6. Phase Classification

6.1 Definition of Orbital, Collapsing, and Flyby Regimes

The dynamical behavior observed in the inertial emergent gravity model naturally separates into three distinct qualitative regimes. These regimes are not imposed a priori but are extracted from the temporal evolution of the system.

• Orbital Regime: Characterized by sustained non-monotonic separation, repeated radial turning points, and a nonzero time-averaged angular momentum proxy. The system exhibits bounded motion over extended time horizons.
• Collapsing Regime: Defined by monotonic decrease of the binary separation $d(t)$, rapid decay of inertial signatures, and convergence toward a merged configuration.
• Flyby (Overdamped) Regime: The separation increases or stabilizes without oscillation, inertia is transient, and the interaction resembles a dissipative scattering event rather than a bound state.

These regimes are mutually exclusive and exhaustive within the explored parameter space. Crucially, the classification depends on temporal structure, not on instantaneous configurations.

6.2 Orbit Count as a Temporal Diagnostic

To quantify orbital behavior in a manner independent of trajectory geometry, a temporal diagnostic is introduced based on radial turning points. Let $d(t)$ denote the binary separation. Each zero-crossing of $\dot{d}(t)$ corresponds to a radial extremum.

The estimated number of orbital cycles is defined as

This definition counts half-cycles of contraction and expansion, making no assumption of circularity, periodicity, or force balance. In the orbital regime, $N_{\text{orbit}} \geq 1$ persists under extension of the integration horizon. In collapsing and flyby regimes, $N_{\text{orbit}} \to 0$.

Orbit count therefore functions as a purely temporal indicator of bounded inertial motion.

6.3 Radial Oscillation Index

Complementary to orbit count, the magnitude of radial variability is captured by a dimensionless oscillation index,

This quantity measures the relative amplitude of radial motion independent of absolute scale. Orbital regimes exhibit $\Delta_r$ values that remain finite and stable across repeats and horizon scaling. Overdamped regimes show $\Delta_r \rightarrow 0$ as $t \rightarrow \infty$.

The pair $(N_{\text{orbit}}, \Delta_r)$ thus provides a robust, scale-independent classification of dynamical phase.

6.4 Independence from Instantaneous Force Laws

A critical feature of the classification scheme is its complete independence from any instantaneous force law. At no point is the system analyzed in terms of acceleration as a function of separation, nor is any effective potential assumed.

Two configurations with identical instantaneous separations and velocities can belong to different phases depending on their temporal history. This demonstrates that phase membership cannot be inferred from local-in-time quantities.

Consequently, gravitational behavior in this framework is not reducible to a force-versus-distance relation. It is instead a global property of the trajectory in time, reinforcing the interpretation of gravity as a dynamical phase rather than an interaction law.

Phase classification therefore marks the transition from phenomenology to ontology: gravity is no longer identified with motion itself, but with the persistence of a specific temporal structure.