  We study the bilinear and higher-order fermion condensates in $4$-dimensional
$SU(N)$ gauge theories with a single Dirac fermion in a general representation.
Augmented with a mixed anomaly between the $0$-form discrete chiral, $1$-form
center, and $0$-form baryon number symmetries (BC anomaly), we sort out
theories that admit higher-order condensates and vanishing fermion bilinears.
Then, the BC anomaly is utilized to prove, in the absence of a topological
quantum field theory, that nonvanishing fermion bilinears are inevitable in
infrared-gapped theories with $2$-index (anti)symmetric fermions. We also
contrast the BC anomaly with the $0$-form anomalies and show that it is the
former anomaly that determines the infrared physics; we argue that the BC
anomaly lurks deep to the infrared while the $0$-form anomalies are just
variations of local terms. We provide evidence of this assertion by studying
the BC anomaly in vector-like theories compactified on a small spacial circle.
These theories are weakly-coupled, under analytical control, and they admit a
dual description in terms of abelian photons that determine the deep infrared
dynamics. We show that the dual photons talk directly to the $1$-form center
symmetry in order to match the BC anomaly, while the $0$-form anomalies are
variations of local terms and are matched by fiat. Finally, we study the fate
of the BC anomaly in the compactified theories when they are held at a finite
temperature. The effective field theory that describes the low-energy physics
is $2$-dimensional. We show that the BC anomaly cascades from $4$ to $2$
dimensions.
