= Solution
Specify the path convention before integrating the <neutral-hydrogen column-density distribution>. Write $f_z=\partial^2\mathcal N/(\partial N\partial z)$ and $f_X=\partial^2\mathcal N/(\partial N\partial X)$, with <absorption distance> satisfying
$$
\frac{dX}{dz}=\frac{H_0}{H(z)}(1+z)^2,\qquad f_z=f_X\frac{dX}{dz}.
$$
The lower panel of the preceding sketch shows the approximate decrease of $f$ with $N=N_{\rm HI}$ near $z=3$. Ordinary forest columns are roughly described by $f\propto N^{-\beta}$, $\beta\sim1.5$, but the slope changes through partially shielded <Lyman limit systems> and high-column <damped Lyman-alpha systems>. No single slope should be extrapolated through all these regimes. In logarithmic bins the number is $N\ln(10)f$, whereas their neutral mass contribution is proportional to $N^2f$; confusing these two weightings would give a wrong baryon inventory.
To derive the <neutral gas density from an absorption distribution>, a redshift interval $dz$ has proper light-path length $d\ell=c\,dz/[(1+z)H]$. The sum of neutral <hydrogen> columns per path length estimates the mean proper number density:
$$
\overline n_{\rm HI}(z)=\frac{(1+z)H(z)}c\int Nf_z(N,z)\,dN.
$$
The proper <mass density> is $m_H\overline n_{\rm HI}$; division by $(1+z)^3$ converts it to a comoving mass density. Thus, with today's <critical density> $\rho_{\rm crit,0}=3H_0^2/(8\pi G)$,
$$
\boxed{\Omega_{\rm HI}(z)=\frac{m_HH(z)}{c\rho_{\rm crit,0}(1+z)^2}\int Nf_z\,dN
=\frac{H_0m_H}{c\rho_{\rm crit,0}}\int Nf_X\,dN.}
$$
A representative finite-sample estimator replaces the last integral by $\sum_j N_j/\Delta X$, with completeness and selection corrections. This defines a comoving inventory divided by today's <critical density>; a fraction of the instantaneous critical density has a different normalization. Neutral gas including accompanying helium is obtained by multiplying by a specified mass correction, approximately $1.3$; hydrogen-only $\Omega_{\rm HI}$ must not silently include it.
At $z\sim3$ the observed neutral inventory is of order $10^{-3}$, far below the <cosmological baryon density> of order $0.04$–$0.05$ for $h\sim0.7$. High-column <damped Lyman-alpha systems> contribute most of this neutral mass although weak forest lines dominate the counts; for $\beta=1.5$ the mass per logarithmic column interval increases as $N^{1/2}$ in the forest regime. The small neutral inventory therefore does not imply a small total gas inventory: the <intergalactic medium> contains a large ionized baryon reservoir invisible to a neutral-only count. Recovering that reservoir requires an ionization correction and a gas-density model. The survey normalization and high-column contribution are described in https://ned.ipac.caltech.edu/level5/Sept05/Wolfe/Wolfe2.html[the neutral-gas inventory analysis].
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