On orthogonality to uniquely ergodic systems

The statement the model attempts to prove is exactly Proposition 5.1 of the paper: u is orthogonal to all uniquely ergodic systems if and only if, for every Furstenberg system κ ∈ V(u), the coordinate function π0 is orthogonal to Fwe,0(Xu, κ, S) . The paper gives a complete two-way proof. In the ⇒ direction, it uses a contradiction argument plus the lifting lemma (Theorem 2.1) and orbital uniquely ergodic models (Theorem 2.4) to realize the required joinings along subsequences with rare switches, yielding a uniquely ergodic model that forces the weighted averages to vanish under u ⟂ UE . In the ⇐ direction, any hypothetical nonzero subsequential limit produces a joining ρ; then Φρ(f) ∈ Fwe,0 implies ⟨π0, Φρ(f)⟩ = 0, contradicting the assumed nonzero limit . By contrast, the model’s proof hinges on an unproved and generally false “quasi-generic representation lemma” (its Step 2) claiming one can glue orbit blocks in an arbitrary uniquely ergodic topological model to represent any joining along the same witnessing subsequence. The paper avoids precisely this pitfall by invoking Theorem 2.1 (to build sequences with sparse changes) and Theorem 2.4 (to pass to an orbital uniquely ergodic model) . In addition, the model’s argument for (B) ⇒ (A) asserts that because one can extract a sub-subsequence whose limit is 0, “every subsequential limit is 0,” which does not follow. The paper’s contradiction proof closes this gap cleanly .