We explore a re-parameterization of the lensing amplitude tension between weak lensing (WL) and cosmic microwave background (CMB) data and its implications for a joint resolution with the Hubble tension. Specifically, we focus on the lensing amplitude over a scale of 12 Mpc in absolute distance units using a derived parameter $S_{12}$ and show its constraints from recent surveys in comparison with Planck 2018. In WL alone, we find that the absolute distance convention correlates $S_{12}$ with $H_0$. Accounting for this correlation in the 3D space $S_{12}\times ω_m \times h$ reproduces the usual levels of $2\sim 3σ$ tension inferred from $S_8\timesΩ_m$. Additionally, we derive scaling relations in the $S_8\times h$ and $S_{12}\times h$ planes that are allowed by $Λ$CDM and extrapolate target scalings needed to solve the $H_0$ and lensing-amplitude tensions jointly in a hypothetical beyond-$Λ$CDM model. As a test example, we quantify how the early dark energy scenario compares with these target scalings. Useful fitting formulae for $S_8$ and $S_{12}$ as a function of other cosmological parameters in $Λ$CDM are provided, with 1% precision.

Matches version published in PRD