# Verification report — OWR-4428-012, note B (genuine witnesses for virtual l2-Betti numbers of Ara–Claramunt; irrational algebraic l2-Betti numbers of Z/p wr Z for every p)

Verification date: 2026-10-15 (UTC). The two results of the note were first obtained separately (Part I: Theorems A
and C; Part II: Theorem D); each part was examined by two independent AI-assisted verification runs, A and B, on
its first written version; the note was then written with their corrections. The final text, and in particular the
general Theorem 3.7 written at that stage, was then examined by a second independent AI-assisted verification run
(section "Second independent verification run (final text)" below), whose five required corrections were made.

**Verdict.** The note gives genuine witnesses for the listed family of virtual l2-Betti numbers of Ara and
Claramunt and answers Grabowski's question (Invent. Math. 2014, §1c) for every p >= 2. Every statement it makes is
proved. The four verification runs re-derived both proofs of both parts, and the second independent verification
run re-derived Theorem 3.7 (both proofs), Proposition 3.8 and all other proofs of the final text; no run found an
error in a theorem, proposition or lemma. **Record-level status: the corpus record OWR-4428-012 ("What is C(Z/2 wr Z)?") remains
PARTIAL** (partially solved): new elements of C(Z/2 wr Z) are given, no description of the set.
- **Settled.** Theorem A: an explicit T in M_2(Q[Z/2 wr Z]) built from five lamp bits and the shift t^5, with 32T
  integral, has dim_vN ker T = (1/16) Σ_N W_N 32^(-N) = (1/4) sqrt(2/7) = sqrt(14)/28, W_N = Σ_k C(N,2k) C(2k,k)
  25^k 14^(N-2k) (a root of 56X^2 − 1). Theorem C: a family T^(n) with n + 2 lamp bits, 2^(n+2) T^(n) integral,
  dim_vN ker T^(n) = (1/8) sqrt(2^n/(2^n − 1)) for every n >= 2 (n = 2: sqrt(3)/12; n = 3: Theorem A). Corollary B:
  these numbers lie in C(Z/2 wr Z) (and in C(Z/p wr Z) for even p); every number (1/8) sqrt(2^t/(2^t − 1)),
  t >= 3, listed by Ara and Claramunt after their Theorem 4.13 as a virtual l2-Betti number is genuine; for
  (1/4) sqrt(2/7) this answers the question of their Remark 4.14 with a concrete matrix over Z Gamma. Theorem D: for
  every integer p >= 2, with rho_j = (1/p)(1 + e_j + ... + e_j^(p−1)) and the two-track counter template, an explicit
  T_p in M_2(Q[Z/p wr Z]) with p^2 T_p integral has dim_vN ker T_p = ((p−1)/p)^2 sqrt((p−1)/(p+3)), irrational
  algebraic of degree 2 since (p−1)(p+3) = (p+1)^2 − 4 is never a square (p = 3: 4 sqrt(3)/27; p = 5:
  8 sqrt(2)/25; p = 2: sqrt(5)/20 again). Corollary E: C(Z/p wr Z) contains irrational algebraic numbers for every
  p >= 2. Primality of p is not used. All three theorems are instances of Theorem 3.7 (the counter template for any
  p, any number r of lamp bits and any partition of the 2^r bit patterns into markers, U, D, I with markers present):
  dim_vN ker T = q_M^2 ((1 − q_I)^2 − 4 q_U q_D)^(−1/2) <= q_M; two proofs (Pontryagin duality with counter
  matrices; Lück's approximation theorem with the exact finite formula of Proposition 3.8).
- **Not new mathematics.** The questions, the definition of C(G) and the frameworks are Grabowski's (Oberwolfach
  Report 40/2010; Invent. Math. 198 (2014); Groups Geom. Dyn. 10 (2016)) and Ara and Claramunt's (J. Algebraic
  Combin. 54 (2021), arXiv:2005.12814). The balanced-word counter (balanced words counted through (1 − 4z)^(−1/2))
  is the mechanism by which Ara and Claramunt obtained the listed family as virtual values (Theorem 4.13); the matrix
  form, the Fourier picture and the decomposition into counter matrices are those of the two companion notes
  (A. Ferudun, doi:10.5281/zenodo.23377729, sqrt(5)/20; doi:10.5281/zenodo.23380620, sqrt 2). The bit [x_j ≠ 0]
  for general p corresponds to Grabowski's splitting of Z/p into {0} and {1, ..., p−1} (GGD 2016, §4). Lück's
  approximation theorem and Pontryagin duality are standard.
- **New, to our knowledge.** Theorems A and C (genuine witnesses for the listed family) and Theorem D for odd p
  (irrational algebraic elements of C(Z/p wr Z)). No priority is claimed.
- **Open.** A description of C(Z/2 wr Z) (the record; first part of GGD Problem 4); whether the other elements of
  G(Z/2 wr Z, Q) of Ara–Claramunt (Proposition 4.11, Theorem 1.1) lie in C(Z/2 wr Z), and whether
  C(G,Q) = G(G,Q) ∩ R_+ in general; a description of C(Z/p wr Z) for odd p; algebraic numbers of odd degree > 1 in
  some C(Z/p wr Z); single self-adjoint elements (Grigorchuk–Simanek, Question 6(a)).

