Alright so I'm creating a realm and a city state let me make sure I have the math correct the realm will have a total number of families =

31,300

this gives me a base urban population of 3,130 which will be a large town class IV , it will be a city state so I decide to bump up the urban population colum by 2 and give it a population of 9,390 families which would be a class IV city in the largest urban settlement column, but I can take the that column and but it up by two steps to a large city and put my city state population at a total of 5,000 families. With that I subtract 5,000 from my total number of families 9,390 which leaves me with a population of 4,390 families to spread around the other urban settlements. Am I understanding this correctly?

this is roughly how I understand it:

Urban population is taken from Realm population, so when it grows, the amount not in it shrinks. Similarly, the Largest settlement's growth comes at the expense of the other population in the realm's other cities.

Right I think I'm on the same track as you I'm increasing the density of the population of one city which will leave a smaller population for the other cites/towns/villages. That's if I'm understanding the example in ACKS core page 232.

yeah, that's how i've always thought of it as well. I think you are correct.

Exactly correct on all fronts!