{
  "schema_version": 1,
  "problem_number": "OWR-14299093-005",
  "title": "A Signed-Weight Counterexample to a Grushin Hardy--Rellich Converse",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a smooth signed radial weight for which a Grushin Hardy--Rellich inequality holds for every compactly supported smooth test function on the punctured unit gauge ball, but the associated Bessel equation has no positive solution. The principal weight is the constant one and the homogeneous dimension is five. The second weight is specified by the lowest eigenvalue of -d^2/dr^2+r^(-4) on (0,1). A quantitative spectral estimate on a codimension-one subspace yields the explicit positive margin 51/8900, and the Grushin spherical decomposition extends the inequality from radial tests to all tests. This is a counterexample to the remainder-free inequality with an unrestricted signed second weight, not to the forward Bessel-pair theorem or to nonnegative-weight variants. The preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "keywords": [
    "OWR-14299093-005",
    "math.AP",
    "math.CA"
  ],
  "manuscript_version_date": "2026-10-22",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14299093-005/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299093-005/paper.pdf?v=491946dad37e",
  "doi": "10.5281/zenodo.23587424",
  "zenodo_record_url": "https://zenodo.org/records/23587424",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For Q=5 and V=1, the smooth signed weight W=lambda1+1/100+2/r^2-1/r^4 has a full punctured-ball Grushin Hardy-Rellich inequality with margin 51/8900, but its Bessel equation has no positive solution; lambda1 is the lowest Friedrichs eigenvalue of -d^2/dr^2+r^-4 on (0,1). This is a self-audited, AI-assisted, unrefereed preprint. No independent certification, formal proof verification or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "491946dad37e567f92b2c9c13663ae59f9aa0e5944a9c971f632df55c19da6ff"
    },
    "source.zip": {
      "sha256": "095e68cf783c48a62b7cc8833559eb10fca30d6fbd6b61249f5104249211c8d7"
    },
    "verification_report.md": {
      "sha256": "9d2a4231f3b882171619acefbf5b8ae9f2d80cb6b7e6af41f20784a2ee8cee23"
    }
  },
  "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."
}