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 _LN(k) ∝ k ^−5/3 must replace in future theory (e.g., in finding the quantum turbulence decay rate) the previously used spectrum E _KS(k) ∝ k ^−7/5, which was recently shown to be inconsistent due to nonlocality of the 6-wave energy cascade.