Uniform continuity of entropy on the regular points endowed with f̄

The paper proves uniform continuity of the entropy map on frequency-typical sequences for the f̄ and d̄ pseudometrics via a careful, block-level (all lengths) construction using finitary isomorphisms and an induced-partition/Abramov-type argument, with precise O(ε) entropy estimates (Theorem 3.3 and Theorems 1.3–1.4) . The candidate solution only controls single-letter marginals via the LCS-based bound and then incorrectly asserts |h(X)−h(Y)| ≤ |H(law(X0))−H(law(Y0))|. This inequality is false in general; equal one-letter marginals can coexist with vastly different entropy rates (e.g., a Bernoulli(1/2) i.i.d. process vs. a 2-periodic process). The key mistake is replacing block-entropy control with single-letter control and reversing an inequality in the subadditivity/infimum step. Hence the model’s proof is invalid.