She promises only Asmodeus-approved uses of whatever she figures out. 

Anyways!  In terms of Law, this whole lecture has now gotten far ahead of itself!  The true Law does not assume that there is some mystical stuff called 'money' floating around, in which everything can already be valued.  The Law begins from people wanting things; a notion of money comes later, beneath the Law and after the Law, not before it and above it.

Going back to where they left off on Law: this classroom has now seen, at this point, that things you want, which by an act of choice or expenditure of time you can swap or trade, need to be consistently orderable among themselves in order to avoid having time or money pumped out of you.  Even indifferences - in the sense of letting other people swap things for you, not just it not being worth the effort of making a trade - must be coherent within the ordering, to avoid this.

They've seen a simple trick for getting dath ilani children or Golarion adults to violate this consistent ordering, just by way of showing that it's not a trivial requirement.  (And hinting at how there must be vastly more terrifying things that high-ranking Keepers can do if not otherwise restrained by ethics, but that's not the main point.)

To take the next step beyond ordering, consider things that are divisible, fungible; things that have quantities over them, and not just unique individual identities.  Let's consider apples, bananas, cherries, and suppose - temporarily - that the relative intensity of your desire for these fruits does not change depending on how many you already have.  (Maybe you're planning to feed those fruits to a large hungry crowd so you actually care more about food-energy-value than a tasty meal with variety, say.)  Suppose, for now, that every fruit is identical to every other fruit of that kind.  Suppose also that you would, for each of these kinds of fruits, prefer to have more rather than less, everything else be equal.

It'd be pretty odd if you valued a cherry equally with a whole banana.  It should take at least two cherries to be worth a banana, right?  Probably more.  If somebody offers to trade you a banana for two cherries, you should probably accept.

But that's not Law; there could be such a thing as a person who would rather have one cherry than two bananas.

What can we say, in this situation, about the patterns of offered trades that you will accept or reject - what property must your trading-rule have to avoid some dreadful thing happening to you, and what indeed is that misfortune?

The class is going to need a moment because they're distracted realizing what Asmodean Keepers will be capable of doing to people. 

Yes yes this is great news Dis has priced us all correctly now answer the question.

"You need to have the same relative valuing of fruits," Tonia says, "no matter what exact bargain is offered, otherwise you'll agree to trades that leave you worse off than you started."

Hm, hm.  Say more about this relative-ordering concept?  What is it to have "the same relative valuing of fruits"?

"Uh, the amount you'd rather have a cherry than a banana can't change with which is being offered, or how many are being offered."

What does 'the amount you'd rather have a cherry than a banana' look like?  92.73 units of wantingness?

"The - amount of cherries that's as good as one banana."

And so long as you keep to constant exchange-rates for apples with bananas, bananas with cherries, and apples with cherries - so long as those three numbers never change, whatever they are - you are safe, are you?

....probably not but I can't think what'd go wrong."

"Two apples for one banana, two bananas for one cherry, two cherries for one apple."

"That does sound like a problematic trading pattern.  What must be true for that to not happen?"

"The constraint-system needs fewer degrees-of-freedom - one fewer than the number of goods being traded.  If there's three goods you can trade, there should only be two trading ratios.  Once you know apples to bananas and bananas to cherries, that gives you apples to cherries.  So if you make up a separate number there that isn't derived from the first two, that's like having two different trading ratios at two different times, or different trading ratios depending on the order you trade things in.  The constraint-system starts with zero degrees-of-freedom for one fruit, and then any time you add a new fruit, you add one degree-of-freedom for a trading ratio between that fruit and any other fruit already in the constraint-system."

Keltham reminds himself once again that these people are not actually six-year-olds; they have any prior mathematics education, it's just been about spellforms rather than the entire rest of Reality.

"Ah, well, in that case, I claim that your condition is exactly equivalent to saying that every fruit in the system other than cherries must have a consistent unchanging price in cherries, and we accept trades whenever the trades are favorable according to those prices, and reject them otherwise.  Do you believe me?"

"Yes."

He was expecting a bit more caution there.  "Why?"

Because your Bluff sucks.  "You seem like a very honest and nice boy who wouldn't lie to an innocent girl about that sort of thing," says alterPeranza.

"Also because any pattern of trading rules that's okay, will have some series of trades you can make, to convert one of any other fruit to some number of cherries, and that series of trades gives you the price of that fruit in cherries.  And if you have the price of everything in cherries, you can figure out which trades are okay, you just compare the value in cherries to see if you're gaining cherries or losing them."

"Hmmm.  That is a compelling argument - the second part, I mean, I'm not commenting on that first argument at all.  But you may recall that I asked, initially, what pattern the system of trading-rules must have, to avoid some dreadful thing happening, and also I asked, what exactly is this terrible misfortune.  What happens to you if the ratios aren't consistent?"

"No matter what fruit you start with, somebody can offer you a series of trades that takes you down to one cherry.  Then more trades to half a cherry, if they still think you've got too much left."

"It seems you trusted that nice boy far too much!  Suppose I'll trade eleven cherries for a banana, or a banana for fourteen cherries, but when it comes to trading a banana for twelve or thirteen cherries, I neither accept nor reverse either of those offers.  You can't pump infinite fruit out of me that way, even if I trade by those rules consistently, and the rules don't correspond to having any single price of a banana in cherries."

"Nope.  You weren't lying before, now you're trying to get me to think you were lying before."

"...though I'm not sure how."

"Don't get so caught up in the metagame that you forget about the object-level math question, which is where all of the important skills to learn here are to be found.  When you are encountering your own problems you will not be able to solve them by guessing a teacher's trolling patterns."

It sounds enough like a rebuke that Peranza loses the thread of alterPeranza just being in class and being good at math.

(Peranza has practice at spellcraft under pressure, answering from memory under pressure, a little doing topology under pressure.  Not so much pretending to be another person thinking about Law the way Keltham does, under pressure.)

...which is not a good look for Cheliax.

"Having one less degree-of-freedom than the number of fruits, with each new fruit after the first having one degree-of-freedom for entering the constraint-system, is definitely equivalent to being able to price all the other fruits in one fruit.  But that's not the same condition as 'trading rules that can't get traded down to one cherry'."

"What's missing... I think is, if somebody else offers you to trade a banana for thirteen cherries, and then to trade twelve of your cherries for a banana, you have to take them up on that.  If you don't, that counts as a terrible misfortune."

"I claim that the two conditions of 'you can't be traded down' and 'you don't miss trades that take you up' is equivalent to being able to price everything in cherries.  Assuming there are any cherries in the system in the first place."