Proof of the Collatz Conjecture

Take any positive integer N. If it is odd, multiply it by three and add one. If it is even, divide it by two. Repeatedly do the same operations to the results, forming a sequence. It is found that, whatever the initial number we choose, the sequence will eventually descend and reach number 1, where it enters a closed loop of 1- 4 - 2 - 1. This is known as the Collatz conjecture which states that the sequence always converges to 1. So far no proof has ever been found that this holds for every positive integer. In this paper, we present a more rigorous, deterministic proof of the Collatz conjecture, guided by heuristic and probabilistic methods. The probabilistic approach provides both structural insight into the iteration and an intuitive explanation for why the deterministic proof should hold.We have noted that the ratio between the number of odd operations and even operations continues to decrease as the sequence length increases, approaching zero for infinite sequence length. This leads to the only possibility that the sequence must eventually decouple from its staring value and enter a cycle, with the only possible cycle being the 1-4-2-1 cycle. We have obtained an equation for the final sequence of infinite length, which is the 1-4-2-1 closed loop: