Saved in:
| Main Author: | |
|---|---|
| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
|
| Online Access: | https://doi.org/10.5281/zenodo.18447604 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- <p>This paper investigates the validity of the Collatz conjecture through the structural properties<br>of the inverse Collatz graph from a measure-theoretic perspective. We propose a generative matrix<br>based on the interaction between the 4x + 1 chain expansion and modulo 3 arithmetic, analyzing<br>the recursive expansiveness of the inverse tree. The key results are as follows: First, by establishing<br>the modular recurrence of the 4x + 1 chain and the principle of ergodic mixing, we demonstrate<br>that the primary tree T (1) achieves an asymptotic density of 1 on the set of natural numbers.<br>Second, we prove that any inverse tree T (x) initiated from an arbitrary seed x possesses a positive<br>measure due to geometric fractal expansion. Finally, we conclude that all natural numbers must<br>connect to 1 by showing that the existence of any disjoint orbit leads to a measure-theoretic<br>contradiction—specifically, that the sum of measures would exceed 1.</p>