Cycle Elimination in Collatz Trajectories via Cross-Constraint System Reduction

Abstract

In this study, trajectory behaviors of odd numbers in the Collatz process are analyzed through the intrinsic operator dependency between k odd steps and n even steps (n = f · k). We introduce the method of Cross-Constraint System Reduction, which bridges forward and reverse trajectory dynamics into a single algebraic identity. Due to prime factor divergence (2^a ≠ 3^b), the cross-constraint equation forces all potential integer cycle solutions to collapse exclusively to the trivial cycle x = 1, thereby mathematically precluding non-trivial cycles (x > 1) in positive integers.