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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.18970809 |
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Table of Contents:
- <p>We establish a rigorous connection between the algebraic structure of second- order linear recurrences and the stability boundaries of Hamiltonian planetary systems.</p> <p>The central result is an algebraic theorem: among all integer recurrences Tn = Tn−1 + c Tn−2 with c ≥ 1, the Jacobsthal case c = 2 is the unique one for which the dominant eigenvalue λ+ equals the coupling coefficient, i.e. λ+ = c. This eigenvalue-coefficient coincidence places the Jacobsthal eigenvalue λ+ = 2 exactly at the trace-stability boundary |Tr(M)| = 2 in symplectic mechanics, where elliptic fixed points become parabolic and resonance overlap initiates chaotic diffusion. </p> <p>We prove that the Jacobsthal sequence is identical to the Lichtenberg sequence(OEIS A000975) and that both are connected to Mersenne numbers by a parity dependent formula. This Lichtenberg–Jacobsthal–Mersenne unification resolves a long-standing terminological ambiguity and reveals a binary duty-cycle structure governing stability transitions. N-body simulations using the REBOUND integrator confirm that the empirical stability boundary for multi-planet systems lies at period ratio P ≈ 1.956, corresponding to δ(log P) ≈ 0.666 ≈ 0.961 ln 2, placing λ = 2 firmly within the stable dynamical regime. MEGNO chaos indicator analysis confirms that Jacobsthal ratios above λ = 2 consistently yield ⟨Y ⟩ → 2, characteristic of quasi-periodic motion. A two-boundary hierarchy is identified: the golden ratio φ ≈ 1.618 marks chaos onset, while λ = 2 marks the resonance-dominated regime boundary.</p> <p>The framework predicts exoplanet period ratio clustering at 11/5 = 2.200 and 43/11 = 3.909 — values unexplained by classical mean-motion resonance theory. Both peaks are observed in Kepler and TESS data. All predictions contain zero free parameters and are falsifiable.</p> <p> </p>