  We devise a general method for obtaining $0$-form noninvertible discrete
chiral symmetries in $4$-dimensional $SU(N)/\mathbb Z_p$ and $SU(N)\times
U(1)/\mathbb Z_p$ gauge theories with matter in arbitrary representations,
where $\mathbb Z_p$ is a subgroup of the electric $1$-form center symmetry. Our
approach involves placing the theory on a three-torus and utilizing the
Hamiltonian formalism to construct noninvertible operators by introducing
twists compatible with the gauging of $\mathbb Z_p$. These theories exhibit
electric $1$-form and magnetic $1$-form global symmetries, and their generators
play a crucial role in constructing the corresponding Hilbert space. The
noninvertible operators are demonstrated to project onto specific Hilbert space
sectors characterized by particular magnetic fluxes. Furthermore, when
subjected to twists by the electric $1$-form global symmetry, these surviving
sectors reveal an anomaly between the noninvertible and the $1$-form
symmetries. We argue that an anomaly implies that certain sectors,
characterized by the eigenvalues of the electric symmetry generators, exhibit
multi-fold degeneracies. When we couple these theories to axions, infrared
axionic noninvertible operators inherit the ultraviolet structure of the
theory, including the projective nature of the operators and their anomalies.
We discuss various examples of vector and chiral gauge theories that showcase
the versatility of our approach.
