{
  "schema_version": 1,
  "problem_number": "OWR-13948-006",
  "title": "Collapsed Boundary Data Do Not Force Sobolev Labeling",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We construct a smooth polynomial, uniformly gradient-convex energy on two-valued maps of the unit disk whose minimizers have completely collapsed zero boundary data but admit no global labeling by two Sobolev functions. The potential is lambda times the squared modulus of y^2-(1-|z|^2)^2z. An unordered square-root competitor has energy 16*pi/5, independently of lambda, whereas scalar minima grow on the order of sqrt(lambda). At lambda=2^20 an explicit energy gap excludes every globally labeled minimizer. The density is quasiconvex in the multiple-valued sense. This answers OWR-13948-006 negatively within the original general value-dependent framework. Autonomous gradient-only energies are not addressed. The argument uses established Sobolev, semicontinuity and orientability methods, with prior work credited. This five-page English preprint is by Alper Ferudun, Mercury Software GmbH. It is AI-assisted, self-audited and unrefereed. No independent review, formal verification or absolute priority is claimed. The source package includes exact arithmetic diagnostics; finite checks do not replace the analytical proof. Version 1.0, 24 October 2026. CC BY 4.0. Source dataset record: OWR-13948-006 (ulamai/UnsolvedMath v1.6.0). The manuscript discloses OpenAI Codex assistance.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "keywords": [
    "OWR-13948-006",
    "multiple-valued maps",
    "Sobolev labeling",
    "quasiconvexity",
    "variational energy"
  ],
  "manuscript_version_date": "2026-10-24",
  "publication_date": "2026-10-24",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-24",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13948-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-13948-006/paper.pdf?v=1eb6f49e50a5",
  "doi": "10.5281/zenodo.23621671",
  "zenodo_record_url": "https://zenodo.org/records/23621671",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The smooth polynomial energy sum_i(|Du_i|^2+lambda|u_i^2-(1-|z|^2)^2z|^2) on the unit disk is Q-quasiconvex with gradient Hessian 2I. At lambda=2^20, all two-valued minimizers with boundary 2[[0]] have no global two-sheet W^{1,2} labeling. Existence and the explicit gap m_2<=16pi/5<517/42<=2m_1 prove the negative answer in the original value-dependent Q-integrand setting. For lambda>=64 scalar minima have matching sqrt(lambda) order, while two-valued minima stay bounded. No gradient-only or autonomous restriction is resolved. The negative answer is for the original value-dependent integrands. Autonomous gradient-only energies are not addressed. 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": "1eb6f49e50a5504864b5895ac9af99206fe16677a84f233dad8dc4180a875f49"
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    "source.zip": {
      "sha256": "e77ff3907689d1978c294d17b296d285b922bbcad84d747344b36815ed8e9c21"
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    "verification_report.md": {
      "sha256": "a2c29d18abe3f5e494acf194efc48cfa6ec635f87e12a3a6bab37e84eac72164"
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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."
}