# Verification report — OWR-4428-012 (Grabowski: is sqrt 2 an l2-Betti number arising from the lamplighter group?)

Verification date: 2026-10-14 (UTC). First round: two independent verification runs on the first written version of
the result; the note was then written with their corrections. Second round: a third independent run on the final
text of the note; its corrections are also made.

**Verdict.** The note answers the concrete question of Problem 4 of Grabowski, Groups Geom. Dyn. 10 (2016), in the
affirmative: sqrt 2 is in C(Z/2 wr Z). Every statement it makes is proved. Three independent AI-assisted verification
runs (two on the first written version, one on the final text) re-derived both proofs and found no 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): no description of the set is given.
- **Settled.** Theorem A: an explicit T' in M_2(Q[Z/2 wr Z]) with 8T' integral has
  dim_vN ker T' = (1/4) Σ_N C(2N,N) 8^(-N) = sqrt(2)/4 (a root of 8t^2 − 1), with two proofs (Pontryagin duality;
  Lück's approximation theorem with an exact formula on the quotients Z/2 wr Z/3P). Corollary B: the integral 8 × 8
  matrix diag(8T', 8T', 8T', 8T') has kernel dimension sqrt 2; k sqrt(2)/4 is in C(Z/2 wr Z) for every k ≥ 1; sqrt 2
  is in C(Z/p wr Z) for every even p. With the companion note and closure under block sums, the numbers
  a sqrt(2)/4 + b sqrt(5)/20 (integers a, b >= 0) are in C(Z/2 wr Z); for a, b >= 1 they have degree 4 over Q.
- **Not new mathematics.** The question, the definition of C(G) and the framework are Grabowski's (Oberwolfach Report
  40/2010; Invent. Math. 198 (2014); Groups Geom. Dyn. 10 (2016)). The mechanism — balanced words counted by central
  binomial coefficients — is the one by which Ara and Claramunt (J. Algebraic Combin. 54 (2021) 1201–1245,
  Theorem 4.13 and the paragraph after it) obtained (1/4) sqrt(2/7) and (1/8) sqrt(2^t/(2^t − 1)), t ≥ 3, as
  *virtual* l2-Betti numbers (differences of two l2-Betti numbers); their Remark 4.14 asks for a genuine witness for
  (1/4) sqrt(2/7), which the note does not provide. The form of the operator, the Fourier picture, the rows, the
  segments and the trace formula are those of the companion note (A. Ferudun, Zenodo, doi:10.5281/zenodo.23377729,
  sqrt(5)/20 in C(Z/2 wr Z)). Lück's approximation theorem and Pontryagin duality are standard. Irrational algebraic
  l2-Betti numbers arising from other groups were known (e.g. every non-negative algebraic number arises from some
  finitely presented group, Grabowski 2014, Theorem 1.4).
- **New, to our knowledge.** Theorem A and Corollary B (the answer to the concrete question of Problem 4). No
  priority is claimed.
- **Open.** A description of C(Z/2 wr Z) (the first part of Problem 4 and the record's question); whether the virtual
  values of Ara and Claramunt are l2-Betti numbers; Question 6(a) of Grigorchuk–Simanek (single self-adjoint
  elements); for odd p, whether C(Z/p wr Z) contains irrational algebraic numbers (nothing is claimed for odd p);
  whether C(Z/2 wr Z) contains algebraic numbers of odd degree greater than 1.

The note is unrefereed.

## Statement checked
- **Corpus record.** ulamai/UnsolvedMath, version 1.6.0 (snapshot c6e7f41b…), 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.
- **The concrete problem.** Grabowski, Irrational l2-invariants arising from the lamplighter group, Groups Geom. Dyn.
  10 (2016) 795–817, doi:10.4171/GGD/366, Problem 4: is there a description of C(Z_2 wr Z) substantially different
  from the definition, and, more concretely, "is sqrt 2 ∈ C(Z_2 wr Z)?". Same number and wording in arXiv:1009.0229v4
  (p. 3) and in the journal version (p. 796; publisher's open PDF, pages rendered and read; the definition with a
  matrix over Z[Γ] on p. 795).
- **Definitions.** OWR p. 2366: r ≥ 0 is an l2-Betti number arising from G iff r = dim_vN ker θ for some
  θ ∈ M_m(QG). GGD p. 795: "a matrix" over the integral group ring (shape not fixed). Invent. Math. §1: trace
  tr ⊗ τ_G on M_k(QG); dim_vN ker T := dim_vN ker T*T for non-self-adjoint T.

