= Solution
Far from the inner edge, use $u_R\simeq-3\nu/(2R)$ and the <alpha disk>[alpha-viscosity prescription]
$$
\nu=\alpha c_sH=\alpha\frac{c_s^2}{\Omega},
\qquad H=\frac{c_s}{\Omega}.
$$
Mass conservation then gives
$$
\Sigma=\frac{\dot m\Omega}{3\pi\alpha c_s^2},
\qquad
\rho\simeq\frac\Sigma H
=\frac{\dot m\Omega^2}{3\pi\alpha c_s^3}.
$$
At the onset of <accretion-disk self-gravity>[disk self-gravity], $\rho=M_{\rm BH}/R^3$. Since $\Omega^2=GM_{\rm BH}/R^3$, this condition reduces to
$$
\boxed{c_s^3=\frac{G\dot m}{3\pi\alpha}}.
$$
The <blackbody> approximation and local radiative balance on one face give
$$
\sigma_{\rm SB}T_{\rm eff}^4
=\frac{3GM_{\rm BH}\dot m}{8\pi R^3},
$$
while the stated ideal-gas estimate is $c_s^2=\gamma k_BT_{\rm eff}/\mu$. Inserting these expressions into the self-gravity condition gives
$$
\left(\frac{\gamma k_B}{\mu}\right)^{3/2}
\left(\frac{3GM_{\rm BH}\dot m}
{8\pi\sigma_{\rm SB}R^3}\right)^{3/8}
=\frac{G\dot m}{3\pi\alpha}.
$$
Solving for $R$ produces
$$
\boxed{
R_{\rm crit}=(3\pi\alpha)^{8/9}
\left(\frac{\gamma k_B}{\mu}\right)^{4/3}
\left(\frac{3M_{\rm BH}}{8\pi\sigma_{\rm SB}}\right)^{1/3}
(G\dot m)^{-5/9}}.
$$
Beyond this radius, the disk's own gravity rivals the central vertical gravity; fragmentation and <star formation> can then invalidate the smooth steady thin-disk model.
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