The measurement of the decay rates $\ecgg$ and $\ebgg$ are part of the BES III and Belle II programmes respectively as tests of the Standard Model. Here we provide, for the first time, precise SM values for these decay rates using lattice QCD. For $\Gec$ we obtain $6.788(61)\keV$ in good agreement with, but much more accurate than, experimental results using $\gamma\gamma\to\eta_c\to K\bar{K}\pi$. Our value is in 4 sigma tension with the PDG global fit result, however. Building on this study, we have been able to predict $\Geb=0.526(30)\keV$. We also compare the ratio of the form factors to the meson decay constants with expectations from NRQCD to assess how well nonrelativistic effective theories work in these two cases.