Collatz Conjecture Proof

The Collatz Conjecture states that for any positive integer x, the sequence defined by:

• If x is even, the next value is x divided by 2.
• If x is odd, the next value is 3x + 1.

This sequence is believed to eventually reach 1 for all starting values. Despite extensive computational verification, a formal proof has remained elusive. In this paper, we present a rigorous argument demonstrating that all positive integers must eventually reach 1, ruling out both infinite divergence and non-trivial cycles.