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Transit timing variations (TTVs) can be induced by a range of physical phenomena, including planet–planet interactions, planet–moon interactions, and stellar activity. Recent work has shown that roughly half of moons would induce fast TTVs with a short period in the range of 2–4 orbits of its host planet around the star. An investigation of the Kepler TTV data in this period range identified one primary target of interest, Kepler-1513 b. Kepler-1513 b is a |$8.05^{+0.58}_{-0.40}$| R⊕ planet orbiting a late G-type dwarf at |$0.53^{+0.04}_{-0.03}$| au. Using Kepler photometry, this initial analysis showed that Kepler-1513 b's TTVs were consistent with a moon. Here, we report photometric observations of two additional transits nearly a decade after the last Kepler transit using both ground-based observations and space-based photometry with TESS. These new transit observations introduce a previously undetected long period TTV, in addition to the original short period TTV signal. Using the complete transit data set, we investigate whether a non-transiting planet, a moon, or stellar activity could induce the observed TTVs. We find that only a non-transiting perturbing planet can reproduce the observed TTVs. We additionally perform transit origami on the Kepler photometry, which independently applies pressure against a moon hypothesis. Specifically, we find that Kepler-1513 b's TTVs are consistent with an exterior non-transiting ∼Saturn mass planet, Kepler-1513 c, on a wide orbit, |$\sim 5~{{\ \rm per \, cent}}$| outside a 5:1 period ratio with Kepler-1513 b. This example introduces a previously unidentified cause for planetary interlopers in the exomoon corridor, namely an insufficient baseline of observations. [ABSTRACT FROM AUTHOR]

An exomoon will produce transit timing variations (TTVs) upon the parent planet and their undersampled nature causes half of such TTVs to manifest within a frequency range of two to four cycles, irrespective of exomoon demographics. Here, we search through published Kepler TTV data for such signals, applying a battery of significance and robustness checks, plus independent light-curve analyses for candidate signals. Using the original transit times, we identify 11 (ostensibly) single-planets with a robust, significant and fast (P TTV < 4 cycles) TTV signal. However, of these, only five are recovered in an independent analysis of the original photometry, underscoring the importance of such checks. The surviving signals are subjected to an additional trifecta of statistical tests to ensure signal significance, predictive capability and consistency with an exomoon. KOI-3678.01, previously validated as Kepler-1513b, is the only case that passes every test, exhibiting a highly significant (>20 σ) TTV signal with a periodicity, amplitude and shape consistent with that caused by an exomoon. Our analysis finds that this planet is |$8.2_{-0.5}^{+0.7}$|   R ⊕ orbiting at |$0.53_{-0.03}^{+0.04}$|  au around a late G-type dwarf. After forecasting the planetary mass, we expect it to be capable of maintaining at least a 0.3  M ⊕ exomoon for 5 Gyr, and the TTV signal corresponds to a moon mass as low as 0.75 Lunar masses. We thus encourage follow-up observations and dynamical analysis of this unique signal, but caution skepticism until such data can be obtained. [ABSTRACT FROM AUTHOR]