What explains the proportionality of global warming to cumulative carbon emissions?

The constant ratio of global warming to cumulative CO₂ emissions underpins the use of remaining carbon budgets as policy tools, and the need to reach net zero CO₂ emissions to stabilize global mean temperature. One requirement for this proportionality is that the temperature response to a pulse emission of CO₂ is independent of the background emissions scenario, and this property has been explained by a balance between the logarithmic dependence of radiative forcing on CO₂ concentration, and the saturation of CO₂ sinks at higher CO₂ levels. Several studies have argued that this proportionality also arises because heat and carbon are mixed into the ocean by similar physical processes, and this argument was echoed in the Intergovernmental Panel on Climate Change Sixth Assessment Report. However, contrary to this hypothesis, atmosphere-ocean fluxes of heat and carbon evolve very differently to each other in abrupt CO₂ increase experiments in five earth system models, and changes in the atmosphere, ocean and land carbon pools all contribute to making warming proportional to cumulative emissions. Moreover, an analytical model only exhibits proportional heat and carbon fluxes and proportional warming to cumulative emissions if the land and atmosphere carbon pools are neglected, among other unrealistic assumptions. These results strongly suggest that this proportionality is not amenable to a simple physical explanation, but rather arises because of the complex interplay of multiple physical and biogeochemical processes.