## Readings
| Reading | Answer | Where |
|---|---|---|
| Unnormalized trace τ_m = Σ_i τ(·)_ii (the sources' convention: "C(G) = N" for torsion-free G in OWR p. 2366 and GGD p. 795; tr ⊗ τ_G in Invent. Math.) — used | sqrt 2 ∈ C(Z/2 wr Z) | Section 1.1, Theorem A, Corollary B |
| Normalized trace (1/m) τ_m | every element of C(G) would lie in [0, 1]; the question would be trivially negative; excluded by the sources | Section 1.1 |
| Matrices over QG (OWR) or over ZG (GGD) | same answer: 8T' is integral and the witness is square | Sections 1.1, 2.2, 3.6 |
| "A matrix" of unspecified shape (GGD) | the witness is square, so it qualifies under every reading | Section 1.1 |
| Left or right multiplication | same dimension (Lemma 2.1, on G and on finite groups) | Lemma 2.1 |
| Non-self-adjoint matrix, dim ker T := dim ker T*T (Invent. Math.) | same number, ker T*T = ker T | Section 2.1 |
| Single self-adjoint element of Q[Z/2 wr Z] (Grigorchuk–Simanek, Question 6(a)) | not answered | Section 5 |
| Z/p wr Z, even p | sqrt 2 ∈ C(Z/p wr Z) (subgroup monotonicity, OWR pp. 2366–2367) | Corollary B, Lemma 2.2(b) |
| Z/p wr Z, odd p | nothing claimed (no element of order 2) | Section 5 |

## Results in the paper
- **Section 2.** Conventions (left regular representation, τ_m); Lemma 2.1 (left and right; [Fer26, Lemma 5.1]);
  Lemma 2.2 ((a) block sums add, (b) C(H) ⊆ C(Γ) for H ≤ Γ); the operator T' = [[χ' − (1−χ')α' + (1−χ')S', −χ'],
  [χ'S', (1−χ')S' − χ'S' − (1−χ')β']] with χ' = π_0, α' = 1 + π'_0π'_1π_2, β' = 1 + π'_0π_1π'_2, S' = t^3, and its
  expansion (2) with coefficients in (1/8)Z.
- **Section 3 (first proof).** Lemma 3.1 (fibre operators, [Fer26, Lemma 5.2]: e_j acts at index k by (−1)^{x_{j−k}},
  t by the shift; a ↦ θ_x(a) multiplicative); Lemma 3.2 (trace formula, [Fer26, Lemma 5.3]); classes r = 0, 1, 2
  with positions k = −3m − r and symbols (x_{3m+r}, x_{3m+r+1}, x_{3m+r+2}): marker 1**, U = 001, D = 010,
  I ∈ {000, 011}; Lemma 3.3 (rows); Definition 3.4 and Lemma 3.5 (counter matrices, kernel [#U = #D],
  [Fer26, Definition 3.2 and Lemma 3.5]); Lemma 3.6 (segments, for every class); Lemma 3.7
  (Σ_{#U=#D} 2^{#I} = C(2N,N), bijective proof via pairs of subsets and Vandermonde); Remark 3.8 (the same numbers
  are the coefficients of the automaton series F_{A_2} of the companion note); proof of Theorem A (P_x on all three
  classes, measurability, μ(E_{N,s,j}) = (1/4) 8^(−N) 2^{#I(s)}, binomial series at y = 1/8); proof of Corollary B.
- **Section 4 (second proof).** Lemma 4.1 (finite Fourier transform, [Fer26, Lemma 6.1]); Proposition 4.2:
  d_{3P}(T') = (1/4) Σ_{N=0}^{P−2} C(2N,N) 8^(−N) + 4 C(2P−2,P−1) 8^(−P) + 2·3^P/(P·8^P); Lück's approximation
  theorem along K = 3·2^i (nested, trivial intersection), in the form of Kirstein–Kremer–Lück, Theorem 6.2 and
  Section 7.1; Remark 4.3 (error identity; |d_{3P} − sqrt(2)/4| ≤ 2^(−P−1) + 2(3/8)^P/P); Table 1 (K = 3, …, 21, 30, 60).
- **Section 5.** sqrt(2)/4 is none of the virtual values of Ara and Claramunt ((sqrt 2/4)^2 = 1/8, their squares lie
  in (1/64, 1/56]); single self-adjoint elements not treated; odd p.

