= Solution
Near the background <quasar>, the <quasar proximity effect> typically reduces the number and strength of <Lyman-alpha forest> features relative to regions farther along the sightline. It is an ionization effect, not evidence that the quasar lies in an empty region. At fixed gas density and <temperature>, <photoionization equilibrium> gives
$$
\Gamma=\Gamma_{\rm bg}+\Gamma_Q,\qquad
\omega=\frac{\Gamma_Q}{\Gamma_{\rm bg}},\qquad
\boxed{N_{\rm HI}=\frac{N_{\rm HI}^{(0)}}{1+\omega}.}
$$
If the unperturbed <neutral-hydrogen column-density distribution> is $AN^{-\beta}$, $\beta>1$, detecting a column above $N_{\min}$ now requires its unperturbed column to exceed $(1+\omega)N_{\min}$. Integrating that distribution shows that the number above threshold is reduced by $(1+\omega)^{1-\beta}$. This supplies an explicit model for the decline of line counts.
The known quasar luminosity estimates its <photoionization rate> at proper separation $R$:
$$
\Gamma_Q(R)=\int_{\nu_{\rm LL}}^\infty
\frac{L_\nu}{4\pi R^2h\nu}\sigma_{\rm HI}(\nu)\,d\nu,
\qquad
\Gamma_{\rm bg}=4\pi\int_{\nu_{\rm LL}}^\infty
\frac{J_\nu\sigma_{\rm HI}(\nu)}{h\nu}\,d\nu.
$$
These expressions assume unattenuated, steady, isotropic quasar emission and a specified ionizing spectrum. Fitting the measured suppression as a function of separation determines the ratio $\Gamma_Q/\Gamma_{\rm bg}$ and hence constrains the <cosmic ionizing background>. A weaker background gives a larger region in which the quasar dominates. For fixed background spectral shape, this constrains its specific intensity near the <Lyman limit system> ionization edge, not an arbitrary spectral function. The <proximity-effect estimator of the ionizing background> is biased if the host gas is overdense, the quasar's past luminosity differs from its observed luminosity, its radiation is anisotropic, or systemic-redshift errors distort the inferred separation; absorption between source and gas also changes $\Gamma_Q$. These effects explain why the method needs environmental and source modeling, as discussed in https://ned.ipac.caltech.edu/level5/Sept01/Rauch/Rauch3_2.html[the proximity-effect measurements].
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