Shiftless & Carryless Collatz Models

A New Visualization Framework for the $3n+1$ Problem via $\text{GF}(2)$ Algebraic Analysis × Fractal Geometry × Information Theory

Hiroshi Harada - April 4, 2026

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🧭 Overview

This repository is a collection of research code for analyzing the unsolved Collatz conjecture ($3n+1$ problem) from a new perspective: information flow, fractal structures, and $\text{GF}(2)$ algebra.

In the traditional Collatz map, observing the internal structure of the trajectory is hindered by:

• Information loss due to right shifts (÷2) in even steps.
• Structural destruction caused by carries (carry propagation) during the $3n+1$ odd steps.

To solve this problem, this repository introduces the following three models.

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✔ Shiftless Model

Constructs a Collatz map that discards zero information by eliminating right shifts and preserving the weight of the LSB.

Update rule: $$n_{k+1} = 3n_k + \text{LSB}(n_k)$$

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✔ Redundant Binary (RB) Representation

Fully records the Shiftless trajectory by retaining carries during addition as polynomial coefficients instead of processing them immediately.

Evaluating the RB polynomial at $x=2$ perfectly matches the Shiftless integer value.

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✔ Carryless Model

Extracts the pure fractal skeleton (Rule 90) by projecting the coefficients of the RB polynomial onto $\text{GF}(2)$ (mod 2), completely eliminating carries.

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🔥 Cellular Automaton (Rule 90) and the Emergence of Chaos

The analysis in this project clarifies that the complexity of the Collatz trajectory can be explained by the interference of three elements: Order × Noise × Carry.

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🟦 1. Order: $3n$ without carries is equivalent to Rule 90

As a $\text{GF}(2)$ polynomial, the carryless $3n$ becomes: $$(x+1)P(x)$$ which perfectly matches the 1D cellular automaton Rule 90.

→ A pure Sierpiński gasket emerges.

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🟧 2. Noise: LSB injection disrupts the fractal

In the Shiftless model, at each step, LSB noise is injected: $$x^{L_k}$$

→ It enters the "holes" of the fractal, causing local disturbances.

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🔴 3. Chaos: Carry avalanches destroy the structure

In the standard Collatz ($3n+1$), LSB noise triggers carries, resulting in global structural destruction (carry avalanches).

→ The collision of Order × Noise × Carry generates the "chaos" of the Collatz trajectory.

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📂 Included Scripts

This repository includes the following analysis code.

File Name	Description
code_01_equivalence_shiftless.py	Verifies equivalence between Shiftless and standard Collatz (odd sequence)
code_02_equivalence_rb.py	Verifies perfect match between RB polynomials and Shiftless
code_03_3n.py	Visualizes fractal destruction with and without $3n$ carries
code_04_carryless.py	Compares Shiftless (noise + carry) and Carryless (noise only)

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🛠 Requirements

• Python 3.8+
• NumPy
• Matplotlib

pip install numpy matplotlib

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▶️ Usage

Each script can be run independently.

Example:

python code_03_3n_carry_vs_carryless.py

The generated figures allow you to visually compare the differences between:

• Fractal structures
• Noise injection
• Carry avalanches

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📜 License

• Python Source Code: MIT License - Research Documents / Theory: CC BY 4.0 - Copyright (c) 2026 Hiroshi Harada

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🌌 About This Project

This repository is designed as a "new platform to play with Collatz."

• Fractals
• Cellular Automata
• $\text{GF}(2)$
• Information Theory
• Number Theory

It is created as a "playground" where these fields intersect, allowing researchers, math enthusiasts, and programmers to explore freely.