$R$ down to 3D. This radius is a modulus and has associated a quantum fluctuation field $\phi$ with canonical kinetic term given by $R=r e^\frac{\phi}{M^{3d}_{\text{p}}\sqrt{2}}$. Here $r$ is any given reference scale to measure $R$ which we set equal to 1 GeV and $M^{3d}_{\text{p}}$ is the 3D Planck mass. Let us concentrate on the deep infrared region, well below the electron threshold, with $R\gg 1/m_e$. As explained in [@Nima] the 3D one-loop effective potential for $R$ is given by the expression $$V(R)\ =\ \ \frac {2\pi r^3 \Lambda_4}{R^2}\       -\ 4 \left( \frac {r^3}{720\pi R^6} \right) \ +\sum_{\nu_e,\nu_\mu,\nu_\tau}\ 
  r^{3} V_{\mathcal{C}}\left[R,m_{\nu_i}\right]\  .
   \label{potuno}$$ The first term comes from the 4D cosmological term $\Lambda_4$ after dimensional reduction (and going to the 3D Einstein frame). The second comes from the one-loop Casimir energy associated to the only two massless particles, the photon and the graviton. The factor 4 gives the number of helicity degrees of freedom of those fields. The remaining term is the contribution to the Casimir energy of the three neutrinos compactified with periodic boundary conditions, and is given by $$V_{\mathcal{C}}\left[R,m_{\nu_i}\right]=\frac{n_{\nu_i}\ m_{\nu_i}^{2}}{8\pi^{4}R^{4}}\sum_{n=1}^{\infty}\frac{K_{2}(2\pi m_{\nu_i} n R )}{n^{2}} .
    \label{potdos}$$ Here $n_{\nu_i}$ is the number of helicities for each neutrino (2 for Majorana and 