Has math ever predicted anything in physics?
I don’t think math makes those kinds of predictions. There are certainly cases, though, where someone worked out certain kinds of mathematics as a theoretical exercise, and then we only later discovered real world applications that the math described.
Mathematics has a very interesting role in theoretical physics.
Let me use as an example the Rydberg equation for calculating the electron energies for the hydrogen atom.
Rydberg observed the spectral lines of hydrogen and found an equation which predicted the various frequencies. This was found with no underlying theoretical basis for choosing the equation.
Many years later Niels Bohr supplied a theory which resulted in that same equation.
This raises the following question: "Does the lack of a theory render the mathematics INCORRECT?" Another question is: "Does the existence of a theory render the mathematics CORRECT?"
I would claim that the answer to both questions is: "No."
My rationale lies in the fact that it would appear that every equation describing the behavior of the physical world is simply an approximation based on various simpifications and assumptions.
For example, so-called "Newtonian physics" offers calculations which would allow one to predict the observed velocity of one object when viewed from the frame of reference of a second, moving object. Would such mathematical calculations be "correct"? The answer is that they are good enough for some circumstances and not for others.
Einstein developed a different mathematical expression for calculating relative velocities. His equation works just as well as Newton's but also is accurate for relative and observed velocities close to the speed of light.
Does this mean Newton's equations are WRONG and Einstein's equations are RIGHT?
The best answer to that question is that "It depends." For many circumstances, Newton is just as good as Einstein. When the day comes that we understand what "dark matter" really is, we just might find Einstein's equations as lacking then as we sometimes find Newton's lacking today.
The point I am trying to make is that it is just possible that mathematics NEVER describes any existing physical system exactly. But for some purposes, the math is really quite good. As for your original question, the Rydberg equation was probably able to predict the frequency of spectral lines that Rydberg was unable to observe.
Yes.
All the time.