarL=sℏ with s∈Z+12s\in\mathbb{Z}+\tfrac12s∈Z+21. (This is the cleanest route; it’s a short calculation if we adopt a simple boundary action.)
# Detailed topological / spinor mapping: - Construct the boundary manifold and show its holonomy / spin structure is nontrivial (i.e., the boundary has spin structure with a sign change on 2π rotation). Demonstrate that the spin bundle over that boundary is a nontrivial double cover. Conclude allowed representations are spinors (half-integer). (This needs a bit of differential geometry language.)

Caveats you should state publicly (so people don’t accuse QAT of loose claims)
* Classical radius manipulation alone does not create spin in orthodox QFT — spin is fundamentally a representation property of the Lorentz/SU(2) group. The above argument is a geometric mechanism for how the boundary domain and phase accumulation could force spinor-type boundary conditions that yield half-integer eigenvalues. It must be backed by an explicit boundary action/wave derivation to be convincing.
* The semiclassical Bohr–Sommerfeld step is an approximation. The rigorous Lagrangian path (above) will convert these heuristics into operator statements.