Mathematicians love theorems; experimental scientists love predictions. Can theorems ever make testable predictions? The answer is yes. In this talk we discuss one powerful kind of predictive theorem (classification of representations). For example, from the basic principles of quantum mechanics and the inh erent spherical symmetry of the hydrogen atom it is possible to predict much of the electronic structure of the hydrogen atom. This is not an isolated example; rather, it is a simple case of a wide-ranging method.