  The dimension of the Hilbert space of QFT scales exponentially with the
volume of the space in which the theory lives, yet in supersymmetric theories,
one can define a graded dimension (such as the supersymmetric index) that
counts just the number of bosonic minus fermionic ground states. Can we make
this observation useful in non-supersymmetric QFTs in four dimensions? In this
work, we construct symmetry graded state sums for a variety of
non-supersymmetric theories. Among the theories we consider is one that is
remarkably close to QCD: Yang--Mills theory with $N_f = N_c$ fundamental Dirac
fermions and one adjoint Weyl fermion, QCD(F/adj). This theory can be obtained
from SQCD by decoupling scalars and carry exactly the same anomalies. Despite
the existence of fundamental fermions, the theory possess an exact 0-form
color-flavor center (CFC) symmetry for a particular grading/twist under which
Polyakov loop is a genuine order parameters. By a two-loop analysis, we prove
that CFC-symmetry remains unbroken at small $\beta $ due to grading. Chiral
symmetry is spontaneously broken within the domain of validity of semi-classics
on $\mathbb R^3 \times S^1$ in a pattern identical to $N_f=N_c$ SQCD on
$\mathbb R^4$ and the two regimes are adiabatically connected. The vacuum
structures of the theory on $\mathbb R^4$ and $\mathbb R^3 \times S^1$ are
controlled by the same mixed 't Hooft anomaly condition, implying a remarkable
persistent order.
