Vortex and Fluxoid Selection Rules on Non-Orientable Superconductors. The loop dichotomy, the closure prohibition, and the pi-shifted Little–Parks ladder: which loops can carry a vortex — with an exact channel-discriminator identity and machine verification

Abstract

Half-quantum fluxoid physics is measured reality: pi-shifted Little–Parks oscillations in beta-Bi2Pd rings, half-quantum features in Sr2RuO4 microrings, and a Ginzburg–Landau theory of the superconducting Möbius strip that found novel states near half-odd flux. What the literature does not provide is the unified selection rulebook: which loops of a non-orientable sample can carry which winding, which vortex-line configurations are forbidden in three dimensions, and how the channel of the condensate is read off a single measurement. This article delivers that rulebook as the superconducting transcription of the framework's selection theorems. Three results. (T1, loop dichotomy) On a non-orientable film the winding of a sector-pure order parameter is integer on every orientation-preserving loop and half-integer precisely on orientation-reversing loops in the twisted channel; a corollary is a no-go: geometry alone creates no local half-vortex core — the half-quantum is a holonomy of the non-contractible reversing loops. (T2, closure prohibition) In a three-dimensional non-orientable sample, circulation carried in the reflected frame cancels over two turns: a single-signed vortex line cannot close around the orientation-reversing handle — it must pair with its mirror or carry only the Z2 residue. (T3, the pi-shifted ladder) The Little–Parks energy ladders of the two channels obey the exact identity E-(phi) = E+(phi + 1/2): the twisted channel's Tc(Phi) parabolas sit at half-integer flux — the phase of the Little–Parks oscillation is a binary readout of the condensate's deck parity. All claims are verified deterministically (15/15), including a two-dimensional magnetic lattice whose ground-state minima shift from integer (cylinder) to half-integer flux (twisted Möbius gluing). No material predictions are made; the contribution is the rulebook and its checkability.