8. The Closure Functional

8.1 Definition of the System History State $\Psi(t)$

The failure of instantaneous and static criteria necessitates a reformulation of gravitational existence in terms of system history rather than momentary configuration.

We define the system history state $\Psi(t)$ as the minimal set of quantities required to characterize the dynamical evolution of the system up to time $t$:

Here:

• $\rho(x,t)$ is the informational density field,
• $\nabla\Phi(x,t)$ encodes the emergent interaction geometry,
• $\gamma$ is the damping parameter controlling inertial decay,
• the integral term represents accumulated memory of past configurations,
• $K(t-\tau)$ is a causal kernel encoding temporal weighting.

Crucially, $\Psi(t)$ is not a state in the Hamiltonian sense. It is a history-bearing object whose definition is intrinsically nonlocal in time.

8.2 Existence Functional $\mathcalC [\Psi(t)]$

Gravitational behavior is now defined through an existence functional acting on $\Psi(t)$:

This functional does not generate dynamics. Instead, it classifies whether the evolving trajectory has achieved temporal closure sufficient to sustain inertial structure.

We define gravitational existence as:

When $\mathcal{C}[\Psi(t)] = 0$, the system may still evolve dynamically, but its behavior is overdamped, collapsing, or transient, and cannot support sustained orbital motion.

The functional $\mathcal{C}$ is therefore neither a force nor a constraint. It is a phase selector acting on histories.

8.3 Empirical Extraction of $\mathcalC$

Rather than postulating a form for $\mathcal{C}$, its structure is extracted empirically from numerical experiments.

Across extensive parameter scans, sustained gravitational behavior is observed if and only if the time-averaged magnitude of the inertial angular momentum proxy exceeds a critical threshold.

This motivates the operational definition:

where:

• $\Theta$ is the Heaviside step function,
• $\langle |L| \rangle = \frac{1}{T}\int_0^T |L(t)|\,dt$,
• $L_{\text{crit}}$ is a history-dependent critical inertial threshold.

This form is not assumed a priori; it is the simplest functional consistent with all observed phase transitions.

8.4 Critical Inertial Threshold L_\textcrit }

The critical threshold $L_{\text{crit}}$ is not a universal constant. It depends on the temporal and structural properties of the system history:

Here:

• $\gamma$ controls inertial dissipation,
• $\tau_{\text{memory}}$ characterizes effective temporal retention,
• $T_{\text{orbit}}$ is the emergent oscillation timescale,
• phase alignment encodes coherence between motion and interaction.

This structure explains the observed windowed behavior:

• Gravity may exist for intermediate values of $\gamma$,
• Disappear for slightly larger or smaller values,
• And reappear when temporal coherence is restored.

Gravity therefore emerges not as a monotonic function of control parameters, but as a temporally closed phase defined by inertial self-consistency over time.

With the closure functional defined, gravity can now be reinterpreted as a set of admissible histories rather than a law of interaction. The next section formalizes this interpretation and establishes gravity as a dynamical phase.