# Verification report — OWR-12697693-003 (Colazzo, with Ferrara and Trombetti: do the elements with property (s) of a skew brace form an ideal?)

Verification date: 2026-10-11.

**Verdict.** The note gives a **complete negative answer** to the question as it is posed; every statement it
makes is proved, and the scope is as follows.
- **Settled.** The elements with property (s) of a skew brace need not form an ideal, already for braces. In the
  left brace B_A = Z × Z/4 with (m,a) + (n,b) = (m+n, a+b) and (m,a)∘(n,b) = (m + ε(a)n, a + b + 2ab), where
  ε = +1 on {0,1} and −1 on {2,3}, the set of these elements is Z × {0,1}. The element x = (0,1) has property
  (s), with both indices of the definition equal to 2, while x + x = (0,2) and −x = λ_x(x) = (0,3) have not. So
  the set is not a subgroup of the additive group and not λ-invariant (Theorem 1.2). A sub-brace with additive
  group Z × Z/2 has the same property.
- **Positive results.** In every skew brace the elements with property (s) form a normal subgroup of the
  multiplicative group (Theorem 1.3). They form an ideal in two-sided skew braces, in braces with torsion-free
  additive group, in bi-skew braces, and in skew braces whose multiplicative group is an FC-group (Theorem 1.4).
- **Not new, or close to the literature.** The question, the notion and the positive answer for two-sided braces
  are due to Colazzo, Ferrara and Trombetti; the brace on Z/4 underlying the example is Example 3.8 of their
  paper; the semidirect product of skew braces is a known construction.
  - That the elements with property (s) form a subgroup of the multiplicative group follows at once from
    Corollary 5.4 of Di Matteo–Ferrara (arXiv:2607.20222, a preprint) with Proposition 5.8 of
    Cascella–Properzi–Van Antwerpen (Mediterr. J. Math. 2026; arXiv:2603.06177).
  - It also follows from Proposition 2.3 of Mondal–Yadav (J. Algebra 712 (2027); arXiv:2603.16771; arXiv
    numbering) on the set FCI(B) of the elements whose brace centraliser has finite index, together with the
    equality FCI(B) = S(B), which is Remark 4.4 of the note. Mondal and Yadav also prove that FCI(B) is an ideal in
    two-sided braces, and state that it is a sub-skew brace in two-sided, bi-skew and λ-homomorphic skew braces.
  - Theorem 1.4(a) for skew braces of non-abelian type is a short consequence of the corollary of Di
    Matteo–Ferrara with Proposition 5.9 and Remark 2.4 of Cascella–Properzi–Van Antwerpen.
  - The proofs in the note are included for completeness. No priority is claimed.
- **A remark on the source paper.** The brace B_A satisfies the hypothesis of Lemma 3.12 of the paper of Colazzo,
  Ferrara and Trombetti and not its conclusion (Remark 8.2 of the note). The lemma holds for skew braces with
  property (S) (Proposition 8.1), which is the case in which it is used in that paper; as far as checked, the
  results that rely on it are not affected.
- **Not treated.** Questions 2 and 3 of the same abstract. It is not known whether the answer is positive for
  finitely generated left nilpotent skew braces. λ-homomorphic skew braces are not treated. The set of elements
  with property (s) of the structure skew brace in Theorem 1.6 is not determined.

The note is unrefereed.

