Local entropy theory and descriptive complexity

The paper proves that for any Cantor space X, Mix(X) ∩ CPE(X) is coanalytic and not Borel, by (i) noting Mix(X) is Borel and CPE(X) is coanalytic, and (ii) applying a Π1_1-rank (the entropy rank from entropy pairs) that is unbounded on mixing CPE systems, yielding non-Borelness; this is stated and proved around Theorem 5.25, invoking Proposition 2.12 (CPE coanalytic), Theorem 2.13 (entropy rank is a Π1_1-rank), and the rank method (Theorem 2.11) . The candidate solution follows the same structure: it establishes Mix(X) is Borel (as in Remark 3.9) and CPE(X) is coanalytic via coding factors, then uses the entropy-rank method together with the existence of mixing CPE systems of arbitrarily high rank to conclude non-Borelness—precisely the paper’s route to Theorem 5.25 . Minor differences (e.g., treating h_top>0 as analytic rather than Borel) do not affect the conclusion.