{
  "schema_version": 1,
  "problem_number": "OWR-3389-011",
  "title": "Intrinsic Negative Sobolev Norms at a Fixed First Eigenvalue",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct smooth positive unit-area conformal metrics on the two-sphere with first nonzero Laplace eigenvalue exactly 6*pi, while the intrinsic H^-1 norm of the conformal factor diverges uniformly over all round conformal parametrizations. The norm uses the scalar-function pairing with the varying metric's area measure. An explicit family of antipodal polynomial caps is small in L^(3/2) but large in L2. A variance-continuity lemma allows exact eigenvalue tuning, and the Mobius Jacobian formula prevents any conformal reparametrization from removing the growth. The five-page manuscript gives a complete proof and exact rational diagnostics. It distinguishes this counterexample from known positive bounds for the area measure in a fixed round-background Sobolev dual. The relation to Nadirashvili's printed 2009 conjecture, represented by UnsolvedMath OWR-3389-011, is explicitly qualified by the choice of pairing; no unqualified original-source closure is claimed. This is an unrefereed preprint, without a claim of absolute priority or formal verification.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DG",
    "math.SP",
    "math.AP"
  ],
  "keywords": [
    "OWR-3389-011",
    "conformal metrics",
    "negative Sobolev norm",
    "fixed first eigenvalue",
    "Mobius transformations",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-3389-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-3389-011/paper.pdf?v=667d80440257",
  "doi": "10.5281/zenodo.23676435",
  "zenodo_record_url": "https://zenodo.org/records/23676435",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the explicitly defined intrinsic scalar-function H^-1 estimate for conformal factors. Smooth positive unit-area sphere metrics have first eigenvalue exactly 6*pi, while the infimum of their intrinsic conformal-factor H^-1 norms over all round conformal parametrizations tends to infinity. A polynomial family and an exact continuity argument give the bound n/(4*pi). The counterexample uses the explicitly defined intrinsic scalar H-minus-one pairing with the varying metric. It does not claim an unqualified closure of Nadirashvili's interpretation-sensitive source conjecture or the frozen dataset record. The positive fixed-round-background estimate is credited to Karpukhin, Nahon, Polterovich and Stern. 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": "667d80440257c67f8e40b74329f7dc1163d652544631f42aa152c31564039dc2"
    },
    "source.zip": {
      "sha256": "60a3099415561395bf85c0621b8f9c0b59a567bde7080c4948ab3f6abe3fabf8"
    },
    "verification_report.md": {
      "sha256": "fb9ae94fc8001867cd45a4bc705be7502fc5e700e6080dda16219db8580ec1f1"
    }
  },
  "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."
}