# Verification report — OWR-14299580-006 (Ferri: Drinfeld twists of set-theoretic Yang-Baxter solutions; must twist-related rack solutions coincide?)

Verification date: 2026-10-20.

**Verdict.** Status: the definite question answered negatively; the classification programme open.
- **The definite question is answered negatively.** Ferri asks (Oberwolfach Rep. 22 (2025), no. 4, 2633-2672,
  p. 2649) whether two rack solutions (X,r), (X,s) on the same set that are related by a Drinfeld twist must
  coincide. Literal equality of maps fails trivially, since isomorphic solutions are always twist-related (Ferri,
  arXiv:2504.21678v2, Remark 2.3); the intended reading (Q) is "must the two racks be isomorphic?". The note shows
  that they need not be: on X = Z/5 the rack solutions r(a,b) = (2b - a, a) of Aff(Z/5,2) and s(a,b) = (3a + 3b, a)
  of Aff(Z/5,3) satisfy s = F r F^-1 for the explicit Drinfeld twist with F(a,b) = (l + 2u^3, l + u^3), u = a - b,
  l = 2a + 4b, Psi = F12^-1 theta, Phi = F23^-1 theta, theta = D_s^-1 (theta_W x id) D_r (Theorem 1.2); the two
  quandles are not isomorphic, since Aff(Z/5,3) is commutative and Aff(Z/5,2) is not (0 |> 1 = 2, 1 |> 0 = 4).
- **Infinite family (Theorem 1.3).** For a prime p and t with t != 0, t^2 != 1, 1 + t + t^2 != 0 and p not dividing
  |G_t| (G_t = <S1, S2> in GL2(F_p)), Aff(Z/p,t) and its dual Aff(Z/p,1/t) are non-isomorphic with twist-related
  rack solutions; the hypotheses hold whenever -t has order 3, 4 or 5 (Proposition 4.5), e.g. (5,2), (7,3), (11,2),
  (13,4), and t = -omega for every prime p = 1 mod 3. Proof: Lemmas 4.1-4.4 (Lemma 4.1 classical; Lemma 4.4 by
  Burnside's theorem on marks, cited) and the orders 12, 24, 60 of the von Dyck groups D(2,3,k) (cited).
- **Exact computations (kept separate from the theorems).** 5 is the smallest order of a counterexample, and at order
  5 the pair of Theorem 1.2 is the only one; order 6: 353 racks in 338 twist classes (13 of size 2, one of size 3),
  16 unordered pairs of non-isomorphic twist-related racks, 15 of them not dual pairs, 5 of them pairs of quandles;
  every rack of order <= 6 is twist-related to its dual, but Aff(Z/7,2) is not (likewise (11,3), (13,2));
  the B4-sets on X^4 separate the pair of Theorem 1.2, while those of Aff(Z/7,3), Aff(Z/7,5) are isomorphic.
- **Not claimed:** the classification of solutions up to Drinfeld twists (the programme of the record) is not
  addressed; whether Theorem 1.3 lists all twist-related affine pairs {t, 1/t} is open (true for p <= 31 by
  computation).

The note is unrefereed. Site result type: COMPLETE_COUNTEREXAMPLE (scoped to the definite question).

## Statement checked
- **Primary source.** D. Ferri, "Drinfeld twists for solutions to the Yang-Baxter equation", extended abstract in
  Oberwolfach Report 49/2025, *Mini-Workshop: The Yang-Baxter Equation and Representations of Braid Groups*
  (organisers I. Colazzo, J. Plavnik, E. Rowell, L. Vendramin), Oberwolfach Rep. 22 (2025), no. 4, 2633-2672
  (issued 2026-03-04 according to Crossref), Ferri's abstract pp. 2645-2649, doi:10.4171/OWR/2025/49. Read in the
  publisher's open-access PDF (text extraction) for this note and by both verification runs.
  - p. 2646: Drinfeld twist = bijections F of X^2 and Phi, Psi of X^3 with F12 Psi = F23 Phi, Psi r12 = r12 Psi,
    Phi r23 = r23 Phi; then F r F^-1 is a solution. Proposition: for a bijection F with F r F^-1 a solution, the
    B3-representations rho_r and rho_{FrF^-1} on X^3 are isomorphic iff F extends to a twist.
  - p. 2648: "Problem. Classify all solutions in Set up to Drinfeld twists"; rack solution = left
    non-degenerate solution with a <- b = a; every right non-degenerate solution can be twisted into a rack
    solution.
  - p. 2649: the question (rack solutions related by a Drinfeld twist: must they coincide?), called open and
    hard to check computationally; a positive answer would imply that racks classify solutions up to twists.
- **Paper version of the definitions.** Ferri, arXiv:2504.21678v2 (23 April 2026) = Ann. Mat. Pura Appl. (online
  first 13 July 2026), doi:10.1007/s10231-026-01703-2: Definition 2.1 ((dt1)-(dt3)), Lemma 2.2 (composition and
  inverse of twists), Remark 2.3 (isomorphic solutions are twist-related, F = f x f, Psi = id x id x f,
  Phi = f x id x id), Convention 2.4, Proposition 2.9 (twist structures on F = isomorphisms of the
  B3-representations), Lemma 3.9, Remark 4.22. All theorem numbers of the note refer to arXiv v2.
- **Corpus record.** UnsolvedMath OWR-14299580-006: "Classify set-theoretic Yang-Baxter solutions up to Drinfeld
  twists. In particular, if two rack solutions on the same set are related by a Drinfeld twist, must they
  coincide?"; label open.

