# Verification report — OWR-12007-010 (Calegari–Walker: a word with universally vanishing rotation number; real-analytic realisation of R(w,r,s))

Verification date: 2026-10-17.

**Verdict.** The note gives a **partial answer** to the record; every statement it makes is proved, and the scope is as
follows.
- **Settled: Question (Q1), negatively.** For every nontrivial w in the commutator subgroup of F = ⟨a,b⟩ there are
  A, B ∈ SL(2,R) with |tr w(A,B)| < 2 (Theorem A). Lifting such a pair to the universal cover of PSL(2,R) gives a
  homomorphism ρ: F → Homeo⁺(S¹)~ with rot~(ρ(w)) ∉ Z; words outside [F,F] are treated with rigid rotations
  (Corollary B). So no nontrivial word has rot~(ρ(w)) = 0 for every ρ. The representations are real-analytic, and
  the rotation number is nonzero modulo 1 as well.
- **Computations for the example of the report.** For the 36-letter word w0 printed on p. 2174, exact rational
  hyperbolic matrices A, B give tr w0(A,B) = 0 and, with the lifts that have fixed points, rot~(ρ(w0)) = −1/2 at
  (r,s) = (0,0) (Proposition C); moreover X(w0,0,0) = [−1,1] and R(w0,0,0) = 1 (Corollary 5.2).
- **Not new.** The questions are Calegari and Walker's. For w = [a,b] the negative answer is in Calegari–Walker
  (2011), Example 1.1; Kim–Koberda–Mj (2019), Theorem 4.21, gives elliptic images with nonzero rotation number for
  words representing peripheral or splitting curves. Proposition D (rot~(ρ(w0)) = 0 when ρ(a) or ρ(b) is conjugate
  to a rotation, in particular when it is elliptic) is essentially Calegari–Walker (2011), Example 4.8 (see also the
  proof of Mann's Lemma 6.18(i)); the scaling of Proposition 5.4 is the cyclic covering construction of their
  Example 4.9. The trace polynomials are classical (Vogt, Fricke); Lemma 3.4 is part of Goldman's description of real
  characters.
- **Open.** Question (Q2), on real-analytic representations attaining R(w,r,s), is not treated. The reading of the
  report's example over discrete faithful representations is not addressed.

The note is unrefereed.

## Statement checked
- **Primary source.** D. Calegari (joint work with A. Walker), "Ziggurats and rotation numbers", in: Geometric Group
  Theory, Hyperbolic Dynamics and Symplectic Geometry, Oberwolfach Report 35/2012, Oberwolfach Reports 9 (2012),
  no. 3, pp. 2173–2175, doi:10.4171/OWR/2012/35 (issue published 2013-05-29 according to Crossref).
  - Read in the open PDF of EMS Press: the text layer of pp. 2173–2175 in full; the 36-letter word was also compared
    letter by letter with the rendered page 2174.
  - Definitions (p. 2173): F free on a, b; Homeo⁺(S¹)~ the universal cover; rot~ a homogeneous quasimorphism;
    X(w,r,s) the set of values rot~(ρ(w)) with rot~(ρ(a)) = r, rot~(ρ(b)) = s; R(w,r,s) = sup X(w,r,s); X(w,r,s) is
    a compact interval with minimum −R(w,−r,−s).
  - Section "5. two open questions" (p. 2174): the report considers representations of F into PSL(2,R), states that
    rot~(ρ(w0)) = 0 for all of them with w0 = ababaabbaBBAABABAbababaabbABBAABABAB, and that this can be proved with
    trace identities and numerical methods. Question 1: is there a nontrivial word w with rot~(ρ(w)) = 0 for every
    ρ: F → Homeo⁺(S¹)~? Question 2: for arbitrary w, is there a real-analytic ρ with rot~(ρ(w)) = R(w,r,s)? The
    report notes that interior points of X(w,r,s), and R(w,r,s) for positive w, are realised by real-analytic
    representations.
  - Capital letters are not defined in the report. They denote inverses in Calegari's work on stable commutator
    length (J. Amer. Math. Soc. 22 (2009), Section 4, read); Calegari–Walker (2011) write a⁻¹, b⁻¹. With this reading
    w0 = [u a u⁻¹, b], u = ababaabb, letter by letter.
- **Corpus record.** ulamai/UnsolvedMath, OWR-12007-010 (record 30002130; dataset version 1.6.0; upstream status
  `open`; literature review of 2026-08-21: both questions open). Its statement reproduces both questions faithfully;
  it omits the standing definitions and the printed example.

