"So why am I trying things this way?"

"Because, if this just works, it will be faster."

"And I am not sure that this won't work.  I haven't tried.  I haven't checked."

"So we're experimenting.  We are looking.  We are going forth and finding out."

"When to that underlying emotional spirit of curiosity, questioning, 'blah blah blah'(*), and glorious discovery, is added the inspiration of Law, it becomes what dath ilani call 'blah blah blah'(**)..."

"Right, uh, let me just pause and tap everyone with Communal Shared Language (Baseline)."

Keltham then goes about from person to person.  He's glad he got two of those spells, today; with this many people, he can only give most researchers and candidates an hour of Baseline apiece per casting.

(*)  'Maniacal-experimentation'.
(**)  'SCIENCE!', lit 'Civilization's-learning', with 'learning' as in a child's learning.  Distinguished from ordinary everyday systematic experimentation using math, in that it is a societywide effort to COMPREHEND and CONQUER the ENTIRE UNIVERSE.

So the entity - Keltham, Keltham, his name is Keltham and he goes by it, it's going to sound so weird if she says "the entity" out loud, even though it probably won't actually cause any wall problems - Keltham actively wants dumb questions, even though the project leaders obviously don't. That's good. It should be easier to think of questions that don't reveal anything if she doesn't have to also filter for being incredibly insightful. And hey, maybe she won't even have to contribute much to classroom chatter in the first place; everyone else is probably just as capable of asking dumb questions as she is. Not that she's going to make any ill-founded yet confident assumptions about what other people are thinking when she can't see it, because she's decided that there's quite enough of that around here, and she's going to at least find less repetitive ways of being constantly wrong.

(Also... apparently Keltham already knows that the way they're mostly taught involves remaining silent, so why do they have to go to so much effort to act like it isn't? But that's a question from her real self, not her alter-world self; she tucks it away and resolves to stick with what the project leaders have told her, because they've obviously had more time to think about the picture they're presenting to him.)

Willa Shilira is sitting in the back of the class and she's very unhappy about it, but you couldn't tell that by looking at her. She has less Chelish dignity and stonefaced composure in her than perhaps any of the other students, but that's still plenty to be unassuming in a crowd of almost two dozen. (She's mentally acute enough to count the twenty-three visible people in the room while barely thinking about it, but not perceptive enough to even notice Broom being the twenty-fourth.)

Willa came into the project bright eyed and excited about Law and learning and superpowers, desperate to be special. She was so desperate for it that when a chance to ask questions came up, she tried to ask one mostly just to show off:

"Of course if the project is so important we should all be getting intelligence headbands, right, and I can put that impertinent demand in fancy probability language to prove I understood the homework too! I didn't stop to think that economic reasons were a trivial concern and that the real thing I should be worried about was fooling Keltham, even though you just gave a giant speech about how the important thing was fooling Keltham. So obviously I'm a complete idiot and you should never trust me with anything at all!"

That's not actually what she said, but it might as well have been. They didn't torture her for it at all which was super weird, but now alter Willa is 'studious, shy, and hesitant.' Forever. And she has to mentally run all her questions by Asmodia now, of course, to make sure they aren't suicidally stupid to say out loud. And she can guess that means if she ever actually has a good one it'll get given to someone else who isn't supposed to be 'shy and hesitant.'

She could choke herself with a rusty spoon.

She messed up and she knows it and she's mostly just sulking, angry at herself. There's a simmering annoyance with Asmodia for forcing alter Willa on her, and still a lingering resentment towards Sevar for capturing most of her soul-profits, but in the end the reason she can't be a model student actor agent is that she's too obsessed with being special and not obsessed enough with winning. It's a terrible failing in her and it's worse that she knows deep down that it's still true, it's not something she knows how to change.

She's not going to be able to distinguish herself to Keltham by asking clever questions, and honestly even if she was she maybe couldn't trust herself to do it safely. Instead she'll have to distinguish herself to Asmodia by thinking useful thoughts. Then maybe eventually Asmodia won't hate her anymore she'll be out of the doghouse and at less risk for washing out entirely.

