A CHARACTERIZATION OF ZERO ENTROPY LOOSELY BERNOULLI FLOWS VIA FK-PSEUDOMETRIC

The paper proves Theorem 1 rigorously: the “⇒” direction passes to an ergodic time‑t map, uses the discrete‑time characterization to obtain a full‑measure set H with ρFK(x,y)=0, and then lifts to the flow via Proposition 3.17; the “⇐” direction uses Ratner’s criterion with a careful reduction to essentially open partitions, Birkhoff/Egorov uniformization, and a comparison lemma to conclude e(Φ,u)=0, hence loosely Kronecker. By contrast, the model’s “⇒” step incorrectly infers that the Ratner criterion yields matchings for µ×µ-almost every pair (x,y), then applies a saturation argument; this Fubini-style leap is not justified. The model’s “⇐” step is directionally right but omits key uniformization and partition-density details that the paper supplies.