= Solution
The required <pressure> integral uses $\int_0^1(1-x^2)^4dx=128/315$. With $\Sigma=(32/35)\rho_0H$ and $p_0/\rho_0=\Omega^2H^2/8$,
$$
\int_{-H}^Hp\,dz=\frac{256}{315}p_0H=\frac{\Sigma\Omega^2H^2}{9}.
$$
The <vertically averaged alpha closure for a mixed-pressure layer> therefore gives
$$
\boxed{\bar\nu=\frac{\alpha H^2\Omega}{9}.}
$$
The height-integrated <dynamic viscosity> is $2H\mu=8cH\lambda/[9\kappa(1+\lambda)]$. Equate it to $\bar\nu\Sigma$ to obtain
$$
\boxed{\frac\lambda{1+\lambda}=\frac{\alpha H\Omega\kappa\Sigma}{8c}.}
$$
These are integrated equalities, not an imposed pointwise relation $\mu(z)\propto p(z)$.
Now keep $\alpha$, the <opacity> and molecular constants independent of radius, and use <Keplerian rotation> $\Omega\propto r^{-3/2}$. In the gas-dominated limit, $\lambda\ll1$ but positive, the last equality gives $\lambda\propto H\Omega\Sigma$. The column-density relation gives $\lambda\Sigma\propto H^7\Omega^6$. Eliminating $\lambda$ yields $H^6\propto\Sigma^2\Omega^{-5}$, whence
$$
\boxed{\bar\nu\propto\Sigma^{2/3}\Omega^{-2/3}\propto r\Sigma^{2/3}\qquad(\lambda\ll1).}
$$
This is the <gas-pressure branch of a mixed-pressure alpha disk>. In the radiation-dominated limit, $\lambda/(1+\lambda)\to1$, so $H\propto(\Omega\Sigma)^{-1}$ and
$$
\boxed{\bar\nu\propto\Omega^{-1}\Sigma^{-2}\propto r^{3/2}\Sigma^{-2}\qquad(\lambda\gg1).}
$$
For <viscous stability of an accretion disk>, linearize the <Keplerian viscous diffusion equation> at fixed radius: a local <mass density> perturbation has diffusion coefficient $3\partial_\Sigma(\bar\nu\Sigma)$. The gas branch has $\bar\nu\Sigma\propto\Sigma^{5/3}$ and a positive derivative, so it smooths perturbations. The radiation branch has $\bar\nu\Sigma\propto\Sigma^{-1}$ and a negative derivative, giving <radiation-pressure viscous instability>. \b[The gas-pressure branch is viscously stable; the radiation-pressure branch is viscously unstable in this closure.] This concerns radial mass-transport stability on wavelengths where the vertically averaged thin-disc model applies.
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