The note is unrefereed.

## Statement checked
- **Corpus record.** ulamai/UnsolvedMath, version 1.6.0, OWR-4428-012 (record 30001586): "What is C(Z/2 wr Z), the
  set of l2-Betti numbers arising from the lamplighter group Z/2 wr Z?", upstream status `partially_solved`. It is
  Question 5 of Grabowski's abstract in Oberwolfach Report 40/2010, p. 2368 (doi:10.4171/OWR/2010/40), and Question 3
  of Grabowski, Invent. Math. 198 (2014), §1c (arXiv v4 numbering; cited as Question 1.7 by Grigorchuk–Simanek and
  Ara–Claramunt 2022).
- **Question of Ara and Claramunt.** J. Algebraic Combin. 54 (2021) 1201–1245, doi:10.1007/s10801-021-01044-8,
  arXiv:2005.12814 (not 2005.12374, which is their Forum Math. 34 (2022) paper). Theorem 4.13: (1/4) sqrt(2/7) is in
  the subgroup G(Gamma, Q) generated by C(Gamma, Q); the paragraph after it: (1/8) sqrt(2^t/(2^t − 1)), t >= 3 (the
  listed family). Remark 4.14: (1/4) sqrt(2/7) is "virtual" (a difference of two l2-Betti numbers); is it itself one,
  and "a concrete matrix over QGamma witnessing this fact"? Same numbers in the arXiv version v2 and in the journal
  version (Theorem 4.13 on p. 1238, Remark 4.14 on p. 1239).
- **Question of Grabowski.** Invent. Math. 198 (2014) 27–69, doi:10.1007/s00222-013-0497-5, §1c (arXiv v4, p. 5;
  cited by Ara–Claramunt as p. 32 of the journal version, not compared): "This raises the question whether
  C(Z/pZ wr Z) contains irrational algebraic numbers." The next sentence announces degree-k numbers in
  C((Z/pZ wr Z)^k), "in a future version" of the GGD paper; this concerns direct powers, and neither GGD arXiv v4 nor
  the journal version contains such a result. The question is not a corpus record. For p = 2 (and every even p) it
  was answered by the companion notes; in the literature it was open for every p.
- **Definitions.** OWR p. 2366: r >= 0 is an l2-Betti number arising from G iff r = dim_vN ker theta for some
  theta in M_m(QG). Invent. Math. §1 (arXiv v4 pp. 2–3, before §1a): trace tr ⊗ tau_G on M_k(QG);
  dim_vN ker T := dim_vN ker T*T for non-self-adjoint T. GGD p. 795: "a matrix" over the integral group ring;
  "C(Gamma) = N" for torsion-free groups. Ara–Claramunt Definition 2.2: A in M_k(KG) acting on the left,
  unnormalized trace.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Unnormalized trace tau_m = Σ_i tau(·)_ii (the sources' convention) — used | the values as stated | §1.2, §2.1 |
| Normalized trace (1/m) tau_m | every value halved (still irrational) | §1.2 |
| Matrices over QG (OWR, Invent. Math., Ara–Claramunt) or ZG (GGD) | same answer: the witnesses are square and integral after scaling by p^r | Cor. B, Thm D |
| Left action on columns or right action on rows | same dimension (Lemma 2.1, on G and on finite groups) | Lemma 2.1 |
| Each entry acting by convolution from the right on columns | conjugate to the entrywise adjoint; the set C(G) is unchanged, the witness is then the entrywise adjoint (a verification run found d_4 = 5/9 instead of 29/81 for T_3 itself) | Remark 2.2 |
| Non-self-adjoint T: dim ker T := dim ker T*T (Invent. Math.) | same number, ker T*T = ker T | §2.1 |
| "Genuine witness" (Ara–Claramunt): one matrix, not a difference | a single square matrix for each value | Thms A, C, D |
| Remark 4.14 for all of G(Gamma, Q) | not answered: only the listed family | §1.1, Remark 4.4 |
| Grabowski's question for some p / every p / primes p | answered for every integer p >= 2 | Cor. E |
| Single self-adjoint element of Q[Z/2 wr Z] (Grigorchuk–Simanek) | not answered | §1.3 |

