Latest Attempts Towards a Proof
The question at hand is to prove that non-trivial branches of x3+1 space hold their own and never have long-term %-membership of zero. On some level, this is really a question about "what's the smallest that a non-trivial branch can get in x3+1 space?" In order to study this, I've spent much of the last year fairly intensely studying the branches.
Each non-trivial branch has its own fascinating fractal-like shape, totally unique like a snowflake (I've been able to prove this). Each branch's shape is entirely dictated by the location of the base of the branch on the Natural Number Line. What's astonishing and mystifying is there seems to be a very deep hidden pattern underlying the shapes of all the Collatz branches next to one another... A mind-blowingly complex pattern hidden within the branches of the great tree. I call it the "Grand Unified Pattern". While I'm not sure that any human mind will ever truly understand it, I've come so far to understand little pieces of it, and perhaps that might be enough to construct a proof out of. In particular, I have been studying the mathematical mechanisms behind what governs the shapes of these Collatz fractal pieces. I'll cover more about this in the next section.
For now, let us restrict our attention to the discussion of the size of these Collatz branches.
How small can a non-trivial Collatz Branch get? This is now the essential question. Suppose we have a x3+1 non-trivial Collatz Branch whose base is at the node n. We've seen experimentally that its
%-membership function seems to settle out into a semi-sinusoidal wiggle. Let us take that wiggle and compute its LimInf and LimSup. We'll then take the average of those two quantities. This new number is a fairly decent approximate to the long-term average "size" of that Collatz branch. Let's give these long-term average sizes a name by defining a new function H(n) := ( LimInf(%(n)) + LimSup(%(n)) ) ÷ 2.
This function is one of the first things I want to aggressively study as soon as college ends. I really want to know, if you were to graph the H(n) function:
Would patterns appear in it? Would there appear to be any kind of logic to which branches are vast and which are tiny? Is it almost random? I'm desperately curious.
But let me offer you this small idea... Consider two particular Collatz branches, the one based at n=10 and the one based at n=1,000,000,000. The branch at n=10 is "big", in a matter of speaking, as it is near the bottom of the entire tree. The branch at n=10^9 is "tiny", as it is way up high in the tree, literally a very small piece of the branch at n=10. Since n=10^9 falls down through n=10 on its way to reaching 1, we already know that H(10) ≥ H(10^9).
But is this really a fair way to discuss the two branches' relative size? Perhaps not. What if the branch at n=10 were moved so that it now started at n=1,000,000,000? What if we left its branching fractal shape exactly as it is now, but simply moved its base point so it coincided with the base point of n=1,000,000,000? We would now be able to, in a much more fair way, tell which branch has a larger fractal shape!
We can do this in a rigorous mathematical way by defining a new function, the relative size of the branch at n, as RS(n) := H(n) * n. This relative size function removes any confusion of which branches are bigger or smaller based on where they are geographically located within the great Collatz tree and lets us strictly compare the sizes of their fractal shapes. Thus perhaps even a more important than The H(n) Graph is The RS(n) Graph:
I'm currently in the process of writing a new fairly advanced Collatz detector that will explore and chart the long-term trajectory of each and every "wiggle" for all the branches under n=10^6. When I finally have this data in my possession, the RS(n) Graph will let me begin to address the following important issue:
It seems so far experimentally that no matter what non-trivial branch you choose, that it always holds its own, i.e. H(n) is never zero on the non-trivial branches. This in turn implies that RS(n) is never zero for them. In a matter of speaking, we could refer to this idea as that each non-trivial branch pointwise holds its own...
But if I'm lucky in my attempts at a proof, I think its more than that. It's possible that the non-trivial branches uniformly hold their own, i.e. that all of them are larger in their fractal shape than some non-zero lower bound. From all my research so far, this seems very plausible. If the RS(n) function appears to have a non-zero lower bound (on the non-trivial branches), then that's great news for the conjecture! It would mean that all I have to do is identify what is the smallest possible Collatz fractal shape a branch can have and then prove that just that one branch holds its own!
Whether the universe is kind (and the branches uniformly hold their own) or the universe is more harsh (the branches only pointwise hold their own), either way to address such questions I'll first need to gain a much deeper understanding of the science of the Collatz fractal shapes, their underlying mechanism so to speak. As I stated earlier, this is no easy task. In the next section let me give you a little intro on how I'm going about it...