Several basic ratios describing the carbon-climate system are observed to adopt relatively steady values. Examples include the CO<sub>2</sub> airborne fraction (the fraction of the total anthropogenic CO<sub>2</sub> emission flux that accumulates in the atmosphere) and the ratio <i>T</i>/<i>Q</i><sub>E</sub> of warming (<i>T</i>) to cumulative total CO<sub>2</sub> emissions (<i>Q</i><sub>E</sub>). This paper explores the reason for such near-constancy in the past, and its likely limitations in future. <br><br> The contemporary carbon-climate system is often approximated as a first-order linear system, for example in response-function descriptions. All such linear systems have exponential eigenfunctions in time (an eigenfunction being one that, if applied to the system as a forcing, produces a response of the same shape). This implies that, if the carbon-climate system is idealised as a linear system (Lin) forced by exponentially growing CO<sub>2</sub> emissions (Exp), then all ratios among fluxes and perturbation state variables are constant. Important cases are the CO<sub>2</sub> airborne fraction (AF), the cumulative airborne fraction (CAF), other CO<sub>2</sub> partition fractions and cumulative partition fractions into land and ocean stores, the CO<sub>2</sub> sink uptake rate (<i>k</i><sub>S</sub>, the combined land and ocean CO<sub>2</sub> sink flux per unit excess atmospheric CO<sub>2</sub>), and the ratio <i>T</i>/<i>Q</i><sub>E</sub>. Further, the AF and the CAF are equal. The Lin and Exp idealisations apply approximately (but not exactly) to the carbon-climate system in the period from the start of industrialisation (nominally 1750) to the present, consistent with the observed near-constancy of the AF, CAF and <i>T</i>/<i>Q</i><sub>E</sub> in this period. <br><br> A nonlinear carbon-climate model is used to explore how the likely future breakdown of both the Lin and Exp idealisations will cause the AF, CAF and <i>k</i><sub>S</sub> to depart significantly from constancy, in ways that depend on CO<sub>2</sub> emissions scenarios. However, <i>T</i>/<i>Q</i><sub>E</sub> remains approximately constant in typical scenarios, because of compensating interactions between emissions trajectories, carbon-cycle dynamics and non-CO<sub>2</sub> gases. This theory assists in establishing both the basis and limits of the widely-assumed proportionality between <i>T</i> and <i>Q</i><sub>E</sub>, at about 2 K per trillion tonnes of carbon.