In terms of psychology, the core personality is the person from whom the other alters were derived or created. In terms of this manual, this would be the source person for all internally sourced residents. There is no guarantee, in this manual or in psychology, that the core personality exists anymore by the time the system is getting treatment, or that anyone is able to find or identify who their personal source person is.
If integration is sought, usually the idea is to find the core personality, and have everyone re-merge into the core. Sometimes the idea is to have everyone merge into the host personality, who presumably is the one coping with the most of the daily grind, but there are many reasons that either of these might not be feasible.