  We compute all $2\to5$ gluon scattering amplitudes in planar $\mathcal{N}=4$
super-Yang-Mills theory in the multi-Regge limit that is sensitive to the
non-trivial ("long") Regge cut. We provide the amplitudes through four loops
and to all logarithmic accuracy at leading power, in terms of single-valued
multiple polylogarithms of two variables. To obtain these results, we leverage
the function-level results for the amplitudes in the Steinmann cluster
bootstrap. To high powers in the series expansion in the two variables, our
results agree with the recently conjectured all-order central emission vertex
used in the Fourier-Mellin representation of amplitudes in multi-Regge
kinematics. Our results therefore provide a resummation of the Fourier-Mellin
residues into single-valued polylogarithms, and constitute an important
cross-check between the bootstrap approach and the all-orders multi-Regge
proposal.

High Energy Physics - Theory