AOC (intersection-based) efficient 3-way join?

I am attempting to improve performance on a highly normalized database wherein I have the following structure.

entity (identity) org (identity) entity-org relation (many to many) org-entity-owner-relation (many to many + column-informed state) vuln (identity) entity-vuln (identity-identity relation 1)

my issue is with the 3-way identity relationship wherein the relationship between entity-vuln relation intersection becomes a subquery of relation between the unrelated org-to-vuln relationship.

If there is an intersection between entity and vuln and there is a relationship between entity and org then I need to be able to show a list and count of org-entity-vuln relation and org-vuln-entity relation. My issue is with sorting and reindexing.

Is there a way to leverage the database for this effectively or must I do my set operations locally in code?