CUBE STRUCTURES OF THE UNIVERSAL MINIMAL SYSTEM, NILSYSTEMS AND APPLICATIONS

The paper’s Theorem 3.7 states exactly the equivalence between proximality of π_d: X → X/RP[d](X) and a unique τ-closure property for HK[d+1](G) on X, matching the SOLVER_QUESTION (HK[d+1](G) has the unique τ-closure on X iff π_d is proximal) . The model’s proof uses the same key mechanisms as the paper: (a) the algebraic identification cl_τ(HK[d+1](G))·x = u^{[d+1]}Q[d+1](X) (paper’s Theorem 3.3) to pass between cubes and τ-closures ; (b) the Host–Kra characterization of RP[d] via (d+1)-cubes (paper’s Lemma 5.9) to manufacture almost-cubes that force uniqueness/non-uniqueness ; and (c) functoriality/Ellis-group facts, including that a proximal extension induces an isomorphism at the level of HK[d+1]-flows, and that π[d] remains proximal (paper’s Proposition 5.27; Theorem 2.9) . The only slight gap in the model’s text is that it states the uniqueness on the order-d factor “by definition,” whereas this is a theorem-level fact in the paper (apply Theorem 3.7 to Y = X/RP[d](X) to deduce the uniqueness property on Y) . Otherwise, the logic and steps align closely with the paper.