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It is believed that for metric-like models in the KPZ class the followingproperty holds: with probability one, starting from any point, there are atmost two semi-infinite geodesics with the same direction that do not coalesce.Until now, such a result was only proved for one model - exponential LPP(Coupier 11') using its inherent connection to the totally asymmetric exclusionprocess. We prove that the above property holds for the directed landscape, theuniversal scaling limit of models in the KPZ class. Our proof reduces theproblem to one on line ensembles and therefore paves the way to show similarresults for other metric-like models in the KPZ class. Finally, combining ourresult with the ones in (Busani, Seppalainen,Sorensen 22', Bhatia 23') weobtain the full qualitative geometric description of infinite geodesics in thedirected landscape.

Non-existence of three non-coalescing infinite geodesics with the same direction in the directed landscape