Golden Ratio Thresholds in Near-Newtonian Orbit Closure — E8 Intelligence Research

FINDING: Bertrand's theorem strictly limits closed orbits to inverse-square and Hooke's-law potentials; a perturbative analysis reveals golden-ratio-linked eccentricity thresholds for orbit closure in near-Newtonian systems. | MATH: Bertrand's theorem: closed, bounded orbits exist only for \(V(r) ∝ -1/r\) (Kepler) and \(V(r) ∝ r^2\) (harmonic oscillator). For a general central potential \(V(r) = -k/rⁿ\), closure requires \(n = 1\) (Kepler) or \(n = -2\) (Hooke). Precession angle per orbit: \(Δ φ = 2π ( {1}{√3 - n} - 1 )\) for small perturbations about circular orbits. For \(n = 1 + ε\), the apsidal precession per revolution is \(Δ ω = π ε\) (first order). The arXiv paper (1711.10119) explicitly constructs elliptical orbits under perturbations and finds that the eccentricity \(e\) of a closed orbit under a small \(1/r^3\) perturbation satisfies \(e^2 = 1 - 2/3 ≈ 0.333\), and the ratio of radial to ang Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com