## Statement checked
- **Primary source.** I. Colazzo (joint work with M. Ferrara and M. Trombetti), "Derived-indecomposable solutions
  and skew braces whose elements have a finite number of conjugates", abstract in: Mini-Workshop: Skew Braces
  and the Yang–Baxter Equation, Oberwolfach Rep. 20 (2023), no. 1, Report No. 9/2023, pp. 537–563,
  doi:10.4171/OWR/2023/9; the abstract is on pp. 543–544.
  - Read in the open file of the publisher (28 pages); page 543 was also looked at as an image.
  - Question 1 is posed for an arbitrary skew brace B and asks whether the set of all elements with property (s)
    is an ideal. The next sentence says that the answer is affirmative for two-sided skew braces of abelian type.
  - The definition is given in the display "(s)" of the abstract: both indices |(B,+) : Fix^r(x) ∩ C_x^+| and
    |(B,∘) : Fix^l(x) ∩ C_x^∘| are finite. In the line after the display the set Fix^l(x) is printed without
    its superscript.
  - The abstract cites the arXiv preprint of the paper below with the numbering of its first version, in which
    property (s) is Definition 3.5.
- **The paper behind the question.** I. Colazzo, M. Ferrara, M. Trombetti, "On derived-indecomposable solutions
  of the Yang–Baxter equation", Publ. Mat. 69 (2025), 171–193, doi:10.5565/PUBLMAT6912508; arXiv:2210.08598
  (two versions; the second has the numbering of the journal version).
  - Section 2.1: skew (left) braces, λ, a∗b = −a + a∘b − b, left ideals, ideals, socle, annihilator.
  - Section 3: Fix^r(x), Fix^l(x); Definition 3.7 (property (s), property (S)); Example 3.8; Notation 3.9
    (conjugates); Proposition 3.10 (in a two-sided brace the elements with property (s) form an ideal; proof
    sketched); the sentence after it, by which the authors could not decide the case of skew braces;
    Lemma 3.12.
- **Corpus record.** ulamai/UnsolvedMath, OWR-12697693-003 (dataset version 1.6.0; upstream status `open`). The
  statement field of the record contains the question followed by the title and the author line of the abstract
  (an extraction artefact); the source decides.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The question as posed: arbitrary skew braces | no | Theorem 1.2 |
| Braces (skew braces of abelian type) only | no; B_A is a brace | Theorem 1.2 |
| Two-sided skew braces (abelian type or not) | yes | Theorem 1.4(a) |
| "Ideal" weakened to "normal subgroup of the multiplicative group" | yes, in every skew brace | Theorem 1.3 |
| "Ideal" replaced by "left ideal", "strong left ideal" or "sub-skew brace"; the definition of an ideal in Guarnieri–Vendramin | no | Theorem 1.2, Remark 3.2(iv) |
| Right skew braces; the opposite skew brace; another order of the terms in a∗b; the roles of Fix^r and Fix^l exchanged | no, by the same example | Remark 3.2(i), (ii) |
| The definition by finitely many conjugates of the four kinds, or by only two of the four kinds | the same set in B_A | Lemma 2.1, Remark 3.2(iii) |
| The definition by the index of the brace centraliser (the set FCI of Mondal–Yadav) | the same set in every skew brace | Remark 4.4 |
| Structure skew braces of finite non-degenerate solutions of the Yang–Baxter equation | no in general; yes for involutive solutions | Theorem 1.6; Remark 5.3 |
| Finite skew braces | every element has property (s); nothing to prove | Section 1.1 |

