Lu Collatz Reverse Pruning Sieve (2026): A Dichotomy Proof of the Collatz Conjecture

This paper presents a complete proof of the Collatz conjecture. We establish three foundational facts: (1) every even number reduces to an odd number; (2) every odd number belongs to exactly one of the three classes 6a + 1, 6a + 3, or 6a + 5; (3) numbers of the form 6a + 3 (odd multiples of 3) serve as the base case, since their forward orbits eventually reach 1. By constructing a “pruned reverse tree” sieve, we prove that any number in the classes 6a+1 or 6a+5, if it is a counterexample, must survive all ffnite levels of the sieve and therefore must belong to the 3-adic limit intersection R∞. We characterize R∞ algebraically as the set of numbers of the form k · 3 n − 1. Then, using the accelerated Collatz map T(n) = (3n +1)/2 v_2(3n+1) , we prove that any such number either strictly collapses to a smaller number, or escapes from R∞ within ffnitely many steps; in both cases, the argument reduces to induction. All rigorous derivations are provided in the full paper.