Termination of Flips for Pseudo-Effective Divisors in Dimension Four

We establish the termination of any sequence of flips for log canonical pairs (X, B) of dimension four, under the assumption that the log canonical divisor KX + B is pseudo-effective. This result constitutes a significant advance in the Minimal Model Program (MMP), generalizing previous termination theorems restricted to the klt category or lower dimensions. The proof methodology synthesizes the theory of special termination with the Ascending Chain Condition (ACC) for log canonical thresholds, as established by Hacon, McKernan, and Xu. We introduce a filtration of the non-klt locus and analyze the asymptotic behavior of generalized log discrepancies along the flipping sequence. Furthermore, we demonstrate that the termination of flips implies the existence of minimal models for all pseudo-effective log canonical 4-folds, thereby completing the MMP in this setting.