  The stationary horizon (SH) is a stochastic process of coupled Brownian
motions indexed by their real-valued drifts. It was first introduced by the
first author as the diffusive scaling limit of the Busemann process of
exponential last-passage percolation. It was independently discovered as the
Busemann process of Brownian last-passage percolation by the second and third
authors. We show that SH is the unique invariant distribution and an attractor
of the KPZ fixed point under conditions on the asymptotic spatial slopes. It
follows that SH describes the Busemann process of the directed landscape. This
gives control of semi-infinite geodesics simultaneously across all initial
points and directions. The countable dense set $\Xi$ of directions of
discontinuity of the Busemann process is the set of directions in which not all
geodesics coalesce and in which there exist at least two distinct geodesics
from each initial point. This creates two distinct families of coalescing
geodesics in each $\Xi$ direction. In $\Xi$ directions, the Busemann difference
profile is distributed like Brownian local time. We describe the point process
of directions $\xi\in\Xi$ and spatial locations where the $\xi\pm$ Busemann
functions separate.
