This manuscript addresses an important issue with a discrepancy between an emergent constraint that held well for CMIP5 and less so for CMIP6. The authors identify the reason, relating to an assumption in the emergent constraint's derivation.

I found this to be a very useful, interesting, and convincing paper; I cannot remember the last time, if ever, that I had so few comments on a manuscript. Nice work.

Minor Comments:

It would have been nice to explain a bit more in the abstract what 'the internal variability parameter' is, if there is space.

It would have been nice to see Spearman or Kendall correlations & also p-values on these correlations, though I understand why the authors chose to stick with the Pearson correlation that's in the literature being compared to.

Even More Minor Comments:

Line 2 – suggest deleting ‘(then latest) state-of-the-art’ – sounds impartial

Line 7 – would say ‘is weaker’ not ‘got weaker’

Line 64 – incomplete sentence. In general lines 64-69 could use some rephrasing.

Line 161 – uncorrelated with what?

Line 268 – would replace ‘good’ with ‘sound’

Line 272 – would replace ‘good’ with ‘strong’

Thanks for the enjoyable read.

B. B. Cael

The manuscript

"Testing the assumptions in emergent constraints; Why does the 'Emergent constraint on equilibrium climate sensitivity from global temperature variability' work for CMIP5 and not CMIP6? by Mark S. Williamson, Peter M. Cox, Chris Huntingford and Femke J. M. M. Nijsse

reviews a standard protocol for the derivation of the equilibriium climate sensitivity (ECS) from annual global mean temperature variability, that was introduced successfully on CMIP5 multimodel ensemble in 2018 (Cox et al. 2018). This emergent constraint proved to be much less robust in CMIP6 multimodel ensemble wrt. CMIP5, and this work addresses specifically the main assumptions underlying it, in order to identify what is not holding anymore in CMIP6 models.

Overall, I acknowledge that the work has a robust, logical structure: the methodology is well established, some important additions, such as testing the 2-boxes conceptual model provide better context for the results obtained with the simpler Hasselmann's model. I do think, as well, that the manuscript falls a bit short in providing an explanation for the reasons why CMIP6 models do not exhibit such a strong emergent constraint as the one found in CMIP5 ensembles. In doing so, I struggle to see how the work could contribute to broader discussions on what use one can make of emergent constraints in the context of detection of the forced signal and attribution. Therefore, I think that the manuscript could substantially improve if at least the following two aspects would be taken into account.

• Why is the internal variability parameter so crucial in defining the ECS emergent constraint in CMIP6. Why is it not the case for CMIP5?
• Is the ECS-Ψ constraint peculiar? What happens with other notable emergent constraints? Do they also differ from CMIP5? If so, do they differ because of similar reasons?

I believe that the first point is of particular relevance, given its implications for the development of synthetic model diagnostics and the usage of historical/paleoclimatic evidences to better understand and predict future climate scenarios.

SPECIFIC COMMENTS

• ll. 56-57: some reference on the usage of simple models for the reproduction of forced global mean temperature response could be useful here and elsewhere in the manuscript;
• ll. 101-104: I do not have clear if the choice of 15 CMIP5 models is just guided by the need for consistency with Geoffrey et al. 2013 work or if there are other practical/theoretical reasons for that. Given that to the best of my knowledge more CMIP5 models should be available, I wonder if it would be possible to have a similar amount of models in both CMIP5 and CMIP6 ensembles. If that is not possible, I wonder if some arguments could be provided on the implications of the size of the ensemble for the robustness of the discussed relation.
• ll. 105-106: a bit in line with my previous comment, I see that it is a common procedure limiting to r1i1p1 or r1i1p1f1 runs, but it might be worthwhile, given that the interpretation strongly relies on the retrieval of the internal variability parameter, to discuss a bit if such parameter holds across different ensemble runs. It is not entirely straightforward to me, whether the choice of the parameter in the chosen runs would be representative of the other runs as well;
• ll. 155-156: provided the discussion above, I do not have clear why the authors opt for linearly detrending the temperature with the 55 years moving window, especially given that in a previous paper (Cox et al. 2018b) they noticed that retaining the external forcing would possibly improve the emergent relationship;
• l. 160: if the correlation value is meant to be the one in the title of the panels of in Figure 2, it would be useful to explicitly mention it here;
• Figures 4 and 6: these figures, showing the relations between the two assumed parameters in CMIP5 and CMIP6, do not seem to add more arguments to the discussion than what already mentioned in the text. Consider whether is possible to remove them;
• Figures 9 and 10: when comparing the PDFs for piControl and historical, the authors evidence their similarities. It is a bit overlooked, though, that at first glance CMIP5 and CMIP6 in the historical runs exhibit substantially different medians and variance. Furthermore, the median for piControl in CMIP6 is possibly negative, whereas it is positive in CMIP5. Can the authors provide an explanation for that?
• l. 288: the authors acknowledge that this remains an unanswered question, but I do think it is crucial to try to provide even a speculative explanation for that, in order to improve the usability of the main result described in the manuscript, at least something that could serve as triggering hypothesis for future work;

TECHNICAL CORRECTIONS

• l. 67: for ‘all practical purposes’ -> ‘for all practical purposes’;
• l. 175: is it actually -0.06, rather than -0.60;
• l. 226: “results” -> “result”;
• l. 238: “RH” -> “rhs”;
• Figures 9 and 10: the legends do not seem to agree with the caption and the text;