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

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 is a selection rule on each of three rungs, sharpened by one layer per rung; the added layers come from one mechanism — the half-integer frequency lattice has no zero mode. At a point the blown-up plane is the Möbius strip and only the first harmonic reaches the divisor; the halving of the circulation is a projective convention. Along a closed curve in a three-manifold with non-orientable normal bundle the contribution of the reflected normal component cancels over two turns, confining a single-valued Burgers vector on a junction-free line to span,_1; on the surface rung, in the twisted-divisor arena, the divisor term collapses under a triple selection to a single term of magnitude A. The arenas follow from the Whitney formula (w_1(N)=w_1(M)|_+w_1()) and Massey's theorem: ordinary ^4 is an arena for the splitting-driven statements but not for the twisted divisor. The curve rung acts in ordinary crystals through the lattice frame: in Frank's Möbius crystal only the screw component of the Burgers vector survives a circuit, and the partials left behind total twice the edge component — a closed form of Sleeswyk's non-conservation. The rules have a superconducting reading: a vortex is a phase dislocation, and the same antiperiodicity underlies the half-quantum fluxoid states predicted for superconducting Möbius strips and, through an internal twist, measured in unconventional-superconductor rings. A deterministic, counter-checked script reproduces every quoted instance. The article assembles the Burgers-selection story from the framework's records and adds two theorems, on the Möbius crystal and on the fractional winding of a locked order parameter.