2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/48051A lower bound on $Δm^{2}$ 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 $Δm^{2}>0.0077$ eV$^{2}$ is easily obtained. Much higher mass scales are implied by the reported energy independence and magnitude of $R$5 pages, 2 figures. The revisions include a full fit to R(E) and a discussion of geomagnetic effectsHigh Energy Physics - PhenomenologyHigh Energy Physics - Experimenttext