The Jacobsthal ratio identity Jn+1/Jn = 2 + (−1) n/Jn predicts that planetary period ratios near the 2: 1 commensurability should be displaced by specific alternating offsets: +1/5 = +0. 200, −1/11 = −0. 091, +1/21 = +0. 048, and so on. We introduce an offset-specific test that measures whether observed displacements from exact 2: 1 cluster at these predicted values, and apply it to 194 planet pairs with period ratios in 1. 8, 2. 3 from 446 Kepler multi-planet systems. The Jacobsthal off-sets match the data better than 99. 85% of random three- offset control sets (p = 0. 0015) ; with sign-matched controls, p = 0. 004. Five comparison sequences—Fibonacci, Mersenne, Pell, Lucas, and Padovan—are tested at the 2: 1 resonance: none reaches significance (all p > 0. 75). We show that the widely used fractional symbolic ratio error (FSRE) test cannot distinguish between these sequences: all produce p < 10−7 due to generic near-2: 1 clustering, and shifting any target set by up to +0. 20 barely changes the signal. REBOUND N-body migration simulations independently cluster at 43/21 = 2. 048, matching the Jacobsthal prediction 2+1/21 at level n = 6 (Kolmogorov–Smirnov p = 0. 002). A Jacobsthal-specific prediction of sign alternation in consecutive near-2: 1 pairs is not supported (13/24 alternating, p = 0. 42). The mapping f (r) = 1 + 1/r carries the Jacobsthal ladder to the 3: 2 region with a derived offset identity (offset from 3/2 = (−1) n+1/2Jn+1), but the predicted offsets (|δ| < 0. 05) are below the detection threshold of the current sample. A secondary finding is that Padovan sequence offsets reach p = 0. 012 near the 4: 3 commensurability (λ ≈ 1. 325), suggesting the pattern may extend beyond the 2: 1. Keywords: exoplanet period ratios, mean-motion res- onance, Jacobsthal sequence, integer recurrences, Ke- pler multi-planet systems, offset-specific test, REBOUND simulations, Padovan sequence, PLATO predictions Intro Multi-planet systems discovered by Kepler exhibit a well-known excess of period ratios just wide of the 2: 1 mean-motion resonance 1, 2, 3. The physical origin of this offset -Tidal dissipation 4, 5, stochastic forcing 7, or overstable librations 6—remains debated. This paper does not adjudicate between these mechanisms. Instead, it asks a narrower question: among the observed displacements from exact 2: 1, do the data prefer specific offset values predicted by the Jacobsthal sequence over those predicted by other integer recurrences? The Jacobsthal sequence 0, 1, 1, 3, 5, 11, 21, 43, 85,. . . satisfies Jn = Jn−1 + 2Jn−2 with dominant eigenvalue λ = 2. Its consecutive ratios obey the identity Jn+1/Jn = 2 + (−1) n/Jn (Lemma 1), which predicts offsets from exact 2: 1 of +1/5, −1/11, +1/21, −1/43,. . . —a specific, alternating, geometrically decaying sequence. The algebraic significance of λ = 2 as the parabolic boundary of the SL (2, R) stability classification is developed in the companion paper 9. We introduce an offset-specific test that directly measures whether observed displacements from 2: 1 cluster at these predicted values. We compare against five other integer-recurrence sequences. We also demonstrate that the fractional symbolic ratio error (FSRE) test, used in earlier work, cannot discriminate between any of these sequences and should not be used for this purpose.
David Coates (Tue,) studied this question.