You're right, I misremembered. Still, I think there is a lot to suggest that interacting directly with one's time-clones (as opposed to waiting with one's eyes closed while a clone drops off a message, for example), particularly in battle, is a bad idea. You would never observe a future-you doing something ineffective in combat---failing to dodge or block, or casting an ineffective spell, for example---since, after observing that mistake, you would be prepared for it in the future. So the only consistent possibilities involve losing or running away before you can go back in time, or winning right away.

But, since you know these are the only consistent possibilities, if you showed up to a battle intending to use a Time Turner and didn't see your time clones appear at the very beginning, you would deduce that you would not win, and therefore (if you could) you would run away. So, I expect that from Time's perspective, the most stable solution for people who intend to use Time Turners in battle is for them to not do battle at all.

He has never interacted directly with himself in a way that involves seeing a copy of himself, or coordinating actions in real-time, as would be the case in a battle. Harry has so far always heeded Professor McGonnagal's advice, and looked the other way when his time clones are around. Though, as pedanterrific points out, I have overstated the severity of McGonnagal's warning.

Seems unlikely that the original prophecy was caused by Dumbledore, at least by the method of the magical clock. As in canon, Trelawney seems to have made the prophecy during a job interview, presumably before she was regularly sleeping with the clock. I expect that if Dumbledore wanted her to make a false prophecy at a specific time, something like an Imperius folled by Obliviation would be more expedient. Furthermore, we have seen Trelawney spontaneously prophecy in the dining hall; this prophecy at least appeared unplanned by Dumbledore.

Sure, though my impression is that people don't think graph isomorphism actually is in P. And integer factorization turned out to be a problem for Harry. But you're right, we can actually just simulate a nondeterministic Turing machine this way: every time you have a choice for which state to visit next, just listen as future you tells you which ones not to visit.

Sorry, I'm really not following your pie argument. Harry would learn about the pies in the near future; since it is his style, he would think about throwing them to frighten the bullies. So, his observation of Harry+1 throwing the pies is not necessary for him to think to throw pies anyway.

What do you mean by "they don't recurse"? Surely the fact that this procedure results in fast graph isomorphism testing shows that it is not a particularly "stable" solution? Or, do fast integer factorization by writing down the first digit for the least factor greater than 1, listen as future you says "no, no, maybe," and change it to whatever, and then figure out the second digit, and so forth. The scenario you've outlined results in nearly instant integer factorization (or password guessing, or whatever), so it is probably illegal.

I hope you write the ending you want, rather than playing games to see which ending we will earn.

that doesn't follow. Where would Harry have gotten the pies if not from Harry+1?

He got them from the breakfast table. Where did he get the idea to get them? Well, he would have seen the pies later on anyway. Just like he would have learned about time turners later on in the day anyway, but a more stable scenario was obtained by learning about them earlier.

The recursion is non-iterative beyond the number of loops actually manifested, however.

I'm not quite sure how to parse this. If you would think about an idea at time T, but don't because future you tells you it won't work out, that means your whole thought process going forward has completely changed. But maybe the thing you start thinking about instead doesn't work out, so someone warns you about the idea at time T+epsilon. And so on. So if you are proposing that Time works by iterating through a number of scenarios until you get to something stable, the situation you've pointed to "requires recursion." (It's worth pointing out that Harry, when he gets his Time Turner, doesn't think this is a likely answer to how Time works.) But perhaps I am not understanding you correctly?

My main objection to the scenario you are proposing, though, is that you are gaining information as a result of some work, but that work is never performed. Try taking your scenario to its logical extreme. You sit in a room with one copy of future-you, and a large composite number N on a sheet of paper. On scrap work hidden from future-you, you write down an integer K. If you are not told that K does not divide N, you check. If it does, you keep track of the factor of N you have found. In any case, you then systematically select a new integer K' to check for divisibility. Once you have a complete factorization, you sit quietly, and at the end of the hour you go back in time. Then, you let past-you know the "keys" for all of the integers that weren't factors. Thus, you must have ended up only trying actual factors. So, you have a slightly more complicated version of Harry's factorization algorithm.

Edit to add: I guess this situation actually still is still an exponential time algorithm, since you still have to consider every possibly integer. But you could do, for example, graph isomorphism testing in polynomial time with a similar method: try constructing a map by first finding what vertex 1 should map to. Future you says "nope, nope, maybe," and on "maybe" you try to figure out where to put vertex 2, given that vertex 1 maps to whatever. If the graphs are not isomorphic, future you will say "no" to every choice for vertex 1, so you have your answer in polynomial time. Otherwise, you will construct the isomorphism in polynomial time.

There is one offhand remark, vengeance, and a practical cold-heartedness favoring Bones. "Why not Bones?" is only a little better than no argument at all.

Also there is the fact (mentioned by someone else, sorry I forget who) that Narcissa's sister, Bellatrix, murdered Bones' brother. Edit: I am an idiot, you already mentioned this.

Bringing in Aberforth is a really interesting idea. Now that I think about it, even given the wizarding wars, it is remarkable that so many siblings have died or nearly died:

• Albus/Aberforth

• Bellatrix/Narcissa

• Bones/her brother (who, exactly?)

• Petunia/Lily

The last one is interesting with the role of survivor exchanged as well, since there is a hint that Petunia may have threatened suicide in order to convince Lily to brew the beauty potion.

I think Lucius may have been playing very high speed chess when he picked the amount. The point isnt to have Harry in debt to him, the point is to afford ex-deatheaters loyal to Lucius the oppertunity to trade in a blood debt to Harry for a monetary one to him.

I don't think so. It's clear from his reaction that he did not want Harry to accept the trade:

It was clear that Lucius Malfoy had not been expecting that reply.

And later:

"I withdraw my offer!" shouted the Lord of Malfoy. "I will not accept the debt to House Potter in payment, not even for a hundred thousand Galleons! The girl's blood debt to House Malfoy stands!"

Indeed, it is a taboo tradeoff from Lucius perspective. He has traded justice for his son for money and the resolution of a political (though not moral) debt to Harry. He picked the amount because he thought there was no possibility that Harry would accept such a large amount.

I don't get how a cunning Slytherin like him can discard such an advantage for pure vengeance over an "attempt" which didn't do any real harm.

On the contrary:

And Draco was getting angry again. "Dumbledore killed Mother, it's not enough to just say it's sad! I don't understand what you think you have to do, but the Malfoys have to take revenge!" Not avenging the deaths of family went beyond weakness, beyond dishonor, you might as well not exist.

Lucius never meant for Harry to accept the monetary bargain. This is clear from his reaction. He wanted his revenge, and Harry was interfering.