OFFLINE EXOFAST used in this work gave:
Linear ephemeris: Tc = [27.46719239 ± 0.00001842](Tc#) + [76.67395760 ± 0.00056218]

ONLINE EXOFAST used in this work gave:
Linear ephemeris: Tc = [27.46719361 ± 0.00001814](Tc#) + [76.67378704 ± 0.00055377]
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One can also assume that the melding of a number of sinusoidal curves (each associated with the gravitational interaction of KOI-6668.01 with other non-transiting planets) additively gives the observed curve.  Thus, guided by Lomb-Scargle periodograms showing credible periodicities (fairly low FAPs) and corresponding sinusoidal curve-fits, the observed OFFLINE-EXOFAST-generated curve (Figure 1a) was deconstructed into 4 sinusoidal arrays yielding the following sets of values:

A. OFFLINE-EXOFAST-generated “TTV-xy” data:
Lomb-Scargle periodogram, candidate P_ttv: 1093.01 days; Power: 8.56; FAP: 0.000553;
Sinusoidal P_ttv: 1120.00 ± 78.32 days; Amp_ttv: 5.03 ± 0.87 minutes.


B. Residuals of A. data:
Lomb-Scargle periodogram, candidate P_ttv: 688.79 days; Power: 11.13; FAP: 0.0000191;
Sinusoidal P_ttv: 692.39 ± 31.89 days; Amp_ttv: 4.31 ± 0.61 minutes.


C. Residuals of B. data:
Lomb-Scargle periodogram, candidate P_ttv: 502.83 days; Power: 9.10; FAP: 0.000488;
Sinusoidal P_ttv: 506.86 ± 18.01 days; Amp_ttv: 2.91 ± 0.49 minutes.


D. Residuals of C. data:
Lomb-Scargle periodogram, candidate P_ttv: 390.41 days; Power: 7.31; FAP: 0.00706;
Sinusoidal P_ttv: 393.37 ± 13.70 days; Amp_ttv: 2.04 ± 0.40 minutes.

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References:
1. http://exoplanetarchive.ipac.caltech.edu/cgi-bin/TblView/nph-tblView?app=ExoTbls&config=cumulative
2. https://cfop.ipac.caltech.edu/home/
3. https://archive.stsci.edu/kepler/eclipsing_binaries.html
4. http://keplerebs.villanova.edu

20 May 2015; updated 22 May 2015
    Kepler KOI-6668 (KIC-6146838): Exoplanetary or Eclipsing Binary System

Discussion:
KOI-6668.01 is currently listed in the NASA Exoplanet Archive (NEA)(ref. 1) and in the Kepler Community Follow-up Observing Program (CFOP)(ref. 2) as an exoplanet candidate.  It is also included among Kepler Eclipsing Binaries, Revision 3 Table (ref. 3), with a morphology classification parameter of 0.03 (i.e.: highly detached) as well as in the more recent version of that tabulation (ref. 4).
KOI-6668.01 shows a very unusual {Time vs. (O-C)}{TTV or ETV} plot (see Figures 1a, Offline EXOFAST, and 1b, Online EXOFAST).  During Q9-Q14, there appears to be a gradual increase in the times between transits.  Then during Q15-Q17, there is a general decrease that seems to become precipitous as the Kepler data runs out.  The ratio of full-width-half-maximum time to the (ingress + egress) time, T_{FWHM} to tau in EXOFAST terms, is exactly 1 for every transit, right through to the end of the recorded data, consistent with a planet that is undergoing a grazing-transit.
Plots of the Transit Depth (Fig.2) and Transit Duration (Fig.3) vs. time show marked increases, which also implies an overall increase in Transit Area.  A plot of the Orbital Period vs. time (Fig.4), as expected, corresponds to the TTV plot.

One likely explanation of the rapid swing in the data is that it is due to a periastron passage of a third star of a heirarchical triple star system (personal communication with Dan Fabrycky).

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