## Readings
| Reading | Answer | Where |
|---|---|---|
| (Q1) as printed: some w ≠ 1 with rot~(ρ(w)) = 0 for every ρ: F → Homeo⁺(S¹)~ | no such word | Theorem A, Corollary B |
| (Q1) for the rotation number in R/Z | no such word (rot~(ρ(w)) ∉ Z) | Corollary B |
| (Q1) for lifts of real-analytic or smooth diffeomorphisms, or for the universal cover of PSL(2,R) | no such word | Corollary B |
| (Q1) restricted to ρ with (rot~ρ(a), rot~ρ(b)) ∉ Z² | no such word | Corollary 5.5(a) |
| (Q1) restricted to ρ with ρ(a), ρ(b) conjugate to rotations (the setting of R(w,r−,s−)) | yes, trivially: w = [a,b] (argument of Proposition D) | Section 1, "Readings of the question" |
| (Q1) at fixed (r,s) with r or s irrational | yes, trivially: X([a,b],r,s) = {0} (Calegari–Walker, Example 4.9) | Section 1 |
| The report's statement on w0, literally: all representations into PSL(2,R) | the word admits a representation with rot~(ρ(w0)) = −1/2 (both generators hyperbolic, (r,s) = (0,0)) | Proposition C |
| The same statement when ρ(a) or ρ(b) is elliptic, i.e. (r,s) ∉ Z² | rot~(ρ(w0)) = 0 | Proposition D, Remark 5.3 |
| The same statement over discrete faithful representations | not addressed (equivalent to tr w0(A,B) ≥ 2 on such pairs; numerical evidence only) | Remark 5.3 |
| Other readings of the string (reversed, inverted, case-swapped) | same trace polynomial; same conclusions up to the sign of rot~ | Remark 5.1 |
| (Q2) | not treated | Section 1 |

