9. Gravity as a Temporally Closed Phase

9.1 Gravity as a Set, Not a Law

Having defined gravitational existence through the closure functional $\mathcal{C}[\Psi(t)]$, gravity can no longer be represented as a force law, field equation, or geometric constraint acting instantaneously.

Instead, gravity is defined as a restricted class of admissible system histories:

This definition introduces a categorical separation:

• Dynamical equations specify how systems evolve.
• Gravitational existence classifies which evolutions are physically realized.

The equations of motion are universal and always well-defined. However, only a subset of their solutions satisfy the temporal closure condition required for gravitational behavior.

Gravity therefore does not act as a force, nor does it arise from geometry by assumption. It exists only as a dynamically selected phase, identified through the full temporal evolution of the system.

This resolves the long-standing ambiguity between force-based and geometry-based interpretations: gravity is neither. It is a temporally closed dynamical phase.

9.2 Phase Windows and Non-Monotonic Emergence

A direct consequence of defining gravity through temporal closure is the existence of phase windows in control-parameter space.

Numerical experiments demonstrate that gravitational behavior appears only within bounded regions of parameters. Outside these regions, systems exhibit monotonic collapse or dispersion without sustained orbital structure.

Formally, closure occurs only when inertial content accumulated over a full history exceeds a history-dependent threshold:

As control parameters are varied:

• Inertia may decay too rapidly for closure to occur,
• Or persist long enough to support coherent oscillatory motion,
• Or be destroyed again as damping overwhelms inertial storage.

The resulting structure is explicitly non-monotonic. Gravitational behavior may emerge, disappear, and re-emerge as parameters are varied, even when the underlying equations remain unchanged.

Such behavior cannot be captured by force laws, static stability criteria, or single-threshold models. It is the defining signature of a phase selected by temporal coherence rather than instantaneous conditions.

9.3 Conditional Existence and History Dependence

In conventional theories, gravity is assumed to exist universally and continuously. Its strength may vary, but its presence is never conditional.

In contrast, the present framework admits that gravity may:

• exist for one system history,
• fail to exist for another with identical instantaneous parameters,
• and reappear when temporal coherence is restored.

This follows directly from the binary nature of the closure functional:

There is no requirement that $\mathcal{C}$ remain invariant under parameter variation, horizon extension, or perturbation of initial histories. Gravitational existence is therefore not a property of matter, fields, or spacetime alone, but a property of closed dynamical histories.

This explains why:

• identical instantaneous configurations can yield distinct outcomes,
• no instantaneous force or field criterion can predict gravity,
• and gravitational behavior may be robust, fragile, or absent under identical control parameters.

Gravity exists only when the system succeeds in closing its own temporal dynamics.

With gravity redefined as a temporally closed phase, the framework admits direct empirical validation. The following section presents large-scale numerical scans demonstrating the repeatability, robustness, and conditional nature of this phase classification.