## 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 or Lück's theorem.
- **The exact program of the note** (`check_note.py`, standard library, exact arithmetic, 109 s; 81,145,766 checks;
  "ALL CHECKS PASSED"): all 29,524 counter matrices M_s, s ∈ {U,D,I}^N, N ≤ 9 (kernels, kernel vectors); the
  automaton A_2 of the companion note over all 625 letters (DFS, 3,197,500 transitions; 2047 accepted words of length
  ≤ 21; recognized matrices of size ≤ 22 = the M_s with s ∈ {U,D}^N, N ≤ 10; coefficients of F_{A_2} up to x^22 =
  1, 2, 6, 20, 70, 252 = B_0, …, B_5); Lemma 3.7 (enumeration N ≤ 10, four letters N ≤ 8, the bijection N ≤ 6,
  Vandermonde and the closed sum N ≤ 300, binomial-series coefficients N ≤ 60); rational enclosures of sqrt(2)/4 and
  sqrt 2; T' from its definition = (2), 8T' integral; left and right multiplication by 8T' on Q[G_K]^2 in the group
  ring for K ≤ 7 (d_1, …, d_7 = 3/2, 1/2, 5/4, 13/32, 31/80, 33/64, 331/896), block sum of four copies for K ≤ 4;
  fibre operators for all x and K ≤ 14 (= group ring for K ≤ 7, = Proposition 4.2 for K = 3, 6, 9, 12, = the values
  recorded by the other programs); all 299,592 cyclic configurations with P ≤ 6, three classes: rows, 2,632,140
  segment blocks = 8 M_s, kernels, totals of Proposition 4.2 (K ≤ 18); 1000 random windows of the line (120,000 rows,
  29,089 blocks); Table 1, the identity of Remark 4.3 (P ≤ 40) and its bound (P ≤ 60, by squaring).
- **Programs with which the result was first obtained** (`original/`): `s1_groupring_direct.py` (group ring, left and
  right, K ≤ 9, two primes, exact over Q for K ≤ 6; block sum K ≤ 4), `s2_fourier_finite.py` (fibre operators,
  K ≤ 18, exact for K ≤ 10; = group ring for K ≤ 9; (5) for K = 3, …, 18), `s3_blocks.py` (rows and segment blocks on
  all configurations, P ≤ 5 three classes, P = 6 class 0), `s4_series.py` (Lemma 3.7, enclosures, Remark 4.3).
- **Verification run A** (`verification_run_A/`): group ring K ≤ 9 (left; right K ≤ 5; exact K ≤ 5); own Fourier
  model K ≤ 16 (= group ring K ≤ 9); class-0 parts for K = 12, 15, 18, 21 (d_21 = 2598067/7340032 = (5)); class-shift
  symmetry; 98,382 segment blocks on random windows of the line; Lemma 3.7, the series, the measure bookkeeping
  (total measure 1), Remark 4.3; the Grigorchuk–Żuk element as a known value (d_K → 1/3).
- **Verification run B** (`verification_run_B/`): group ring K ≤ 9 (left and right, exact over Q for K ≤ 6);
  eigenspaces of right multiplication by t, K ≤ 12; fibre operators K ≤ 20 (exact K ≤ 10); structure on all
  configurations P ≤ 5 (three classes) and P = 6 (class 0); Lemma 3.5 on all 3,280 words of length ≤ 7; Lemma 3.7,
  enclosures to 2·10^(−31), (5) for P ≤ 40; three deliberately altered operators (coefficient, shift t^2, relabelled
  symbol), all detected by the structure checker, the two value-changing ones also by the comparison with (5);
  Monte-Carlo of the trace integrand on the line without markers or segments (100,000 samples: 0.35354 ± 0.00130);
  re-runs of the four original programs (identical outputs up to timings).
- **Re-runs.** At the writing (2026-10-14) every program of the package with a recorded output (23 runs) was run
  again from an extracted copy of a preliminary archive of the package, at most three at a time: all outputs are
  identical to the recorded ones up to the fields that record running times (`reproducibility/RERUN_LOG.txt`).
  Verification run B had already re-run the four original programs from a copy (`verification_run_B/claim_rerun_*.out`).
  The verification run of the second round re-ran all 24 runs with a recorded output, `check_note.py` included, from
  an extracted copy of the archive of the first version of the package: all 24 outputs reproduced (8 byte for byte,
  16 up to running times; `reproducibility/independent_run_2/rerun_package_v1.txt`).
- **Verification run 2** (second round, `independent_run_2/`; programs written for that run, no code reused):
  `run2_groupring.py` (group law; T' = (2); left multiplication by 8T' on Q[G_K]^2 in the group ring, K ≤ 9, exact
  over Q for K ≤ 5, right multiplication K ≤ 8, d_3, d_6, d_9 = (5); block sum of four copies; the image of T' in
  Q[Z/4 wr Z/K]^2, K ≤ 4; controls: a relabelled letter keeps the values, other letter probabilities and the shift t^2
  change them; 3,390 checks), `run2_fibres.py` (the change of basis of Lemma 4.1 built explicitly for K ≤ 5, a wrong
  sign rule detected; all fibre operators K ≤ 15, exact K ≤ 10, = group ring K ≤ 9 and = (5); class parts for all x
  and all three classes, K ≤ 21, = (5); rows and 1,059,276 segment blocks on all configurations P ≤ 5 (three classes)
  and P = 6 (class 0); the finite trace formula with exact kernel projections, K ≤ 9; 2,871,128 checks),
  `run2_series_line.py` (Lemma 3.5 for all 9,841 words of length ≤ 8; Lemma 3.7; the series and an enclosure of
  sqrt 2; the bookkeeping of Section 3.5 with exact kernel projections of M_s, N ≤ 7; Remark 3.8; Table 1 as printed;
  Remark 4.3 for P ≤ 120; Section 5; rows and 128,246 segment blocks on 1,500 windows of the line, three classes;
  560,429 checks). All checks passed.

