Mixing Properties and Entropy Bounds of a Family of Pisot Random Substitutions

The paper proves that for every random noble Pisa substitution ψ_{n,p} (n>2), the subshift (X_{n,p},S) is W-semi-mixing with W = ⋃_{i=1}^{n-1} ψ_{n,p}(α_i), via a numeration-system argument based on the canonical Γ-construction and the inflation-length recursion L_m, together with Lemmas 24–26, culminating in Theorem 19 . The candidate solution claims the same result but hinges on a “corridor lemma” that applies ψ to arbitrary blocks α_1^R to create α_2 α_1^L α_2 for any L, thereby asserting α_1^L is legal for all L. This step is invalid: it implicitly assumes α_1^R is legal for arbitrarily large R, which need not hold in the language, and would in fact force the periodic configuration α_1^ℤ to lie in X_{n,p}, contradicting the paper’s statement that X_{n,p} has no periodic points (since λ_{n,p}∉ℕ) . The model also leaves unjustified a key alignment step (“absorb α_n until a non-α_n appears”), whereas the paper supplies precise length-control via [N]_{n,p} digits and Γ-based lemmas. Hence, the paper’s proof is sound; the model’s proof is not.