Is There Anything Like That in Mathematics?

I was just wondering, is there anything like that maybe in pure math where if one were to solve it or do something to it, I guess some property of math will be fulfilled forever? So in C star algebra, what is that one thing or is there that one thing? Well, there's this thing called the classification program. You want to understand when two of them are fundamentally the same. But then you basically get to come up with the criteria for what it means to be in one of these baskets.