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Even still, I would expect a recursive prediction to converge at a point. An 'ideal response' as it were. The first iteration would dictate the event, the second and so on would dictate the response. There would come a point at which the responses predicted would probably converge to the point where it would be pretty much determined. Especially if the output explained the reason for doing the dictated actions.
Any Predict-O-Matic that does actually work within a universe that can be changed by the output of the Predict-O-Matic is what I'd term a 'flip-flop' one (or possibly one in which a new branch of time-line is created by the actions of the POM[1]).
It seems much neater to me (i.e. I have absolutely no proof) that the tapestry of time is laid out in a self-consistent manner, which means that any 'looping' threads are thoroughly integrated into their own 'historical' part of the warp and weft already. 'Tapestry time' is the length-wise direction, along which most 'threads of existence' occur, time-travelers (and time-looped lines of information) are represented by threads looping back and over into the 'past' parts of the tapestry (and there's nothing to stop a thread being a never-ending closed loop.
The external time, during which the tapestry is constructed, is 'meta-' to the tapestry's own form of time, i.e. anything that the universe-inside-the-tapestry 'experiences', and speculative in nature. If there's a Universe-Weaver (sentient or otherwise) busy rearranging threads until it works out, it's not a proper stable universe until the threads are all settled by the rules the warp and weft require. An "always was thus, and always will be" tapestry (no inconsistent versions actually being possible in the meta-space) is a possibility, apart from various philosophical objections you might have.
A stable/valid/etc tapestry is thus laid out, from start of time to its end, any loops being part of the weave, and anyone 'traveling' (i.e. beign part of) the loops "go back and do what they've already been back and done. POMs derive their information from the loops[2]. POMs in a stable universal-tapestry therefore either:
1) Don't work, or are just lucky beyond normal statistical probability,
2) Do work, just are 'creatively misinterpretive' so that nobody can complain that they are wrong, or
3) Work perfectly, but only providing such predictions that can be given to self-support the future reality that the predictions came from (e.g. with that example where it helps set up the bungled inspection on the aircraft engine to create the problem it predicts).
A variation on 2 is of the "If you go into battle against your enemy, today, a great empire will fall!" variety. Producing maxims that are true (and, in that case, in an unexpected way) if followed but don't reveal anything about the future if not followed. (Whether you follow the advice or not might well be determinable, so a valid response could be any suitably portentous waffle designed to off-putting to the recipient, sending them down the null-change route, but with enough correlation with null-change events (sidelined due to ignoring) to make it look as if there could have been something in it after all, after the fact.
A variation on 3 is easily be one that says "Not telling you" all the time, on the grounds that anything it did say would invalidate causality and produce futures that weren't its own. Which would mean its own future existence wouldn't be too extensive, even if it can 'prove' it knew about it, after the fact, but causality remains consistent.
IYSWIM.
[1] If the Universe sums up as zero in its energy/mass equations then creation of a new branch may not need anything as drastic as the complete collapse of the old branch, but otherwise I'd expect the whole (potential) time-line of the original branch just past the point of prediction to be deconstructed for its mass/energy complement to be used in the creation of the new branch.
[2] Given the computational difficulty/impossibility (never mind the measurement aspect), and the far from disallowed nature of actual time-loops, within physics, I'm plumping for POMs getting a 'history' from the future rather than deriving (or 'divining') the future from ever-outdated and inaccurate 'current' information.