These relate to the Collatz conjecture.
1. There might be some surprising results when looking at large numbers of elements simultaneously, studying the graph the same way one might study a gas. There could be some analogue to pressure and temperature which stays constant as a set of elements evolves through time.
There might also be an analogue to entropy. If one could demonstrate that entropy must increase over time as a well-ordered set evolves (say for instance all integers from 1 to 2^n), then the existence of a loop might disturb this entropy.
2. There is an almost-but-not-quite predictive relationship between the binary expansion of a number and its parity sequence. By parity sequence, I mean the sequence of ups and downs that happen to an element x under the accelerated Collatz function. For instance, the number 3 has an up (to (3(3) + 1)/2 = 5) then another up (to (3(5) + 1)/2 = 8) then a down (to 8/2 = 4). If we label ups with 1s and downs with 0s, then the parity sequence of 3 starts with 110. 3s binary expansion is 011. In fact, for all numbers from 1 to 4, the binary representation is a mirror image of the first two terms of the parity sequence. However, as you start examining more terms, random errors start cropping up. For instance, the parity sequence of 5 is 100, but the binary expansion is 101. At any given step, only a fraction of elements exhibit a new parity error. For the life of me I can't find a pattern. If I could demonstrate that the sequence of elements that develop a new parity error is random (i.e. irreducibly complex), then I could demonstrate that a proof of the conjecture would be impossible.
There are two sequences that never develop errors. Anything with ...000000 as a binary expansion always has parity sequence 000000..., and anything with ...1111111 as a binary expansion always has parity sequence 1111111... It appears that parity errors happen more regularly on elements with binary expansions with roughly equal numbers of 0s and 1s.
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