# Verification report — OWR-1738-011 (Gonçalves–del Río: a uniform bound for free pairs of bicyclic units and for free points at algebraic integers)

Verification date: 2026-10-14.

**Verdict.** The note answers both questions of the source positively, with complete proofs.
- **Settled.** (Q2) For every nonzero algebraic integer z, kz is a free point for every integer k with |k| ≥ 4;
  hence m(z) ≤ 4, and the bound is sharp: m(1) = m(−1) = 4 (Theorem A). (Q1) For every finite group G and all
  a, b ∈ ZG with a² = b² = 0 and ab not nilpotent, (1 + ma, 1 + nb) is a free pair whenever |mn| ≥ 4
  (Theorem B); hence m(G) ≤ M(G) ≤ 4 for every finite non-Hamiltonian group G (Corollary C). The implication
  "(Q2) positive ⇒ (Q1) positive" stated in the source holds as stated, with m(G) ≤ lcm(1, …, K) when
  m(z) ≤ K for all z (Remark 3.3).
- **An elementary combination of known facts.** The theorem of Sanov (1947) and Brenner (1955) that every w with
  |w| ≥ 4 is a free point; the invariance of freeness under Galois conjugation (Chorna–Geller–Shpilrain 2017,
  Section 2; Kim–Koberda 2022, Section 1.1); the two-dimensional reduction in the proof of Theorem 1.1 of
  Gonçalves–del Río (2008). Exponents for each pair separately were known (Salwa 1999; Gonçalves–del Río 2008,
  Corollary 3.1(2)). What the note adds is the remark that integrality gives an eigenvalue of ab (or a Galois
  conjugate of z) of modulus at least 1, which makes the exponent uniform. No priority is claimed.
- **Computed values.** Exact: m(S3) = 1, M(S3) = 2, m(S4) = M(S4) = 2 (Proposition 4.2; the first three values
  are published, M(S4) = 2 is not stated in the sources read). Tested (exact characteristic polynomials,
  numerically located roots): m(S5) = M(S5) = 2, and the values of Table 2 for twelve further groups; at m = 1
  the value is undecided in D24 (same type; depends on whether 2√3 is free), GL(2,3) (same type; 2 + i√2) and
  C7 ⋊ C3 (mixed types; (7 + i√7)/2). The third verification run found that every irreducible factor that occurs
  in these computations has degree 1 or 2 and decided every case by exact comparisons of integers, with the same
  results; the note keeps the label "tested".
- **Open.** The second half of Question 5.4 of Gonçalves–del Río (2008): is m(G) ≤ 2 for every finite
  non-Hamiltonian group? The note shows only sup m(G) ∈ {2, 3, 4}.
- **Priority caveat.** No answer was found in the literature accessible to us; a search that finds nothing is not
  a proof of novelty. Four later sources that could contain the observation were not read (closed access):
  the survey of Gonçalves–del Río (J. Algebra Appl. 12 (2013) 1350004), Chapter 14 of Jespers–del Río,
  *Group Ring Groups* Vol. 2 (2016), Gonçalves, Contemp. Math. 688 (2017) 93–102, and Kimmerle, Jahresber. DMV
  115 (2013) 101–112.

The note is unrefereed.