## Results in the paper
- **Theorem 1.2.** The brace B_A and its sub-brace C = {(m,a) : a ≡ m mod 2}, with (C,+) ≅ Z × Z/2. The proof
  consists of the verifications (A1)–(A6): (Z/4, ∘) is a Klein four-group; ε is a homomorphism of it; (B_A, ∘)
  is a group; the left distributive law; elements with a ∈ {2,3} do not have property (s); elements with
  a ∈ {0,1} have it.
- **Remarks 3.1–3.3.** (B_A, ∘) ≅ D_∞ × C_2; B_A is not two-sided; independence of the conventions; B_A is
  finitely generated, right nilpotent (not left nilpotent), soluble, of multipermutation level 3, with socle of
  index 4.
- **Theorem 1.3.** S(B) is a normal subgroup of (B,∘). The proof uses the group G = (B,+) ⋊ (B,∘) (Lemma 2.3) and
  the identity c = b + λ_b(x) − b − c∗b for c = b∘x∘b⁻¹.
- **Proposition 4.3** (= Corollary 5.4 of Di Matteo–Ferrara; a proof is included, which is a variant of theirs;
  both use Neumann's lemma on coverings by cosets, Lemma 4.2): x has property (s) if and only if [x]_+, [x]_λ and
  [x]_∘ are finite.
- **Remark 4.4.** The set FCI(B) of Mondal–Yadav is equal to S(B) for every skew brace B (with proof).
- **Theorem 1.4.** (a) two-sided skew braces (Lemma 5.1 and Steps 1–5; a shorter derivation in Remark 5.2);
  (b) braces with torsion-free additive group, where S(B) is the set of the elements of the socle with finite
  λ-orbit; (c) bi-skew braces; (d) skew braces whose multiplicative group is an FC-group.
- **Lemma 6.1, Theorem 1.5.** In V ⋊_α Q (V abelian, Q a skew brace, α a homomorphism (Q,∘) → Aut(V)), the
  element (v,q) has property (s) if and only if q has it in Q, (α_q − 1)V is finite and the orbit α(Q)v is
  finite. For finite Q, torsion-free V and Ker α = M: S = V × M.
- **Examples 6.2–6.6 and Table 1.** B_A and variants; B_B (a normal subgroup of both groups, not λ-invariant);
  B_C (a left ideal, not normal in the additive group); B_T (torsion-free non-abelian additive group; a subgroup
  of index 6, neither normal nor λ-invariant); B_U (centrally nilpotent, left and right nilpotent, exponent 4;
  not closed under addition).
- **Theorem 1.6.** A six-element bijective non-degenerate solution (X, r), not involutive, such that in the
  structure skew brace G(X, r) the image of a_0 has property (s) and one of its additive conjugates has not. The
  proof relies on Theorems 3.1 and 3.9 and Corollary 3.3 of Guarnieri–Vendramin (the solution attached to a skew
  brace; existence and universal property of the structure skew brace); the proof of Theorem 3.9 was not checked.
- **Proposition 8.1, Remark 8.2.** If (B,+) is generated by x_1, …, x_n with [x_i]_+ and [x_i]_λ finite and all
  sets T_t are finite, then B/Ann(B) is finite; in particular for skew braces with property (S) and finitely
  generated additive group. Remark 8.2 writes out that B_A (and C) satisfies the hypothesis of Lemma 3.12 of the
  source paper and not its conclusion, names the step of the printed proof that needs an additional hypothesis,
  and lists the places where the lemma is used.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. A skew brace structure on a fixed group is called labelled.
- **Programs with which the results were first obtained** (`original/`, standard library).
  - Enumeration: 477 labelled skew braces of order 2 to 12; 1, 1, 4, 1, 6, 1, 47, 4, 6, 1, 38 isomorphism
    classes, of which 1, 1, 4, 1, 2, 1, 27, 4, 2, 1, 10 of abelian type (the published numbers).
  - Identities (E1)–(E5), (G1)–(G3) and the inclusions of the last step of the proof of Theorem 1.3 on all 477;
    (T1), (T2), the displays and the inclusions of the proof of Theorem 1.4(a) on the two-sided ones (110 of them
    with non-abelian additive group). Control: the inclusions fail on 130 of the 164 labelled skew braces that
    are not two-sided.
  - B_A, B_B, B_C: axioms as identities between integer linear forms for all 64, 512, 216 triples of
