= Solution
{c}
Use the complex <electric field> convention $e^{-i\omega t}$ and take the locked <wave amplitudes> $a(\omega_p)$ to be real and nonnegative after extracting their common <phase>. Linear <superposition> gives
$$
E(t)=\sum_p a(\omega_p)e^{-i\omega_pt+i\phi_0}
=e^{-i(\omega_{\rm ce}t-\phi_0)}\sum_p a(\omega_p)e^{-ip\Delta\omega t}.
$$
The leading factor has unit modulus, so it does not change the <light intensity>. The <triangle inequality> bounds the remaining modulus by $\sum_pa(\omega_p)$. Consecutive occupied <normal modes> attain this bound together at
$$
\boxed{t_m=mT,\qquad T=\frac{2\pi}{\Delta\omega},\qquad m=1,2,\ldots.}
$$
There is also a peak at $t=0$. The <electric field> obeys $E(t+T)=e^{-i\omega_{\rm ce}T}E(t)$, although the <light intensity> is periodic.
The <carrier-envelope phase> specifies where the carrier oscillation lies relative to the maximum of an <optical pulse>'s envelope. With $s=t-t_m$ and $E=F(s)e^{-i\omega_0s+i\psi_m}$, the signed complex-field <phase> at the envelope maximum is $\psi_m$. Consequently,
$$
\boxed{\psi_{m+1}-\psi_m=-\omega_{\rm ce}T
=-\frac{2\pi\omega_{\rm ce}}{\Delta\omega}\pmod{2\pi}.}
$$
If instead the real <electric field> is written $F(s)\cos(\omega_0s+\phi_{{\rm CE},m})$, then $\phi_{{\rm CE},m}=-\psi_m$ and the conventionally quoted slip is $\phi_{\rm slip}=+2\pi\omega_{\rm ce}/\Delta\omega$. These are the same physical shift with opposite phase definitions.
The <carrier-envelope phase> matters for few-cycle <optical pulses>, when the largest instantaneous <electric field> affects strong-field dynamics, for <high harmonic generation>, and for stabilizing an <optical frequency comb>. An ordinary cycle-averaged <light intensity> measurement does not determine it.
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