Warning, big swath of text coming through.
I've recently been rereading the story from the beginning. By now the whole thing has a bit of a halo effect and judging things without bias is getting tricky. So kudos on accomplishing that... but there are a few issues that I think harm the piece overall. They didn't hurt my enjoyment of it, but they end up limiting it to a smaller audience. There's a lot of smart people who would love this fic if there weren't certain things that turned them off to it.
The main problem is that Harry too absurdly intelligent to believable at first glance. In the first few chapters people tend to assume that the "primary change" is simply that Petunia married someone different, which isn't enough to justify him not only being saner but being genuinely smarter than the original Harry was. My sister was particularly annoyed by this. I'm not sure how much of that had do with her reading it before you updated the intro-text to say "multiple points of departure." But by now she's internalized Harry as a creepily overintelligent jerk and I can't get her to give it a second chance, despite the clues you've dropped more recently about why he is the way he is. And while you've put a lot of effort into showing that Harry is not always right (and is in fact, often grossly wrong), the bleed between your own opinions and writing style and Harry's opinions is often difficult to distinguish.
Something about the early chapters makes it particularly hard to tell, and if I had to pick the single biggest issue, it would be your portrayal of Dumbledore. I know that you're writing from Harry's perspective, and Harry's perspective is flawed. I suspect you also want us to keep guessing about what precise changes you've made to Dumbledore's character. Is he insane? Why/did he kill Narcissa? Etc. I'm sure there's other plot elements that you're foreshadowing that we haven't picked up on yet. But even when I try to compensate for that... the character simply reads... off. Too weird, too dim, too much of a different character than the one we remember. My suspicion is that Dumbledore's personality IS pretty different from your own, so writing him convincingly is simply difficult. If that's true, well, fair enough.
I understand what you were trying to do in chapter 39, but he really comes across as a strawman. Not because his reasons for deathism are poorly thought out (it's fine for him to either have undisclosed information or just lack that kind of rationality) but because he seems to fail to understand how angry/disappointed Harry is in him. Dumbledore should NOT be bad at that kind of empathizing, and that's what makes him look genuinely stupid instead of simply having a different viewpoint. If he's only pretending to be bad at understanding Harry.... eh, I'll withhold judgment in case you have an awesome resolution to that scene planned out for later, but right now it doesn't read as clever foreshadowing to me, it just reads as a character who's written oddly.
I think chapter 39 would work a lot better, though, if it wasn't set up by chapter 17. This is our main introduction to Dumbledore, and it's just... too weird. Too insane in ways that Dumbledore was not traditionally insane, too dim sounding when Harry starts talking about locating Hypothesis. (I confess I'm biased a bit simply because the "detector box" thought experiment has never been compelling for me. Yes, it makes sense, but it doesn't register at all on an emotional level, so taking a few paragraphs to describe it just feels weird and offputting).
I'm not a skilled enough writer to do it better, especially without knowing all of your longterm goals for the story. But I think chapter 17 is something you should revisit at some point. If there's a way to accomplish the same goals without feeling so different from Canon!Dumbledore, and without making Harry look genuinely smarter, then the character would feel more real and you'd probably hold on to a wider audience.
By chapter 62, it's become clear that this IS the Dumbledore we know and love, just a little more realistic when it comes to fighting a war and perhaps a tad more whimsical. But the audience has to wait a long time to find that out, and I can't see any reason that we had to wait for that. I appreciate the way you force us to think "hmm... DO we really trust this guy in this alternate universe?" but the overall way you go about it just feels... off.
Dumbledore plays a deep game. He's not "the Dumbledore we know and love" in many, many ways, and perhaps that big change is the largest cause of the issues.
Update: Discussion has moved on to a new thread.
After 61 chapters of Harry Potter and the Methods of Rationality and 5 discussion threads with over 500 comments each, HPMOR discussion has graduated from the main page and moved into the Less Wrong discussion section (which seems like a more appropriate location). You can post all of your insights, speculation, and, well, discussion about Eliezer Yudkowsky's Harry Potter fanfic here.
Previous threads are available under the harry_potter tag on the main page (or: one, two, three, four, five); this and future threads will be found under the discussion section tag (since there is a separate tag system for the discussion section). See also the author page for (almost) all things HPMOR, and AdeleneDawner's Author's Notes archive for one thing that the author page is missing.
As a reminder, it's useful to indicate at the start of your comment which chapter you are commenting on. Time passes but your comment stays the same.
Spoiler Warning: this thread is full of spoilers. With few exceptions, spoilers for MOR and canon are fair game to post, without warning or rot13. More specifically: