{
  "schema_version": 1,
  "problem_number": "AMR-010-0302",
  "title": "No Uniform Hanna Neumann Bound for Convex-Core Volume",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "There is no universal constant K such that V(A intersection B) <= K V(A)V(B) for torsion-free convex-cocompact Kleinian groups, where V is three-dimensional convex-core volume. An explicit rank-two classical Schottky deformation has positive core volume tending to zero and constant systole arcosh(31). Independent parity kernels are distinct proper index-two subgroups of rank three, with rank-five intersection of index four; their volume ratio is exactly 1/V(G). Fixed ranks, fixed indices and a fixed positive systole therefore do not restore a uniform bound. We also give the exact finite-cover double-coset identity and prove a neighborhood-volume inequality by counting lifts. This is a complete counterexample to the universal-constant interpretation of Swarup's Question 3.2 in Bestvina's list, recorded as AMR-010-0302. It does not settle arbitrary ambient-dependent bounds for all infinite-index subgroups of one fixed G, so the broad source record is not claimed wholly resolved. Classical Schottky theory, Bonahon's volume regularity and covering theory are credited. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority certification is claimed.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.GT",
    "math.GR"
  ],
  "keywords": [
    "AMR-010-0302",
    "convex core",
    "Schottky group",
    "Hanna Neumann",
    "volume inequality",
    "counterexample"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-010-0302/",
  "pdf_url": "https://eulersolve.org/papers/amr-010-0302/paper.pdf?v=93e917148bbe",
  "doi": "10.5281/zenodo.23636651",
  "zenodo_record_url": "https://zenodo.org/records/23636651",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every K>0 there are rank-two classical Schottky groups G with systole arcosh(31), and distinct proper normal index-two subgroups A,B with rank-three and index-four rank-five intersection, such that all core volumes are positive and V(A intersection B)>K V(A)V(B). The ratio is exactly 1/V(G), which diverges in an explicit deformation. The finite-cover double-coset identity and a thickened-core multiplicity bound are also proved. The five-page paper gives a complete counterexample to a universal convex-core volume Hanna Neumann bound, with fixed ranks, indices and systole. Arbitrary ambient-dependent bounds for all infinite-index subgroups of one fixed group remain outside the result. The broad frozen record stays PARTIALLY_SOLVED; no whole-source closure is added. Classical Schottky theory, Bonahon regularity and covering theory are credited. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "93e917148bbe970af807f89d7afce6206d09023f3cbff8290dd4405524046aa8"
    },
    "source.zip": {
      "sha256": "4f6ca5a9701a495c8bed8534de148d1004c198c042827fde9de6eaebeb996503"
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    "verification_report.md": {
      "sha256": "bc226946c6aa03edb5e639e7e005cbcfa46eb34d20e6410f4a3ac104188f7028"
    }
  },
  "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."
}