with low-scale violation of lepton number [@Joshipura:1992hp] deserve special interest in this context, because of their potential in changing the structure of the vacuum as well as their phenomenological impact upon Higgs boson decays, specially the presence of invisible Higgs decays [@Joshipura:1992hp]. An analysis of the resulting sensitivities of Higgs Boson searches at the ATLAS and CMS experiments at the LHC was also performed in the presence of the Higgs boson triplet associated to the type-II seesaw mechanism [@Bonilla:2015eha; @Bonilla:2015jdf].

Here we examine more closely the issue of the consistency of the Higgs vacuum within the more conventional high-scale type-I seesaw extensions of the Standard Model and their dark matter completions. We focus specifically on 21 type-I seesaw extensions both with explicit [@Schechter:1980gr] as well as spontaneous violation of lepton number [@Chikashige:1980ui; @Schechter:1981cv]. The latter contains a physical Nambu-Goldstone boson, dubbed Majoron. We show how the associated scalar couplings can easily restore stability of the electroweak symmetry breaking even if the lepton number violation scale is high.

We stress that 21 seesaw extensions are the most general ones, as they can be formulated with any number of “right-handed” neutrinos, as noted in Ref. [@Schechter:1980gr], who proposed seesaw models with an 