# Verification report — OWR-2040-002 and OWR-2040-003 (Knudson's Theorem 1 and Conjectures 2, 3 on persistence pairings and discrete gradients)

Verification date: 2026-10-02 (two independent verification runs, both AI-assisted).

**Verdict.** Conjecture 2 of the source (record OWR-2040-002) is false: there are filtrations in which a non-incident persistence pair is joined by
no gradient path in V_P (7 simplices is the minimum, and there are exactly two such filtrations), and by two or by three paths (examples with 17 and 19
simplices; none with at most 13 simplices). The first public refutation is due to Kriebel (Zenodo, DOI 10.5281/zenodo.22929556, September 2026,
unrefereed); this is not a new result of the note. Theorem 1 of the source (V_P is a discrete gradient) is false, also in the source's own formulation
by the modified Hasse diagram; the least counterexample has 17 simplices, and V_P is a gradient for every filtration of every complex with at most 16
simplices. Conjecture 3 (record OWR-2040-003) is false even when V_P is acyclic, for every integer function and every tie-breaking (17 simplices, with a
hand proof), and holds for all filtrations with at most 11 simplices. A corrected statement is proved, as 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 is odd and sigma is the youngest
critical simplex reached an odd number of times. The note is unrefereed.

## Statement checked
- **Primary source.** K. P. Knudson, "Persistent homology and discrete Morse theory", Oberwolfach Reports 5(3) (2008), pp. 1628–1630 (Report No. 29/2008,
  *Computational Algebraic Topology*), doi:10.4171/owr/2008/29.
  - The PDF was fetched anonymously (EMS Press) and read; sha256 63a63560…e17d5b7f (the same file in both verification runs).
  - The pairing is described on pp. 1628–1629. Theorem 1 ("V_P is a gradient vector field") and Conjecture 2 (a single gradient path from tau to sigma in V_P for
    every pair of the persistence pairing with sigma not a face of tau) are on p. 1629. Conjecture 3 (with the correct choice of f, P_{V_P,f} = P) is on p. 1630.
  - Definitions as printed: filtration by one simplex at a time; Z/2 pairing of Edelsbrunner–Letscher–Zomorodian; V_P = pairs {sigma < tau} of P with sigma a facet of tau;
    V-paths; G = Hasse diagram (arrows beta -> alpha) with the arrows of V reversed; f integer-valued and strictly decreasing along directed paths; K_i generated by
    f^{-1}(−∞, i]; several simplices may enter at one level and are ordered "in such a way that we may still talk about the persistence pairing".
  - The printed definition of a V-path omits the condition alpha_{i+1} ≠ alpha_i; read literally every non-empty V would be non-gradient. The note uses the standard condition
    (printed in the author's slides) and also verifies the failure of Theorem 1 in the Hasse-diagram formulation, where the issue does not arise.
- **Slides.** K. P. Knudson, slides dated February 23, 2013 (102 pages, sha256 cf38f857…30dbb8): the same statements; slide 84 calls the picture proof of Theorem 1 a
  "gross oversimplification"; slide 89 states the conjecture of a unique gradient path; slide 92 says the filtration by a discrete Morse function lets several simplices
  enter at once; slide 99 lists "nail down the proper definition of P_V" as future work.
- **Corpus records.** ulamai/UnsolvedMath v1.6.0: OWR-2040-002 (record_id 30000990) and OWR-2040-003 (30000991), both with status `open` (literature check dated 2026-08-21, before Kriebel's deposit).
  Their statements match the abstract's Conjectures 2 and 3.

