{
  "schema_version": 1,
  "problem_number": "OWR-5152-009",
  "title": "Weighted Switchings and Linear Corner Growth in Uniform Contingency Tables",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a uniformly chosen nonnegative integer n-by-n matrix with both margin vectors (3n,n,...,n), the corner expectation has order n. An explicit finite weighted-switching bound gives liminf E[X_11]/n >= 2-sqrt(2). The same corner law has total variation distance at least 1/4096 from every geometric distribution, including n-dependent choices of parameter. A general bound covers margins (L,t,...,t). The proof does not use an independence approximation or a Markov-chain mixing theorem. This answers the linear-order question and gives a counterexample to unrestricted geometric entry approximation in OWR-5152-009. It does not assert the exact limiting mean, a Gaussian limit, or the full phase diagram. The fixed-base cloning results are not contradicted. English unrefereed preprint. Author affiliation: Mercury Software GmbH. OpenAI Codex assisted with derivation, literature review, computational checks and preparation. No independent review, formal verification, novelty certification or absolute priority claim. CC BY 4.0.",
  "result_type": "COMPLETE_PROOF_WITH_NON_GEOMETRIC_COUNTEREXAMPLE",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "keywords": [
    "OWR-5152-009",
    "math.PR",
    "math.CO"
  ],
  "manuscript_version_date": "2026-10-21",
  "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-5152-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-5152-009/paper.pdf?v=9b2b4797a11d",
  "doi": "10.5281/zenodo.23586367",
  "zenodo_record_url": "https://zenodo.org/records/23586367",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For a uniform nonnegative integer n-by-n table with margins (3n,n,...,n), the corner expectation is Theta(n), with explicit finite lower bounds and liminf E[X11]/n >= 2-sqrt(2). Its total variation distance from every geometric law exceeds 1/4096 for n>=2, even with n-dependent parameters. A weighted 2x2 bijection proves a finite side-sum bound for margins (L,t,...,t). This answers the source's linear-order question and refutes unrestricted geometric approximation, but does not prove an exact mean limit, Gaussian limit, subcritical phase or complete margin classification. Fixed-base cloning results are not contradicted. 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": "9b2b4797a11dcd9426edf20ec72c1843508a6d41a5d66ba57933bc3d55c96a92"
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    "source.zip": {
      "sha256": "b51f5b60f879a86aae3ee84fd2a84e02e3e5c33f0854a45e5462f93b0518601c"
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    "verification_report.md": {
      "sha256": "cc34db2ec9181585485e3d7965ad2a08e95477e64fde4bf46eed18c591b7d4c9"
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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."
}