The Paddraean Archaeopolis
The sole reason we came here is to collect one specific fragment, and then we're outta here. You start in the camp to the northeast of the map, and your objective lies in The Pass of Paddra, right in the southwest portion of the map in the area with the searchlights. Grab a red chocobo for riding - while these birds behave exactly the same as the yellow ones, red choccies continuously drain your reserves of Gysahl Greens . And they will always run away from you once you dismount.
Also, while near Chocolina's shop at the entrance to The Paddraean Archaeopolis, take a look down from the top of the ramp to find another treasure cube. Toss Mog at it to collect the ' Reminiscence' Gate Seal which enables Time Reversal in AF100 Yaschas Massif.
Now to hunt for the last graviton core's location!
Graviton Core Epsilon
Once you make it to the searchlight area, look at the map. There are three triangular cracks in the ground, and the piece you're looking for is near the largest crack. Mog's bobble also lights up when you're in close vicinity. Sniff for a concealed item, use R1 to reveal it and nab AF100 YASCHAS MASSIF FRAGMENT - Graviton Core Epsilon (01/07) + 500CP !
Five graviton cores have been collected! Return to AF4XX Academia.