After skimming through five recipes in the seventh-year book, Harry had read the sixth recipe, for a potion of fire breathing, which required Ashwinder eggs... and the book warned that the resulting fire could be no hotter than the magical fire which had spawned the Ashwinder which had laid the eggs.
Harry had shouted "Eureka!" right in the middle of the Ravenclaw common room, and been severely rebuked by a nearby prefect, who'd thought Mr. Potter was trying to cast a spell. Nobody in the wizarding world knew or cared about some ancient Muggle named Archimedes, nor the ur-physicist's realization that the water displaced from a bathtub would equal the volume of the object entering the bathtub...
Conservation laws. They'd been the critical insight in more Muggle discoveries than Harry could easily count. In Muggle technology you couldn't raise a feather one meter off the ground without the power coming from somewhere. If you looked at molten lava spilling from a volcano and asked where the heat came from, a physicist would tell you about radioactive heavy metals in the center of the Earth's molten core. If you asked where the energy to power the radioactivity came from, the physicist would point to an era before the Earth had formed, and a primordial supernova in the early days of the galaxy which had baked atomic nuclei heavier than the natural limit, the supernova compressing protons and neutrons into a tight unstable package that yielded back some of the supernova's energy when it split. A light bulb was fueled by electricity, fueled by a nuclear power plant, fueled by a supernova... You could play the game all the way back to the Big Bang.
Magic did not appear to work like this, to put it mildly. Magic's attitude toward laws like Conservation of Energy was somewhere between a giant extended middle finger, and a shrug of total indifference. Aguamenti created water out of nothingness, so far as anyone knew; there was no known lake whose water level went down each time. That was a simple fifth-year spell, not considered impressive by wizards, because creating a mere glass of water didn't seem amazing to them. They didn't have the wacky notion that mass ought to be conserved, or that creating a gram of mass was somehow equivalent to creating 90,000,000,000,000 joules of energy. There was an upper-year spell Harry had run across whose literal incantation was 'Arresto Momentum!' and when Harry had asked if the momentum went anywhere else he'd just gotten a puzzled look. Harry had kept an increasingly desperate eye out for some kind of conservation principle in magic, anywhere whatsoever...
...and the whole time it had been right in front of him in every Potions class. Potions-Making didn't create magic, it preserved magic, that was why every potion needed at least one magical ingredient. And by following instructions like 'stir four times counterclockwise and once clockwise' - Harry had hypothesized - you were doing something like casting a small spell that reshaped the magic in the ingredients. (And unbound the physical form so that ingredients like porcupine quills dissolved smoothly into a drinkable liquid; Harry strongly suspected that a Muggle following exactly the same recipe would end up with nothing but a spiny mess.) That was what Potions-Making really was, the art of transforming existing magical essences. So you were a little tired after Potions class, but not much, because you weren't empowering the potions yourself, you were just reshaping magic that was already there. And that was why a second-year witch could brew Polyjuice, or at least get close.
Harry had kept scanning through Moste Potente Potions, looking for something that might disprove his shiny new theory. After five minutes he'd flipped the older boy another Sickle (over his protests) and kept going.
The potion of giant strength required a Re'em to trample the mashed Dugbogs you stirred into the potion. It was odd, Harry had realized after a moment, because crushed Dugbogs weren't strong themselves, they were just... very, very crushed after the Re'em got through with them.
Another recipe said to 'touch with forged bronze', i.e., grasp a Knut in pliers so you could skim the potion's surface; and if you dropped the Knut all the way in, the book warned, the potion would instantly superheat and boil over the cauldron.
Harry had stared at the recipes and their warnings, forming a second and stranger hypothesis. Of course it wouldn't be as simple as Potions-Making using magical potentials imbued in the ingredients, like Muggle cars fueled by the combustion potential of gasoline. Magic would never be as sensible as that...
And then Harry had gone to Professor Flitwick - since he didn't want to approach Professor Snape outside of class - and Harry had told Professor Flitwick that he wanted to invent a new potion, and he knew what the ingredients ought to be and what the potion should do, but he didn't know how to deduce the required stirring pattern -
After Professor Flitwick had stopped screaming in horror and running in little circles, and Professor McGonagall had been called into the ensuing fierce interrogation to promise Harry that in this case it was both acceptable and important for him to reveal his underlying theory, it had developed that Harry had not made an original magical discovery, but rediscovered a law so ancient that nobody knew who had first formulated it:
A potion spends that which is invested in the creation of its ingredients.
The heat of goblin forges that had cast the bronze Knut, the Re'em's strength that had crushed the Dugbogs, the magical fire that had spawned the Ashwinder: all these potencies could be recalled, unlocked, and restructured by the spell-like process of stirring the ingredients in exact patterns.
(From a Muggle standpoint it was just odd, a deranged version of thermodynamics invented by someone who thought life ought to be fair. From a Muggle standpoint, the heat expended in forging the Knut hadn't gone into the bronze, the heat had left and dissipated into the environment, becoming permanently less available. Energy was conserved, could be neither created nor destroyed; entropy always increased. But wizards didn't think that way: from their perspective, if you'd put some amount of work into making a Knut, it stood to reason that you could get exactly the same work back out. Harry had tried to explain why this sounded a bit odd if you'd been raised by Muggles, and Professor McGonagall had asked bemusedly why the Muggle perspective was any better than the wizarding one.)
The fundamental principle of Potions-Making had no name and no standard phrasing, since then you might be tempted to write it down.
And someone who wasn't wise enough to figure out the principle themselves might read it.
And they would start having all sorts of bright ideas for inventing new Potions.
And then they would be turned into catgirls.
It had been made very clear to Harry that he wasn't going to be sharing this particular discovery with Neville, or Hermione either after the next armies' battle. Harry had tried to say something about Hermione seeming really off lately and this being just the sort of thing that might cheer her up. Professor McGonagall had said flatly that he wasn't even to think it, and Professor Flitwick had raised his little hands and made a gesture as of snapping a wand in half.
Although the two Professors had been kind enough to suggest that if Mr. Potter thought he knew what the potion's ingredients should be, he might be able to find an already-existing recipe that did the same thing; and Professor Flitwick had mentioned several volumes in the Hogwarts library that might be useful...