Solution (source code)

= Solution

Subtracting the <deep-inelastic structure functions> in <2/e> gives
$$
F_2^p-F_2^n=\frac{x}{3}(U-D).
$$
Thus at moderate and large $x$, the charge-weighted <up quark> content of the <proton> exceeds its <down quark> content. In the <valence quark> dominated regime this is consistent with the $uud$ valence structure, compared with $udd$ for the <neutron>. The larger $2/3$ <up quark> <electric charge> makes the <proton> scatter more strongly. This observation alone does not establish $u^p(x)=2d^p(x)$ pointwise: the valence-number sum rules constrain integrals, not every $x$ bin.

If the small-$x$ observation means equality to good \i[relative] accuracy, it supports dominance by approximately charge-symmetric <sea quarks>, with $U\simeq D$ and negligible relative valence excess. <Gluon splitting> creates <quark>-<antiquark> pairs, and the leading <strong interaction> treats light <quark flavors> similarly. It does not follow that every <sea quark> flavor has exactly the same distribution at every $x$.

\b[Logical qualification.] Equality of the limiting values alone is weaker than equality of their ratio. For example $U=2$, $D=1$, $S=0$ near $x=0$ gives $F_2^p=x$ and $F_2^n=2x/3$: both tend to zero, although $U\ne D$ and $F_2^n/F_2^p=2/3$. The usual inference about a symmetric sea needs $F_2^n/F_2^p\to1$ or equivalent information about relative differences, rather than merely coincident zero limits.