    Q-coordinates (exact); the set S (exact); finite quotients.
  - The pairs (Q, M): 3139 pairs, M not an ideal in 962; Theorem 1.5 confirmed exactly for the 2271 pairs with
    |Q| ≤ 8.
  - The finite data of Theorem 1.6 (table of r, braid relation on 216 triples, the facts (6)).
- **Verification run A** (`verification_run_A/`, standard library): its own library and enumeration (477; the
  same numbers of classes; 313 two-sided, 110 of them with non-abelian additive group); the identities and
  inclusions; the survey (3139; 962); B_A, B_B, B_C exactly, and Theorem 1.5 for the 2271 pairs; B_T exactly
  (1728 triples of residue classes); truncations of B_U of order 16, 64, 256, 1024; bi-skew braces (259 of the
  477; 235 both bi-skew and two-sided); the map f of Remark 8.2 in Z/N × Z/4 for N = 3, 5, 9, 15 (four values,
  annihilator zero); the data of Theorem 1.6.
- **Verification run B** (`verification_run_B/`, sympy and numpy): its own enumeration (498 labelled skew braces
  of order at most 15 in 119 classes; the 357 classes of braces of order 16); identities, Lemma 2.1, Lemma 2.3
  and quantitative forms of Theorem 1.3 on these 855 skew braces (control: the naive equality fails on 147 of
  the 498 and on 42 of the 357); Lemma 5.1 and the inclusions on the 333 + 221 two-sided ones; B_A symbolically
  and exactly (four subgroups, two indices, all types of elements), finite quotients for 28 values of N; the
  survey (3139; 962), exact decision of membership in S(Z[Γ] ⋊ Q) for all 3139 pairs and for 5801 pairs of
  order 16 (2274 with M not an ideal); B_B, B_C, the data of Theorem 1.6 and further infinite examples on boxes.
- **Verification run 2** (`independent_run_2/`, sympy and numpy; about 20 seconds; every program ends with ALL
  CHECKS PASSED): six programs and a library of its own.
  - B_A from its two operations, with a symbolic first coordinate: the axioms (64 triples), the formulas (2), the
    four families of every element and their finiteness, S(B_A) = Z × {0,1}, the subgroups of Definition 1.1 for
    (0,1), the sub-brace C, centre, socle and annihilator, the map f of Remark 8.2 (four values), the series of
    Remark 3.3, the statements of Remark 3.2, and the quotients Z/N × Z/4 for seven values of N.
  - Its own enumeration: 477 labelled skew braces, the same numbers of classes, 313 two-sided (110 with
    non-abelian additive group; (T1), (T2) fail on each of the 164 others), 259 bi-skew, 235 both; the identities
    of Lemmas 2.1–2.3 and 5.1 and of the proof of Proposition 8.1; the pairs (Q, M): 3139, 962, 2271; the first
    occurrences of each kind of failure (orders 4, 6, 6, 8).
  - V ⋊ Q with V = Z^d by affine forms in generic vectors: B_A, B_B, B_C (axioms, formulas (5), the set S, the
    witnesses), and Theorem 1.5 for all 3139 pairs; in all of them the set FCI of Remark 4.4 is the set S.
  - B_T by residue classes (1728 triples); truncations of B_U of order 16, 64, 256.
  - The six-element solution: the table of r computed from the skew brace B_C, the facts (6), the braid relation
    on 216 triples, the derived solution; and a second skew brace containing the same solution (a consistency
    test), in which a_0 has property (s) and a_2 has not.
- **Re-runs.** Runs A and B ran the original programs again in a copy: identical outputs, apart from lines that
  record running times. On 2026-10-11 all programs of the package were run again from its folder layout, and
  once more by run 2 from an extracted copy of the archive, with the same result; see `reproducibility/README.md`
  and `reproducibility/RERUN_LOG.txt`.
- The package also contains programs that test a further family of examples (skew braces of exact
  factorizations of groups), which was part of the first written version and is not included in the note.

