Arithmetician

A young student slumps over a number of open tomes. The sun rises after a long night of study and failed experimentation. As she stands to finally retire, the equation finally clicks. It all comes together. She holds her tome in one hand and draws sigils in the air with the other. A bolt of arcane energy blasts forth across the room, impacting with a bookshelf on the other side.

An elderly professor walks a stone path through the woods, reciting arcane theory out loud to no one in particular. Several bandits drop from the trees and pull their knives on the man. In a flash of practiced finesse, the professor instinctively drops to his knees and draws a sigil on the earth, and a companion of rock and stone emerges from the ground to protect him.

A nervous gnome scholar defends his home from invaders. His brethren fight valiantly to hold off the horde of goblins, but they're outnumbered 5 to 1. The gnome searches the pages of his tome for an answer, and finds the calculations to summmon an ally. A sylph is conjured from seemingly nothing, and grants the gnomes the strength they need to succeed.

Arithmeticians wield arcane magics through the use of solving the right archaic equations. Their ability to call on the Weave is tied exclusively to their mastery of arcane calculations. Arithmeticians vary widely between their cultures and intentions, but they all share belief in the adage of "brains over brawn."

Ruinous Blast

You begin your arithmetician career knowing the most basic of mathematical arcane applications in the form of the Ruinous Blast cantrip. This cantrip is an arithmetician cantrip for you, and does not count toward your cantrips known.

Expeditious Evaluation

At 2nd level, you can quickly analyze and react to a bad situation. You may add your Intelligence modifier, rather than your Dexterity modifier, to initiative checks.

Aetherflow

Also at 2nd level, you have deciphered the equations necessary to command the stream of ambient aether around you to various effects. You have a pool of aetherflow dice—which are d6s—equal to your proficiency bonus. An aetherflow die is expended when you use it. You regain all of your expended aetherflow dice when you finish a short or long rest.

You know how to wield aether in the following ways:

• Outburst. When you deal damage with Ruinous Blast, you may expend one aetherflow die to increase its intensity. If you do, roll the die. The target and each creature of your choice within 5 feet of the target take force damage equal to the number rolled.
• Adloquium. When you cast a spell of 1st level or higher, you may use your reaction to expend an aethflow die to stiffen the spell's strands of aether into an invisible shield. If you do, roll the aetherflow die and grant one creature within 30 feet temporary hit points equal to the number rolled.
• Brainstorm. When you make an Intelligence check, expend one aetherflow die to sharpen your mind. If you do, roll the die and add the number rolled to your total. You can use this ability once per check.

Academic Specialization

At 3rd level, you choose the type of specialist you are: Convoker, Theurgist, or Binder, each of which is detailed at the end of the class’s description. Your choice grants you features at 3rd, 5th, 9th, and 15th level.

Ability Score Improvement

When you reach 4th, 8th, 12th, 16th, and 19th level, you can increase one ability score of your choice by 2, or you can increase two ability scores of your choice by 1. As normal, you can’t increase an ability score above 20 using this feature.

Improved Aetherflow

At 6th level, your aetherflow dice turn into d8s.

Top of the Class

At 7th level, your studies have made you an academic force to be reckoned with. Choose one skill from Arcana, History, Nature, or Religion. You gain proficiency in that skill if you don't already have it, or expertise if you do.

Theory of Probability

At 10th level, you learn to alter the the reality behind probability. If you would roll a d20 for an attack roll, ability check, or saving throw, you may choose to roll 5d4 instead. If you do, you cannot critically fail or critically succeed on the check. You can do this a number of times per long rest equal to your proficiency bonus.

Improved Aetherflow

At 11th level, your aetherflow dice turn into d10s.

Theory of Relativity

At 14th level, you learn how to manipulate the laws of physics to your advantage. You can use each of the following once per short or long rest:

• An object in motion remains in motion. When you move on your turn in combat, you can double your speed until the end of the turn.
• Force equals mass times acceleration. When a creature within 60 feet is damaged by a weapon attack, you can use your reaction to either accelerate or decelerate the weapon. If you accelerate the weapon, it deals an additional die of damage equal to your aetherflow die. If you decelerate the weapon, reduce the damage dealt by one roll of your aetherflow die.
• An equal and opposite reaction. When a creature within 60 feet casts a spell that deals damage, you may use your reaction to deal damage to that caster equal to the damage they rolled for the spell. The target takes half as much damage on a successful Dexterity saving throw.

Improved Aetherflow

At 18th level, your aetherflow dice turn into d12s.

Divide by Zero

At 20th level, you may cast Dark Star (Explorer's Guide to Wildemount) once per long rest without expending a spell slot or material components. You may regain 1 aetherflow die for each creature that is reduced to 0 hit points while within the radius of this spell.

Amun-Her Khepesef, The Desecrated Scion

Amun-her Khepeshef was the heir to the throne, son of the king, and leader of the army. The empire kept thousands of slaves, who they used to build numerous pyramids, tombs, and palaces. When the slaves revolted, the empire imposed harsh strictures, limiting their religious freedoms. Eventually, the people rebelled, calling on their long suppressed god to smite their captors. Furious, the god struck the empire with numerous plagues—fire, vermin, disease, drought, and worse. The final plague was the most terrible even in the empire's history—the foreign god struck down the firstborn sons of each family in the empire, including Amun-her Khepeshef.

The loss of the first-born sons was tragic, bringing an entire generation to its knees with depression and desperation. To make matters worse, grave robbers and rival invaders desecrated the numerous new tombs, destroying the sanctity of the afterlife of the dead sons. These spirits became furious: angry at the foreign god and his people, enraged at those who betrayed them in their complacency, livid at the tomb robbers who interrupted their after-life, the spirits appealed to the god of the dead, who agreed to banish them beyond space and time, so that they would suffer no longer. These spirits were removed from existence as a single bound entity known as Amun-her Khepeshef.

• Sign. You take on the look of the dead: jaundiced skin, sunken cheeks, bags under the eyes, and stiff movements.
• Minor Arcana: Halt Death. You can cast the Spare the Dying cantrip at will.
• Minor Arcana: Firstborn Guardian. You have advantage on saving throws against poison, and you have resistance against poison damage.
• Major Arcana: Burst of Life. Once per short or long rest as an action or attack, you can expend an aetherflow die to release a 30-foot radius burst of positive energy centered on yourself. Choose any number of creatures in this radius, including yourself. Each creature restores X hit points.

Andras, the Gray Knight

Andras was once an elf paladin famed for his prowess in battle and his implacable dedication to doing what was right and good for all. A series of misjudgments and misfortunes broke Andras's faith in both himself and his deity, however, and he became a blackguard. During his subsequent service to the dark gods, his infamy rapidly outgrew his fame, and his name was whispered in fear.

After nearly three hundred years of almost constant battle on behalf of both good and evil, Andras grew tired of both causes. In the midst of a duel in the key battle of a great war, he simply dropped his weapon and left, never to be seen alive again. Sages speculate that after his betrayal of both causes, he was no longer welcome in any god's realm, and thus his soul was condemned to to the aether.

• Sign. You sprout two useless, gray-feathered wings from your back. The wings are small enough to be hidden beneath a shirt or cloak, but doing so makes you appear hunchbacked.
• Minor Arcana: Saddle Sure. You gain proficiency in the Animal Handling skill.
• Minor Arcana: Sure Blows. Your weapon attacks score a critical hit on a roll of 19 or 20
• Major Arcana: Sow Discord. Once per turn as an action or attack, you can expend an aetherflow die to sow discord among your enemies. One creature within 30 feet must make a Wisdom saving throw. On a failure, the creature takes X psychic damage and becomes charmed by you until the end of its next turn. While charmed this way, the target must use its action or first attack on its next turn, if possible, to attack its closest ally within reach or range of its currently held weapon. On a success, it takes half as much damage and suffers no other effect.

Andromalius, The Repentant Rogue

Once the herald of the trickery god Olidammara, Andromalius foreswore theft and mischief on his deathbed, repenting all the actions he had taken on behalf of his god during his life. By this means, he hoped to steal his soul from his deity, thus accomplishing his greatest theft and prank in history, and proving himself the most worthy of his god's favor.

