{
  "schema_version": 1,
  "problem_number": "OWR-3385-008",
  "title": "A Proof of the Below–Krummeck–Richter-Gebert Conjecture on Cyclic Determinant Quotients",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For points A_1, …, A_n, A'_1, …, A'_n in K², where K is a field, write [XY] for the 2 × 2 determinant and let α(π) be the cyclic quotient ∏_k [A_{π_k} A_{π_{k+1}}] / ∏_k [A'_{π_k} A'_{π_{k+1}}], with indices read cyclically. Below, Krummeck and Richter-Gebert conjectured in 2003 that Σ sgn(π) α(π) = 0, where the sum runs over the permutations π of {1, …, n} with π_1 = 1. Richter-Gebert posed the problem again at Oberwolfach in 2009. It had been proved for n = 5 and checked by computer algebra for n = 6, and it follows from a symmetry argument when n ≡ 0, 3 (mod 4). We prove the identity for every n ≥ 3 and every field, assuming only that [A'_iA'_j] ≠ 0 for i ≠ j. Equivalently, the associated bracket polynomial vanishes identically. The proof writes the numerator as the trace of a product of traceless 2 × 2 matrices. The reciprocals of the denominators are Parke–Taylor factors. They satisfy a shuffle identity, known from the Kleiss–Kuijf relations of gauge theory, and hence are the coefficients of a Lie polynomial. Mapping this Lie polynomial into 2 × 2 matrices over an exterior algebra, where the commutators of the generators are central, gives zero. The same argument shows that Σ_σ sgn(σ) M_{σ_2} ⋯ M_{σ_n} / ([A'_1 A'_{σ_2}] ⋯ [A'_{σ_n} A'_1]) = 0 for all traceless 2 × 2 matrices M_2, …, M_n when n ≥ 4, and it recovers the vanishing of the same sum without the matrices. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.RA"
  ],
  "keywords": [
    "cyclic determinant quotients",
    "bracket polynomials",
    "alternating sums",
    "phirotopes",
    "complex matroids",
    "Parke–Taylor factors",
    "shuffle identity",
    "Kleiss–Kuijf relations",
    "Ree's theorem",
    "free Lie algebra",
    "exterior algebra",
    "Oberwolfach Reports",
    "OWR-3385-008",
    "math.CO",
    "math.RA",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-3385-008/",
  "pdf_url": "https://eulersolve.org/papers/owr-3385-008/paper.pdf?v=c077c306bbb9",
  "doi": "10.5281/zenodo.23064792",
  "zenodo_record_url": "https://zenodo.org/records/23064792",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proves the Below–Krummeck–Richter-Gebert conjecture (OWR 2/2009, Problem 6) for every n ≥ 3 over every field; for n = 2 the sum is a nonzero square and for n = 1 it is undefined. The shuffle identity (Kleiss–Kuijf) and Ree's theorem are known; the sign identities of Proposition 5.2 are not new in substance.",
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      "sha256": "c077c306bbb96cfd7a5c6f2e3595206a48a9efee3da4252461337fc1356a04c5"
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    "source.zip": {
      "sha256": "0e02423cc539c874d1ee64589e82c25b7498a4f42c5b9585d280e3470e8e28b5"
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    "verification_report.md": {
      "sha256": "966c3e011f001bfec9805e7bc5ec26c59e7406187feaf61570c19758cfced173"
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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."
}