It's a perfectly stable closed time-loop. Despite all attempts to go back and prevent the incident, based on future knowledge (and knowledge sent
to the future, via that time-delayed voicemail thingummy, or whatever it was again), the immutable time-stream
keeps on... has kept on... will and has and always will have been being doing the same things[1].
Including all parts of the loop containing the time-traveller and the information he possesses, as 'already' witnessed (by his younger self) in his eventual end.
The 'mission' (as planned in the future) is based off information that the mission (as enacted in the past) inadvertently planted. Had there not been a pursuit of the Twelve Monkeys meme, the Twelve Monkeys meme would never have been thought to have mattered, with no pursuit. It happened because of itself. A paradox of 'amemogenesis', to coin a term (originally I'd considered "amathemogenesis", from "knowledge", however...), but this makes the (pan-temporal) universe self-consistent and fully deterministic. (Which, I'll admit, covers my favoured hypotheses about the nature of causality, so I may be biased.)
And, to my mind, it shows how Time Travel movies don't have to have a
deus ex machina of a time-traveller (from the 'wrong' ending, or of one from the 'right' ending that managed to survive the malicious flip to the wrong timeline) who moves heaven and earth to "put right what once went wrong"[2], in order to get drama and a story with revelations and surprises... Yes, that's the intent by the protagonist and his backers (changing things, or at least mitigating things in the (future) present based on classically irretrievable information), but the tapestry of the Universe is already woven, folks... The interest in this film (on subsequent watchings, at least, if in the first you weren't aware of where we are being led) is in tracing how beautifully various threads are intricately wound around each other...
As for Peters (if that's who I think it is), he's something that happened. Had happened (from the future perspective), was still to happen (from the past perspective), did happen despite (because..? no, just despite, IIRC) the attempts of the future-now-past to stop it. Because the future-now-past's efforts had no effect (or exactly as much effect as any non-'paradoxical' occurrence had... because future-now-past is and always was part of the contemporary event-chain).
To put the crux in somewhat 'real-world' terms, someone decides to cheat the Lottery thanks to time-travel. He arranges that a future person shall communicate back to him, here in the past, the Lottery numbers that came up on a Triple-Rollover night in which no-one won... Assuming, of course, that having received the precise numbers and date that
he would now win, and with no other winners to share with (but of course now changing things from "no-one won", to "one person won"). There are many "temporal protection" situations that could mean this plan is/was/always will be doomed to fail, but in the Twelve Monkeys context I choose to believe that the information he received was the correct numbers, the correct date and the traveller did
not fail to buy the ticket... but he just had failed to specify
which lottery the future end of the deal was to tell him about. "The Lottery" means one thing to him, but by the time someone has put together the wherewithall to construct the means to communicate a competing venture has risen from obscurity whilst the current major holder of the title (also possibly complicated by the future-end's differing locality/nationality, again also perhaps a quirk of intervening events) falls into state where there's relatively, or even absolutely, a lack of information about it. Future-end thus sends back the (perfectly good, 'triple-rollover winnable', uncontested) numbers, but these do not reward our Chronoentrepreneur in the way he expected they should.
(Which is not to say he couldn't have won something... But he was
always going to win £10 (or local equivalent) on three of the numbers he plucked out of the time-loop... If the Universe has a sense of humour he might even have won due to a
highly unlikely (except that it obviously must have happened, in this scenario) circumstances that the other-lottery numbers were also good for the intended target lottery... But maybe had to share with thousands of others who had
also been contacted by the same (incompetent?) informant from the future. Still, the
other (more obscure) Lottery remains unclaimed... And then word spreads about the coincidence, contemporary awareness is artificially raised about how the Lottery appeared to duplicate OtherLottery's numbers, people now keep an eye on OtherLottery, while the utterly inconceivable (to non-statisticians) situation of the coincidence sparks off probes of enquiry into the practices of both businesses, resulting in totally unconnected revelations of skullduggery within Lottery (while further enhancing the image of OtherLottery)... and
that's why, after enough time has passed to make the business of retro-information a practical possibility, the error in Lottery identification was made....)
But maybe I have written too much about this.
TL;DR;: The film has been crafted to represent a consistent time-loop, under the constraints thus required, where a
lot of films with time-travel involve fading Marty McFlys or bad-guys self-destructing (both of themselves) when they touch their younger equivalent, or sending the timeline
into a "Nazis win WW2" universe, or bring the universe
out of that track, or back out due to previously accidental/malevolent intervention which had switched it the 'wrong' way...
And, because TL;DR; is getting TL;DR; in itself (and I
like alternate histories... but perhaps less so when prompted by an actual temporal intervention), I shall now stop my Apolegism for the film of your scorn. Too late, I hear you cry! Well, go back in time and stop me, then! I bet you can't, and in
this timeline, you didn't (and in any other that somehow exists I will have never even have made this challenge, to lose), so pay up.