  We, for the first time, report a first-principle proof of the equations of
state used in the hydrodynamic theory for integrable systems, termed
generalized hydrodynamics (GHD). The proof makes full use of the graph
theoretic approach to Thermodynamic Bethe ansatz (TBA) that was proposed
recently. This approach is purely combinatorial and relies only on common
structures shared among Bethe solvable models, suggesting universal
applicability of the method. To illustrate the idea of the proof, we focus on
relativistic integrable quantum field theories with diagonal scatterings and
without bound states such as strings.
