Rigidity of strong and weak foliations

The paper’s Theorem 1.1 establishes exactly the chain (1)⇔(2)⇔(3)⇒(5)⇒(4) and, under density of all Lyapunov subfoliations of W^ws, full equivalence of the five statements, with detailed proofs via normal forms, holonomy invariance, and a constructive straightening map h̃. The candidate solution states the correct implications but its proof replaces the paper’s core normal-form/holonomy machinery with (i) an unsubstantiated upgrade from leaf-conjugacy to a global C∞ diffeomorphism in (2)⇒(3), and (ii) a cohomological/Livšic argument for (3)⇒(5) and for (4)⇒(1) that presumes a compact hyperbolic base or a well-defined quotient leaf space—assumptions not justified and generally false when the u+ss leaves are dense. Consequently, while the candidate’s high-level roadmap matches the theorem, its arguments have serious gaps, whereas the paper’s arguments are complete and internally consistent.