Nontrivial 3x + 1 cycles have at least four runs of unit valuations

We study nontrivial cycles of the accelerated Collatz map on odd positive integers, which sends an odd number to the odd part of three times the number plus one. In such a cycle, the steps where the division by two is performed exactly once group into maximal cyclic runs. Under the shortcut Collatz map, these runs correspond to ascents of at least two consecutive odd steps. We prove that every nontrivial cycle has at least four such runs: no nontrivial cycle has one, two or three of them, whatever their lengths and without any restriction on the number of local minima. The proof compares two bounds. The first is an upper bound for the carry term of the cycle closure equation, in which the contribution of the runs other than the initial one is controlled explicitly. The second is a lower bound for the distance between the number of halvings times log 2 and the cycle length times log 3. That lower bound is obtained from Eliahou's lower bound for the length of a nontrivial cycle, from a computation with the continued fraction expansion of log 2 / log 3 certified in exact rational arithmetic, and from Matveev's theorem on linear forms in logarithms. As intermediate results, we obtain an explicit upper bound for the length of each run in terms of the total number of unit valuations, and an upper bound for the length of the shortest run. The general cycle problem remains open.