  We construct a general class of pseudo-Goldstone composite Higgs models,
within the minimal SO(5)/SO(4) coset structure, that are not necessarily of
moose-type. We characterize the main properties these models should have in
order to give rise to a Higgs mass around 125 GeV. We assume the existence of
relatively light and weakly coupled spin 1 and 1/2 resonances. In absence of a
symmetry principle, we introduce the Minimal Higgs Potential (MHP) hypothesis:
the Higgs potential is assumed to be one-loop dominated by the SM fields and
the above resonances, with a contribution that is made calculable by imposing
suitable generalizations of the first and second Weinberg sum rules. We show
that a 125 GeV Higgs requires light, often sub-TeV, fermion resonances. Their
presence can also be important for the models to successfully pass the
electroweak precision tests. Interestingly enough, the latter can also be
passed by models with a heavy Higgs around 320 GeV. The composite Higgs models
of the moose-type considered in the literature can be seen as particular limits
of our class of models.