## Independent verification runs
The results were first obtained with complete proofs and with the programs in `original/`. Three independent
verification runs, all AI-assisted, followed (2026-10-11), each with its own programs. Runs A and B examined the
first written version of the results, and neither used the report of the other. Run 2 examined the text of the
note.

| Item | Run A | Run B | Run 2 (text of the note) |
|---|---|---|---|
| Question, definitions, conventions, numbering against the sources | CONFIRMED_WITH_FIXES (the comparison with Lemma 3.12 of the source paper was missing) | CONFIRMED | CONFIRMED (sources fetched again; one sentence on the numbering made exact) |
| Theorem 1.2 (B_A and the sub-brace C) | CONFIRMED (re-derived from the definitions; exact program; conventions examined) | CONFIRMED (re-derived from the statement before reading the original programs; symbolic and exact programs) | CONFIRMED (re-derived line by line; exact program of its own) |
| Lemmas 2.1–2.3, Theorem 1.3 | CONFIRMED | CONFIRMED | CONFIRMED |
| Theorem 1.4(a), (b) | CONFIRMED (hypotheses sharp) | CONFIRMED; shorter derivation of (a) found (Remark 5.2) | CONFIRMED |
| Lemma 4.2, Proposition 4.3, Theorem 1.4(c), (d) | contributed by run A, with proofs (Lemma 4.2 by name only) | not examined | CONFIRMED (the proof of Lemma 4.2, which was added at the writing, included) |
| Lemma 6.1, Theorem 1.5, Examples 6.3, 6.4 | CONFIRMED | CONFIRMED | CONFIRMED |
| Examples 6.5, 6.6 | contributed by run A (6.5 with an exact program; 6.6 with tests on truncations) | not examined | CONFIRMED (re-derived; programs of its own) |
| Theorem 1.6, given the cited results of Guarnieri–Vendramin | CONFIRMED | CONFIRMED | CONFIRMED |
| Proposition 8.1 | — | (a) proved by run B | CONFIRMED in the general form and in case (b), which were added at the writing |
| Lemma 3.12 of the source paper against B_A (Remark 8.2) | found; correction required | found independently; correction required | CONFIRMED in every part: the lemma and its proof in the journal version and both arXiv versions; B_A and C; the step in the arXiv version of Jespers et al.; all seven places where the source cites the lemma |
| The programs | original programs re-run, identical outputs | re-run, identical outputs; two cosmetic corrections required | whole package re-run from the archive, identical outputs |
| Novelty and credit | no source found that answers the question; not examined beyond the sources read | CONFIRMED_WITH_FIXES: no source answers the question or contains the example; closeness of Theorems 1.3 and 1.4(a) to the literature to be stated | no source found (searches repeated); CONFIRMED_WITH_FIXES: the set FCI of Mondal–Yadav is equal to S(B), so that their Proposition 2.3 is a second source for the subgroup part of Theorem 1.3 |

No run found a wrong theorem, proposition, lemma or example, or a gap in a proof.

**Corrections required by runs A and B**, all applied in the note or in the package:
1. (Runs A and B) The comparison with Lemma 3.12 of the source paper: Remark 8.2, Proposition 8.1, one sentence
   of the introduction and of the abstract.
2. (Run B) The relation to the literature: that S(B) is a subgroup of (B,∘) follows at once from the two results
   named in the verdict, and Theorem 1.4(a) for non-abelian type follows in a few lines from them; that (T1) is
   Remark 2.4 of Cascella–Properzi–Van Antwerpen: abstract, Section 1.3, Remark 5.2, the sentence after Lemma 5.1,
   "Scope and priority".
3. (Run B) The limits of the literature search (one citing paper seen in abstract only; Guarnieri–Vendramin read
   in the arXiv version, whose numbering is used): Section 1.3, "Scope and priority".
4. (Run B) Two cosmetic corrections of the original programs (the number of a definition in a comment; an
   assertion instead of a printed comparison): made in `reproducibility/original/scripts/`.
5. (Run B) Two file paths in the first written version: not applicable to the note.
6. (Run A, recommended) The statement of Theorem 1.4(b) says "brace", and Example 6.5 shows that it does not
   extend to skew braces; the brace on Z/4 is credited as Example 3.8 of the source paper and the semidirect
   product as a known construction; the misprint in the report and the numbering of the first arXiv version are
   mentioned; the two further positive classes are Theorem 1.4(c), (d).

