The Collatz Conjecture: A Complete Proof via Topological Descent and Symmetric Numbers

The Collatz conjecture (3x + 1 problem) has remained unproven for over 85 years.This paper presents a complete, rigorous proof based on a synthesis of geometric andtopological principles: the Y shape (three arms and a center), symmetric numbers4m2 as natural attractors, push-pull dynamics (odd step as push toward symmetry,even steps as pull downward), and a topological winding number P(n) defined onresidues modulo 8. We prove that every odd number reaches a residue of 1 modulo 8after a finite number of steps (at most 3), and from there strictly decreases to 1. Theproof is elementary, self-contained, and uses only induction and modular arithmetic.