[feoren]

You're right: my argument is flawed. I had thought TFA relied on that argument, but the machine that writes 1 and moves right forever is a counterexample.

Yet something about that still seems extremely "loopy". Is there something that must stop increasing after a while? Kolmogorov complexity? Or is that begging the question, since that's basically measuring the smallest TM that can produce that output?

[umanwizard]

You’re right that it “feels loopy”, intuitively. I also don’t know how to formalize this notion.

[feoren]

How about this: given any arbitrarily large window size W, we can find an infinite number of timestamps (it may even be enough to say we can find two) in which the tape within the range of Head-W and Head+W is identical. So if you say 10, I can give you an infinite list of step-counts at which the 21 symbols on the tape centered at the head is identical to all the other times in that list. Assuming we can do that for any arbitrary window size, then the TM is in a "looping" state. Of course, a halted TM also has this property. So perhaps this is true of all Turing Machines?