My recent work suggests that these problems can be analyzed using binary length evolution as a key property for distinguishing between convergence and divergence.

If a sequence has a systematic tendency to reduce its binary length, it converges.

If binary length increases persistently, it diverges.

The methodology extends to large-scale computational experiments, showing universal patterns in stepwise binary length variations across different k-values.

I made available the following preprints on zenodo:

COLLATZ

A Simple binary analysis of the Collatz Conjecture

Preprint A Simple binary analysis of the Collatz Conjecture

Numerical Analysis of Collatz Sequences Using Binary Representation

https://doi.org/10.5281/zenodo.14219218

Statistical Analysis of Binary Length Evolution in Collatz Sequences

https://doi.org/10.5281/zenodo.14252942

K X + 1

Divergent binary length growth in the 5x+1 problem

https://doi.org/10.5281/zenodo.14711287

Analysis of the 7x + 1 problem as a boundary case

https://doi.org/10.5281/zenodo.14778057

Analysis of the kx+1 Problem for k>7

https://doi.org/10.5281/zenodo.14789415

DIDACTICS

Exploring the Collatz Conjecture with an arduino collatzer

https://doi.org/10.5281/zenodo.14696864

The arduino based K-Collatzer A Hardware Implementation for Exploring the 5x+1 and kx+1 sequences

https://doi.org/10.5281/zenodo.14852361

Any feedback, critiques, or discussion would be greatly appreciated!