Mathematical relationship between phase modulated (PM) signal and the complex phase/magnitude of the frequency component at the carrier frequency?
What is the mathematical relationship between phase modulated (PM) signal and the complex phase/magnitude of the frequency component at the carrier frequency used?

The phase of the carrier wave is your data-carrying signal. It also is the phase in complex equivalent baseband.
So, that's the mathematical relationship: equality.
Magnitude doesn't matter to PM. It might be the same for all signals you consider, but it doesn't have to be; it's simply irrelevant to the information signal.