Deterministic Structures in the Stopping Time Dynamics of the 3x+1 Problem

Abstract

The $3x+1$ problem concerns the iteration of the map $T:\mathbb{Z}\to\mathbb{Z}$ defined by $T(x)=x/2$ for even $x$ and $T(x)=(3x+1)/2$ for odd $x$. We study the \emph{coefficient stopping time} dynamics of $T$ (in the sense of Terras) by relating parity vectors of Collatz trajectories to exponential Diophantine equations. We construct a recursively generated tree of congruence classes $\bmod\,2^{\sigma_N}$ that characterizes the sets of integers with equal coefficient stopping time $\sigma^\ast(x)=\sigma_N$. We show that these classes satisfy a deterministic recursion and derive arithmetic transition rules between neighboring congruence classes based on differences of the associated Diophantine sums. Finally, we prove that the union of coefficient stopping time congruence classes generated up to a fixed order $N$ is periodic and establish a computable finite-range coverage bound. These results do not resolve the $3x+1$ conjecture, since it remains unproved that the coefficient stopping time coincides with the classical stopping time.