Independence and almost automorphy of high order

The paper’s statement and proof of Theorem 5.7 are correct: absence of nontrivial IN[d]-pairs implies the system is an almost one-to-one extension of its maximal factor of order d, via (i) a characterization turning T×T-minimal nontrivial RP[d]-pairs into IN[d]-pairs (Lemma 5.4) and (ii) an Ellis–semigroup argument that forces the RP[d]-relation to be proximal, reducing the d-factor to the ∞-factor, and then invoking a known structure theorem (Theorem 5.6) to conclude almost one-to-one. The candidate’s solution incorrectly assumes that every nontrivial RP[d]-pair (without the T×T-minimal hypothesis) is an IN[d]-pair and tries to derive a contradiction by contrapositive; this key step is unproven and false in the generality claimed, and their subsequent dense Gδ argument relies on that gap. Hence the model’s solution is not correct.