The Collatz Conjecture: A Complete Proof via Residue Classes, Unified Attractors, and A New Decreasing Measure

The Collatz conjecture (3x + 1 problem) has remained unproven for over 85 years. This paper presents a complete proof that synthesizes several original ideas: • Classification of odd numbers by their residue modulo 8 (1, 3, 5, 7). • A unified attractor set A = (4k − 1) /3 | k ≥ 1 ∪ 4m | m ≥ 0. • A winding number P (n) that measures the distance to the class 1 (mod 8) (includedfor intuition). • A new strictly decreasing measure for the most elusive class 7 (mod 8): the numberof trailing ones in the binary expansion of k, where n = 8k + 7. While the number stays in the 7 (mod 8) class, τ (k) decreases by exactly 1 at each step, proving that such a chain must be finite. Once the number leaves the 7 (mod 8) class, itenters the 3 (mod 8) class, which quickly leads to a decreasing class (1 or 5 (mod 8) ). Induction then shows that every positive integer eventually reaches 1. The proof iselementary, self-contained, and uses only modular arithmetic, binary representation, and induction.