Having tapped everyone with Share Language - including Broom, once Broom reminded Keltham of his existence - Keltham speaks forth again:

"The first thing to do in this experiment... is something I expect you'll all find way too natural, namely, don't say anything and control your facial expressions.  That's not the usual rule in this classroom, but in this case I want you to confront a problem separately and without leaking information to each other, so no comments out loud on what I'm about to do or say here.  If you want to say something to me, or have a question, say 'Message' and then use Message."

Keltham goes to the wall, casts Prestidigitation, and draws three Baseline numbers via color-changing the wall's surface.  Possibly using Prestidigitation all the time will help train magic use to where it becomes more innate and reflexive for him?  Though he should actually check that concept with Carissa at some point, because it's a time-costly experiment.

The Baseline numbers that Keltham draws are 2, 4, 6.

(It should be immediately apparent that Baseline numbers are both easier to draw, and more visually distinguishable, than Taldane digits.  Just one more little reminder that almost everything about dath ilan was designed and not just allowed to happen.)

That is not, actually, immediately apparent to Korva; maybe it would be if she were focused on it, but right now she's busy trying to control her heartbeat even though she's pretty sure nobody can hear it leaking evidence that she's freaking out a little about whatever bizarre new kind of math this is going to be.

"I am now the - world? Nature? - 'environment' in Baseline," in case any of the concepts of 'thing that interacts with the agent and is separated from it only by a false conceptual boundary' make it across in translation.

"On each round, you input to me a sequence of any three integers, and I output to you either 'yes' or 'no'."

"You are learning me, unraveling me, theorizing about me and experimenting on me; you are trying to predict what makes me say 'yes' or 'no' on each round.  To be clear, I wouldn't particularly suggest that you think of these symbols 'yes' or 'no' as really meaning anything, except insofar as you think you've determined experimentally what makes me say them."

"You will now write down 2, 4, 6 on your paper - Taldane is fine, I can still read that.  When you're done, raise your hand, and I'll mark down on your paper whether this initial input causes me to output 'yes' or 'no' to you."

"If you think you've guessed the Law that governs my outputs, you can write that down, and I'll write 'correct' or 'wrong' next to that, where those words do have their standard meanings."

Willa's still a little jumpy.

When he first asks them to raise their hands just for the '2, 4, 6' she suspects it's a trap, and writes just slowly enough to make sure she sees a couple other students raise their hands first. Turns out it isn't a trap?

2, 4, 6: YES

8, 10, 12: YES

Willa would ordinarily guess here, right away, if she hadn't just been admonished to be careful. But she was, so she stopped, thought for a second, and then-

-6, -8, -10: YES

3, 1, -1: YES

The law is that the absolute value of the number must cha- Willa crosses that one out, realizing something.

5, 7, 5: YES

And there's a triumphant expression that's too subtle for Keltham to see but not too subtle for any Chelish students that might happen to be looking at her.

The law is that the the absolute value of the change of the number must be 2 each time.: WRONG

She experiences existential anguish, and then pauses in terror, expecting Asmodia to scream at her in indignation.

Asmodia does in fact yell at her, on relay through Security:  Control your facial expressions and don't fucking mess up Keltham's test, you fucking moron.  He may be able to tell from the results he gets if we didn't all follow instructions.  Luckily I don't think anybody who wasn't Security was stupid enough to be looking at you.

Don't do anything alterWilla isn't doing, FUCKING PERIOD.  If you prove unable to fucking understand that, alterWilla will fail to keep up and sadly resign from this Project.

Of course she messed up again, apparently she can't do anything right. She fiercely pushes everything down, and at least for a little while, her expression will remain thoroughly blank.

Idly translating Keltham's pre-Share Language sentences, Alexandre Esquerra, second-circle wizard and new Project Lawful researcher, discovers that Baseline has a word for "a societywide effort to COMPREHEND and CONQUER the ENTIRE UNIVERSE," and Alexandre Esquerra has never been so happy in his life. 

Then, Keltham's test. Alexandre will, of course, do better than his rivals; his first instinct is that he needs to solve it in as few guesses as possible, which he immediately crushes, stamps down on. He should not win the contest, he should learn, getting as much information as he can from Keltham before he dares hazard any guess. Once he's solved it, he can dawdle around and do extra checks and answer everything.

2,4,6: YES.

All right, think of this as spellcraft, to make sure magic items or spells stabilize; a ratio of 2,4,6 is in balance, is 2,4,7?

