  We report new results for lattice gauge theories with twelve fermion flavors
in the fundamental representation and two fermion flavors in the two-index
symmetric (sextet) representation of the SU(3) color gauge group. Both models
are important in searching for a viable composite Higgs mechanism in the Beyond
the Standard Model (BSM) paradigm. We subject both models to opposite
hypotheses inside and outside of the conformal window. In the first hypothesis
we test chiral symmetry breaking ($\chi{\rm SB}$) with its Goldstone spectrum,
$F_\pi$, the $\chi{\rm SB}$ condensate, and several composite hadron states as
the fermion mass is varied in a limited range with our best effort to control
finite volume effects. Supporting results for $\chi{\rm SB}$ from the running
coupling based on the force between static sources is also presented. In the
second test for the alternate hypothesis we probe conformal behavior driven by
a single anomalous mass dimension under the assumption of unbroken chiral
symmetry. Our results show very low level of confidence in the conformal
scenario.
