Genuine Witnesses for Virtual ℓ²-Betti Numbers of Ara and Claramunt and Irrational Algebraic ℓ²-Betti Numbers of Z/p ≀ Z

Grabowski asked what the set C(Z/2 ≀ Z) of ℓ²-Betti numbers arising from the lamplighter group is, that is, of von Neumann dimensions of kernels of square matrices over its rational group ring, and whether C(Z/p ≀ Z) contains irrational algebraic numbers (Invent. Math. 2014). Ara and Claramunt obtained (1/4)√(2/7) and (1/8)√(2^t/(2^t − 1)), t ≥ 3, as virtual ℓ²-Betti numbers of Z/2 ≀ Z, that is, as differences of two ℓ²-Betti numbers, and asked for a concrete matrix whose kernel has dimension (1/4)√(2/7) (J. Algebraic Combin. 2021, Remark 4.14). We give genuine witnesses for these numbers. An explicit T ∈ M_2(Q[Z/2 ≀ Z]) built from five lamp bits and the shift t^5, with 32T integral, has dim_vN ker T = (1/16) Σ_{N≥0} W_N 32^(−N) = (1/4)√(2/7), where W_N = Σ_k C(N,2k) C(2k,k) 25^k 14^(N−2k); a family T^(n) with n + 2 lamp bits gives (1/8)√(2^n/(2^n − 1)) for every n ≥ 2. So every number of the family listed by Ara and Claramunt lies in C(Z/2 ≀ Z); the other elements of their group of virtual values are not treated. With ρ_j = (1/p)(1 + e_j + ⋯ + e_j^(p−1)) the same template gives, for every integer p ≥ 2, an explicit T_p ∈ M_2(Q[Z/p ≀ Z]) with p²T_p integral and dim_vN ker T_p = ((p−1)/p)² √((p−1)/(p+3)), which is irrational of degree 2 because (p−1)(p+3) = (p+1)² − 4 is never a square; for p = 3 it is 4√3/27. All values are proved twice: by Pontryagin duality with a decomposition into counter matrices, and by Lück's approximation theorem with an exact formula on finite quotients. The questions and the frameworks are Grabowski's and Ara and Claramunt's; the balanced-word counter is the mechanism of Ara and Claramunt and of our two companion notes, which gave √5/20 and √2 for p = 2. Apart from these notes, the sources we searched contain no earlier non-virtual irrational algebraic ℓ²-Betti number of a group Z/p ≀ Z; a search that finds nothing is not a proof of novelty. A description of C(Z/2 ≀ Z) remains open. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-4428-012 (Grabowski, On the Atiyah problem for the lamplighter groups, Oberwolfach Report 40/2010, Question 5; the note answers Remark 4.14 of Ara–Claramunt, J. Algebraic Combin. 54 (2021), and the question of Grabowski, Invent. Math. 198 (2014), §1c, for every p ≥ 2, and the record remains partially solved; companion notes doi:10.5281/zenodo.23377729 and doi:10.5281/zenodo.23380620).