{
  "schema_version": 1,
  "problem_number": "AIM-OTHER-0060",
  "title": "Finite Bipartite Graphs as Induced Subgraphs of Subfactor Principal Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Every finite bipartite simple graph occurs as a vertex-induced unrooted subgraph of the principal graph of an irreducible finite-depth inclusion of hyperfinite type II1 factors. For part sizes m and n, an explicit neighborhood-multiplicity parameter q gives index q3^n, with q at most m+1, and depth at most four. The selected vertices have depths two and three. The complete depth ranks and adjacency spectrum are computed. A separate amplification realizes every finite bipartite multigraph as an ordinary subgraph, with edge deletion allowed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.OA",
    "math.CO"
  ],
  "keywords": [
    "subfactor",
    "principal graph",
    "induced subgraph",
    "finite group",
    "restriction matrix",
    "AIM-OTHER-0060",
    "math.OA",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-other-0060/",
  "pdf_url": "https://eulersolve.org/papers/aim-other-0060/paper.pdf?v=253a2c0e87f6",
  "doi": "10.5281/zenodo.23048180",
  "zenodo_record_url": "https://zenodo.org/records/23048180",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Variable-index finite unrooted scope, not the full AIM polymer programme, fixed-index rooted extension, infinite subgraphs or induced multigraph universality. Classical ingredients credited; low historical novelty confidence, possibly folklore; no absolute priority, independent review or formal verification. Self-audited and unrefereed, with AI assistance; no independent peer review or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "253a2c0e87f68df0de2c6f5146beeeaefd889d43f13f09a53d83a9b7ae07ae72"
    },
    "source.zip": {
      "sha256": "f211450c7cbc19939af7da48c91eaf539798b4a710c3d96cfcfc1d4b9fdcae06"
    },
    "verification_report.md": {
      "sha256": "e5bee78f2d093b703e352a3db6e13a10c7a5514f9d378237b72ef57567dac014"
    }
  },
  "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."
}