## Independent verification runs
The result was first obtained, with both proofs, together with the programs in `original/`. Two rounds of independent
verification, all AI-assisted, followed (2026-10-14); each run wrote its own programs from the definitions. In the
first round two runs (A, B) examined the first written version, and neither saw the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement of Problem 4 and of the definitions against the sources (OWR 40/2010; GGD arXiv v4 and journal version; Invent. Math.) | CONFIRMED; journal version compared (fix 1); evidence "C(Γ) = N" for the unnormalized trace (fix 3); "a matrix" in GGD (fix 4) | CONFIRMED; journal version read (fix 2) |
| Expansion of T', 8T' integral | CONFIRMED (re-derived; program) | CONFIRMED (program) |
| Lemma 2.1 (left/right), Lemma 2.2 (block sums, subgroups) | CONFIRMED (re-derived) | CONFIRMED |
| Lemma 3.1 (Fourier conventions), Lemma 3.2 (trace formula) | CONFIRMED (re-derived; group ring = Fourier model for K ≤ 9) | CONFIRMED (derived independently in finite form; three decompositions agree) |
| Lemmas 3.3, 3.5, 3.6 (rows, counter matrices, segments) | CONFIRMED (re-derived; 98,382 line segments) | CONFIRMED (all configurations P ≤ 6); P_x on all three cosets to be stated (fix 4) |
| Lemma 3.7 and the measure bookkeeping | CONFIRMED (two proofs; total measure 1) | CONFIRMED (enumeration) |
| Value sqrt(2)/4, Corollary B (k sqrt(2)/4, even p) | CONFIRMED | CONFIRMED |
| Proposition 4.2 and the second proof (Lück) | CONFIRMED (re-derived; (5) to K = 21) | CONFIRMED (re-derived; (5) to K = 18) |
| Literature, closest prior work | not stale; JACO Remark 4.14 does not cite Problem 4 (fix 2); Semantic Scholar "newest" (fix 5) | not stale; Ara–Claramunt's Theorem 4.13 uses the SAME mechanism for virtual values; to be said plainly (fix 1); bibliographic completions (fix 3) |

No run of the first round found an error in a theorem, proposition or lemma.

