{
  "abstract": "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: = 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.",
  "abstract_shown": true,
  "affiliations": [
    "Independent Researcher"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "László Márk",
      "orcid": "0009-0006-5033-633X"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cdy9582qnvbc5.bib",
  "cited_by": 34,
  "doi": "10.5281/zenodo.21532594",
  "fulltext_url": null,
  "id": "cdy9582qnvbc5",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.21532594",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7171110224",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-24",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/cdy9582qnvbc5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7171110224",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.21532594"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Mathematical Analysis and Transform Methods",
      "subfield": "general"
    }
  ],
  "title": "The Möbius and Klein Transforms: Spectral Analysis on Non-Orientable Manifolds",
  "title_text": "The Möbius and Klein Transforms: Spectral Analysis on Non-Orientable Manifolds",
  "type": "preprint",
  "updated_at": "2026-11-07T13:20:41Z",
  "url": "https://research.draconitic.com/paper/cdy9582qnvbc5",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}