{
  "schema_version": 1,
  "problem_number": "OWR-734-007",
  "title": "Algebraic Shifting Can Destroy Every Polynomial HVC Bound",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Kalai asked whether algebraic shifting preserves homological VC dimension, a uniform polynomial bound on total Betti numbers in a family closed under induced subcomplexes and links. We give a full negative answer to the unrestricted assertion in Oberwolfach Report 32/2004, p1730, Conjecture 3, recorded as OWR-734-007 in UnsolvedMath v1.6.0. A strongly hereditary family of minimum polynomial HVC exponent exactly 1, with constant 3, has after either exterior or symmetric algebraic shifting over the rationals a strongly hereditary closure with no finite polynomial HVC exponent. For a shift of the r-dimensional cross-polytope boundary, the link of the first s vertices has 2r-s vertices and top reduced Betti number binom(r,s); central links therefore grow exponentially. The proof uses established shifting properties and the classical shifted facet interpretation of h-vectors, which are credited. The example has unbounded simplicial dimension and does not resolve a fixed-dimensional modification or neighboring fractional-Helly and nerve conjectures. This four-page English preprint includes a standard-library checker with 4,317 exact checks. Developed with AI assistance and self-audited, it is unrefereed, without independent review, proof-assistant verification or absolute-priority certification.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO",
    "math.AT",
    "math.AC"
  ],
  "keywords": [
    "OWR-734-007",
    "homological VC dimension",
    "algebraic shifting",
    "cross-polytope",
    "simplicial links",
    "h-vector"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-734-007/",
  "pdf_url": "https://eulersolve.org/papers/owr-734-007/paper.pdf?v=de61a783a9f7",
  "doi": "10.5281/zenodo.23697440",
  "zenodo_record_url": "https://zenodo.org/records/23697440",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "A strongly hereditary family of minimum polynomial HVC exponent exactly1 has, under either standard exterior or symmetric algebraic shifting over Q, a strongly hereditary closure with no finite polynomial HVC exponent. The initial s-link of a shifted r-dimensional cross-polytope boundary has2r-s vertices and top reduced Betti binom(r,s). Full negative answer to unrestricted HVC preservation. The input family has unbounded simplicial dimension; no fixed-dimensional variant or neighboring fractional-Helly or nerve conjecture is resolved. 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": "de61a783a9f7aa009258959625b17bbeab861b3c331515d6f483a7f79548932a"
    },
    "source.zip": {
      "sha256": "6cae55ce98f0a3fd3f2d22c0c59976ed39d864ba3b000e1feba4454abde530cf"
    },
    "verification_report.md": {
      "sha256": "0930c2896c5a9905039426ab6a6264a9eaa17409d97e39e1427d1d5e564989fb"
    }
  },
  "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."
}