Law-Magic

Regular magic is prescriptive and scientific. Perform a series of actions, (typically say some words, make a bunch of hand signs, and present a material component and let the spell-ferret fly out of your head.) and you create a desired effect.
Law-Magic is the restrictive. Magic tells a mage "Do _____, and you can ____", while Law-Magic tells a population area "If you do_____, bad things will happen."
The difference between magic and law-magic is like the difference between horses and zebras. In the same family, but a whole different animal.

Manifestation

Magical arithmetic and runes that inflict whatever punishment the magic deems necessary. Color changes with the law mage. Almost always violent- body parts explode or are chopped off or turn into a swarm of doves or some other weird effect that the Axios (law spirits) decides