  We calculate the intrinsic quark spin contribution to the total proton spin
using overlap valence quarks on three ensembles of $2+1$-flavor RBC/UKQCD
domain-wall configurations with different lattice spacings. The lowest pion
mass of the ensembles is around 171 MeV which is close to the physical point.
With overlap fermions and topological charge derived from the overlap operator,
we verify the anomalous Ward identity between nucleon states with momentum
transfer. Both the connected and disconnected insertions of the axial-vector
current are calculated. For the disconnected-insertion part, the
cluster-decomposition error reduction (CDER) technique is utilized for the
lattice with the largest volume and the error can be reduced by $10\%\sim40\%$.
Nonperturbative renormalization is carried out and the final results are all
reported in the $\overline{{\rm MS}}$ scheme at 2 GeV. We determine the total
quark spin contribution to the nucleon spin to be $\Delta\Sigma=0.401(25)(37)$,
which is consistent with the recent global fitting result of experimental data.
The isovector axial coupling we obtain in this study is $g_A^3=1.256(16)(30)$,
which agrees well with the experimental value of 1.2723(23).
