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tokens: this figure; both parameters are prior dominated. 5.1 CMB lensing with <italic>A</italic><sub>s</sub> prior and supernova Ω<sub>m</sub> constraint With a conservative external A s prior, the L eq – A s degeneracy can be broken, leading to a tight constraint on L eq . Since L eq depends only on Ω m and H 0 , given an additional Ω m constraint, H 0 can then be determined. As described in Section 4 , our conservative A s prior (10 9 A s = 2.19 ± 0.09) is motivated by current constraints from the primary CMB, while our Ω m constraint (Ω m = 0.3 ± 0.012) is forecast for future supernovae measurements. Both the A s and Ω m constraints are independent of assumptions about r s . In Fig. 3 , we show (blue curves) the results of imposing the CMB-motivated A s prior on the CMB lensing constraints from Fig. 2 . Imposing this prior results in tighter constraints on L eq than in Fig. 2 , as expected. The results of additionally imposing the supernova prior on Ω m are shown with the red curves in Fig. 3 . We find that an uncertainty on H 0 of 3.5 km s −1 Mpc −1 is achieved. We will refer to this as our pessimistic forecast, since it assumes a conservative A s prior and a modestly improved Ω m from future surveys. Figure 3. Forecast constraints for a CMB-S4 measurement of the CMB lensing power spectrum upon including external A s and Ω m information. By imposing a conservative, CMB-motivated (but r s independent) prior on A s we break the A s – L eq degeneracy, leading to a constraint on L eq ≡ k eq χ * (blue curves; compare to Fig. 2 ). Since , the resultant constraints on Ω m and H 0 are degenerate (the black dashed curve illustrates the expected degeneracy). By including an additional Ω m constraint from supernovae we break the Ω m – H 0 degeneracy, and obtain a constraint on H 0 (red curve, with marginalized parameter uncertainties reported along the diagonal). As an alternative to a supernova constraint on Ω m , we also consider adding information from an LSST-like measurement of cosmic shear (orange curve); in combination with CMB lensing, galaxy lensing also provides Ω m information that serves to break the H 0 –Ω m degeneracy. Because of large-scale information in the CMB lensing power spectrum, our constraints are not very sensitive to the A s prior. If this prior is completely removed, we still obtain a constraint on H 0 (grey dashed curve). We note that the constraints on H 0 are not very sensitive to the A s prior. As can be seen in Fig. 2 , even in the absence of any A s prior, we obtain some constraint on L eq . This is likely because low- L information in the lensing power spectrum breaks the degeneracy between A s and L eq . In the absence of the A s prior, the combination of CMB lensing and the Ω m constraint yields an H 0 uncertainty of roughly 6 km s −1 Mpc −1 (grey dashed curves in Fig. 3 ). It is also interesting to consider how the constraints on H 0 could improve with more optimistic constraints on A s , Ω m , and Ω b . For