= Solution
In the limiting model with negligible <gas pressure>, $p=p_r=4\sigma T^4/(3c)$. Its gradient is $p_z=16\sigma T^3T_z/(3c)$. Combining <radiative diffusion> and <hydrostatic equilibrium> gives
$$
F=-\frac{c}{\kappa\rho}p_z=\frac{c\Omega^2}{\kappa}z.
$$
Because $\kappa$ and $\Omega$ are constant with height at the fixed disk radius,
$$
\frac{dF}{dz}=\frac{c\Omega^2}{\kappa}.
$$
Equating this to the viscous heating rate $9\mu\Omega^2/4$ forces
$$
\boxed{\mu=\frac{4c}{9\kappa}.}
$$
This proves <constant viscosity in radiation-supported vertical balance>. It fixes the local effective <dynamic viscosity> uniquely, independent of height, <surface density> and radius within this ideal limiting model. It does not fix a height-independent <kinematic viscosity>: $\nu(z)=4c/(9\kappa\rho(z))$ still depends on the local <mass density>.
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