## Readings
| Reading | Refuted? | Witness |
|---|---|---|
| Conjecture 2, existence of the gradient path (standard V-path convention) | yes | S7 = (0,1,2,3,23,13,03) and Pi7 = (0,1,2,3,23,13,02): the pair {1,03} resp. {1,02} has no path |
| the same with "directed path in G" instead of "gradient path" | yes | the same filtrations (the vertex 1 is not reachable) |
| Conjecture 2, uniqueness | yes | M17: pair {01,034} has two paths; M19: pair {02,134} has three paths |
| Theorem 1 with the standard V-path convention | yes | T17: closed V-path 14<134>34<234>24<124>14 |
| Theorem 1 as acyclicity of the modified Hasse diagram G | yes | T17: directed cycle 14→134→34→234→24→124→14 |
| Theorem 1 with the V-path definition as printed (no alpha_{i+1} ≠ alpha_i) | trivially false whenever V_P ≠ ∅ | not used |
| Conjecture 3, weakest reading: some V_P-decreasing f and some filtration L in which every K^f_i is an initial segment, with pairing P | yes | B17 (hand proof); A17 (computation) |
| Conjecture 3, strict reading (additionally: pairs of V_P adjacent in L) | yes | B17, A17 (computation) |
| Conjecture 3 when V_P has a closed V-path | meaningless (no f exists) | T17 |

## Results in the paper
- **Proposition 3.1.** S7 and Pi7 have P = {(3,23),(2,13),(1,x)}, V_P = {{3<23}}; the pair {2,13} has one path, the pair {1,x} has none (not even a directed path in G).
  Pi7 is Kriebel's filtration a,b,c,d,cd,bd,ac (a,b,c,d = 0,1,2,3); S7 is the other 7-simplex filtration violating Conjecture 2 and is not Kriebel's.
- **Proposition 3.3 (computation).** All filtrations with n simplices, vertices named in order of first appearance: n = 7: 522 filtrations, 2 violating; n = 8, …, 13: 24; 441; 6,372; 108,130;
  1,854,016; 35,028,376 violating filtrations (709,593,015 filtrations at n = 13); none with a pair of two or more paths; no closed V_P-path.
- **Proposition 3.4.** Smallest examples found with two paths (17 simplices) and three paths (19 simplices); reductions in the appendix of the note.
- **Proposition 4.1, Lemma 4.2, Theorem 4.3.** T17 = (0,1,2,3,4,03,04,12,13,14,23,24,34,124,234,034,134). Lemma 4.2: a closed V_P-path forces at least 16 simplices, and with exactly 16
  the complex is connected, has 5 vertices and at least 3 triangles. Exhaustive check of all 2,484,335,648 filtrations of the 17 isomorphism classes of complexes on at most 5 vertices
  with at least 3 triangles and 13–16 simplices: no closed V_P-path. Positive control: the complex of T17 has 1,412,014,592 filtrations, 437,696 with a closed V_P-path.
  Remark 4.4: of the 11 classes with 17 simplices on 5 vertices only that of T17 has a closed V_P-path (30,582,025,120 filtrations).
- **Theorem 5.1.** B17 = (4,1,0,14,3,13,01,2,23,02,04,12,24,124,024,123,012) has V_P acyclic and no V_P-decreasing f with a total refinement of the sublevel filtration having pairing P.
  Hand proof: F1 (12 precedes 02 and 24, from the arrows), F2 (024 precedes 123), F3 (123 precedes 024, by the eight cosets). Remark 5.2: A17 fails in the strict reading
  and, by computation with the block criterion of the reproducibility README, also in the weak reading.
- **Proposition 5.3 (computation).** Conjecture 3 (strict reading) holds for all 4,005,434 filtrations with at most 11 simplices.
- **Theorem 6.1 (proved).** For the earliest non-incident death tau, with V' the pairs dying before tau, assumed to be a gradient, sigma is the youngest critical simplex c with an
  odd number of V'-paths tau → c. The proof is elementary (an elimination map on chains and the persistence basis); it is a corollary of algebraic Morse theory (Forman, Kozlov,
  Skoldberg). The three-path example shows that "odd" cannot be improved to "one". Observations A and B (general pairs; the Morse complex of an order with adjacent pairs) are tested only.
- **Remark 6.4.** An argument (not a computation) about the dunce hat; the known failure of full cancellation outside surfaces is due to Bauer–Lange–Wardetzky.