## Results in the paper
- **Section 1.** The questions (Grabowski's quoted from Invent. Math. §1c with the following sentence; Ara–Claramunt's
  Remark 4.14 with the listed family, their Proposition 4.11 and Theorem 1.1 not treated); Theorems A, C, D,
  Corollaries B, E; the template formula (2); credits and open questions.
- **Section 2.** Conventions; Lemma 2.1 (left and right); Remark 2.2 (third reading); Lemma 2.3 (block sums,
  subgroups, Z/2 wr Z in Z/p wr Z for even p); Lemma 2.4 (fibre operators for Z/p wr Z, characters zeta^(Σ g_j x_j),
  theta_x(rho_j) = multiplication by [x_{j−k} = 0]); Lemma 2.5 (trace formula); Lemma 2.6 (finite Fourier transform).
- **Section 3.** Lemma 3.1 (entries; p^r T integral); classes modulo r, symbols (nu(x_{rm+c}), ..., nu(x_{rm+c+r−1})),
  Lemma 3.2 (rows); Definition 3.3 and Lemma 3.4 (counter matrices, kernel [#U = #D]); Lemma 3.5 (segments);
  Lemma 3.6 (omega_N, the series, (1−q_I)^2 − 4q_Uq_D = q_M^2 + 2q_M(q_U+q_D) + (q_U−q_D)^2 >= q_M^2);
  Theorem 3.7 (value and bound; first proof by the direct integral with the cells E_{N,s,j}, indices 2j−1, 2j
  explained); Proposition 3.8 (exact formula for d_{rP}); Lück's theorem in the form used (7); second proof;
  Remark 3.9 (error bound).
- **Section 4.** Lemma 4.1 (the 5-bit pattern sets, counts 8, 5, 5, 14); proof of Theorem A; formula (8) and the worked
  value d_5 = 23/16; Table 1 (d_5, ..., d_200); Lemma 4.2 (family pattern sets); proof of Theorem C; proof of
  Corollary B; Remark 4.3 (five bits is the least number, not forced; six and seven bits also work); Remark 4.4
  (virtual and genuine values; only the listed family; value <= q_M <= 1).
- **Section 5.** Theorem D as the partition M = {11}, U = {01}, D = {10}, I = {00}; proof (expansion, value,
  irrationality); Corollary E (values for p = 3, 5, 7); formula (9) and values for p = 3; signs of the error
  (examined for p = 2, 3, 5, 7, 11 only); Remark 5.1 (primality not used; subgroups); Remark 5.2 (the encoding
  matters; tied to the choice of the idle letter). That the 2014 question is not a corpus record and that Theorem D
  adds no new element of C(Z/2 wr Z) is said in Section 1.2.
- **Section 6.** Computations, Verification, Scope and priority, AI disclosure.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. The computations do not test the direct integral of Section 3.4 or Lück's
theorem.
- **The exact program of the note** (`check_note.py`, standard library, exact arithmetic over Q and Q(zeta_p),
  about five minutes, "ALL CHECKS PASSED"): for T (Theorem A), T^(2), T^(4) (Theorem C), T_3, T_5 (Theorem D): the
  operator built from its definition in Q[Z/p wr Z] with the group law; chi, alpha − 1, beta − 1 idempotent and
  orthogonal, their values at all characters of the lamps 0..r−1 (pattern sets, counts (8,5,5,14), (4,2,2,8),
  (16,11,11,26)); Lemma 3.1; integrality; the expansion of Theorem D; left and right multiplication on
  Q[Z/p wr Z/K]^2 directly in the group ring for K <= 7, 7, 6, 4, 3; the fibre operators for all characters and
  K <= 15, 16, 12, 10, 6 with entries in Q(zeta_p), equal to the group-ring values and to Proposition 3.8 for
