= Solution
Equating marginal mitigation cost to the previous marginal damage gives
$$
\frac{M_0W}{E}=\frac{\kappa\gamma D_0WT^{\gamma-1}}{r-g},\quad
E=\frac{M_0(r-g)}{\kappa\gamma D_0T^{\gamma-1}}.
$$
Using $\dot T=\kappa E$ from the cumulative-emissions approximation,
$$
T^{\gamma-1}\dot T=\frac{M_0(r-g)}{\gamma D_0},\qquad
\boxed{T(t)^\gamma=T(t_0)^\gamma+\frac{M_0(r-g)}{D_0}(t-t_0).}
$$
Thus $T\propto t^{1/\gamma}$ at late times and $E\propto t^{-(\gamma-1)/\gamma}$. Warming slows but never stops, and cumulative emissions remain unbounded. This follows from a cost model in which zero emissions is infinitely costly and from a marginal-damage rule with constant positive $r-g$.
The approximation eventually fails if cumulative emissions leave the range of constant <TCRE>, if large damages suppress <economic growth>, if technological change alters $M_0$, or if fossil resources and mitigation options impose different limits. The slowly varying-<temperature> approximation also needs $\dot T/[T(r-g)]\ll1$. Thresholds, adaptation, uncertainty and unequal impacts are omitted; the formula is a property of this model rather than a universal policy trajectory.
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