  Conventionally, the strong-$CP$ problem is assumed to be a naturalness
puzzle, with the axion solution sometimes viewed as an ad hoc fix. Gravity is
either ignored or taken as a threat for the global Peccei-Quinn symmetry. We
explain that the situation is fundamentally different. In gravity, axion is a
matter of consistency imposed by the $S$-matrix: Each gauge sector must include
axion with exact relaxation of the corresponding $\bar{\theta}$. We show that
this favors an alternative and remarkably simple formulation of the axion,
fully fixed by the gauge redundancy of QCD, without involvement of a global
symmetry. The axion mechanism is a Higgs effect for the QCD $3$-form, ensuring
that physics is independent of $\bar{\theta}$ to all orders in operator
expansion. A near-future experimental detection of the neutron EDM will be an
unambiguous signal of $CP$-violating physics beyond the Standard Model. The
axion coupling is tied to the scale of gravity.
