= Solution
The ratio
$$
\frac{\Gamma_W}{H}\simeq\left(\frac{k_BT}{\kappa}\right)^3
$$
falls rapidly as the universe cools. Below $k_BT\simeq0.7\,\mathrm{MeV}$, weak reactions occur less than once per expansion time and can no longer track the falling equilibrium ratio. The neutron-to-proton ratio therefore freezes at approximately $1/6$.
Before nuclei can form efficiently, free neutrons continue to beta-decay,
$$
n\to p+e^-+\bar\nu_e.
$$
During the <deuterium bottleneck> this lowers the ratio to about $1/7$ by $k_BT\simeq0.07\,\mathrm{MeV}$.
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