= Solution
The second box represents a reservoir exchanging carbon with the atmosphere, with equal capacity in the chosen concentration units. For a unit atmospheric pulse, $C(0^+)=1$, $C_1(0^+)=0$ and no subsequent emissions, total $C+C_1=1$ is conserved while $d(C-C_1)/dt=-2k(C-C_1)$. Thus
$$
\boxed{C(t)=\tfrac12(1+e^{-2kt}),\qquad C_1(t)=\tfrac12(1-e^{-2kt}).}
$$
The atmospheric burden falls from one to one half with <time constant> $(2k)^{-1}$, not $k^{-1}$. The model has a permanent airborne fraction because neither box contains an external sink. The carbon-response panel of the original figure in S12 shows this explicitly.
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