Does the distinction between a prior analytic & synthetic truths apply to quasi-empirical propositions in math or logic?

I'm not sure I understand the background history. I notice the statement "a set is entirely defined by its members" is a priori analytic, however the proposition "The universe is a fluctuation in a quantum vacuum" isn't easily distinguishable. I guess a simple comparison is "a constant set is entirely its elements & data members." The entire argument & facts I'm alluding to are in the book A Universe From Nothing. Physicist Lawrence Krauss's cosmological contextualization helps explain how the universe is understood by reliable experts. I can't help but point out the book's arguments and interests don't answer the other relevant philosophical question of why the universe is ontological & natural.

But digress. To me this quantum vacuum proposition includes a mathematical realist position which could meet the criteria for a priori truths. but on the subject of definitions, I can read the proposition through a lens of an empirical model's extensional meaning that measures a 'fluctuation' as primordial background noise. I can also interpret the intensional meaning simply in rule-like words and their connotations of physical phenomena. Altogether the statement recognizes a systems patterning with regularity. I guess the latter is a closer construal of meaningful scientific exploration. But it's not exactly a priori. So I'm not sure how to judge the strengths of mathematical realism or whether it supports an a priori or a posterior distinction.