On Systems Disjoint from All Minimal Systems

The uploaded paper’s abstract and proofs establish the equivalence with a dense union of minimal subsets, not with a full union, but Theorem A (and Theorem 6.3) as typeset incorrectly state ⋃Mi = X. The abstract says “whose union is dense in X” and the proof of Theorem A actually assumes/proves density, not equality, confirming a presentation error in the paper. Meanwhile, the model correctly identifies the dense-union version as the valid statement and explains why the full-union claim would contradict known transitive, non‑minimal examples in M^⊥; however, its (2) ⇒ (1) argument hinges on a “key lemma” about joinings that is false as stated (e.g., the diagonal joining on a minimal rotation gives a counterexample), so the proof needs repair. Thus, the paper’s statement is flawed (though the intended result is supported by its own abstract and proof), and the model’s correction is conceptually right but contains a critical gap in its proof. See the paper’s abstract for the dense-union formulation and Section 6.1 for the proof using density, contrasted with the equality written in Theorem A and in the transitive case Theorem 6.3 (where (1)⇒(2) is cited from Theorem 3.3, which only yields density). These inconsistencies are visible in the file at the abstract (dense union), Theorem A as written (full union), DDMS-property (dense union), the proof of Theorem A (dense union), and the transitive simplification (states full union). The existence of transitive, non‑minimal systems in M^⊥ is also acknowledged in the paper (e.g., the Bernoulli shift), which contradicts the full‑union claim for transitive systems.