Path

"So you can't help but think that maybe the Subjects of Ymir are all connected by something invisible. One successor said they saw paths. Paths invisible to the eyes. The blood and bones that formed a Titan were sent to them through those paths. At times, those paths even carried memories or the will to act. And all of those paths cross at a single coordinate. In other words, that... is the Founding Titan. All Titans... all Subjects of Ymir are connected to that coordinate. Through paths that transcend physical space."

- Eren Kruger explains the secrets of Titans to Grisha Yeager

Paths (道) are channels that connect all Titans and Subjects of Ymir together.

Overview
The idea of paths came from noticing that the Nine Titans are sent to a random baby among the Subjects of Ymir, regardless of distance or blood relations, if they die without being eaten. This implies that the Subjects of Ymir are all connected by something invisible, and what connects them are called paths.

Paths serve as channels for the transportation of several things, such as memories and wills from other Subjects of Ymir, flesh and bones that make up a Titan's body, commands from a Titan's scream, and the power of the Titans.

Paths seem to be strengthened by blood relations: memories are transferred far easier between family members. Karl Fritz's vow renouncing war is only passed down to his own family. The Ackerman clan, who are also connected to each other by a path, gain the battle experiences of all past Ackermans when they awaken.

Paths transcend physical space, and they appear to transcend time as well, for Eren Kruger had "memories" of Mikasa Ackerman and Armin Arlert before they were even born. Grisha Yeager likens them to magic.

World of the path
Although paths are invisible in the physical world, they can be seen by entering the world of the paths (道の世界). There, white soil covers the ground and paths manifest as trails of light in the sky, similar in appearance to an aurora. Time is irrelevant and conflicting there. Zeke Yeager says that it felt like he was there for years, but it was also all over in an instant.

Ymir Fritz inhabits the land as a little girl, often carrying a pail of water. She is capable of creating anything out of the earth. She is the cause of Titan transformation and regeneration, as she is the one who molds their bodies whenever the power of the Titans is invoked. Those with royal blood, her descendants, are able to command Ymir Fritz; Zeke considers her a slave with no free will of her own.

Ymir's death created the world of the paths after gaining the power of the Titans. While dying, Zeke Yeager also enters the realm when a nearby Titan shoves his body into its belly. Eren Yeager enters this land when his decapitated head touches Zeke. Zeke speculates that those who inherit the Founding Titan enter the land in order to command Ymir Fritz.

Coordinate
When paths cross, the point is known as a "coordinate" (座標). The Founding Titan is the Coordinate where the paths of all Subjects of Ymir and Titans cross. It has the ability to alter the memories of Subjects of Ymir, change the body composition of Subjects of Ymir, transform Subjects of Ymir into Titans (even Colossus Titans), and command all other Titans. According to Tom Ksaver, the Subjects of Ymir are like a part of the Founding Titan's body, no matter where they are. In the unknown land, the Coordinate is manifested as a giant pillar of light that branches out like a tree.

Other Coordinates
While the term "Coordinate" is most commonly used for the Founding Titan's Coordinate, Zeke Yeager also uses it to describe his own abilities with the Beast Titan. If Zeke's spinal fluid enters the body of a Subject of Ymir, that Subject of Ymir becomes connected to Zeke's Beast Titan by a path and is marked with a "coordinate." Zeke's Beast Titan can act like a faux Founding Titan to those connected to him: He can transform them into Titans, and he can give those Titans commands.

History
At some point, Ymir Fritz came to inhabit the unknown land where paths are manifested. Throughout history, she molded the bodies of the Titans and sent them along the paths.

According to Eren Yeager and Eren Kruger, one of their predecessors saw paths.

During the year 845, after Ymir devoured Marcel Galliard and gained the power of the Titans, she awoke in the unknown land. When she saw the paths in the sky, she called them "freedom" spread out before her. She vowed to live a life without regrets afterward.

War for Paradis arc
As Zeke Yeager is dying from the wounds inflicted by a Thunder Spear explosion, he has a vision of Ymir, in tattered clothes, in a world full of sand. A Titan approaches him and opens its abdomen to put his body inside. Zeke awakens inside the unknown land to Ymir Fritz rebuilding his body using the surrounding earth, and he sees trails of light in the sky above. Zeke later comes out of the Titan, fully recovered, and speculates that what he just saw are paths.

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