An ultralight pseudoscalar boson
  Using a fundamental discrete symmetry, $\mathbb{Z}_N$, we construct a
two-axion model with the QCD axion solving the strong-$CP$ problem, and an
ultralight axion (ULA) with $m_{\rm ULA}\approx 10^{-22}\text{ eV}$ providing
the dominant form of dark matter (DM). The ULA is light enough to be detectable
in cosmology from its imprints on structure formation, and may resolve the
small-scale problems of cold DM. The necessary relative DM abundances occur
without fine tuning in constructions with decay constants $f_{\rm ULA}\sim
10^{17}\text{ GeV}$, and $f_{\rm QCD}\sim 10^{11}\text{ GeV}$. An example model
achieving this has $N=24$, and we construct a range of other possibilities. We
compute the ULA couplings to the Standard Model, and discuss prospects for
direct detection. The QCD axion may be detectable in standard experiments
through the $\vec{E}\cdot\vec{B}$ and $G\tilde{G}$ couplings. In the simplest
models, however, the ULA has identically zero coupling to both $G\tilde{G}$ of
QCD and $\vec{E}\cdot\vec{B}$ of electromagnetism due to vanishing
electromagnetic and color anomalies. The ULA couples to fermions with strength
$g\propto 1/f_{\rm ULA}$. This coupling causes spin precession of nucleons and
electrons with respect to the DM wind with period $t\sim$months. Current limits
do not exclude the predicted coupling strength, and our model is within reach
of the CASPEr-Wind experiment, using nuclear magnetic resonance.
