so, mana physics

[xax]

you start off emitting a pulse of 'basic' mana, which i'm going to codify as M 2 1 1 for reasons that will become clear soon.

the first tool is a beam splitter, which separates out your mana polarities. so M 2 1 1 splits into two mana bands of M 1 1 0 (hard mana) and M 1 0 1 (soft mana). the hard mana bounces off the splitter surface whereas the soft mana passes through, allowing you to isolate the components.

the second tool is a mana collider, which merges two beams of mana. it operates as such: M g a b + M h c d = M (g + h) (a + c) (b - d), and can only be performed if g == h. this means soft + hard = base. but this isn't commutative, so hard + soft = ???, a different mana type w/ the equation M 2 1 -1. i'm gonna call this kind of mana 'anti-mana'. (you can also do soft + soft to get M 2 0 0 ('active' mana) and hard + hard to get M 2 2 0 ('still' mana)).

you can then combine that generation of mana, in sixteen different ways (base + base, base + anti, base + active, base + still, anti + base, anti + anti, etc etc and so on), which will get you nine distinct frequencies of third-generation mana. and there are eighty-one ways to combine those nine frequencies, with twenty-five distinct frequencies of fourth-generation mana. and so forth onto infinity.

but also, anti-mana is interesting, because if you decompose it you get M 1 1 0 (so, hard) as well as something new: M 1 0 -1. that's a new fundamental mana frequency, which we'll call 'antisoft'. with antisoft you can create some new mana flavors, by combining a soft/antisoft pair or an antisoft/soft pair. this means you can build a total of six second-generation mana frequencies: base, anti, active, still, and 'sharp' and 'dull', from the soft/antisoft pairs. (an 'antihard' mana frequency theoretically exists but isn't reachable, because the combination equation is noncommutative in the second value, and so it can only produce antisoft. if it was noncommutative in the first value, it would only be able to produce antihard and not antisoft.)

what's interesting about that, is that still mana (M 2 2 0) matches up with hard mana (M 1 1 0), and sharp mana (M 2 0 2) matches up with soft mana (M 1 0 1). then dull mana (M 2 0 -2) rounds out the triad of basic mana flavors by lining up with antisoft (M 1 0 -1). so those are kind of the basic building blocks of all mana frequencies.

(splitting mana is actually more complex than above, since there's a general rule that basic mana is kind of a special case for. you can't split anything that has a 0 in one of its components, since that's already split. a hard mana pulse hitting a splitter will just bounce with no decrease in intensity; likewise a soft mana pulse hitting a splitter will just go through. if both components are nonzero, then it decreases generation and splits the beam into pairs. there's also a fourth case, where both components are already zero (active mana is M 2 0 0). in that case, the beam intensity is halved, but the output is just two beams of the same generation. so where basic mana decomposes, active mana takes both paths without decomposing.)

the REAL question is how to make the mana flavors do interesting things without making them entirely arbitrary