Self-Referential Scattering and the Birth of Fermions: Riccati Square Roots, Spinor Double Cover, and a $\mathbb{Z

Standard quantum field theory explains the relation between spin and statistics through the spin--statistics theorem, derived from Lorentz covariance and microcausality on a continuous spacetime background. In topological approaches, the antisymmetry of fermionic wavefunctions can also be understood via the nontrivial topology of configuration spaces and associated line bundles, as in the Finkelstein--Rubinstein construction for solitons in nonlinear field theories. However, these frameworks typ