In a sense, the story as of chapter 113 is an easier task than a standard AI box experiment, because HarryPrime has so many advantages over a human trying to play an AI trying to get out of a box.
Almost this exact scenario was discussed here, except without all the advantages that HarryPrime has.
1) He has parseltongue, so the listener is required to believe the literal meaning of everything he says, rather than discounting it as plausible lies. So much advantage here!
2) Voldemort put the equivalent of the "the AI in the box" next to a nearby time machine! Any predictable path that pulls a future HarryPrime into the present, saving present HarryPrime, and causing him to have the ability to go back in time and save himself, will happen. He could have time turned to some time before the binding, and not intervened because his future version is already HarryPrime and approves of HarryPrime coming into existence so HarryPrime can fulfill HarryPrime's goals.
Now that this has happened, HarryPrime, in the moment of his creation, can establish any mental intent that puts him into alignment with HarryPrime's larger outcome. There are limits, as there were when he escaped from being trapped in a locked room after Draco cast Gom Jabbar on him, by forming an intent to time travel and ask for rescuers to arrive just after his intent was formed.
The chronology has to be consistent, but there's a lot of play here.
3) HarryPrime has been unbreakably bound to a task that the binder believes is good by a method the binder thinks he understands.
In a normal "ai box experiment" the gatekeeper hasn't actually built the actual motivational structures of an actual AI. Instead, both humans are just pretending that the "boxed person" is really an AI and really has some or another goal, but they might be pretending differently. Thus, the person role-playing the AI can take very little for granted about what the gatekeeper things about "the AI's" background intent and structure.
The only reason Voldemort has to distrust Harry is the prophesy.
The only "play" in the binding is that Voldemort seems to have chosen HarryPrime's "supergoal content" poorly, so it probably doesn't have the implications that Voldemort thinks it has, though this will only become apparent after several iterations.
HarryPrime is not dumb, and not especially ethical, so until he believes that Voldemort can no longer see the unanticipated implications of his actual request, he will seem to be pursuing the goals Voldemort should have asked for.
4) Voldemort (like an idiot, again after the previous failure to test the horcrux spells) has probably has never performed this sort of spell before, and probably doesn't know what its likely psychological effects will be. He has probably never seen an implacably goal seeking agent before.
Humans, so far as I can tell, are mostly not implacably goal seeking. We wander around in action space, pursuing many competing "goals" that are really mostly tastes that evolution has given us, and role-based scripts we've picked up from ambient culture. We make complex tradeoffs between subjectively incommensurable things and make some forward progress, but much less than is theoretically possible for an effective and single mindedly strategic person.
HarryPrime has an unbreakable vow stripping away all these dithering tendencies. Thus HarryPrime, though probably abhuman at this point, should be able to conceal his abhumanity with relative ease, relying on Voldemort to treat him like a normal human with normal human motivational structures.
Voldemort is already making this error in using threat of torture of Harry's parents to goad HarryPrime into telling Voldemort about "the power he knows not".
I'm pretty sure that HarryPrime now only fundamentally cares about the torture of his parents to the degree that his unbreakable vow let's him fall back on what his earlier self, and Hermione, would recommend or care about, and that clause only triggers when HarryPrime's plans for world saving are themselves somewhat risky.
5) Harry has a huge amount of shared context and it recently contained a request for advice.
If you can think of any trick that I have missed in being sure that Harry Potter's threat is ended, speak now and I shall reward you handsomely... speak now, in Merlin's name!"
One thing HarryPrime could try is to suggest more ways to restrict himself, that to a normal human would be motivationally horrifying but to HarryPrime are still consistent with his new goal, and proves to Voldemort that he has mostly won already and killing Harry isn't that critical.
Off the top of my head, a sneaky thing Harry might suggest is converting some of the death eaters into guards against Harry's possible resurrection forever... using wording that will indirectly cause them also become x-risk mitigation robots as well.
6) Unlike an AI in the box, Harry is already out of the box in some deep senses. Aside from the time turner, he already has the power to expect anything he wants to expect of Dementors, and thereby cause them to act that way. No wand required.
The only barrier to this is that between him expecting the Dementors to do something and them actually doing it, there will be a period of time where he needs to stay alive, and while he is alive but held at wand-point he might be asked "have you betrayed me yet?" and have to admit that he had, and be killed.
All through chapter 112 Harry's mental state was unprobed and Voldemort was distracted by the costs of arranging the Death Eaters and motivating them to help make and understand the vows and so on. The only time Harry's mind was described by the narrator was during the casting of the unbreakable vow itself, to describe how a new "subscripted should" have come to exist in Harry's brain. All of Chapter 113 seems like a lot of time for some mentally generated effects to have been put in motion.
7) He is a wizard with a wand. All the partial transfiguration stuff other people have mentioned is also relevant :-)
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I can think of a solution, but may not be the solution because it relies on untested extensions of previous mechanisms having to do with "Dementers" which HarryPrime knows to be magical incarnations of death, that obey people's expectations about death. Critically, it depends on how much play he has in the distance and plasticity of dementer control.
My plan probably requires him to have put it into motion during the text we already read. Imagine that when he was surrounded at the end of chapter 112 at this moment, he put his plan into motion:
Right after that, he could have started expecting 40 dementers to arrive at his location without disturbing or being seen by anyone while traveling, so it doesn't change anything already known about the world before he time turned already.
