This is a well-written and interesting manuscript presenting a novel analytical framework for analyzing coupled carbon–climate simulations. The approach is creative, the results are relevant, and the manuscript merits publication. I have provided several detailed comments in the annotated PDF, but here I summarize my major points:

1) I suggest using delta (Δ) notation for variables such as temperature and carbon pools in the ocean, land, and atmosphere. This would make it clear that the quantities represent changes from pre-industrial values.

2) In eqn (3) it becomes clear only later in the text why (k–1) is used instead of k. This reasoning can be explained upfront.

3) Equation (4) is certainly a crude approximation. I feel this needs to be acknowledged a bit more openly with a few references. For example, see Gillett (2023) (https://www.nature.com/articles/s41467-023-42111-x).

4) Consider revising the titles of Sections 2.1 and 2.2 to better reflect the distinction between the two. 2.1 uses linear approximation for F  = f(Δ CO2) whereas 2.2 does not. Fortunately in the case of section 2.1 linearity leads to an analytical solution but linearity does not always guarantee an analytical solution.



I have made some suggestions in the PDF, but more descriptive titles could be chosen. Figure 4 panels (a) and (b) could be retitled similarly. In addition, an extra panel showing the actual airborne fraction from the flat10 simulations would help readers directly compare the analytical solutions to the model airborne fraction results.

5) The asymptotic airborne fraction of ~0.3 in Figure 4a contrasts with observation-based estimates of ~0.5. In Figure 4b, the model-mean airborne fraction appears to rise toward ~0.5 after about 100 years. Including actual model-simulated airborne fraction in a panel (c) could clarify how well the analytical approaches capture this. Also, is an airborne fraction ~0.5 is an emergent property of the real Earth system?

6) I can't help comparing Figure 4a, b show airborne fraction (AF) under continued emissions to Figure 5a of Torres Mendonça et al. (2024) (https://bg.copernicus.org/articles/21/1923/2024/) which shows the response to a pulse emission. I realize the distinction between continued and pulse emissions. Can this distinction be made explicit so that readers don't directly compare Figure 4a to figures similar to Figure 5a of Torres Mendonça et al. (2024).

7) In Section 2.1 (Equation 13), AF = 1 at t = 0, which makes sense for an instantaneous pulse but seems less realistic for continuous emissions. This should be clarified. Again including actual model AF in a new panel 4(c) would be helpful.

8) On p. 12 (lines ~185–200), processes that slow carbon uptake at higher CO₂ for land and ocean are discussed. However, the analytical model is unaware of these processes in Section 2.2. So how does AF actually increase in Figure 4b?

9) The discussion on page 13 (lines ~220 onward) is insightful but would be stronger if introduced earlier. Also, note that TCRE is constant in Section 2.1 (where F is linear function of atmospheric CO₂ change) but also in Section 2.2. It appears some loose ends need to be tied here.

Overall, this is an interesting manuscript and additional clarifications will allow readers to gain insight into the underlying properties which lead to emergent behaviour even in this simple framework.



I will be happy to read a revised version of this manuscript.

Summary

The paper proposes two tractable models that allow modelling the response of the globally averaged temperature to carbon dioxide forcing. The authors use an energy balance model to derive a model that is in agreement with the hypothesis of linear response of global mean annual temperature to cumulative carbon emissions. The authors derive an analytical solution from the differential equation of the EBM allowing to represent atmospheric carbon content Ca(t) and global annual mean temperature T(t). This requires assuming a linear relationship between carbon content relative to (preindustrial) equilibrium and greenhouse effect radiative forcing as well as a linear relationship between ocean heat and carbon uptake. A numerically solvable model is proposed to when the standard log relationship empirically demonstrate by Myhre et al. The results of the simplified model are compared with a set of eight 3D ESMs. These results show that within a range of annual emissions the linear response T(t) to cumulative emissions obtained by simplified models are in good agreement with those of the set of ESMs chosen.

General comments

The manuscript is written in a clear fashion. The research question proposed, the methodology and results obtained are clearly presented and the assumptions taken are well argued. The results obtained by the paper provide a useful tractable model that can be used for research in areas beyond the climate sciences, for instance in economics and finance where tractable models are needed for coupling with the state-of-the-art models used. The paper is worthy of publication. It could could however be improved by addressing the following points:

1. The results presented show that the analytically solvable model is in agreement with the results obtained with the ESMs up to a point. Although mentioned, could the authors further stress the range of cumulative emissions where the analytically solvable model is in good agreement and when it is no longer reliable?.

1. While the authors focus on the flat10MIP experiment to carry out comparison and validation of their model that supports the TCRE heuristic of linear response to cumulative emissions, it would be useful for the reader to (briefly) mention that the climate system is inherently nonlinear and that the linearity assumption holds within a certain range of temperature and carbon which is still uncertain today. Several studies also cited in the same AR6 report (section 7.4.3.1 State-dependence of Feedbacks in Models) stress the limits of this proportionality relationship which no longer holds at higher atmospheric carbon levels and global temperature.

1. It would be useful the understand whether (and why) the subset of 8 ESMs used in flat10MIP are representative of the over 45 models included in IPCC reports and participating in CMIP experiments. Do the 8 models cover the spread of CMIP6 models? If not, several of the conclusions should be tempered to note that they appear robust across the set of tested models under flat emissions.

1. Concerning the discussion made and the justification to exclude ACCESS from the analysis of the climate-carbon dynamics, could the authors detail whether ACCESS uses a different land surface model than the 7 other ESMs to support the hypothesis made?

1. In Table 3, the authors present “adjusted parameters” for analytical and numerical solutions. Could the authors further comment (already started in lines 175-179) on the large wedge between estimated and adjusted parameter for certain models (UKESM, CESM2) in contrast with the good fit with MPI-ESM?

1. Could the authors elaborate on the reason for only showing results with the MPI model for the CDR experiments in section 2.3. Is it a question of data availability which may be? If so, please do say so, and otherwise could results with all other models be also presented to support the claim made.

Detailed comments

• The size of the x- and y-axis labels and ticks should be increased for readability (as as well as the font size of the legends in each of the subplots across figures).
• In Fig 6 the label should be NorESM2-LM not NORESM2-LM
• Fig A3: Could the authors write in the figure text what GCB stands for?
• Across the whole article, I read “GCB” in the legend of figures, however it should be “GCP” I think.
• Fig 2 text could eliminate “analogous to Figure 1”
• In Fig 7 it would be useful to use different colors for the flat10 and the cdr experiments that look very similar. Also, why is there the bisectrice in the left panel and not on the right panel.
• There seems to be a typo in Winker et al (2024) in Appendix A2 – which should be Winkler.
• Line 60, “zeroes” should be “zeros”
• Line 107: “Because the later term is proportional” should be “Because the latter term is proportional”
• The title of section 2.1 should be “Analytical solution for the dynamical system”