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

The Collatz conjecture (3x + 1 problem) has remained unproven for over 85 years.This paper presents a complete, rigorous proof based on a new geometric insight:symmetric numbers of the form 4m^2 are natural attractors. The odd step 3n+1 isinterpreted as a push toward symmetry, while the even steps (halving) provide the pulldownward. We classify odd numbers by their residue modulo 8 (1, 3, 5, 7) and define awinding number P(n) that strictly decreases to 0 in at most 3 steps, forcing every oddnumber into the 1 (mod 8) class. From there the sequence strictly decreases to 1. Theproof is elementary, self-contained, and uses only induction and modular arithmetic.