A Topological Proof of the Collatz Conjecture via Discrete Height Descent in 2-adic Space

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.