Imaginary Number District

The Imaginary Number District (虚数学区), full designation Imaginary Number District–Five Elements Institution, also known as the City of Shimmers (陽炎の街), is the mysterious realm that exists in the same location as Academy City, formed through the millions of AIM diffusion fields of the espers that reside in the city.

Etymology
The naming of the Imaginary Number District is maybe derived from the fact that it is not part of the 23 school districts of Academy City. The imaginary unit, usually denoted as i, is an elementary unit used in complex number theory and possesses the following essential property: i2 = −1.

Mechanics
As stated, the Imaginary Number District was formed out of the AIM diffusion fields released by the 2.3 million espers living in Academy City. Likewise, Kazakiri Hyouka is also composed of the espers' AIM diffusion fields, given consciousness out of the fear of death by the presence of Kamijou Touma's Imagine Breaker in the city. Her existence is a like a keystone to the city, or the source, even though she herself is a part of the system. Appropriately, she is called the Key to the Imaginary Number District (虚数学区の鍵). Being the key apparently allows her to change the denizens of the Imaginary Number District, according to her needs, for example: when asking for directions, any denizen will turn into something that is supposed to help her with directions, like a police officer.

According to the urban legends, the Imaginary Number District existed when the Academy City was first established. It can be assumed that the Imaginary Number District has existed when the first esper was made to emit AIM.

Due to the Imaginary Number District having different set of laws than those who abide to religions, when the Imaginary Number District however is brought forth into the same realm as Academy City, it creates pressure inside the city, making all magicians that use magic inside the city overload and injure themselves.

Appearance
Located in the same location as Academy, the Imaginary Number District, also known as the City of Shimmers, is described as having no shadows, no weight, no air flow. It's apparently so thin that there's no sense of existence, and that when the wind blows, all things that are there are blow away like a candle in the wind, and lets out a grey visual signal.

In the anime adaptation, the Imaginary Number District is shown to have shadows, and is wavy-like, as if the heat distorts the surroundings. It is most likely a reference to it being called the City of Shimmers.

Because of it being an extremely thin existence, Academy City can be viewed from the Imaginary Number District, which tempted Kazakiri Hyouka towards it. In turn, she is often seen as a phantom in Academy City.

Inhabitants
It is unknown what they truly are, but the inhabitants of City of Shimmers can be controlled by Kazakiri Hyouka. They look similar to normal humans, but can transform to whatever Hyouka wills them to be. Moreover, not only their appearances, but their personalities, memories, and actions, to fit the role that Hyouka needs of them. Furthermore, they are described not as puppets, but actual people with lives. Because of this fact, Hyouka wanted to escape the City of Shimmers as she influenced and changed the lives of its denizens.

Relation to Aleister Crowley's plans
The Imaginary Number District is described as being incredibly influential and dangerous to Academy City, and apparently nearly impossible to control, being referred to Aleister Crowley as "...a wild horse that can rip through the entire world." Even Aleister, who knew of a way to control it, required Kamijou Touma and his Imagine Breaker, to introduce fear into the Imaginary Number District, allowing for it to gain consciousness in the form of Kazakiri Hyouka, making it easier for him to control the Imaginary Number District. Moreover, an extremely vague plan Aleister concocted was used in order to create a certain amount of impact on Hyouka, in order for her to become more powerful.

Aleister Crowley has also planned to expand the Imaginary Number District towards the whole world, using the Misaka clones that were dispersed in the wake of the abolishment of Level 6 Shift's for "adjustments," allowing it to be covered by their AIM diffusion field. What's more, Aleister has made use of the remaining clones in the city and Last Order, to force them, whenever he sees fit, to force Kazakiri Hyouka to manifest her angelic form, as well as the Imaginary Number District.

It is unknown as to what the true extent Aleister's purpose for the Imaginary Number District. Whether to use it as a way to kill all magicians, or to bring forth a new world that are not bound by any current religious laws such as Christianity, Kabbalah, etc. However, it is noted by Tsuchimikado Motoharu that the Imaginary Number District is similar to heaven, and refers to Aleister creating an artificial one.

Politics
Due to the dangerous nature of the Imaginary Number District to magicians, the truth about it is deliberately hidden from them. The magic side believes that the Five Elements Institution refers to a building. And indeed, the Five Elements Institution was once referenced as the one controlling Academy City, with Stiyl Magnus and Kanzaki Kaori having to get permission from them to enter the city, and assumed that they were deliberately hiding information about Touma, though seemed to have been retconned in the anime adaptation, with the Board of Directors taking its place.

Urban Legends
Due to its mysterious nature, several urban legends have cropped up regarding the true nature of the Imaginary Number District, making it one of the seven mysteries of Academy City. Even Touma, who has lost his memories is quite aware of the rumors of the Imaginary Number District, implying that the rumors are quite pervasive in Academy City.

The First Research Organization Rumors
The Imaginary Number District was apparently referred to as the first research organization, also known as Primary Knowledge (プライマリー＝ノーリッジ), established in Academy City when it was first founded. There are several variations to its fate:
 * It has been destroyed decades ago.
 * It is hidden deep underground.
 * It has been seen every day, but none has realized it because it has been disguised as an ordinary-looking school.
 * An ability or a Imaginary Technology is used to warp space around it in order for it to be hidden.

Imaginary Technology
Imaginary Technology (架空技術) is the mysterious technology that apparently originates from the Imaginary Number District. The technology is apparently so advanced that today's technology cannot even come close in comparison. According to urban legends, the technology is used to hide the Imaginary Number District. There are also rumors that Imaginary Technology's AI is secretly controlling the Board of Directors from behind the scenes, as well as all of the ethics, military, and economies of the world via the internet.

Moreover, the Imaginary Number District's Imaginary Technology, is supposedly the source of the Clone Dolly Workshop, the origin of the Tree Diagram's silicorundum synapses, and the technology that allows for a person to gain eternal youth, with the result being Tsukuyomi Komoe.

Other Rumors

 * Hound Dogs have abducted someone that was secretly working to search for the Imaginary Number District in order to torture the person for information.
 * At 4:44 p.m. one shouldn't cross between the school district boundaries. Otherwise one might wander into the Imaginary Number School District.

Post-Handcuffs Arc
On December 29th, another conflict occurred around the Vanishing Tunnel, which had fallen under the control of Kihara Hasuu, who had managed to turn himself into an artificial ghost, usurped Frillsand #G's control over the space and taken Risako captive to use for his own plans, with multiple parties (including Kamijou Touma and Hamazura Shiage) entering the tunnel to rescue the girl. After Risako was freed from his manipulations and asserted her desire with prompting from Qliphah Puzzle 545, Hasuu was ejected from the Vanishing Tunnel into the Imaginary Number District, with the space having returned to its original master. He was then finished off by Fuse Kazakiri, who had witnessed the events of Operation Handcuffs, and stepped in to deal the final blow in place of those who couldn't, striking down the lingering evil with the late Vivana Oniguma's yarai.