The c = 2 Boundary in Second-Order Recurrences: Eigenvalue Analysis, Hamiltonian Isomorphism, and Orbital Classification

We establish that among positive integer coefficients c in the second-order linear recurrence an = an−1 + c · an−2, the value c = 2 is unique in producing marginal stability where |λ2| = 1. This boundary separates the stable Fibonacci regime (c = 1, |λ2| < 1) from unstable regimes (c ≥ 3, |λ2| > 1). We demonstrate that this recurrence boundary maps isomorphically onto the trace stability condition of symplectic Jacobian matrices in Hamiltonian dynamics, establishing that c = 2 corresponds precisely to the Chirikov resonance overlap threshold. Through seven theorems with complete proofs, we derive: (1) the uniqueness of the c = 2 boundary; (2) that φ equals the Fibonacci eigenvalue while τ /π = 2 equals the Jacobsthal eigenvalue; (3) the Harmonic Offset Constant HOC = 18 = J6 − J3 from Jacobsthal self-reference. We develop the FSRE (Finite-Sample Ratio Extrapolation) algorithm for classifying planetary systems by period-ratio regularity, deriving the coefficient of variation threshold CV ≈ 0.25 directly from eigenvalue ratios at the c = 2 boundary. Validation against Kepler/TESS multi-planet systems confirms that CV effectively discriminates between systems retaining primordial resonant structure and those that have undergone dynamical relaxation. The framework provides theoretical grounding for observed geometric spacing in planetary systems through dynamical constraints rather than numerological interpretation.