TTV-Not-So-Fast: Uniqueness and Degeneracy in Perturbing Planet Parameters

Nontransiting planets can reveal themselves through transit timing variations (TTVs), but inferring the properties of the perturbing planet is a highly degenerate inverse problem. We present a systematic reassessment of all 12 published cases in which a nontransiting planet was claimed to have been uniquely characterized using TTVs. Two systems (KOI-142 and Kepler-419) stand out clearly with compelling evidence for unique solutions. Two other systems (KOI-872 and KOI-884) exhibit complex degeneracies, but the data are just precise enough to single out a best solution. Six systems (Kepler-82, Kepler-411, Kepler-725, KOI-134, Kepler-138, and TOI-4562) admit multiple viable solutions involving very different perturbing planets. In the remaining two systems (WASP-18 and WASP-126), the evidence for any perturbing planet is weak. We find that a necessary (but not sufficient) condition for a unique solution is the detection of short-timescale TTV structure associated with conjunctions, either in the near-resonant "chopping" regime or in eccentric systems with phase-dependent close approaches. In some systems, aliasing of the synodic period leads to ambiguities in associating observed TTV timescales with physical timescales, threatening uniqueness. Our results highlight the difficulty of achieving unique solutions in TTV inversions and underscore the need for long time baselines, accurate timing uncertainties, and complementary constraints from radial velocities or other observations when characterizing nontransiting planets.