{
  "schema_version": 1,
  "problem_number": "AMR-014-0005",
  "title": "Cube Ranks of Monomials",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let M = product x_i^{a_i} have degree 3d > 0, and let o be the number of odd exponents. We determine the least number of cubes of homogeneous degree-d forms whose sum is M. Over C, this number is one if every exponent is divisible by three, three if M is not a cube and o <= d, and four if o > d. The same formula holds over R. The new lower bound uses an exponent-one coordinate, which must exist when o > d. Restricting a hypothetical three-cube expression to its coordinate hyperplane gives proportional forms, because the Fermat plane cubic has no nonconstant rational parametrization. Differentiation then forces a degree-2d square divisor in a monomial whose square capacity is smaller than d. Classical constructions give the matching upper bounds. Version 2 removes the support restriction of version 1 (DOI 10.5281/zenodo.23425441); it is a strengthening of the same manuscript, not a new paper. This completes the k = 3 case of Problem C in Algebraic Stories, but does not determine general monomial k-ranks for k >= 4. This is an AI-assisted, self-audited, unrefereed preprint; no independent human review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG",
    "math.AC"
  ],
  "keywords": [
    "monomial rank",
    "sums of cubes",
    "Fermat cubic",
    "normal derivative",
    "AMR-014-0005"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "2.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "2.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-014-0005/",
  "pdf_url": "https://eulersolve.org/papers/amr-014-0005/paper.pdf?v=fb2613cd26d0",
  "doi": "10.5281/zenodo.23425851",
  "zenodo_record_url": "https://zenodo.org/records/23425851",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The theorem determines the cube rank of every complex or real monomial, with no degree, support or variable-count restriction. It completes the k = 3 case of Problem C, not its general k >= 4 cases. Classical upper bounds and the non-rank-two theorem are credited. AI-assisted, self-audited and unrefereed; no independent human review, formal verification or absolute priority is claimed.",
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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.",
  "concept_doi": "10.5281/zenodo.23425440",
  "concept_url": "https://doi.org/10.5281/zenodo.23425440",
  "version_history": [
    {
      "version": "1.0",
      "title": "A Support Obstruction for Three-Cube Decompositions of Monomials",
      "doi": "10.5281/zenodo.23425441",
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  "revision_note": "Version 2 removes the support restriction and completes the k = 3 classification over C and R. Same manuscript; no new whole-source closure.",
  "original_problem_resolved": false,
  "theorem_scope_resolved": true,
  "review_disclosure": "AI-assisted self-audit only; no independent human review or formal verification claimed."
}