Complete Results for Positronium Energy Levels at Ordermα6

We have completed theoretical predictions for positronium energy levels through order $m{\alpha }^6$ by the calculation of the spin independent radiative recoil correction. This contribution is significant and amounts to 10.64 MHz for the 1S state. We further perform a detailed comparison of theoretical predictions to experimental results for 1S-2S and 2S-2P transitions. There is a serious discrepancy between previous theoretical results for the hfs of the ground state and with corresponding experiments. This problem remains to be resolved.