= Solution
The required <energy> increment is 9.9 TeV. If the usable longitudinal <gradient> is $\eta E_0$, with $E_0=m_ec\omega_p/e$, the channel length is
$$
\boxed{L_p=\frac{9.9\ \mathrm{TeV}}{\eta m_ec\omega_p}.}
$$
Under the \i[illustrative] preceding $\sigma_{zf}=20\ \mu$m and $k_p\sigma_{zf}=1$ choices, $E_0\simeq25.6$ GV m$^{-1}$ and $L_p\simeq390/\eta$ m. \b[TODO — a unique numerical length inherits the missing compression/<density> specification and additionally requires a usable wake <amplitude>/loading factor $\eta$ or driver information.] A driver <energy> per <proton> alone does not fix its bunch population or accelerating <gradient>.
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