The rest of the time Harry thinks up a clever explanation, and once the explanation is clever enough to solve all the odd constraints placed on it, (1) he stops looking for other explanations, and (2) he doesn't check to see if he is actually right.

Examples:

Comed-tea in ch. 14

Hariezer decides in this chapter that comed-tea MUST work by causing you to drink it right before something spit-take worthy happens. The tea predicts the humor, and then magics you into drinking it. Of course, he does no experiments to test this hypothesis at all (ironic that just a few chapters ago he lecture Hermione about only doing 1 experiment to test her idea).

Wizards losing their power in chap. 22

Here is the thing about science, step 0 needs to be make sure you’re trying to explain a real phenomena. Hariezer knows this, he tells the story of N-rays earlier in the chapter, but completely fails to understand the point.

Hariezer and Draco have decided, based on one anecdote (the founders of Hogwarts were the best wizards ever, supposedly) that wizards are weaker today than in the past. The first thing they should do is find out if wizards are actually getting weaker. After all, the two most dangerous dark wizards ever were both recent, Grindelwald and Voldemort. Dumbledore is no slouch. Even four students were able to make the marauders map just one generation before Harry. (Incidentally, this is exactly where neoreactionaries often go wrong- they assume things are getting worse without actually checking, and then create elaborate explanations for non-existent facts).

Anyway, for the purposes of the story, I’m sure it’ll turn out that wizards are getting weaker, because Yudkoswky wrote it. But this would have been a great chance to teach an actually useful lesson, and it would make the N-ray story told earlier a useful example, and not a random factoid.

Atlantis in chap. 24

Using literally the exact same logic that Intelligent Design proponents use (and doing exactly 0 experiments), Hariezer decides while thinking over breakfast:

Some intelligent engineer, then, had created the Source of Magic, and told it to pay attention to a particular DNA marker.

The obvious next thought was that this had something to do with “Atlantis”.

Gateway to the after life in chap. 39:

Here is Hariezer’s response to the gateway to the afterlife:

That doesn’t even sound like an interesting fraud,“ Harry said, his voice calmer now that there was nothing there to make him hope, or make him angry for having hopes dashed. "Someone built a stone archway, made a little black rippling surface between it that Vanished anything it touched, and enchanted it to whisper to people and hypnotize them.”

Do you see how incurious Hariezer is? If someone told me there was a LITERAL GATEWAY TO THE AFTERLIFE I’d want to see it. I’d want to test it, see it. Can we try to record and amplify the whispers? Are things being said?

Laws of magic in chap. 85:

No surprise, then, that the wizarding world lived in a conceptual universe bounded - not by fundamental laws of magic that nobody even knew - but just by the surface rules of known Charms and enchantments…Even if Harry’s first guess had been mistaken, one way or another it was still inconceivable that the fundamental laws of the universe contained a special case for human lips shaping the phrase ‘Wingardium Leviosa’. …What were theultimate possibilities of invention, if the underlying laws of the universe permitted an eleven-year-old with a stick to violate almost every constraint in the Muggle version of physics?

You know what would be awesome? IF YOU GOT AROUND TO DOING SOME EXPERIMENTS AND EXPLORING THIS IDEA. The absolute essence of science is NOT asking these questions, it’s deciding to try to find out the fucking answers! You can’t be content to just wonder about things, you have to put the work in! Hariezer’s wonderment never gets past the stoned-college-kid wondering aloud and into ACTUAL exploration, and its getting really frustrating.

Vision and the invisibility cloak in chap. 95:

arry had set the alarm upon his mechanical watch to tell him when it was lunchtime, since he couldn’t actually look at his wrist, being invisible and all that. It raised the question of how his eyeglasses worked while he was wearing the Cloak. For that matter the Law of the Excluded Middle seemed to imply that either the rhodopsin complexes in his retina were absorbing photons and transducing them to neural spikes, or alternatively, those photons were going straight through his body and out the other side, but not both. It really did seem increasingly likely that invisibility cloaks let you see outward while being invisible yourself because, on some fundamental level, that was how the caster had - not wanted - but implicitly believed - that invisibility should work.

This would be an excellent fucking question to explore, maybe via some experiments. But no. I’ve totally given up on this story exploring the magic world in any detail at all. Anyway, Hariezer skips straight from “I wonder how this works” to “it must work this way, how could we exploit it"

Centaurs and astrology in chap. 101

Still in the woods, Hariezer encounters a centaur who tries to kill him, because he divines that Hariezer is going to make all the stars die.

There are some standard anti-astrology arguments, which again seems to be fighting the actual situation because the centaurs successfully use astrology to divine things.

We get this:

“Cometary orbits are also set thousands of years in advance so they shouldn’t correlate much to current events. And the light of the stars takes years to travel from the stars to Earth, and the stars don’t move much at all, not visibly. So the obvious hypothesis is that centaurs have a native magical talent for Divination which you just, well, project onto the night sky.”

There are so, so many other hypothesis Hariezer. Maybe starlight has a magical component that waxes and wanes as stars align into different magical symbols or some such. The HPMOR scientific method:

observation -> generate 1 hypothesis -> assume you are right -> it turns out that you are right.

I think you're missing the point of the Many Worlds posts in the Sequences. I'll link to my response here.

Regarding HPMoR, Eliezer would agree that Harry's success rate is absurdly unrealistic (even for a story about witchcraft and wizardry). He wrote about this point in the essay "Level 2 Intelligent Characters":

In the real world, everything is harder than it is for characters in stories; clever insights are less likely to be true and clever strategems are overwhelmingly less likely to work. In real life, I have to try literally ten good ideas before one of them works at all, often putting in years of effort before giving up or succeeding. Yes, I’ve been known to pull off implausible tricks like “Write a Harry Potter fanfiction good enough to recruit International Mathematical Olympiad gold medalists” but that’s not the only implausible-sounding thing I’ve ever tried to do. You just don’t hear as much about the clever ideas that didn’t work, over the many years I’ve been trying weird and nonweird ways to get my task done.

In fiction you as the author can decide that the bright ideas do work, being careful to accompany this by an appropriate amount of sweat and pain and unintended consequence so that the reader feels the character has earned it. You cannot evade the curse of building your story out of clever ideas that would be far less likely to work in real life, not just because you have no way to test the ideas to find the ones that actually work, but because in real life we’re talking about a 10:1 ratio of failures to successes. We get to see Harry fail once in Ch. 22, because I felt like I had to make the point about clever ideas not always working. A more realistic story with eight more failed ideas passing before Harry’s first original discovery in Ch. 28 would not have been fun to read, or write.

I would agree with you, however, that HPMoR lets Harry intuit the right answer on the first guess too much. I would much prefer that the book prioritize pedagogy over literary directness, and in any case I have a taste for stories that meander and hit a lot of dead ends. (Though I'll grant that this is an idiosyncratic taste on my part.)

As a last resort, I think HPMoR could just have told us, in narration, about a bunch of times Harry failed, before describing in more detail the time he succeeded. A few sentences like this scattered throughout the story could at least reduce the message to system 2 that rationalist plans should consistently succeed, even if the different amounts of vividness mean that it still won't get through to system 1. But this is a band-aid; the deeper solution is just to find lots of interesting lessons and new developments you can tell about while Harry fails in various ways, so you aren't just reciting a litany of undifferentiated failures.

You have exhausted all of the examples that I can recall from the entire series. That's what's wrong.

Without rereading I can recall experiment with time tuners where Harry finds out "Don't mess with time".

But what might be missing is a detailed exploration of "learning from mistakes". Harry get's things right through being smart.