  PHENIX measurements are presented for the cross section and double-helicity
asymmetry ($A_{LL}$) in inclusive $\pi^0$ production at midrapidity from
$p$$+$$p$ collisions at $\sqrt{s}=510$~GeV from data taken in 2012 and 2013 at
the Relativistic Heavy Ion Collider. The next-to-leading-order
perturbative-quantum-chromodynamics theory calculation is in excellent
agreement with the presented cross section results. The calculation utilized
parton-to-pion fragmentation functions from the recent DSS14 global analysis,
which prefer a smaller gluon-to-pion fragmentation function. The
$\pi^{0}A_{LL}$ results follow an increasingly positive asymmetry trend with
$p_T$ and $\sqrt{s}$ with respect to the predictions and are in excellent
agreement with the latest global analysis results. This analysis incorporated
earlier results on $\pi^0$ and jet $A_{LL}$, and suggested a positive
contribution of gluon polarization to the spin of the proton $\Delta G$ for the
gluon momentum fraction range $x>0.05$. The data presented here extend to a
currently unexplored region, down to $x\sim0.01$, and thus provide additional
constraints on the value of $\Delta G$. The results confirm the evidence for
nonzero $\Delta G$ using a different production channel in a complementary
kinematic region.
