= Solution
No measured normalization of the <neutral-hydrogen column-density distribution>, ionizing field or gas <temperature> is supplied, so the paper's formulas do not determine numerical <densities> on their own. With observational data and suitable ionization corrections, the high-redshift <Lyman-alpha forest> gives a total gas <density> of order a few $10^{-2}$ relative to today's <critical density>, comparable to the <cosmological baryon density> $\Omega_b\sim0.04$–$0.05$ for $H_0\sim70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$. The conclusion is that a large fraction of baryons at $z\simeq3$ reside in diffuse ionized gas, rather than that every baryon has been individually accounted for. https://arxiv.org/abs/astro-ph/0104272[Schaye's analysis] and https://arxiv.org/abs/astro-ph/0101005[Kim and collaborators' survey] support that interpretation under their stated gas and ionizing-background assumptions.
The neutral-gas reservoir traced predominantly by <damped Lyman-alpha systems> is much smaller, of order $10^{-3}$ in the same convention. It is broadly comparable in order of magnitude to today's luminous stellar matter, whose <cosmic stellar mass density> is a few $10^{-3}$, but exact comparisons depend on cosmology, <helium> correction and which stellar components are counted. https://arxiv.org/abs/astro-ph/9608147[Storrie-Lombardi, McMahon and Irwin's neutral-gas census] finds especially close comparison with present stellar disks. \b[Most high-redshift gas is ionized; the smaller neutral reservoir has a mass of roughly the same order as today's visible stellar population.] This comparison does not imply that the measured neutral reservoir alone simply becomes all present-day stars: gas accretion, recycling, ionization changes and outflows continue during <galaxy> evolution.
The <neutral gas density from an absorption distribution> is comparatively robust for high columns where the hydrogen is mostly neutral, but saturated lines, dust-obscured quasars, incomplete surveys and small numbers of rare high-column systems affect the integral. The <ionized gas density from an absorption distribution> has additional systematic uncertainty in the <cosmic ionizing background>, <temperature>, absorber depth, <helium> treatment and small-scale <density> inhomogeneity. At low columns, line blending and incompleteness matter; at high columns, attenuation makes an optically thin ionization correction invalid. A single <power law> extrapolated outside the measured column range can also produce a misleading mass integral. Hot gas, molecular gas and gas missed by the selected absorption population are not included automatically. These are observational and modelling qualifications to a baryon census, not failures of the column-mass accounting itself.
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