This description of neutrinos as spheres, with radius one quarter of their Compton wavelength and thickness of the Planck length, surrounding a central core along their rotation axis, with diameter of the Planck length, allows only three neutrino mass eigenstates. Describing neutrinos as excitations of cosmic dark energy responsible for observed accelerating expansion of the universe and equating electron neutrino energy density to vacuum energy density (i.e., cosmic dark energy density) then predicts a neutrino mass sum consistent with observations.

Neutrinos (and charged Standard Model fermions) can be reliably described as spheres with radius $r$, mass $m$, and density $\rho =m/\left(\frac{4}{3}\pi r^3\right)$ consisting of spherical shells (with mass $m_S$, thickness of the Planck length $l_P=\sqrt{\frac{\hslash G}{c^3}}$, and matter density ${\rho }_S{l}_P$ per unit area) enclosing cylinders along their rotation axis (with mass $m_A$, diameter $l_P$, and matter density ${\rho }_A{l}_P^2$ per unit length). The resulting equation for neutrino radius

$\begin{array}{c}\frac{4}{3}\pi \rho r^3=m_S+m_A\\ ={\rho }_S{l}_{P}4\pi r^2+{\rho }_A\pi l_P^2\left(2r\right)\end{array}$

can be rewritten as

$\rho r^3-3{\rho }_S{l}_P{r}^2-\frac{3}{2}{\rho }_A{l}_P^{2}r=a{r}^3+b{r}^2+cr=0$

with $a=\rho$, $b=-3{\rho }_S{l}_P$ and $c=-\frac{3}{2}{\rho }_A{l}_P^2$. The discriminant $b^2{c}^2-4a{c}^3$ of the cubic equation is positive and the three real roots of the equation correspond to the radii of three neutrino states.

Characteristic lengths of neutrinos are their Compton wavelengths $\lambda =\frac{\hslash }{mc}$. If electron neutrinos are spheres with radius $r=\frac{1}{4}\frac{\hslash }{mc}$ and the lowest energy density in the universe (cosmic vacuum energy density ${\rho }_v=5.83\times {10}^{-30}\text{g}/{\text{cm}}^{\text{3}}$), they have mass $m_1={\left[\frac{\pi }{6}{\rho }_v{\left(\frac{\hslash }{c}\right)}^3\right]}^{\frac{1}{4}}=2.02\times {10}^{-36}\text{g}=0.0013\text{eV}$. Neutrino oscillation data [1] [2] predict $m_2=\sqrt{m_1^2+7.42\times {10}^{-5}{\left(\text{eV}\right)}^2}=0.00871\text{eV}$ and $m_3=\sqrt{0.5{\left(m_1+m_2\right)}^2+2.517\times {10}^{-3}{\left(\text{eV}\right)}^2}=0.0507\text{eV}$, resulting in neutrino mass sum = 0.0607 eV consistent with the minimum neutrino mass sum presented in Reference 3.

The description of neutrinos presented above allows only three neutrino mass eigenstates and predicts a neutrino mass sum consistent with observations [3] . As background, the Appendix reviews holographic analysis of Standard Model fermions described as spheres that: