Can we build the math from the ground up? (And does Vancian help or hurt that?)

[Pex]

I don't really get how this is responsive. Sure, you could normalize everything to one damage progression (you should not actually do that, but you can). But that doesn't change the fact that Burning Hands is a melee-range AoE, Scorching Ray is a close-range single target attack, Fireball is a long-range AoE, and Wall of Fire is a medium-range BFC effect that's not even instantaneous. How are you going to set things up so those are all modes of the same underlying "Fire" spell in a way that isn't more trouble than just having four different spells?

It's a means to limit magic knowledge without sacrificing diversity. The various effects are physically printed under the heading of Fire Attack. The spell caster gets the Fire Attack ability and with it comes all these different things he can do with it based on the spell slot used or mana spent depending on the system. There will be Electricity Attack, Acid Attack, Healing, Mind Control, Alteration, Travel, etc., various types of magical forms. A spellcaster gets a limited number of them so he can't do them all, but within the ones he does get he has variety of things he can do. How many magic forms he gets and when to be determined in the hypothetical.