Return to Alyssa and talk to her five times to hand over Graviton Cores Alpha, Beta, Gamma, Delta and Epsilon. After Epsilon is given, a scene plays, in which you get the Vagabond Artefact that activates the shiny gate along the east side.
PARADOX ENDING - After you beat the game, visit the Serendipity timeline. Enter the leftmost building and talk to the girl dressed as chocobo to unlock Paradox Scope fragment skill. MAKE SURE TO TURN IT ON! Return to AF4XX Academia and do Time Reversal. Replay the map up to the point where you give Alyssa the Graviton Cores. DO NOT SKIP THE EVENTS DURING THIS TIME. Eventually you’re offered a Live Trigger choice! Press Triangle for paradox ending (05/08).
After getting the Artefact, talk to Noel who’s standing near the door. Exit the building completely and grab a chocobo, ride it south, turn east and up the ramp. While on the ramp, you’ll spot the activated shiny gate on the northeast platform of Central Bridge.
Dismount the chocster, examine the shiny gate and use the Vagabond Artefact for a Live Trigger: (Caius could be waiting on the other side of the gate. Are you ready to face him? / Triangle), which in turn unlocks… Hmm, things seem to go very wrong for both Serah and Noel…
You have three AF??? nodes, and the one you need to go to is the middle one.