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The $γ^\star γ^\star \to ππ$ scattering amplitude plays a key role in a wide range of phenomena, including understanding the inner structure of scalar resonances as well as constraining the hadronic contributions to the anomalous magnetic moment of the muon. In this work, we explain how the infinite-volume Minkowski amplitude can be constrained from finite-volume Euclidean correlation functions. The relationship between the finite-volume Euclidean correlation functions and the desired amplitude holds up to energies where $3π$ states can go on shell, and is exact up to exponentially small corrections that scale like $\mathcal{O}(e^{-m_πL})$, where $L$ is the spatial extent of the cubic volume and $m_π$ is the pion mass. In order to implement this formalism and remove all power-law finite volume errors, it is necessary to first obtain $ππ\to ππ$, $πγ^\star \to π$, $γ^\star \toππ$, and $ππγ^\star \toππ$ amplitudes; all of which can be determined via lattice quantum chromodynamic calculations.