The Underhive is definitely the low(est) area of a Hive City.
Unfortunatly we haven’t been there yet.
For there being three Hives on Atoma, I dunno,
but “Tertium” could also mean “Third Attempt” ( for the third time - tertium, tertio → Tertium latin english - Google Search) as in:
Maybe the first two just sank into the swamp. 
