Shiftless & Carryless Collatz-3 Model

Extraction of the Ternary Gasket and Emergent Chaos via Redundant Ternary Representation

Hiroshi Harada — May 5, 2026

This repository presents a research project that decodes how complex trajectories (chaos) emerge in the 3-adic Collatz map (Collatz-3). It explains this complexity as the interference of three core elements: Order × Noise × Carry Avalanche.

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📘 Overview

The primary obstacles in analyzing the 3-adic Collatz map are two-fold:

• Nonlinear information loss due to right shifts (division by 3)
• Intense carry propagation inherent to ternary addition

To isolate and extract these destructive mechanisms, this study introduces the following three abstractions:

1. Shiftless Model
   Eliminates right shifts and preserves all trit-level information.

2. Redundant Ternary (RT) Representation
   A redundant ternary representation using polynomials $P(x)$.
   It retains carries as coefficients ("delayed carries") instead of resolving them immediately.

3. Carryless Model
   Projects coefficients onto the finite field GF(3), creating a pure algebraic model that completely eliminates carries.

By utilizing these models, we successfully extract the pure ternary Sierpiński gasket, which is normally hidden by carries in standard integer arithmetic. This framework provides a visual and algebraic explanation for the origin of chaos in 3-adic dynamics.

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🔍 Key Findings

1. Worldview Contrast: 2-adic vs 3-adic

• 2-adic Collatz:
  Order (Rule 90) dominates; noise "floats on a sea of structure."

• 3-adic Collatz:
  Noise and carry avalanches dominate; order "sinks beneath the noise."

This inversion of dominance is the defining characteristic of Collatz-3 complexity.

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2. Anatomy of Chaos: Order × Noise × Carry

The complexity of the Collatz-3 trajectory emerges from the interaction of the following factors:

• Order:
  The fractal structure (ternary gasket) generated by the geometric expansion $(x+1)$.

• Noise:
  Asymmetric noise injected by the lowest non-zero trit (NZT):
  "+1" (triggering avalanches) and "-1" (silent erasure).

• Carry Avalanche:
  Asymmetric ternary carries that propagate upward and destroy large-scale structure.

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📁 Repository Contents

📄 Documents

• TITLE_EN.pdf
  The title page (English) of this study.
• REPORT_EN.pdf / REPORT_JP.pdf
  Research reports summarizing the theoretical background, visualization results, and conclusions of this study.

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💻 Source Code

A suite of Python scripts designed to verify and visualize the theory presented in the study.

code_01.py — Shiftless Model Verification

Confirms that the non-3 number subsequence of the standard Collatz-3 map exactly matches the Shiftless model under NZT normalization.

code_02.py — RT Polynomial Equivalence

Demonstrates that Shiftless integer arithmetic perfectly matches the evaluation of the redundant ternary polynomial $P(x)$ at $x = 3$.

code_03.py — Standard $4^n$ vs Carryless $(x+1)^n$

Visually compares how carry propagation destroys the fractal versus the emergence of a perfect ternary Sierpiński gasket over GF(3).

code_04_3adic.py — Visualizing Emergent Chaos (Seed 1 & 28)

Compares the Shiftless model (Noise + Carry) with the Carryless model (Noise only) to visualize the exact moment chaos emerges from the "Order × Noise × Carry" interference.

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🧩 Requirements

pip install numpy matplotlib

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📜 License

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

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