On dense orbits in the space of subequivalence relations

The paper proves the equivalence between aperiodicity plus everywhere infinite index and having a dense [R0]- or Aut(R0)-orbit in Sub(R0), via two key ingredients: (1) a characterization of when the orbit closure hits the diagonal (Theorem 4.7) and (2) a reduction showing that, under these hypotheses, the [R]-orbit closure contains all hyperfinite subequivalence relations (Theorem 4.11), yielding Corollary 4.12 and Corollary 4.13 for R0. These steps are clearly stated and coherently tied together in the text. By contrast, the candidate solution’s “tower” proof outline contains substantive gaps. Most critically, its necessity direction attempts to uniformly separate an Aut(R0)-orbit from a “discrete pattern” using a nonstandard metric that samples only finitely many T^k-edges at a time; this fails because the separation can be pushed to large k so that the proposed metric sees arbitrarily little discrepancy, defeating the claimed uniform lower bound. There are also technical omissions in the extension of partial orbit maps (global injectivity/measurability and piecewise T^k structure) that are nontrivial to justify. Hence, while the model’s ideas are thematically aligned with the paper’s result, they do not constitute a correct proof as written.