{
  "schema_version": 1,
  "problem_number": "OWR-14299911-029",
  "title": "A Counterexample to a Corner-Peeling Conjecture for Subcomplexes of Z^3",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let G be a finite connected subgraph of the grid Z^3, not necessarily induced, and let X(G) be the cube complex whose cells are the squares and 3-cubes of Z^3 all of whose vertices and edges lie in G. A corner is a vertex that lies in exactly one maximal cell, and a corner peeling is an ordering v_1,...,v_n of the vertices in which each v_i is a corner of the subgraph induced by v_1,...,v_i. In the problem session of the 2026 Oberwolfach workshop Median Geometry and Applications, V. Chepoi, from discussions with J. Chalopin and M. Kokkou, conjectured that G admits a corner peeling whenever X(G) is simply connected and every connected component of every section of X(G) by a coordinate plane is simply connected. We show that the conjecture is false. We give a subgraph G_62 of Z^3 with 62 vertices in [0,3]^3 and an induced subgraph H_73 with 73 vertices in [0,4]^3 such that the cube complex and all its sections are collapsible, but there is no corner at all. Both examples have a symmetry group of order 6. The proofs can be checked by hand; explicit sequences of elementary collapses are supplied as machine-checkable certificates. A subdivision construction turns every finite subgraph into an induced subgraph whose pieces and corners are dilates of the original ones, so the partial- and induced-subgraph formulations are equivalent. Uncertified solver computations indicate that no counterexample fits into a 3 x 3 x 3 box of lattice points.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.GT"
  ],
  "keywords": [
    "OWR-14299911-029",
    "corner peeling",
    "cube complex",
    "cubical complex",
    "subgraphs of the grid Z^3",
    "collapsible complex",
    "sections by coordinate planes",
    "counterexample",
    "Oberwolfach Reports",
    "math.CO",
    "math.GT"
  ],
  "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/owr-14299911-029/",
  "pdf_url": "https://eulersolve.org/papers/owr-14299911-029/paper.pdf?v=09d7bb785560",
  "doi": "10.5281/zenodo.23049729",
  "zenodo_record_url": "https://zenodo.org/records/23049729",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Scope: This paper supplies exact counterexamples to the partial- and induced-grid-graph versions of the corner-peeling conjecture corresponding to corpus record OWR-14299911-029. The 62- and 73-vertex examples and their collapse certificates are verified. Small-box solver results are explicitly uncertified; no smallest-size counterexample is proved, and the least size remains unknown. A formulation with the additional isometry requirement (3) is not refuted here. No absolute priority claim is made. Self-audited, AI-assisted and unrefereed; no independent peer review is claimed.",
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      "sha256": "09d7bb785560d05bbd87bf5ab0da260a04c67ed1391d6afcd4bc53e5b652e2cd"
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    "source.zip": {
      "sha256": "5f3abac5ecb027dd36950f4150c1f3516fc9f2b01038047bf7e4f4aeaa256bfe"
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    "verification_report.md": {
      "sha256": "3329c4d6e9fe7d56b68f4e0ea6433136fcb15a0ad9e32d31d45ece4e5d1d7dcc"
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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."
}