# Verification report — OWR-4428-011 and OWR-4428-012 (Grabowski: automaton series; the l2-Betti numbers of the lamplighter group)

Verification date: 2026-10-14 (first round: two runs on the first written version; second round: one run on the
final text).

**Verdict.** The note gives **partial answers**; every statement it makes is proved. Three independent AI-assisted
verification runs found no error in a theorem, proposition or lemma. **Record-level status: both corpus records
remain PARTIAL** (partially solved): OWR-4428-011 bundles Questions 4, 5 and 6 of Grabowski's Oberwolfach abstract,
and OWR-4428-012 is Question 5 alone.
- **Settled.** The "in particular" part of Question 6: there are an automaton A and a rational q with F_A(q)
  irrational and algebraic (Theorem 1: the automaton A_2 with n = 2, C = 2; F_{A_2}(x) = x^2 (1 - 4x^4)^(-1/2);
  F_{A_2}(1/2) = sqrt(3)/6, a root of 12t^2 - 1).
- **New element of C(Z/2 wr Z).** An explicit T in M_2(Q[Z/2 wr Z]) with 4T integral and
  dim_vN ker T = sqrt(5)/20, a root of 80t^2 - 1 (Theorem 2), with two proofs. This answers, for p = 2, Grabowski's
  question (Invent. Math. 198 (2014), Section 1c) whether C(Z/p wr Z) contains irrational algebraic numbers; by the
  monotonicity of C under subgroups it also gives sqrt(5)/20 in C(Z/p wr Z) for even p (Remark 7.1).
- **Partial.** The "possible values" part of Question 6: every non-negative rational number is a value
  (Proposition 4.1); regular languages with one linear letter-count constraint give algebraic values
  (Proposition 4.2, Corollary 4.3, using Furstenberg's theorem on diagonals as a citation); the value set is not
  determined.
- **Not new mathematics.** The questions and the automaton framework are Grabowski's (OWR 40/2010; Invent. Math.
  2014; Groups Geom. Dyn. 2016). The Fourier/component method for kernel dimensions is the one of Dicks–Schick,
  Austin and Grabowski; the letter-count idea is the one Ara and Claramunt used for irrational algebraic *virtual*
  l2-Betti numbers of Z/2 wr Z (J. Algebraic Combin. 54 (2021), Theorem 4.13 and Remark 4.14; Forum Math. 34
  (2022)). Lück's approximation theorem, Pontryagin duality and Furstenberg's theorem are standard. Irrational
  algebraic l2-Betti numbers arising from *other* groups were known: every non-negative real number arises from some
  finitely generated group (Grabowski, Invent. Math. 2014, Theorem 1.3; Pichot–Schick–Żuk), and every non-negative
  number with computable binary expansion, in particular every non-negative algebraic number, from some finitely
  presented group (Grabowski 2014, Theorem 1.4).
- **Open.** Question 4; a description of C(Z/2 wr Z), in particular Problem 4 of Grabowski (GGD 2016:
  is sqrt(2) in C(Z_2 wr Z)?); Question 6(a) of Grigorchuk–Simanek (single self-adjoint elements); the set of
  values of F_A (in particular, whether it contains transcendental numbers); odd p.

The note is unrefereed.

