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Suppose I have a complex pulse in time $\sim{S}(t)$. This pulse represents an electric field, therefore $\sim{S} = |S|e^{i\varphi(t)}e^{i\omega_ct}$ where $\varphi$ is the temporal phase of the pulse (which in this case I know to be a constant), and $\omega_c$ being the pulse center frequency.

If I can measure $|S|$, then given that the spectral phase is a constant, all that is required to know is $\omega_c$ to fully reconstuct $S$. Is it possible to recover the center frequency from the FFT of the envelope of S?

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No, it's not possible. You just need to look at your formula to see that $|S|$ is completely independent of $\omega_c$.

Other than your formula: you know that complex sinusoids $e^{ix},\,x\in\mathbb R$ are always $|·|= 1$! So, the actual oscillation contributes no change to your envelope – if your modulating signal is constant-magnitude, you have a constant-envelope modulation.

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