Constructive Approaches to QP-Time-Dependent KAM Theory for Lagrangian Tori in Hamiltonian Systems

The paper proves an a‑posteriori KAM theorem for quasi‑periodic time‑dependent Hamiltonian systems using a quasi‑Newton scheme in an adapted symplectic frame and an automatic reducibility identity. The candidate solution implements the same scheme: linearization in the [L N] frame, torsion 〈T〉 nondegeneracy, Diophantine cohomological solves with strip‑loss, quadratic error reduction, and iteration. Minor bookkeeping differences (where the torsion T appears in the triangularized system) do not affect solvability or estimates. Both arguments close under the stated hypotheses (H1–H5), achieve quadratic convergence and the closeness bound, and conclude the Lagrangian property of the limit torus.