We present ALMA observations of the CO(6-5) and [CII] emission lines and the sub-millimeter continuum of the $zsim6$ quasi-stellar object (QSO) SDSS J231038.88+185519.7. Compared to previous studies, we have analyzed a synthetic beam that is ten times smaller in angular size, we have achieved ten times better sensitivity in the CO(6-5) line, and two and half times better sensitivity in the [CII] line, enabling us to resolve the molecular gas emission. We obtain a size of the dense molecular gas of $2.9pm0.5$ kpc, and of $1.4pm0.2$ kpc for the 91.5 GHz dust continuum. By assuming that CO(6-5) is thermalized, and by adopting a CO--to--$H_2$ conversion factor $rm alpha_{CO} = 0.8~ M_{odot}~K^{-1}~ (km/s)^{-1} ~pc^{2}$, we infer a molecular gas mass of $rm M(H_2)=(3.2 pm0.2) times 10^{10}rm M_{odot}$. Assuming that the observed CO velocity gradient is due to an inclined rotating disk, we derive a dynamical mass of $rm M_{dyn}~sin^2(i) = (2.4pm0.5) times 10^{10}~ M_{odot}$, which is a factor of approximately two smaller than the previously reported estimate based on [CII]. Regarding the central black hole, we provide a new estimate of the black hole mass based on the C~IV emission line detected in the X-SHOOTER/VLT spectrum: $rm M_{BH}=(1.8pm 0.5) times 10^{9}~ M_{odot}$. We find a molecular gas fraction of $rm mu=M(H_2)/M^*sim4.4$, where $rm M^*approx M_{dyn} - M(H_2)-M(BH)$. We derive a ratio $v_{rot}/sigma approx 1-2$ suggesting high gas turbulence, outflows/inflows and/or complex kinematics due to a merger event. We estimate a global Toomre parameter $Qsim 0.2-0.5$, indicating likely cloud fragmentation. We compare, at the same angular resolution, the CO(6-5) and [CII] distributions, finding that dense molecular gas is more centrally concentrated with respect to [CII]. We find that the current BH growth rate is similar to that of its host galaxy.
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