Blink blink. 

"... I regret to inform you that the present state of documented mathematical pedagogy is not actually very much better here than in the institutions with which you are familiar, I think. That said I will cheerfully owe you a favor if you ask Spellman that fascinating question and then bring me a high fidelity mental image of his facial expression." 

"Deal!"

The problem with Arithmancy - 

(well, there are a lot of problems with Arithmancy, actually, but the one that's relevant right now) 

- is that when you put too many numbers in close proximity, while being a wizard, the numbers start having opinions. This drove Ted absolutely up the wall as a child, as he was the type of kid who had all his times tables down before most of his yearmates could add, rocketed past middle-grade numeracy without slowing down, and was badgering his local public librarian for algebra textbooks by the time he was nine. It wasn't until he got his Hogwarts letter that he realized he wasn't just dyslexic and was doing this with a much deeper and stupider handicap, to whit, the numbers having opinions is not the same thing as the numbers having logical or sensible or remotely correct opinions. 

With arithmetic you just need to double-check your work regularly, though the fact that this is necessary at all is why wizard accountants who successfully compete with goblin ones are vanishingly rare, but as soon as you start introducing symbolic variables you start getting all sorts of nonsense. Define a force vector of varying magnitude and try to use that value downstream to structure a spell that needs to rotate that vector widdershins to scale the counterforce appropriately and other parts of your spell start helpfully suggesting that perhaps you could calculate it as the hypotenuse of the legs of this other triangle you defined over here which is not actually related, and if you don't watch yourself very closely while you're defining the gestures based on the steps of the geometric proof you'll end up with steps in the middle that individually sound perfectly reasonable and aren't, actually, valid. This can and does happen by itself with no enemy action, which also bricks your spell, but it does not usually look exactly like it's going to work and then not work and instead explode without even been cast per se. You'd only get that if you told the numbers something stupid and they were trying very hard to arrive at a stable result anyway and this didn't work because they don't actually understand themselves. 

"It looks like," says Ted, slowly, and very dubiously, shaking numbers out of his hair and pointing at the offending section of the diagram, "gamma-chi(1) went below zero when you touched it?" 

(1) Written Γ(χ), this is in magical theory a solution for finding a finite value of the ambient magical field during an ongoing spellcast by defining it as the integral over r=0 to infinity of various exponential and power functions related to recent spells cast, the behavior of your expected target, where you're currently standing, etc., which if you do it right you can then reduce to a gamma function that intakes the measured magical temperature χ at a specific point in space. I think. I'm enthusiastically making things up here, you understand. 

This is the face of someone who has taken just enough Arithmancy classes to know that this is insane. The gamma function goes to infinity at zero and the only reason you can safely use it in spell design is that in real life things always have at least a little magic in them. 

This is the face of someone who has taken no Arithmancy classes and has an unrelated deeply embarrassing fact he's just realized might be relevant. 

"...long-duration calming charm emits negative thaums." 

"It what? How? WHY?" 

There will now follow an argument that this margin does not know enough math to contain.

Does Ophelia have thoughts on the political ramifications of this revelation? 

...Why is there a long-duration calming charm involved in this clusterfuck?  Oh.  Oh dear.  Hm.

"...But how, exactly, did they know anything like that might be involved?"

"Which is to say, that is surely private information and not just anyone would have it.  Either they have been observing Professor Weasley in enough detail that I'd hope he'd have noticed, or they suborned whosoever told him about long-duration calming charms."

"Great question." The wizard cultural understanding of medical confidentiality is 'what's that?', but... "You can't very well tell by looking, at least not with any spell I've heard--" 

"Course there isn't, wouldn't that be like figuring out how to count exactly how many photons a nox spell doesn't emit?"

Shrug. "...I'm still not totally clear why you can't do that either, but it's not important. My point is that I think even if somebody managed not to be noticed spying that's not enough. An observer would have to be reading the nonverbal incantations in my morning routine and if there's another one of you this generation I'll eat my hat. Even two this close together in age-" (gesture between Ophelia and Cedrella) "-is already really unlikely--" 

"Gambler's fallacy! Either it's independent and therefore exactly as likely as before or it's related somehow and a higher than expected rate should tell you to expect more!" pipes up Nelya, from the floor. 

"What?" 

Wince. ".... please don't talk unless it's really important, it'll jostle your skull and then take longer to fix." Why are wizards all like this. No, she knows why, it's that it doesn't kill them,  but augh.  

:( but statistics :( okay she will shut up now :( :( 

"Okay. I.... guess maybe it's only regular unlikely then,  but that's still not very. Much more likely it's spread of secondhand information, somebody casually mentioned it to a friend because they didn't realize it has any combat implications... ugh, I'm going to have to assume it's not safe to keep doing, aren't I." 

"...Very possibly.  There are, depending on the reasons it's needed, some possible nonmagical alternatives...?  I don't know your medical history, and I don't rightly want to, but - well, without magic available, Muggle doctors do some very inventive things with chemistry that might be relevant."

"...If a nox spell is a more generalized darkness, wouldn't it be possible to measure the energy of the - but photons are their own antiparticle --  But that's math, not counting --  I really don't know enough about Arithmancy to have any useful input to that whole line of theory, I don't think, but the physics does suggest...  Well, if it emits, then surely it has to be emitting?"

"...But then what does the annihilation reaction emit, or are we just completely throwing conservation out the window -- probably the latter but we don't know, do we? --"

"...But.  Magic theory can probably wait until after the assassination plot."

She carefully tucks away the train of thought running wild on speculation of how one could think about and possibly manipulate anti-thaums for later.

"Right! It's like saying you have a negative temperature! Why is magic like this! That's not how anything--" 

"What? There's negative temperatures all the time. It's negative degrees in Scandinavia for like half the winter." 

"Not if you use a SANE MEASURING SYS--" 

"Boys. Can we focus on the security implications please." Appreciative nod at Ophelia for restraining herself.