CLASSIFICATION OF RANK-ONE ACTIONS VIA THE CUTTING-AND-STACKING PARAMETERS

The paper’s Theorem 2.1 gives a correct necessary-and-sufficient criterion for isomorphism of (C,F)-actions via interleaved indices and finite linking sets Jn, J̃n, with a rigorous proof: the “only if” direction constructs Jn, J̃n by cylinder approximations; the “if” direction passes through the auxiliary (C,F)-sequences V := (Fkn, J̃nJn+1) and W := (F̃ln, Jn+1J̃n+1), then uses reductions, telescoping, and chain equivalence to obtain an isomorphism between T and T̃ . By contrast, the model’s “if” proof attempts to build a direct conjugacy by assigning the next-block coordinate in X̃ as jn+1(x)·j̃n+1(x). This assignment is generally invalid: (iii)′ only guarantees that Jn+1J̃n+1 approximates C̃ln+1,ln+1 in counting measure, not that each chosen product lies inside C̃ln+1,ln+1; without performing the paper’s reduction/telescoping step (or an equivalent pruning), the constructed point may fall outside X̃. Hence the model’s construction does not define a bona fide (C,F)-point in X̃ and the claimed conjugacy is not established, while the paper’s proof is complete and correct.