Appendix A: Full Mathematical Derivations

This appendix collects the mathematical derivations underlying the Inertial Emergent Gravity framework. No additional assumptions are introduced beyond those stated explicitly in the main text. All quantities presented here are either primitive dynamical variables or diagnostics extracted directly from the numerical evolution.

A.1 Continuity Equation

The conservation of mass (or density) is imposed through the standard continuity equation:

This equation is not interpreted as a gravitational postulate, but purely as a transport constraint on the density field.

A.2 Inertial Equation of Motion

The velocity field evolves according to an inertial equation with linear damping:

The presence of the inertial term $\partial_t \mathbf{v}$ is essential. In the overdamped limit ($\partial_t \mathbf{v} \to 0$), the system reduces to gradient flow and cannot support orbital motion. All non-trivial phenomenology described in this work relies on retaining this term.

A.3 Screened Poisson Equation

The emergent potential $\Phi$ is determined instantaneously from the density field via a screened Poisson equation:

The subtraction of the spatial mean $\langle \rho \rangle$ enforces global neutrality and prevents spurious large-scale drift. The screening parameter $\mu$ controls the interaction range but does not, by itself, determine the existence of gravitational behavior.

A.4 Centers of Mass

For each localized body $i$, the center of mass is defined as:

This definition is applied to dynamically evolving density distributions, without assuming point particles or rigid bodies.

A.5 Binary Separation and Radial Velocity

The instantaneous separation between two bodies is:

The radial velocity is then obtained by temporal differentiation:

Zero-crossings of $\dot d(t)$ are used to diagnose turning points in the relative motion.

A.6 Angular Momentum Proxy

An effective angular momentum diagnostic is defined as:

where $\mathbf{v}_{\text{eff}}(t)$ is the spatially averaged velocity field associated with one body.

This quantity is not assumed to be conserved. Its persistence or decay is an empirical diagnostic of inertial storage in the system.

The time-averaged absolute angular momentum is defined as:

A.7 Orbit Count

The number of completed orbital cycles is estimated from the number of radial turning points:

This definition is purely temporal and does not rely on geometric closure or predefined orbital shapes.

A.8 Radial Oscillation Index

The relative amplitude of radial oscillations is quantified by:

This dimensionless quantity distinguishes monotonic collapse from sustained non-monotonic motion.

A.9 System History State

The complete dynamical state of the system is represented as a history-dependent object:

This representation makes explicit that instantaneous configurations are insufficient to characterize the system’s behavior.

A.10 Existence Functional

Gravitational behavior is characterized by an existence functional:

A minimal empirical realization consistent with the numerical data is:

where $\Theta$ is the Heaviside function and $L_{\text{crit}}$ is a history-dependent inertial threshold.

A.11 Definition of Gravity

Within this framework, gravity is defined as the set of system histories for which temporal closure is achieved:

No force law or geometric structure is assumed at any stage of this derivation.