The Universality of the 2/3 Virial Factor in Keplerian Systems: From the Harmonic Oscillator to General Relativity

Abstract This work presents a unified analysis of the radial dynamic response in bound systems under central inverse potentials (1/r), bridging the gap between classical celestial mechanics, atomic physics, and General Relativity. By defining a "Reference Virial Rigidity," we demonstrate that orbital stability is not unique but bifurcates into two fundamental dynamic regimes determined by the system's symmetry constraints. We analytically identify two dimensionless response factors: The Flexible Mode (Factor 2): Characteristic of classical Keplerian systems with strict angular momentum conservation (L), where the orbital frequency adapts adiabatically. The Rigid Mode (Factor 2/3): Characteristic of systems subject to a rigid intrinsic frequency (constant w). We demonstrate that the linear Stark Effect in the hydrogen atom follows exactly this 2/3 factor, suggesting that quantum degeneracy acts mechanically as a frequency rigidity constraint. Finally, we extend this framework to strong gravity. We show that the precession of Mercury's perihelion and orbital instability near black holes (ISCO) can be physically interpreted as the breakdown of this Newtonian virial rigidity due to anharmonic corrections in the effective potential. We conclude that the 2/3 factor represents a scale invariant for the stability of degenerate virial systems, providing a new structural intuition for phenomena typically restricted to quantum or relativistic mecanics.