  K = rP (T: d_5, d_10, d_15 = 23/16, 537/1024, 14227/49152; T^(2): d_4, ..., d_16; T^(4): d_6, d_12; T_3: d_2, ..., d_10;
  T_5: d_2, d_4, d_6) and to the values recorded by the other programs; for all configurations with K = rP <= 15, 16,
  12, 10, 6 and every class: no cross-class entries, the rows of Lemma 3.2, the segment blocks equal to p^r M_s,
  closed and partitioning (125,480 blocks for T), the kernel counts of Proposition 3.8; random windows of the line
  (300 for T, 200 for T_3); Proposition 3.8 for 24 random partitions (p in {2,3}, r <= 3) and for the two encodings
  of Remark 5.2; Lemma 3.4 for all 9,841 words of length <= 8; Lemma 3.6 (enumeration, generating function up to
  N = 40); rational enclosures of the five values; the bound of Remark 3.9 for P <= 40; the signs of §5.1; (p−1)(p+3)
  not a square for p <= 10^5; the counts of Remark 4.3 for r <= 7. Output `check_note_output.txt`.
- **Programs with which the results were first obtained.** Part I (`part1_original/`): group ring for K <= 8 and an
  eigenspace decomposition for K <= 10; fibre operators K <= 16 and the class-0 part at K = 20; rows and segment
  blocks on all configurations P <= 3 and class 0 for P = 4; series; the family for n = 2, 4. Part II
  (`part2_original/`): group ring p = 2, 3, 5, 7, K <= 8, 6, 4, 3; fibre operators K <= 16, 12, 8, 6 (exact in
  Q(zeta_p)); blocks on all characters P <= 8, 6, 4, 3; series; four altered operators; other encodings.
- **Verification runs.** See the next section; programs and outputs in `part1_verification_run_A/`,
  `part1_verification_run_B/`, `part2_verification_run_A/`, `part2_verification_run_B/`.
- **Re-runs.** At the writing (2026-10-15) all 75 runs of the package with a recorded output (all but
  `check_note.py`) were repeated from an extracted copy of a preliminary archive of the package, at most three at a
  time (about 75 minutes of computing time): 70 outputs are identical to the recorded ones (15 byte for byte, the
  others up to running times); the other five agree in every exact value and differ only in printed diagnostics
  (the floating-point singular-value gaps in three outputs of run A of Part I; one additional field printed by a
  later revision of `vA4_window_mc.py`; three header lines printed by `own_fourier.py` when `own_mutations.py` loads
  it) (`reproducibility/RERUN_LOG.txt`). Before that, run B of each part had re-run the original programs of its
  part from a copy, with identical outputs up to running times.
- **Second independent verification run** (final text; `reproducibility/independent_run_2/`, about five minutes):
  its own programs `r2_direct.py`, `r2_fibres.py`, `r2_line.py`, `r2_symbols.py`, `r2_series.py` (described in the
  section "Second independent verification run (final text)" and in the README of that folder), and a second
  re-run of all 76 runs of the package from an extracted copy of `source.zip` (three at a time, about 80 minutes of
  computing time): all exit 0; 14 outputs byte-identical, 60 identical up to running times, 2 differing only in the
  printed diagnostics described in `RERUN_LOG.txt`; `check_note.py` again "ALL CHECKS PASSED".

