  We present the four-loop remainder function for six-gluon scattering with
maximal helicity violation in planar N=4 super-Yang-Mills theory, as an
analytic function of three dual-conformal cross ratios. The function is
constructed entirely from its analytic properties, without ever inspecting any
multi-loop integrand. We employ the same approach used at three loops, writing
an ansatz in terms of hexagon functions, and fixing coefficients in the ansatz
using the multi-Regge limit and the operator product expansion in the
near-collinear limit. We express the result in terms of multiple
polylogarithms, and in terms of the coproduct for the associated Hopf algebra.
From the remainder function, we extract the BFKL eigenvalue at
next-to-next-to-leading logarithmic accuracy (NNLLA), and the impact factor at
NNNLLA. We plot the remainder function along various lines and on one surface,
studying ratios of successive loop orders. As seen previously through three
loops, these ratios are surprisingly constant over large regions in the space
of cross ratios, and they are not far from the value expected at asymptotically
large orders of perturbation theory.
