Correlation sum and recurrence determinism for interval maps

The paper proves that for interval maps with an order‑preserving metric, the asymptotic correlation sum exists (i) for all ε>0 when the ω-limit set is solenoidal, and (ii) for all but finitely many ε when the ω-limit set is finite, via Theorem A and its components Theorem 7 (solenoidal case) and Theorem 8 (finite case). The proofs hinge on a combinatorial bound on pairs of intervals (Proposition 5) and the order‑preserving metric, not on unique ergodicity. In contrast, the candidate solution’s solenoidal argument incorrectly asserts unique ergodicity of the solenoidal set Q and treats Q×Q as zero‑dimensional; but in the paper’s construction some K_α can be nondegenerate intervals, so Q need not be zero‑dimensional, and uniqueness of invariant measures fails in general. The finite ω‑limit set part of the candidate solution matches the paper’s statement and formula (4.1). Therefore, the paper’s argument is correct, while the model’s argument is flawed in the solenoidal case. Key definitions and statements are given in the PDF (definition of Cm and cm/c̄m, Theorem A, Theorem 7, Theorem 8, and the counting bound), e.g., see the correlation sum definition and asymptotics and Theorem A summary, the solenoidal case proof (Theorem 7) and its interval‑counting backbone, and the finite case (Theorem 8 with the explicit formula).