consistent as an effective theory, see Fig. \[RG-31a-Running\].

![[]{data-label="RG-31a-Running"}](plots/HighScaleConsistensi.pdf){width="75.00000%"}

.

Stability in higher (3,n) type-I seesaw mechanism, n $\geq$ 2 {#sec:3gen}
-------------------------------------------------------------

Since right-handed neutrinos appear as singlets, there are no theory constraints on their number, so the (3,n) type-I seesaw mechanism, with n $\geq$ 2 is perfectly viable. Let us now briefly discuss this case. The (3,2) case leads to two light neutrinos getting small masses [@Schechter:1980gr], while in the “sequential” (3,3) case, all three light neutrinos get small masses. Hence these schemes are phenomenologically viable even in the absence of radiative corrections present, for example, in the scotogenic dark matter completion proposed in [@Rojas:2018wym].

As already mentioned, the problem of Higgs vacuum stability in type-I seesaw extensions only gets worse with the addition of extra right-handed neutrinos. This fact is clearly illustrated in Fig. \[robustness\], where we compare the Higgs quartic self coupling $\lambda$ evolution within the [[Standard Model ]{}]{}with (3,n) seesaw completions, with n = 1 and n = 3. For simplicity, in Fig. \[robustness\], we have fixed the benchmark value of $Y^{aj}_\nu = 0.5$; $a=j= 1,2,3$ and taken the off-diagonal terms to be zero for the $(3,3)$ case.

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

As can be seen from Fig. \[robustness\], 