  Gildener-Weinberg (GW) models of electroweak symmetry breaking are especially
interesting because the low mass and nearly Standard Model couplings of the
$125\,{\rm GeV}$ Higgs boson, $H$, are protected by approximate scale symmetry.
Another important but so far under-appreciated feature of these models is that
a sum rule bounds the masses of the new charged and neutral Higgs bosons
appearing in {\em all} these models to be below about $500\,{\rm GeV}$.
Therefore, they are within reach of LHC data currently or soon to be in hand.
Also so far unnoticed of these models, certain cubic and quartic Higgs scalar
couplings vanish at the classical level. This is due to spontaneous breaking of
the scale symmetry. These couplings become nonzero from explicit scale breaking
in the Coleman-Weinberg loop expansion of the effective potential. In a
two-Higgs doublet GW model, we calculate $\lambda_{HHH} \simeq
2(\lambda_{HHH})_{\rm SM} = 64\,{\rm GeV}$. This corresponds to $\sigma(pp \to
HH) \cong 15$--$20\,{\rm fb}$, its {\em minimum} value for $\sqrt{s} =
13$--$14\,{\rm TeV}$ at the LHC. It will require at least the $27\,{\rm TeV}$
HE-LHC to observe this cross section. We also find $\lambda_{HHHH} \simeq
4(\lambda_{HHHH})_{\rm SM} = 0.129$, whose observation in $pp \to HHH$ requires
a $100\,{\rm TeV}$ collider. Because of the above-mentioned sum rule, these
results apply to {\em all} GW models. In view of this unpromising forecast, we
stress that LHC searches for the new relatively light Higgs bosons of GW models
are by far the surest way to test them in this decade.