## Statement checked
- **Primary source.** J. Gonçalves, Á. del Río, Problem 28 of the problem session of Oberwolfach Report
  No. 55/2007, *Mini-Workshop: Arithmetik von Gruppenringen* (25 November – 1 December 2007), organised by
  E. Jespers, Z. Marciniak, G. Nebe and W. Kimmerle; Oberwolfach Reports 4 (2007), issue 4,
  doi:10.4171/OWR/2007/55.
  - Read in the open PDF on the publisher's site (EMS Press, 32 pages, fetched on 2026-10-14). The report is
    printed on pp. 3149–3180; Problem 28 is on p. 3176 (PDF page 28). The DOI metadata give pp. 3209–3240.
  - The source defines the bicyclic unit 1 + (1 − g)x(1 + g + ⋯ + g^(n−1)) (g, x ∈ G, n the order of g), the set
    B of these units, recalls the dichotomy (either ⟨b1, b2⟩ is nilpotent, or ⟨b1, b2^m⟩ is free non-cyclic for some m > 0),
    defines m(G) = min{m : ⟨b1, b2^m⟩ free for all b1, b2 ∈ B with ⟨b1, b2⟩ not nilpotent} for non-Hamiltonian
    G, and asks whether {m(G)} is bounded; it adds that symmetric groups suffice and that m(S3) = 1, m(S4) = 2.
  - It calls z ≠ 0 a free point if the group generated by (1 1; 0 1) and (1 0; z 1) is free, sets
    m(z) = min{k ∈ N : kz is a free point}, asks whether {m(z) : z a nonzero algebraic integer} is bounded,
    and states that a positive answer to it gives a positive answer to the first question.
- **The paper behind the problem.** J. Z. Gonçalves, Á. del Río, "Bicyclic units, Bass cyclic units and free
  groups", J. Group Theory 11 (2008), no. 2, 247–265, doi:10.1515/JGT.2008.014; read as arXiv:math/0612091v2,
  Sections 1–5 and the references. Used: the units β_{x,h}, γ_{x,h} and the notion of free pair (Section 1);
  free points and the normalisation remark (footnote 1); Theorem 1.1; Corollaries 3.1–3.4; Section 5 (M(G), m(G),
  Examples 5.2 and 5.3, the results on dihedral groups, Question 5.4 in two parts). Question 5.4 asks only about
  m(G) (a uniform bound; m(G) ≤ 2?); the free-point question appears only in the problem session.
- **Corpus record.** ulamai/UnsolvedMath, record OWR-1738-011 (dataset version 1.6.0, upstream status `open`). It
  differs from the source in three points: (1) it prints the second matrix as [[1, z], [0, 1]]; this matrix
  commutes with (1 1; 0 1), so the literal question is degenerate (the group is abelian, never free of rank two);
  (2) it joins the two questions by "Equivalently", while the source states only the implication from the second
  question to the first; (3) it gives pp. 3209–3240 (the DOI metadata), while the report is printed on
  pp. 3149–3180. The note follows the source and says so in Section 1.1.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (Q2) as in the source: z a nonzero algebraic integer, matrices (1 1; 0 1), (1 0; z 1) | bounded; the maximum of m(z) is 4, at z = ±1 | Theorem A |
| (Q2) with first matrix (1 2; 0 1) (normalisation used by some authors) | bounded by 2, attained at z = 1 | Remark 2.5 |
| (Q2) as printed in the record, with (1 z; 0 1) | degenerate (abelian group); not the source's question | Section 1.1 |
| (Q1) as in the source: units of type β, ⟨b1, b2⟩ not nilpotent | m(G) ≤ 4 | Corollary C |
| m(G) of Gonçalves–del Río: pairs of the same type, ab not nilpotent | the same number as in the source | Remark 3.4 |
| M(G): pairs of arbitrary types | M(G) ≤ 4 | Corollary C |
| "free", "free non-cyclic", "free pair" for ⟨b1, b2^m⟩ | agree when ⟨b1, b2⟩ is not nilpotent | Section 1.1 (ii), Remark 3.4(b) |
| "Hamiltonian": all subgroups normal (abelian groups included), or the usage in which abelian groups are not Hamiltonian | no change: in such groups every bicyclic unit is 1; under the second usage abelian groups contribute the value 1 | Section 1.1 (i) |
| Second half of Question 5.4 of Gonçalves–del Río: m(G) ≤ 2 for all G? | not answered; sup m(G) ∈ {2, 3, 4} | Section 4 |

