The 4 Elements Monk setup of spellcasting is, if I recall correctly, actually effectively the same as a half-caster using spell point system and a standard adventuring day. Is the 4EM the points level, recovery, and spend cost that we're going with?
I'm assuming that there will be abilities than convert points into damage on a hit. How much baseline damage should a point be worth? D8 would fit with the paladin smite model. Obviously riders and benefits would change this.
Are you thinking of an actual extra attack at 5th level, or a different style of damage boost? or would that be subclass-dependent: (Circle fighter gets extra attack, swordcaster gets to add +2d8 elemental damage etc.)
The two aren’t mutually exclusive.
And the concept is exactly a combatant that uses complex mathematics and anatomical knowledge as part of its martial prowess. It’s The Magic Circle, but actually magic.
Other than the involvement of "magic" they sounds like two different archetypes.
One uses mental mathematics, sacred geometry and anatomical knowledge to move effectively, defend against attacks and deal damage. - Sounds very "internal". (Actually sounds very similar to the Monk class already.)
The other uses teleporting, shields of force and blasts fire out of their weapon to move effectively, defend against attacks, and deal damage. - Much more flashy, externally focused and elemental. (This sounds like the 4e swordmage.)
They're both cool concepts, but they sound different enough to be separate archetypes at least.
edit: in the draft of the introductory fluff I posted, I even describe a character Drawing The Circle in his mind and gaining basically a magical HUD that shows him the math of battle.
Indeed. But he is not judging the posiitiion and potential movements of the battlefield with a keen eye and acquired skill, he is casting a spell that does all that for him.
To my mind that is Arcana, not Investigation.