At first angered by Andromalius's betrayal, Olidammara quickly realized the irony of the moment and burst into laughter. Yet the god's good humor was short-lived, because he realized that to accept Andromalius's soul would be to prevent the theft and ruin the joke. Since Olidammara was loath to let such a clever servant to go to the realm of some other god, he repaid his servant's honor a hundredfold—he stole Andromalius's soul from the cosmos. Whether Andromalius deemed this result an honor or not remains unclear.

• Sign. You gain an extra digit on each limb. This appendage prevents you from wearing normal gloves or gauntlets, but magic gloves and gauntlets reshape to fit you.
• Minor Arcana: Sense Trickery. You gain proficiency in the Insight skill.
• Minor Arcana: Detect the Unseen. You gain a +5 bonus to your passive perception. This does not stack with other bonuses to your passive perception.
• Major Arcana: Jester's Mirth. Once per turn as an action or attack, you can expend an aetherflow die to psychically assault a target within 30 ft. The target must make a Wisdom saving throw. On a failure, the target takes X psychic damage and then perceives everything as hilariously funny and falls prone in fits of laughter, becoming incapacitated and unable to stand up until the end of its next turn. On a success, the target takes half as much damage and suffers no other effect. A creature with an Intelligence score of 4 suffers damage but no other effect.

Ansitif, the Befouler

Ansitif the Befouler is an ancient tanar'ri lord who rose to power in the wake of the obyriths' fall and quickly seized control of the 21st layer of the Abyss, now known as the Sixth Pyre. Although most demon lords hunger for the power of true gods, Ansitif was enraged by the very existence of divine beings. Unable to challenge most deities directly, the Befouler vented his fury through the corruption of religious sites and relics dedicated to true gods. At the height of his power, Ansitif's demesne was littered with the shattered relics of countless faiths and the ruins of desecrated temples drawn into the Abyss.

Ansitif joined with six tanar'ri allies to hunt down and destroy a powerful obyrith called the Malgoth. The affair took Ansitif and his allies away from their centers of power, and opportunistic demon lords assassinated or imprisoned each member of the alliance in turn. In the case of the Befouler, his most powerful general, a balor named Kardum, betrayed Ansitif and had him imprisoned in the Wells of Darkness on the 73rd layer of the Abyss.

• Sign. Your hair and skin become blackened and scorched, as if briefly touched by fire.
• Minor Arcana: Demon's Defense. You have resistance to fire damage.
• Minor Arcana: Know Thy Enemy. You have advantage on Religion checks.
• Major Arcana: Blasphemous Strike. Once per turn as an action or attack, you may expend one aetherflow die to strike a foe in a blaze of smoke and fire. If you do, make a melee spell attack against one creature within 5 feet. On a hit, the creature takes X fire damage and must make a Constitution saving throw or become stunned until the end of its next turn.

Ashardalon, Pyre of the Unborn

Ashardalon was a red dragon of unusual greed and power. He sought to control a vast stretch of land, and ravaged it in cruel hunts for food, sport, and power. So great was Ashardalon's power that cults grew to revere him as a deity and followed him into what they believed were holy wars. In time, however, an alliance of rangers and elves managed to defeat Ashardalon's armies. Shortly thereafter, the dragon faced a powerful druid who had risen to defend the land, and was nearly slain.

As a near god, however, Ashardalon was a force to be reckoned with, even in defeat. He hid on the Outer Planes for millennia and, through complex machinations, managed to replace his heart with that of a Balor, becoming a creature of even greater supernatural evil. Renewed, the dragon made an assault on the Bastion of Souls in an effort to destroy all living creatures. There, Ashardalon faced a distant descendant of a druid who had vanquished him once before and was truly killed.

Still, his death at the birthplace of all souls (combined with his nearly deific status) allowed him to resist a permanent resting place. His essence lacked the power to become a true immortal, leaving him only the nebulous existence of a vestige.

• Sign. A patch of skin over your heart is marked by a deep-hued crimson sigil of a curled red dragon.
• Minor Arcana: Ashardalon's Vigor. When you bind this vestige, you gain temporary hit points equal to your arithmetician level.
• Minor Arcana: Ashardalon's Vengeance. You have advantage on opportunity attacks.
• Major Arcana: Ashardalon's Presence. Once per turn as an action or attack, you may expend one aetherflow die to strike at a foe while unleashing the roar of a mighty dragon. If you do, make a melee spell attack against one creature within 5 feet. On a hit, the creature takes X thunder damage and must make a Wisdom saving throw or become frightened of you until the end of its next turn.

