Three-photon-annihilation contributions to positronium energies at order $m \alpha^7$

Positronium spectroscopy (n=1 hyperfine splitting, n=2 fine structure, and the 1S-2S interval) has reached a precision of order 1 MHz. Vigorous ongoing efforts to improve the experimental results motivate the calculation of the positronium energy levels at order $m \alpha^7$. In this work we present the result for a complete class of such contributions--those involving virtual annihilation of positronium to three photons in an intermediate state. We find an energy shift of $2.6216(11) m \alpha^7/(n \pi)^3$ for the spin-triplet S state with principal quantum number n.