Maximizing entropy for power-free languages

The paper proves that for every d ≥ 2 and β > 12, both X^d_β and X^d_{β+} have a unique MME (Theorem 1.1) via a concrete language decomposition and a variable-length specification property on a “good” set G, together with a strict entropy gap for the prefix/suffix obstruction families Cp, Cs defined using 4-powers. This is established in Theorem 2.1 (properties (I)–(III)) and Theorem 2.3 (a general uniqueness theorem applying when (I), (II′), (III), (IV′) hold), with the key combinatorial lemmas (e.g., Lemma 3.9) verifying (III) for d ≥ 3, β ≥ 12 (T = 1) and for d = 2, β > 12 (T = 2) and the entropy gap proved by counting G4 or G8 to show h(Cp ∪ Cs) < h(X) (Section 3.3.3) . In contrast, the model asserts the problem is likely open and claims h_top(X^d_β) = log d for all β > 1, based on a loose union bound that does not respect extendability and vastly overcounts “bad” words; this contradicts the paper’s framework and is not supported by the literature on power-free languages.