One Twist, Two Views — the connection between the tensor-invariant system and the Möbius–Klein transform

Two independently built frameworks – the unified tensor-invariant system of vector analysis (jet/jad/dev operators, dimensional ladder) and the spectral theory of non-orientable surfaces (Möbius and Klein transforms, sector decomposition) – rest on a single common actor: orientation. In the tensor system orientation hides inside the formulas (the sign flip of the epsilon sorter under reflection: the pseudotensor birthmark); in the Möbius-Klein framework orientation is built into the space itself (the space carries a mirror that cannot be removed). In steps a student can follow, the paper shows that the two frameworks are two views of the same twist – a local-differential and a global-spectral view – writes down the precise dictionary between them, and derives the machine-verified result standing at their meeting point: on the Klein bottle the curl switches sector while div and dev stay in sector, hence the stream function of the Helmholtz decomposition lives in the twisted sector. It discusses the physical meaning (mirror behaviour of the magnetic field, sourcelessness and monopoles, fermionic sectors) and the engineering use (type-correct spectral computation, closed-form Helmholtz projection). The cosmological chapter states the conjecture of an E-B sign signature in a non-orientable universe, discusses topological Casimir energy and the program of twisted (Möbius-type) Kaluza-Klein compactification; the closing part explores three senses of passage between dimensions and the question of time, from the Lorentz boost to the Matsubara temperature. The quantum-mechanical section shows that the two sectors are the two inequivalent quantizations of the Klein bottle (the twisted sector having a zero-point gap E=1/8), and that a selection rule follows for the axial angular momentum: <L> = 0 identically in any sector-pure state, so L is not an observable. One row of the dictionary is also sharpened: rot and jad are the same slice in different readings – the tensor form of jad contains no epsilon and is therefore single-valued on the Klein bottle, while rot as the dual reading switches sector; on a non-orientable space it is thus better to compute in jad. From this a decidable criterion follows for the whole ladder: a rung carries over unchanged if and only if its selector contains an even number of epsilons. Finally, a consequence for materials physics on the third-order rung: the Burgers vector changes sign on the orientation-reversing loop of the bottle, so it is not a global constant but a sector-dependent quantity.