DanArmak comments on Harry Potter and the Methods of Rationality discussion thread, part 10 - Less Wrong
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Harry and co. have one untapped potential ally: Lucius Malfoy. If they gave him all their clues, he may be convinced, just as they have been. And he has a powerful motive to find out who really tried to kill his son, even if he goes through with the trial against Hermione to avoid losing face.
The problem is how to approach him. He would not trust Dumbledore (his political enemy), Harry (he believes he is Voldemort and will soon hate him for 'turning' Draco), or Snape or Minerva (Dumbledore's agents).
I nominate Quirrel (to be sent by Harry) - known (or at least publicly displaying himself to be) free of Dumbledore's influence, a powerful Slytherin, and the one who actually saved Draco's life. Lucius would listen to him. Whether Quirrel would want to cooperate is another matter, but he should have some difficulty saying no to Harry and Dumbledore at once.
For Lucius to trust them, some of them might have to volunteer to testify in front of him under Veritaserum that they really believe the theory that Hermione didn't do it. Dumbledore is a known Occlumens, Snape and Quirrel would at least be suspected of being such, Harry told Draco he is so now Lucius knows as well. A weaker character like Minerva would be useless because Lucius could easily believe Dumbledore misled her. This is a problem...
Lucius would obey a direct order from Voldemort, I should think. Maybe not from Harrymort, but if Quirrell could reveal himself...
If revealed as Voldemort, yes. But Harry doesn't know that Quirrel can do this, and Quirrel isn't going to volunteer that bit of information for Hermione's sake.
Or do you mean exploiting Lucius' apparent belief in Harrymort? (Edit, see you added that.) I don't think he's going to obey an order from him. Maybe consider a request. Probably not even that, given his threat of destroying Harrymort if he hurt Draco, which by Lucius' lights he probably has while 'turning' him.
Harry needn't know how Quirrell convinces Lucius. So if the professor really is innocent, and Harry threatens to end their relationship if Hermione is sentenced, then Quirrell might arrange a highly secret meeting with Lucius in order to preserve his grasp on the Boy-Who-Lived.
If the professor is Hat and Cloak, he's probably going to be certain he's detained until the trial ends.
Why would Harry make such a demand unless he believed Quirrel could deliver? Do you imagine Harry asking Quirrel to Imperius Lucius, or mess with his memory? Harry's knowledge of such (presumed) acts would give him a way too big lever against Quirrel for as long as he continues under his Quirrel identity (which Harry would think is forever). Lucius or his allies might also find out and try to retaliate in the long term. And Quirrel has no personal motive to help Hermione.
So while Harry would ask Quirrel for help (if he could reach him), he would not demand that particular help, not with a threat of severing their relationship. And I imagine that even if he did Quirrelmort would probably refuse. Agreeing to such a huge price just to keep their relationship would not be in character.
Edit: I see now that you didn't specify Harry would ask Quirrel to act against Lucius, just to resolve the situation somehow. Still, if the observed result was a change in Lucius's behavior, and if later Lucius (through Draco) didn't supply a convenient excuse for it, Harry would forever wonder what presumably-political levers Quirrel could have at that level.
Also, Quirrel clearly isn't leaving Auror custody until the trial and so Harry can't ask him and Quirrel can't act.
Now that Dumbledore is using the map (and perhaps other means?) to search for, presumably, Tom Riddle on the grounds of Hogwarts, it's not clear how Quirrel can ever risk going back. I'd expect Dumbledore to keep checking the map once or twice a day, indefinitely, since he seems to believe it can find Voldermort if he's not in the Chamber of Secrets.
Harry already has (he would think) an ultimate lever: The breakout from Azkaban. If he so wished, he could inform Dumbledore and the DMLE and bring all the wrath of Britain down on Quirrell. It would mean incredible costs for Harry, but he could do it. Although, my mental model of Harry says that he would never actually do that.
Even as an occlumens, Harry could prove he and Quirrell did it. Harry has knowledge of Azkaban that no 10 year old should know. He can communicate with dementors to identify himself. He could recreate the rocket he used. Dumbledore can identify his patronus, etc. He's also got a good chance of getting off without many repercussions due to being a minor under someone else's influence and being the boy-who-lived.
Now, Harry doesn't seem like the trust authority like that, but he could pull it off. Heck, even if Quirrell wouldn't agree to help Harry, Harry could probably just lie and say Quirrell planned this against Hermoine and get this crime pinned on Quirrel if he really wanted to. I don't think Harry would actually do this, but it's a possibility. Harry has an untapped resource to save her that nobody knows about (Knowledge of the Azkaban heist), but he either wouldn't think of it or he'd consider it not worth it.
Harry also knows that Quirrell is an unregistered animagus.
The thing is, If Quirrell is behind all this and is being detained unnecessarily is his plan, then Harry has got to realize that everything that has transpired has been according to his design. In fact, if he were H&C I'd expect him to be present so he could put on a show of doing his best to save Hermione. So it's very possible that the real culprit has planned for the professor's absence in order to turn Harry against him. That might be exactly what Quirrell will tell Harry when he is released post-trial, but frankly I don't see why he'd go through the extra trouble and cast doubt on himself just so as to avoid pretending to help.
Unless, as you say, he is avoiding the map. However, the battle-map in 78 demonstrates that the Marauders did not invent the device. Thus, there is almost definitely one at the Ministry, where security is paramount. So if Quirrell did show up as Tom Riddle on such maps, he'd be assaulted by a hundred Aurors the moment he stepped foot on the premises.
Therefore, I conclude that Quirrell is innocent.
The founders of Hogwarts created the map. Dumbledore considers the wards at Hogwarts stronger than those in the Department of Mysteries, so it stands to reason that all the aspects of the Hogwarts security system are stronger than those at the Ministry, including the map.
No, Fred and George think that Salazar might have created the map.
Dumbledore asked for the map, and used the sorting hat to procure a crystal rod which allowed him to manipulate it. It is an artifact of the founders.
Dumbledore didn't order the map to obey on his authority as Headmaster, he summoned a tool with which to fiddle with it. How do you know that crystal rods aren't a standard way to affect enchanted objects, and the one kept in the Sorting Hat is just unusually high-quality?
I mean, if it was created by the founders, the Marauders would have had to do something very similar in the first place. So either Dumbledore undid the Marauders' alterations, or he altered their (or whoever's) creation the same way they would have had to, counterfactually. How do you distinguish between the options?
Actually, if in MoR it doesn't mention Moody, Wormtail, Padfoot and Prongs, then the founders are a good guess.
Well, it doesn't stand to reason exactly, but it stands to guess given a lack of contradictory evidence, at least.