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Take the retained disc mass to be the cosmic baryon fraction of the total dark matter halo mass: $M_d=f_b{M}_h$, where $f_b=0.05/0.25=0.2$. Adopt $v_{\mathrm{vir}}^2=G{M}_h/r_{\mathrm{vir}}$ and describe the disc gravity by $v_d^2=C_{d}G{M}_d/R_d$. The structure coefficient $C_d$ must be stated: the scale-free Mestel disc gives $C_d=1$ when its flat rotation speed and enclosed mass are used, while a finite disc requires boundary and thickness information. The equality of edge specific angular momentum gives $R_d{v}_d=q{r}_{\mathrm{vir}}{v}_{\mathrm{vir}}$, with $q=0.1$. Combining these relations gives the angular-momentum-conserving self-gravitating disc radius:With the commonly intended $C_d\simeq 1$ approximation,The corresponding masses are $M_h\simeq 2.49\times 10^{11}{M}_{\odot }$ and $M_d\simeq 4.97\times 10^{10}{M}_{\odot }$, where $M_{\odot }$ is the solar mass.

For a definite formation estimate, assume the dark matter halo acquired this mass and radius at virial equilibrium, retained them afterward, and had mean density ${\Delta }_v{\bar{\rho }}_m(z_f)$ with ${\Delta }_v=18{\pi }^2$. This uses the matter-dominated spherical-collapse model as an approximation to the specified matter-plus-vacuum cosmology. Since ${\bar{\rho }}_m(z_f)=3H_0^2{\Omega }_{m0}(1+z_f{)}^3/(8\pi G)$,The approximate distance therefore gives formation modestly before observation. Using the numerical rather than approximate distance integral gives $z_f\simeq 1.22$, within the accuracy of these structural assumptions. The calculation also assumes inclination-corrected spectroscopy, negligible disc pressure support, conserved edge specific angular momentum, no substantial later accretion or mergers, and no significant baryon loss.

This formation redshift is model-dependent, not uniquely fixed by the stated disc data. Keeping $C_d$ explicit gives $1+z_f\simeq 2.209C_d^{-4/3}({\Delta }_v/18{\pi }^2{)}^{-1/3}$. In particular, replacing the mean-density convention by $200$ times the critical density gives $H(z_f)=v_{\mathrm{vir}}/(10r_{\mathrm{vir}})$ and $z_f\simeq 0.87$, later than the observation. That inconsistent chronological result cannot be silently used as the formation epoch.

There is a further idealization in applying $C_d=1$ exactly at the edge: an abruptly truncated razor-thin disc with nonzero edge surface density of a disk has a logarithmically divergent in-plane force at that edge. Locally the available mass occupies a half-plane, and its radial contribution is proportional to $G{\Sigma }_d{\int }_{ϵ}^{\ell }ds/s$, with no opposite exterior half-plane to cancel it. Finite thickness or a smooth taper removes that sharp-edge force singularity of a thin disk. Thus $C_d=1$ is an explicitly qualified Mestel disc or monopole-scale approximation, not an exact finite-disc calculation; enclosed mass does not determine a disc rotation curve.

Treat the features in the QSO spectrum as Lyman-alpha absorption and use the cosmological wavelength-redshift relation to convert their spacing into $\Delta z=304/1216=0.25$. In the specified flat matter-plus-cosmological-constant model,A local radial interval has comoving length $d\chi =cdz/H(z)$ and proper length $dl=d\chi /(1+z)$. Evaluating the mean interval locally at $z=3$ gives the proper absorber separation from a wavelength intervalThe corresponding comoving interval is about $262\mathrm{Mpc}$. We neglect peculiar-velocity shifts and treat $H$ as approximately constant over one mean interval.

The proper number density is $n_p=(1+z{)}^3{n}_{\mathrm{gal}}=179.2\mathrm{Mp}{\mathrm{c}}^{-3}$. To compute the mean projected area of a randomly oriented thin disc, note that a disc whose normal makes angle $i$ to the sightline has projected area $\pi R_d^2|\cos i|$. Isotropic normals haveFor independent intersections with fully covering thin discs, the mean free path through randomly oriented disc absorbers is ${\ell }^{-1}=n_p\pi R_d^2/2$. ThusAll radii here are proper radii at the absorber epoch. The projected-area average is over the underlying orientation distribution; it already accounts for the larger intersection probability of face-on discs.

The cosmic baryon fraction is $f_b={\Omega }_b/{\Omega }_m=0.2$, so the disc mass is $M_d=f_d{M}_h$ with $f_d=f_b/4=0.05$. Define $v_{\mathrm{vir}}^2=G{M}_h/R_{\mathrm{vir}}$. Disc gravity depends on its radial mass profile, which has not been specified. Parameterize its edge circular speed by $v_d^2=C_{d}G{M}_d/R_d$ and use $C_d=1$ for a centrally concentrated disc in a monopole approximation at its outer edge. The angular-momentum condition givesCombining the equations yields the angular-momentum-conserving self-gravitating disc radiusFor the stated $C_d=1$ choice, $R_{\mathrm{vir}}\simeq 36.8\mathrm{kpc}$ and $v_d=0.5v_{\mathrm{vir}}$. The factor $C_d$ makes explicit the otherwise undetermined disc-profile dependence; an enclosed-mass formula is not exact for an arbitrary flattened disc.

To determine the halo speed, a virial-density convention is also needed. Choose a spherical overdensity ${\Delta }_v=200$ relative to the critical density at $z=3$:The virial velocity of a spherical-overdensity halo then obeysThe nearly matter-dominated spherical-collapse choice ${\Delta }_v\simeq 18{\pi }^2$ instead gives about $99.5\mathrm{km}{\mathrm{s}}^{-1}$. More generally the numerical radius and velocity scale respectively as $C_d$ and $C_d\sqrt{{\Delta }_v/200}$. With $G\simeq 4.30\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$, the adopted model has $M_h\simeq 9.5\times 10^{10}{M}_{\odot }$ and $M_d\simeq 4.8\times 10^9{M}_{\odot }$; these reproduce the edge speed and the specified angular momentum.

The calculation assumes homogeneous, unclustered intersections, one detectable absorption feature per disc crossing, unit neutral-gas covering factor, isotropic orientations, universal halo baryon fraction and conservation of the specified edge specific angular momentum. It assumes the halo is virialized at the absorber epoch and neglects its inner gravity as instructed. The extra profile and overdensity choices are necessary to turn the geometric and angular-momentum constraints into unique numerical halo properties.