DIRECTIONAL WEAK MIXING AND SEQUENCE ENTROPY N-TUPLES FOR A MEASURE FOR Zq-ACTIONS

The paper’s Theorem 3.3 proves the equivalence among (a) v-weak mixing (K^v_μ = {X,∅}), (b) positivity of directional sequence entropy for every nontrivial two-atom partition {B,B^c} along some infinite S ⊂ Λ_v(b), and (c) the same for any finite nontrivial partition, via Lemma 3.2 which identifies h^S_μ(T,α) with H_μ(α | K^v_μ) for an appropriate S and bounds it above for all S. This matches the candidate solution’s target exactly. The paper defines Λ_v(b) and K^v_μ in the same way and notes independence of b, aligning with the candidate’s setup. The candidate provides a different, self-contained proof: (i) compactness of the L2-orbit of 1_B along Λ_v(b) implies h^S_μ(T,{B,B^c}) = 0 for every infinite S, and (ii) non-compactness yields an S with linearly growing joined entropies via an L2–entropy lower bound and a geometric selection argument in Hilbert space. These arguments reproduce the paper’s equivalences without appealing to Lemma 3.2. No substantive conflicts were found; only minor expositional details (e.g., absorbing finitely many outliers into a finite-dimensional subspace) would benefit from explicit mention. Overall, the paper and the model are both correct; the proofs are different in flavor but consistent with each other. See Theorem 3.3 and Lemma 3.2 for the paper’s statements and strategy, and the definitions in Section 2 for Λ_v(b) and K^v_μ independence of b.