I'm still thinking about your modular perspective. I believe that is an accurate restatement of one of my cases, but it is not all of them and would not be suff

I'm still thinking about your modular perspective. I believe that is an accurate restatement of one of my cases, but it is not all of them and would not be sufficient alone. Whereas all the cases together, however, are exhaustive. To summarize, my above comments/claims boil down to: 1) any even number input taken will produce a sub-sequence ending in an even output, by exhaustion (see above), that is guaranteed to be smaller after a maximum number of steps 2) this new even can be taken again, as often as necessary, to reach the smallest integer in that sequence greater than 8 3) the next even number (at most 1 odd away) is by definition below 8 and, while it may not have a next even by my arguement of necessity, it will still be below 8, and all inputs below 8 terminate 4) there is nothing about this reasoning that limits it to a particular sequence/input (all initially-odd inputs adhere after 1 iteration), so it is true of every sequence Am I saying anything untrue or ambiguous? Thanks for taking the time!