School of Magic Theory

You finally understood the absolute theorotical limit of magic, you learned to break the frontier between the different type. You understanding of this phenomenal is the most perfected, it has no longer limit for you.

Docent

You are able to more easily recover spent magical energy, due to your close study of the theory of magic. 
Starting at 2nd level, whenever you use your Arcane Recovery feature, you may regain an additional amount of spell slots with a total level equal to your proficiency bonus.

Additionally, you copy spells from different class spell lists into your spellbook but doin so takes twice of the time and gold as normal. As long as this spell is in your spellbook, it counts as a wizard spell for you.

Also, if you take the spellcrafter feat, you may make one original spell for each spell slot you have accouridng to the spell creation guidelines and may do so again each time you gain a spell slot of a new level.

Margin of Error

You study of the theory of magic has allowed you to estimate at the correct way to cast a spell at a higher level without expending the extra energy to do so. 

At 2nd level, whenever you cast a spell you may attempt to upcast it without expending a spell slot of a higher level. The spell may not already be upcast. You may attempt an upcast an amount up to half your proficiency bonus rounded down (max total level is 9) in this manner.

To do so make an Arcana check with a DC equal to 13 + twice the amount you are upcasting. On a success, you cast the spell at the higher level while expending the normal spell slot. On a failure, you instead downcast the spell by the amount you are trying to upcast while expending the normla spell slot.

You may use this feature an amount of times equal to your intelligence modifier, and regain all spent uses on a long rest.

Improved Docent

Your constant study of theoretical magic has improved your ability to comprehend the notes of others as well as streamlining your own spellbook.

At 6th level, the time and money spend to copy any spell into your spellbook is halved.

In addition, you may prepare an additional amount of spells per day equal to your proficiency bonus.

Counterflow

By observing perturbations in the Gygax Field, you may partially counter this pertubration with your own.

Starting at 10th level, when you would tkae damage from a spell, you may use your reaction to expend a spell slot, reducing the damage by an amount equal to the spell slot's level multiplied by your Intelligence modifier.

Aditionally, you gain advantage on saving throws made against the effects of spells.

Theoretical Limit

At 14th level you learn to terst the bounds of your theories.

When casting a spell, you may either :
- Increase the size of your spells' damage die by 2,
- Roll max damage on all dice,
- Add twice your proficiency bonus to the damage,
- Change a single target attack spell to an 30ft sphere requiring a Dexterity saving throw against half damage,
- Change an area of effect with a radius of 40ft or less into a single-target attack.

You may use only one of these effects per spell and may only use each effect once per long rest.