{
  "schema_version": 1,
  "problem_number": "OWR-4136-004",
  "title": "A Centering Counterexample for Prekopa-Leindler Stability",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A conjecture printed in the 2009 Oberwolfach report on convex geometry proposes quantitative Prekopa-Leindler stability in the unshifted L1 distance of probability densities with zero mean. We give an explicit counterexample in every dimension. Strictly positive real-analytic densities with mean zero and covariance I_n have L1 distance at least 11/8, while an admissible midpoint majorant has integral 1+2k/(1+k^2), for every integer k>=3. The same densities admit majorants with vanishing deficit for every fixed interpolation parameter in (0,1), uniformly on compact subintervals. Small remote Gaussian components cancel the barycenters without aligning the main mass. An explicit translation makes the distance at most 2/(1+k^2), consistent with established stability modulo translation. This English preprint addresses the literal UnsolvedMath OWR-4136-004 formulation; it does not contradict log-concave or translated stability theorems. The elementary argument is AI-assisted and self-audited, not independently peer reviewed or formally verified. No historical-priority claim is made.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.FA",
    "math.PR",
    "math.MG"
  ],
  "keywords": [
    "OWR-4136-004",
    "Prekopa-Leindler inequality",
    "stability",
    "barycenter",
    "isotropic density",
    "Gaussian mixture",
    "counterexample"
  ],
  "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-4136-004/",
  "pdf_url": "https://eulersolve.org/papers/owr-4136-004/paper.pdf?v=31bdcabe9eab",
  "doi": "10.5281/zenodo.23665906",
  "zenodo_record_url": "https://zenodo.org/records/23665906",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The centered, unshifted Prékopa-Leindler stability conjecture printed in OWR2009/53, p2863, Conjecture1 is false. In every dimension and at lambda=1/2, explicit strictly positive real-analytic isotropic probability densities have L1 distance at least11/8 and an admissible majorant of integral1+2k/(1+k^2), k>=3. A related majorant gives vanishing deficit for every fixed lambda in(0,1), uniformly on compact subintervals. Complete counterexample to the literal centered, unshifted stability conjecture printed in OWR2009/53. Modern stability theorems modulo translation and results with extra log-concavity assumptions remain valid. The elementary construction is self-audited; 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": "31bdcabe9eabffa2ade51faad1f4da7a05e6ab7fe7ab717992323aa1c0298ccd"
    },
    "source.zip": {
      "sha256": "5cdeb8e3ed827ee726e9014ac0532a830c84328b1a78961bb22ef4aa7e9659b4"
    },
    "verification_report.md": {
      "sha256": "eaaee1279af9788d5c2d145c9019ca57e2b0f0af40778d06d8cea6c6eb44dfc0"
    }
  },
  "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."
}