Collatz Dynamics: Topological, Combinatorial, and Analytic Program

This repository gathers the numerical experiments, visualizations, and formal documents related to the study of the accelerated Collatz dynamics.

Overview

This project investigates the Collatz conjecture through a synthesis of empirical observation and mathematical structure.
It combines three complementary perspectives:

1. Topological — the natural extension of the accelerated Collatz map on the compact product space $\mathbb{Z}_2 \times \mathbb{Z}_3$, where the integer dynamics lies on the invariant diagonal;
2. Combinatorial — the construction of a backward automaton with canonical horizon, ensuring exhaustivity of admissible predecessors and reduction by modular constraints;
3. Analytic — a hybrid Lyapunov potential combining an Archimedean component and a 3-adic valuation to establish a pointwise negative drift outside a compact region.

The framework unifies these layers into a coherent proof program demonstrating the uniqueness of the periodic orbit (the fixed point {1} for the accelerated map, and {1,2} for the non-accelerated version).

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Genesis and Intuition

This project began from numerical exploration.
By plotting the number of steps to reach 1 for large ranges of integers (e.g., numbers $n \equiv 4 \pmod{6}$), clear banded and layered structures appeared, suggesting hidden regularities.

These patterns revealed a deeper structure when interpreted through the arithmetic lenses of:

• Parity (mod 2), linked to the 2-adic topology;
• Divisibility by 3, linked to the 3-adic topology.

Their product, $\mathbb{Z}_2 \times \mathbb{Z}_3$, provides a compact, continuous state space where the Collatz dynamics extends smoothly.
In this setting, the apparently chaotic integer trajectories become projections of an ordered motion on the product space.

Collatz on $\mathbb{Z}$ is the shadow of a regular map on $\mathbb{Z}_2 \times \mathbb{Z}_3$.

The accompanying visualizations illustrate this empirical intuition and serve as a bridge between numerical patterns and rigorous formulation.

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Papers

The repository includes two complementary papers:

• Collatz.pdf: Uniqueness of the Cycle for the Accelerated Collatz Dynamics — a structured academic study combining topological, combinatorial, and analytic perspectives on the Collatz map.
• Intro.pdf: Genesis and Intuition in the Collatz Dynamics — a bilingual (English/French) narrative describing the empirical origin of the research and the intuition behind the topological model on $\mathbb{Z}_2 \times \mathbb{Z}_3$.

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Visual Highlights

Layered patterns for n ≡ 4 (mod 6)

Toroidal representation in $\mathbb{Z}_2 \times \mathbb{Z}_3$

Logarithmic scale visualization

Flight time for selected integers

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Repository Structure

.
├── latex
│   ├── Collatz.pdf        # Formal mathematical paper: "Uniqueness of the Cycle for the Accelerated Collatz Dynamics"
│   ├── Collatz.tex        # LaTeX source for the paper
│   ├── Intro.pdf          # Genesis and intuition (bilingual EN/FR exposition)
│   └── Intro.tex          # LaTeX source for the introduction
├── images
│   ├── modulus.png        # Layered patterns for n ≡ 4 (mod 6)
│   ├── toroidal.png       # Toroidal representation in Z2 × Z3
│   ├── log-scale.png      # Logarithmic scale visualization
│   └── flight-time.png    # Flight time for selected integers
├── notebooks              # Jupyter notebooks for numerical experiments and 
└── README.md              # This document

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Citation

If you use this repository or reference its content, please cite it as:

@misc{boyer2025_collatz_dynamics,
  author       = {Alexandre Boyer},
  title        = {Collatz Dynamics: Topological, Combinatorial, and Analytic Program},
  year         = {2025},
  note         = {Zenodo repository},
  url          = {https://github.com/ng-galien/collatz}
}

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License

Distributed under the CC BY 4.0 License.
You are free to share and adapt the material, provided proper attribution is given.

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Keywords

Collatz conjecture · Z2 × Z3 · p-adic analysis · dynamical systems ·
Lyapunov function · backward automaton · topological dynamics · empirical mathematics

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Alexandre Boyer — October 2025