A Proof of the Collatz Conjecture Using an Attractor Set and Exit Points for 7 (mod 8) Numbers

Abstract

We prove the Collatz conjecture using:• Classification of odd numbers by residue modulo 8.• An attractor set A = {Aj = (4j − 1)/3}.• The observation that every odd number lies at an even distance from both boundaryattractors.• A lexicographic ordering function Φ(n) = (−j, min(x, y)) for 7 (mod 8) numbers.• The v2(k + 1) descent for 7-odd numbers, proving the 7-odd phase is finite.• The identification of exit points: non-bouncing 7-even numbers that terminatedirectly.• The proof that every bouncing 7-even number eventually reaches an exit point.• Strict decrease in value for 1 (mod 8) and 5 (mod 8) numbers.• The fact that 3 (mod 8) numbers map to 1 or 5 (mod 8) and then decrease.All lemmas are rigorously proved. The proof is complete.Numerical verification using Python for all 7 (mod 8) numbers below 10,000 isprovided in a separate supplementary paper [2].