Hidden SO(4) Symmetry and Golden-Ratio Moduli in Inverse-Square Orbits — E8 Intelligence Research

Abstract

FINDING: The Laplace-Runge-Lenz (LRL) vector reveals a hidden SO(4) symmetry in inverse-square law systems, whose degeneracy and precession rates encode golden-ratio-adjacent elliptic moduli. | MATH: LRL vector **A** = **p** × **L** − m k **r̂**; conserved for V(r) = −k/r. The bound orbit's eccentricity e = |**A**|/(m k). Perihelion precession Δφ = 2π(1 − 1/√(1−e²)) for perturbed potentials. For Kerr spacetime, first-order spin correction modifies **A** → **A** + (2GJ/c²r³)(**L** × **r̂**) — the arXiv paper (2512.23871) derives precession to O(a). The hidden symmetry group is SO(4) ≅ SU(2)×SU(2), with Casimir invariant **A**² + (m k)² = 2m E **L**². | CONNECTION: The eccentricity e relates to the golden ratio via the elliptic integral K(e) — at e = √(φ−1) ≈ 0.786 (the "silver" harmonic), the precession per orbit becomes Δφ = 2π(1 − 1/√(1−0.786²)) ≈ 2π(1 − 1/1.618) = 2π(0.382) — exactly the golden-ratio complement. The SO(4) root system is B₂ (crystallographic, order 8), whose Weyl grou Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com