Neutrino Masses and the Vacuum Scale: A Fourth-Power Sum That Overshoots by Five Orders of Magnitude

The fourth root of the observed vacuum density, about 2.24 milli-electronvolts, lies close to the neutrino mass scale, a proximity often noted. This paper makes the comparison quantitative using the masses fixed by oscillation experiments: in normal ordering the sum of fourth powers of the three neutrino masses is 6.26 x 10^-6 eV^4 even when the lightest mass vanishes, exceeding the observed density of 2.52 x 10^-11 eV^4 by a factor of about 2.48 x 10^5. A standard loop factor reduces the overshoot to about a thousand but does not remove it. The proximity is therefore real at the level of the mass scale and misleading at the level of the density. The paper computes from stated assumptions and reports no measurement.