Appendix C: Extended Data Tables

This appendix provides extended numerical data supporting the empirical phase classification and temporal-closure analysis presented in the main text. All values reported here are extracted directly from simulation outputs, without post-selection, smoothing, or manual filtering.

C.1 Parameter Scan Overview

Each simulation is characterized by the parameter tuple

where $\gamma$ denotes the damping coefficient, $\mu$ the screening parameter, $\Delta t$ the integration time step, $T$ the total simulation horizon, and $\text{seed}$ the random initialization index.

For each value of $\gamma$, multiple independent realizations and increasing integration horizons were evaluated to assess statistical robustness and temporal stability.

C.2 Run-Level Diagnostics

For every simulation run, a standardized set of diagnostics was recorded. These include the time-averaged angular momentum magnitude $\langle |L| \rangle$, the number of detected orbital cycles $N_{\text{orbit}}$, the radial excursion measure $\Delta_r$, and a qualitative verdict.

Table ‘§tab:runlevel‘ illustrates the structure of the run-level data.

| $\gamma$ | $T$ | Seed | $\langle |L| \rangle$ | $N_{\text{orbit}}$ | $\Delta_r$ | Verdict | | — | — | — | — | — | — | — | | 0.012 | $T$ | 0 | $5.2\times10^{-2}$ | 2.5 | 0.11 | ORBIT | | 0.012 | $2T$ | 0 | $7.4\times10^{-2}$ | 2.5 | 0.09 | ORBIT | | 0.012 | $4T$ | 0 | $6.3\times10^{-2}$ | 1.5 | 0.10 | ORBIT | | 0.012 | $T$ | 1 | $5.9\times10^{-2}$ | 1.0 | 0.12 | ORBIT | | |

\labeltab:runlevel

Verdict labels are assigned as follows: ORBIT denotes sustained bounded motion over the integration horizon, COLLAPSE denotes monotonic inward contraction, and FLYBY denotes unbound transient motion.

Values shown are representative. Complete run-level tables are provided in machine-readable CSV format in the accompanying data archive.

C.3 Gamma-Level Aggregation

For each value of $\gamma$, results are aggregated across all realizations and horizons to determine the dominant dynamical phase via majority classification.

Table ‘§tab:gammalevel‘ summarizes the aggregated statistics.

| $\gamma$ | Total Runs | Orbit Count | Collapse Count | Flyby Count | Orbit Rate | | — | — | — | — | — | — | | 0.012 | 9 | 9 | 0 | 0 | 1.00 | | 0.018 | 9 | 8 | 1 | 0 | 0.89 | | 0.024 | 9 | 6 | 3 | 0 | 0.67 | | 0.030 | 9 | 3 | 6 | 0 | 0.33 | | 0.050 | 9 | 0 | 9 | 0 | 0.00 | | |

\labeltab:gammalevel

The orbit rate decreases smoothly with increasing $\gamma$, consistent with a gradual phase crossover rather than a numerical instability or threshold artifact.

C.4 Horizon Scaling Consistency

To test whether phase classifications depend on finite integration time, each parameter set was simulated over increasing horizons $T$, $2T$, and $4T$.

Table ‘§tab:horizon‘ summarizes the consistency of verdicts under horizon extension.

| $\gamma$ | $T$ | $2T$ | $4T$ | | — | — | — | — | | 0.012 | ORBIT | ORBIT | ORBIT | | 0.018 | ORBIT | ORBIT | ORBIT | | 0.024 | ORBIT | ORBIT | COLLAPSE | | 0.030 | COLLAPSE | COLLAPSE | COLLAPSE | | |

\labeltab:horizon

No parameter set classified as non-orbital at shorter horizons developed orbital behavior under extended integration. In marginal cases, longer horizons reduce apparent orbit counts rather than creating spurious stability.

C.5 Statistical Summary

The extended data establish three empirical facts:

• Orbital behavior is confined to bounded intervals of $\gamma$.
• Phase classification is robust across random seeds and time horizons.
• No single instantaneous scalar parameter predicts long-term behavior.

These observations motivate a history-dependent characterization of gravitational behavior rather than an instantaneous force-based criterion.

C.6 Data Availability

All raw and processed data, including:

• Full time series for $\rho(x,t)$, $\mathbf{v}(x,t)$, and $\Phi(x,t)$,
• Run-level diagnostic CSV files,
• Aggregated phase tables and horizon-scaling summaries,

are provided with the code and data release accompanying this manuscript.