Appendix I: Multistability, Temporal Hysteresis, and Stratified Phase Boundaries

I.1 Motivation

While the main text establishes gravity as a temporally closed dynamical phase, large-scale validation data from the Big Orbit Validator reveal several nontrivial structural properties that extend and strengthen the framework. These properties were not assumed a priori and emerge only after systematic horizon scaling and replication.

This appendix documents five closely related discoveries:

1. Genuine multistability within fixed control parameters,
2. Temporal hysteresis of gravitational existence,
3. Failure of time-averaged angular momentum as a unique order parameter,
4. Thick, stratified phase boundaries in parameter space,
5. Robustness classes under horizon extension.

None of these phenomena contradict the core theory. Instead, they provide independent confirmation that gravity is a history-dependent phase of motion.

I.2 Multistability at Fixed Control Parameters

Across extensive runs at fixed damping strength $\gamma$, we observe the coexistence of distinct long-time outcomes:

• Sustained orbital (temporally closed) regimes,
• Monotonic collapse (non-closed) regimes.

Crucially, these outcomes occur under:

• Identical governing equations,
• Identical numerical schemes,
• Identical control parameters.

They differ only in microscopic initial history:

Definition (Multistability).

A parameter value $\gamma$ is said to exhibit multistability if there exist distinct histories $\Psi^{(1)}(t), \Psi^{(2)}(t)$ such that

under identical $\gamma$.

Numerical evidence shows multistability robustly near

This immediately excludes force-law interpretations, where outcomes are single-valued functions of parameters.

I.3 Temporal Hysteresis of Gravitational Existence

Beyond multistability, the data reveal a stronger phenomenon: temporal hysteresis.

Once a trajectory enters a temporally closed (orbital) phase, it may remain closed under parameter values for which fresh initializations fail to close.

Formally, there exist histories such that:

for generic initial data $\tilde{\Psi}$ at the same $\gamma$.

This establishes that:

• Gravitational existence depends on path, not just location in parameter space,
• The system possesses dynamical memory,
• Closure is not an instantaneous property.

Temporal hysteresis is a defining feature of phase transitions with internal storage and cannot arise in overdamped gradient systems.

I.4 Failure of $\langle |L| \rangle$ as a Unique Order Parameter

Initial intuition might suggest that the time-averaged angular momentum $\langle |L| \rangle$ serves as an order parameter for gravitational existence. The validation data falsify this hypothesis.

Observed numerically:

• Some collapsing trajectories exhibit larger $\langle |L| \rangle$ than certain orbital ones,
• Some orbital regimes persist with comparatively small $\langle |L| \rangle$.

Thus, there is no scalar function

that uniquely predicts closure.

Conclusion.

Gravitational existence is determined by the temporal structure of $L(t)$, not by its instantaneous or averaged magnitude.

This reinforces the necessity of the closure functional:

I.5 Stratified (Thick) Phase Boundaries

Instead of a sharp critical value $\gamma_c$, the data reveal extended stratified transition regions.

Within these regions:

• Some histories close rapidly and robustly,
• Others close weakly and decay under horizon extension,
• Others never close at all.

We therefore define a thick phase boundary:

I.6 Robustness Classes Under Horizon Scaling

Horizon scaling ($\times 1$, $\times 2$, $\times 4$) reveals distinct robustness classes:

1. Strongly closed: orbital behavior persists unchanged,
2. Weakly closed: orbital behavior degrades but survives,
3. Non-closed: collapse persists under all horizons.

Importantly:

Instead, weak closure corresponds to marginal temporal coherence near phase boundaries.

I.7 Implications for the Core Framework

The phenomena documented here imply that:

• Gravity is a multistable, history-dependent phase,
• Its existence exhibits hysteresis and memory,
• No local or scalar diagnostic suffices,
• Phase boundaries are extended and structured.

All of these properties are predicted qualitatively by the temporal-closure framework and quantitatively confirmed by the validator data.

I.8 Final Conclusion

Key Result.

This appendix elevates the framework from a novel interpretation to a structurally rich dynamical theory. The observed phenomena are not anomalies; they are signatures of a new organizing principle for gravitational existence.

Once these properties are acknowledged, force-based and purely geometric descriptions of gravity become mathematically insufficient.

Temporal closure remains the only invariant criterion consistent with the full numerical evidence.