A Stepwise Relation in the Collatz Function and the Termination of Numbers

This paper presents a formal theorem concerning the behavior of the Collatz function for all even integers. By establishing that for any even integer k=2x, the number of steps required to reach 1 satisfies the relation, steps(k) = steps(2) + steps(x). We provide a recursive structure that reduces the complexity of analyzing Collatz sequences. The proof combines inductive reasoning with computational verification to demonstrate that this relation holds across a wide range of values. These results contribute to a deeper understanding of the sequence dynamics and offer a framework that may assist in future efforts toward resolving the general Collatz conjecture.