# Independent priority audit Audit date: 2026-09-23 UTC. Checkpoint: 03:30 UTC. Completion estimate for this bounded literature audit: 100%. This estimate concerns completion of the search, not a probability that the result is new. No person was contacted. This audit was performed independently of the mathematical verifier. ## Verdict **Scoped priority pass: no earlier explicit refutation or matching four-vertex construction was located in the searched public literature.** This supports preparing a short, carefully attributed preprint. It does **not** certify absolute novelty, first discovery, or the current open status of an online catalogue. The public record contains close predecessors concerning successive cancellation and algebraic Morse theory, so the claim should be narrowly framed as an explicit counterexample to **Conjecture 2 as printed in Knudson's 2008 report**, for the fixed incidence-restricted matching. Suggested priority wording: “We give an explicit four-vertex counterexample to Conjecture 2 in Knudson's 2008 Oberwolfach report. A public-literature search completed on 23 September 2026 located no earlier explicit refutation; this is not a claim to have excluded unpublished or unindexed prior work.” Two material gaps remain: Knudson's author bibliography lists a related work with Ulrich Bauer as being in preparation, without an accessible manuscript; and the full 2015 book and 2011 Bauer thesis were not obtained for complete text inspection. These are limitations of the priority audit, not mathematical objections to the supplied counterexample. ## Precisely audited claim The filtration inserts `a,b,c,d,cd,bd,ac`. Its finite persistence pairs are `(d,cd),(c,bd),(b,ac)`. The matching obtained by retaining incident pairs is exactly `{(d,cd)}`. The critical edge `ac` cannot reach critical vertex `b` by a gradient path in that fixed matching. This refutes the universal existence-and-uniqueness assertion; it does not refute statements about a matching modified by earlier cancellations, or about an algebraic gradient after a change of basis. ## Primary evidence inspected 1. **Kevin P. Knudson, “Persistent homology and discrete Morse theory,” Oberwolfach Report 29/2008, pp. 1628–1630**, within *Computational Algebraic Topology*, DOI [10.4171/OWR/2008/29](https://doi.org/10.4171/OWR/2008/29). [Publisher PDF](https://ems.press/content/serial-article-files/46173), especially printed p. 1629 / PDF p. 27. The report states a simplex-wise filtration over mod-two coefficients and defines the field from incident persistence pairs. Conjecture 2 concerns paths in that field itself. This is the primary target; no restriction to positive-dimensional births or lower-star filtrations was found in its setup. 2. **Knudson, “Discrete Morse Theory and Persistent Homology,” slides dated February 23, 2013.** [Author's talk hosted by Florida State](https://www.math.fsu.edu/~hironaka/FSUUF/knudson.pdf). The title-page date was read directly; search-engine crawl/publication labels were not used as the talk date. PDF p. 89 repeats the unique-path conjecture for persistence pairs absent from the incidence matching. PDF pp. 29–31 give the path and modified-Hasse-diagram conventions. This is evidence that the conjecture was publicly reiterated in 2013, not evidence that it remained unresolved throughout 2026. 3. **Knudson's current public research bibliography.** [Author page](https://kpknudson.com/math), read on the audit date. A listed collaboration with U. Bauer, *Persistence and discrete Morse theory*, has preparation status and no linked manuscript. The page also lists the 2015 book *Morse Theory: Smooth and Discrete*. The listing may be stale and establishes neither what the unpublished work contains nor whether it was completed. No accessible manuscript under that coauthored title was located. 4. **Ulrich Bauer, Carsten Lange, Max Wardetzky, “Optimal topological simplification of discrete functions on surfaces.”** [Author preprint arXiv:1001.1269](https://arxiv.org/abs/1001.1269), [PDF](https://arxiv.org/pdf/1001.1269); published in *Discrete & Computational Geometry* 47 (2012), 347–377, [DOI 10.1007/s00454-011-9350-z](https://doi.org/10.1007/s00454-011-9350-z). The key inspected material is Section 3.3, especially Lemma 9 and Theorem 13. Unique connecting paths are guaranteed at an appropriate stage of a cancellation sequence after descendant pairs have been cancelled. This is a close structural predecessor and should be cited when explaining the fixed-versus-updated-field distinction. It neither asserts the printed fixed-field conjecture nor supplies the candidate's four-vertex example in the inspected passages. The paper also has a counterexample section for general two-complexes; that concerns its simplification framework, and is not evidence of this particular refutation. 5. **Ulrich Bauer and Fabian Roll, “Wrapping Cycles in Delaunay Complexes: Bridging Persistent Homology and Discrete Morse Theory.”** [arXiv:2212.02345](https://arxiv.org/abs/2212.02345), [v3 PDF read](https://arxiv.org/pdf/2212.02345); [SoCG 2024 publisher record, DOI 10.4230/LIPIcs.SoCG.2024.15](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.15). Section 4.1, Proposition 13, builds a reduction gradient using a changed chain basis. This is directly relevant to the difference between algebraic elimination and gradient paths on the original simplices. It is not the same matching as the one conjectured about. A full-text search of the available PDF found no occurrence of Knudson's name. 6. **Tamal K. Dey, Jiayuan Wang, Yusu Wang, “Improved Road Network Reconstruction using Discrete Morse Theory,” SIGSPATIAL 2017.