A Light in the Darkness
For the first time ever we may actually be holding in our hands a legitimate attack on one half of the Collatz Conjecture!
A sketch of a proof...
Suppose for a moment the my conjecture about "The Wiggle" is true. Suppose that indeed in x3+1 space every non-trivial branch's % membership function %(n) has LimSup > 0 aka all non-trivial branches hold their own, as n goes to infinity. Then we get the following:
Suppose additionally that x3+1 space contains divergence. The existence of a divergent element implies that there is a divergent tree. As the orbit of this divergent element goes to infinity, we know it must utilize steps to the right, x3+1 steps, to reach infinity. But with every x3+1 step to the right, we know that the Collatz tree splits right there into two branches, as you could also reach that same value by a ÷2 step. Thus the divergent tree necessarily contains non-trivial branches.
These non-trivial branches have % membership LimSup > 0, so the overall divergent tree has % membership LimSup > 0. Thus the collection of all divergent trees put together has % membership LimSup > 0. Thus the long term fraction of hailstones which are divergent does not go to zero as n goes to infinity.
Which violates Terras Theorem! >>> Contradiction!
Thus if I can show my conjecture to be true about non-trivial branches holding their own, I'll have proven one half of the Collatz Conjecture!
But how do we prove that non-trivial branches hold their own and "wiggle"? This is now paramount! When I first realized the importance of this statement to the Collatz Conjecture, I set out with a maniacal burst of enthusiasm to prove it.
... Instead it proved itself ... difficult.
Ultimately, who is surprised? This is still the Collatz Conjecture we're talking about. It's a notoriously very hard math problem.
My recent life has been dedicated (amongst other things) to a very active attempt to prove this interesting new version of the divergence half of the Collatz Conjecture. In the next chapter I'll walk you through my main two plans of attack on the problem.