{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0023",
  "title": "A Cycle Criterion for CAT(0) Signed Permutation Mapping Tori",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let F be freely generated by a,b_1,...,b_m and let phi(a)=a^epsilon and phi(b_i)=a^{p_i} b_{sigma(i)}^{delta_i} a^{q_i}, where epsilon and delta_i are signs, sigma is a permutation and p_i,q_i are integers. We give an explicit necessary and sufficient cycle criterion for F semidirect_phi Z to be CAT(0). A positive cycle has return data (P_C,Q_C,l_C). If epsilon=1, the ratios (P_C-Q_C)/l_C must agree on positive cycles with nonzero P_C+Q_C. If epsilon=-1, each even positive cycle must satisfy P_C^2=Q_C^2. Negative cycles impose no additional condition. In the positive cases a compact two-dimensional locally CAT(0) graph of spaces realizes the original group; failure obstructs every proper semisimple CAT(0) action. Positive cycles become HNN edges of a torus or Klein-bottle group, and negative cycles become square-root attachments. We also prove power-invariance within this family and give a polynomial-time arithmetic test. The unsigned identity-permutation subcase is the theorem of Wu and Ye. The full AIM characterization of CAT(0) free-by-cyclic groups remains open. This is an AI-assisted, self-audited, unrefereed preprint; no absolute priority is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.GR",
    "math.MG"
  ],
  "keywords": [
    "AIM-GEOMETRIC_GROUP_THEORY-0023",
    "free-by-cyclic groups",
    "CAT(0) groups",
    "signed permutation",
    "mapping torus",
    "Klein bottle"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0023/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0023/paper.pdf?v=05b4c5cd9cd0",
  "doi": "10.5281/zenodo.23449986",
  "zenodo_record_url": "https://zenodo.org/records/23449986",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete criterion for the displayed signed-permutation family, with a direct CAT(0) model of the original group. The full AIM Problem 5.1 remains open. The unsigned identity-permutation case is due to Wu and Ye. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "05b4c5cd9cd0c803e2e6e992a54196b2a5faaada7d2c9417d449c209f3e53851"
    },
    "source.zip": {
      "sha256": "00446005b32091b77cbfc07677f1ce7397c3c38479d6295198fa4dba2d98fce8"
    },
    "verification_report.md": {
      "sha256": "c616b9954eac3a043aedd46b55dfab75bd1db9b9365680d3dc98aa2c49c293d3"
    }
  },
  "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."
}