This is essentially what you get in String Theory. The problem is that String Theory requires you to define the "shape" of the extra dimensions. Once you do thi

This is essentially what you get in String Theory. The problem is that String Theory requires you to define the "shape" of the extra dimensions. Once you do this, all of the fundamental constants naturally follow. However, there are so many potential shapes that are mathematically valid that this essentially moves the problem from out universe having arbitrary constants to our universe being an arbitrary shape. Unfortunately, it is possible that there is literally to elegant solution to this. For example, consider our solar system. At one point, people tried to explain the distance of the planets from the sun without invoking arbitrary constants. We now know that the distance between the planets and the sun are in largly arbitrary, because we are but one of many solar systems and other solar systems have different distances. Similarly, it is possible (although potentially non-provable and non-falsifiable). That we live in but one of many universes, and that other universes have different constants. If this is true then there is no way to avoid the fact that our universe has some arbitrary constant (even if we obscure it by calling it an arbitrary shape).