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The stationary horizon (SH) is a stochastic process of coupled Brownianmotions indexed by their real-valued drifts. It was first introduced by thefirst author as the diffusive scaling limit of the Busemann process ofexponential last-passage percolation. It was independently discovered as theBusemann process of Brownian last-passage percolation by the second and thirdauthors. We show that SH is the unique invariant distribution and an attractorof the KPZ fixed point under conditions on the asymptotic spatial slopes. Itfollows that SH describes the Busemann process of the directed landscape. Thisgives control of semi-infinite geodesics simultaneously across all initialpoints and directions. The countable dense set $\Xi$ of directions ofdiscontinuity of the Busemann process is the set of directions in which not allgeodesics coalesce and in which there exist at least two distinct geodesicsfrom each initial point. This creates two distinct families of coalescinggeodesics in each $\Xi$ direction. In $\Xi$ directions, the Busemann differenceprofile is distributed like Brownian local time. We describe the point processof directions $\xi\in\Xi$ and spatial locations where the $\xi\pm$ Busemannfunctions separate.

The stationary horizon and semi-infinite geodesics in the directed landscape