BOUNDARY ENTROPY SPECTRA AS FINITE SUBSUMS

Oppelmayer’s note proves Theorem 1.1 cleanly: for Γ = Z[1/p1,...,1/pl] ⋊ S and any positive sequence (βj), there is a finitely supported, generating, Λ-absorbing measure τ with negative pi-drifts φpi(τ) = −βi, so that Poi(Γ,τ) ≅ ∏i Qpi, every τ-boundary is a coordinate projection ∏j∈J Qpj, and the Furstenberg entropy of that factor equals ∑j∈J βj. The key steps are: (i) Brofferio’s Poisson-boundary identification under negative drift, giving X = ∏ Qpi with unique stationary ν (Theorem 4.1) ; (ii) passing to the homogeneous G = (∏ Qpi) ⋊ S and classifying G-factors of X as coordinate projections (Lemma 4.2), using the density of Z[1/p1,...,1/pl] in each Qpi , ; (iii) showing every τ-boundary is such a G-factor when τ is Λ-absorbing (Corollary 4.5) , and (iv) computing entropy additively via Proposition 3.4 and the Radon–Nikodym derivative d(r,s)−1 mQp / dmQp = |s|p, yielding hτ(YJ,ηJ) = ∑j∈J −φpj(τ) = ∑j∈J βj (Section 3/Prop. 3.4; Section 4.2) , . The construction of such τ (Λ-absorbing, finitely supported, generating, with prescribed drifts) is given in Lemmas 4.6–4.7 , . By contrast, the model’s outline has two critical gaps: (a) it does not enforce the Λ-absorbing property needed to identify all τ-boundaries with G-factors, and instead asserts—without justification—that every Γ-equivariant quotient of the Poisson boundary comes from a closed S-invariant additive subgroup; (b) it mis-specifies the Radon–Nikodym factor on the boundary (taking Δj(g)=|s|−1p and then using −logΔj to get +βj), which contradicts the paper’s correct computation with Haar measure on Qp where d(g−1mQp)/dmQp=|s|p . These issues render the model’s proof invalid even though the final statement matches the paper’s result.