## Results in the paper
- **Section 2.** Lemma 2.1: basic properties of rot~ (with the defect bound |rot~(fg) − rot~ f − rot~ g| ≤ 1 quoted from
  Calegari–Walker). Lemma 2.2: lifts of matrices to R; the group PSL(2,R)~. Lemma 2.3: lifts of elliptic matrices
  have non-integral rotation numbers, of non-elliptic ones integral, of trace-zero ones half-integral. Lemma 2.4: for
  w ∈ [F,F], ρ(w) does not depend on the lifts; rot~ is even if tr w(A,B) ≥ 2 and odd if ≤ −2.
- **Section 3.** Fricke–Vogt trace polynomials t_w (quoted). Lemma 3.1: the cubic V = {κ = 2} of reducible
  characters, its parametrisation, t_w = 2 on V for w ∈ [F,F]. Lemma 3.2: divisibility by κ − 2. Lemma 3.3: no
  vanishing on open pieces of the parametrisation. Lemma 3.4: realisation of real characters by SL(2,R) or SU(2)
  pairs. Lemma 3.5: a ping-pong pair with |tr w| > 2 for all w ≠ 1, so t_w − 2 ≠ 0. Lemma 3.6: singular points of V.
- **Section 4.** Theorem A (sign comparison of t_w − 2 = (κ − 2)^m h on both sides of V near the compact part and
  near an unbounded sheet). Corollary B. Corollary 4.1: the image of the word map contains every element of trace
  in (2 − δ, 2). Remark 4.2: comparison with Gordeev–Kunyavskiĭ–Plotkin (2018), Proposition 2.6 and Question 2.5
  (Theorem A does not answer Question 2.5, not even for words in F), and with Hui–Larsen–Shalev (2015).
- **Section 5.** Structure of w0. Remark 5.1 (readings). Proposition C (exact certificate; rot~ = −1/2). Corollary 5.2
  (X(w0,0,0) = [−1,1]). Proposition D (after Calegari–Walker, Example 4.8). Remark 5.3 (the statement of the report:
  holds when a generator is elliptic, fails as literally printed when both generators are hyperbolic with integer
  rotation numbers, though not at every such representation; the argument of Proposition D gives rot~ = 0 for a
  commutator one of whose entries is mapped to a conjugate of a translation, so Proposition D holds for every
  commutator of a conjugate of a^{±1} with a conjugate of b^{±1} and does not single out w0; discrete faithful
  reading open). Proposition 5.4 and Corollary 5.5 (scaling; the answer to (Q1) stays negative for (r,s) ∉ Z²; the
  action of −A, −B on rays has rotation numbers (1/2, 1/2) and rot~(ρ(w0)) = −1/4).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on floating-point computation. Exact rational arithmetic enters in Proposition C (the matrix
w0(A,B) and the integers h_k that fix the sign −1/2), Corollary 5.2 (the pair with tr w0 < −2 and the sign +1) and
Remark 5.3 (the pair with character (3,3,3)).
- **The program of the note** (`check_note.py`, standard library, about 0.4 s; ALL CHECKS PASSED): word structure;
  certificate (also through the commutator structure), the four readings, w0(−A,−B) = w0(A,B); exact winding count
  for five start vectors (−1/2, and +1/2 after conjugation by diag(1,−1)); the pair (A2, B2) (tr w0 ≈ −56.89,
  rot~ = +1, and −1 after conjugation); the pair (A3, B3) (tr w0 = 247165558, rot~ = 0); the polynomial identities of
  Lemmas 3.1, 3.2, 3.4 and identity (2); the singular points of Lemma 3.6; the data of Lemma 3.5 and |tr w(A0,B0)| > 2
  for all 29,540 cyclically reduced words of length ≤ 9; Lemma 3.1(iv) and tr[A,B] = κ on random exact instances; an
  exact elliptic value for each of 14 words of [F,F] (a test, not part of any proof); Proposition D on 180 random
  exact pairs with an elliptic generator.
- **Programs with which the results were first obtained** (`original/`): exact certificate and commutator structure;
  the trace polynomial t_w0 by sympy (degree 24, 249 monomials, κ − 2 divides t_w0 − 2 exactly once, cofactor with
  151 monomials); floating-point explorations of elliptic values, of explicit lifts and of Proposition D; symbolic
  checks of the identities; a second exact certificate near an unbounded sheet of V(R).
- **Verification run A** (`verification_run_A/`): exact certificate under four readings; t_w0 by another method
  (identical); rotation number −1/2 to 50 digits; the pair (A2, B2); exact tests of 10,630 marked Schottky
  representations (tr w0 > 2 in all); exact re-verification of elliptic values for 43 words.
- **Verification run B** (`verification_run_B/`): exact certificate and exact winding count (−1/2); t_w0 with its own
  sparse polynomial arithmetic (identical); the sign map of the cofactor h on the unbounded sheets (h changes sign
  between the two families of sheets); exact tests of Proposition D on 800 pairs; elliptic values for 68 words;
  exact pairs with rot~ = ±1; Lemma 3.5 on 39,364 reduced words; a degree cross-check with Kapovich's formula;
  a re-run of the original programs (`claim_rerun_outputs/`, identical outputs).
- **Verification run on the final text** (`independent_run_2/`): the certificates and rotation numbers of
  Proposition C, Corollary 5.2 and Remark 5.3 exactly, by counting signed crossings of a fixed ray (another method
  than the integers h_k), and with 60 digits by iterating the lifts; t_w0 by the trace recursion
  tr(XgYg) = tr(Xg) tr(Yg) − tr(XY⁻¹) (identical degree, 249 monomials, m = 1, cofactor with 151 monomials);
  Lemma 3.5 for all 88,592 cyclically reduced words of length ≤ 10; Proposition D on 600 exact pairs; exact elliptic
  values for 40 words of [F,F] built by the mechanism of the proof of Theorem A (one of them, [[a,b],[a,b⁻¹]], has
  m = 2 and is handled next to the compact part); an exact pair at which [a, bab⁻¹] is elliptic although B is
  elliptic.
- **Re-runs.** On 2026-10-17, while the package was assembled, every program with a recorded output was run again
  from a copy of the package, and later the same day the verification run on the final text ran all of them again
  from an extracted copy of `source.zip`, with identical outputs; see `reproducibility/RERUN_LOG.txt` and
  `independent_run_2/rerun_report.txt`.

