{
  "schema_version": 1,
  "problem_number": "OWR-14299097-014",
  "title": "Linear-Length Mixed Identities for Finite Projective Unitary Groups",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let q=p^e be a prime power and n>=2. We construct a nontrivial singular mixed identity for PSU_n(q) of variable length at most 16q+8p+12. The bound is uniform in the dimension and includes characteristic two without an exception. A product of two unitary transvections has order dividing one of q-1, q+1 or 2p. A nested commutator combines these alternatives, while separate free conjugators ensure formal nontriviality. Together with the lower bound of Bradford, Schneider and Thom, this gives Theta(q) for the length of a shortest mixed identity in the projective special unitary family. This is a complete theorem for the stated unitary-family scope, not a solution of all remaining Lie-type families or shortest nonsingular identities in the associated Oberwolfach questions. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR"
  ],
  "keywords": [
    "finite unitary groups",
    "mixed identities",
    "transvections",
    "free products",
    "word length",
    "OWR-14299097-014"
  ],
  "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-14299097-014/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299097-014/paper.pdf?v=7af1c98f3e4a",
  "doi": "10.5281/zenodo.23507216",
  "zenodo_record_url": "https://zenodo.org/records/23507216",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete uniform linear upper bound for mixed identities in the projective special unitary family. The Theta(q) conclusion uses the explicitly credited Bradford-Schneider-Thom lower bound. The broader remaining-Lie-type and shortest-nonsingular-identity questions are not resolved; frozen OWR-14299097-014 remains PARTIALLY_SOLVED, with zero whole-source closure increment. Finite tests are diagnostics, not the general proof. Novelty and absolute priority are not certified. AI-assisted, self-audited and unrefereed; no independent review or formal verification.",
  "files": {
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      "sha256": "7af1c98f3e4add53a7909b2aa89fb68ce87b387518d2f8545532001bbbce07f8"
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    "source.zip": {
      "sha256": "57f59db416a54cac03eae0e67d98356a9d02eee0277a1252de6e0d8e1bf18f6c"
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    "verification_report.md": {
      "sha256": "58a1d589019a0d11a14e02fdfa86454427a84b9da78547a4be0dd38c6c3a6449"
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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."
}