Quasi-isometric center action in dimension 3

The candidate solution reproduces, step by step, the paper’s argument: (i) quasi‑isometric center action ⇒ dynamical coherence (Theorem 2.1) and uniqueness of the center foliation; (ii) virtually solvable π1 classification reduces to skew‑product or discretized Anosov flow (Lemma 3.2); (iii) in the non‑virtually solvable case, chain‑recurrence or dense center leaf ⇒ minimal W^{cs}, W^{cu}, then Gromov‑hyperbolic leaves and Hausdorff center leaf spaces (Lemmas 3.3–3.6) ⇒ quasi‑geodesic center and collapsed Anosov flow, and finally the Barthelmé–Gogolev periodic self‑orbit equivalence criterion yields a DAF iterate; (iv) the branching case is debranched (Proposition 4.2) and the same classification holds (Theorem A'). These are exactly the paper’s statements and proof strategy, with only minor reference differences. See Theorem A and its proof outline, including Theorems 2.1–2.3, Lemma 3.2, Lemmas 3.5–3.6, and Section 4 on branching foliations .