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Choose the incident photon along the $x$ direction. Conservation of four-momentum preserves the squared total energy invariant $\mathcal{S}=E_{\mathrm{tot}}^2-c^2|{\mathbf{p}}_{\mathrm{tot}}{|}^2$. Initially,In the centre-of-momentum frame let the outgoing photon energy be $e_{\ast }>0$; the larger particle has opposite momentum magnitude $e_{\ast }/c$ and energy $\sqrt{M^2{c}^4+e_{\ast }^2}$. ThusConversely, whenever $\mathcal{S}>M^2{c}^4$, the positive value $e_{\ast }=(\mathcal{S}-M^2{c}^4)/(2\sqrt{\mathcal{S}})$ constructs a final state with the required invariant, which can be boosted to the laboratory frame. The limiting photon excitation threshold with a soft final photon is thereforeStrictly, an outgoing physical photon requires $E>E_{\mathrm{th}}$; the printed minimum is the threshold infimum approached as the final photon becomes soft.