**Corrections required by the runs of the first round**, all applied in the note (details in `audit/fixes_applied.md` of the working
folder): (A1) the journal version compared — same Problem 4 and definitions; (A2) Ara–Claramunt JACO Remark 4.14
refers to the question of Invent. Math. (and "[18, Question 3]"), Problem 4 is cited in the Forum paper; (A3) the
evidence "C(G) = N" for the unnormalized trace (OWR p. 2366, GGD p. 795) and the triviality of the normalized
reading; (A4) GGD says "a matrix" over the integral group ring; (A5) the Semantic Scholar remark; (A6, optional)
d_21 added to Table 1; (B1) the same balanced-word / central-binomial mechanism of Ara–Claramunt, Theorem 4.13, for
the virtual values (1/4) sqrt(2/7) and (1/8) sqrt(2^t/(2^t − 1)), t ≥ 3, and Remark 4.14's request for a genuine
witness, stated plainly in the abstract, Section 1.3 and "Scope and priority"; (B2) the journal page of Problem 4;
(B3) bibliographic completions; (B4) P_x defined on all three cosets and bounded.

**Page of Problem 4.** Run B gave journal p. 797, run A p. 796. The publisher's PDF shows Problem 4 on the page with
the running head "796" (PDF page 2; p. 797 begins with Remarks 5(ii)), as the run of the second round confirmed by
rendering the page; the note says p. 796.

**Not included.** Run A's observation that symbols of five lamp bits would give (1/4) sqrt(2/7) by the same template
is not part of the note (not proved here); neither is the general formula of the first write-up.

**Changes made when the note was written** (examined first by the run of the second round, and confirmed): the
multiplicativity of θ_x in Lemma 3.1; the classes r = 0, 1, 2 written out; Remark 3.8; the sharper bound in
Remark 4.3; Section 5 ("Virtual values"); `check_note.py`; the reproducibility package and the re-runs.

### Second round: verification run 2 (the final text)
A third AI-assisted run examined the final text line by line, the parts changed at the writing first, as if nothing
had been checked. It downloaded the sources again (anonymous requests): OWR 40/2010 (pp. 2366–2369), GGD 10 (2016)
in the journal version (pages 795–797 rendered and read) and arXiv v4, Invent. Math. 2014 in arXiv v4 (the journal
version is not openly accessible), Ara–Claramunt JACO 2021 in the journal version (open access) and arXiv v2, the
Forum paper (introduction), Grigorchuk–Simanek (Section 7), Kirstein–Kremer–Lück (Sections 6 and 7.1) and the
published companion note; it checked the bibliographic data with Crossref and DataCite, re-derived every proof and
read `check_note.py` completely.

| Item | Run 2 |
|---|---|
| Statement of Problem 4 (journal p. 796, arXiv v4 p. 3), definitions, scope (record partially solved) | CONFIRMED |
| Expansion (2), Lemmas 2.1–2.2, Lemma 3.1 (incl. multiplicativity) and 3.2, rows, counter matrices, segments, Lemma 3.7, Section 3.5, Corollary B | CONFIRMED (re-derived; programs) |
| Proposition 4.2, second proof, Remark 4.3, Table 1, Section 5 | CONFIRMED (re-derived; (5) for all x and all classes up to K = 21) |
| Statements cited from the companion note (Section 5.1, (2), Definition 3.2, Lemmas 3.4, 3.5, 5.1–5.5, 6.1, Theorems 1, 2, Remark 7.1, Section 8) | CONFIRMED against the published PDF |
| Cited external results | CONFIRMED; two precision fixes (2, 3 below) |
| Literature | not stale (searches repeated; one web search) |

