Topological Mild Mixing of All Orders Along Polynomials

The paper’s Theorem 3.2 defines Δ–G^*_A-transitivity (Definition 3.1) and proves that for abelian Γ, if each nontrivial T ∈ Γ acts minimally and is mixing along a fixed IP-set A, then (X,Γ) is {g1,…,gk}Δ–G^*_A-transitive for any polynomial word family with all singletons and pairwise differences nontrivial in n. The proof is topological, via PET induction and the Bergelson–Leibman topological polynomial recurrence (Theorem 2.11), together with an IP-filter argument (Lemma 3.4) to pass from single-coordinate to multi-coordinate returns . By contrast, the model’s solution attempts a measure-theoretic proof: it builds a Γ-invariant full-support measure by Følner averaging (valid for amenable Γ), then invokes an “IP-polynomial multiple-recurrence theorem for commuting actions” to obtain positivity along an arbitrary prescribed IP-ring. The crucial step is unsupported: the cited Bergelson–McCutcheon results that deliver IP*-large polynomial return sets require mild mixing of the measure-preserving action, not merely invariance; the constructed measure need not be mildly mixing, nor even ergodic. The model also does not actually use the paper’s mixing-along-A hypothesis, which is essential for the topological PET route and for achieving the G^*_A conclusion for every IP-ring generated by A . Consequently, the model’s argument has a fatal gap at its Step 3, while the paper’s PET-based proof is logically coherent and self-contained (see the proof outline and Step 1/Step 2 in Section 6) .