Shiftless Collatz-3 Model

Hiroshi Harada — May 2, 2026

Overview

This repository contains a collection of Python scripts for simulating and visualizing the Shiftless 3-adic Collatz (Collatz-3) model. This is an information-preserving model that eliminates the "information loss caused by right shifts" inherent in the standard 3-adic Collatz map.

The Shiftless model aims to visualize the following:

• The geometric structure of carry avalanches
• The interference moiré between Term A (pure expansion) and Term B (complement injection)
• Fractal structures (gaskets) unique to 3-adic spaces

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Contents

Documents

• TITLE_EN (English title page for the research report)
• REPORT_JP / REPORT_EN
  Research reports detailing the theoretical background, the linear decomposition of the Shiftless Collatz-3 trajectory, and in-depth explanations for each Figure.

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Source Code

Executing each script will automatically generate the corresponding Figure (PNG format).

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code_01.py

Verifies equivalence with the standard 3-adic Collatz model.
Extracts the non‑multiple‑of‑3 part (u_k) of the trajectory to demonstrate perfect alignment with the standard model.
Outputs: figure1.png, figure1.csv

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code_02.py

Visualizes the compact trajectory of the Shiftless Collatz-3 model.
Captures the intense carry avalanches triggered by NZT=2.
Output: figure2.png

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code_03.py

Visualizes the full trajectory by decomposing a single step of the Shiftless model into two distinct phases:

• Reach (Left shift + self-superposition in ternary notation)
• Fill (Local distortion via NZT injection)

Output: figure3.png

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code_04.py

A time-synchronized comparison of three distinct dynamics:

• Pure Expansion: 4n
• Fixed Addition: 4n + 1
• Shiftless: 4n + c_k * 3^v_k

Visualizes the structural differences side-by-side.
Output: figure4.png

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code_05.py

Performs a perfect linear decomposition of the trajectory into the following components to visualize their spatial interference (moiré):

• Term A (Pure Expansion)
• Term B (Complement Interference)

Output: figure5.png

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Requirements

• Python 3.x
• NumPy
• Matplotlib

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Usage

Run each script from your terminal or command prompt:

python code_01.py
python code_02.py
python code_03.py
python code_04.py
python code_05.py

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License

• Research Document: CC BY 4.0
• Python Source Code: MIT License
• © 2026 Hiroshi Harada

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