## Independent verification runs
The results were first obtained with proofs and with the programs in `original/`. Two independent verification runs,
both AI-assisted, followed (2026-10-17); they examined the first written version of the results, and each wrote its
own programs before looking at those of the first stage. A third independent run examined the final text (see the
section "Verification run on the final text" below).

| Item | Run A | Run B |
|---|---|---|
| Transcription of the word and of the questions (p. 2174) | CONFIRMED (text layer) | CONFIRMED (text layer) |
| Lifts irrelevant for w0 (exponent sums 0) | CONFIRMED | CONFIRMED |
| Lemmas 3.1–3.6 | CONFIRMED (re-derived; identities checked symbolically) | CONFIRMED (re-derived; own symbolic checks) |
| Theorem A | CONFIRMED (re-derived, no gap) | CONFIRMED (re-derived, no gap); tested on 68 words |
| Corollary B, answer to (Q1) | CONFIRMED | CONFIRMED |
| Proposition C | CONFIRMED (exact; rot~ = −1/2 to 50 digits) | CONFIRMED (exact, including an exact winding count) |
| Description of the report's statement | CONFIRMED WITH FIXES (qualified wording; discrete faithful reading open) | CONFIRMED WITH FIXES (failures only at (r,s) ∈ Z²) |
| Proposition D | CONFIRMED; generic, holds for [a,b] too | CONFIRMED; not new in substance (Calegari–Walker, Example 4.8) |
| Remark on the sign of h on the unbounded sheets (first write-up) | not examined | REFUTED (h < 0 on the sheets z = l/m + m/l); not used by any proof |
| R(w0,0,0) | = 1, X(w0,0,0) = [−1,1] (proved; pair with tr w0 < −2) | = 1 (exact pairs with rot~ = ±1) |
| Literature and novelty | no source states Theorem A; one author list wrong | closest work found: Calegari–Walker Ex. 1.1/4.8, Kim–Koberda–Mj Thm 4.21, Gordeev–Kunyavskiĭ–Plotkin Prop. 2.6 |

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

**Corrections required by run A**, all applied in the note:
1. The description of the report's statement is qualified: it fails as literally printed (all representations into
   PSL(2,R)) at a representation with both generators hyperbolic, (r,s) = (0,0), which is not faithful; its failures
   need both generators non-elliptic, and there it need not fail (tr w0 = 247165558 at the pair with character
   (3,3,3); no failure on 10,630 Schottky representations tested exactly); "no misprint can repair it" is restricted to
   the literal reading; the discrete faithful reading is stated as open (Remark 5.3).
2. Proposition D is presented as a statement that holds for every commutator of a conjugate of a^{±1} with a
   conjugate of b^{±1}, including [a,b], so it does not explain the choice of w0; not as "the correct statement
   behind" the report (Remark 5.3). (The text first said "with c or k conjugate to a generator or its inverse"; the
   run on the final text showed that this is too broad, see below.)
3. Words outside [F,F] are treated with rotations (e_a(w)t ∉ Z), so that the rotation number is nonzero modulo 1 as
   well (proof of Corollary B).
4. The variant with generators conjugate to rotations is listed as not covered by Theorem A (it has the trivial
   positive answer [a,b]) (Section 1, "Readings of the question").
5. arXiv:1106.1619 is by T. Bandman and S. Garion (reference [BG12]).
6. The capital-letter convention: the report does not define it; it is the convention of Calegari's scl work
   (checked in J. Amer. Math. Soc. 22 (2009), Section 4); Calegari–Walker (2011) write a⁻¹, b⁻¹; the other readings
   have the same trace polynomial (Section 1, Remark 5.1).
7. The zbMATH query behind "3 documents" ("ziggurat rotation number"; the plain query "ziggurat" gives 88) and the date
   of the OpenAlex count (29 citing works on 17 October 2026) are stated ("Scope and priority").
8. (Optional, adopted) X(w0,0,0) = [−1,1], R(w0,0,0) = 1 (Corollary 5.2); the parity statement (Lemma 2.4); the
   SL(2,C) and finite-field word-map results are mentioned as concerning other questions.

**Corrections required by run B**, all applied in the note:
1. As run A, item 1: failures only at (r,s) ∈ Z², truth for (r,s) ∉ Z² (Proposition D), restriction of the
   "no misprint" sentence (Remark 5.3).
2. The remark on the sign of h after Theorem A and the corresponding sentence of the first write-up are removed; the
   note makes no statement about the sign of h.
3. Credits: Calegari–Walker (2011), Example 4.8, and the proof of Mann's Lemma 6.18(i) for Proposition D;
   Calegari–Walker (2011), Example 1.1, for w = [a,b].
4. Gordeev–Kunyavskiĭ–Plotkin (2018, Proposition 2.6 and Question 2.5), Hui–Larsen–Shalev (2015) and
   Kim–Koberda–Mj (2019, Theorem 4.21) are cited, and Theorem A is positioned against Proposition 2.6; the note says
   that Theorem A does not answer Question 2.5, not even for words in F (Remark 4.2, Section 1).
5. arXiv:1106.1619: Bandman–Garion.
6. (Optional, adopted) R(w0,0,0) = 1; the scaling remark, with a complete proof (Proposition 5.4, Corollary 5.5).

