 Discounted one-capacity climate pulse damage

A persistent forcing increment $\alpha$ from a unit emission pulse drives $C\dot{T}=\alpha -\lambda T$, with $T(0)=0$. Linear annual damage $\beta T$ discounted at positive rate $k_D$ integrates to the displayed value. It assumes constant coefficients, a permanent forcing step and damage valued relative to the emission date. A decaying carbon burden or a different damage law changes the result.