## Statement checked
- **Primary source.** Ł. Grabowski, "On the Atiyah problem for the lamplighter groups", abstract in Oberwolfach
  Report 40/2010, *Actions and Invariants of Residually Finite Groups: Asymptotic Methods* (workshop of 5–11
  September 2010), Oberwolfach Rep. 7 (2010), no. 3, pp. 2335–2391, doi:10.4171/OWR/2010/40; the abstract is on
  pp. 2366–2369.
  - Read in the publisher's open PDF (https://ems.press/content/serial-article-files/46298; 575,583 bytes, 58 pages,
    md5 ec910147fe8cae5a1335e355f7f1873c), fetched anonymously at the writing and again by the run of the second
    round (same file); pages 2366–2369 rendered as images and read, the title page read.
  - p. 2366: a non-negative real r is an l2-Betti number arising from G iff there is θ in M_m(QG) with
    dim_vN ker θ = r; C(G) is the set of these numbers; known results recalled there: C(finitely generated groups)
    = R≥0 and C(finitely presented groups) ⊄ Q (Grabowski; Pichot–Schick–Żuk), C((Z/2 wr Z)^3) ⊄ Q (Grabowski),
    irrational algebraic numbers in C(Z/p wr F_d), d ≥ 2, p ≥ 2d − 1 (Lehner–Wagner).
  - p. 2367: Theorem 2 (C(Z/p wr Z) contains transcendental numbers, p ≥ 2); Questions 3 and 4.
  - p. 2368: Question 5 ("What is C(Z/2 wr Z)?"); the remark that the component families of the transcendental
    example are encoded by a regular language; the definitions for Question 6 (integers n and C; an automaton whose
    letters are n × n integer matrices with entries in {−C, …, C}; a matrix of dimension at least n is recognized
    iff its entries vanish outside the n-neighbourhood of the diagonal and the automaton recognizes the word of
    subsequent n × n minors along the diagonal; F_A(x) = Σ dim ker M_i · x^{d(M_i)}); Question 6.
  - The front matter gives the workshop dates 5–11 September 2010 and the organisers M. Abért, D. Gaboriau,
    F. Grunewald; Crossref lists "George Glauberman" instead of Abért among the contributors of the report (a
    metadata discrepancy; the note cites the report by the PDF).
- **Corpus records.** ulamai/UnsolvedMath, version 1.6.0 (snapshot c6e7f41b…): OWR-4428-011 (record 30001585) bundles
  Questions 4, 5, 6 (Question 4 written "for some prime p", equivalent to the source's "for some p", since
  Z/q wr K ⊆ Z/p wr K for a prime q dividing p); OWR-4428-012 (record 30001586) is Question 5. Both carry the
  upstream status `partially_solved`.
- **Later statements of the questions.** Grabowski, Invent. Math. 198 (2014), Section 1c (arXiv 1004.2030v4):
  Question 3 "What is C(Z/2Z wr Z)?" (cited as Question 1.7 by Grigorchuk–Simanek and Ara–Claramunt) and the
  question whether C(Z/pZ wr Z) contains irrational algebraic numbers; Grabowski, Groups Geom. Dyn. 10 (2016)
  (arXiv 1009.0229v4 and the journal version): Problem 4 (a description of C(Z_2 wr Z) different from the definition;
  is sqrt(2) in it?); Grigorchuk–Simanek, Trans. AMS 374 (2021), Section 7, Question 6(a) (the values μ_m(0) for
  self-adjoint m in Q[L]).

## Readings
| Reading | Answer | Where |
|---|---|---|
| Sliding n × n windows W_1, …, W_{d−n+1}; band \|j−k\| ≤ n−1 (the entries seen by the windows) — used | yes: F_{A_2}(1/2) = sqrt(3)/6 | Definition 2.1, Theorem 1 |
| Block-diagonal reading (disjoint blocks, block-diagonal matrices) | every value rational, answer no; rejected (the source allows every dimension ≥ n, speaks of a neighbourhood of the diagonal, and the question would be trivial) | Section 2.2 |
| Disjoint blocks with a band-shaped region | band entries between blocks unconstrained, series in general undefined | Section 2.2 |
| Band \|j−k\| ≤ n | entries at distance n in no window, coefficients in general infinite; if bounded by C, not determined by the windows — not treated | Section 2.2 |
| n = 1 | kernel dimension additive, F_A rational, values rational | Section 2.2 |
| Window size n ≥ 3 | same conclusion: F(1/2) = sqrt(3)/6 minus a rational | Remark 3.6 |
| Exponent = length of the window word | value sqrt(3)/3 | Remark 3.7 |
| Kernel over Q, R or C | the same | Definition 2.1 |
| C(G) with right instead of left multiplication | the same number | Lemma 5.1 |
| C(G) with matrices over ZG (GGD 2016) | 4T is integral, same kernel | Section 5.2 |
| Normalized trace | sqrt(5)/40 (still irrational algebraic) | Section 5.1 |
| Question 5 for single self-adjoint elements (Grigorchuk–Simanek Q6(a)) | not answered (T is a 2 × 2 non-self-adjoint matrix) | Section 7 |
| Grabowski's question for odd p | not answered; even p follows from p = 2 | Remark 7.1 |

