It is common practice for a wizard to specialize in a certain school or aspect of magic, as a way to both further increase their prowess in the magical arts, as well as create new discoveries in the field to make a name for themselves. However, there is a small sect of magic philosophers that specialize in a field that serves as the basic foundations for both the magical and material world; mathematics. The curation of magical formulas and mutagens does not stem solely from the powers of The Weave alone. Arcane Arithmeticians seek to further enhance their spells by using arcane formulas and theories to alter their effects, as well breaking down the components of creating a spell in order gain access to unique spells that a typical wizard does not know.

Algebraic Anthropology

When you select this tradition at 2nd level, you gain the tools to access the studies of many cultures. You learn the comprehend languages spell and add it to your spell book if you do not already know it. This spell is always prepared and does not count towards your number of prepared wizard spells.

In addition, you gain advantage on any skill checks that have to do with solving equations, puzzles, coded languages and ciphers.

Arcane Theories

Also at 2nd level, you have honed the art of modifying your repertoire with advanced effects. Choose a total of two Arcane Theories from the list below. Once per turn, before you cast a spell, you may choose to modify the properties of the spell with one of your known Theories and add it's special effect to the spell, as long as the conditions to the theory are met. You can only modify spells cast from your spell book and cannot modify a spell with more than a single theory per turn. Theories that require a saving throw use your spell save DC.

You can modify a spell with your Arcane Theories a number of times equal to your proficiency bonus and regain all uses after a long rest.

At 6th level, you learn an additional Arcane Theory from the list below. At 14th level, you learn another Arcane Theory.

Corrosion

When you cast a spell that damages a creature, you may use this theory to force the affected creatures to make a Constitution saving throw. Any creatures that fail cannot regain hit points or take reactions until the start of your next turn.

Evolution

(Prerequisite: 6th Level) When you cast a 1st level spell or higher, you may use this theory to cast the spell as if it was cast 1 spell level higher than the spell slot you used to cast it. You cannot use this theory if you do not have access to the level of casting the spell was modified to. For example, A 6th level Wizard casting fireball at 3rd level cannot use this theory as they do not have access to 4th level spell slots. However, if they were to cast scorching ray at 2nd level, they would be able to use this theory to modify the spell.

Exertion

(Prerequisite: 6th Level) When you cast a spell that forces a creature to make a saving throw to resist its effects, you may use this theory to gain a bonus to your spell save DC equal to half your proficiency bonus (rounded down) until the start of your next turn. 

Immunization

When you cast a spell, you may use this theory to grant either yourself or a single creature affected by your spell temporary hit points equal to your proficiency bonus plus your Intelligence modifier.

Momentum

When you cast a spell that damages a creature, you may use this theory to force the affected creatures to make a Strength saving throw or be pushed back 10 feet from you and knocked prone.

Relativity

(Prerequisite: 6th Level) When you cast a spell, you may use this theory to envelop yourself or a single creature affected by your spell in an arcane disruption field. Enemies have disadvantage on all attack rolls against the affected creature until the start of your next turn.

Velocity

When you cast a spell, you may use this theory to increase the movement speed of all affected creatures by 10 feet until the start of your next turn. Alternatively, you may use this theory to force all affected creatures to make a Dexterity saving throw or have their speeds be reduced by 10 feet until the start of your next turn.

Spell Analysis

At 6th level, you have learned to deconstruct the very building blocks of magic formulae and have learned a unique spell you normally wouldn't have the skills to cast.

Choose a 1st level spell or higher from either the Artificer, Bard, Druid, Ranger, Sorcerer or Warlock spell list and add that spell to your spell book. This spell counts as a Wizard spell to you, is always prepared, and does not counts towards your prepared number of Wizard spells. The chosen spell must be of a level you can cast, as shown on the Wizard Table.

Calculate the Odds

At 10th level, your mastery of battlefield analysis has grown to such a level that you can estimate the best way to maneuver sticky situations. Once per long rest, when you are rolling a saving throw but before you are given the results of the roll, you may use your reaction to add your Intelligence modifier to the number rolled.

In addition, you no longer require sleep in order to complete a long rest, as your long nights of study have hardened your resolve. However, you cannot receive the benefits of a long rest while partaking in rigorous work.

Advanced Spell Analysis

At 14th level,  your mastery of the foundation of magic has granted you another boon in unlocking the secrets of the arcane.

You may choose another spell of 1st level or higher from either the Artificer, Bard, Druid, Ranger, Sorcerer or Warlock spell list and add that spell to your spell book. This spell counts as a Wizard spell to you, is always prepared, and does not count towards your prepared number of Wizard spells. The chosen spell must be of a level you can cast, as shown on the Wizard Table.