Weak mixing behavior for the projectivized derivative extension

The paper proves, via a quantitative AbC construction, the existence of C^∞ diffeomorphisms f ∈ A_α(M) (and a real-analytic torus example with entire complexification) whose projectivized derivative extension (f,df) is weakly mixing with respect to a fiber–absolutely-continuous invariant measure µ̄, giving Theorems 1.1 and 1.2. The candidate solution outlines essentially the same AbC scheme: Liouville approximations, commuting conjugacies, construction of an f-invariant measurable Riemannian metric, and a weak-mixing criterion on PTM via well-distributed partitions and controlled shears. Minor inaccuracies include describing dS_{α} as a constant rotation on tangent fibers (it is trivial in the tangent direction for torus translations, and more generally treated through the isometric nature of the action) and an imprecise choice of the mixing times m_n; these do not alter the core method. Overall, the model follows the paper’s approach and arrives at the same conclusions.