Similarity Signature Curves for Forming Periodic Orbits in the Lorenz System

The paper’s Theorem 3.2 is plausibly true but the written proof is logically incomplete and cites an inapposite uniqueness claim; e.g., it asserts x also satisfies R^{n1}(x)=x from R^{n2}(x)=x without addressing divisibility and gives no rigorous bridge from symbolic uniqueness to signature non-coincidence . The model’s core idea—using equality of similarity-signature images to align the similarity arc-length parameter and then reconstruct curvature/torsion to infer curve similarity—is solidly grounded in the paper’s Definition 2.2 and the general signature-equivalence principle , but its final contradiction step incorrectly infers a shorter-period closure on P1 from a global similarity, leaving a gap. Hence, both arguments need repair.