From the list of 2321 transiting planet candidates announced by the Kepler Mission, we select seven targets with favorable properties for the capacity to dynamically maintain an exomoon and present a detectable signal. These seven candidates were identified through our automatic target selection (TSA) algorithm and target selection prioritization (TSP) filtering, whereby we excluded systems exhibiting significant time-correlated noise and focussed on those with a single transiting planet candidate of radius less than 6 Earth radii. We find no compelling evidence for an exomoon around any of the seven KOIs but constrain the satellite-to-planet mass ratios for each. For four of the seven KOIs, we estimate a 95% upper quantile of M_S/M_P<0.04, which given the radii of the candidates, likely probes down to sub-Earth masses. We also derive precise transit times and durations for each candidate and find no evidence for dynamical variations in any of the KOIs. With just a few systems analyzed thus far in the in-going HEK project, projections on eta-moon would be premature, but a high frequency of large moons around Super-Earths/Mini-Neptunes would appear to be incommensurable with our results so far.

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