## Independent verification runs
Each part was examined by two independent AI-assisted verification runs on its first written version; each wrote
its own programs from the definitions, and the two runs of a part did not see each other's reports.

| Item | Part I run A | Part I run B | Part II run A | Part II run B |
|---|---|---|---|---|
| Statement and definitions against the sources | CONFIRMED (OWR publisher PDF, GGD arXiv v4 and journal, AC21 arXiv v2, Invent. Math. arXiv v4); fixes 1-6 | CONFIRMED; citation fixes | CONFIRMED (Invent. Math. §1 and §1c, GGD both versions, OWR, AC21/AC22, KKL, GS21); fixes 1, 2, 7 | CONFIRMED |
| Operator algebra, integrality | CONFIRMED (re-derived; program) | CONFIRMED (program) | CONFIRMED | CONFIRMED (p = 2..7) |
| Fourier conventions, trace formula | CONFIRMED (two derivations) | CONFIRMED (own eigenspace decomposition) | CONFIRMED; third reading found (fix 4) | CONFIRMED (own derivation) |
| Rows, counter matrices, segments | CONFIRMED | CONFIRMED (300 windows of the line) | CONFIRMED | CONFIRMED (all characters) |
| Value, series, irrationality | CONFIRMED; "five bits forced" false (fix 1) | CONFIRMED | CONFIRMED; encoding remark (fix 3), sign change (fix 5) | CONFIRMED |
| Exact finite formula, second proof (Lück) | CONFIRMED (formula for K <= 40); KKL numbering checked | CONFIRMED (K = 5, 10, 15, 20) | CONFIRMED (every even K computed) | CONFIRMED |
| Independent computations | group ring K <= 8, 10; fibre model K <= 15; class sequences K <= 40; Monte Carlo d_200 = 0.13381 ± 0.00029 | group ring K <= 8; eigenspaces K <= 16, 20; family n = 2, 4, 5; re-run of the originals | group ring p = 2,3,5,7; Fourier model to K = 16; Monte Carlo on the line (p = 3: 0.25657 ± 0.00140) | group ring p = 2..7; Q(zeta_p) fibres (p = 3 to K = 12); symbol model; five altered operators; re-run of the originals |
| Literature | not stale | not stale | not stale | not stale |

No run found an error in a theorem, proposition or lemma. **Corrections required by the runs**, all applied (details
in `audit/fixes_applied.md` of the working folder): five lamp bits are the least number, not forced (Remark 4.3);
the statement about Ara–Claramunt restricted to the listed family, with their Proposition 4.11 and Theorem 1.1
mentioned and no expectation attributed to them (§1.1, Remark 4.4); the template formula and the bound <= q_M stated
and proved in the note (Lemma 3.6, Theorem 3.7); Grabowski's question quoted directly, with
its location, the following sentence and the note that the journal version was not compared (§1.1); the location
of the definitions in Invent. Math. §1 (§2.1); Remark 4.14 cited to arXiv:2005.12814 / JACO 54 (2021) and the sqrt 2
problem as GGD Problem 4, p. 796; the form of Lück's theorem with its references (§3.5); the third reading
(Remark 2.2); the encoding remark tied to the idle letter (Remark 5.2); the sign change stated for the examined p
only (§5.1); the wording of Lemma 2.6; the record status of the 2014 question (§1.2); primality not used
(Remark 5.1); the indices 2j−1, 2j (proof of Theorem 3.7); "irreducible over Q"; bibliometric details dropped.

