The Symplectic Stability Framework

This monograph establishes a rigorous connection between the algebraic properties of linear recurrences and the stability of Hamiltonian planetary systems within the symplectic period-doubling universality class. We prove that λ = 2 emerges as a boundary constant through multiple independent derivations: the eigenvalue-coecient uniqueness theorem,the trace-stability criterion |T| = 2, and the xed-point structure of the trace map recursionTn+1 = T2n − 2. The central mathematical result demonstrates that the Jacobsthal sequence characterstic equation r 2 − r − 2 = 0 is identical to the xed-point equation of the trace recursion, establishing Jacobsthal eigenvalues {2, −1} as the natural scaling factors near marginal stability. We prove the equivalence of Jacobsthal and Lichtenberg sequences, both deriving from Mersenne numbers via J(n) = (2n − (−1)n)/3, and establish the 50% Duty Cycle Theorem governing the binary structure at stability boundaries. We construct explicit derived maps (Keplerian Map, Migration Map) and validate predic- tions numerically. The framework generates falsiable predictions for period ratio clustering at 2.2 and 3.87, with applications to Saturn rings, galactic pattern speeds, and X-ray binary QPOs.