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The Enigmatic and Ephemeral M Dwarf System KOI 6705: Cheshire Cat or Wild Goose?

We confirm a 0.995 d periodic planetary transit-like signal, KOI 6705.01, in the Kepler lightcurve of the star KIC 6423922. Optical and infrared spectra show that this star is a mid M-type dwarf with an effective temperature $= 3327 \pm 60$K, metallicity [Fe/H] $= -0.08 \pm 0.10$, radius $= 0.31 \pm 0.03 R_{\odot}$, and mass $= 0.28 \pm 0.05 M_{\odot}$. The star is $\approx 70$ pc away and its space motion, rotation period, and lack of H$α$ emission indicate it is an older member of the "thin disk" population. On the other hand, the star exhibits excess infrared emission suggesting a dust disk more typical of a very young star. If the KOI 6705.01 signal is produced by a planet, the transit depth of 60 ppm means its radius is only $0.26^{+0.034}_{-0.029} R_{\oplus}$, or about the size of the Moon. However, the duration ($\gtrsim 3$~hr) and time variation of KOI 6705.01 are anomalous: the signal was undetected in the first two years of the mission and increased through the latter two years. These characteristics require implausible orbits and material properties for any planet and rule out such an explanation, although a dust cloud is possible. We excluded several false positive scenarios including background stars, scattered light from stars that are nearby on the sky, and electronic cross-talk between detector readout channels. We find the most likely explanation to be that KOI 6705.01 is a false positive created by charge transfer inefficiency in a detector column on which KIC 6423922 and a 1.99 d eclipsing binary both happened to fall.