Profinite approach to S-adic shift spaces I: Saturating directive sequences

The paper’s Theorem 1.1 establishes (i) recognizable ⇒ S-saturating (Theorem 10.10), (ii) pure ⇒ recognizable (via pure ⇒ S-saturating, then Corollary 10.14/10.17 under the global eventual recognizability assumption), (iii) saturating + encoding ⇒ recognizable (Theorem 10.13/Cor. 10.14 with eventual recognizability and V-recognizable images), and (iv) recurrent + bounded + encoding ⇒ recognizable (Theorem 10.18). The candidate solution omits essential hypotheses and makes incorrect algebraic claims: notably, it assumes injectivity of σn^S from mere “encoding” (false in S; the paper requires H-encodings to invoke Theorem 3.8), and it tries to deduce recognizability directly from purity by a code-factorization argument, bypassing the eventual recognizability and V-level machinery used in the paper. Several steps are hand-wavy or unjustified (e.g., isomorphisms between maximal subgroups under σn^S). Hence, the paper’s results and proofs are correct, while the model’s arguments are incomplete and in parts incorrect.