Clearing up the Strong $CP$ problem

The absence of a neutron electric dipole moment (EDM) constrains the quantum chromodynamics (QCD) theta angle to be less than one part in ten billion, posing the Strong $CP$ problem. We revisit two classes of proposed solutions. First, we show that when $P$ or $CP$ is realized as a gauged discrete symmetry - as can arise in quantum gravity - the vacuum necessarily preserves $CP$, contrary to recent claims that discrete-symmetry solutions fail. Gauged discrete models face model-building challenges, such as avoiding contributions to the neutron EDM after spontaneous $P$ or $CP$ breaking, but in principle have no fundamental obstructions. Second, we critically examine recent arguments that the Strong $CP$ problem is illusory, demonstrating that a nonzero neutron EDM at finite $\bar\theta$ follows directly from well-understood QCD dynamics. Taken together, our results reinforce the reality of the Strong $CP$ problem and highlight gauged discrete-symmetry realizations of $P$ or $CP$ as plausible solutions.