{
  "schema_version": 1,
  "problem_number": "AIM-TOPOLOGY-0101",
  "title": "Binary Interior Excision and Nonuniqueness of Graph Homology",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give an explicit nonuniqueness example for the (J1, categorical-product) Eilenberg-Steenrod axioms on Cech closure spaces. Applying the known clique and closed neighborhood complex constructions to the mutual singleton graph gives two integral relative homology theories, with subspace complexes formed intrinsically. For every binary interior cover, both constructions preserve the relevant union and intersection, so excision is an actual relative chain isomorphism. The theories share point coefficient group Z, homotopy invariance, the long exact sequence of a pair and arbitrary coproduct additivity, but on the undirected four-cycle they give the homology of S1 and S2. Their restrictions to finite graph pairs also respect ordinary interval homotopies. The complexes and general closure-space nonuniqueness are established prior work; the focus is the explicit relative binary-cover argument and its graph consequence. This self-audited, unrefereed note does not settle every strengthened axiom variant, and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AT",
    "math.CO"
  ],
  "keywords": [
    "Cech closure space",
    "Eilenberg-Steenrod axioms",
    "graph homology",
    "excision",
    "closed neighborhood complex",
    "clique complex",
    "relative homology"
  ],
  "manuscript_version_date": "2026-10-13",
  "publication_date": "2026-10-13",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-13",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-topology-0101/",
  "pdf_url": "https://eulersolve.org/papers/aim-topology-0101/paper.pdf?v=9b2bb66c6d53",
  "doi": "10.5281/zenodo.23327483",
  "zenodo_record_url": "https://zenodo.org/records/23327483",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete nonuniqueness counterexample for the specified binary interior-cover excision and (J1, categorical-product) homotopy axioms, with an ordinary-interval corollary only on finite graph pairs. The broader AIM question about suitable variants remains unresolved. Known complexes and prior general nonuniqueness are credited; no certified novelty, independent review or formal verification is claimed.",
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      "sha256": "9b2bb66c6d531b5cc7f469a66064836a7bd56bdbd6917acb7d0fc221db75e220"
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    "source.zip": {
      "sha256": "7f3bbe88d1504c578ed8fbd7c789392779db42e5f0615740565760d94b825194"
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    "verification_report.md": {
      "sha256": "673f517aa3746c2e4d134b9da3b63484bb2cdaac2eeef67bfb28188f393e946d"
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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."
}