= Solution
True <thermal equilibrium> requires a thermal ionizing <radiation> field as well as chemical balance; equality of <photoionization> and recombination rates alone would require a specified external <radiation> field. Under the stronger equilibrium assumption, apply the <Saha equation> to pure <hydrogen>. The electron spin factor cancels the ground-state electronic degeneracy, giving
$$
\frac{n_en_p}{n_{\rm HI}}=S(T),\qquad
S(T)=\left(\frac{2\pi m_ek_BT}{h^2}\right)^{3/2}e^{-\chi/(k_BT)}.
$$
At $T=10^4\ \mathrm K$, $k_BT=0.862\ \mathrm{eV}$ and $S=3.38\times10^{20}\ \mathrm{m^{-3}}$. Charge neutrality gives $n_e=n_p$. Importantly, the known quantity from <1/d> is the \i[neutral] <number density>, not the total <number density>; use <Saha neutral fraction from a measured neutral density> to obtain $n_p=\sqrt{Sn_{\rm HI}}=5.31\times10^7\ \mathrm{m^{-3}}$. Therefore
$$
\boxed{1-x_e=\frac{n_{\rm HI}}{n_p+n_{\rm HI}}=\frac{1}{1+\sqrt{S/n_{\rm HI}}}\simeq1.57\times10^{-13}.}
$$
This extraordinarily small neutral fraction is the result of applying <thermal equilibrium> to dilute intergalactic gas; it is not a realistic inference from <photoionization equilibrium> with a nonthermal cosmic background. For the axial geometry of <1/d>, the equilibrium neutral fraction is smaller by $\sqrt{20}$.
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