{
  "schema_version": 1,
  "problem_number": "OWR-13490-013",
  "title": "Component Symmetry and Tightness in Dimension Two",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite nonempty simplicial complex of dimension at most two, we prove that tightness over a field of characteristic two is equivalent to a complete one-skeleton together with symmetry of the link-component counts in every induced subcomplex. The proof adapts the cycle-link argument of Bagchi, Datta and Spreer and fills induced link cycles using mod-two fundamental chains of the resulting closed surfaces. Consequently, tightness over any field implies tightness over F2 in dimension two. A four-vertex disk refutes the literal component-symmetry conjecture in the 2015 Oberwolfach report when neighbourliness is omitted; the theorem proves its neighbourly repair. This is an AI-assisted, self-audited, unrefereed preprint, not a claim of higher-dimensional field independence or certified absolute priority.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "keywords": [
    "tight simplicial complex",
    "component symmetry",
    "induced subcomplex",
    "characteristic two",
    "simplicial homology",
    "OWR-13490-013"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-13490-013/",
  "pdf_url": "https://eulersolve.org/papers/owr-13490-013/paper.pdf?v=fa4f1ec2e248",
  "doi": "10.5281/zenodo.23477149",
  "zenodo_record_url": "https://zenodo.org/records/23477149",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A complete four-vertex counterexample to the literal 2015 component-symmetry conjecture, whose statement omits neighbourliness, and a complete proof of the neighbourly characterization in dimension at most two. The consequence is field transfer to characteristic two in this dimension only. The cycle-link method of Bagchi, Datta and Spreer is explicitly credited. Higher-dimensional field independence is not proved. Exhaustive finite tests supplement, not replace, the proof. AI-assisted, self-audited and unrefereed; no independent review, formal certification, dataset-maintainer acceptance or absolute priority is claimed.",
  "files": {
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      "sha256": "fa4f1ec2e248cde326f17f71c0edb9127e7f131993512b954d8718a80e3cfd68"
    },
    "source.zip": {
      "sha256": "5a6043f72797a79968f67f948e0c9f182e9e08f7252e25c86a6b8d19a0ca8de8"
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    "verification_report.md": {
      "sha256": "c1d31a31cbd227d8c354cfa863014ba8509a1cdccff865dca9dc4652b6b6af95"
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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."
}