Trope-a-Day: Stable Time Loop
Stable Time Loop: The Chronological Consistency Protection Theorem (see: You Already Changed The Past) does appear to permit these kinds of predestination paradoxes (at least, the informational kind; the object paradox may also be possible, but testing it is dependent on finding an object that won’t age during the period of the loop, including in all the aspects that the universe will check that current technology may not be able to).
It is also somewhat limited by the main people who can make use of it for Retroactive Precognition/Retroactive Preparation being the transcendent/weakly godlike artificial intelligences. Experiments by lesser intellects… well, they don’t actually produce messages reading “YOU ARE NOT SMART ENOUGH TO BUGGER ABOUT WITH THE TEMPORAL CONTINUUM”, but the gist of the results appears to be fairly plain.