The minimal SO(5)/SO(4) linear $\sigma$-model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, for instance, if the Higgs has a Goldstone boson parenthood. Varying the $\sigma$ mass allows to sweep from the perturbative regime to the usual non-linear implementations. A complete renormalizable Lagrangian -including gauge bosons and fermions- is considered, and the implications for LHC data and precision observables are explored. Furthermore, the effective low-energy benchmark non-linear Lagrangian is determined, including bosonic and fermionic operators and the leading $1/m_\sigma$ corrections.