## Results in the paper
- **Theorem 1.** The automaton A_2 (n = 2, C = 2; six live letters X_2, X_1, P_2, P_1, E, Z; states q0, qA, qB,
  qE, qacc, qdead; eight live transitions). Lemma 3.1 (language), Definition 3.2 (counter matrices M_s; rows
  u_0 − v_0, u_{i+1} − a_i u_i, v_{i+1} − b_i v_i, u_N − v_N), Lemma 3.3 (windows), Lemma 3.4 (recognized
  matrices = M_s, s in {U,D}^N), Lemma 3.5 (dim ker = [Π a = Π b], i.e. [#U = #D]). Count: binom(2k,k) balanced
  words of length 2k; F_{A_2}(x) = x^2 (1 − 4x^4)^(−1/2); value sqrt(3)/6.
- **Section 4.** Proposition 4.1 (all non-negative rationals); Proposition 4.2 (one letter-count constraint:
  F = x^2 G_{L,c}(x^2)); Corollary 4.3 (G_{L,c} algebraic, by Furstenberg's diagonal theorem for c ≢ 0, rational
  for c ≡ 0); Example 4.4 (central trinomial coefficients, F(1/2) = 1/sqrt(5)); Remark 4.5 (open questions).
- **Theorem 2.** T in M_2(Q[Z/2 wr Z]) built from π_j = (1 − e_j)/2, χ = π_0 π_1, α = 1 + π'_0 π_1,
  β = 1 + π_0 π'_1, S = t^2 (formula (2)), expanded in (3); Lemma 5.1 (left and right multiplication give the
  same dimension); Lemma 5.2 (Fourier picture: lamp e_j at fibre index k acts by (−1)^{x_{j−k}}, t shifts the index);
  Lemma 5.3 (trace formula for an explicit measurable field of kernel projections); Lemma 5.4 (rows at a position:
  marker 11, letters U = 01, D = 10, I = 00); Lemma 5.5 (segments between consecutive markers are closed blocks
  equal to M_s); proof: cylinders E_{N,s,j} of measure 4^{−(N+2)}, dim_vN ker T = Σ T_N 4^{−N−2}, generating
  function (1 − 2y − 3y^2)^(−1/2), value sqrt(5)/20; Remark 5.6 (the same series from Grabowski's Proposition 13,
  which applies to T*T, not to T itself).
- **Section 6 (second proof).** Lemma 6.1 (finite Fourier transform on G_K = Z/2 wr Z/K); Proposition 6.2:
  d_{2P}(T) = Σ_{N=0}^{P−2} T_N 4^{−N−2} + T_{P−1} 4^{−P} + 2^{1−P}/P; Lück's approximation theorem along K = 2^i
  (in the form for matrices over ZG and finite-index normal chains, pointer to Kirstein–Kremer–Lück 2025); Remark 6.3
  (error terms: single-marker configurations and truncated tail, each ≤ (1/4)(3/4)^P; marker-free configurations
  only 2^{1−P}/P); Table 2.
- **Section 7.** Relation to Grabowski 2014 (answered for p = 2), Remark 7.1 (even p; nothing for odd p),
  GGD Problem 4 (not answered), Grigorchuk–Simanek Q6(a) (single self-adjoint elements; not literally answered),
  virtual versus genuine values (Ara–Claramunt; sqrt(5)/20 differs from their numbers), the identity
  dim_vN ker T = (1/4) F(1/2) for the automaton of Example 4.4.

