  The possibility that gauge theories with chiral symmetry breaking below the
conformal window exhibit an infrared fixed point is explored. With this
assumption three aspects of pion physics are reproduced if the the quark mass
anomalous dimension at the infrared fixed point is $\gamma_* = 1$: First, by
matching the long-distance scalar adjoint correlation function. Second, by
perturbing the fixed point by a small quark mass, the $m_q$-dependence of the
pion mass is reproduced by renormalisation group arguments. Third, consistency
of the trace anomaly and the Feynman-Hellmann theorem, for small $m_q$, imply
the same result once more. This suggests the following picture for the
conformal window: close to its upper boundary $\gamma_*$ is zero and grows as
the number of fermions is reduced until its lower boundary $\gamma_*=1$ is
reached, where chiral symmetry breaking sets in. Below, the strongly coupled
gauge theory with $\gamma_*=1$ is infrared dual to the free theory of pions. A
possible dilaton sector of the scenario will be addressed in a companion paper.
