A Symplectic framework

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.