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

TL;DR

This paper investigates the structure of the stopping time dynamics in the 3x+1 problem by analyzing parity vectors, constructing a recursive tree of congruence classes, and proving periodicity properties, without resolving the conjecture.

Contribution

It introduces a recursive framework for classifying integers by coefficient stopping time using congruence classes and Diophantine equations, revealing deterministic patterns.

Findings

Constructed a tree of congruence classes modulo powers of two.

Derived arithmetic transition rules between classes.

Proved periodicity of union of classes up to fixed order.

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.