wobster109 comments on Harry Potter and the Methods of Rationality discussion thread, March 2015, chapter 119 - Less Wrong
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If I remember well, it's not just "person", but information. I can't use a Time Turner to go 6 hours back to the past, give a piece of paper to someone (or an information to that person), and have that person goes back for 6 more hours.
So while it is an interesting hypothesis, it would require no information to be carried... and isn't the fact that the Stone still exists and works an information in itself ? Or that's nitpicking ?
My feeling is things that are overwhelmingly likely do not get treated as information. For example, Harry's clothes go with him, but "Time" doesn't consider that to be information of his clothes still existing. It feels like that there's a Deus ex Machina aspect to how "Time" works and deals with information. Sometimes when you try to time-turn you just encounter Paradox.
So based on that I'd predict that if you try to time-turn with intention to get more uses out of the stone, you will encounter Paradox.