Kolmogorov four-fifths law ($⟨(\Delta u_L{)}^3⟩=-4ϵr/5$)

The signed third-order longitudinal structure function has this leading inertial range form under homogeneity, isotropy and the usual local-equilibrium scale separation. Its coefficient follows from the Kármán-Howarth equation and the mean kinetic energy budget, without assuming the Kolmogorov two-thirds law. Its negative sign encodes a forward energy cascade; it does not establish every higher-order similarity exponent.