Astaroth, Diabolus of Baator

This once-powerful demon lord who is said to have once rule a layer of the Abyss know as the Terminal Archives, with a gift for prophecy offered to fight alongside the Queen of Chaos during her war with the Wind Dukes of Aaqa, but was rejected. In the early days of the Blood War, Astaroth earned the name Diabolus by infiltrating the legions of Baator and rising to the rank of Treasurer of Hell. His true nature was eventually exposed by the arch-devil Gargauth, forcing Astaroth to flee the wrath of Asmodeus, but his spying caused incalculable harm to the devil's war effort and prevented the legions of hell from winning a clear and decisive victory against the hordes of the abyss.

Upon his return to the Abyss, Astaroth retreated to a steam-filled layer filled with floating chunks of burning stone. In preparation for the inevitable retaliation of the Lords of Nine, Astaroth began cultivating mortal cults on countless worlds through the use of his prophetic powers, in hopes of transforming himself into a god. He must have succeeded in some fashion, although not enough to forestall his fate. Astaroth was eventually slain by Gargauth, at the command of Asmodeus. In addition to claiming his predecessor's name, the Tenth Lord of Nine is said to have seized the mantle of divinity from Astaroth as well.

• Sign. You acquire the stench of brimstone and a cloudy film covers your eyes.
• Minor Arcana: Serpentine Tongue. You are proficient in the Deception skill.
• Minor Arcana: Blackflame Blast. When you cast Ruinous Blast, you may have it deal fire damage instead of force damage. If you do, it ignores resistance to fire damage and you can treat any 1 on a damage die as a 2 instead.
• Major Arcana: Gift of Prophecy. Once per turn, when you roll a weapon attack, skill, or saving throw, you may expend one aetherflow die. If you do, roll the die and add it to the total.

Eligor, Dragon's Slayer

Eligor was a great half-elf horeseman and dragonslayer. After Eligor's death, Tiamat sent her draconic minions against the followers of both the human and the elven deities, demanding that they release his soul to her. Despite Eligor's services, the deities gave up his soul to stave off the dragon attacks against their living followers.

Only one deity argued against this injustice, the details of whom differs with the teller. Whoever it was, this god set off alone to face Tiamat and wrest Eligor's soul from her grasp. Upon arrival, however, the deity found Eligor in the service of Tiamat rather than in bondage. Unbeknownst to the other gods, Tiamat had raised him from death to be her champion and enforcer, using his abandonment by the other gods to win his loyalty. Eligor and the nameless deity fought, and Eligor lost his life yet again. This time, no deity laid claim to his soul.

• Sign. One of your hands becomes thickly scaled.
• Minor Arcana: Eligor's Resilience. Once per short or long rest, you may cast Absorb Elements without expending a spell slot. The level of this spell is equal to your proficiency bonus.
• Minor Arcana: Eligor's Skill in the Saddle. When you or your controlled mount use an action to Dash, you can use a bonus action to make one melee weapon attack or to shove a creature.
• Major Arcana: Chromatic Strike. Once per turn as an action or attack, you may expend one aetherflow die to strike with power of Tiamat. If you do, make a melee spell attack against one creature within 5 feet. On a hit, the creature takes X damage of your choice of acid, cold, electricity, or fire.

Eurynome, Mother of the Material

Before recorded time, the gods and titans battled on the Outer Planes. Tired of the struggle, the titan Eurynome fled to the roiling chaos that made up the Material Plane. She divided the world into sky and sea, and she danced alone upon the waves. Incensed by her impertinence in meddling with a world as yet unformed, the gods struck Eurynome down. Angered by her abandonment of their fight, her fellow titans refused to come to her aid. Eurynome's body became the first island, her blood became the first river, and her soul became a vestige.

• Sign. Your skin becomes clammy, and you leave moist prints on any object your body touches, even if clothing blocks direct contact.
• Minor Arcana: Water Dancing. You can move on liquid as if it were firm ground.
• Minor Arcana: Fish Friend. Through sounds and gestures, you can communicate simple ideas with aquatic beasts.
• Major Arcana: Sea Conjuring. Once per short or long rest, you may expend one aetherflow die to cast Summon Beast without expending a spell slot and without material components. You summon a water beast, and the duration is reduced to 1 minute. The level of this spell is equal to your proficiency bonus.

