There does not exist an infinite sequence of 4-fold canonical flips. §1. Introduction One of the most important conjectures in the minimal model program is (log) Flip Conjecture II. It claims that any sequence of (log) flips: has to terminate after finitely many steps. In this paper, we prove it for 4dimensional canonical pairs. For the details of the log minimal model program, see [KMM, Introduction] or [KM, §3.7]. The following is the main theorem of this paper: Theorem 1.1 (Termination of

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... old canonical flips). Let X be a normal projective 4-fold and B an effective Q-divisor such that (X, B) is canonical. Consider a sequence of log flips (see Definition 2.2) starting from (X, B) = (X 0 , B 0 ): (X 0 ,B 0 ) (