5. Orbital Phenomenology

5.1 Emergence of Orbits Without Central Forces

A defining result of this work is the appearance of sustained orbital motion in the absence of any imposed central force, prescribed potential, or geometric constraint. The governing equations contain no inverse-square law, no angular momentum conservation law, and no predefined notion of attraction.

Despite this, for a finite range of the damping parameter $\gamma$, the two density concentrations exhibit long-lived, non-radial motion characterized by repeated changes in relative orientation and separation. These trajectories are not circular in the classical sense; rather, they are dynamically maintained paths arising from the interplay of inertia, transport, and the emergent potential field.

Crucially, the orbital behavior is not encoded in the equations themselves but arises as a collective, time-extended phenomenon. This distinguishes the observed dynamics from effective-force models, where orbital motion is assumed by construction.

5.2 Radial Oscillations and Phase Structure

The primary diagnostic of orbital behavior is the time-dependent binary separation $d(t)$. In overdamped regimes, $d(t)$ evolves monotonically, indicating direct collapse or dispersive flyby. In contrast, the inertial regime exhibits pronounced radial oscillations:

These oscillations are not numerical noise. They persist under refinement of the grid, extension of the integration horizon, and variation of initial perturbations. The normalized radial fluctuation,

serves as a phase-sensitive order parameter distinguishing orbital dynamics from collapse.

The existence of repeated turning points in $d(t)$ demonstrates that the system cannot be described by a gradient flow. Any purely dissipative dynamics would forbid such reversals.

5.3 Angular Momentum as an Emergent Diagnostic

Although no angular momentum is imposed or conserved by the equations, an effective angular momentum proxy can be constructed from the relative configuration and the averaged velocity field:

In the orbital regime, $L(t)$ fluctuates around a nonzero mean value, while in overdamped regimes it decays rapidly toward zero. The time-averaged magnitude,

emerges as a robust inertial diagnostic.

Importantly, $\langle |L| \rangle$ is not conserved, nor is it a fundamental quantity. It is a derived measure capturing the persistence of rotational motion across time, and its sustained nonzero value signals the presence of an inertial phase.

5.4 Breakdown of Overdamped Dynamics

As the damping parameter $\gamma$ is increased, a sharp qualitative transition occurs. Beyond a critical range, radial oscillations vanish, $\langle |L| \rangle$ collapses, and trajectories become monotonic. The system re-enters an overdamped regime where all motion is slaved to instantaneous gradients.

This transition is not gradual in trajectory shape alone, but structural in time. The same initial configuration can produce either orbital motion or collapse depending solely on whether inertial memory survives long enough to close dynamically.

The failure of overdamped dynamics to support orbits underscores the central thesis of this work: gravitational behavior does not arise from static interactions, but from temporally extended dynamical closure enabled by inertia.

Orbital phenomenology therefore serves as the empirical bridge between the microscopic equations and the existence-based formulation introduced in later sections.