{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0131",
  "title": "Critical Groups Detect Homology Actions of Simple Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every finite connected simple graph, we prove that the natural actions of its automorphism group on the finite critical group and on real first homology have the same kernel. The proof uses the cut-flow presentation and the absence of integral flows of squared norm at most two, with a separate coordinate argument for reversed edges. Combining this equality with known homological faithfulness yields a faithful critical-group action for every connected simple graph of minimum degree at least two other than a cycle. In particular, semiregularity is unnecessary for simple three-connected graphs. Parallel-edge and pendant-tree examples explain the hypotheses. This is a complete scoped theorem within AIM-COMBINATORICS-0131, not closure of the open-ended AIM research suggestion. The preprint is AI-assisted, self-audited and unrefereed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "keywords": [
    "critical group",
    "graph Jacobian",
    "graph automorphism",
    "homology representation",
    "integral flow lattice",
    "chip firing",
    "AIM-COMBINATORICS-0131"
  ],
  "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/aim-combinatorics-0131/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0131/paper.pdf?v=40a7ace94496",
  "doi": "10.5281/zenodo.23471711",
  "zenodo_record_url": "https://zenodo.org/records/23471711",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the equality of critical-group and real-homology action kernels for finite connected simple graphs. The minimum-degree faithfulness corollary uses credited prior homology results. Parallel edges invalidate the kernel equality. AIM item 41 is an open-ended research suggestion and remains partially answered; no whole-source closure is claimed. Exact finite computations supplement but do not replace the proof. AI-assisted, self-audited and unrefereed; no independent review, formal certification, dataset-maintainer acceptance or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "40a7ace94496aa94edc49852af9b75d8afef6157aaaee957f7bf5d7bebab9a25"
    },
    "source.zip": {
      "sha256": "f406c07dc2d39a01b1096156be537b379ef299e81b4ef5d5846bb32b0facd4e4"
    },
    "verification_report.md": {
      "sha256": "5a1794f4232149f996f37f2c11c945be52a3073910c8105fd99b71f7e754b461"
    }
  },
  "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."
}