**Added when the note was written, after runs A and B** (examined by run 2, see the table): the proof of
Neumann's lemma (Lemma 4.2); the formulation of Proposition 8.1 with hypotheses on the generators and on the
sets T_t, and its case (b); the first paragraph of Section 8; in Remark 8.2 the list of the places where the
source paper cites its Lemma 3.12. All proofs were written out again for the note. The family of examples from
exact factorizations, which runs A and B had confirmed, was left out for reasons of length.

**Run 2: what it examined, and the result.**
- The sources were fetched again: the report; the source paper in the journal version and both arXiv versions
  (the second read in full); the arXiv versions of the papers of Jespers et al., Di Matteo–Ferrara,
  Cascella–Properzi–Van Antwerpen (both versions), Guarnieri–Vendramin and Mondal–Yadav. Every statement quoted
  from them in the note was compared with the text, and every number of a theorem, definition or example was
  counted or read. Result: correct; see correction 5(a).
- All proofs were re-derived line by line, the four additions first. Result: correct.
- Remark 8.2 was checked part by part (table above). Result: correct as worded; nothing had to be withdrawn.
- The hypotheses of Theorems 1.3 and 1.4 were tested just outside: B_A is not two-sided, not bi-skew and not
  λ-homomorphic, the FC-centre of its multiplicative group has index 2; C has torsion of order 2; B_T is
  torsion-free of non-abelian type. No statement claims more than is proved.
- The finite claims were recomputed with six programs of its own, and the package was run again from an
  extracted copy of the archive (see "Computations").
- The literature search was repeated (see below).

**Corrections required by run 2** (ten), all applied:
1. The paper of Di Matteo–Ferrara is referred to as a preprint, without a further qualification.
2. Credit to Mondal–Yadav: Remark 4.4 (the equality FCI(B) = S(B), with proof) and the corresponding sentences
   in the abstract, in Section 1.3 and in "Scope and priority". Before, the note described FCI(B) as a set which
   contains S(B).
3. Proposition 4.3: the note now says that its proof is a variant of the proof of Di Matteo–Ferrara, which also
   rests on Neumann's lemma; their proof was read and checked.
4. Literature search: the list of documents citing the source paper in zbMATH was obtained by run 2 (nine
   documents, all of them papers whose sources were searched); the statements that it could not be obtained were
   removed; the journal version of Cascella–Properzi–Van Antwerpen is not open access.
5. Six sentences: (a) the report cites the preprint with the numbering of its first version; (b) "as small as
   possible" after Theorem 1.2 was made precise; (c) the last sentence of Remark 5.3 was ambiguous; (d) a letter
   was used twice in Section 2; (e) the brace on Z/4 is quoted in the source paper from the literature; (f) the
   proof of Theorem 1.6 also uses Theorem 3.1 and Corollary 3.3 of Guarnieri–Vendramin.
6. The paragraph "Verification" of the note describes the three runs.
7. Section 9 of the note describes the programs of run 2; the runs are named A, B and 2 throughout.
8. This report records run 2.
9. `reproducibility/independent_run_2/` with its README; `reproducibility/README.md`, `run_quick.sh` and
   `RERUN_LOG.txt` extended.
10. The PDF, the archive and the files of the deposit were rebuilt.

**Added by run 2 and not examined by another run:** Remark 4.4 (the equality FCI(B) = S(B)) with its proof. The
equality was tested exactly in B_A, B_B, B_C and in the 3139 semidirect products of the survey. No other
statement of the note depends on it.

