Order at the Nodes: A Unified Geometric Law for Gravitational Hierarchy Decay

This paper presents the Parks Node Ejection Protocol (PNEP), a real-time geometric framework for predicting stability, ejection timing, decay regime, and escaping body identity in hierarchical gravitational N-body systems. The central finding: while trajectories in the three-body problem are chaotic, the geometry at mirror symmetry nodes — the instants when any pair of bodies reaches closest approach and radial velocity vanishes — is not. A single scalar, the Hierarchy Index H = σ²/(1+σ²), computed from pairwise distance variance at these geometrically honest moments, classifies stability, forecasts ejection timing, and identifies the escaping body across multiple topologies. Validated across three topologies: hierarchical triples (92.0% accuracy, 100% specificity, n=87), 3+1 quadruples (93.5%, n=200), and 2+2 peer binary quadruples (99.2%, n=120). Two novel pre-ejection signals are reported: the Liberation Energy Signature (100% coverage, median 26.2t advance warning) and Nodal Breathing (detects marginal instability thousands of time units before ejection while H remains flat). Validated on Burrau's Pythagorean Problem with correct body identity prediction and 4.7t lead time. Computational overhead approximately 1% of full N-body evaluation. All code open source: github.com/alikamp/Parks-Node-Ejection-Protocol Interactive overview: alikamp.github.io/Parks-Node-Ejection-Protocol