Belinskaya’s theorem is optimal

The paper proves the claimed existence result rigorously: for any ergodic T1 and any sublinear ϕ, there exists an ergodic T2 and an orbit equivalence S such that the associated cocycle c1 is ϕ-integrable, yet T1 and T2 are not flip-conjugate (Theorem 1.2, proved via Theorem 4.14 and, in the case T1n ergodic for some n>2, via Theorem 3.10/Corollary 3.11) . The construction proceeds through ϕ-integrable full groups and a carefully controlled cycle construction plus a Baire category argument, not through a direct Dye back-and-forth with quantitative cocycle bounds. By contrast, the candidate solution asserts a direct refinement of Dye’s tower construction that forces c1(x)=1 off a union of stage-top sets Ek and |c1(x)|≤Hk on Ek, from which ∫ϕ(|c1|)dμ<∞ follows. However, the key quantitative claims are not justified: in particular, it is not shown that later stages can be arranged so that the cocycle at each stage-k top is bounded by Hk in the limit, nor that no additional “bad” sets arise beyond the proposed Ek. Without a precise back-and-forth scheme proving these bounds, the integrability estimate is unproven. The paper’s argument is complete and correct as written; the model’s proof is not.