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Quadratum is a unique world introduced in Kingdom Hearts III. Unlike other worlds before, Quadratum is located in a plane of existence not present among the realms in the series, described to be a world of neither light, dark or in-between, but rather on the other side of existence.

Setting

Modeled after real-life Shibuya, Quadratum is a large expansive city with massive skyscrapers. While it is implied that the city is populated by others excluding the known characters, no one has been seen inhabiting the world, as the roads around the city are littered with abandoned vehicles. A prominent location shown in Quadratum is a large crosswalk at the base of a building with the numbers "104" on the front.

As seen in Verum Rex: Beat of Lead, Quadratum has water access on the city outskirts, and features a sizable oil platform where the minigame is set on.

Story

Kingdom Hearts III

Quadratum is the main setting of Verum Rex, a video game originating from the world Toy Box.

In the secret ending "Yozora", both Riku and Sora wander the streets of Quadratum. While Riku walks around the city, he is closely observed by Yozora atop a skyscraper. Elsewhere in the city, the Master of Masters stands atop another skyscraper and frames his hands to form a heart over the moon, alluding to Kingdom Hearts.

Kingdom Hearts III Re Mind

After Master Xehanort's ultimate defeat, Riku begins to have recurring dreams of him being in Quadratum while searching for Sora, who was lost after misusing the power of waking to save Kairi. While wandering the streets of the city, Riku felt as though he was being watched by someone. The Fairy Godmother is then brought in by Merlin and Yen Sid to meet with Riku to investigate, and determines that him and his dreams are one of three keys needed to save Sora.

Meanwhile, Sora awakens in a night version of The Final World and encounters Yozora. After a brief talk on the reality of the situation and their identities, he challenges Sora into a fight and warps them to the 104 building’s rooftop in Quadratum.

If he wins, he crystallizes Sora and vows to save him, but if he loses, he calmly accepts defeat, stating his powers aren't needed yet, and vanishes. Regardless, Yozora wakes up in the backseat of a car, accompanied by a driver who asks if what he had seen was impressive.

Kingdom Hearts: Melody of Memory

After Kairi awakens from her year long sleep, she recalls what she had learned from her memories of her past, revealing that Xehanort had previously mentioned of a world belonging to neither light or darkness, but to "the other side"; Ansem the Wise hypothesizes that the world was not a part of their reality, calling it an "unreality" and deems it to be fictional. Once Riku and later the Fairy Godmother arrive, Kairi explains that Xehanort's message was a clue to Sora's location. The Fairy Godmother reveals that they now have two of the three keys needed to find Sora; Riku's dreams and Kairi's memories respectively.

To locate the final key, the Fairy Godmother warps Kairi and Riku with her to The Final World, where they come into contact with a Nameless Star who previously interacted with Sora. The Fairy Godmother explains that the star originates from the world on the other side, and as such was the final key. When questioned on what she knows of a place described as a large city with skyscrapers, the star namedrops Quadratum, and the group concludes that it was where Sora was. Using the power of waking on the star, Riku forms a portal to Quadratum and travels through it.

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