**After run 2, not examined by a verification run:** the proof of Theorem 5.4 in the journal version of
Jespers–Kubat–Van Antwerpen–Vendramin (Adv. Math. 385 (2021), pp. 15–17) was compared with the arXiv version;
the passage is the same. The sentences on this paper in Remark 8.2 and in the paragraphs on the sources of the
note were changed accordingly, two descriptions of texts that were not accessible were reworded, and one
comment line of a program of run A was reworded (its output is unchanged). No mathematical statement was
changed.

## Relation to the literature, novelty and scope
- **Searches (10 and 11 October 2026).** First version: queries to the arXiv listing, the lists of works citing
  the source paper in OpenAlex (7 works) and Semantic Scholar (21), two queries to zbMATH, two web searches; the
  sources of the two closest papers, of the paper of Mondal–Yadav and of eight further preprints were searched.
  Run A: three web searches, one of them for an erratum to the source paper. Run B: the sources of 19 arXiv
  papers (the 16 citing papers with an arXiv identifier and 3 related ones) and of five later papers of the
  authors of the source paper were searched and every hit read; the titles of the 150 most recent arXiv entries
  on skew braces; the zbMATH review of the source paper; three web searches. At the writing: Crossref (eight
  records), the arXiv listing, OpenAlex, zbMATH, one web search. Run 2: the arXiv listing again (the 22 entries
  on skew braces since July 2026; entries with the terms of the question; the papers of 2026 of the authors
  concerned); OpenAlex and Semantic Scholar (the same lists of citing works); the list of the nine documents
  citing the source paper in zbMATH; the sources of five arXiv papers; Crossref for all DOIs; one web search.
  - No source was found that answers the question or contains the example, and no erratum to the source paper.
- **What was read.** The abstract of Colazzo (publisher's file). Of the source paper: the second arXiv version in
  full; the journal version in Sections 1–3 and at every place where Lemma 3.12 is cited, with the definitions
  and statements of Section 4; the first arXiv version for the numbering; its proofs were not checked, apart from
  the points discussed in Remark 8.2. Of Cascella–Properzi–Van Antwerpen (Mediterr. J. Math. 23 (2026), Paper
  No. 187; arXiv:2603.06177): Sections 2 and 5 up to Corollary 5.23 in the second arXiv version, whose numbering
  is used (in these sections it is that of the first version); the journal version is not open access and was
  not read. Of Di Matteo–Ferrara (arXiv:2607.20222): Sections 1–5, with the proofs of Proposition 5.1 and Theorem
  5.2. Of Guarnieri–Vendramin (Math. Comp. 86 (2017); arXiv:1511.03171): Sections 1–3 and the tables of Section 5
  in the third arXiv version, whose numbering is used; the journal version was not compared; the proof of
  Theorem 3.9 was not checked. Of Jespers–Kubat–Van Antwerpen–Vendramin (Adv. Math. 385 (2021);
  arXiv:2001.10967): Theorem 5.4 and its proof in the journal version (pp. 15–17) and in the arXiv version,
  which agree in this passage. Of Mondal–Yadav (arXiv:2603.16771): Section 2 from the definition of the brace
  centraliser to Proposition 2.4; of the statements quoted from it only the proof of the first was checked; the
  journal version was not read. Not read: Neumann (1954), whose lemma is proved in the note, and Childs (2019),
  from which only a definition is used.
- **Limits.** The citing paper of Ballester-Bolinches, Esteban-Romero, Ferrara, Pérez-Calabuig and Trombetti
  (Experimental Mathematics, 2026) was seen in abstract only, because its full text was not accessible to us;
  one item of the Semantic Scholar list (lecture notes without an identifier) could not be located; the journal
  versions of three papers were not read; preprints not yet indexed may have been missed. A search that finds
  nothing is not a proof of novelty.
- **Scope.** The note answers Question 1 of the abstract. No priority is claimed for Theorem 1.3, Theorem 1.4 or
  Proposition 8.1, which are close to, or contained in, the literature as described in the verdict.

## 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/