## Readings
- "Coincide" = equal maps: answered negatively and trivially by Ferri's Remark 2.3 (Remark 2.4 of the note: the
  transposition (0 1) relabels Aff(Z/5,2) into a different map by a twist).
- "Coincide" = isomorphic (reading (Q)): the only reading under which a positive answer would make racks classify
  solutions up to twists (one rack per twist class). For rack solutions, isomorphism of solutions = isomorphism of
  racks. Answered negatively by Theorems 1.2 and 1.3.
- Left versus right rack conventions (report: r(a,b) = (a |> b, a); Ferri, Remark 4.22: (b, a <| b)): conjugation by
  the flip and the reversal of X^3 carries twists to twists (Lemma 2.5); the answer does not depend on the
  convention. Racks versus quandles: the counterexamples are quandles.
- Not the question: D-isomorphisms (Ferri, Definition 2.5), group Drinfeld twists of braided groups, the
  classification programme itself.

## Results in the paper
- Proposition 2.2 (Ferri, Proposition 2.9, direction used; re-proved, not new) and Corollary 2.3 (finite X:
  twist-related iff the B3-sets on X^3 are isomorphic; the cycle type of r on X^2 is forced).
- Theorem 1.2 (Section 3): steps (a)-(f); Table 1 with the twelve identities S_i'(T(w)) = d^3 T(w_2).
- Theorem 1.3 (Section 4): Lemma 4.1 (Aff(Z/p,t) determines t; classical, a case of Nelson's classification),
  Lemma 4.2 (splitting W_t x F_p), Lemma 4.3 (P S_i^-T = S_i(1/t) P), Lemma 4.4 (coprime case, via Burnside's
  theorem on marks, proof sketch), Proposition 4.5 (ord(-t) in {3,4,5}; the identification of
  <x, y | x^2, y^3, (y^-1 x)^k> with D(2,3,k) via y' = y^-1, and |G_t| divides (p-1)|D(2,3,k)|).
- Section 5 (exact computations): Remark 5.1, Proposition 5.2, Remark 5.3.

## Computations (sanity checks; programs and outputs in reproducibility/)
- `check_note.py` (standard library; exact): 45 checks, 0 failures, about 9 s. It covers every identity of the proof
  of Theorem 1.2 on all points of X^2 and X^3 (Table 1 as printed, (dt1)-(dt3) literally, all 120 bijections),
  Remark 2.4 and Lemma 2.5 on this example, Lemmas 4.1-4.3 on small cases (Lemma 4.3 as an identity of rational
  functions, by evaluation of degree-2 polynomial identities at 8 points), Proposition 4.5 on all 46 pairs (p,t)
  with p <= 61, explicit twists for six pairs (p,t), Remark 5.1, Proposition 5.2 (own enumeration of the 1, 2, 13,
  114, 1708, 36538 labelled racks of order <= 6; all 16 pairs of order 6 certified) and Remark 5.3 (own B4
  computation, written for the note before the remark was included).
- The programs with which the results were first obtained (`original/`) and the programs of the three verification
  runs (`verification_run_A/`, `verification_run_B/`, `independent_run_2/`) are included with their recorded
  outputs. A complete re-run
  from a copy of the package on 2026-10-20 reproduced every recorded output (identical, or identical except
  timings; `RERUN_LOG.txt`). In the copies of `original/cert_z5.py` and `original/enum_racks.py` the output file is
  written into the current directory instead of the program's directory (README).

