Orbit

What is usually left out

Orbits can be periodic, eventually periodic, or neither, and which of those occurs can depend on the starting state in ways no formula captures. The Collatz conjecture is precisely a claim about orbits: that every positive integer’s orbit under its rule reaches the cycle containing 1. Note that it does not claim orbits are short. The orbit of 27 climbs to 9,232 and takes 111 steps, so any argument has to survive excursions far above the starting value.