## Results in the paper
- **Lemma 2.1 (Sanov, Brenner).** Every w ∈ C with |w| ≥ 4 is a free point; ping-pong proof with
  X1 = {|z| > 1/2} ∪ {∞}, X2 = {|z| < 1/2}, including |w| = 4.
- **Lemma 2.2.** (a) Invariance under field embeddings of Q(w) for algebraic w; (b) w free iff −w free iff w̄
  free; (c) w free ⇒ jw free for every j ≠ 0.
- **Lemma 2.3.** A nonzero algebraic integer has a conjugate of modulus ≥ 1 (norm argument).
- **Proposition 2.4.** A B_{−k} = (1−k 1; −k 1) has order 6, 4, 3 for k = 1, 2, 3; so ±1, ±2, ±3 are not free.
- **Theorem A** and **Remark 2.5** (m(±2) = m(±3) = 2, m(±4) = 1; the other normalisation).
- **Lemma 3.1.** If a² = b² = 0 and ab is nilpotent, then ⟨1 + a, 1 + b⟩ is nilpotent (part of Gonçalves–del Río,
  Corollary 3.1(3)).
- **Proposition 3.2.** X² = Y² = 0, λ ≠ 0 an eigenvalue of XY, m ≠ 0 and mnλ a free point ⇒ (I + mX, I + nY) is
  a free pair (the two-dimensional reduction).
- **Theorem B** (integer matrix L_ab, eigenvalue of modulus ≥ 1) and **Corollary C** (m(G) ≤ M(G) ≤ 4; also
  (b1², b2²) free).
- **Remark 3.3** (the implication of the source, with lcm(1, …, K)); **Remark 3.4** (nilpotency of ⟨b1, b2⟩ iff of
  ab; free vs free pair; types and the involution g ↦ g^(−1); Theorem B for all square-zero elements).
- **Lemma 4.1.** |Im w| > 2 ⇒ w free (ping-pong with vertical strips).
- **Proposition 4.2** with Table 1 (exact values for S3, S4); **Proposition 4.3** (a pair in Z S5 whose nonzero
  eigenvalues all have modulus < 2 and which is nevertheless free at m = 2).
- **Section 4, tested:** m(S5) = M(S5) = 2; M(H) ≤ 2 for every non-Hamiltonian H ≤ S5; Table 2 (twelve groups,
  M(G) ≤ 2, three undecided values at m = 1).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The exact program of the note** (`check_note.py`, standard library, 0.2 s; TOTAL failures: 0). Part 1:
  Proposition 2.4 and the formulas for A^n, B_w^n. Part 2: all bicyclic units of Z S3 (6 + 6) and Z S4
  (156 + 156); for all four type orders, first unit over orbit representatives, second over all units; the
  nonzero eigenvalues of ab are certified by p(ab) = 0 and (p/(t − λ))(ab) ≠ 0 with p(t) = t² ∏(t − λ); every
  pair decided at m = 1, 2, 3, 4; Table 1 (S3 same/mixed: 8/12 pairs with ab not nilpotent, 0/8 not free at
  m = 1; S4: 1,676/1,948, 24/96 not free at m = 1; all free at m = 2); m(S3) = 1, M(S3) = 2, m(S4) = M(S4) = 2.
  Part 3: Proposition 4.3, minimal polynomial t(t−1)(t+1)(t²−2t+3)(t²+2t+3) certified exactly.
- **Programs with which the results were first obtained** (`original/`): Proposition 2.4; all ordered pairs of
  Z S3, Z S4 (exact); S4 and S5 with exact characteristic polynomials from Murnaghan–Nakayama characters
  (115,250 (β,β) and 125,410 (β,γ) pairs with ab not nilpotent in S5, first unit over the 36 orbit
  representatives; minimum of the largest eigenvalue modulus √3; all free at m = 2); the minimal polynomial of
  Proposition 4.3; eleven further groups in floating point.
- **Verification run A** (`verification_run_A/`): exact identities (Proposition 2.4, the block of
  Proposition 3.2 on 40 random exact examples, Proposition 4.3 in both product conventions); spectra of all
  pairs in S3, S4 and S5 from six integer representations (60-digit roots); cross-check by the regular
  representation on 600 random pairs; the count 838.
- **Verification run B** (`verification_run_B/`): exact minimal polynomials in S3 and S4 (both product
  conventions); counts over all ordered pairs; S5 by four integral representations with Faddeev–LeVerrier
  modulo primes and Chinese remaindering (230,500 same-type and 250,820 mixed pairs with ab not nilpotent,
  both orders; at m = 1, 1,768 + 2,672 not free and 2,424 + 4,512 undecided; at m = 2 all free, non-integer
  certificates with margin ≥ 0.47); the twelve groups of Table 2 exactly; the spectra of the undecided pairs;
  detector tests; the explicit pairs of Gonçalves–del Río, Examples 5.2 and 5.3; Hamiltonian groups; general
  square-zero elements of Z S3.
- **Third verification run** (`independent_run_2/`): Young's natural representations of S3, S4, S5 (Specht
  modules over Z, checked to be representations with orthonormal characters), characteristic polynomials of
  ρ(ab) from principal minors for every pair (S4 also over all ordered pairs), exact factorisation and exact
  decisions; exact minimal polynomials of ab for the twelve groups of Table 2 (built as permutation, matrix or
  affine groups, compared with all 156 subgroups of S5); Proposition 2.4, Proposition 4.3 in both product
  conventions, detectors. All numbers of the note were reproduced: Table 1; 230,500 / 250,820 pairs in S5 with
  ab not nilpotent, minimum √3, at m = 1 1,768 / 2,672 not free and 2,424 / 4,512 undecided, at m = 2 all free
  (smallest non-integer margin 2√5 − 4 ≈ 0.472); every entry of Table 2 and the three undecided spectra (64,
  128 and 160 pairs).
- **Re-runs.** On 2026-10-14 all 25 programs with a recorded output were run again from a copy of the package;
  every output agrees with the recorded one except for lines that report running times. The third verification
  run ran all 26 command lines again from the extracted archive `source.zip` (17 outputs byte-identical, 9
  differing only in timing values), and its own programs from the archive of this version. Details and command
  lines are in `reproducibility/README.md`.

