Completeness of our Y-band imaging of JKCS 041. The Y-band magnitude within a 0.5 arcsec aperture compared with the peak surface brightness. There are galaxies with a peak surface brightness fainter than the detection threshold (~26.5 mag arcsec^-2, vertical line), but they are undetectable with these data. Extrapolating the trend seen at brighter magnitudes (slanted solid line), the completeness limit (horizontal dashed line) is 27.15 mag. Owing to uncertainties inherent in this extrapolation, we conservatively adopt a limit that is 0.5 mag brighter, which leads to a limiting magnitude of Y = 26.6.