A Proof of the Collatz Conjecture via Modulo‑4 Congruence Classification

The Collatz conjecture, also known as the Hailstone conjecture, states that for any positive integer n, define the iterative transformation rule: divide an even number by 2 directly; if n is odd, calculate 3n+1. All positive integers will converge to 1 after finite iterations.This paper classifies positive integers by modulo‑4 congruence, analyzes the iteration law when odd numbers trigger the formula 3n+1. With the infinite descent method in number theory, we prove iteration values strictly decrease finitely. Restricted by the minimum positive integer 1, iterations converge to 1 ultimately, which completes the proof of Collatz conjecture.