This is the data I gathered searching the websites about it.
Each of the 3 Remnants are used to spawn a certain NM in Promivion-Vhazl. You find a ??? and examine it, the text it gives clues you in to which Remnant to use.
This one will spawn the NM that can drop the Recollection of Anxeity. Take that Recollection to Harith and trade with 2000 gil and you will get Stone Gorget.
The Remnant of Satiator will spawn the NM that can drop the Recollection of Animosity. Trade this one to Harith and you'll get the Dia Wand.
The last Remnant "Remnant of Cerberator" will spawn the NM that can drop the Recollection of Suffering. Trade this one to Harith and you'll get the Hamayumi.
EDIT:
Unfortunatly I haven't found any info telling which remnant goes with what text you get from the ??? to spawn the NMs.
Well the only thing I can verify is what this item is not for: You can not trade it to Harith. You can not trade it to the Memory Flux.
There is a lot of speculation about the use of this and the other two remnants from the NMs. I posted on somepage and someone said you can get items from it, the stonskin gorget, the dia wand/staff(not sure which it is), and the Ranger Bow. However the information on how you actually get the items is still uncertain.