## Computations (exact; scripts and outputs in reproducibility/)
- `python/verify_paper_examples.py` (standard library only, a few seconds) recomputes every explicit example and asserts every stated fact: pairings by column reduction and by the rank formula
  (they agree), V_P, non-incident pairs, path counts and lists, the closed path and directed cycle of T17, acyclicity of G for B17, the eight cosets, and all total orders of B17 and A17
  with pairing P (1,852 and 302; none with adjacent V_P-pairs; none of B17 with 12 before 02 and 24).
- Enumeration of filtrations (Python for n ≤ 11, C++ for n ≤ 13); Python search `thm1_minsearch.py`; C++ brute force over the 17 classes; independent C++ program (`cyc_class.cpp`) with the
  same totals and per-class counts; size-17 classes; B17: unpruned enumeration of 1,412,014,592 filtrations and pruned enumeration (1,852 orders with pairing P); Conjecture 3 for n ≤ 11 in Python
  (12 parts) and C++; random tests of Theorem 6.1 (60,000 trials, 36,267 admissible cases, 0 failures; an independent program: 25,677 cases, 0 failures); random tests of Observations A
  (43,661 pairs) and B (19,976 instances), 0 failures.
- `independent_run_2/`: the programs and outputs of the second verification run (see below).

## Independent verification runs
Two independent verification runs (themselves AI-assisted) checked the note. They are not peer review.

**Run 1** re-read the source, re-derived the pairings step by step and with separately written code, re-proved Lemma 4.2 line by line, reran and extended the
enumerations (to n = 12, 13), reproduced the totals of Theorem 4.3, enumerated the filtrations of B17, read Kriebel's deposit, and searched the literature. Its required
fixes were applied before run 2: (1) the star S7 is not Kriebel's filtration; (2) "exactly two with 7 simplices" and "smallest examples found" for two and three paths;
(3) the sentence on the Hasse-diagram formulation of Theorem 1; (4) Theorem 6.1 as a corollary of algebraic Morse theory with citations, Observations A and B marked
"tested, not proved"; (5) bounded priority wording; (6) the dunce-hat remark as an argument.

**Run 2** (2026-10-02, after the writing pass) re-fetched the source and the slides, read Kriebel's deposit (Zenodo landing page through a fetch tool; the files of the
author's public mirror have the same md5 sums), searched the literature again, checked every proof line by line (Lemma 2.1, Proposition 4.1, Lemma 4.2, Theorem 5.1 with
F1–F3, Lemma 6.2 and Theorem 6.1) and wrote new programs from scratch (`independent_run_2/`):
- every explicit example (S7, Pi7, M17, M19, T17, B17, A17) with two pairing algorithms, including all reduction tables of the note, the eight cosets, the arrows of F1 and the
  Morse boundary of B17;
- all filtrations with n ≤ 13 simplices (Table 1) and Conjecture 3 (strict reading) for all 4,005,434 filtrations with n ≤ 11;
- its own generation of the isomorphism classes (1, 2, 6, 8 classes of sizes 13–16; 11 of size 17) and an exhaustive search of all 2,484,335,648 and all 30,582,025,120
  filtrations: no closed V_P-path below 17 simplices; at 17 only in the class of T17 (437,696 of 1,412,014,592);
- all orders of B17 and A17 with pairing P (1,852 and 302; 186 orders of B17 with adjacent V_P-pairs): none adjacent, none satisfying F1 (B17), none satisfying the block
  condition (B17 and A17);
- a random test of Theorem 6.1 on 92,955 cases (0 failures);
- a rerun of the package from an extracted copy of `source.zip` (15 Python runs and 8 C++ runs reproduce the saved outputs byte for byte).

| Item | Run 1 | Run 2 |
|---|---|---|
| Statement fidelity | CONFIRMED | CONFIRMED |
| Proofs | CONFIRMED | CONFIRMED (Lemma 4.2, Theorem 5.1, Theorem 6.1 and Lemma 6.2 line by line; no gap) |
| Computations | CONFIRMED | CONFIRMED_WITH_FIXES (all numbers reproduced except one: Table 1, n = 13 has 709,593,015 filtrations, not 709,593,014) |
| Answer as posed | CONFIRMED | CONFIRMED (negative; the first public refutation of Conjecture 2 is Kriebel's) |
| Novelty | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES (no earlier statement or refutation of Theorem 1 or Conjecture 3 found) |
| Presentation | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES |

