Rank-one systems, flexible classes and Shannon orbit equivalence

The paper proves the φ-integrable orbit equivalence for every flexible class via a carefully staged construction of S while building T, precise quantitative bounds on the cocycles cT and cS (Lemmas 5.20 and 5.21), and parameter-growth constraints (notably Lemma 5.11 and the Qn-inequalities (16)–(23)), culminating in Theorem 3.9. The candidate solution replaces these with an unproven 'three-scale carry compression' estimate |Δn| ≲ h_{n−3} and assumes the cocycle decomposes into stage-n corrections with support μ(Δn≠0) ≤ σn/hn+1; neither is established in the paper. It also relies on choosing σn=0 off a sparse subsequence and tuning ∑ h_{n−3}^p σn/hn+1<∞ without checking whether the flexible-class axioms allow such freedom at every step (particularly under Condition (C1)). These gaps undercut the claimed L^p bound and the derivation of the 1/3-threshold.