**Changes made when the note was written.** The three constructions are presented as instances of Theorem 3.7, whose
two proofs are the common proofs of the instances written once for all p, r and partitions (run A of Part I had
re-derived the value formula for general probabilities, run A of Part II that for other encodings); `check_note.py`
tests Proposition 3.8 on 24 random partitions. Proofs literally identical to those of the companion notes are
shortened with a reference. `check_note.py` and the reproducibility package were written.

## Second independent verification run (final text)
After the note was written, its final text was examined by a second independent verification run, AI-assisted,
which had not seen the note before. It was the first run to examine Theorem 3.7, the general counter template
(written at the writing stage), and it began with it.

| Item | Verdict of the second run |
|---|---|
| Statement fidelity: OWR 40/2010 pp. 2366–2369, Invent. Math. arXiv v4 §1 and §1c, GGD (journal pp. 795–797, arXiv v4 §4), Ara–Claramunt 2021 (Definition 2.2, Theorem 1.1, Proposition 4.11, Theorem 4.13 and the following paragraph, Remark 4.14; journal pp. 1237–1239), KKL25 arXiv v1 (§5, Theorem 6.2, Conjecture 7.1(1), Remark 7.2), GS21 and AC22 (numbering of the questions), the companion notes (all cited numbers), the corpus record | CONFIRMED (read anew from newly fetched copies) |
| Theorem 3.7: both proofs, the formula q_M^2((1 − q_I)^2 − 4 q_U q_D)^(−1/2), the bound <= q_M, all hypotheses (M non-empty gives q_M > 0, q_I < 1, the full measure of X_0) | CORRECT (re-derived line by line) |
| Proposition 3.8, Remark 3.9, Lemmas 2.1–2.6, 3.1–3.6, 4.1, 4.2, Remarks 2.2, 4.3, 4.4, 5.1, 5.2 | CORRECT |
| Theorems A, C, D as instances; Corollaries B, E | CORRECT |
| Own programs (`reproducibility/independent_run_2/`) | all finite claims reproduced: kernels without Fourier transform (T, T^(2), T^(4) for K <= 7; T_3 for K <= 5; T_5 for K <= 3); fibre operators of all characters after a numerical check of the conjugation of Lemma 2.6 (T: K <= 15; T^(2): K <= 16; T^(4): K <= 12; T_3: K <= 10 and 12; T_5: K <= 6), equal to the group ring and to Proposition 3.8; the integrand of the first proof cell by cell on windows of the line; Proposition 3.8 for 24 random partitions with p in {2, 3, 4, 5, 6}, for T_4, T_6 and the encodings of Remark 5.2; Table 1, the values of Sections 4.2 and 5.1, errors, signs, Remarks 3.9, 4.3, 5.2 |
| Re-run of the package from an extracted copy of `source.zip` | all 76 runs exit 0; 14 outputs byte-identical, 60 identical up to running times, 2 differing only in the printed diagnostics described in `RERUN_LOG.txt`; `check_note.py`: ALL CHECKS PASSED (29,577,739 checks) |
| Literature | not stale (searches in the next section) |
| Bibliography | the 11 DOIs checked in Crossref, the two companion DOIs in the Zenodo API: correct |

The run found no error in a theorem, proposition or lemma. **Required corrections** (all made in the final text):
(1) Section 1.3 attributed the use of Pontryagin duality also to Dicks and Schick, which the sources do not support
(their paper uses elementary methods); the attribution is now to Austin (see the Oberwolfach abstract, p. 2366) and
Grabowski, and Dicks–Schick is no longer cited; (2) the Verification paragraph of the note records this run and no
longer says that the final text was not examined; (3) the paragraph "Scope and priority" includes the searches of
this run (thirteen web searches in all); (4) the caption of Table 1 no longer uses the undefined label "Part I";
(5) Remark 2.2 says that the other values of T_3 under the third reading are values on the finite quotients
(5/9 instead of d_4 = 29/81, confirmed by this run). Programs, outputs, the re-run log and the comparison are in
`reproducibility/independent_run_2/`.