## Independent verification runs
The results were first obtained, with proofs and with the programs in `original/`. Three independent verification
runs, all AI-assisted, followed on 2026-10-14; each wrote its own programs. The first two examined the first
written version of the results; the third examined the final text of the note. Run A examined the reading of the
source, re-derived the proofs and recomputed the finite data; run B recomputed every finite statement with its
own exact programs, re-derived the proofs, re-ran the original programs, and examined the literature and the
bibliography.

| Item | Run A | Run B |
|---|---|---|
| Reading of the source (matrix (1 0; z 1), one-way implication, definition of m(G)) | CONFIRMED; three defects of the record found | CONFIRMED; record defects confirmed against the PDF |
| Lemma 2.1 (ping-pong, including \|w\| = 4) | CONFIRMED (re-derived) | CONFIRMED (re-derived) |
| Lemma 2.2(a), Lemma 2.3, Theorem A | CONFIRMED (re-derived) | CONFIRMED (re-derived; Galois step checked in Kim–Koberda) |
| Proposition 2.4, m(±1) = 4 | CONFIRMED (re-derived; exact orders computed) | CONFIRMED (exact relations; short-word search as a detector) |
| Lemma 3.1, Proposition 3.2, Theorem B, Corollary C | CONFIRMED (re-derived; block identities on random exact examples) | CONFIRMED (re-derived; Step 2 asserted on every tested pair) |
| The implication (Q2) ⇒ (Q1) of the source | CONFIRMED_WITH_FIXES: kz free ⇒ jkz free, bound lcm(1, …, K) | the same fix |
| Equality of the two definitions of m(G) | FIX: one sentence on types (involution) | CONFIRMED (involution argument) |
| Lemma 4.1, Proposition 4.3 | CONFIRMED (re-derived; minimal polynomial in both conventions) | CONFIRMED (both conventions; wrong-polynomial detector) |
| Published values (157 units, m(S3) = 1, M(S3) = 2, m(S4) = 2) and M(S4) = 2 | CONFIRMED (own exact computation) | CONFIRMED (three exact methods) |
| m(S5) = M(S5) = 2 (tested) | CONFIRMED (own exact characteristic polynomials) | CONFIRMED (own method; margins ≥ 0.47) |
| Further groups (tested) | not checked (only that the eleven groups do not embed in S5) | CONFIRMED exactly for twelve groups; FIX: three undecided cases at m = 1, not only D24 |
| Sources and bibliography | mostly CONFIRMED; citation fixes | CONFIRMED; FIX: journal version of Kim–Koberda |
| Novelty | not established; closed sources unread | no prior answer found; ingredients published; closed sources unread |

