$\Delta m^{2}$ Limits from $R(E_{\nu})$
  A lower bound on $\Delta m^{2}$ for atmospheric neutrino oscillations can be obtained from the global value of $R$ at a given energy. This bound may be more reliable than one based on the observed isotropy. Some global values of $R$ imply an anisotropy in the neutrino flux that may, or may not be manifest in the observed neutrino interactions. A limit of $\Delta m^{2}>0.0077$ eV$^{2}$ is easily obtained. Much higher mass scales are implied by the reported energy independence and magnitude of $R$ 
J.M. LoSecco