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Astazha comments on Harry Potter and the Methods of Rationality discussion thread, February 2015, chapter 108 - Less Wrong Discussion
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Indeed. And how did the Ministry even key the time-turner to Harry? He wasn't physically present and "magic" seems to think his name is Tom Riddle.
You could ask the same question of the Hogwarts letter.
The simplest explanation seems to be that the Marauders' Map is keying off of something different than the other person-specific magic we know about. Perhaps the Ministry's running something like a magical Social Security database that associates names with bodies at birth, and the Map (and, by extension, the Hogwarts security system) is taking MD5 hashes of people's souls and associating them with names on admission.
Which raises the interesting question of whether Harry counts as a professor where the wards are concerned.
And given that MD5 is broken that could lead to interesting consequences... :-D
Well, that's a collision attack; I don't think we've seen a way to create an arbitrary pair of souls, and if we had it'd probably be obvious on their admission. Though I'd read the story of someone breaking preimage resistance on the Hogwarts wards and using that to, say, get a toad credentialed as the Muggle Studies professor. It'd probably do a better job...
...actually, I kind of want to write that now.