  Both parameters in the Higgs field's potential, its mass and quartic
coupling, appear fine-tuned to near-critical values, which gives rise to the
hierarchy problem and the metastability of the electroweak vacuum. Whereas such
behavior appears puzzling in the context of particle physics, it is a common
feature of dynamical systems, which has led to the suggestion that the
parameters of the Higgs potential could be set through some dynamical process.
In this article, we discuss how this notion could be extended to physics beyond
the Standard Model (SM). We first review in which sense the SM Higgs parameters
can be understood as near-critical and show that this notion can be
extrapolated in a unique way for a generic class of SM extensions. Our main
result is a prediction for the parameters of such models in terms of their
corresponding Standard Model effective field theory Wilson coefficients and
corresponding matching scale. For generic models, our result suggests that the
scale of new (bosonic) physics lies close to the instability scale. We explore
the potentially observable consequences of this connection, and illustrate
aspects of our analysis with a concrete example. Lastly, we discuss
implications of our results for several mechanisms of dynamical vacuum
selection associated with various Beyond-Standard-Model (BSM) constructions.
