Another day, another automaton

Created another automaton to model an internal property of numbers when they go through the collatz function. There's a clear pattern that emerges. For every (3x + 1)/2 step, a 4 is appended to the end. Lemme explain.

The automata looks at the value k in base 3, where k = either x/2 or (3x + 1)/2, depending if x is even or odd, respectively. However, I insist that there are only even numbers in the base 3 representation of k, so every time there's a 1 or a 3, I subtract 1 from that spot and add 3 on the next trit down (trit = 3-valued bit). This continues until all values are even or a 3 squirts out the end. If the 3 squirts out (this is a technical term) then it means that x/2 isn't an integer, and therefore the next number in the sequence will be 3x + 1. In base 3, this is the same as adding a 4 to the end of the sequence, which is even, so we can proceed.

Now, every time that there is a 3x + 1 step, the 4 gets appended. However, the cool part is that any 4 that existed previous will stay there. So, a sequence which alternates multiplication and division steps might generate this sequence for k values:

2404

4244

20444

24444

44444

After this, there follow several division steps.

It seems that if a number gets to a sequence of 4s, then it will fall precipitously for the next several steps. There seem to be opposing forces at work here. If the number is getting bigger (by alternating multiplication and division steps) then this sequence of 4s builds up on the end. However, the 4s act like sandbags in a hot air balloon - the more there are, the more likely it will crash back to the ground.

Maybe this thing is provable yet.