Required fixes of run 2, all applied: (1) Table 1, n = 13: 709,593,015 (the C++ program of run 1, `cpp/enum_conj2.cpp`, admits at most 12 vertices and missed the one
filtration consisting of 13 vertices; the other columns are unaffected); (2) the hypothesis "the field of the earlier pairs is a gradient" added to the corrected statement in
the abstract and the introduction, which now also call it a consequence of algebraic Morse theory; (3) Remark 5.2: A17 fails also in the weak reading (computation), instead of
"not examined"; (4) Knudson's book (2015) listed among the sources not accessed, and the citing-items sentence corrected (the 2021 software paper listed by OpenAlex cites
lecture notes of the same title); (5) the Verification paragraph describes both runs; (6) this report updated; (7) README note on `enum_conj2.cpp`, run 2's code and outputs in
`independent_run_2/`, comment numbering in `cpp/run_cpp_checks.sh`; (8) Remark 3.2 credits Kriebel's all-fields check and relates his sequential remark to Observation A;
(9) minor: page range of the pairing description, Lemmas 3.3 and 3.5 of the Ripser paper, wording "a replacement of (K1) that holds in general".

## Relation to the literature, novelty and scope
- **Kriebel** (Zenodo, DOI 10.5281/zenodo.22929556, DataCite date 2026-09-23, unrefereed): refutes Conjecture 2 with the filtration a,b,c,d,cd,bd,ac (7 simplices); three boundary reductions;
  also over any field; no minimality claim; no statement on Theorem 1 or Conjecture 3; remarks that after reversing the path of the earlier pair the missing path appears. The Zenodo API
  answered scripted requests with HTTP 403; the record page (read through a fetch tool) lists paper.pdf, SHA256SUMS and source-and-verification.zip with the same md5 sums as the files in
  the author's public repository mirror, from which the text was read.
- **Searches (2026-10-02).** Crossref, OpenAlex, DataCite, zbMATH, the arXiv API and two web searches (one per run); queries on persistence pairs and discrete gradients, Knudson's abstract,
  gradient paths, Morse matchings, apparent pairs. OpenAlex lists three works citing the report: Kriebel's two Zenodo records and giotto-ph (2021), which in fact cites V. Nanda's lecture
  notes "Computational Algebraic Topology" (a title collision). No statement or refutation of Theorem 1 or Conjecture 3 was found.
- **Related correct results.** Bauer–Lange–Wardetzky (DCG 2012): on combinatorial surfaces persistence pairs can be cancelled in a nested order along unique paths (Lemma 9), and this fails for
  non-manifold 2-complexes and in higher dimensions (Sect. 1.1; example in Sect. 6.5). Bauer–Roll (SoCG 2024): in the reduction basis all persistence pairs form an algebraic gradient
  (Prop. 13); apparent pairs are persistence pairs and form a discrete gradient (Bauer, Ripser, Lemmas 3.3 and 3.5). Mischaikow–Nanda (DCG 2013): Morse matchings compatible with
  filtrations (not read beyond the abstract).
- **Caveats.** Not accessible or not read: the manuscript "Persistence and discrete Morse theory" of Knudson and Bauer (listed as in preparation on Knudson's pages), Bauer's 2011 thesis
  (HTTP 403 / bot protection, not circumvented), Knudson's book *Morse Theory: Smooth and Discrete* (World Scientific 2015; its zbMATH review describes the applications chapter as
  evasiveness and a homology reduction algorithm), textbook chapters on persistence and discrete Morse theory (Scoville 2019, Kozlov 2021), Mischaikow–Nanda. This negative search is not a
  proof of priority.
- **Scope.** The note answers statements of a workshop abstract. Conjecture 3 is under-defined in the source (the author's slides list the definition of P_V as future work) and the proof sketch of Theorem 1 is called a "gross oversimplification" there; the note refutes the weakest reading of Conjecture 3.
  Minimality is proved for Conjecture 2 (7) and Theorem 1 (17) only; for Conjecture 3 the least counterexample found has 17 simplices and the threshold is open (between 12 and 17).

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/