{
  "schema_version": 1,
  "problem_number": "OWR-13750335-005",
  "title": "A Counterexample to a Homogeneous Fractional Hardy Conjecture on the Heisenberg Group",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A conjecture in the 2023 Oberwolfach report on hypoelliptic operators proposes replacing the homogeneous weight in a fractional Hardy inequality by the power of the standard Heisenberg homogeneous norm while retaining the displayed constant. We give a complete counterexample to this replacement. On the first Heisenberg group, the explicit real function rho^(-1) sin^2(log(rho)/3), supported where 1 < rho < exp(3 pi), violates the proposed inequality for every fractional order 99/100 <= s < 1. Its fractional Rayleigh quotient is less than 273/284, whereas the proposed constant is greater than 98/101. The proof combines an exact radial energy calculation, elementary Gamma-ratio bounds and spectral concavity, keeping the conformal fractional operator distinct from the ordinary spectral power. A single smooth compactly supported witness also exists for the whole interval. This four-page English preprint addresses UnsolvedMath OWR-13750335-005. It does not rule out a smaller Hardy constant or assert failure for all smaller orders. Classical horizontal-gradient obstructions are credited. The work is AI-assisted and self-audited, not independently peer reviewed or formally verified; bounded literature searches do not certify absolute priority.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "keywords": [
    "OWR-13750335-005",
    "Heisenberg group",
    "fractional Hardy inequality",
    "conformal fractional sublaplacian",
    "counterexample",
    "Gamma ratio",
    "spectral concavity"
  ],
  "manuscript_version_date": "2026-10-26",
  "publication_date": "2026-10-26",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-26",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13750335-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-13750335-005/paper.pdf?v=710a6e9b2400",
  "doi": "10.5281/zenodo.23664734",
  "zenodo_record_url": "https://zenodo.org/records/23664734",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The homogeneous replacement with the constant printed in OWR 24/2023, p1373, formula (9), is false on the first Heisenberg group. One explicit compactly supported real function in D(L) violates it for every 99/100 <= s < 1. A single smooth compactly supported witness also exists on the entire interval. Complete counterexample to the homogeneous replacement with the constant printed in the source conjecture, for every 99/100 <= s < 1. It does not rule out a smaller Hardy constant or determine the optimum. Classical horizontal-gradient obstructions are credited; absolute historical priority is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "710a6e9b240038cfd31ac6752332176fd7b18353ac41a14703590ed22fe62124"
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    "source.zip": {
      "sha256": "78b7dd5dea5c2666cc582b792018da398dcd433bd9dedaf4530bd36d3737e270"
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    "verification_report.md": {
      "sha256": "2a7b92afb88454151699b56663471066a74d93a54e85c0f7d5b401ad42a1f597"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}