= Linear three-reservoir carbon model
{title2=$D\dot{\mathbf C}=E\mathbf e_a-L\mathbf C$}
A capacity-weighted exchange model for atmospheric, fast surface and deep <ocean> <carbon dioxide> anomalies. Positive diagonal capacities form $D$, while $L$ is the weighted path-graph <Laplacian matrix> with exchange conductances $r,s$. Then $\mathbf C^TL\mathbf C=r(C_a-C_b)^2+s(C_b-C_d)^2\geq0$. The homogeneous generator $-D^{-1}L$ is similar to a symmetric negative-semidefinite <matrix>, with two decaying <normal modes> and one zero <normal mode> for equal concentrations. Summing all budgets proves that the capacity-weighted total is conserved after emissions stop. The long-time concentration from total input $M$ is $M/(R_a+R_b+R_d)$, rather than zero.
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