## 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, exact arithmetic, about one minute; 14,386
  checks, ALL CHECKS PASSED): the transition table over all 625 letters; DFS over all 625 letters (3,197,500
  transitions): the 2047 accepted words of length ≤ 21 are those of Lemma 3.1; brute force over all 9,844,380
  tridiagonal matrices with entries in {−2..2} of size ≤ 4; the 2047 recognized matrices of size ≤ 22 equal the M_s
  and have kernel [#U = #D]; coefficients of x^2(1−4x^4)^(−1/2); sqrt(3)/6 enclosed; Proposition 4.1 (p/s = 1) and
  Example 4.4 over all 625 letters; the odd-label kernel rule of Proposition 4.2 (2047 words); T built in the group
  ring equals (3), 4T integral; left and right multiplication by 4T on Q[G_K]^2 in the group ring for K ≤ 5 (exact);
  fibre operators for all x and K ≤ 14 (agree with the direct values for K ≤ 5 and with Proposition 6.2 for even K);
  block structure on all 21,844 configurations with P ≤ 7 (36,409 segment blocks equal to 4 M_s); T_N, the
  generating function, sqrt(5)/20 enclosed, the identity of Remark 6.3 for P ≤ 60.
- **Programs with which the results were first obtained** (`original/`): `verify_construction.py` (41 s),
  `possible_values_check.py` (8 min), `lamplighter_check.py` (27 s).
- **Verification run A** (`verification_run_A/`): `vA_automaton.py` (recognized matrices up to size 26, 12.8 million
  transitions), `vA_direct_groupring.py` (group ring, left and right, K ≤ 8, exact for K ≤ 6), `vA_fourier_model.py`
  (own Fourier model, K ≤ 17), `vA_blocks.py` (87,380 configurations, 167,481 segment blocks).
- **Verification run B** (`verification_run_B/`): `automaton_enum.py` (brute force size ≤ 4; enumeration up to
  size 30), `lamp_regular.py` (group ring, K ≤ 9, exact for K ≤ 5), `lamp_fourier.py` (fibre operators, K ≤ 22,
  0 block mismatches), self-tests with deliberately wrong automata and operator (all detected), and re-runs of the
  three original programs.
- **Independent run of the second round** (`independent_run_2/`, programs written for that run):
  `run2_automaton.py` (52 s; 32 checks): all tridiagonal matrices accepted by A_2 up to size 28 (16,383 = the M_s),
  windows, exact kernels, coefficients, sqrt(3)/6; Remark 3.6 for n = 3 (size ≤ 16; every letter passed to the
  transition function) and n = 4 (size ≤ 12; every letter without non-zero entries at distance ≥ 2, the others
  being rejected by the first rule); Remark 3.7; Proposition 4.1 for four rationals; Proposition 4.2 for the
  language of the words over {a,b,c} without two consecutive a's (weights (2,−1,0), 258 recognized matrices up to
  size 12); the diagonal identity of Corollary 4.3 and a cubic example. `run2_groupring.py` (52 s; 271 checks):
  T from (2) = (3), 4T integral; left and right multiplication on Q[G_K]^2 for K ≤ 8 (exact for K ≤ 6): d_K = 3/2,
  5/4, 1/3, 3/8, 1/5, 5/24, 71/448, 19/128; Remark 7.1 on Q[Z/4 wr Z/K], K ≤ 4; a wrong operator detected.
  `run2_fibres.py` (12 min; 66 checks): the change of basis of Lemma 6.1 built explicitly for K ≤ 4 (block diagonal
  with the blocks of rule (4); a wrong sign rule fails); fibre operators for all x and K ≤ 20 (exact for K ≤ 10),
  equal to the group-ring values and to Proposition 6.2 for every even K ≤ 20 (d_18 = 1039/9216,
  d_20 = 294457/2621440); the trace formula on the finite quotients with exact kernel projections, K ≤ 10; Table 2;
  Remark 6.3 for P ≤ 80; sqrt(5)/20.
- **Re-runs.** At the writing (2026-10-14) all programs with a recorded output except `lamp_regular.py 9` and
  `lamp_fourier.py 22` were run again from a copy (`reproducibility/RERUN_LOG.txt`). The run of the second round ran
  all of them again from an extracted copy of the archive, the two long runs included: all 16 recorded outputs were
  reproduced up to running times (`reproducibility/independent_run_2/rerun_package_v1.txt`); the recorded
  `lamp_regular_K9.out` corresponds to the option `--exact-upto 0` (README corrected).

## Independent verification runs
The results were first obtained, with proofs, together with the programs in `original/`. Two rounds of
independent verification, all AI-assisted, followed; each run wrote its own programs. In the first round runs A and
B examined the first written version, each from the definitions of the source and without seeing the other's
report. In the second round a further run examined the final text, the parts written at the writing stage first.