Neither run found a mathematical error.

**Corrections required by runs A and B**, all applied in the note:
1. (A, B) The corpus record: the second matrix is (1 0; z 1), not (1 z; 0 1); the source states a one-way
   implication, not "Equivalently"; the page ranges. Section 1.1, "The corpus record"; also Section 1.1 for the
   matrix of the source.
2. (A, B) The remark on the implication stated in the source: if kz is free then jkz is free (j ≠ 0), so the
   implication holds as stated, with m(G) ≤ lcm(1, …, K). Lemma 2.2(c) and Remark 3.3.
3. (A) The symmetry between the type orders: the involution turns ab into a*b* = (ba)*, not (ab)*. The note does
   not use this reduction: `check_note.py` and both runs compute all four type orders, and Remark 3.4(c) uses the
   involution only on pairs of units.
4. (A) "m(H) ≤ 2 for every non-Hamiltonian H ≤ S5". Section 4.
5. (A) Labels of the counts (pairs examined vs. pairs with ab not nilpotent). Table 1 has both columns.
6. (A) The problems of Jespers (2021) are Problems 2.3, 2.4, 8.1, 8.4 and 8.5. Section 1.3.
7. (A) The citation of m(S3) = 1: the source, and the case of dihedral groups D_2n with n prime quoted in
   Gonçalves–del Río (2008), Section 5, not Example 5.2. Remark after Proposition 4.2.
8. (A) Why the m(G) of the source (type β only) equals that of Gonçalves–del Río (pairs of the same type).
   Section 1.1 (iii) and Remark 3.4.
9. (A, B) The sources that could not be read: the 2013 survey, Chapter 14 of *Group Ring Groups* Vol. 2,
   Gonçalves (2017), Kimmerle (2013). Abstract, Section 1.3, "Scope and priority".
10. (A) The final statement names the matrix (1 0; z 1) of the source. Section 1.1.
11. (B) All three groups with an undecided value at m = 1: D24 (2√3), GL(2,3) (2 + i√2), C7 ⋊ C3, mixed types
    ((7 + i√7)/2). Section 4 and Table 2.
12. (B) The journal version of Kim–Koberda, Michigan Math. J. 71 (2022) 809–833. Bibliography.
13. (B, cosmetic) Three recorded outputs ended with a line written by the shell's `time` command; it was removed.
    `reproducibility/README.md`.

Suggestions of run B that were adopted: the route from the strong form of Theorem A to the bound 4
(Remark 3.3), and the remark that Theorem B holds for all square-zero elements (Remark 3.4(d)).

