It sounds like you’re answering a much more sophisticated question than I meant to ask. I had discovered the final loop (which goes 5->26->8->65->122->5) and proved that no matter where you start you have to end up in that loop. It sounds like you’re asking the additional questions of which number (of 5, 8, and 11) you hit first, and how long it takes to arrive. I haven’t given any thought to that, though it strikes me as a subtle and potentially interesting question.

It may be simply a matter of timing. If you look at the loop mod 9, you see that you always end up in the 5->8->2 (or 11) loop. It may be that if you just have to pick number that are larger to end up further along in that loop. In other words, higher tier may change your conjectures (maybe 8 is a common root at certain tiers).

For example, look at what happens when you square the following numbers and then add 1, and reduce mod 9

1->2->5->8->2
2->5
3->1->2
4->8
5->8
6->1->2
7->5
8->2
0->1->2

If you end quickly (i.e., if you’re starting with a small number), then it seems like you’ll often hit an 11 (i.e., 2) or a 5 first. On the other hand, if you start a little larger, you might give yourself time to loop around a little more. I’m pretty sure your multiple-of-three observation won’t hold for larger numbers. Both 87 and 93 hit 11 before 5 or 8.

Short answer: I don’t have any good way off the top of my head to predict root or tier either. My guess is that you can approximate tier decently by the size of a number, but that it’s a hard problem to get a precise formula for it.