Atomic Two-Body $(Z\alpha )^6$ Predictions

For muonic hydrogen, positronium, and ordinary hydrogen, we show that the existence of a relativistic two-body wave equation whose energy levels are physically accurate to order $(Z\alpha )^4$ (where $Z\alpha$ is the binding coupling constant) implies a previously unknown two-body Sommerfeld energy formula which can be used to predict energy terms to order $(Z\alpha )^6$ using simple algebra. Two such terms are verified to be physically correct by earlier $(Z\alpha )^6$ calculations for positronium. For muonic hydrogen and ordinary hydrogen, these terms are predictions.