{
  "schema_version": 1,
  "problem_number": "AIM-GEOMETRIC_GROUP_THEORY-0042",
  "title": "Sharp Degree Bounds for Multipartite Lorentzian Coxeter Forms",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Partition at least four vertices into at least two nonempty parts. Assign Coxeter exponent 2 within each part, a fixed exponent m >= 5 between parts, and replace selected cross-part exponents by 3. Suppose each vertex has at most one replacement neighbor in any other part. If the replacement graph has maximum degree Delta and (cos(pi/m) - 1/2) Delta < 1, the normalized Coxeter form has exactly one negative eigenvalue and no zero eigenvalue, with an explicit lower bound for its positive eigenvalues. The uniform degree caps are 3 for m = 5 and 2 for every finite m >= 6; both are sharp. A paired-part construction gives the entire obstruction spectrum. The diagrams avoid A3 and B3, so the established rewriting theorem of Blasco-Garcia, Cumplido, Holt, Morris-Wright and Rees supplies a quadratic word algorithm for each fixed Artin presentation. Nonstar examples have unboundedly many exponent-3 pairs. This is a complete spectral construction and an application of an existing algorithm, not a new general Artin word algorithm or a resolution of the small-type problem. The manuscript is AI-assisted, self-audited and unrefereed; no absolute priority claim is made.",
  "result_type": "COMPLETE_SCOPED_MULTIPARTITE_SPECTRAL_THEOREM",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "Coxeter forms",
    "Artin groups",
    "Lorentzian signature",
    "multipartite graphs",
    "spectral bounds",
    "word problem",
    "sharp degree criterion"
  ],
  "manuscript_version_date": "2026-10-11",
  "publication_date": "2026-10-11",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-11",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0042/",
  "pdf_url": "https://eulersolve.org/papers/aim-geometric-group-theory-0042/paper.pdf?v=7ff1ad188d23",
  "doi": "10.5281/zenodo.23292974",
  "zenodo_record_url": "https://zenodo.org/records/23292974",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete spectral degree criterion and exact sharpness family for the stated multipartite construction. The quadratic Artin word algorithm is an application of an established theorem, not a new algorithm. The general Artin word problem, the small-type problem and the broad AIM source remain unresolved. Self-audited and unrefereed; no independent review, formal verification or certified priority.",
  "files": {
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      "sha256": "7ff1ad188d234020cd3db3b1ca8e445015a11288f2fb65316d1eda095bfc4609"
    },
    "source.zip": {
      "sha256": "dd72315999ff9ee273389ca85744bd1aebfb9c7d22401b2606dfa034debaeac4"
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    "verification_report.md": {
      "sha256": "9c54b6dca3dfca3a440bf3ff9d18db1ac4f789683dc882d84cd10d81e483ec2b"
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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."
}