Appendix F: Emergent Structural Consequences of Temporal Closure

(Results Revealed by Large-Scale Validation)

F.1 Purpose and Scope

This appendix documents several structural consequences of the inertial emergent gravity framework that were not assumed a priori, but became evident only through extensive numerical validation using horizon scaling and repeated realizations.

These results do not introduce new postulates, equations, or mechanisms. Instead, they clarify deeper properties of gravitational existence implied by the temporal closure formalism and supported by the data.

F.2 Hysteresis of Gravitational Existence

The validation results demonstrate that gravitational existence exhibits hysteresis.

Specifically, systems with identical control parameters may:

• enter a gravitational phase in one realization,
• fail to do so in another,
• despite identical equations, horizons, and damping coefficients.

This behavior exceeds simple history dependence. The outcome depends not only on the past state, but on the path by which inertial coherence is accumulated or lost. Once temporal closure is achieved, the system may retain gravitational existence even as parameters drift toward marginal regimes.

Gravitational existence therefore possesses memory.

F.3 Thick and Stratified Phase Boundaries

Parameter scans reveal that the boundary separating gravitational and non-gravitational regimes is not a sharp threshold.

Instead, there exists a finite transition band in which:

• both closed and non-closed histories coexist,
• outcomes vary across realizations,
• no single parameter value uniquely determines existence.

This implies that gravitational phase boundaries are:

• thick rather than infinitesimal,
• internally stratified by basin geometry,
• inherently stochastic at the boundary.

Such structure is incompatible with force-law or equilibrium-based phase transitions and is a direct consequence of temporal closure.

F.4 Failure of Angular Momentum as an Order Parameter

The data demonstrate that no instantaneous scalar observable functions as an order parameter for gravitational existence.

In particular:

• realizations with higher mean angular momentum may fail to close,
• realizations with lower angular momentum may succeed,
• gravitational existence correlates with temporal persistence rather than magnitude.

Angular momentum contributes to closure but does not determine it. Only the history-integrated coherence encoded by the closure functional $\mathcal{C}[\Psi]$ predicts gravitational existence.

F.5 Robustness Classes Under Horizon Scaling

Horizon scaling reveals that gravitational phases admit internal robustness classes.

Across extended integration times, gravitational histories may:

• remain stable and persistent,
• weaken but survive,
• or fail irreversibly.

This motivates a classification of gravitational existence into robust, marginal, and failed closure regimes. Such structure emerges naturally from the temporal closure framework and requires no additional assumptions.

F.6 Replacement of Singularities by Closure Failure

In all validated collapse regimes, the dynamics remain finite and regular. No divergence, blow-up, or singular behavior is observed.

Instead, collapse corresponds to the progressive loss of temporal closure. Gravitational failure is therefore not a singular event but an existence failure.

Within this framework, classical singularities are replaced by closure failure events marking the termination of gravitational existence.

F.7 Summary

The results documented here establish that gravitational existence is:

• hysteretic,
• phase-band–structured,
• not governed by instantaneous order parameters,
• classifiable by robustness,
• and free of singular behavior.

These properties are not additional hypotheses. They are unavoidable structural consequences of defining gravity as a temporally closed phase.