 Solution

Use a positive stabilizing feedback coefficient: the net flux is $N=F_{\mathrm{ext}}-\lambda \Delta T$. The Planck feedback increases outgoing thermal radiation as temperature rises and therefore contributes positively to $\lambda$. Water-vapor feedback raises infrared opacity as a warmer atmosphere holds more water, tending to reduce $\lambda$; ice-albedo feedback also tends to reduce it as bright ice is lost. These signs depend on the explicitly stated convention.

For doubled carbon dioxide forcing $F_{2\times }>0$, equilibrium givesA symmetric positive interval $[{\lambda }_0-{\sigma }_{\lambda },{\lambda }_0+{\sigma }_{\lambda }]$ maps toThe upward excursion is larger than the downward one because inversion is nonlinear. If the interval reaches zero, it supplies no finite upper bound on equilibrium warming within this linear stable model.