## Independent verification runs
The results were first obtained, with proofs, by the programs in `original/`. Three independent verification runs,
all AI-assisted, followed on 2026-10-20, each with its own programs; each re-ran the original programs and
reproduced their recorded outputs. Runs A and B examined the first written version of the result; the third run
examined the final text of the note (section "Third run" below).

| Item | Run A | Run B |
|---|---|---|
| Statement against the source (definitions, composition order, rack solution, the question, reading of "coincide") | CONFIRMED (report PDF and Ferri arXiv v2 read) | CONFIRMED (both sources downloaded and read independently) |
| Proposition 2.2, Corollary 2.3, Remark 2.4, Lemma 2.5 | PROVED (re-derived) | PROVED |
| Theorem 1.2 | PROVED step by step (all 12 rows of Table 1 recomputed); CONFIRMED by a generic orbit search not using the table, the closed formulas and the recorded tables | PROVED; CONFIRMED by its own twist (cycle and orbit matching) and the closed formulas re-implemented from the text; negative controls rejected |
| Theorem 1.3, Lemmas 4.1-4.4, Proposition 4.5 | PROVED (all lemmas and the von Dyck argument re-derived); explicit twists for the six pairs with p <= 13 | PROVED (cited facts: Burnside's marks, von Dyck orders); scan for all p <= 31 with explicit twists; Lemmas 4.2, 4.3 symbolically over Q(t) |
| Remark 5.1 | CONFIRMED (49/7, 121/11, 169/13) | CONFIRMED |
| Proposition 5.2 | CONFIRMED by its own enumeration (16 pairs certified, 5 of quandles) | CONFIRMED by its own enumeration (16 pairs certified); invariant self-test on random relabellings of all 353 racks of order 6 |
| Remark 5.3 (B4) | — | found and computed (Z/5 pair separated; Aff(Z/7,3), Aff(Z/7,5) isomorphic) |
| Literature | arXiv search, one web search | Semantic Scholar citations (one citing paper), Crossref, arXiv, zbMATH; OpenAlex unavailable |

Neither run found a mathematical error. Required corrections (all made): the order-6 count (16 unordered pairs inside the 14 classes, 15 of them not dual
pairs; the first program had certified the 15 pairs formed with the first member of each class, both runs and
`check_note.py` certified all 16); the identification with the von Dyck group in Proposition 4.5(d) and the
divisibility |G_t| | (p-1)|D(2,3,k)|; the status wording "the definite question answered negatively; the
classification programme open" everywhere; the published versions of Ferri's paper and of Lebed-Vendramin in the
bibliography; the novelty paragraph with the citation search (Semantic Scholar: one citing paper; OpenAlex
unavailable; Crossref: no citations of the published paper).

### Third run (on the final text of the note)
An AI-assisted verification run examined the final text of the note line by line on 2026-10-20, before any change,
with programs of its own written from the text (`independent_run_2/`, README there).

| Item | Verdict |
|---|---|
| Statement against the sources (OWR 49/2025 pp. 2645-2649 re-downloaded and re-read; Ferri arXiv v2 Sect. 2, 3.2, Remark 4.22; Lebed-Vendramin v2 Def. 1.1, Examples 1.2-1.5, Thm. 1.8, Cor. 1.9; Ghobadi Sect. 2; Martin-Rowell-Torzewska v3 (7.3), Example 7.28; Crossref data of all DOIs) | CONFIRMED; the record wording agrees with the source; scope stated correctly |
| Reading of "coincide", rack solutions = left racks, Lemma 2.5 (flip, reversal) | CONFIRMED (re-derived) |
| Proposition 2.2, Corollary 2.3 (both directions, cycle types) | PROVED (re-derived); precision fix: bijective solutions |
| Theorem 1.2 | PROVED (all constants re-derived: l_r, l_s, D^-1, S1, S2, S1', S2', C_s^-1, (3u)^3 = 2u^3); all 12 rows of Table 1 recomputed and identical to the printed table; own program: (dt1)-(dt3), F r F^-1 = s, bijectivity on all points, no isomorphism among 120 bijections |
| Theorem 1.3, Lemmas 4.1-4.4, Proposition 4.5 | PROVED (re-derived; P S_i^-T = S_i(1/t) P multiplied out; von Dyck identification via y' = y^-1); own program: Lemmas 4.2, 4.3 over Q(t), abs(G_t) = 96, 24, 600, 600, 24, 96 recomputed, all 46 pairs with p <= 61, coset enumeration 12, 24, 60 |
| Remark 5.1 | CONFIRMED (49/7, 121/11, 169/13; G_t = GL2 for (11,3), (13,2); scan p <= 19) |
| Proposition 5.2 | CONFIRMED by an own enumeration in C (1, 2, 13, 114, 1708, 36538 labelled racks) and own B3-set code: 73 and 338 twist classes, 16 pairs at n = 6, one dual pair, 5 quandle pairs, all 16 certified |
| Remark 5.3 | CONFIRMED (125-point orbit, 25 versus 5 maps; Aff(Z/7,3), Aff(Z/7,5) isomorphic) |
| Package | re-run in full from an extracted copy of `source.zip`: all recorded outputs reproduced, up to running times (`independent_run_2/rerun_package_log.txt`) |
| Literature | arXiv API (10 queries), zbMATH (8 documents), Crossref, one web search: no answer to the question found; Semantic Scholar HTTP 429; OpenAlex: no citing work of the published Ferri paper, search HTTP 429 |

