There are 6 cores in total. The ones available through story missions do not count toward this trophy. Cauldron Iota is accessible early in the story and does not have additional requirements. In general, it is best to leave the cauldrons until after you complete the story. One of the cauldrons will require the breathing apparatus from a story mission.
I made videos showing the full walkthroughs, minus most of the fights. Hopefully these videos help. There is no audio commentary, but I did chop them up into chapters and added overlays when explanations are needed.
Cauldron Iota:
Cauldron Mu:
Cauldron Chi:
Cauldron Kappa: