{
  "schema_version": 1,
  "problem_number": "OWR-2654831-006",
  "title": "Five Possible Centers for a Cube Move of a 10-Realization",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Melotti, Ramassamy and Thevenin proposed that a cube move of an alpha-realization, for alpha>1, has at most three possible central points. We give a counterexample at alpha=10: two construction curves with a common focus have at least five distinct intersection points. All foci and boundary vertices of the cube configuration are rational. The proof reduces the intersection equations to a continuous scalar function and certifies five sign changes using rational enclosures of fifth roots. A reflection construction supplies a rational input center and verifies the noncollinearity conditions. The same boundary has at least five output centers for exponents in an open neighborhood of 10. This refutes the three-center bound in Melotti's contribution to Oberwolfach Report 35/2020, p1785, and Remark4.8 of Cube moves for s-embeddings and alpha-realizations (2023), exactly frozen UnsolvedMath record OWR-2654831-006. We do not assert that five is the maximum or that the configuration is a proper embedding; crossed edges are allowed by the source definition. This four-page English preprint includes a standard-library exact arithmetic checker with 67 checks. Developed with AI assistance and self-audited, it is unrefereed, without independent review, proof-assistant verification or absolute-priority certification.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "keywords": [
    "OWR-2654831-006",
    "cube moves",
    "alpha-realizations",
    "construction curves",
    "exact rational certificate"
  ],
  "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-2654831-006/",
  "pdf_url": "https://eulersolve.org/papers/owr-2654831-006/paper.pdf?v=71f457fe819a",
  "doi": "10.5281/zenodo.23692375",
  "zenodo_record_url": "https://zenodo.org/records/23692375",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "An explicit rational boundary hexagon with noncollinear same-color triples has a valid rational input center and at least five distinct output centers for a cube move at alpha=10. The property persists for alpha in some open neighborhood of 10. This refutes the universal three-center upper bound in OWR35/2020 p1785 and Melotti-Ramassamy-Thevenin Remark4.8. At least five output centers at alpha=10, with persistence near 10. The source permits crossed edges. No claim that exactly five centers exist or that five is a maximum, and no conclusion about proper embeddings or other source questions. 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": "71f457fe819a82aedb89dfa309e1c3fada0dd79f4494f101ed48bf6dd07b7389"
    },
    "source.zip": {
      "sha256": "24ec24bbe87940c9d3cbd6c1a9c344672993cd2c31201085769ec307209810e7"
    },
    "verification_report.md": {
      "sha256": "0fb8bc79e7e32a0d35f1471d87a49765b506a3d474fcd1f141fbb035ad749dd8"
    }
  },
  "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."
}