= Solution
Use the <parton model>, neglect <antiquarks>, binding and target <mass> corrections, <quark masses>, <muon mass>, <QCD scaling violation> and the <W boson> propagator correction. The struck <quark> is effectively free during the short interaction and carries <parton momentum fraction> $x$, giving $\hat s\simeq xs$. The <charged current> reaction is $\bar\nu_\mu u_i\to\mu^+d_j$. Its <helicity amplitude> has the same $(1-y)$ factor as in <1/a>. Define the effective <parton distribution function> by $q(x)=\sum_{i,j}|V_{ij}|^2u_i(x)$, summing only kinematically accessible final <quarks>; for valence-only <protons> and accessible $d,s$ this is approximately $u(x)$. It is not the unweighted sum $u+d$.
$$
\frac{d^2\sigma}{dx\,dy}=\sum_{i,j}u_i(x)|V_{ij}|^2\frac{G_F^2xs}{\pi}(1-y)^2=\frac{G_F^2sx}{\pi}(1-y)^2q(x).
$$
Here $y=p\cdot(k-k')/(p\cdot k)$ is the inclusive <deep inelastic scattering> variable, and $x=Q^2/(2p\cdot(k-k'))$ in the scaling limit. Integrating $0<y<1$ gives $1/3$, hence
$$
\sigma(\bar\nu_\mu p)=\frac{G_F^2s}{3\pi}\int_0^1xq(x)\,dx\simeq\frac{2G_F^2m_pE}{3\pi}\int_0^1xq(x)\,dx\propto E.
$$
The moment measures the <momentum> fraction in the active <quarks>. Approximate linearity is evidence for scattering from pointlike spin-one-half constituents with approximately scale-independent <parton distribution functions>, rather than coherent scattering from an extended <proton>. It is an intermediate-energy prediction: thresholds, experimental <deep inelastic scattering> cuts, <QCD scaling violation> and eventually the finite <W boson> propagator modify it.
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