The Underlying Mechanics of the Collatz Branches
If our goal is to fully understand the % memberships of x3+1 branches, then we'll need to understand more of the mechanics of how Collatz branches work. We'll need to lift the car hood and study the inner workings of the engine.
The true science of the Collatz branches is very intricate and advanced. Even with all I've learned, I'm only so far exploring the very tip of a vast deep iceberg. It would take far too long (and be too complex) for the purposes of this website for me to explain to you everything. Let me, however, show you a little of what I know in the following video:
If the video was a little long or confusing, I apologize. It's just that we're now starting to talk about Collatz ideas that are much more advanced and involved than we've done formerly on the website. Basically I've invented a way to categorize Collatz fractal shapes, labeling each one with a unique sequence of 0's, 1's, and 2's... a sequence I call it's "Blue Name Sequence".
As I stated before in the video, when you encounter orange branching nodes of the different colors, blue, red, and white, it tells you a little something about the size of that Collatz branch. Since the sequence of 0's, 1's, and 2's dictates the colors, it turns out that branches whose blue name sequence have lots of 0's tend to be huge! Ones with lots of 1's are sort of average sized. Ones with lots of 2's are tiny. As an example the famously enormous branch at n=9232 has a blue name sequence which begins with 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 0... On some level the sequence determines the size.
What I really badly want to explore is something I call the "Idealized" RS Graph for Sequences. I want to take tons and tons of these sequences of 0's, 1's, and 2's and carefully construct their corresponding idealized Collatz fractal shapes. (Refer to Chapter 5.8 if you forget what idealized means.) I then want to determine the long-term % membership Relative Size for these idealized branches. This I will plot on the y-axis. ... On the x-axis I want to take that sequence of 0's, 1's, and 2's and convert it into a base 3 decimal. This all, put together, should produce a graph which, given a particular idealized Collatz Fractal shape, shows what its relative size is.
Considering that there is a strong correlation between the relative size of a branch and how many 0's are in its blue name sequence, I would not be at all surprised if the above graph proves to have some strong obvious patterns in it. Hopefully, if I'm lucky, I'll be able to use the technique of these blue name sequences (my own homemade kind of symbolic dynamics) and actually be able to determine (with proof!) literally exactly which Collatz fractal shape takes up the least amount of room. This will reduce the problem of showing that all Collatz non-trivial branches hold their own to showing that precisely one smallest non-trivial branch holds its own.
It's been a fascinating endeavor. I'll keep you posted on the progress!