Validating the Jacobsthal Eigenvalue Framework for Orbital Stability

We present a comprehensive, multi-disciplinary assessment of the Universal Boundary framework's proposition that λ = 2 the dominant eigenvalue of the Jacobsthal recurrence relation serves as a fundamental constant governing orbital stability transitions in planetary systems. This report evaluates the hypothesis that the distribution of period ratios in multi-planet systems is imprinted with Jacobsthal consecutive ratios Jn+1/Jn, speci cally predicting peaks at 2.2 (11/5) and 1.667 (5/3). We rigorously assess these claims against independent datasets from the Kepler and TESS missions and contrast them with competing dynamical theories, including resonant migration, packed stability, and geometric progression models. A central nding of this analysis is the independent con rmation of the 2.2 period ratio peak in TESS data at greater than 90% con dence, a feature previously identi ed in Kepler data but lacking a de nitive dynamical explanation in standard formation models. Furthermore, we clarify the distinction between the predicted Jacobsthal peak at 1.667 and the classical 3:2 mean-motion resonance (1.5), demonstrating that while the former corresponds to the 5:3 second-order resonance, its prominence and placement are predicted by the bilateral oscillation of the Jacobsthal sequence rather than purely by capture probability. We articulate a physical mechanism linking the mathematical properties of eigenvalue-2 grounded in the MEGNO quasi-periodic limit (⟨Y ⟩ → 2.0) and the symplectic trace condition (|Tr(M)| = 2) to the dynamical sculpting of planetary architectures via sequential resonance sweeping and stability ltering. We detail a robust Bayesian validation protocol using nested sampling, with explicit parameterisation against competing models. This framework provides a uni ed, eigenvaluebased explanation for anomalous features in planetary architectures, moving beyond descriptions based solely on integer commensurability toward a deeper numbertheoretic stability criterion.