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

Mike Winkler

TL;DR

The paper addresses the stopping time problem in the Collatz map by encoding trajectories with parity vectors and linking them to exponential Diophantine equations. It constructs a deterministic, recursively generated tree of stopping-time congruence classes modulo $2^{ abla_n}$ and derives explicit arithmetic transition rules between neighboring classes based on Diophantine sums. It proves that the union of stopping-time congruence classes up to order $N$ is periodic with period $2^{ abla_N}$, yielding a computable finite-range coverage bound $K(N)$. While not resolving the global conjecture, this framework constrains the possible starting residues and provides a rigorous method for finite-range verification and structural insights into Collatz dynamics.

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 $x$. This paper investigates the stopping time dynamics associated with $T$ within a deterministic and algebraic framework. By relating the parity vectors of Collatz trajectories to exponential Diophantine equations, we construct a recursively generated tree of congruence classes $\bmod\, 2^{σ_n}$ that characterizes the stopping time classes $σ(x)=σ_n$. We demonstrate that the generation of these classes follows an explicit deterministic recursion and derive arithmetic transition rules between neighboring congruence classes, based on the differences of the associated Diophantine sums. Finally, we prove that the union of stopping time congruence classes generated up to a fixed order $N$ is periodic, establishing a computable finite-range coverage bound.

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

TL;DR

The paper addresses the stopping time problem in the Collatz map by encoding trajectories with parity vectors and linking them to exponential Diophantine equations. It constructs a deterministic, recursively generated tree of stopping-time congruence classes modulo and derives explicit arithmetic transition rules between neighboring classes based on Diophantine sums. It proves that the union of stopping-time congruence classes up to order is periodic with period , yielding a computable finite-range coverage bound . While not resolving the global conjecture, this framework constrains the possible starting residues and provides a rigorous method for finite-range verification and structural insights into Collatz dynamics.

Abstract

The problem concerns the iteration of the map defined by for even and for odd . This paper investigates the stopping time dynamics associated with within a deterministic and algebraic framework. By relating the parity vectors of Collatz trajectories to exponential Diophantine equations, we construct a recursively generated tree of congruence classes that characterizes the stopping time classes . We demonstrate that the generation of these classes follows an explicit deterministic recursion and derive arithmetic transition rules between neighboring congruence classes, based on the differences of the associated Diophantine sums. Finally, we prove that the union of stopping time congruence classes generated up to a fixed order is periodic, establishing a computable finite-range coverage bound.

Paper Structure

This paper contains 22 sections, 10 theorems, 46 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Stopping Time and Congruence Classes
  3. Subsequences and a stopping time term formula
  4. Parity Vectors and Parity Vector Sets
  5. Generating the parity vectors
  6. Order of the generated parity vectors
  7. Diophantine Equations and Integer Solutions
  8. Scope and limitations
  9. Finite-Range Coverage and a Computable Bound
  10. Conclusion and outlook
  11. Appendix
  12. Algorithms in PARI/GP
  13. Algorithm for Theorem \ref{['thm_A100982_A076227']} and A100982
  14. Algorithm for Theorem \ref{['thm_A100982_A076227']} and A076227
  15. Algorithm for Theorem \ref{['thm_algo_parity_vectors']}
  16. ...and 7 more sections

Key Result

Theorem 3

For each $n\geq 1$, there exists a finite set $\mathcal{C}_n\subseteq\{0,1,\dots,2^{\sigma_n}-1\}$ of representatives for congruence classes $\bmod\, 2^{\sigma_n}$ such that for every integer $x>1$, For each $n\geq 3$, the cardinality $|\mathcal{C}_n|$ (listed in https://oeis.org/A100982) and the number of remaining congruence classes $\bmod\, 2^k$ (listed in https://oeis.org/A076227) can be gene

Figures (3)

  • Figure 1: Initial levels of the directed rooted tree of parity vectors, shown up to $\mathbb{V}(4)$.
  • Figure 2: Visualization of the two-dimensional growth of the tree structure.
  • Figure 3: Initial structure of the directed rooted tree of congruence classes $\bmod\, 2^{\sigma_n}$, up to $\mathbb{V}(4)$.

Theorems & Definitions (29)

  • Definition 1
  • Remark 2
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • proof
  • ...and 19 more