I think it's very obvious no such answer is even possible in principle. Mathematics has no limits, you can describe anything you like by picking some axioms. Do you want to make sense of the expression 1+1=3? I can find axioms in which this is true.

So, there is no way to start from mathematics and find something that must exist in some way, such as "there can only be 100 types of universe". Any such discovery is contingent upon some arbitrary choice of axioms. You can choose axioms that appeal to some ultimately esthetic sense of elegance or simplicity, and that can explain our universe more or less uniquely, but this doesn't mean that they are right to any extent more than the SM is.