**Changes made when the note was written.** The proofs were rewritten in a self-contained form: the lifts to the
universal cover of PSL(2,R) are constructed directly (Lemma 2.2), the parity statement of run A is proved
(Lemma 2.4), and Proposition D is proved by the fixed-point argument of Calegari–Walker instead of the trace identity
and path-lifting argument of the first write-up. The sign of the rotation number in Proposition C is determined by an
exact winding count (integers h_k computed from signs of determinants) in the program of the note. Corollary 4.1 and
Remark 4.2 (image of word maps) were added. The program `check_note.py` was written with the text.

## Verification run on the final text
A third independent verification run, AI-assisted, examined the final text on 2026-10-17, and first the parts written
after the two earlier runs: Section 2 (lifts, dictionary, parity), the proof of Proposition D, Corollary 4.1,
Remark 4.2, Proposition 5.4 and Corollary 5.5. Its programs and outputs are in `reproducibility/independent_run_2/`;
they were written from the statements of the note before the other programs of the package were opened.

| Item | Verdict |
|---|---|
| Statement fidelity | CONFIRMED: Oberwolfach Report 35/2012 re-read (text layer of pp. 2173–2175); the 36-letter word compared letter by letter on the page rendered at three times the normal size; definitions, both questions and the printed statement correctly paraphrased; the corpus record faithful |
| Cited results | CONFIRMED against the sources read again: Calegari–Walker (2011, journal version) §1.4, Example 1.1, §2 (Proposition 2.1, proof of Lemma 2.11, Proposition 2.16), Definition 3.23, Definition 4.3, Examples 4.8 and 4.9; Calegari (2009) §4; Gordeev–Kunyavskiĭ–Plotkin arXiv v1 §2.2 (Question 2.5, Proposition 2.6 and its proof); Kim–Koberda–Mj arXiv v4 Theorem 4.21; Mann arXiv v1 Lemma 6.18 and its proof; Goldman (2009) Theorem A and (2.2.8); Goldman (2003) §2.6. Two citations were imprecise (fixes 2 and 3) |
| Section 2 (Lemmas 2.1–2.4, the group PSL~(2,R)) | CONFIRMED (re-derived; a one-line justification that each lift belongs to exactly one matrix was added) |
| Section 3, Theorem A, Corollary B | CONFIRMED (re-derived; exact elliptic values for 40 words by the mechanism of the proof, both branches m odd and m even occur) |
| Corollary 4.1, Remark 4.2 | CONFIRMED (the image of a word map is invariant under GL(2,R)-conjugation; Theorem A does not answer GKP Question 2.5) |
| Proposition C, Corollary 5.2, Remark 5.3 (A3, B3) | CONFIRMED: the matrices exactly; −1/2 (+1/2 after conjugation) and ±1 by exact crossing counts; all values, including 0 for (A3, B3) (which follows from the exact trace by Lemma 2.4), again by 60-digit iteration |
| Proposition D (fixed-point argument after Calegari–Walker, Example 4.8) | CONFIRMED |
| Remark 5.3, general form of Proposition D | REFUTED as written: "for every commutator [c,k] in which c or k is conjugate to a^{±1} or b^{±1}" is too broad. For [a, bab⁻¹] (both entries conjugate to a), A = [[1, 5/2], [−7/4, −27/8]] (hyperbolic) and B = [[−97/18, −628/135], [221/48, 341/90]] (elliptic) give tr = 463036521146401/348285173760000 ≈ 1.3295, so rot~ is not an integer although ρ(b) is elliptic. Corrected (fix 1) |
| Proposition 5.4, Corollary 5.5, the paragraph on −A | CONFIRMED (rotation number −1/4 on the circle of rays recomputed) |
| Package | all 28 programs re-run from the archive with identical outputs |
| Literature | no printed answer to (Q1), no statement of Theorem A for arbitrary words; credits adequate |