Gaia, Soul of the Land

Gaia was both the nature goddess of a lost continent and the body of that land itself. She was the patron of all creatures and the defender of sacred life. When the land was injured, she too was hurt, her life-essence pumping in time with the earth. When Tkhaluuljin, a mammoth alien squidlike entity, attacked the land, Gaia fought, hoping to sacrifice her essence to protect the lives of her beloved residents. Her attempts were a failure and Tkhaluuljin swallowed her along with all life that lived on her surface.

Even in the gargantuan stomach of the alien squid, Gaia continued to fight for her people, raking Tkhaluuljin with sharp branches and rocks and inciting her surviving followers to use their magic against the entity. Gaia and those that lived upon her could not save themselves, but they did destroy their captor. After a long drawn-out fight, Tkhaluuljin was so injured that it could no longer control its flight. It plummeted into the ocean, killing itself, Gaia, and her charges in the process.

• Sign. The hair on your face, head, and body takes on a greenish hue, as do the eyes and skin.
• Minor Arcana: Earth's Voice. You can understand, but not speak, all spoken languages.
• Minor Arcana: Earthsense. You gain tremorsense out to 15 feet.
• Major Arcana: Planetary Healing. Once per short or long rest as an action, you may expend an aetherflow die to start a regeneration effect on your body for 1 minute or until you lose concentration (as if concentrating on a spell). When you use this ability and at the start of each of your turns, you regain hit points equal to one roll of your aetherflow die.

Karsus, Hubris in the Blood

Karsus was a potent mortal spellcaster who attempted to steal the powers of a deity that had jurisdiction over magic. He succeeded, but realized too late that his mortal frame and soul could not contain the power. He died, and his soul remained tied to the Material Plane for ages, never becoming a petitioner. Some claim that part of it somehow still lingers there. With no planar home and no deity who would claim him, Karsus became a vestige.

• Sign. You bleed more than normal from wounds. Even a small scratch releases a sanguine flood. This effect does not deal extra damage.
• Minor Arcana: Karsus's Senses. You can cast Detect Magic once per short or long rest without expending a spell slot.
• Minor Arcana: Karsus's Will. You may attune to items that require attunement by a Wizard.
• Major Arcana: Karsus's Touch. Once per turn as an action or attack, you may expend one aetherflow die to strike with spell-breaking magic. If you do, make a melee spell attack against one creature within 5 feet. On a hit, the creature takes X force damage, and you select up to one spell effect on the target. Make a spellcasting ability check. The DC equals 10 + the spell's level. On a successful check, the spell ends.

Marchosias, King of Killers

In life, Marchosias was a human who brought death to others. His favorite targets were other assassins and murderers, but this choice of foes had nothing to do with morals. Despicably evil, Marchosias was obsessed with improving his skill as a killer, and ending the lives of other professional slayers seemed the best challenge.

When Marchosias met his death, his soul traveled to the Nine Hells. The devils gleefully accepted his powerful spirit, but others took note of his arrival and were not pleased. The spirits of hundreds of thugs, slaughterers, executioners, and assassins banded together and rebelled against their devilish captors—intending not to escape or take control, but to attack Marchosias. Although the devils were loath to allow such lawlessness, they let the souls of the damned fight it out, thinking to step in and punish all the spirits when the battle was over. Marchosias fought well, but he could not prevail against so many foes at once, and he fell under the onslaught. When the devils pulled back the attackers, nothing was left—Marchosias's soul had been torn to pieces.

• Sign. The pupils of your eyes glow with a red-orange light.
• Minor Arcana: Death Attack. Once per short or long rest, when you reduce an enemy creature to 0 hit points, you can restore 1 aetherflow die.
• Minor Arcana: Silent and Sure. You are proficient in the Stealth skill.
• Major Arcana: Smoke Attack. Once per turn as an action or attack, you may expend one aetherflow die to assume the form of a smoke cloud. You can immediately move up to 15 feet in any direction, and your movement doesn't provoke opportunity attacks. You then reassume your original form and can make a melee spell attack with advantage against one creature within 5 feet. On a hit, the creature takes X force damage.

