{
  "schema_version": 1,
  "problem_number": "AIM-TOPOLOGY-0106",
  "title": "Induced-Cover Excision Characterizes Dismantlable Hom Tests",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix a nonempty finite graph T, with loops permitted, and a nonzero commutative coefficient ring R with identity. We classify when ordinary relative homology of Hom(T,-) satisfies excision for every induced graph cover. This occurs exactly when T dismantles to a single looped vertex. Exactness of the degree-zero Mayer-Vietoris sequence for all such covers already characterizes these tests. Otherwise, an explicit obstruction uses at most |V(T)|+1 target vertices and its defect contains a split copy of R. The proof combines the established fold theorem and isolated identity of a stiff graph with an unlooped-twin or nonedge-deletion construction. For the edge test K2, the two-edge cover of a three-vertex loopless path is a counterexample. Exact finite checks cover 1,098 tests, 502 obstructions and 46,758 induced clique covers; a separate cellular-boundary implementation replays 31 small negative cases. This is a complete classification of the specified fixed-test family, not a full resolution of the broader AIM graph-homology program. Classical inputs are credited. The preprint is self-audited and unrefereed, with no independent-review, formal-verification or absolute-priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "keywords": [
    "graph homology",
    "Hom complex",
    "excision",
    "dismantlable graph",
    "stiff graph",
    "Mayer-Vietoris",
    "induced cover",
    "AIM-TOPOLOGY-0106"
  ],
  "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-topology-0106/",
  "pdf_url": "https://eulersolve.org/papers/aim-topology-0106/paper.pdf?v=2c1d279913d4",
  "doi": "10.5281/zenodo.23382370",
  "zenodo_record_url": "https://zenodo.org/records/23382370",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete classification of fixed nonempty finite Hom tests under all induced graph covers, with optional loops and nonzero commutative unital coefficients. Not a full resolution of AIM Problem 5.2, nor a prohibition on other homology theories, restricted covers or derived gluings. Classical inputs are credited. Self-audited, unrefereed and AI-assisted; no absolute novelty, independent review or formal verification claimed.",
  "files": {
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      "sha256": "2c1d279913d4317958b8dafe8f5dfddb790cc648e1233c4973dcc295f0067238"
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    "source.zip": {
      "sha256": "0cb50532f33c3cf0373da55ab4e93175c2c1d6a13ce083a1200acaf6cf4f0b04"
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    "verification_report.md": {
      "sha256": "e4aefe5db67f97fa501f3a6a6e5a9114af29381ac7764558152c2622037e214b"
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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."
}