Constraining the Gravitational Lensing of $z\gtrsim6$ Quasars from their
  Proximity Zones
  Since their discovery twenty years ago, the observed luminosity function of
$z\gtrsim6$ quasars has been suspected to be biased by gravitational lensing.
Apart from the recent discovery of UHS J0439+1634 at $z\approx6.52$, no other
strongly lensed $z\gtrsim6$ quasar has been conclusively identified. The
hyperluminous $z\approx6.33$ quasar SDSS J0100+2802, believed to host a
supermassive black hole of $\sim10^{10} M_\odot$, has recently been claimed to
be lensed by a factor of $\sim450$, which would negate both its extreme
luminosity and black hole mass. However, its Ly$\alpha$-transparent proximity
zone is the largest known at $z>6$, suggesting an intrinsically extreme
ionizing luminosity. Here we show that the lensing hypothesis of $z\gtrsim6$
quasars can be quantitatively constrained by their proximity zones. We first
show that our proximity zone analysis can recover the strongly lensed nature of
UHS J0439+1634, with an estimated magnification
$\mu=28.0^{+18.4}_{-11.7}(^{+44.9}_{-18.3})$ at 68% (95%) credibility that is
consistent with previously published lensing models. We then show that the
large proximity zone of SDSS J0100+2802 rules out lensing magnifications of
$\mu>4.9$ at 95% probability, and conclusively rule out the proposed $\mu>100$
scenario. Future proximity zone analyses of existing $z\gtrsim6$ quasar samples
have the potential to identify promising strongly lensed candidates, constrain
the distribution of $z\gtrsim6$ quasar lensing, and improve our knowledge of
the shape of the intrinsic quasar luminosity function.
