I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping.
For example, If I was interested in this sequence:
$S_1$ = 10, 21, 32, 23, 14, 25, 36, …
And I also have the sequence:
$S_2$ = 0, 1, 2, 3, 4, 5, 6, 7, ...
I argue that the latter sequence is off by sequence $S_2$. If I respected the order of both of these sequences and subtract the first term of $S_2$ from $S_1$, then I would create a periodic sequence.
($S_1$ - $S_2$) = (10-0), (21-1), (32-2), (23-3), …
($S_1$ - $S_2$) = 10, 20, 30, 20, 10, 20, 30, ...
I am most interested in how this applies to the Collatz Conjecture, because being able to measure a sequence and determine how close its behavior resembles a loop could be used to argue how larger and larger trajectories either point to there being a counter example or suggest that no counter examples can exist.
I am assuming someone else already came up with my idea and I want to find pre-existing work that explores this approach. I believe it is possible I came across a published work already discussing this topic, but I dismissed it because I did not have enough of an understanding of mathematics to understand what they did. I wanted to at least make sure my understanding of the language describing this topic is correct so I can then research the mathematical methods and tools other people are using.