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Gorak comments on Harry Potter and the Methods of Rationality discussion thread, July 2014, chapter 102 - Less Wrong Discussion
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Hypothesis: Harry lives in a close to post-singularity universe. The source of magic is a boxed AI, created by the Atlanteans and Harry himself.
Currently, the boxed AI can be manipulated by means of spells, potions and rituals and is acting as a limited outcome pump. Harry, in his quest to end death, will release the AI from the box, thus bringing about a true singularity end state and the foretold end of the universe as we know it.
The method through which Harry will achieve this (releasing the boxed AI while simultaneously being partly responsible for its creation by the Atlanteans in the past) will be a trick using multiple time turners and something like the algorithm Harry invented in chapter 17 for solving an NP complete problem coupled with Timeless Decision Theory.
If you build a device capable of factorising in less than polynomial time, you have a major building block of a supercomputer. If you can do it in negative time (solution before input) you've built an AI. Harry attempted this in chapter 17 but was stopped by a future Harry with the admonishment "DO NOT MESS WITH TIME" which Harry resolves to obey until age 15. That intervention could have been from an even further future Harry.
The theft of Hermione's body could also have been conducted by an even further future Harry which allows him to pass the questioning of the headmaster.
If Harry can figure out how to bypass the limitations of the time turner, that is all he needs to do to bring about the scenario I outlined, and build an outcome pump that will restore Hermione, be established as the source of magic in the past and the bringer of the singularity in the future.
I assign 60% confidence to this hypothesis.
So your hypothesis is that Harry will win by doing the thing he already tried and failed to do, and got a stern warning from the Universe for trying. The one thing that he can't do, that's the thing he'll do.
Okay, except you've not done the work. If Eliezer puts a huge road block in front of an obvious solution to the story — "NOPE, can't do that, sorry" — and your hypothesis is "No he'll do it anyway", then the actual work is not just saying he'll do it (since the story explicitly states the insane power Harry would have if it did work; it's not like that's a big discovery itself) but rather saying how we get from our current state of impossibility to the state where Harry pulls it off.
"If Harry can figure out how to bypass the limitations of the time turner" — that dismissive 'If' is the entire problem you should be trying to solve.