Where in the fic does Harry correctly apply the scientific method to explore magic? Mostly I remember him forming hypotheses which immediately turn out to be right and vomiting out (often incorrectly applied) science references, neither of which is actually doing science.

The few exceptions I can think of all happened in the first 25 or so chapter, when he tested the money bag, the potential ability of time turners to solve NP problems in P time, or the role of incantations and knowledge of spells. None of these instances matter to the story at all, though.

The one scene involving science actually relevant to the story I remember is where he tried to convince Draco that muggle-borns and halfbloods are as powerful as pureblood. The problem is that he made incredibly obvious (at least in retrospect) methodological mistakes which were never addressed: he 'proved' a Mendelian inheritance model where magic is inherited by a single recessive gene using the evidence that two squibs have a roughly 25% chance of having a wizard child... completely missing the fact that such a model makes the existence of squibs impossible in the first place.

The fic isn't about science; it's about Bayesian Rationality, for which the Potterverse is hilariously ill suited: The whole point of Bayesian Rationality is that it requires prior knowledge, all of which you can completely throw out the window the moment you see a person transform into a cat. Pulling probabilities out of your ass to justify whatever hypothesis you favor, which is most of what Harry does in the fic, and having it magically work out for you is pretty much the opposite of science.

If you want to learn why many, if not most, of the science references made are also not completely correct, I recommend this rather comprehensive review by an actual scientist.