This paper provides a formal proof of the Collatz Conjecture (the 3n + 1 problem) by employing a topological approach within the 2-adic metric space. By defining a discrete“height” function h (n) based on 2-adic valuation, we demonstrate that the Collatz map C (n) induces a deterministic descent dynamics. We prove that every forward orbit Ck (n) is eventually contained within the compact invariant subspace K = 1, 2, 4. This convergence is shown to be a topological necessity arising from the connectivity of the 2-adic binary tree, ensuring that all trajectories terminate in the 1, 4, 2 cycle.
Yuya Saito (Sun,) studied this question.