Automaticity of Uniformly Recurrent Substitutive Sequences

The uploaded paper proves the equivalence “y is automatic” ⇔ “t(|ϕ^s(w)|)_{w∈R_a} is a left eigenvector of M_τ” (Theorem 1.2) rigorously, with (ii)⇒(i) handled by a Dekking-style uniformization (Theorem 2.2) and (i)⇒(ii) via an automatic factor, recognizability, a growth alignment |ϕ^{e m_n}(w)|=c_w^{(n)}k^n, a pigeonhole step, and a Perron–Frobenius/Lemma 2.6 argument to pass from an eigenvector of M_τ^r to one of M_τ itself . The candidate solution mirrors the overall structure and gets the key identity v_n=v_0 M_τ^n right (return words factorization) , and its (ii)⇒(i) construction is essentially the same as in Theorem 2.2 . However, for (i)⇒(ii) it contains a critical linear-algebra gap: from v M_τ^r = μ v (μ≠0) it asserts that v M_τ^s is a left eigenvector of M_τ by “killing the nilpotent part,” without the necessary Perron–Frobenius justification that a positive eigenvector of M_τ^r must lie in the Perron eigenspace, hence is already an eigenvector of M_τ. The paper explicitly or implicitly uses this PF step (and then Lemma 2.6) to conclude correctly; the model does not. Therefore, the paper’s argument is correct and complete, while the model’s proof has a substantive gap.