every Hailstone sequence should decrease in the long run, although this is not evidence against other cycles, only against divergence. The argument is not a proof because it assumes that Hailstone sequences are assembled from uncorrelated probabilistic events. (It does rigorously establish that the 2-adic extension of the Collatz process has two division steps for every multiplication step for almost all 2-adic starting values.) Stopping times As proven by Riho Terras, almost every positive integer has a finite stopping time. In other words, almost every Collatz sequence reaches a point that is strictly below its initial value. The proof is based on the distribution of parity vectors and uses the central limit theorem. In 2019, Terence Tao improved this result by showing, using logarithmic density, that almost all Collatz orbits are descending below any given function of the starting point, provided that this function diverges to infinity, no matter how slowly. Responding to this work, Quanta Magazine wrote that Tao "came away with one of the most significant results on the Collatz conjecture in decades". Lower bounds In a computer-aided proof, Krasikov and Lagarias showed that the number of integers in the interval that eventually reach 1 is at least equal to for all sufficiently large . Cycles In this part, consider the shortcut form of the Collatz function A cycle is a sequence of distinct positive integers where , , ..., and . The only known cycle is of period 2, called the trivial cycle. Cycle length The length of a non-trivial cycle is known to be at least . In fact, Eliahou (1993) proved that the period of any non-trivial cycle is of the form where , and are non-negative integers, and . This result is based on the continued fraction expansion of . -cycles A -cycle is a cycle that can be partitioned into contiguous subsequences: increasing sequences of odd numbers alternating with decreasing sequences of even numbers. For instance, if the cycle consists of a single increasing sequence of odd numbers followed by a decreasing sequence of even numbers, it is called a 1-cycle. Steiner (1977) proved that there is no 1-cycle other than the trivial . Simons (2005) used Steiner's method to prove that there is no 2-cycle. Simons & de Weger (2005) extended this proof up to 68-cycles: there is no -cycle up to . For each beyond 68, this method gives an upper