Non-existence of Non-trivial Positive Integer Cycles in the 3n+1 Iteration

The 3n+1 problem asserts that for any positive integer n, repeatedly applying the operation "divide by 2 if even, multiply by 3 and add 1 if odd" will eventually reach 1. The problem is divided into two parts: (a) there are no non-trivial cycles; (b) there are no divergent trajectories. This paper investigates part (a) within the framework of the 4n+1 axis. We transform the 3n+1 iteration into the composition of three types of linear transformations am+b => cm+d, introducing a fixed-point parameter S = (D-B)/Delta (requiring S to be a non-negative integer for a cycle) and a composite gap G = A+B-C-D. The core result is: if A > C, then G > 0 (Section 4), which implies -1 < S < 1, thus forcing S = 0. Using modular reduction, size arguments, and Diophantine approximation, we prove S != 0, thereby excluding all non-trivial positive integer cycles.