= Solution
The <transient climate response to cumulative carbon emissions> is $\boxed{\kappa\simeq\Delta T/\mathcal E}$, where $\mathcal E$ is cumulative emitted carbon dioxide in a stated <mass> unit. Here the unit is tonnes of $\mathrm{CO_2}$, rather than tonnes of carbon. Increasing concentration weakens the incremental <radiative forcing> per added atmospheric tonne because forcing is approximately logarithmic. Over a substantial range this is partly offset by a weakening of carbon sinks and by the changing ratio between transient warming and forcing as ocean heat uptake evolves. These approximate compensations make warming nearly proportional to cumulative emissions; no exact cancellation follows from the logarithmic forcing law alone. The approximation is range- and pathway-dependent, and is not a theorem valid for arbitrary emissions histories.
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