  We derive a type of kinetic equation for Kelvin waves on quantized vortex
filaments with random large-scale curvature, that describes step-by-step
(local) energy cascade over scales caused by 4-wave interactions. Resulting new
energy spectrum $E\Sb{LN}(k)\propto k^{-5/3}$ must replace in future theory
(e.g. in finding the quantum turbulence decay rate) the previously used
spectrum $E\Sb {KS}(k)\propto k^{-7/5}$, which was recently shown to be
inconsistent due to nonlocality of the 6-wave energy cascade.
