  In the present work, we systematically study the $\mathcal{\alpha}$ decay
preformation factors $P_{\alpha}$ within the cluster-formation model and
$\mathcal{\alpha}$ decay half-lives by the proximity potential 1977 formalism
for nuclei around $Z=82$, $N=126$ closed shells. The calculations show that the
realistic $P_{\alpha}$ is linearly dependent on the product of valance protons
(holes) and valance neutrons (holes) $N_pN_n$. It is consistent with our
previous works [X.-D. Sun \textit{et al.},
\href{https://journals.aps.org/prc/abstract/10.1103/PhysRevC.94.024338}{ Phys.
Rev. C 94, 024338 (2016)}, J.-G. Deng \textit{et al.},
\href{https://journals.aps.org/prc/abstract/10.1103/PhysRevC.96.024318}{Phys.
Rev. C 96, 024318 (2017)}], which $P_{\alpha}$ are model-dependent and
extracted from the ratios of calculated $\mathcal{\alpha}$ half-lives to
experimental data. Combining with our previous works, we confirm that the
valance proton-neutron interaction plays a key role in the $\mathcal{\alpha}$
preformation for nuclei around $Z=82$, $N=126$ shell closures whether the
$P_{\alpha}$ is model-dependent or microcosmic. In addition, our calculated
$\mathcal{\alpha}$ decay half-lives by using the proximity potential 1977
formalism taking $P_{\alpha}$ evaluated by the cluster-formation model can well
reproduce the experimental data and significantly reduce the errors.
