Which Loops Can Carry a Burgers Vector? Selection Rules on Non-Orientable Geometry

Abstract

The strength of a dislocation is its Burgers vector: the circulation of the distortion field along a loop encircling the defect line. This article pursues a single question: what happens to this circulation when the carrier geometry is non-orientable? The answer reduces to one mechanism — the half-integer frequency lattice has no zero mode — and this mechanism yields the same selection rule on three successive rungs: at a point resolution the circulation is halved; along a curve with non-orientable normal bundle the longitudinal, Mobius-direction Burgers component cancels over two turns, confining the Burgers vector to span{t, e1}; on the surface rung the divisor term collapses under a triple selection to a single term (pi·A). The arenas are fixed by the Whitney formula (w1(N) = w1(M)|_Sigma + w1(Sigma)), among them the Euclidean arena: a non-orientable surface in R^4 with vanishing normal Euler number. The rules have a direct superconducting reading: a vortex is a phase dislocation, and the same gluing condition yields the half-quantum fluxoid physics measured on Mobius-type geometries. Every claim is verified by a deterministic, counter-checked script (16/16). The article extracts and assembles into a single whole the Burgers-selection story from the framework's published records.