I have a weird long-shot idea for GPT to make a new discovery in physics: Ask it to find a mathematical relationship between some combination of the fundamental physical constants[1]. If it finds (for example) a formula that relates electron mass, Bohr radius, and speed of light to a high degree of precision, that might indicate an area of physics to explore further if those constants were thought to be independent.
[1] https://en.wikipedia.org/wiki/List_of_physical_constants
My dream is that powerful agents trawl through all the research papers looking for diamonds in the rough.
They evaluate papers that look interesting and should be looked at more deeply. Then, research ideas as much as they can.
Then flag for human review the real possible breakthroughs.
The Bohr radius is the result of a simple classical physics calculation (a common exercise for undergraduates in their first year). It depends only on the electron mass and the fine structure constant which is the strength of the electromagnetic interaction. In the SI system, the speed of light has a fixed value which defines the unit of length.
There are known mathematical relationships between almost all fundamental physical constants? In particular, in your example, Bohr radius is calculated from electron mass and the speed of light in vacuum... I don't think this path is as promising as it sounds.
"Please derive and unify all of quantum mechanics and general relativity starting only with the Fine Structure Constant."
;)