YES. 1,4,6? YES. 2,3,6? YES.

So much for the 'ratios' idea. That's thoroughly disproved. Well, Keltham's knowledge is strange and alien, and he should not expect it to match that of his own civilization. 6,4,2? YES.

... All right, so even reversing the order works; he's starting to get annoyed, he doesn't have good alternate possibilities and so he's stabbing in the dark. He isn't letting it show, of course, he's not a child. Does 1,1,1 stabilize? YES...

There has to be some spell structure that doesn't stabilize under these absurd laws, doesn't there? How about if he tries to break the bounds with -6000,7000,0: YES. Smoldering annoyance. Desire to burn everything. He can manage, while hating, if he could not manage he would not be alive. 500,0,500? YES. 100000000000,0,0. YES.

All right, he knew that was going to happen, his actual expectation was that one hundred billion would stabilize, which, per Keltham's rules, means he already knew it. The actual goddamn law is 'any three numbers', isn't it? There has to be something that isn't accepted unless Keltham is a -

Of course. Yes. Keltham is a sadist. He should not have considered this question as 'the output of a superior civilization bent on teaching its students', he should have considered the question as 'the output of a sadistic teacher intent on inflicting suffering on his class'. Hell is the destruction of hope, such as his hope of successful learning. He'd like to let security know he thinks he has a correct guess, but wants to make sure he isn't the first one to write it down.

(And, while he does that, he'll continue writing random strings of numbers so Keltham doesn't guess. 313-496-386? YES, of course.)

Security relay from Asmodia:  Stop.  Stop trying to act in ways that your alterCheliax self wouldn't.  Just stop.  Don't do anything fancy.  Just.  Be.  AlterAlexandre.  He'd go ahead and guess, so go ahead and guess before Keltham notices you're acting weird.

(...is this what it's like for Aspexia Rugatonn when she tries to explain to anyone about what probably seems to her like an incredibly simple and straightforward concept of just following orders?  It totally is, isn't it.  No, Asmodia isn't feeling even slightly sympathetic, because Aspexia Rugatonn.)

... All right, yes, acknowledged, he rather deserved that. The law is that any sequence of three integers will get YES.

WRONG.

This sounds even worse than normal math!! 

She writes down 2, 4, 6 on her paper in Taldane, as instructed. She raises her hand, being careful not to be the first one.

2, 4, 6: YES.

1, 1, 1: YES.

Huh. Okay, different numbers, all odd, same pattern. 

3, 5, 7: YES.

Opposite order.

7, 5, 3: YES.

Uh, branch out more, so you can see some no’s. 

7, 8, 9: YES.

Her heart is hammering out of its chest, and it shouldn’t be. She’s so scared of being the last one to figure this out. Realistically, not everyone will, but they can kick as many people off the project as they need to, so -

1, 2, 3? No - 1, 9, 2, that’s more different. 

1, 9, 2: YES.

Nothing to be gained from not guessing, probably? Guess the obvious. 

Guess: Always outputs “YES”.  And she adds another number guess, so as not to waste Keltham’s time when her first obviously-wrong-guess is wrong.

2, 5, 1: YES. WRONG.

Maybe it can be any difference?

101, 102, 101: YES

... or maybe it can just be anything at all?

101, 102, 100: YES

Let's just make sure.

1, 10, 107:

This time, Keltham fails to notice her for a round, because she's hiding in the back like a mouse and she hates alter Willa. Eventually though he notices her again.

1, 10, 107:YES

Let's be really sure.

pi, 5.67, 78/7: INVALID INPUT

And now she remembers they were supposed to be integers. She needs to get it together. It's probably wording and she's just phrasing it wrong, it has to be.

The law is that the three numbers must be integers.:WRONG

The law is that nature always says yes.:WRONG

Maybe if she throws enough things at the wall he will eventually tell her no.

-5, 10, -7:YES

1 is a pretty special number.

1, 1, 1:YES

Zero is too.

0, 1, 2:YES

Maybe the numbers just need to be really large.

10^20, 10^20, 2:YES

Is it really just a tricky wording after all?

The law is that if you give 3 valid inputs nature always says YES.: WRONG

No. YES to everything, absolutely everything, except something she can't think of. She's just stupid, obviously.

