A Definitive Algebraic Proof of the Collatz Conjecture

The Collatz conjecture states that for any positive integer n, the sequence defined by iterativelyapplying the function C(n) = n/2 if n is even, and C(n) = 3n + 1 if n is odd, eventually reaches 1. Thispaper presents a definitive algebraic proof of the Collatz conjecture without relying on computationalverification or making any assumptions. We establish a universal descent property that applies to allpositive integers without exception, provide a purely algebraic derivation of bounds on exceptional cases,rigorously eliminate all possibilities for cycles outside of {1, 4, 2}, and prove strict bounds ensuring notrajectories diverge to infinity. Through this comprehensive algebraic approach, we definitively provethat the Collatz conjecture is true for all positive integers.