Double magicity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>Pb</mml:mi> <mml:mprescripts/> <mml:none/> <mml:mn>164</mml:mn> </mml:mmultiscripts> </mml:math> within the deformed relativistic Hartree-Bogoliu

The self-conjugate nucleus $^{164}\mathrm{Pb}$, with equal neutron and proton numbers ($N=Z=82$), lies far beyond the proton dripline and at a traditional magic number. Its possible doubly magic nature is investigated based on the state-of-the-art deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc), which incorporates nuclear deformation, pairing correlations, and continuum effects in a fully self-consistent framework. The experimentally known binding energies of neutron-deficient $Z=82$ isotopes and $N=82$ isotones are well reproduced. Pronounced two-neutron and two-proton gaps, together with sizable single-particle shell gaps, support the persistence of the $N=Z=82$ shell closures and point to mutually enhanced magicity in $^{164}\mathrm{Pb}$. Potential decay modes, including one-proton, two-proton, and multiproton emissions, are analyzed using the direct decay model with quantified uncertainties, predicting a dominant one-proton emission with a half-life ranging from $22\phantom{\rule{0.16em}{0ex}}\mathrm{ps}$ to $1.8\phantom{\rule{0.16em}{0ex}}\textmu{}\mathrm{s}$. These results suggest that $^{164}\mathrm{Pb}$ may define the outermost reach of experimentally accessible nuclei beyond the proton dripline, making it a compelling target for future experiments.