= Kolmogorov four-fifths law
{c}
{title2=$\langle(\Delta u_L)^3\rangle=-4\epsilon r/5$}
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.
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