Symmetry Breaking in the Möbius Klein Transformation: Sector Mixing, Twisted Walls and Multiplet Splitting

The Möbius Klein Transformation (MKT) sorts the fields on a non-orientable space into the untwisted and the twisted sector of its orientable double cover. This article asks what each rule does when the symmetry holds only approximately. The data that make the sector split exact are discrete. A small breaking enters through a field that is not sector-pure, a nearly invariant operator on the cover, a detuned twist, curvature on a flat carrier, or a broken antilinear gluing. A protection that rests on commutation is broken only by the deck-odd part of a perturbation, and one that rests on anticommutation only by the deck-even part. A twisted factor thus keeps the curl spectrum symmetric. A crossing of an untwisted and a twisted level opens linearly, while a level away from crossings moves only at second order under the deck-odd part. Detuned twisted lattice sums are stationary in every twisted sector and have a linear cusp, from the lifted zero mode, in the untwisted one. On a fundamental domain of the Möbius strip or the Klein bottle the divergence theorem fails for an impure field by exactly twice the flux of the twisted admixture through the cut. The circulation theorem fails by twice the line integral of the genuine admixture, and the ε-dual exchanges the 2 laws. The helicity of an impure field is exactly twice the mixed helicity of its parts. On non-abelian covers the levels of a real operator follow the count of Wigner and Herring, and a breaking to a subgroup splits a multiplet by restriction and by the type of each piece. Under a conformal change of the flat Klein bottle the harmonic 1-forms stay and the parallel ones are lifted at second order. Deterministic scripts check every measured statement.