...people sure are making some assumptions that dath ilani wouldn't make!  Possibly human beings just naturally make a lot of assumptions, and naturally stay in small mental boxes, that dath ilani kids have already been implicitly jailbroken from by the time they first encounter this problem?

Alexandre will burn this useless scrap of paper! (No, he will write -3, -6, -9 and get back a YES.)

... alterAlexandre would snap even alterAlexandre is not weak. 0,0,0. YES. He continues his search for NO in the endless sea of YES; 2,4,8? (Yes.) 3,9,27? (Yes.) 8,4,2? (Yes.) -2,4,-8? (Yes.)

Mindful of Asmodia's warning not to do anything alterAlexandre wouldn't, he begins letting some strain show on his face. It is rather less than he's feeling, but he's never been outside Cheliax and doesn't know how much Taldorians conceal their facial expressions. He's used to having some idea of where to search.

Next step: 300, 400, 500. Is there some band where the usual ones don't apply? (YES, that is, SCREW YOU.)

4444,3333,2222... (YES.) wait, Baseline sometimes uses Base-12. 12-144-1728? (YES.) So really, Yes and No are inverted, and he can't find any point where he can stabilize this. 59326-78442-19848? YES. 16,4,2? YES.

... Hmm. Alexandre is starting to notice that Keltham is taking longer and longer to answer.

... ... Let's think about this logically. He's tried ascending order and descending order and powers and in the opposite order. He's been told to only try integers or he'd move into fractions. He's tried to break it from the top and from the bottom and input three of the same number and...
 
 ... Wait. Hmm. Odd. 2,2,4.

(YES, but with a long delay.)

That's... odd. 2-4-2. (YES)
 
Probably someone else has stumbled on the secret, and whatever the secret is involves Keltham having to do a lot of complicated mental math.

So he's trying prime factors, and seeing if there's a slot that the 4 should go in. 4-2-2? (YES.)

... 4-2-2 took less time than 2-2-4. Keltham may have just been distracted, or be waiting pauses of ambiguous length. 3-3-3? YES. Yes, yes, as expected, but if it was 'all primes and we don't count 1 as a prime' he doesn't know how he could have lived with himself. (In a dry, slightly ironic tone.) He hesitates a bit to do some quick arithmetic of his own, then: 79-11-879?

An immediate Yes.

Well, that should be interesting.

2, 4, 8. Those numbers at least have a relationship.

2, 4, 8: YES.

Did he say the integers have to be less than ten? He didn’t, did he? Okay. 

1, 247, 6: YES.

Okay, try negatives. 

1, -1, 0: YES.

Try all negatives?? 

-5, -247, -8: YES.

Try all zeroes. 

0, 0, 0: YES.

This exercise is STUPID.

5, 3, 15: YES.

Stupid and AWFUL. 

Okay, what other traits do integers have. They can be even or odd. They can be positive or negative. They can be factors of one another. They can be… addition problems, the first one was an addition problem. They can be division problems. 

Try writing down STUPIDLY LARGE numbers, even though this won’t get her anywhere. 

568, 9840348, 653203: YES.

Obviously.

She can’t think of anything but she’s obviously not allowed to give up, even though her heart is dropping like a rock. She gets a sense of curious worry from the butterfly familiar still in her room, over the empathic link, presumably from all of the intense emotions she’s having. She tries to tamp down her current emotions enough to send a coherent message back. The feeling of seeing a warm meal when hungry, the feeling of being chained up alone in a dark room, the feeling of seeing a warm meal again. This is supposed to indicate to Pretender that she’s fine, actually. She’s not entirely sure that she is.

……..okay, wait. When Korva had reason to design a code with things sort of like digits, she did - alternating differently-valenced things? Try alternating things that are different. 

1, 2, 1: YES. 

-1, 2, -3. YES.

 This is obviously doomed, but be thorough anyway. 

1, -4, 2: YES.

She hates this. They’re going to send her to hell right away, and she won’t even have learned anything, and she will therefore be starting at a disadvantage relative to everyone else in hell because she died young and didn’t even get to learn the things that mortals are supposed to learn, so she’ll be worthless for her entire existence and have a boring and fundamentally useless eternity, not even because she was inherently worthless, but because someone sent her into this stupid, ridiculous class for how to be an axiomite made of math just because she knows how HISTORY works.

