Conway proved it's undecidable. Any FRACTRAN program can be converted to a Collatz-like form, thus the general problem of predicting cycles is as hard as the Halting problem.

Good point. I agree it applies here, but ONLY "for some value of general"! So, eschewing despondency, I'm not yet willing to curl up on the proposition.

Of course, that doesn't say anything about the original Collatz problem, and I'd be very surprised if it's universal for computation (only one known non-halting program!)

True! And, moreover, this DOESN'T say that some suitably chosen SUBSETS of "Collatz-like" forms (eg those in some as-yet-unspecified parametric family) MUST contain undecidable instances. That is, it's not an all-or-nothing situation. We're not forced to deal with JUST the "classic" Collatz instance simply because the MOST general case has been proven undecidable (as neat and useful as that result is). For example the FRACTRAN programs with Collatz images where the multipliers, are, say, prime (to pick an arbitrary property) MIGHT in fact be a decidable subset. We can ONLY conclude that we should study JUST the 3N+1 instance IF we can establish that ALL more general properties P satisfied by 3N+1 do in fact necessarily lead to classes with undecidable members. This seems MUCH harder and less plausible than the original problem--thus at least engendering the faint hope that embedding Collatz in SOME more general setting might enable some progress. That, of course, is analogous to what happened with FLT.