Nonsingular Transformations That Are Ergodic with Isometric Coefficients and Not Weakly Doubly Ergodic

The paper establishes all three claims for the nonsingular HK(+1, λ) system: type III_λ (Theorem 4.1), failure of WDE (Theorem 4.2), and EIC (Theorem 4.3), with coherent proofs grounded in the rank‑one construction and a dense‑orbit lemma for isometric factors . The candidate’s ratio‑set and non‑WDE arguments are essentially correct (though the “unique representation” of times needs a stronger growth condition than stated), but the EIC part omits the crucial dense‑orbit hypothesis and quantitative overlap estimate used in the paper; the asserted “pigeonhole along a long chain” does not by itself yield closeness in the isometric factor. Thus the paper is correct, while the model’s EIC proof is flawed/incomplete.