These aren’t things that alter-Korva would be thinking except that OH WAIT YES THEY QUITE POSSIBLY ARE, GIVEN THAT EVEN IN ALTER CHELIAX SHE ALSO NEVER GETS TO GO HOME UNTIL THE PROJECT IS OVER.

1, 2, 3, 4, she writes down, angrily, in case disobeying instructions is the POINT.

1, 2, 3, 4: INVALID INPUT.

Okay. Not that.

That technically breaks the “always outputs yes” rule. 

3, 2, 1, she writes, and a guess. Guess: Always outputs "YES" when given a valid input sequence.

3, 2, 1: YES. WRONG.

Of course.

What other things can you do with integers. Division problems, sure, that won’t fucking be it but she can test it. 

…wait. 

Maybe she'll just try prime numbers. It's something, at least.

2, 3, 5: YES

It seems to take Keltham a while to answer this time. Maybe it really is primes, somehow?

23, 19, 41: YES

And it's faster this time. Her primes have betrayed her! Maybe the numbers just need to be pretty small somehow, in some confusing way.

2, 3, 6: YES

But it's slow again! Somehow she has something. Maybe.

Sense Motive check, is Keltham:

A) just messing with her, or
B) legitimately having trouble computing this quickly, or 
C) is he wondering why she's so stupid?

Also, Willa's trying to very subtly look around the room at this point to see if everyone else is finishing and she's alone in her pathetic stupidity.

Results:  Keltham is patient but wondering - more like he's wondering about everyone, not wondering about her in particular, his facial expression doesn't change much as he moves away from her.  Possibly he wasn't expecting this problem to be as hard for everyone as it seems to be?

Some frustration here and there is leaking through other students' expressions in a way that Willa can detect, despite Keltham's instruction.

A number of the other previous hires look finished, but such is only to be expected.  Oh, and what's-her-name - Korva - seems to be finished too.

(The Chosen of Asmodeus finished almost immediately, of course, but that was hardly worth noting.)

The more senior researchers don't shake her much. They've had more time. It's only fair.

But Korva's just as new as Willa, and she's already better. It's a dagger in her heart, but she has to keep going. Whatever happens she can't let herself be last. It'd be the end of the world.

Ok. Go time.

Is she absolutely sure it's not a wording thing?

Nature's output to Willa is YES when she writes three valid inputs.:WRONG

She's sure enough for now. Let's try other small numbers, that won't look stupid to Keltham at all, no siree.

3, 4, 6: YES

It takes him a while again, and she has a terrible, horrible suspicion now.

879-97-11? Immediate YES. -1,1,-1. Slow YES. ... Oh? -1,1,2? 1,-1,-1. Both swift YES. Very interesting. 1,-1,1? YES, very slowly. 

What on Golarion could this possibly be? Now he really wants to know. 1,1,-1. Another slow YES. -1,-1,1 YES, slowly again.

... Hmm. He hypothesizes that -1,-1,-1 is quick? He'll test that hypothesis. YES but a slow yes.

Fascinating. He really just wants to know the answer... Wait. -3,0,0? YES but a slow YES.

... He's imagining that Keltham has to do a lot of mental math here because he's trying to calculate - in his model, everything is either positive once Keltham has done some calculations, and that gets YES, or negative, and that gets NO, and that suggests to him that's multiplication of some sort but the different places meaning different things but he isn't sure which place means which...

-2,1,-2?

Keltham gazes at the paper, eyes flickering a bit, and then writes YES.  It took him about as much time as before; relatively slow.

- Wait, actually, no, there's an even dumber test he should do: -2,1,-2

An immediate NO.

... This cannot be that simple. 2,4,6?

NO.

... Alexandre now sees his fault. He had been, fundamentally, assuming that these were standard tests, independent tests, each question isolated from the last. He had not realized... the meaning of what Keltham had said. He had heard the word, and not thought about it, so only now, does he see the truth: That he cannot consider all his steps in isolation. He does not know the Law, yet, but he is one step closer to finding it.

If he was not in Cheliax, he would be smiling.

Among the first things one does with unfamiliar measuring instruments, in standard tests, is check that equipment's reliability-width by measuring the same thing multiple times!  Any child would know that!