AbstractCuprate superconductors have the highest critical temperatures (T_c) at ambient pressure, yet a consensus on the superconducting mechanism remains to be established. Finding an empirical parameter that limits the highest reachable T_c can provide crucial insight into this outstanding problem. Here, in the first two Ruddlesden-Popper members of the model Hg-family of cuprates, which are chemically nearly identical and have the highest T_c among all cuprate families, we use inelastic photon scattering to reveal that the energy of magnetic fluctuations may play such a role. In particular, we observe the single-paramagnon spectra to be nearly identical between the two compounds, apart from an energy scale difference of ~30% which matches their difference in T_c. The empirical correlation between paramagnon energy and maximal T_c is further found to extend to other cuprate families with relatively high T_c’s, hinting at a fundamental connection between them.