## Relation to the literature, novelty and scope
- **Searches** (15 October 2026, by the two runs that obtained the results and by the four verification runs;
  repeated in part at the writing): the arXiv API ("lamplighter" with Betti numbers, the Atiyah problem, von Neumann
  dimension or spectral measures; "irrational algebraic" with Betti numbers; wreath products; author searches for
  Ara, Claramunt and Grabowski; the most recent titles with "lamplighter", up to 6 October 2026), OpenAlex (works
  citing Grabowski 2014 and 2016, Ara–Claramunt 2021 and 2022, Lehner–Wagner 2013, Grigorchuk–Simanek 2021),
  Semantic Scholar (partly refused for the rate limit), Crossref, zbMATH Open, the texts of Ara–Claramunt 2022 and
  Kirstein–Kremer–Lück 2025, and thirteen web searches. At the writing: Crossref (11 DOIs) and DataCite (the two
  companion notes) for the bibliographic data, two arXiv listings and one web search; nothing new. By the second
  independent verification run: the arXiv API (five queries, the abstracts of the newest hits), OpenAlex (works
  citing Grabowski 2014 and 2016, Ara–Claramunt 2021 and 2022, Lehner–Wagner 2013, and the abstracts of the
  post-2021 ones), Crossref (two queries and the 11 DOIs), zbMATH Open (two queries), the Zenodo API (the two
  companion notes) and one web search; nothing new (newest related items: Löh–Uschold 2023 on computability of
  l2-Betti numbers, Claramunt 2026 on separated graphs; neither gives values for lamplighter groups).
  - No source found answers Remark 4.14 of Ara–Claramunt or gives a genuine witness for any number of the listed
    family.
  - No earlier non-virtual irrational algebraic l2-Betti number of a lamplighter group Z/p wr Z was found other
    than those of the companion notes (p = 2). Known there: transcendental and other irrational numbers (Grabowski
    2016; Ara–Claramunt 2021), irrational algebraic numbers for the free lamplighters Z/p wr F_d (Lehner–Wagner
    2013), irrational algebraic virtual l2-Betti numbers of Z/2 wr Z (Ara–Claramunt 2021, 2022).
- **What was read.** OWR 40/2010 pp. 2366–2369 (publisher's PDF); Invent. Math. arXiv v4 Section 1 (the journal
  version not openly accessible, not compared); GGD arXiv v4 pp. 1–3 and §4 (beginning), journal pp. 795–797;
  Ara–Claramunt 2021 arXiv v2 Sections 1, 2.2, 4.3 and the journal version Section 4.3; Ara–Claramunt 2022 arXiv v2
  introduction; Grigorchuk–Simanek arXiv v2 Section 7; Kirstein–Kremer–Lück arXiv v1 Sections 5, 6.4, 7.1; both
  companion notes. The second independent verification run read the same places again from newly fetched copies
  (open EMS PDFs of OWR 40/2010 and GGD, the open journal PDF and arXiv v2 of Ara–Claramunt 2021, arXiv versions of
  the others; the companion notes from files whose checksums equal the Zenodo records), and the arXiv version of
  Dicks–Schick 2002 to check an attribution (it uses elementary methods, not Pontryagin duality; the attribution
  was removed and the paper is no longer cited). Not read: Lück 1994 (cited as a standard result); Lehner–Wagner,
  Austin (statements taken from Grabowski's texts).
- **Caveats.** The literature search is not a proof of novelty (theses, talks and unindexed preprints are not
  covered). Lück's theorem is used only for the second proof. The witnesses are 2 x 2 matrices, not single elements.
- **Scope.** Theorems A and C and Theorem D for odd p are, to our knowledge, new; no priority is claimed. The record
  stays 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/