Orthos, Sovereign of the Howling Dark

Orthos might well be the original vestige. Pact magic texts always mention this entity, and persistent explorers can find its seal represented in art or architecture on most planes, as well as in the ruins of many ancient civilizations. Binder scholars have a thousand theories about Orthos's origins, but none is more than mere supposition. All agree that Orthos is inestimably old, and it has long since shed whatever form and persona it might once have had, becoming an alien and distant being. In deference to its great age and the hallmark of its appearance, binder scholars have dubbed Orthos the Sovereign of the Howling Dark.

• Sign. You always seem to be buffeted by a breeze that no one else can feel, even when you're indoors. The eerie wind makes no noise, but it tousles your hair and belongings, frequently changing direction.

• Minor Arcana: Displacement. Once per short or long rest, you may cast Blur without expending a spell slot.
• Minor Arcana: Whispering Wind. You can cast the Message cantrip at will.
• Major Arcana: Whirlwind Breath. Once per turn as an action or attack, you may expend one aetherflow die to exhale a forceful gale. Each creature in a 15 cone originating from you must make a Dexterity saving throw. On a failure, a creature takes X bludgeoning damage and is knocked prone. On a success, they take half as much damage and are not knocked prone.

Savnok, the Instigator

Savnok served Hextor and Heironeous before the two half-brothers came to blows. The gods were charged with guarding their mother's arms and armor while she met with her lovers. Both Hextor and Heironeous were awed and tempted by their mother's implements of war, but neither son dared disobey his mother.

Savnok convinced Hextor and Heironeous into letting him guard their mother's armory for them. But once his gaze fell on the goddess's armor, Savnok could not resist donning it. Just touching the metal made him drunk with power. After putting it on, he knew he could never take it off, so he fled the godly realms with the divine armor. Shocked at his betrayal and horrified by their own failure to perform their duties, Hextor and Heironeous appeared before Savnok and ordered him to relinquish their mother's armor. Their former servant responded by attacking, and although he could not harm them, neither could they harm him.

Heironeous flew into the sky and tore thunderbolts from the clouds to hurl at Savnok, but Hextor, realizing that they needed deific weapons to defeat the armor, fled back to his mother's armory. There, he found a mighty bow. Hextor barely had the strength to draw his mother's bow, but draw it he did. With each arrow he fired, a dozen missiles streaked down to strike Savnok. Though the arrows had little power behind them, they did pierce the armor, and Savnok slowly bled to death from dozens of small wounds.

• Sign. A piece of an arrow appears under your skin somewhere on your body. It looks as though your skin has healed over a broken-off arrow that had previously wounded you.
• Minor Arcana: Deific Armor. While you are wearing light or medium armor, bludgeoning and slashing damage that you take is reduced by 3.
• Minor Arcana: Call Armor. You can don and doff armor as an action.
• Major Arcana: Pincushion. If an ally within 30 feet that you can see would be damaged by a ranged weapon attack, you may use your reaction and expend an aetherflow die to redirect that missile to you instead. If you do, the attack automatically hits you, and you may reduce the damage dealt by the attack by one roll of your aetherflow die plus your proficiency bonus.

• Arithmetician, Convoker, and Theurgist are inspired by the arithmetician, Summoner, and Scholar from Final Fantasy XIV.
• Binder is my interpretation of the Binder class from DnD 3.5.
• Vestige lore and information are taken from the following website: Link
• Waterstains are supplied by Jared Ondricek (/u/flamableconcrete): Link
• Vitruvian Aeon by Andromatonrecursion
• Unbound by Todd Lockwood
• Teferi's Tutelage by Ilse Gort
• FFXIV Summoner Alphinaud Leveilleur & Bahamut Fanart by Bom Yeon
• FFXIV Summoner by Akihiko Yoshida; Square Enix
• しょうりょう の 魔法師 by pingffy*
• Airy by Akihiko Yoshisa; Square Enix
• Fairy ring by LiigaKlavina
• Binder; Wizard of the Coast
• Emergent by Chaos Fissure
• Spellbook by Candra
• Void Dragon by Peter Lin