  We use a coarse-graining approach to prove that inter-scale transfer of
kinetic energy in compressible turbulence is dominated by local interactions.
Locality here means that interactions between disparate scales decay at least
as fast as a power-law function of the scale-disparity ratio. In particular,
our results preclude transfer of kinetic energy from large-scales directly to
dissipation scales, such as into shocks, in the limit of high Reynolds number
turbulence as is commonly believed. The results hold in broad generality, at
any Mach number, for any equation of state, and without the requirement of
homogeneity or isotropy. The assumptions we make in our proofs on the scaling
of velocity, pressure, and density structure functions are weak and enjoy
compelling empirical support. Under a stronger assumption on pressure
dilatation co-spectrum, we show that \emph{mean} kinetic and internal energy
budgets statistically decouple beyond a transitional "conversion" range. Our
analysis demonstrates the existence of an ensuing inertial scale-range over
which mean SGS kinetic energy flux becomes constant, independent of scale. Over
this inertial range, mean kinetic energy cascades locally and in a conservative
fashion, despite not being an invariant.
