  Focusing on three-pion states with maximal isospin ($\pi^+\pi^+\pi^+$), we
present the first non-perturbative determination of an energy-dependent
three-hadron scattering amplitude from first-principles QCD. The calculation
combines finite-volume three-hadron energies, extracted using numerical lattice
QCD, with a relativistic finite-volume formalism, required to interpret the
results. To fully implement the latter, we also solve integral equations that
relate an intermediate three-body K matrix to the physical three-hadron
scattering amplitude. The resulting amplitude shows rich analytic structure and
a complicated dependence on the two-pion invariant masses, represented here via
Dalitz-like plots of the scattering rate.
