{
  "schema_version": 1,
  "problem_number": "OWR-4425-011",
  "title": "A Four-Color-Theorem Proof of the Duplicate-Leaf Parse-Word Conjecture",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Cooper, Rowland and Zeilberger conjectured that two ordered full binary trees sharing a pair of consecutive uncle-nephew leaves have a common parse word with equal letters at that prescribed pair. We prove this existence assertion using the four-color theorem. Gluing the two corresponding polygon triangulations gives a planar graph whose specified boundary vertex has degree at most four. Contracting a length-two path, four-coloring the minor, and restoring the vertex forces the desired equal labels. A padding argument, together with the known equivalence for unrestricted parsing, shows that the duplicate-leaf assertion is equivalent to the four-color theorem. This is a proof using that established theorem, not a new proof of it or a four-color-free reduction to the particular smaller tree pair. Exact finite diagnostics and reproducible source are included. This unrefereed preprint makes no absolute novelty claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "OWR-4425-011",
    "binary trees",
    "parse words",
    "four-color theorem",
    "polygon triangulations"
  ],
  "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-4425-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-4425-011/paper.pdf?v=9a584cfa1ae9",
  "doi": "10.5281/zenodo.23626716",
  "zenodo_record_url": "https://zenodo.org/records/23626716",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Every pair of rooted ordered full binary trees with a specified consecutive uncle-nephew leaf pair in both trees has a common parse word whose letters at that pair agree. Proof uses the established four-color theorem. A padding argument gives equivalence with the four-color theorem through the known unrestricted-parsing equivalence. The frozen existential duplicate-leaf assertion is proved using the established four-color theorem. This is not a new proof of that theorem or a four-color-free inductive reduction. The common word may depend on the prescribed eligible position. 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": "9a584cfa1ae972b4265e265ab4dd3fc8af8b36de7f2d5be4864bcff7864ed1c2"
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    "source.zip": {
      "sha256": "2bdcce55cb656a1be894374aa943a2f1c1dfc45836c642f2b08591944d2f9038"
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    "verification_report.md": {
      "sha256": "f10501e9a4126f222292301434dc0778252769287020b9019d8839402dec0426"
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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."
}