Geometric Corroboration of the Earliest Lensed Galaxy at z~10.8 from
  Robust Free-Form Modelling
  A multiply-lensed galaxy, MACS0647-JD, with a probable photometric redshift
of $z\simeq 10.7^{+0.6}_{-0.4}$ is claimed to constitute one of the very
earliest known galaxies, formed well before reionization was completed.
However, spectral evidence that MACS0647-JD lies at high redshift has proven
infeasible and so here we seek an independent lensing based "geometric
redshift" derived from the angles between the three lensed images of
MACS0647-JD, using our free-form mass model (WSLAP+) for the lensing cluster
MACSJ0647.7+7015 (at $z=0.591$). Our lens model uses the 9 sets of multiple
images, including those of MACS0647-JD, identified by the CLASH survey towards
this cluster. We convincingly exclude the low redshift regime of $z<3$, for
which convoluted critical curves are generated by our method, as the solution
bends to accommodate the wide angles of MACS0647-JD for this low redshift.
Instead, a best fit to all sets of lensed galaxy positions and redshifts
provides a geometric redshift of $z\simeq 10.8^{+0.3}_{-0.4}$ for MACS0647-JD,
strongly supporting the higher photometric redshift solution. Importantly, we
find a tight linear relation between the relative brightnesses of all 9 sets of
multiply lensed images and their relative magnifications as predicted by our
model. This agreement provides a benchmark for the quality of the lens model,
and establishes the robustness of our free-form lensing method for measuring
model-independent geometric source distances and for deriving objective central
cluster mass distributions. After correcting for its magnification the
luminosity of MACS0647-JD remains relatively high at $M_{UV}=-19.4$, which is
within a factor of a few in flux of some surprisingly luminous $z\simeq
10$--$11$ candidates discovered recently in Hubble black field surveys.
