Core memory fragments are fragments of the core memories obtained in the Hall of Memories by catching the NPC with the same name. These items should be placed on an empty plinth in the north-east or north-west of the hall for some Divination experience. When all 6 plinths are filled, the memory bud in the centre of the hall opens and plays a memory at random.
This article on an item has an associated dialogue page. Click here for the transcript.