Abstract. Several basic ratios of responses to forcings in the carbon-climate system are observed to be relatively steady. Examples include the CO 2 airborne fraction (the fraction of the total anthropogenic CO 2 emission flux that accumulates in the atmosphere) and the ratio T /Q E of warming (T ) to cumulative total CO 2 emissions (Q E ). This paper explores the reason for such near-constancy in the past, and its likely limitations in future.The contemporary carbon-climate system is often approximated as a set of first-order linear systems, 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 carbonclimate system is idealised as a linear system (Lin) forced by exponentially growing CO 2 emissions (Exp), then all ratios of responses to forcings are constant. Important cases are the CO 2 airborne fraction (AF), the cumulative airborne fraction (CAF), other CO 2 partition fractions and cumulative partition fractions into land and ocean stores, the CO 2 sink uptake rate (k S , the combined land and ocean CO 2 sink flux per unit excess atmospheric CO 2 ), and the ratio T /Q E . Further, the AF and the CAF are equal. Since the Lin and Exp idealisations apply approximately to the carbon-climate system over the past two centuries, the theory explains the observed near-constancy of the AF, CAF and T /Q E in this period.A nonlinear carbon-climate model is used to explore how future breakdown of both the Lin and Exp idealisations will cause the AF, CAF and k S to depart significantly from constancy, in ways that depend on CO 2 emissions scenarios. However, T /Q E remains approximately constant in typical scenarios, because of compensating interactions between CO 2 emissions trajectories, carbon-climate nonlinearities (in land-air and ocean-air carbon exchanges and CO 2 radiative forcing), and emissions trajectories for non-CO 2 gases. This theory establishes a basis for the widely assumed proportionality between T and Q E , and identifies the limits of this relationship.