= Solution
Work at one radius and write $\mathcal R=k/(\mu_m m_p)$. Use the <surface density> to fix the <mass density> normalization and introduce the height scale
$$
h^6=\frac{3\alpha\kappa\Sigma^2\mathcal R^4}{16\sigma\Omega^5}.
$$
Choose the dimensionless variables
$$
z=h\zeta,\qquad \rho=\frac\Sigma h D,\qquad p=\Sigma\Omega^2hP,\qquad T=\frac{\Omega^2h^2}{\mathcal R}\Theta,\qquad F=\alpha\Sigma\Omega^3h^2Q.
$$
The <pressure> and <temperature> scales make the hydrostatic and <ideal gas> equations parameter-free. The <flux> scale fixes the heating coefficient at $9/4$, and the choice of $h$ makes the diffusion coefficient unity. Direct substitution gives the universal <ordinary differential equations>
$$
\boxed{\frac{dP}{d\zeta}=-D\zeta,\qquad \frac{dQ}{d\zeta}=\frac94P,\qquad \frac{d\Theta}{d\zeta}=-\frac{DQ}{\Theta^3},\qquad P=D\Theta.}
$$
For example, the dimensionless coefficient in the <temperature> equation is $3\alpha\kappa\Sigma^2\mathcal R^4/(16\sigma\Omega^5h^6)=1$.
Let the upper free surface be at $\zeta=\zeta_s$, so its physical semithickness is $H=h\zeta_s$. Midplane <symmetry> and the <radiative-zero disk surface> conditions give
$$
\boxed{Q(0)=0,\qquad P(\zeta_s)=\Theta(\zeta_s)=0,\qquad 2\int_0^{\zeta_s}D\,d\zeta=1.}
$$
The last condition is precisely $\Sigma=\int_{-H}^H\rho\,dz$. The unknown surface position and the two positive midplane values $P(0),\Theta(0)$ are determined as part of the normalized <boundary value problem>. <Symmetry> makes $P,D,\Theta$ even and $Q$ odd; $P'(0)=\Theta'(0)=0$ already follow from the equations and are not additional independent <boundary conditions>.
The zero conditions do not mean zero surface <flux>. Integrating the heating equation gives $Q(\zeta_s)=(9/4)\int_0^{\zeta_s}P\,d\zeta>0$. Near a regular surface with finite outgoing <flux>, dividing the hydrostatic equation by the <temperature> equation yields $dP/d\Theta\simeq\zeta_s\Theta^3/Q_s$. Thus $P\propto\Theta^4$, $D\propto\Theta^3$ and $\Theta\propto\zeta_s-\zeta$, so <mass density> also vanishes. The apparent singular quotient in the diffusion equation has a finite regular limit. All <boundary conditions> and equations are independent of the physical parameters: this is <homologous vertical structure of a constant-opacity alpha disk>.
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