  In this paper we show that an $H$-Brownian Gibbsian line ensemble is
completely characterized by the finite-dimensional marginals of its lowest
indexed curve for a large class of interaction Hamiltonians $H$. A particular
consequence of our result is that the KPZ line ensemble is the unique line
ensemble that satisfies the $H_1$-Brownian Gibbs property with $H_1(x) = e^x$
and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow
wedge KPZ equation.
