{
  "schema_version": 1,
  "problem_number": "AIM-PROBABILITY-0167",
  "title": "A Cubic-Link Obstruction to Manifold Skeleta of Generalized Cluster Complexes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every finite root system Phi of rank n>=3, we prove that the generalized cluster complex Delta^2(Phi) is not the full (n-1)-skeleton of a finite polytopal PL manifold, with or without boundary, in any dimension. A face consisting of negative simple roots has link K3,3. A connected cubic nonplanar graph cannot be the full graph of a polytopal sphere or ball: the graph of any three-dimensional cell would have minimum degree three and hence exhaust the nonplanar graph. The type-A3 example consists of partial quadrangulations of a decagon. This gives a complete negative answer to the universal manifold-skeleton suggestion in AIM Problem 6.3 and Fomin-Reading Remark 4.5, but does not classify other parameters or close the entire dataset record. Cluster compatibility and parabolic rules, rank-two projective-plane examples and the separate known homological results are credited. The exact checker uses two independent integer crossing predicates. This is an unrefereed, AI-assisted preprint; absolute priority is not certified.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "keywords": [
    "AIM-PROBABILITY-0167",
    "generalized cluster complex",
    "polytopal manifold",
    "skeleton",
    "local link",
    "quadrangulation"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-probability-0167/",
  "pdf_url": "https://eulersolve.org/papers/aim-probability-0167/paper.pdf?v=17dce8b58d8a",
  "doi": "10.5281/zenodo.23633552",
  "zenodo_record_url": "https://zenodo.org/records/23633552",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every finite root system Phi of rank n>=3, Delta^2(Phi) is not the full (n-1)-skeleton of a finite polytopal PL manifold, with or without boundary, in any dimension. A negative-simple-root face has local graph K3,3; a connected cubic nonplanar graph cannot be the full graph of a polytopal sphere or ball. Delta^2(A3), partial quadrangulations of a decagon, is an explicit counterexample. This refutes the universal geometric realization suggestion in AIM Problem 6.3 and Fomin-Reading Remark 4.5, but does not classify other parameters or re-prove the separate known Cohen-Macaulay assertions. Complete negative answer to the universal geometric realization suggestion in AIM Problem 6.3 and Fomin-Reading Remark 4.5: at parameter two, every finite root system of rank at least three has a forbidden K3,3 local link. The broader classification is not closed. Established cluster rules, rank-two projective-plane examples and positive homological results are credited. Absolute priority is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "17dce8b58d8abf1db903bb98b842d0aaa3f2be4da8d23aa95c99763dc85ed52f"
    },
    "source.zip": {
      "sha256": "866d967a62f3079910bde9c2caa549a11395431c783d5eab832bbe6fbbd9a5e5"
    },
    "verification_report.md": {
      "sha256": "eb53cd708c5e884d78bf984071014b277aab4912c524c96cd66392f47946176f"
    }
  },
  "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."
}