Abstract This monograph proposes a connection between the algebraic properties of linear recurrences and the stability of Hamiltonian planetary systems. We establish that λ = 2 emerges as a candidate boundary constant through a proven algebraic property: c = 2 is the unique positive integer for which the dominant eigenvalue of an order-2 linear recurrence equals its coupling coefficient. This mathematical fact connects to the trace-stability criterion |T| = 2 in symplectic mechanics, suggesting a possible first-principles basis for orbital stability thresholds. The framework generates falsifiable predictions for exoplanetary period ratio clustering, notably peaks at 2.2 (= 11/5) and 3.87 (≈ 43/11) observed in Kepler and TESS data—values unexplained by classical mean-motion resonance theory. We present numerical validation using N-body simulations with the REBOUND integrator, demonstrating that the practical stability boundary lies at approximately 96% of ln(2) in log-period space. New results include energy conservation analysis revealing that the golden ratio φ ≈ 1.618 marks a chaos boundary where energy errors transition by approximately seven orders of magnitude, while λ = 2 exhibits resonance effects that intensify with increasing planetary multiplicity. These findings suggest a hierarchical structure: φ as a chaos onset boundary, and λ = 2 as a resonance-dominated regime boundary. We also note a structural coincidence with Lyapunov exponent theory: at full chaos in paradigmatic one-dimensional maps (logistic at r = 4, tent at μ = 2), the average stretching factor exp(λ_Lyap) = 2, matching the Jacobsthal eigenvalue. This traces to the universal role of period-doubling in both recurrence stability and routes to chaos. Status of this work: The mathematical results (Theorem 2.3, trace criterion) are proven. The connection to orbital dynamics is a hypothesis supported by numerical simulations and empirical correlations, but the causal mechanism remains undemonstrated. This paper presents the framework as a testable proposal with explicit falsification criteria, not as an established theory. This work emphasizes statistical rigor through proper treatment of the look-elsewhere effect, explicit falsifiability criteria, and transparent acknowledgment of limitations.