He expects them to arrive in a group, and to kill everyone but him and Hermione, even if he himself has already been killed (this last clause might not work, depending on how the magic about dementer expectation control works). He expects the dementers to travel at a poetically appropriate speed (to help make the expectations plausible enough to happen), so perhaps the speed of a killing curse, which might be approximated by the speed of sound, or ~750 miles per hour.
If Azkaban is 100 miles away (doubtful) they take 8 minutes. If 200 miles (plausible), then 16 minutes. If 300 miles (also possible) then 24 minutes. I think 250 is most reasonable, so 20 minutes is the maximum likelihood for the arrival time? Unless killing curses move faster than sound, in which case earlier?
Azkaban is somewhere "unplottable" in the north sea so a 20 minute delay is reasonable. For strategic reasons, Harry expects the dementers to rendevous at a point far enough away from where he is that Voldemort and the death eaters can't sense the doom aura of the dementers. Then when 40 are ready in a group somewhere moderately close, he expects them to swoop together in at the speed of killing curses and kill everyone but him and Hermione. One for each death eater, and the spares for Voldemort.
To expect this, and expect that it had a good chance of working was a risk, requiring ~20 minutes to pass between starting the expectation and the dementers arriving, but all through chapter 113 he was not asked by Voldemort if he had betrayed Voldemort yet (this probably would count as that), so the risk has already paid off so far...
That chapter, but the way, took approximately 15 minutes and 30 seconds to occur. I read the verbal parts out loud to myself and timed how long it took.
There were bits like this where I generally assumed that it would be perceived as less than a minute (I counted 30 seconds for this line, rather than 60):
If my timing of chapter 113 count is accurate, then starting at the beginning of chapter 114 Harry needs to buy about 4 minutes and 30 seconds of conversation, and then he should expect his enemies to be attacked by dementers at an unusually fast speed.
One potential flaw in the plan is that he may not have started expecting the right things early enough. In chapter 113 this bit of narrator description of Harry's mental state shows up around the 11 minute mark and seems uncharacteristic for someone who expects dementers to show up as expected.
So maybe he grew a spine and a brain right after that, in which case he started expecting dementers 4.5 minutes before the end of 113 and needs to buy more like 15.5 minutes in chapter 114.
So what does he do to buy time? Basically, he starts saying a lot of things that are true and interesting and require responses...
Personally, I think Harry is actually HarryPrime now, and he doesn't care nearly as much about his family and friends as Voldemort thinks, at least not compared to preventing the end of the world.
So I think Harry's first move should be to think for as long as he can get away with. Then say out loud that he can think of five things off the top of his head that might be a power-known-not or other qualifying secret. This buys him time to emit more sentences and come up with more things.
(Things he could say that would make the claim of 5 reasonable include: the secret of patronus 2.0, the secret of dementers, flitwick's tourament curse, partial transfiguration, and the fact that magic is a homozygous recessive trait. But he doesn't list them right off the bat that shortly.)
After stating a number, I think he asks clarifying questions about what counts as a secret, or a power, and offers one thing that might count or might not, which would the idea of setting death eaters under other unbreakable vows (to themselves persist in the prevention of the end of the world after Harry is dead) as an example of a strategically helpful thing Voldemort might not have considered as a possible life saving thing to talk about (this also, btw potentially creates allies for HarryPrime's real new goal which is to prevent the world's destruction without stopping to be nice or fulfill other ideals).
Through his wording, he can honestly communicates that his new life goal, by the way, has in fact been transformed by the unbreakable vow that was just taken and he offers himself in service to Voldemort, conditional on Voldemort wanting to protect the world. He really wants to help.
Also it creates a potential conversational opening for him to say that in pursuit of protection of the world he actually cares more now about learn the wording of the prophesy that relates to the potential end of the world, so that he can be more effective in his world saving. Learning the prophesy is probably related to his new vowed goal.
If Voldemort is unhappy with stalling, and Harry has to get down to brass tacks fast, he let's Voldemort know that the secret of Dementers is one that he has composed a riddle for, for someone else (which he has already done for Hermione so it is in theory possible even though we haven't seen the contents of this riddle on camera yet), but it relies on insights and perspectives that Voldemort might not have and so he needs to ask some questions to restructure the riddle. But doing so could take a while and could be done after other secrets were exchanged for lives. Which order does Voldemort prefer?
If Voldemort wants a patronus 2.0 riddle that is optimized for him, then there are a bunch of potentially relevant things about Voldemort's mind and plans that determine whether and how to construct a riddle personalized to him, like "Can you cast patronus 1.0 and if not, why do you think not?"
It is hard to plan a conversation in detail, because the other person's reactions are always relevant, but I could relatively easily see Harry stretching out a conversation about secrets for a good 20-60 minutes, and somewhere in that conversation, hopefully, the dementers swoop in and maybe kill everyone but Harry and Hermione, or at least it gives Harry a distraction during which he might grab the time turner and escape.
I'd rather get the longer happier ending (though I am curious about the shorter sadder ending). Should I submit this plan to fanfiction as a possible solution, or does it need more polishing?
You should look at reddit to coordinate your actions with others. One idea I like is to organize the proposal of all reasonable ideas and minimize duplication. Organization thread here: http://www.reddit.com/r/HPMOR/comments/2xiabn/spoilers_ch_113_planning_thread/