Modular Structure Analysis of 3-adic Collatz Trajectories

Hiroshi Harada - April 12, 2026

This research aims to elucidate the trajectory transitions of the Collatz-type mapping (Collatz-3) in an integer space with a 3-adic structural topology, using base-3 logarithmic spiral plots and modular arithmetic.

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Overview

In contrast to the classical binary Collatz conjecture (3n+1 problem), this study focuses on the Collatz-3 mapping based on modulo 3 arithmetic.

By introducing the perspective of Balanced Ternary, we reveal:

• The hidden ±1 mirror symmetry in trajectory transitions
• The magazine (series) structure formed by non-multiples of 3
• The hierarchical confluence of these series into specific launcher axes (a·3^b)

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Theoretical Pillars

1. Definition of the Collatz-3 Mapping (Balanced Ternary)

Based on balanced ternary, the mapping f(n) is unified as follows:

• When n ≡ 0 (mod 3)

  f(n) = n / 3
  

• When n ≡ c (mod 3), c ∈ {1, -1}

  f(n) = (4n - c) / 3
  

This formulation eliminates the asymmetry of the traditional “3n+1 / 3n−1” rules and reveals the ±1 symmetric structure.

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2. 9m±2 Twin-Engine Magazines (Cooling Injectors)

Among non-multiples of 3, the 9m±2 type is a special series that always lands on a multiple of 3 in exactly one step (cooling).

• Cooling: Landing on a multiple of 3
• Heating: Jumping to another non-multiple of 3

In the inverse mapping, the sign always toggles:

9m + 2  ↔  9m - 2

forming a twin-engine toggle structure.

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3. a·3^b Glide Launchers

For any non-multiple of 3, a, the sequence:

a, 3a, 9a, 27a, ...

glides radially inward while maintaining a fixed azimuth angle, forming the “launcher axes” toward which trajectories converge.

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Visualization Method: Base-3 Logarithmic Spiral Plot

Any natural number n is mapped to the following polar coordinates:

r = log_3(n)
θ = 2π · frac(log_3(n))   # frac(x) is the fractional part

This visualizes the 3-adic hierarchical structure of integers as a logarithmic spiral on a radar-like chart.

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File Structure

• TITLE_EN.pdf
• REPORT_EN.pdf / REPORT_JP.pdf
• code_01_collatz3_radar.py
  • A script that visualizes the full trajectory from an initial value (e.g., n = 8) to 1 on a base-3 logarithmic spiral.

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Usage

Requires Python (NumPy, Matplotlib).

python code_01_collatz3_radar.py

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Case Study: Trajectory Transition for n = 8

The initial value 8 converges through the following deterministic phases:

1. 8 (Heating) → 11 (Cooling)

   • Merges into the 11·3^b launcher

2. 15 (Falling) → 5 (Heating)

   • Merges into the 5·3^1 launcher

3. 7 (Cooling) → 9 (Falling)

   • From the 7·3^0 launcher to 1·3^2

4. 3 → 1

   • Final convergence

This transition illustrates the hierarchical modular structure:
Magazine → Launcher → Falling → Next Launcher.

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License

• Reports and Documents:
  Creative Commons Attribution 4.0 International (CC BY 4.0)

• Source Code:
  MIT License

© 2026 Hiroshi Harada