The Rubicon is a district in The World Ends With You. It is the second part of the Shibuya River and connects the Trail of the Sinner with the Trail of the Bygone over a small river with a bridge.
The Rubicon appears in Game (Week 3, Day 7). Neku Sakuraba and Beat confront a taboo Sho Minamimoto who declares his intent to become Composer. It is where Mitsuki Konishi reveals herself from Beat's shadow, and makes a deal with Minamimoto to let down the Rubicon's wall if, when he becomes Composer, he makes her Conductor. She fights Neku and Beat in her Noise form Tigris Cantus but is defeated and erased. After defeating Tigris Cantus, a bridge forms over the river between Rubicon and Trail of the Bygone. When Neku and Beat cross it, Rubicon is sealed off.
The Rubicon is most likely named after the Rubicon river in Rome, Italy. Famously, Julius Caesar's army crossing this river in January 49 BC marked the beginning of the civil war that would result in the Roman Republic being reorganized into the Roman Empire and Caesar becoming dictator of Rome. The phrase "crossing the Rubicon" refers to a point of no return. The Rubicon in The World Ends with You functions as a point of no return - once the player enters Trail of the Bygone in Game they are barred from entering the rest of Shibuya until they complete the game. Minamimoto could be seen as a reference to Caesar because he has to move through the Rubicon to fight and usurp the Composer, however he is crushed in Trail of the Bygone.