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We outline a proof and give a discussion at a physical level of an assertion of Onsager's: namely, that a solution of incompressible Euler equations with Hölder continuous velocity of order h> 1 3 conserves kinetic energy, but not necessarily if h≤ 1 3. We prove the result under a “∗-Hölder condition” which is somewhat stronger than usual Hölder continuity. Our argument establishes also the fundamental result that the instantaneous (sub-scale) energy transfer is dominated by local triadic interactions for a∗-Hölder solution with exponenth in the range 0< h< 1. However, we must use a “band-averaged” energy flux for the proof: as we explain, the ordinary definition of the flux fails to adequately measure transport in wavenumber space (scale), since it is insensitive to the distance through which energy is displaced by individual interactions. We discuss some connections of the results with phenomenological theories of …