Quadratum

Quadratum, also known as Shibuya (シブヤ), is a world introduced in Kingdom Hearts III. According to Ansem the Wise, unlike other worlds in the Kingdom Hearts universe, it is not a world of light or dark, but a world of "fiction", that exists on the other side.

It is a modern, urban world based on the real-life city of Tokyo, primarily its Shibuya district with some distinct variations and differences. Quadratum is the main setting of Verum Rex, a fictional video game that exists in Toy Box.

Settings and Areas
Terra-Xehanort describes that the worlds of the other reality, including Quadratum, are not part of either light or darkness. Ansem the Wise infers that it is an unreal universe, a world of "fiction", beyond their comprehension.

The only part of Quadratum extensively glimpsed so far is a crosswalk located in front of a building with "104" displayed. Quadratum is populated by clothing and tech shops and retailers. In the distance, many high-rise buildings and skyscrapers extend out towards the horizon. Curiously, despite the presence of many automobiles, the city appears to be almost entirely abandoned and desolate.

Some distance away, a large twin building with red aviation lights can be seen.

The minigame Verum Rex: Beat of Lead takes place on a sort of oil-drilling platform located some miles offshore. The platform acts as a battle arena against numerous Gigas. Scattered throughout the platform are spherical fuel containers, and in the center an electric energy field that discharged in continued intervals. Above the energy trap is a suspended glass platform that can be destroyed. In the distance, the city along with a bridge and a golden-lit tower can be seen.

Before Kingdom Hearts χ
Sometime before the Keyblade War, the Master of Masters and discussed the existence - or lack thereof - of a fictional universe that is found outside reality. The Master admitted that the discovery of such an inconceivable reality caught even him off-guard, and compared it to something one step beyond the afterlife - worlds where neither darkness nor light could exist, where neither eyes nor voice could reach.

Before Kingdom Hearts III
Yozora and the original persona of the Nameless Star were inhabitants of Quadratum.

The Master of Masters would later come to reside in this world, as neither light nor darkness would be able to follow him, while he waits for his plans to unfold.

During his time as an apprentice under Ansem, Terra-Xehanort researched the worlds on the other side of reality, where the world of Quadratum is also present.

Kingdom Hearts III
Quadratum is first seen when Rex and Hamm watch a Verum Rex video game and action figure trailer upon first entering the Toy Box.

Some time later, Sora and Woody's gang find Rex in the game store looking for Verum Rex merchandise, convinced that Sora is its hero Yozora and looking to prove it. As they are about to leave, Young Xehanort appears and has a Marionette possess Buzz to experiment with the toys' hearts. Young Xehanort then transports Sora into Verum Rex's offshore platform arena to battle the Gigas, but he's able to escape and return to Galaxy Toys.

After the defeat of Master Xehanort, Riku starts having a dream in which he and Sora awaken separated and disoriented in Quadratum. While Sora finds himself at the crosswalk staring at a building with the number "104" displayed, Riku walks past a large building, being watched by Yozora from above. Meanwhile, The Master of Masters watches the moon from a faraway building and compares it to Kingdom Hearts.

Kingdom Hearts III Re Mind
One year after Xehanort's ultimate defeat, Sora wakes up in a nighttime version of The Final World, and desperately calls out to whoever he can. He encounters Yozora, whom he learned more about from a girl he met before, and is surprised to discover that Yozora is actually a real person. Yozora appears to know exactly who Sora is and claims that he is tasked to "save" him, but suddenly questions whether that is truly Sora at all. To Sora's shock, he summons his crossbow and sword and the location suddenly changes to the "104" building, where a fearsome battle ensues.

After Sora defeats Yozora, Yozora fades away into light and Sora ends up in the daytime version of The Final World. Meanwhile, Yozora is woken up from his slumber in the back seat of a car by his driver, who asks him, "Impressive, is it not?" Sora and Yozora both wonder whether or not their encounter was real or just a weird dream. Sora then states that none of this makes any sense to him.

Kingdom Hearts Melody of Memory
The name Quadratum is revealed when the Nameless Star in The Final World surmises it must be what Riku is talking about, after he explains his dream about a large city with tall buildings. Using the power of waking, Riku opens a to travel there.

Kingdom Hearts 3D: Dream Drop Distance
Neku Sakuraba, Joshua, Daisukenojo Bito, Rhyme, and Shiki all hail from Shibuya. However, Neku's Shibuya and Yozora's are two completely separate worlds.

Kingdom Hearts IV
In the Kingdom Hearts IV announcement trailer, Strelitzia‎ greets Sora as he wakes up and explains to him that he is in Quadratum, describing it as similar to an "afterlife" to the both of them.‎ As a ball of darkness form in the street and a gigantic creature appear from it, Sora rush to the street to fight it, all while being observed by two beings in black coat.

Character Design
Sora retains his normal outfit, but his clothes are more detailed and textured, along with having a darker shade.