{
  "schema_version": 1,
  "problem_number": "AIM-OTHER-0067",
  "title": "Sharp Tensor Nilpotence Bounds for Finite Crossed Products",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let a finite group G act on an additive braided rigid monoidal category by additive strong monoidal autoequivalences. Every tensor-nilpotent object of the crossed product has nilpotency index at most the order of the subgroup generated by its nonzero homogeneous degrees. The bound applies to arbitrary sums of homogeneous objects, without a Noetherianity or support-theory hypothesis. In characteristic p, explicit permutation modules over (C_p)^G give stable matrix units and realize every directed graph on G. A Hamilton path attains the bound |G| for every finite group; a four-group example has index four despite group exponent two. This yields a bound for the first projective tensor power in the corresponding finite tensor category. The result is a scoped contribution to AIM-OTHER-0067, not a solution of the broad AIM transfer problem. Known cyclic examples and categorical constructions are credited. This preprint is unrefereed and makes no certified absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.RT",
    "math.CT",
    "math.RA"
  ],
  "keywords": [
    "tensor nilpotence",
    "crossed product",
    "finite tensor category",
    "modular representation",
    "projective tensor powers",
    "directed graph",
    "AIM-OTHER-0067"
  ],
  "manuscript_version_date": "2026-10-09",
  "publication_date": "2026-10-09",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-09",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-other-0067/",
  "pdf_url": "https://eulersolve.org/papers/aim-other-0067/paper.pdf?v=145c2149ef3f",
  "doi": "10.5281/zenodo.23269705",
  "zenodo_record_url": "https://zenodo.org/records/23269705",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for a finite group acting by additive strong monoidal autoequivalences on an additive braided rigid monoidal category with biadditive tensor product: every tensor-nilpotent crossed-product object has index at most the order of the subgroup generated by its nonzero homogeneous degrees. Explicit modular permutation modules attain the group-order bound for every finite group and model all directed graphs on the group. This is a scoped contribution to AIM-OTHER-0067, not a solution of Gelaki's broad transfer problem (AIM 2012 Problem 9.20). Known cyclic projective-power examples, the braided duality-retract argument and categorical constructions are credited. Unrefereed and self-audited; no independent review, formal verification or certified absolute priority is claimed.",
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      "sha256": "145c2149ef3f96fe160a2afc249a2c31c92f657d830dde962675a360500e0b40"
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    "source.zip": {
      "sha256": "c235941c3172f0cb95e6dc49738e89f55295520a2eb5dccecd080089cdf4eec8"
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    "verification_report.md": {
      "sha256": "9c0c3a3241bde1de9208f66c62f29c28808fafbb7cf213c548a4c81e566a11d8"
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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."
}