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Strong three-body interactions above threshold govern the dynamics of manyexotics and conventional excited mesons and baryons. Three-body finite-volumeenergies calculated from lattice QCD promise an ab-initio understanding ofthese systems. We calculate the three-$\pi^+$ spectrum unraveling thethree-body dynamics that is tightly intertwined with the $S$-matrix principleof three-body unitarity and compare it with recent lattice QCD results. Forthis purpose, we develop a formalism for three-body systems in moving framesand apply it numerically.

Three-body unitarity versus finite-volume $\pi^+\pi^+\pi^+$ spectrum from lattice QCD