Bonus Chapter:
Strange Little Pyramids
Okay. So as we drive down the xa+1 Collatz highway, we spot these "strange little pyramids" along the way. But what are they exactly? The most sensible obvious ones are at x2+1, x4+1, etc... These exist because in these worlds at these locations every odd number is its own loop_seed. At values very close to such a pyramid, perhaps x4.003+1 for instance, n=1 follows the exact same path as it does in the x4+1 world. In fact for n=1 all the way up to n=333, these operate just like they do in x4+1. But for n > 333, these hailstones now once again enter the random chaotic madness of the Collatz systems. The closer you get to x4+1, the higher up the values of n you have to go to see any deviation. This helps explain the pyramid shape that we are seeing.
However, what about all the other strange little pyramids that we are observing? What causes them? Let's take a particular close look at one in particular... the one at x11.314+1.
We see the familiar Pyramid structure reveal itself amid the loop_seeds of these worlds. We see the gradual build-up from the first appearance of convergence at x11.236+1 with a single loop_seed at n=3, up to a roaring long list of loop_seeds at x11.314+1 (the peak), dwindling down gently to a single loop_seed at x11.399+1, and then a return to worlds that are all divergent. But is x11.314+1 actually the peak of this pyramid? If not, what is the peak? Let's zoom in with Loop_Seed Graphs.
This is the Loop_Seed Graph for the pyramid as zoomed-in as we have gone so far:
Graphing x11.200+1 to x11.435+1 /\ = 0.001
We see that it appears to have a great "spike" right on top. One that if we zoom in on far enough might even have its peak reaching directly up towards infinity (like the infinitely tall spike at x2+1). Let's zoom in further...
Graphing x11.313+1 to x11.315+1 /\ = 0.0001
The spike seems outrageously tall. Let's zoom in even further:
Graphing x11.313708+1 to x11.313709+1 (in ten steps)
And even further:
Graphing x11.3137084989845+1 to x11.3137084989850+1 (in fifty steps)
Alright! We seem to be well on our way to a better understanding of some of the properties of these strange little pyramids. Just like the pyramid spike at x2+1 and at x4+1, the peaks of the strange little pyramids appear to rise all the way to infinity, spiking at some particular real number xa+1. It's bloody fascinating.
And here's one more absolutely mind-blowing detail... These strange little pyramids are all over the place on Path A. No matter where you look on xa+1, so long as a is greater than 2 and a is not "near" to a multiple of 2 (these exceptions are two regions of particularly notable stability), there always seem to be hidden strange little pyramids (tiny ones) within any arbitrary distance away.
I myself would gently conjecture that these strange little pyramids are dense in Path A... on most domains anyway. I have some absolutely fascinating evidence for this conjecture, but it is incredibly technical and is outside the scope of this presentation. Suffice it to say that proof or disproof of this fact is very much so an active area of research.
I hope you enjoyed this bonus page on the marvelous "strange little pyramids".
Cheers...