{
  "schema_version": 1,
  "problem_number": "AIM-OTHER-0057",
  "title": "Brauer-Picard Groups and Module Orbits for Twisted Cyclic Categories",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every cyclic group C_n and every associator class omega_t, this preprint explicitly computes the Brauer-Picard group of Vec_(C_n)^(omega_t) and its action on all indecomposable module categories. Module labels are the divisors of gcd(n,t). For p^k exactly dividing n and s=min(v_p(t),k), let U be the units modulo p^k satisfying a^2=1 modulo p^(k-s). The local group is U when s=0, and the inversion semidirect product U with C_2 when s>0. The units fix all module labels; the involution sends the local exponent j to s-j. There are product_p(floor(s/2)+1) orbits, with transitivity exactly when gcd(n,t) is squarefree. The proof computes the center metric group, all Lagrangian subgroups, group generators, orbit invariants and stabilizers, including every 2-primary case. The calculation uses established categorical theorems and credits earlier cyclic-Lagrangian counts and untwisted orbit formulas. It is a complete theorem for the cyclic family, not a resolution of the broad AIM-OTHER-0057 classification program. The unrestricted proof is supplemented by 383 exact finite isometry cases and 1,556,863 categorical checks in each of normal and optimized Python. This is an AI-assisted, unrefereed preprint; no independent review, formal proof verification or absolute priority is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.QA",
    "math.CT"
  ],
  "keywords": [
    "AIM-OTHER-0057",
    "fusion categories",
    "Brauer-Picard group",
    "cyclic group",
    "associator",
    "metric group",
    "Lagrangian subgroup",
    "module category"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-other-0057/",
  "pdf_url": "https://eulersolve.org/papers/aim-other-0057/paper.pdf?v=e794477c3c85",
  "doi": "10.5281/zenodo.23393763",
  "zenodo_record_url": "https://zenodo.org/records/23393763",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete theorem for all complex cyclic pointed fusion categories and all associator classes, including every 2-primary case. The broad AIM-OTHER-0057 classification program remains open. Established categorical theorems, earlier cyclic-Lagrangian counts and untwisted orbit formulas are credited. Self-audited, unrefereed and AI-assisted; no independent review, formal proof verification or absolute priority is claimed.",
  "files": {
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      "sha256": "e794477c3c85c2cef5162bb4f0cac7a36ffafdf50b066c4ff7bc661fd1ddd88b"
    },
    "source.zip": {
      "sha256": "603dcf8eceae3d9eb28114a0e229c06be777987b6b0e0c60c10730e4770a4fd6"
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    "verification_report.md": {
      "sha256": "f5837a24f0268682be08bc347eeb114e3d17fcfe0019f3a583fe86c26f54f10a"
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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."
}