Independence and Mean Sensitivity in Minimal Systems under Group Actions

The paper’s Theorem 1.3 states exactly the solver’s target claim and proves it by: (i) showing a dense Gδ/full-measure set of good fibers via the upper semicontinuous fiber map and Lusin’s theorem (Lemma 3.2) ; (ii) establishing an interior-saturation property for minimal centers under either incontractibility or local Bronstein with an invariant measure (Proposition 3.4 and Lemma 3.5) ; and (iii) building infinite independence sets using the Ellis group of the maximal equicontinuous factor together with an open evaluation map and a measure/thickness lemma (Theorem 2.10, Lemma 4.4, and Claim 4.5) . The candidate solution follows the same three pillars (good fibers, interior saturation, Ellis-group construction). Its “Haar thickness” lemma and lifting argument are close in spirit to the paper’s Lemma 4.4 + Claim 4.5 mechanism, differing only in presentation. Minor phrasing imprecisions (e.g., describing E(Y) vs. Y) do not affect correctness. Overall, both are correct and substantially the same approach.