**Changes made when the note was written.**
- The exact program `check_note.py` was added. It certifies the eigenvalue sets of Proposition 4.2 without using
  characters and checks Propositions 2.4 and 4.3; its counts agree with those of the three earlier groups of
  programs.
- Proposition 3.2 is stated for free points mnλ, not only for |mnλ| ≥ 4, as Remark 3.3 and Proposition 4.3 need.
- Lemma 2.2(b) (sign and complex conjugation) was added; it reduces the undecided cases to the three points named.
- Lemma 3.1 uses products of 2r + 1 factors (both runs noted that 2r + 1 suffices); Remark 3.4(a) proves the
  converse through Theorem B.
- At the writing, the following were read again or for the first time: Problem 28 of the source (p. 3176);
  Gonçalves–del Río (2008), Sections 1–5 and the references; Kim–Koberda, Section 1 up to the statements of the main results, and the
  references (from which the bibliographic data of Sanov and Brenner are taken); Chorna–Geller–Shpilrain, the beginning of
  Section 1 (Sanov's theorem for (1 2; 0 1), (1 0; 2 1)) and the passage of Section 2 on Galois conjugation;
  Jespers (2021), Proposition 3.3, Theorem 3.5 and the problems; the abstract of Salwa (1999) in the Crossref
  record; the Crossref record of Jiménez López (2008), whose published title differs from the preliminary title
  cited in arXiv:math/0612091.

### Third independent verification run (on the final text)

The third run, AI-assisted, examined the final text of the note on 2026-10-14, before any change, and wrote its
own programs (`reproducibility/independent_run_2/`, with a README).

| Item | Third run |
|---|---|
| Reading of the source and of Gonçalves–del Río (2008), Section 5 (read again: OWR p. 3176, arXiv v2 Sections 1–5) | CONFIRMED; only the first question is part of Question 5.4 (abstract corrected) |
| Lemmas 2.1–2.3, Proposition 2.4, Theorem A, Remark 2.5 | CONFIRMED (re-derived line by line; Proposition 2.4 by exact matrices) |
| Lemma 3.1, Proposition 3.2, Theorem B, Corollary C, Remarks 3.3 and 3.4 | CONFIRMED (re-derived; block identities on random exact instances) |
| Lemma 4.1, Propositions 4.2 and 4.3 | CONFIRMED (re-derived; Proposition 4.3 from scratch in both product conventions) |
| Table 1, m(S3) = 1, M(S3) = 2, m(S4) = M(S4) = 2 | CONFIRMED (Specht modules; also over all ordered pairs of Z S4) |
| S5: all counts, minimum √3, margins, m(S5) = M(S5) = 2 | CONFIRMED, every case decided exactly (all irreducible factors of degree ≤ 2) |
| Table 2 and the three undecided spectra | CONFIRMED with exact minimal polynomials, every case decided exactly |
| Cited results (Theorem 1.1, Corollary 3.1, Examples 5.2, 5.3, footnote 1 of Gonçalves–del Río; Chorna–Geller–Shpilrain §2; Kim–Koberda §1, §1.1; Jespers Proposition 3.3, Theorem 3.5, problems; abstracts of Salwa, Marciniak–Sehgal, Jiménez López) | CONFIRMED, except the attribution of Lemma 2.1 (corrected) |
| Bibliography (13 DOIs) | CONFIRMED with Crossref |
| Re-run of the package from `source.zip` | all recorded outputs reproduced (timing values excepted) |
| Novelty | no prior answer found (search repeated; see below) |

It found no mathematical error. **Corrections required by the third run**, all applied in the note:
1. Abstract: only the first question is (the first half of) Question 5.4 of Gonçalves–del Río (2008); the
   free-point question is only in the problem session.
2. Attribution of Lemma 2.1: Kim–Koberda (Section 1) attribute |w| = 4 to Sanov and |w| > 4 to Brenner for real w;
   Gonçalves–del Río (Section 1) and Jespers (Proposition 3.3(2)) state the complex statement as Sanov's theorem.
   Section 2 now says exactly this; Section 1.3 says "goes back to".
3. Section 1.2: sup_G m(G) ∈ {2, 3, 4} follows from m(S4) = 2 and Corollary C (the sentence followed Proposition
   4.3 with "Hence").
4. Corollary C: the positive answer to (Q2) comes from Theorem A ("with Theorem A").
5. Section 4: the original programs computed the type orders (β, β) and (β, γ) of Z S5, the first two verification
   runs all four.
6. Section 4: the second verification run used exact characteristic polynomials and numerically located roots
   for the twelve groups (not "exactly"); the third run's exact decisions are stated, the label "tested" is kept.
7. Table 2: in (C3 × C3) ⋊ C4 a generator of C4 acts by (x, y) ↦ (−y, x), faithfully.

