  We derive model-independent quantization conditions on the axion couplings
(sometimes known as the anomaly coefficients) to the Standard Model gauge group
$[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_q$ with $q=1,2,3,6$. Using these
quantization conditions, we prove that any QCD axion model to the right of the
$E/N=8/3$ line on the $|g_{a\gamma\gamma}|$-$m_a$ plot must necessarily face
the axion domain wall problem in a post-inflationary scenario. We further
demonstrate the higher-group and non-invertible global symmetries in the
Standard Model coupled to a single axion. These generalized global symmetries
lead to universal bounds on the axion string tension and the monopole mass. If
the axion were discovered in the future, our quantization conditions could be
used to constrain the global form of the Standard Model gauge group.
