minimum $\lambda_{H\sigma}$ are 0.2, 0.22, 0.30 and 0.34 respectively. Note, however, that a large Yukawa coupling $Y_{\nu}\geq 0.6$ is not allowed. The reason is two-fold: $\lambda$ becomes negative for $\lambda_{H\sigma}\leq 0.34$, while non-perturbative effects arise for $\lambda_{H\sigma}\geq 0.34$.

To sum up, these results show how, in contrast to the type-I seesaw with explicit breaking of lepton number, this Majoron version can have stable electroweak vacuum all the way upto Planck scale for adequate Yukawa coupling choices. Thus, the Majoron completion of the (3,1) type-I seesaw can be considered as a full theory, unlike the simplest version which at best can be considered as a effective theory.

Going to the (3,3) Majoron type-I seesaw with sequential right handed neutrino assignment, we find that the Higgs vacuum can be still kept stable up to Planck scale for appreciable Yukawa couplings. Of course, the presence of two more new fermions means that the maximum values of $Y^{ai}_\nu$, for which Higgs vacuum stability can be achieved up to Planck scale, is somewhat reduced. In Fig. \[fig:maj-robustness\] we compare the Higgs vacuum stability of the (3,3) Majoron seesaw case with the (3,1) analogue as well as with Standard Model.

![[]{data-label="fig:maj-robustness"}](plots/MajoronThreeGen.pdf){width="70.00000%"}

In plotting Fig. \[fig:maj-robustness\] we 