It found no error in a theorem, proposition or lemma. **Corrections required by run 2**, all applied: (1) Section 1.3
said "we do not even know whether C(Z/2 wr Z) contains algebraic numbers of degree greater than 2"; this was false,
since sqrt(2)/4 + sqrt(5)/20 is in C(Z/2 wr Z) (Lemma 2.2(a), Theorem A, companion note) and has degree 4; the
sentence now states this and leaves open whether algebraic numbers of odd degree > 1 occur; the odd-p item is made
precise; (2) Question 3 of Invent. Math. §1c is the arXiv numbering; GS21 and AC22 cite it as Question 1.7 (added);
(3) Lück's theorem: the matrix form is Kirstein–Kremer–Lück, Conjecture 7.1(1), which holds for this chain by their
Remark 7.2 (G amenable, hence sofic; finite quotients); cited so; (4) Section 6 said that every program with a
recorded output had been re-run from a copy at the writing, but `check_note.py` had only been run in place;
corrected, and this run's re-run of all 24 recorded runs recorded; (5) this run recorded in the note ("Verification",
"What was read", the searches, Table 1 caption, Section 6) and in this report; programs and outputs in
`reproducibility/independent_run_2/`; (6) the reproducibility README explains the internal labels "round 23" and
"lead T" left in the comments of `verification_run_A/`.

## Relation to the literature, novelty and scope
- **Searches** (14 October 2026, when the result was obtained and by the two verification runs): the arXiv API
  (lamplighter with Betti numbers, the Atiyah problem, von Neumann dimension, kernels, spectral measures; Betti numbers
  of wreath products with "algebraic"/"irrational"; "Atiyah" with "square root"/"irrational algebraic"; author
  searches; the 100 most recent titles with "lamplighter"), OpenAlex (works citing OWR 40/2010, Invent. Math. 2014,
  GGD 2016, Ara–Claramunt 2021 and 2022, Lehner–Wagner 2013, Grigorchuk–Simanek 2021), Semantic Scholar (20 works
  citing arXiv:1009.0229, the most recent from 2026; 2 citing arXiv:2005.12814), Crossref, zbMATH Open, four web
  searches; texts of Ara–Claramunt (both papers), Grigorchuk–Simanek and Kirstein–Kremer–Lück searched for the
  lamplighter group and sqrt 2. At the writing only the bibliographic data were checked again (Crossref: 14 DOIs;
  DataCite: the companion note). Run 2 repeated part of the searches on the same day (arXiv API: abstracts with
  "lamplighter" and Betti numbers, the Atiyah problem, von Neumann algebras or spectral measures; "Betti" with
  "irrational algebraic"; author searches; the 40 most recent titles with "lamplighter", up to September 2026;
  Crossref; OpenAlex: works citing GGD 2016, Ara–Claramunt 2021 and 2022, and works on lamplighter groups and Betti
  numbers since 2024; zbMATH Open; one web search) and checked the bibliographic data again (Crossref: 11 DOIs;
  DataCite: the companion note); it found nothing new.
  - No source found states, proves or cites sqrt 2 ∈ C(Z/2 wr Z).
  - No earlier non-virtual irrational algebraic l2-Betti number of a lamplighter group Z/p wr Z was found other
    than sqrt(5)/20 of the companion note. Known 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), 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 PDF); GGD arXiv v4 pp. 1–5 and the journal version
  pp. 795–798; Invent. Math. arXiv v4 Section 1 (the journal version is not openly accessible and was not read);
  Ara–Claramunt JACO arXiv v2 (introduction, Section 4.3: Theorem 4.13, Remark 4.14, Question 4.15) and, by run 2, the
  journal version (open access; same numbers and text, pp. 1238–1239);
  Ara–Claramunt Forum arXiv v2 (introduction); Grigorchuk–Simanek arXiv v2 (Section 7); Kirstein–Kremer–Lück arXiv v1
  (Sections 6 and 7.1); the companion note completely. Not read: Lück (1994) (cited as a standard result); Austin,
  Grigorchuk–Żuk, Lehner–Wagner (statements taken from Grabowski's texts and Ara–Claramunt 2022).
- **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 note does not reduce T' to a single self-adjoint
  element.
- **Scope.** Theorem A and Corollary B 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/