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?
This is partially related to the question that I asked on phase noise, but it seems that the complex magnitude of this component is the same for all PM signals using that carrier, just that the phase is rotated. Is it something like the average instantaneous phase difference?
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.