Doppel
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A Doppel or Doppel Witch is a phenomenon caused by the Automatic Purification System in Magia Record.
Typically, a Magical Girl becomes a Witch when their Soul Gem becomes completely black from developing too many impurities. Under the influence of the APS however, the "Doppel" power is invoked instead. When a Soul Gem becomes completely black, a manifestation of a Magical Girl's Witch form is summoned, taking on a variety of forms for each individual girl.
Doppels are very powerful, making them useful in battle, but can come with high risks with overuse and depending on the specific nature of the Doppel. In severe cases of despair, the Doppel can envelop the Magical Girl and trap them in a dream world of their psyche, similar to the case of Rebellion. Even others can be pulled inside this dream labyrinth and the Doppel, or even the complete Witch, must be defeated to break out of it.
Since the Doppel phenomenon uses the Magical Girl's body as a medium to disperse the curse, it is unknown what would happen if a Soul Gem separated from its owner's body becomes black under the influence of the APS, but it can be guessed that it would either shatter, become a Grief Seed, or hatch into a Witch.
From their in-game description, a Doppel's manifestation depends on several factors. For instance, though this is not always the case, older Magical Girls (or potentially, those who have been contracted longer, not necessarily one's actual age) experience a narrower range of emotions than younger Magical Girls, and as such, their Doppels are more likely to appear fused to their master's body rather than as separate entities.
Later in the anime, it is revealed that the formal name of the Doppel is Doppel Witch by Mitama Yakumo. It reflects negative emotions in a magical girl and when a Doppel takes control of a magical girl, they contract Doppel Syndrome.
Exclusive to the anime adaptation's timeline, due to the flawed Doppel System, when a Doppel is summoned, a Magical Girl's face will be covered by a mask to appear completely white with solid black eyes and mouth. This face does not appear on those who gain enough experience summoning their Doppel that they can retain consciousness while using it, but may reappear if their inner Witch gains the upper hand and risks taking control of their body. During moments when the inner Witch and the girl are in agreement, only the left half of the mask will appear.
Doppel Syndrome
In the timeline of the anime, due to the Doppel System being incomplete, overuse of the Doppel by weak, or unexperienced girls leads to it taking a toll on them, making them sickly, their soul gems appear clean on the outside, causing the Doppel to emerge more often and ultimately contracting Doppel Syndrome. The only differences between Doppel Syndrome and becoming a Witch are that Doppel Syndrome victims' Witches appear fused to their bodies and the victims are left in a state akin to a coma. The condition is preventable by regularly being coordinated and it is reversible by using Connect on them and directly entering the victim's psyche to comfort them, thus requiring a much more taxing and intimate approach. While in Doppel Syndrome, their Doppel makes an inner barrier like in the Rebellion story, and the girl will usually fall into a continuous nightmare.
Doppel Fusion
When initially summoned, a subconscious confrontation occurs between the Doppel user and their Witch self, in which the Witch attempts to convince the girl to give in to their inner darkness and give the Witch full control. Usually, the Doppel user will simply reject their inner darkness, and the Doppel will be semi-autonomous; however, by achieving true catharsis, a fusion referred to as a Doppel Version can be achieved. In a Doppel Version, the user is fully in control; how permanent this fusion state is is somewhat unclear, and may vary on a user-to-user basis.