| Item | Run A | Run B | Run of the second round |
|---|---|---|---|
| Statement of Question 6 and definition of C(G) against the source (OWR pp. 2366–2368) | CONFIRMED (publisher PDF read) | CONFIRMED (text of the publisher PDF) | CONFIRMED (publisher PDF rendered and read); the known results recalled on p. 2366 to be stated (fix 1) |
| Readings (sliding windows, band) | CONFIRMED_WITH_FIXES (block-diagonal reading to be stated and rejected) | CONFIRMED | CONFIRMED; two overstatements softened (fix 9) |
| Lemmas 3.1–3.5, Theorem 1 | CONFIRMED (re-derived; recognized matrices up to size 26) | CONFIRMED (re-derived; brute force size ≤ 4; enumeration up to size 30) | CONFIRMED (re-derived; all accepted tridiagonal matrices up to size 28) |
| Remark 3.6 (other window sizes) | checked by reading only | checked by reading only | CONFIRMED (re-derived; exhaustive computation for n = 3, 4) |
| Propositions 4.1, 4.2, Corollary 4.3 | CONFIRMED; case c ≡ 0 to be treated; Furstenberg's theorem number not checked | CONFIRMED (Furstenberg cited, not re-proved) | CONFIRMED (re-derived; Proposition 4.2 computed for a non-trivial language; Furstenberg cited) |
| Lemma 5.1 (left/right), Lemma 5.3 (trace formula) | left/right checked; Lemma 5.3 in its first form | standard facts checked | CONFIRMED (re-derived; finite trace formula computed exactly for K ≤ 10) |
| Theorem 2: Fourier picture, rows, segments, measure bookkeeping, generating function | CONFIRMED (re-derived; also via Grabowski's Proposition 13) | CONFIRMED (re-derived) | CONFIRMED (re-derived; Lemma 6.1 and rule (4) checked by an explicit change of basis); Remark 5.6: Proposition 13 applies to T*T, not to T (fix 2) |
| Theorem 2: computations on finite quotients | group ring K ≤ 8 (left = right), own Fourier model K ≤ 17, blocks P ≤ 8 | group ring K ≤ 9, fibre operators K ≤ 22, 0 mismatches | group ring K ≤ 8 (left = right) and Z/4 wr Z/K, K ≤ 4; fibre operators K ≤ 20 |
| Second proof via finite quotients and Lück's theorem | derived the exact formula d_{2P} | derived the same formula independently; suggested it as a second proof | CONFIRMED (re-derived; hypotheses of Lück's theorem checked against Kirstein–Kremer–Lück; pointer added, fix 6) |
| Remark 7.1 (even p) | suggested | — | CONFIRMED (re-derived; finite check) |
| Cited statements (Gra14, Gra16, AC21, AC22, GS21) | CONFIRMED | CONFIRMED | CONFIRMED, Gra16 and AC21 also in their journal versions (fixes 3–5) |
| `check_note.py` | — | — | read completely; output reproduced |
| Novelty | plausible, not refuted; GGD Problem 4 omitted in the first write-up | no prior work found; wording to name the sources | no prior work found; context of other groups added (fix 1) |

No run found an error in a theorem, proposition or lemma.

**Corrections required by runs A and B**, all applied in the note (details in `audit/fixes_applied.md` of the
working folder): (A1) the block-diagonal reading is stated and rejected, Section 2.2; (A2) the record-level status
is stated at the top; (A3) GGD Problem 4 is cited and declared not answered; (A4) Grigorchuk–Simanek Question 6(a)
concerns single self-adjoint elements; (A5) the wording on works citing Grabowski's papers and the number of Remark
14(ii); (A6) the case c ≡ 0 in Corollary 4.3, no unchecked theorem number of Furstenberg (1967); (A7) the source of
the error on the finite quotients, Remark 6.3; (A8) even p, Remark 7.1, and the novelty wording; (B1) the sources of
the novelty statement, the Grabowski–Virág manuscript, Ara–Claramunt's family (1/8) sqrt(2^t/(2^t − 1)); (B2)
p = 2 only; (B3) the remark on the final GGD paper; (B4) the second proof, Section 6.

**Corrections required by the run of the second round**, all applied:
1. The known irrational algebraic l2-Betti numbers of other groups (Grabowski 2014, Theorems 1.3 and 1.4;
   Pichot–Schick–Żuk; C((Z/2 wr Z)^3) ⊄ Q) are stated: Sections 1.1 and 1.3, "Scope and priority".
2. Remark 5.6: Grabowski's Proposition 13 integrates over the points and needs finite components; for T the
   components are infinite; it applies to T*T (same kernel), whose fibre operators are the direct sums of M_s^*M_s.
3. Section 7: neither the arXiv version v4 nor the journal version of GGD contains the announced statement on
   algebraic numbers of degree k (journal version read by this run); the numbering of Problem 4, Proposition 13,
   Remark 14 and Lemma 18 is the same in both; "What was read" updated.
4. "What was read": the journal version of Ara–Claramunt 2021 (open access) has the same numbering of Theorem 4.13,
   Remark 4.14 and Question 4.15.
5. "[AC22, p. 2]" (a page of the arXiv version) replaced by "[AC22, Introduction]".
6. Second proof: a pointer to a statement of Lück's theorem in the form used (matrices over ZG, nested normal
   subgroups of finite index with trivial intersection): Kirstein–Kremer–Lück, Contemp. Math. 816 (2025),
   Theorem 6.2 and Section 7.1 (arXiv numbering), new bibliography entry checked with Crossref.
7. Section 8: "Proposition 4.1 for r = 1" now "for p/s = 1 (C = 2, q = 1/2)"; the same in the README.
8. Section 3.2: the letter I is used "from Section 4 on" (it is also used in Section 7).
9. Section 2.2: for the band |j−k| ≤ n, only window words of length ≥ 2 belong to infinitely many matrices, and the
   coefficients are "in general" infinite; the variant with disjoint blocks is "in general" undefined.
10. The Verification paragraph and this report record the second round; the late parts are no longer described
    as unexamined; the search count is seven web searches; `independent_run_2/` added with a README.
11. README: the command line of `lamp_regular_K9.out` (`--exact-upto 0`) corrected; `independent_run_2/` listed.
In addition the PDF title metadata now reads "ℓ²-Betti" (it read "2-Betti").

