= Solution
Albedo reduction increases absorbed short-wave flux by $S_0(a_1T+a_2T^2)/4$. Define $A=\lambda_P-1.5-S_0a_1/4$ and $B=S_0a_2/4$. The perturbation <energy balance> is $N(T)=F-AT+BT^2$, where $F=3.7$ W m$^{-2}$ for doubling. With the preceding constants, $A\simeq1.220$ W m$^{-2}$ K$^{-1}$ and $B=0.068$ W m$^{-2}$ K$^{-2}$. The two roots are
$$
\boxed{T_{\mathrm{stable}}\simeq3.86\ \mathrm K,\qquad T_{\mathrm{unstable}}\simeq14.1\ \mathrm K.}
$$
Stability follows from $N'(T)=-A+2BT<0$ at the lower root. They merge at $T=A/(2B)\simeq8.97$ K and $F_{\max}=A^2/(4B)\simeq5.47$ W m$^{-2}$; at equality the <equilibrium> is marginal, not <linearly stable>.
Assuming the usual logarithmic forcing extension $F=3.7\log(C/280)/\log2$ gives a stable branch only for $\boxed{C<C_{\mathrm{crit}}\simeq7.8\times10^2\ \mathrm{ppm}}$. This “maximum” is the supremum of stable concentrations in the stated quadratic, linearized model. It is not a physical prediction of a real-world runaway threshold; extrapolating its feedback laws far beyond their small-change regime is an additional assumption.
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