A Relational Stability Reformulation of the N-Body Problem Through the ∆E Field of the Relative Cosmic Equilibrium (RCE) Theory

The classical N‑body problem has resisted solution for centuries due to the chaotic sensitivity inherent in pairwise Newtonian forces. This work applies the Relative Cosmic Equilibrium (RCE) framework—a relational field theory in which motion emerges from energetic imbalance rather than mutual forces—to reformulate the N‑body problem from first principles. In RCE, bodies move to minimize their deviation from a global equilibrium field ∆E, seeking local energetic basins rather than responding to direct force interactions. The ∆E field generates smooth, self‑organizing landscapes that suppress chaotic divergence and channel trajectories into quasi‑equilibrium flows. Chaos appears only when the field narrows beyond a critical threshold, triggering a transition to a new stable basin, thereby replacing sensitivity to initial conditions with relational reorganization. High‑resolution spectral simulations of 3‑, 5‑, and general N‑body configurations under the RCE formalism demonstrate robust stability: near‑zero Lyapunov exponents, smooth trajectories, and consistent energetic profiles. The effective pairwise potential extracted from ∆E matches the modified Bessel form K0(αr), analytically confirming the field‑governed dynamics predicted by RCE. These results indicate that the RCE relational framework provides a physically consistent, numerically stable alternative to force‑based descriptions. By shifting the paradigm from pairwise forces to a unified energetic landscape, the RCE formulation offers a promising, non‑chaotic pathway toward a general solution of the classical N‑body problem in closed systems.