  We present a survey of 41 Kepler Objects of Interest (KOIs) for exomoons
using Bayesian photodynamics, more than tripling the number of KOIs surveyed
with this technique. We find no compelling evidence for exomoons although
thirteen KOIs yield spurious detections driven by instrumental artifacts,
stellar activity and/or perturbations from unseen bodies. Regarding the latter,
we find seven KOIs exhibiting >5 sigma evidence of transit timing variations,
including the 'mega-Earth' Kepler-10c, likely indicating an additional planet
in that system. We exploit the moderately large sample of 57 unique KOIs
surveyed to date to infer several useful statistics. For example, although
there is a diverse range in sensitivities, we find that we are sensitive to
Pluto-Charon mass-ratio systems for ~40% of KOIs studied and Earth-Moon
mass-ratios for 1 in 8 cases. In terms of absolute mass, our limits probe down
to 1.7 Ganymede masses, with a sensitivity to Earth-mass moons for 1 in 3 cases
studied and to the smallest moons capable of sustaining an Earth-like
atmosphere (0.3 Earth masses) for 1 in 4. Despite the lack of positive
detections to date, we caution against drawing conclusions yet, since our most
interesting objects remain under analysis. Finally, we point out that had we
searched for the photometric transit signals of exomoons alone, rather than
using photodynamics, we estimate that 1 in 4 KOIs would have erroneously been
concluded to harbor exomoons due to residual time correlated noise in the
Kepler data, posing a serious problem for alternative methods.
