precisely, axial $N_u +N_d$) is broken intrinsically in the quantum theory by an anomaly, and to a small extent by quark masses, and spontaneously by $\bar q q$ condensation. Simulations with large artifactual quark masses mangle this quasi-symmetry, just as they mangle chiral $SU(2)\times
    SU(2)$. For all the associated questions of $\eta^\prime$ phenomenology, squeezing out of instantons and approximate $U(1)_A$ restoration at high temperature, and axion cosmology, accurate implementation of [*classical*]{} $U(1)_A$ is essential;

-   tests of the instanton liquid model. This model predicts significant dependence of certain hadronic quantities on light quark masses.

### Explore QCD Variants

Besides enabling the noble bread-and-butter work of computing properties of real-world QCD, lattice gauge theory offers us the opportunity to explore variants, for example with different quark mass spectra, different representations, or different gauge groups. I have already mentioned many possibilities for the creative use of such flexibility, in quantitative anthropics, in testing our understanding of the phase structure, and in testing instanton liquid ideas. This list is far from exhaustive. As a general remark, we can test proposed qualitative explanations of phenomena in QCD by seeing whether they predict the correct direction of change in the phenomena as the underlying parameters 