Chaos and Undecidability in Near-Commensurable Planetary Resonances — E8 Intelligence Research

FINDING: Small divisor sensitivity in near-commensurable planetary resonances reveals a formal link between Hamiltonian chaos and undecidability in dynamical systems. MATH: The Jupiter-Saturn 2:5 near-commensurability (ratio ~0.4, close to 0.382) generates small divisors in the perturbation series. The Hamiltonian expansion to 3rd order in masses and 8th degree in eccentricities/inclinations shows the divisor denominator \(2n_J - 5n_S\) becomes arbitrarily small, causing the perturbation series to diverge. This is structurally analogous to the small divisor problem in KAM theory, where the Diophantine condition \(\left| k \cdot \omega \right| > \gamma |k|^{-\tau}\) fails for resonant tori. The undecidability arises because determining whether a given rational approximation leads to a convergent or divergent series is equivalent to solving a Diophantine equation, which is undecidable in general (Matiyasevich's theorem). CONNECTION: The 2:5 ratio (0.4) is near the golden ratio conjugat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com