Danylo comments on Harry Potter and the Methods of Rationality discussion thread, part 7 - Less Wrong
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Let's speculate.
Say, Harry Potter tried, failed, and sent a note to the past. What happens to the Harry in the future? He presumably continues to exist, in an alternate universe where he didn't get a note and went on with the plan.
Thus, we have a scenario where, if the test was planned for, Harry must have both Gone on the mission and Not Gone on the mission, and we're merely following the one that did in the narrative.
That's not how time-turners work in canon, nor in this fic (other fics notwithstanding). See TVTropes:Stable Time Loop.