Destruction of CPE-Normality Along Deterministic Sequences

The paper’s main theorem states that for a non‑Bernoulli shift with completely positive entropy (CPE), every essential deterministic sequence (lower density < 1 and upper density > 0) destroys µ‑normality, and the proof hinges on product joinings with zero‑entropy derived measures and a strict “spread decreases measure” lemma (mixing of all orders) yielding a deficit in selected block frequencies; see Theorem 4 and Lemma 8 with the joining argument in the proof of Theorem 4 . The candidate solution reproduces the core joining and convex‑combination mechanism correctly, but (i) replaces the paper’s precise hypothesis “essential deterministic” by an unreferenced notion of “non‑superficial deterministic” and asserts that “superficial sets never destroy normality,” which is neither stated nor supported in the paper, and conflicts with the paper’s necessary condition framed exactly in terms of essential sets ; and (ii) claims a stronger inequality comparing all sufficiently spread cylinders directly to the contiguous cylinder [B], while the paper’s Lemma 8 establishes strict decrease relative to [B_{\bar p_0}] for some threshold vector \bar p_0, not necessarily the zero vector . Hence the paper is correct and complete for its stated result, while the model’s solution overclaims beyond the paper’s scope and introduces unsupported conditions.