{
  "schema_version": 1,
  "problem_number": "OWR-14299904-009",
  "title": "No Uniform Linear Threshold for Hook Monotonicity in Self-Conjugate and Doubled Distinct Partitions",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We disprove the uniform linear threshold for adjacent-hook monotonicity in self-conjugate and doubled distinct partitions proposed in Conjecture 3 of Ae Ja Yee's 2026 Oberwolfach report. For every fixed integer K>=3, the self-conjugate ratio H_(t+1)(Kt)/H_t(Kt) tends to infinity along even t; for doubled distinct partitions the same holds along odd t when K>=4 is even. More strongly, counterexamples persist at n=alpha*t^2/(log t)^2+O(1) for every 0<alpha<pi^2/24, with the necessary parity. Consequently any eventual threshold has a corresponding logarithmically subquadratic lower bound. The six-page proof combines credited exact hook formulas, classical partition asymptotics and an elementary diagonal-hook injection. Exact integer checks and source files accompany the manuscript. The known fixed-t eventual inequalities and the proved opposite parity cases remain valid. This is a complete negative answer to the explicitly t-independent linear assertion represented by UnsolvedMath OWR-14299904-009, not a claim that all hook monotonicity statements are false. This is an unrefereed preprint developed with AI assistance, without formal proof-assistant verification or an absolute-priority claim.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "keywords": [
    "OWR-14299904-009",
    "integer partitions",
    "hook lengths",
    "uniform threshold",
    "self-conjugate partitions",
    "doubled distinct partitions",
    "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-14299904-009/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299904-009/paper.pdf?v=2d332e4e79c8",
  "doi": "10.5281/zenodo.23677491",
  "zenodo_record_url": "https://zenodo.org/records/23677491",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete counterexample to the t-independent linear threshold in Conjecture 3, Yee, OWR 2/2026, p142. Both self-conjugate even-to-odd and doubled-distinct odd-to-even hook inequalities fail beyond every fixed linear scale. Counterexamples persist at n=alpha*t^2/log(t)^2+O(1) for every 0<alpha<pi^2/24, giving an eventual-threshold lower bound. The t-independent linear threshold in the printed Conjecture 3 is disproved for both partition families. The known fixed-t eventual inequalities and opposite-parity theorems are retained; the lower bound is not asserted to be sharp. 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": "2d332e4e79c8c265deade36ee459c3a910a001ce0d15cd31dc5eac9b676bafc7"
    },
    "source.zip": {
      "sha256": "8f7c59e8b88bf3d45adf70d4380a44d0bdd8f11ed7a736c9d79103e67258faf7"
    },
    "verification_report.md": {
      "sha256": "1cfbe43fc256e31701cf05bd7332a6502991296f6147aaf276a46086b322cc24"
    }
  },
  "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."
}