{
  "title": "A four-vertex counterexample to Knudson's persistence-gradient conjecture",
  "upload_type": "publication",
  "publication_type": "preprint",
  "description": "<p>Knudson conjectured that every nonincident persistence pair is joined by exactly one gradient path in the discrete vector field obtained by retaining the incident persistence pairs. This note gives a simplex-wise filtration of a triangulated interval with four vertices and three edges for which one such pair has no gradient path. It disproves Conjecture 2 in Knudson's contribution to Oberwolfach Report 29/2008, p. 1629. The proof consists of three boundary reductions and an inspection of the matching, and the same example works over any field.</p><p>The deposit includes a three-page manuscript, LaTeX source, two exact standard-library Python verifiers, original inputs, and mathematical and priority audit notes. A bounded literature search located no earlier explicit refutation, without guaranteeing historical novelty. This is an unrefereed, AI-assisted preprint; the audits are not external human peer review. Text is CC BY 4.0; code is MIT.</p>",
  "creators": [
    {
      "name": "Kriebel, Alec",
      "orcid": "0009-0001-9320-500X",
      "affiliation": "Independent researcher"
    }
  ],
  "publication_date": "2026-09-23",
  "version": "1.0.1",
  "access_right": "open",
  "license": "cc-by-4.0",
  "keywords": [
    "persistent homology",
    "discrete Morse theory",
    "persistence pairing",
    "gradient paths",
    "counterexample",
    "Knudson conjecture"
  ],
  "language": "eng",
  "related_identifiers": [
    {
      "identifier": "10.4171/OWR/2008/29",
      "relation": "references",
      "scheme": "doi"
    },
    {
      "identifier": "https://github.com/AlecKriebel/Math/tree/main/knudson_gradient_path_counterexample",
      "relation": "isSupplementedBy",
      "scheme": "url"
    }
  ],
  "notes": "Prepared with OpenAI Codex and independent AI audit agents from a candidate supplied by the author from prior AI-assisted work. Mathematical verification passes; priority is a bounded search with explicit limitations. No minimality claim is made."
}