Appendix E: Negative Results and Failed Regimes

(Demonstration of Non-Triviality)

E.1 Purpose of This Appendix

This appendix documents parameter regimes in which gravitational behavior does not emerge within the inertial emergent gravity framework. The inclusion of negative and failed regimes serves three essential scientific purposes:

• To demonstrate that orbital behavior is not generic or automatic.
• To rule out trivial explanations based on numerical artifacts or force-law tuning.
• To establish that gravitational existence is conditional, history-dependent, and phase-like.

All results reported here were obtained using the same equations, numerical schemes, and analysis pipeline described in the main text.

E.2 Definition of Failure Regimes

A simulation run is classified as a failed gravitational regime if it satisfies one or more of the following criteria:

1. Absence of sustained radial oscillations:

2. Vanishing or rapidly decaying inertial content:

3. Monotonic decrease of separation:

4. Fewer than one completed orbital cycle:

Such regimes are labeled COLLAPSE or FLYBY depending on their asymptotic behavior.

E.3 Observed Failure at Fixed Control Parameters

A central result of this work is the observation that identical control parameters can yield different physical outcomes.

E.3.1 Case Study: gamma = 0.014

At damping strength $\gamma = 0.014$, the system exhibits mixed outcomes across repeated runs with identical numerical settings but differing initial histories (noise seeds and symmetry shifts):

• Some realizations undergo direct collapse with no sustained oscillations.
• Other realizations enter orbital regimes completing one or more cycles.
• The same $\gamma$ value therefore supports both $\mathcal{C}[\Psi]=0$ and $\mathcal{C}[\Psi]=1$ outcomes.

This behavior is incompatible with:

• Force-law interpretations, where outcomes are uniquely determined by parameters.
• Static threshold models, where a single critical value separates phases.

Instead, the outcome depends on the full system history $\Psi(t)$.

E.4 Horizon Scaling and Failure Persistence

Failed regimes persist under horizon scaling. Increasing the integration time by factors of $\times2$ and $\times4$ does not convert collapsing trajectories into orbital ones. Instead, longer horizons reveal:

• Continued decay of inertial content,
• Progressive damping of residual oscillations,
• Eventual monotonic convergence.

This confirms that failed regimes are not premature truncations of orbital dynamics but genuine non-gravitational phases.

E.5 Implications for Non-Triviality

The existence of negative results establishes that:

1. Gravitational behavior is not an automatic consequence of the equations of motion.
2. No function of the form $F(\gamma)$ or $L(\gamma)$ can predict existence.
3. Gravity is neither guaranteed nor forbidden by parameters alone.

The framework therefore defines conditions of existence, not a force law.

E.6 Role in the Overall Argument

This appendix plays a critical logical role in the manuscript:

• It excludes trivial explanations based on numerical tuning.
• It justifies the introduction of a temporal closure functional.
• It demonstrates that gravitational emergence is phase-like and conditional.

The negative regimes documented here are not failures of the framework; they are essential evidence for its correctness.

E.7 Summary

Gravitational behavior within the inertial emergent framework exists only within restricted, history-dependent regions of phase space. The documented failed regimes confirm that gravity is not a universal outcome of the dynamics but a temporally closed phase that may or may not be realized.

E.8 Non-Monotonic Reappearance of Gravitational Phases

Large-scale parameter scans further reveal that gravitational behavior may reappear at higher damping values after being suppressed at intermediate ones. In particular, orbital regimes are observed at damping strengths larger than those exhibiting collapse, demonstrating that gravitational existence is not a monotonic function of $\gamma$.

This non-monotonic structure rules out interpretations based on simple damping suppression or bifurcation thresholds. Instead, it confirms that gravitational existence is governed by history-dependent temporal closure rather than by control parameters alone.