= Solution
Typical weak <Lyman-alpha forest> absorbers have neutral <hydrogen> <column densities> roughly $10^{12}$–$10^{16}\,\mathrm{cm}^{-2}$, with stronger forest absorption extending towards $10^{17}\,\mathrm{cm}^{-2}$. Common <Doppler broadening> parameters are a few tens of $\mathrm{km\,s}^{-1}$, consistent with gas at approximately $10^4$ K together with bulk velocity gradients. The thermal contribution is
$$
b_{\rm th}=\sqrt{\frac{2k_BT}{m_H}}=12.9\left(\frac{T}{10^4\,\mathrm K}\right)^{1/2}\mathrm{km\,s}^{-1}.
$$
A line width is consequently a temperature upper bound if all other broadening is neglected, not a unique thermometer. The gas commonly lies in diffuse sheets and filaments at modest <density contrast>; only the strongest systems need substantial self-shielding or a galactic environment.
For optically thin gas illuminated by the <cosmic ionizing background>, <photoionization equilibrium> requires
$$
n_{\rm HI}\Gamma_{\rm HI}=n_en_p\alpha(T),\qquad
x_{\rm HI}\equiv\frac{n_{\rm HI}}{n_H}\simeq\frac{\alpha(T)n_H}{\Gamma_{\rm HI}},
$$
where the last approximation uses almost complete hydrogen ionization. For example, $n_H=10^{-5}\,\mathrm{cm}^{-3}$, $\alpha\simeq4\times10^{-13}\,\mathrm{cm}^3\mathrm s^{-1}$ and $\Gamma_{\rm HI}=10^{-12}\,\mathrm s^{-1}$ give $x_{\rm HI}\simeq4\times10^{-6}$. A $100$ kpc path then has $N_{\rm HI}\sim10^{13}\,\mathrm{cm}^{-2}$ despite its tiny neutral fraction. This illustrates the <Lyman-alpha forest photoionization scaling>: low density makes the two-body <radiative recombination> rate weak relative to the one-body <photoionization> rate, but a long path and a strong resonance still produce measurable absorption.
Independent information is important because a <column density> alone is not a volume density. Similar absorption in close pairs of <quasars> indicates substantial transverse coherence, while an ionizing <optical depth> $N_{\rm HI}\sigma_{912}\ll1$ shows that these weak systems do not shield themselves from the background. The continuum-shielding threshold is $1/\sigma_{912}\simeq1.6\times10^{17}\,\mathrm{cm}^{-2}$. Together with the observed line widths and ionization balance, these facts favor \b[extended, low-density, highly ionized gas] over small cold neutral clouds. The physical interpretation and characteristic scales are consistent with https://arxiv.org/abs/astro-ph/9806286[Rauch's forest analysis] and https://arxiv.org/abs/astro-ph/9507047[the measured column and line-width distributions].
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