At 1st level, your senses and mind are opened to the repeating mathematical patterns and probabilities inherent to the universe. Whenever a hostile creature makes an attack roll that deals damage to a friendly creature within 100 feet of you, you can use your reaction to record the result. If you make an attack roll that is of the same range and type, you can choose to use the recorded result in place of your roll. You can only have one result recorded at a time. Once you use a result, it is lost and a new one must be recorded.

The Unifying Equation lets you choose from an expanded list of spells when you learn a warlock spell. The following spells are added to the warlock spell list for you.

You have stumbled across the secret of a universal constant that lies at the core of all of creation. Through your research and search for truth, you have tapped into the ultimate culmination of all principles of mathematics and logic in the universe, a numeric formula with such complexity that it exhibits a form of artificial sentience completely alien to physical organic life. By aligning your mind and will with the absolute logical and geometric order of the Unifying Equation, you have learned to see the world as it really is, unimpeded by imperfections in your own perceptions and thoughts. Its motives are foreign and difficult for mortals to process, but one thing is for certain; no matter the seeming disorder of things, they will inevitably be submitted to the power of the Equation. It’s not a boast or a threat, it’s just the way things are.

Acute Counter-Strike

At 6th level, the geometric order of your patron flows through you to fend off attackers that would interrupt your research. When you are attacked by a creature that you can see, you can use your reaction to create an area of warped space and razor-sharp angles around you immediately before the attack roll is resolved. The creature must make a Dexterity saving throw with a DC equal to your warlock spell save DC. If they fail, they take force damage equal to twice your warlock level plus your Charisma modifier and are thrust 10 feet directly away from you. On a successful save, they take half as much damage and are not moved. If the movement would cause them to no longer be able to attack you, or if they are reduced to 0 hit points, their attack misses. Once you use this feature, you can't do so again until you finish a short or long rest.

Precision Reflexes

At 10th level, your reaction times are enhanced by the power of the Equation. You gain advantage on Dexterity saving throws, and become proficient in the perception skill.

Eye of the Mathematician

At 14th level, you gain the ability to see the underlying mathematical order of the world around you. As a bonus action, you may channel your patron’s power to gain one of the following effects. The number of times you use this feature is equal to your Charisma modifier, and you regain these uses upon completing a long rest.

Pinpoint Weakness: Chose a creature you can see. Until the beginning of your next turn, the next successful attack you make against this creature is automatically considered a critical hit.

Pierce Illusion: You are affected as if by the true seeing spell for 1 minute.

Calculate Ricochet: When you make a ranged weapon attack this turn, you can treat half cover as no cover, and three quarters cover as half cover.

Eldritch Invocations

Binary Targeting

Prerequisite: Unifying Equation patron, Pact of the Chain feature

If you have a polyhedral assistant as your familiar, it gains additional hit points equal to you warlock level, and you gain the ability to cast the true strike cantrip.

Formulae of Algebraic Awareness

Prerequisite: Unifying Equation patron, Pact of the Tome feature

You gain the ability to cast the arcane eye spell as a ritual with the following changes. The spell's casting time is increased to ten minutes, and its range is increased to 150 feet. You have advantage on concentration checks to maintain this spell.

Fractal Burst

Prerequisite: Unifying Equation patron, 5th level

You can cast hypnotic pattern once without expending a spell slot. You regain the ability to do so when you finish a long rest.

Omega Blast

Prerequisite: Unifying Equation patron, 7th level, eldritch blast cantrip

When attacking a creature with eldritch blast, that creature must make a Dexterity saving throw. On a failed save, your attack ignores half cover and any AC bonus given by shields.

Vector Broadheads

Prerequisite: Unifying Equation patron, Pact of the Blade feature

You can create a shortbow that fires arrows with triangular, nearly 2-dimensional tips using your Pact of the Blade feature. This weapon ignores resistance to piercing damage, and treats immunity to piercing damage as resistance. You may use your bonus action to summon an additional 20 ammunition.

Vector Dirk

Prerequisite: Unifying Equation patron, Pact of the Blade feature

You can create a dagger with a triangular, nearly 2-dimensional blade using your Pact of the Blade feature. This weapon deals slashing damage, ignores resistance to slashing damage, and treats immunity to slashing damage as resistance.

Traits of the Euclidian

Consider adding one or more of these traits when creating a warlock or after becoming more aligned with the Equation.