Chapter 7.3:
One Last Surprise
If for any reason all my research on trying to understand the intimacies of the science of the Collatz branches does not prove to be helpful... There is another way. One last major route to try and solve the divergence half of the Collatz Conjecture.
It's possible, at the end of the day, that studying the branches at all is the wrong direction to go in. It's possible that I really should be spending my time directly attacking the wiggle. For the legendary "wiggle" has one last surprise in store for us:
As you can see from that video, the wiggle is not just "semi-sinusoidal", it's actually just about periodic! To be more precise, it appears that if you look out far enough that the wiggle begins to asymptotically approach a particular fractal-like periodic shape. This occurs precisely on powers of (4/3)^n. This wild, unexpected level of periodic behavior gives us a whole new attack on the problem.
If we stop considering the full %(n) function (whose graph produces the wiggle), and instead limit our attention to cross sections of it at intervals of powers of (4/3)^n, it becomes a lot lot nicer. What I mean by this is, if we let Xn be the sequence (4/3)^n (rounded down to the nearest integer), and we now consider the sequence %(Xn) ... This new sequence appears to very rapidly approach a non-zero Limit!
What this means is that instead of getting lost out on the branches trying to understand their logic and structure, all we really have to do is to finally fully understand what is causing the Tiling Phenomenon discussed in Chapter 5.3.
In Chapter 5.8 we discussed why the Stopping Times Graph has these bands at all... but what I would need to do now is fully understand the science and mechanism behind exactly how many dots are in each horizontal band. There's clearly some kind of very strong pattern going on here. I just need to find it. If I can do so, then I'll be holding a proof of the divergence half of the Collatz Conjecture in my hands.