Also adopted: the credit to Corollary 3.1(1) of Gonçalves–del Río (2008) for the dependence on mn only
(Section 1.3); a change of letter in Remark 3.4(c); a note in the bibliography entry of Jiménez López (2008) on the
title under which Gonçalves–del Río cite it; the record of this run in the Verification paragraph, in Section 5
and in "Scope and priority".

## Relation to the literature, novelty and scope
- **Searches (all on 2026-10-14).** arXiv API (bicyclic units, free pairs, free points, parabolic generators, the
  authors of the problem), zbMATH Open, Crossref, OpenAlex and Semantic Scholar (works citing Gonçalves–del Río
  2008, the Oberwolfach report, Salwa 1999 and the 2013 survey; later works of the two proposers), publisher
  pages, and seven web searches, made when the results were first obtained and by the first two verification
  runs. At the writing: Crossref (all DOIs), zbMATH (pages of Gonçalves–del Río 2008), the arXiv API (the arXiv
  version of Chorna–Geller–Shpilrain), and the open texts listed below. The third verification run: Crossref (all
  13 DOIs), OpenAlex (the works citing Gonçalves–del Río 2008, the Oberwolfach report, the 2013 survey, Salwa 1999,
  Kimmerle 2013 and Jiménez López 2008, and the open-access status of the four unread sources: all closed), the
  arXiv API (five queries, among them "bicyclic units" by date), zbMATH Open ("bicyclic & free", 2009–2026: 61
  records; the reviews of the 2013 survey, of *Group Ring Groups* Vol. 2 and of Kimmerle 2013 read), and one
  web search; eight web searches in all.
  - None of the works found states a uniform bound for m(G), M(G) or m(z). For the 54 works citing one of the
    six items above, the titles and the 35 abstracts available in OpenAlex were read; the full texts only for
    the open sources listed below.
- **What was read.** The source (p. 3176); Gonçalves–del Río (2008), arXiv v2, Sections 1–5; Kim–Koberda (2022),
  arXiv version, Section 1 up to the statements of the main results, and the references; Chorna–Geller–Shpilrain (2017), arXiv v4, Sections 1 and 2 in the
  parts named above; Jespers (2021), arXiv version, Proposition 3.3, Theorem 3.5 and the statements of all its
  problems; Salwa (1999), abstract. The verification runs also read Jespers–Olteanu–del Río (2012, Remark 5.5),
  Gonçalves–Passman (2010, pp. 1–5), and the abstracts of later works on free products in unit groups; none of
  them concerns m(G).
- **Not read.** Sanov (1947) and Brenner (1955) (their content is taken from Chorna–Geller–Shpilrain,
  Kim–Koberda, Gonçalves–del Río and Jespers); the full texts of Marciniak–Sehgal (1997) (cited after the source;
  abstract read) and Jiménez López (2008) (its results are cited after Gonçalves–del Río, Section 5; the abstract,
  read by the third run, states them). Closed access, and therefore the remaining
  caveat on novelty: Gonçalves–del Río, J. Algebra Appl. 12 (2013) 1350004; Jespers–del Río, *Group Ring Groups*
  Vol. 2, Chapter 14 "Free Groups"; Gonçalves, Contemp. Math. 688 (2017) 93–102; Kimmerle, Jahresber. DMV 115
  (2013) 101–112. If one of them contains the observation, priority belongs there.
- **Bibliography.** All DOIs were checked with Crossref on 2026-10-14; the pages of Gonçalves–del Río (2008)
  (247–265) with zbMATH (Zbl 1144.16024).
- **Scope.** Both questions of the source are answered as posed. The second half of Question 5.4 of
  Gonçalves–del Río (2008) is not answered. The tested statements of Section 4 are labelled as such.

## 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/