Chapter 4.6:

How Close to x3+1?

             So one absolutely fascinating result of all this is that we have finally come closer to answering the question of "how close" is x3+1 to having divergence or to having alternative loops? We seem to have found (at least experimentally) that the x3+1 world lives in a small "partial neighborhood" on its right side such that for all tiny values of s and t, x(3+s)+(1+t) is entirely convergent and has only 1 loop, i.e. the famous loop at n=1. Let's for now focus our attention to Collatz Worlds along Path A, i.e. worlds of the form xa+1 and ask two very specific questions:

                          For a ≥ 3, what's the lowest value of a such that xa+1 has divergence?

                          For a ≥ 3, what's the lowest value of a such that xa+1 has more than one loop?

             It might be relevant to point out that the lowest value of a that I've ever seen which results in a world containing divergence is x3.2+2.3. But along Path A, the lowest value pretty decisively seems to be x3.6+1. It stands out well and beyond as the earliest one. Maybe there is something special about the values x3.2 and x3.6 that results in them having some kind of feedback loop which sends certain odd numbers vaulting up higher and higher in a controlled and non-chaotic fashion. Maybe they are fundamentally different than the values around them and should be viewed as some kind of interesting outliers? I do not know. 

             Past x3.6+1, the earliest next case where we spotted divergence was x3.984+1. But let's take a closer look at that world:

             See those little pink dots at the top of the Stopping Times Graph? Those are the divergent entries. But are they really actually divergent? Recall that when investigating Collatz Worlds with Python, you cannot actually "find" divergence. All you can find are hailstones which shoot up incredibly far. We currently have the divergence cutoff set to 10^100. Those pink dots rise above that. But are they actually diverging? Let's take a look at a few and see if we can figure out what they are really doing...

             The left-most pink dot is at n=673. When tossed into my Python engine, it currently returns as divergent. But the divergence cutoff is set to 10^100. Let's relax that to 10^1000. Suddenly it returns as convergent, looping at n=19. The problem was that its max value along the way was a number with 207 digits. So n=673 does not really diverge at all. Testing down the line, each of those pink dots that seemed to diverge... they're all actually convergent hailstones, just ones that fly astronomically high to values which have hundreds and then later on thousands of digits before falling back down. It seems a lot like x3.984+1 Collatz Space is actually entirely convergent. 

             Keeping the divergence cutoff super high (I have now it set to 10^100000) makes my python engines run a lot slower and is impractical for creating the kinds of graphs we were doing before. But we're serious now. When, after x3.6+1, does divergence again appear on Path A?

 

             Exploring down Path A with the divergence cutoff set to 10^100000, we find that the first world which even then still has divergence is x4.015+1. The input n=437 is the first one in that world which appears to diverge. This one hailstone seems to be a strong potential candidate to be the actual first appearance of divergence between x3.6+1 and x5+1 spaces. I will add, though, that it appears to diverge diverge very slowly, ridiculously slower than the hailstones in x5+1 space do. 

             Conjecture: On Path A, between x3.6+1 and x5+1 spaces, the first appearance of divergence is somewhere right after x4+1. Perhaps even the infimum of the values of a which result in divergence is 4. 

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             Ok. Now to discuss the second question: when along the way do alternative loops than n=1 appear? Ready for this? This one's kinda wild...

             So when I was first exploring non-integer Collatz realms, back before I had learned Python, I was working just by pencil and paper. I checked out x3+1, x3.1+1, x3.01+1... and found them all to be 1-Looped with a loop at the bottom of n=1. I kinda gently figured that near to x3.000+1, there was always just the single loop at n=1. 

             Boy was I wrong. All I would have had to do was go just one notch further to find that the x3.001+1 world has an alternate loop at n=103. What a strange surprising fact that is. Check out how gorgeous this Collatz World is!!

             I can't get enough of this particular pair of images. I find this Collatz World beautiful and a half. Anyhow, this discovery really sent me down the rabbit hole of finding exactly when does Path A first encounter an alternate loop than just n=1? For the longest time my Python engines just didn't have the kind of deep deep precision needed to handle such a question. The closest I could get was the knowledge that x3.0065+1 has two loops, one at n=1 and one at n=31, and yet x3.0064+1 (or lower) has only the loop at n=1. For the longest time the precise value eluded me. 

             In a fairly recent programming epiphany one night I figured out how to generate literally arbitrary precision! And so my roommate John and I stayed up deep into the night once to calculate the exact value at which, as we travel towards x3+1, what's the last time we see an alternate loop? That night we discovered that the precise value of the bifurcation is approximately:

 

x3.0006499837504062398440039+1

 

             Let me show you what I mean... 

             This particular constant appears to be exactly where the bifurcation happens past which, as you travel towards x3+1, there is never again any loop other than n=1. The graph will forever thereafter be entirely blue. It's bloody fascinating, and I cannot tell you for the life of me why this is true. 

             Conjecture: On Path A, for values of a with

3 ≤ a ≤ 3.000649983750406239844038, all worlds xa+1 are 1-Looped. 

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             Alright. And with that, we conclude Chapter 4 on the Collatz A-B Plane. The next chapter is radically different from this one. We stop exploring new stuff and start very carefully analyzing what we have already found. What clues and patterns can we find in the data? And what Collatz knowledge can we learn from these clues?

The x3.984+1 World

The x3.001+1 World

The x3.000649983750406239844039+1 World

The x3.000649983750406239844038+1 World

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