**Corrections required by the run on the final text**, all applied:
1. Remark 5.3: the general form of Proposition D is restricted to commutators [c,k] with c conjugate to a^{±1} and
   k to b^{±1}, or vice versa; the argument is stated in the form "rot~(ρ([c,k])) = 0 whenever ρ(c) or ρ(k) is
   conjugate to a translation".
2. Mann's Lemma 6.18 is in Section 6.2 of arXiv:1510.00728v1 (the note said Section 6.1); the note now says that the
   numbering of Mann and of Gordeev–Kunyavskiĭ–Plotkin is that of the arXiv versions read.
3. Section 1: "its centre is {T_k} [CW11, §2]" replaced by what §2 of Calegari–Walker states (the kernel of the
   projection to Homeo⁺(S¹) is the central group of integer translations); "universal central extension" is cited
   to their §1.4.
4. Proof of Proposition C: "four programs written independently of each other" replaced by "the program of this
   note and, independently of it, the programs of the first stage and of each verification run".
5. The Verification paragraph says which version the first two runs examined and records this run; this report
   likewise.
6. The record of the searches includes this run's queries (ten web searches in all).
7. Package and metadata: `independent_run_2/`, README, re-run log; Zenodo date 2026-10-17 and new checksums.

## Relation to the literature, novelty and scope
- **Searches (17 October 2026; by the first stage, the three runs and at the writing).** OpenAlex (29 works citing
  Calegari–Walker 2011, titles; the count was confirmed again by the run on the final text), Semantic Scholar (37
  citing works, titles and available abstracts), zbMATH Open ("ziggurat rotation number": 3 documents —
  Calegari–Walker 2011, Gordenko 2015, Chowdhury 2017; "ziggurat": 88, mostly on a sampling algorithm; further
  queries on word maps, trace polynomials and rotation numbers of words, among them title queries that return
  Bandman–Zarhin (2016) and Rani (2026) on PSL(2,C), SL(2,C) and SU(2), and Gnutov–Gordeev (2020)), arXiv API and
  listings, Crossref, and ten web searches. The arXiv API, OpenAlex and Semantic Scholar refused part of the requests
  (HTTP 429; OpenAlex refused all list queries of the last run because the free daily budget of the shared address
  was used up); two AMS meeting-abstract pages returned by a web search refused automated access (HTTP 403) and were
  not read.
  - No printed answer to Question (Q1) was found, and no statement of Theorem A for arbitrary words.
  - Closest results: Calegari–Walker (2011), Example 1.1 ([a,b]); Kim–Koberda–Mj (2019), Theorem 4.21 (peripheral or
    splitting curves); Gordeev–Kunyavskiĭ–Plotkin (2018), Proposition 2.6 (split semisimple elements always in the
    image of a word map on PSL(2,R); all semisimple elements if an involution is). Word maps over finite fields
    (Bandman–Garion 2012) and over algebraically closed fields concern other questions.
  - The problem list of Nariman–Yazdi (2025) cites Calegari–Walker for ziggurats only and lists neither question.
- **What was read.** Oberwolfach Report 35/2012, pp. 2173–2175 (and its problem session, text layer); Calegari–Walker
  (2011), journal version, Sections 1, 2 and 4.2 (numbering used in the note; arXiv v2 read by run A); Mann,
  arXiv:1510.00728v1, Section 6.2 with Lemma 6.18 (numbering used); Gordeev–Kunyavskiĭ–Plotkin, arXiv:1808.02303v1,
  Section 2 with the proof of Proposition 2.6 (numbering used); Kim–Koberda–Mj, arXiv:1610.04098v4, Section 4.4
  (numbering used); Goldman (2009),
  Theorem A and Section 2.2; Goldman (2003), Section 2.6; Calegari (2009), Section 4. Abstracts only: Chowdhury
  (2017), Gordenko (2015), Matsuda (2015); text searched: Chowdhury's thesis (2018), Nariman–Yazdi (2025).
  Hui–Larsen–Shalev (2015) is cited as stated in Gordeev–Kunyavskiĭ–Plotkin; Horowitz (1972) for background only.
  All DOIs were checked with Crossref on 17 October 2026.
- **Caveats.** A search that finds nothing is not a proof of novelty; Theorem A is elementary and may be known. The
  proofs of the cited results (the defect bound for rot~, the interval property of X(w,r,s), the Fricke–Vogt theorem)
  were not re-checked.
- **Scope.** (Q1) is answered negatively in its literal reading; (Q2) is open as far as we know. Suggested status of
  the record: partially solved (Q1 answered no; Q2 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/