** [Author-hosted PDF](https://www.cs.purdue.edu/homes/tamaldey/paper/DiscreteMorse/MorseGIS.pdf). Algorithm 1 starts with the trivial field and updates it at each cancellation; Lemma 3.1 addresses cancellability at each loop stage on a triangulated orientable two-manifold. This is another explicit successive-cancellation precedent, not a result about the candidate's fixed field. 7. **Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, Dmitriy Morozov, “Topological Simplification Guided by Forbidden Regions.”** [SoCG 2026 record, DOI 10.4230/LIPIcs.SoCG.2026.72](https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.72); [full-version arXiv:2603.16416v1](https://arxiv.org/html/2603.16416v1). The public publisher record dates publication to 2026-05-27. The inspected full version makes reversibility / a unique gradient path a hypothesis of its simplification procedure, then changes the field by reversal. A text search found no mention of Knudson. This recent work is compatible with the counterexample and does not establish priority for it. 8. **Bauer's author bibliography and thesis metadata.** [Author page](https://ulrich-bauer.org/) and [TUM thesis record](https://portal.fis.tum.de/en/publications/persistence-in-discrete-morse-theory/). *Persistence in discrete Morse theory* is a 2011 doctoral thesis, DOI [10.53846/goediss-2536](https://doi.org/10.53846/goediss-2536). The author-linked PDF was identified as `https://webdoc.sub.gwdg.de/diss/2011/bauer_u/bauer_u.pdf`. Browser retrieval failed; a direct public-file retrieval also failed. Therefore its full text is not certified as searched. Knudson's 2013 repetition gives some historical context but cannot close this gap. ## Search record The following exact web-search strings were issued during this audit. Results were used to locate primary sources; snippets from aggregators and unofficial book mirrors were not treated as proof of a mathematical claim. Quoted strings here are search syntax, not quotations from source documents. ```text "Knudson" "Conjecture 2" "gradient" "Persistent homology and discrete Morse theory" counterexample "OWR-2040-002" "persistent homology" "gradient path" conjecture Knudson "Persistent homology and discrete Morse theory" "Knudson" "2008" "Knudson" "unique gradient" counterexample "Knudson" "conjecture" persistence "OWR/2008/29" "Bauer" "Knudson" "Persistence and discrete Morse theory" "Knudson" "persistence" "counterexample" "persistence pairs" "unique gradient path" "gradient paths" "Knudson" "conjecture" Ulrich Bauer Persistence in discrete Morse theory thesis 2011 pdf "Knudson" "conjecture" "Bauer" "persistence pairing" "nonincident" "Topological simplification guided by forbidden regions" site:ediss.uni-goettingen.de Bauer "Persistence in discrete Morse theory" site:webdoc.sub.gwdg.de Bauer "Persistence" "Persistence in discrete Morse theory" "counterexample" "Knudson" "1629" "Bauer" "dissertation" "2536" "Persistence in discrete Morse theory" "conjecture" "Morse Theory Smooth and Discrete" "conjecture" persistence "persistence" "Knudson" "counterexamples" "Knudson" "gradient path" "counterexample" "persistence" "zero gradient paths" "persistence" "non-incident" "Morse" "persistence" "incident pairs" "Knudson" "30000990" mathematics "OWR-2040-002" conjecture "Knudson" "Conjecture 2" "Morse" "gradient path" "persistence" "four vertices" "Persistent homology and discrete Morse theory" "Conjecture" "Knudson" "gradient" "false" "Knudson" "Morse" "counterexample" "persistence" after:2023-01-01 site:arxiv.org Knudson "persistence" "gradient" site:mathoverflow.net Knudson "Morse" "Persistence in discrete Morse theory" "bauer_u.pdf" "Persistent homology and discrete Morse theory" refuted ``` Additional primary pages opened: the [University of Florida 2026 faculty-publications list](https://math.ufl.edu/people/faculty/publications2026/) and the publisher/author links above. The bibliography and citations in close predecessor papers were inspected selectively to identify plausible competing mechanisms. This was not a complete forward-citation crawl across subscription databases. ## Coverage limits and release guidance - The [user's problem page](https://www.unsolvedmath.com/problems/OWR-2040-002) was inaccessible to the web tool. Its current catalogue status and the alternate identifier `30000990` were not independently confirmed. Ground the paper's assertion in the original report and treat catalogue identifiers only as supplied cross-references. - No comprehensive MathSciNet, zbMATH, Google Scholar, or proprietary citation-index export was obtained. Search engines can miss short notes, mathematical notation, dissertations, books, unindexed corrections and private communications. - Full texts of Knudson's 2015 book and Bauer's 2011 dissertation were not inspected. An attempted publisher-book URL did not retrieve content; it must not be used as verified bibliographic identification. - The inaccessible coauthored preparation item is a concrete priority uncertainty. Independent researchers should record this uncertainty; no outreach was prepared or initiated, in accordance with project policy. - Correctness of the explicit construction is separable from historical novelty. Prefer a factual title such as *A four-vertex counterexample to a persistence–gradient-path conjecture*. Do not assert “first,” “previously unknown,” or an eighteen-year open status as established facts. - No additional supporting paper is needed for priority. Preserve this dated audit alongside the short proof and executable certificate. If earlier work is later identified, update the attribution without changing the checkable counterexample.