{
  "schema_version": 1,
  "problem_number": "OWR-2040-002",
  "title": "Failures of Theorem 1 and Conjecture 3 in Knudson's Persistence–Gradient Abstract, and a Corrected Form of Conjecture 2 after Kriebel",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In an Oberwolfach abstract, Knudson considered the discrete vector field V_P formed by the incident pairs of the persistence pairing P of a simplexwise filtration. He stated that V_P is a discrete gradient (his Theorem 1), conjectured a single gradient path from τ to σ in V_P for every non-incident pair {σ, τ} of P (Conjecture 2), and conjectured that a filtration by a function decreasing along the modified Hasse diagram reproduces P (Conjecture 3). Conjecture 2 was first refuted publicly by Kriebel (Zenodo, September 2026) with a filtration of 7 simplices; we credit this and add that there are exactly two such filtrations with 7 simplices, none with fewer, that no filtration with at most 13 simplices has a pair joined by two or more paths, and examples with two and three paths on 17 and 19 simplices. The parts of this note that we did not find elsewhere are the following two. Theorem 1 is false, also in the abstract's own formulation by the modified Hasse diagram: a filtration of 17 simplices has a closed V-path, and for every filtration of every complex with at most 16 simplices V_P is a gradient (a short lemma and an exhaustive search over 2,484,335,648 filtrations). Conjecture 3 is false even when V_P is acyclic, for every choice of the function and of tie-breaking: a filtration of 17 simplices with a hand-checkable proof; it holds for all 4,005,434 filtrations with at most 11 simplices. Finally, we prove a corrected form of Conjecture 2, which is a consequence of algebraic Morse theory: for the earliest non-incident pair, if the field formed by the earlier pairs is a gradient, the number of gradient paths in it from τ to σ is odd, and σ is the youngest critical simplex reached an odd number of times. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.AT"
  ],
  "keywords": [
    "persistent homology",
    "discrete Morse theory",
    "persistence pairing",
    "discrete gradient vector field",
    "Knudson",
    "Kriebel",
    "counterexample",
    "Oberwolfach Reports",
    "OWR-2040-002",
    "OWR-2040-003",
    "math.AT",
    "cs.CG",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-2040-002/",
  "pdf_url": "https://eulersolve.org/papers/owr-2040-002/paper.pdf?v=ffe1450c844f",
  "doi": "10.5281/zenodo.23110481",
  "zenodo_record_url": "https://zenodo.org/records/23110481",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Refutes Theorem 1 and Conjecture 3 of Knudson's abstract (OWR 29/2008, pp. 1628-1630): a 17-simplex filtration has a closed V-path, and a 17-simplex filtration with acyclic V_P violates Conjecture 3 for every admissible function and tie-breaking; exhaustive searches show V_P is a gradient for all filtrations with at most 16 simplices. Conjecture 2 (record OWR-2040-002) was first refuted publicly by Kriebel (Zenodo, September 2026); the note adds minimality and multi-path examples and proves a parity statement that follows from algebraic Morse theory. Conjecture 3 is under-defined in the source; its weakest reading is refuted. Minimal sizes and the sequential form of Conjecture 2 remain open.",
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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."
}