= Solution
In the cold underdense model, $1-v_g/c\simeq n/(2n_c)$ and $\gamma_\phi^2\simeq n_c/n=500$. A relativistic <electron> slips through half an accelerating sinusoidal wavelength over $L_d\simeq\gamma_\phi^2\lambda_p\simeq8.35$ mm. For $E_z=E_0\sin\phi$ and injection at the start of that half-cycle, <integration> over $0<\phi<\pi$ gives
$$
\boxed{\Delta E_{\max}\simeq4\gamma_\phi^2m_ec^2\simeq1.0\ \mathrm{GeV}.}
$$
Injection at the field crest and <acceleration> only to its next zero gives half this value, about 0.51 GeV; this explains a common factor-two convention in dephasing estimates. Multiplying the peak field by the entire dephasing length is instead an <upper bound>, not the sinusoid's actual <integral>. These are ideal wave-amplitude estimates: $a_0<1$ need not excite $E_0$, and finite pump depletion, pulse <evolution> and injection physics may reduce the gain. Without a specified nonlinear wake shape, the data warrant an order-GeV cold-wave upper scale, not a unique precision maximum for an actual experiment.
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