Collatz high cycles do not exist

The Collatz function takes odd n to $(3n+1)/2$ and even n to $n/2$. Under the iterated Collatz function, every positive integer is conjectured to end up in the trivial cycle 1-2-1. Two types of rational Collatz cycles are of special interest. Consider the set $S(k,x)$ consisting of the smallest members of k-length cycles with x odd terms. The circuit contains the smallest member of $S(k,x)$, while the high cycle contains the largest. It is known that no circuits of positive integers exist (except 1-2-1); this paper shows that there are likewise no high cycles of positive integers.