**Changes made when the note was written** (examined by the run of the second round): Lemma 5.3 is proved directly
for the explicit measurable field of kernel projections; Remark 3.6 (other window sizes) is written out with a
construction of the automaton; Lemma 5.1 is written out; Remark 7.1 is added; `check_note.py` is new. The sqrt(2)
variant mentioned in run A's report is not part of the note.

## Relation to the literature, novelty and scope
- **Searches** (14 October 2026: when the results were first obtained, by the runs of the first round, in part at
  the writing, and again by the run of the second round): the arXiv API (keyword searches combining "lamplighter" with
  Betti numbers, the Atiyah problem, von Neumann dimension or spectral measures; "irrational algebraic" or "virtual"
  with Betti numbers; regular languages or automata with Betti numbers and kernels; wreath products with l2-Betti
  numbers; author searches; all titles containing "lamplighter"), Crossref, zbMATH Open, OpenAlex and Semantic
  Scholar (works citing Grabowski's three items, Ara–Claramunt's two papers, Lehner–Wagner, Pichot–Schick–Żuk and
  Grigorchuk–Simanek, by title and abstract; 41 citing works in OpenAlex at the second round), and seven web
  searches. Crossref checks of all 15 DOIs of the bibliography (all data agree).
  - Question 6 appears only in the Oberwolfach abstract; GGD mentions regular languages only in Remark 14(ii); no
    citing work found treats the series F_A.
  - No earlier non-virtual irrational algebraic l2-Betti number of a group Z/p wr Z was found in these sources. The
    known values there: transcendental and other irrational numbers (Grabowski 2016; Ara–Claramunt 2021, Theorem
    4.5), irrational algebraic numbers for the free lamplighters Z/p wr F_d (Lehner–Wagner 2013), and irrational
    algebraic virtual l2-Betti numbers of Z/2 wr Z (Ara–Claramunt 2021, Theorem 4.13 and Remark 4.14; 2022). For
    other groups irrational algebraic values were known (Grabowski 2014, Theorems 1.3 and 1.4).
  - An unpublished manuscript of Grabowski and Virág ("Random walks on lamplighters via random Schrödinger
    operators", cited in Grigorchuk–Simanek) could not be inspected.
- **What was read.** The Oberwolfach report (publisher PDF: title page, pp. 2366–2369); GGD in arXiv v4
  (introduction, Sections 1, 3, 4) and in the journal version (publisher's PDF; same numbering of Problem 4,
  Proposition 13, Remark 14, Lemma 18; no statement on algebraic numbers of degree k); Invent. Math. in arXiv v4
  (Section 1; its numbering is used); Ara–Claramunt, J. Algebraic Combin., arXiv v2 (Sections 1–4, the proof of
  Theorem 4.5 in outline; Section 5 not read) and the journal version (open access; Theorem 4.13, Remark 4.14,
  Question 4.15 with the same numbers); Ara–Claramunt, Forum Math., arXiv v2 (introduction); Grigorchuk–Simanek,
  arXiv v2 (Section 7); Kirstein–Kremer–Lück, arXiv v1 (Sections 6.1–6.4 and 7.1). Lehner–Wagner: abstract and
  the summary in the report. Not read: Lück (1994, 2002) and Furstenberg (1967) (cited as standard; ScienceDirect
  refused automated access to Furstenberg's paper, HTTP 403); Dicks–Schick, Grigorchuk–Żuk, Austin,
  Pichot–Schick–Żuk (statements taken from Grabowski's texts).
- **Caveats.** The literature search is not a proof of novelty. Furstenberg's theorem is used only for Corollary 4.3;
  Lück's theorem only for the second proof of Theorem 2. The note does not reduce T to a single self-adjoint element.
- **Scope.** Theorem 1 (the "in particular" part of Question 6) and Theorem 2 (sqrt(5)/20 in C(Z/2 wr Z)) are, to
  our knowledge, new; no priority is claimed. Both records stay partially solved.

## 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/