$\mathcal{O}(1)$ and a neutrino mass parameter of order $10^{-1}$ eV i.e. of the order of the atmospheric scale [@deSalas:2017kay]. Hence, with $v_{\sigma}\approx 10^{14}$ GeV, $Y_{R}$ will be of $\mathcal{O}(1)$. But for large $Y_{R}$, $\lambda_{\sigma}$ would become negative very quickly when RG-evolved, so the scalar potential will not be bounded, as $\beta_{\lambda_{\sigma}}\simeq -Y_{R}^{4}$. Hence, we have chosen $v_{\sigma}\approx 10^{16}$ GeV i.e. at the Grand Unified Theory (GUT) scale, so that $Y_{R}\approx 10^{-2}$ and its effects will be small in the running of $\lambda_{\sigma}$. For such very large $v_{\sigma}\approx 10^{16}$ GeV the light and heavy Higgs sectors will be almost decoupled, though we can still allow appreciable $\lambda_{H\sigma}$ with very small mixing angle $\alpha$, see Appendix \[app:higgs\].

Our results for the (3,1) type-I seesaw mechanism with Majoron are shown in Fig. \[RG-31aJ-Running\], where we have taken $\lambda_{\sigma}=0.1$ at the initial mass scale $m_{t}$. The RG evolution in Fig. \[RG-31aJ-Running\] is shown for four Yukawa coupling values $Y_{\nu}=0.1,\,0.3,\,0.4\,\text{and}\,0.5$. It shows that, indeed, the stability properties can substantially improve due to the presence of the new scalar. In fact, for appreciable Yukawa couplings, one can have positive $\lambda$ all the way up to Planck scale. For the Yukawa couplings $Y_{\nu}=0.1,\,0.3,\,0.4\,\text{and}\,0.5$, the required values of 