The Möbius and Klein Transforms: Spectral Analysis on Non-Orientable Manifolds

Construction of Fourier-analogous unitary transform pairs on the Möbius strip and the Klein bottle. The guiding principle: the topological twist is carried not by a phase mask glued onto the basis, but by a constraint on the function space – the orthogonal basis follows from it. The construction works on the double cover (a circle of circumference 4pi, respectively the covering torus), where half-integer-frequency exponentials form the natural orthonormal system; the function space splits into two sectors under the deck transformation (invariant and twisted), and the pairing rule of the Klein transform (cosine terms with integer, sine terms with half-integer theta-frequency) spans exactly the invariant sector. The paper contains full derivations, proofs of unitarity and orthogonality, the extension to tensor fields, a fast (FFT-based) implementation, numerical verification (Gram matrix, reconstruction, counter-tests) and applications ranging from signal processing to topological Casimir energy. A new section of this edition gives the sector rule for the slices of the derivative tensor: the curl switches sector while div and dev stay in sector, hence the stream function of the Helmholtz decomposition is a twisted quantity – with a closed spectral solution and machine-precision verification. A further new section of this edition is the quantum-mechanical application: the basis is the eigensystem of the Klein-bottle Hamiltonian (the twisted sector has a zero-point gap E=1/8), the two sectors are the two inequivalent quantizations of the space, the split-step MKT solver is unitary and sector-preserving (measured Strang order 2.004), and a selection rule follows for the axial angular momentum: <L> = 0 identically in any sector-pure state – 20 of 20 tests at machine precision. The new Section 9.11 records the time-axis reading: half-wave symmetry is the P- sector in time (push-pull amplifier), the fermionic Matsubara frequencies are the rescaled half-integer lattice (the tau-circle can enter the Casimir machinery as a third factor), the I/Q pair of conjugate-periodic signals realizes the Klein pairing rule, and Misner space marks what the time reading does not mean. The section closes with three numerical demos (signed envelope/BPSK, Mathieu divide-by-two, glide selection rule), the Z4 tower of PSK/QAM (quarter-integer carrier lattice) and an eight-row sector dictionary showing the common cover/quotient skeleton of existing radio techniques, from the squaring loop through DDMA MIMO radar to the Alamouti code and the Stokes formalism.