No mathematical error was found. Required corrections (all made): bijectivity in Definition 2.1 and Corollary 2.3;
the attribution, as in Ferri's Remark 2.3, of the observation "isomorphic solutions are twist-related" to Ghobadi;
the updated account of the searches; the description of this run in the note and in this report; the scan
statement of Remark 5.1 ("the verification runs found the same, one of them up to p = 31") and the Crossref issue
date of the report in the bibliography. An observation that changes nothing in the note: Lemma 3.11 of Ferri's
arXiv v2 (not cited by the note) seems, as literally stated, to need an extra hypothesis (for F = id, or p = q, it
conflicts with Remark 2.3); the published version was not compared.

## Relation to the literature, novelty and scope
- **Credits.** The question, the definitions ((dt1)-(dt3), rack solution) and the B3 criterion are Ferri's
  (OWR 49/2025; arXiv:2504.21678v2 = Ann. Mat. Pura Appl. 2026): Definition 2.1, Lemma 2.2, Remark 2.3 (the literal
  reading fails), Proposition 2.9 (twist structures = B3-set isomorphisms, used as Proposition 2.2 of the note,
  re-proved, not new). Ferri credits the definition of twists to Kulish-Mudrov in Ghobadi's formulation, the
  observation of his Remark 2.3 for set-theoretic solutions to Ghobadi (arXiv:2105.03286, Sect. 2) and the proof of
  his Lemma 2.2 to Ghobadi's Theorem 2.2; the note says so. Affine (Alexander) quandles (Joyce 1982; Fenn-Rourke 1992) and Lemma 4.1 (a case of Nelson's
  classification of finite Alexander quandles, Topology Proc. 27 (2003)) are classical. Burnside's theorem on marks
  (tom Dieck, Transformation Groups, 1987) and the orders of D(2,3,k) (Coxeter-Moser, 4th ed., 1980) are cited. Rack
  and quandle counts agree with OEIS A181770 and A181769 (checked on oeis.org on 2026-10-20).
- **Related work (none settles the question).** Lebed-Vendramin, J. Algebra 607 (2022) 360-380 (arXiv:2008.01752):
  k-derived solutions have isomorphic braid actions (Theorem 1.8, Corollary 1.9); their Examples 1.4 and 1.5 of
  non-isomorphic solutions with isomorphic actions are not pairs of rack solutions. Ghobadi, arXiv:2105.03286:
  twists between skew braces come from families of isomorphisms of the additive groups. Ferri, Lemma 3.9
  (permutation solutions) and Remark 4.22 (group Drinfeld twists between braided groups of rack type are of type
  (I)) do not cover the example (Aff(Z/5,2) is not a permutation rack; no braided group is involved).
  Martin-Rowell-Torzewska, arXiv:2506.07950 (v3): the only citing paper of Ferri's arXiv paper listed by Semantic
  Scholar; it cites Proposition 2.9 for the equivalence of 3-equivalence with the existence of a Drinfeld twist,
  asks whether 3-equivalence implies infinity-equivalence for linear braid representations, and sorts the 73
  solutions on a 3-element set in Example 7.28; it does not settle the question.
- **Searches.** arXiv API (Ferri's papers; "Drinfeld twist" with rack, quandle, set-theoretic, braid group; Lebed or
  Vendramin with twists; guitar maps; Ghobadi; recent "Drinfeld twist" abstracts), zbMATH, Crossref (no citations of
  the published Ferri paper, none of the report), Semantic Scholar (citations of arXiv:2504.21678, obtained in run B;
  repeated requests for the note and in the third run were refused with HTTP 429), OpenAlex (the record of the
  published Ferri paper lists no citing work; the search was refused with HTTP 429), a few web searches; repeated
  on 2026-10-20 by the third run (ten arXiv API queries, zbMATH, Crossref, one web search) with the same outcome.
  No printed answer to the question was found in the sources searched; no source states that two non-isomorphic
  racks can be twist-related. A search that finds nothing is not a proof of novelty: the phenomenon may be known in
  the language of Hurwitz actions or colouring invariants, which were searched by keywords only. No priority claim.
- **Scope.** The note answers only the definite question (both readings), plus Theorem 1.3, Remarks 5.1, 5.3 and the
  data of Proposition 5.2. The classification programme of the record stays open.

## 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/