  The recent Pulsar Timing Arrays (PTAs) nano-Hz gravitational wave (GW)
background signal can be naturally induced by the annihilation of domain walls
(DWs) formed at a symmetry-breaking scale $f\simeq 200$~TeV in the clockwork
axion framework. Based on our first successful and precise prediction, we for
the first time suggest that the recent PTA observations strongly support the
novel mechanism of the QCD instanton-induced DW annihilation in the clockwork
axion framework. We also for the first time discover a novel correlation
between dark matter (DM) relic abundance and nano-Hz GW background, which in
turn indicates a natural connection between the axion decay constant and the
symmetry-breaking scale in the clockwork framework. We find that the GW signal
has a peak $h^2\Omega_{\rm GW}\simeq 10^{-6.6}-10^{-6.1}$ at about 50~nHz,
which is definite and testable for future PTA data at frequencies $\gtrsim
25$~nHz and CMB-S4 experiment. We also propose various phenomena that may
appear in PTAs and future GW interferometers.
