{
  "schema_version": 1,
  "problem_number": "OWR-17294-007",
  "title": "Orbit Lattice Formulas for Second Homology of Finite Abelian Quandles",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let Q be a finite quandle whose right translations commute, with r orbits and stabilizer lattices H_i in L = Z^r. We give an explicit presentation T_i = wedge^2 L / (sum_j e_j wedge H_j + wedge^2 H_i) for the torsion summand of each orbit in second integral rack and quandle homology. The free ranks are r^2 and r(r-1). Each orbit size annihilates its summand, and the bound is sharp; a twenty-element example disproves the stronger orbit-exponent bound. Smith normal form computes the groups. The universal coefficient theorem gives second cohomology for every constant abelian coefficient group, with integral second cohomology torsion-free. This makes explicit consequences of the credited methods of Darné, Eisermann and Lebed–Mortier. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AT",
    "math.GR"
  ],
  "keywords": [
    "quandle homology",
    "rack homology",
    "orbit lattices",
    "Smith normal form",
    "OWR-17294-007"
  ],
  "manuscript_version_date": "2026-10-20",
  "publication_date": "2026-10-20",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-20",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-17294-007/",
  "pdf_url": "https://eulersolve.org/papers/owr-17294-007/paper.pdf?v=3a93c2e6aa40",
  "doi": "10.5281/zenodo.23496647",
  "zenodo_record_url": "https://zenodo.org/records/23496647",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete explicit calculation for finite nonempty quandles with commuting right translations, integral second rack and normalized quandle homology, and second cohomology with any constant abelian coefficient group. The source asks about cohomology without specifying coefficients; integral second cohomology is torsion-free and is not confused with integral homology. The result uses and credits Darné, Eisermann and Lebed–Mortier. It does not cover arbitrary medial quandles, twisted or nonabelian coefficients, or higher homology. Novelty and absolute priority are not certified. AI-assisted, self-audited and unrefereed; no independent review or formal verification.",
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      "sha256": "3a93c2e6aa40ee46767125c1d6988f651dd8c394ad0b819319641afd3e32b42c"
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    "source.zip": {
      "sha256": "dd0f5c9e6a7a2b12f2af2e8460fab818d70bfdee127e4881f9e398ed612ad9d5"
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    "verification_report.md": {
      "sha256": "e9b28e48f032fb84310a9ac3bd3b0462e27eae33d5193dc60296f6f9194905b0"
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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."
}