= Solution
Use the leading-order <parton model>: scatter incoherently from an approximately on-shell collinear <quark> carrying nucleon <momentum> fraction $x$, neglect <quark> and <lepton> <masses>, use an unpolarized target, and neglect nuclear binding and higher-twist corrections. The condition $Q^2\gg m_p^2$ suppresses target-mass effects, while $Q^2\ll M_W^2$ permits the local <Fermi interaction> instead of the full W propagator. CKM-weighted accessible final flavours are approximated by their leading light-quark sums. The struck-quark <centre-of-momentum frame>[centre-of-mass] <energy> is $\widehat s\simeq xs$, with $s=(k+p)^2\simeq2m_NE_\nu$.
A <neutrino> scatters through $\nu d\to\mu^-u$ or $\nu s\to\mu^-c$, and through the corresponding crossed <antiquark> channels such as $\nu\bar u\to\mu^-\bar d$. The chiral <weak charged current>[charged current] makes the <massless particle>[massless] <quark> contribution independent of $y$ and the <antiquark> contribution proportional to $(1-y)^2$. The scattering probability is weighted by the <parton distribution functions>; the explicit factor $x$ comes from $\widehat s=xs$. In the usual convention $q(x)dx$ is a number probability density, while $xq(x)$ is a <momentum> density. Approximate <Bjorken scaling> leaves mainly $x$ dependence; <Quantum chromodynamics>[QCD] evolution adds a logarithmic $Q^2$ dependence.
\b[The printed proton-density formula is not an isoscalar formula in the standard PDF convention.] For a <proton>, neglecting incoming charm and taking accessible CKM sums as unity, the conventional leading-order result is
$$
\frac{d^2\sigma^{\nu p}}{dxdy}=\frac{G_F^2s}{\pi}\,x\left[d_p+s_p+(1-y)^2\bar u_p\right].
$$
For an <isoscalar target> per nucleon, average <proton> and <neutron> and use <charge symmetry of nucleon parton distributions>: $d_n=u_p$, $\bar u_n=\bar d_p$, $s_n=s_p$. This gives
$$
\boxed{\frac{d^2\sigma^{\nu N}}{dxdy}=\frac{G_F^2s}{2\pi}\,x
\left[u_p+d_p+2s_p+(1-y)^2(\bar u_p+\bar d_p)\right].}
$$
The standard <normalization> follows equivalently by inserting $F_2^{\nu p}=2x(d+s+\bar u)$ and $xF_3^{\nu p}=2x(d+s-\bar u)$ into the <weak charged current>[charged-current] structure-function formula.<sources/structure functions by Particle Data Group (2024)>[Section 18.1, Eqs. (18.8) and (18.19): <weak charged current>[charged-current] <normalization> and <proton> <quark> combinations]{ref}
A concrete discrepancy remains even with a simple valence model: take $u_p=2h(x)$, $d_p=h(x)$ and all sea densities zero. The correctly averaged isoscalar bracket is $3h(x)$, while the printed bracket is only $h(x)$ with the same prefactor. Thus the intended <helicity> and $x$ dependence can be explained, but the literal printed isoscalar claim cannot be derived. If the symbols were instead defined as <momentum> densities, the extra explicit $x$ would also need removal.
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