Inverse Compton backscattering maximum (${\epsilon }_{\max }^{\prime }={\gamma }^2(1+\beta {)}^2\epsilon /[1+2\gamma (1+\beta )\epsilon /(m{c}^2)]$)

For a head-on photon and massive-particle collision, boost into the initial particle rest frame, apply the Compton recoil formula for backward photon scattering, then boost the outgoing photon forward. For incident photon energy below the particle momentum times $c$, this backward-scattering endpoint is the maximal outgoing laboratory energy. The result in the title approaches $4{\gamma }^2\epsilon$ only when the particle is ultrarelativistic and its rest-frame incident photon energy is small compared with $m{c}^2$. Recoil limits amplification at larger incident energies and preserves the total-energy bound.