Phase Structure and Empirical Classification

\labelapp:phase_structure

This appendix provides a purely empirical description of the dynamical regimes observed in the numerical simulations. Its purpose is strictly descriptive: to document what appears in the data, how the behavior changes with the control parameter $\gamma$, and how the observed outcomes separate into distinct classes.

No interpretation, physical mechanism, or theoretical explanation is introduced in this appendix. All statements are constrained to quantities directly measured and reported by the validator output.

Control Parameter and Diagnostics

Each simulation run is characterized by:

• Dissipation parameter $\gamma$
• Independent realization index (‘rep‘)
• Simulation horizon (number of steps)

For each run, the following diagnostics are extracted:

• Mean angular momentum $\langle |L| \rangle$
• Radial oscillation amplitude

• Qualitative phase label from the validator (‘ORBIT‘ or ‘COLLAPSE‘)

No derived quantities beyond these diagnostics are used in the analysis below.

Empirical Phase Diagram

Figure ‘§fig:phase_diagram‘ shows the distribution of mean angular momentum $\langle |L| \rangle$ as a function of the dissipation parameter $\gamma$ for all validated runs.

\beginfigure[h!]

[figure: see original PDF] fig1_phase_diagram.png Figure: Phase diagram showing mean angular momentum $\langle |L| \rangle$ as a function of dissipation $\gamma$. Each point corresponds to a single simulation run. The data separate into two clearly distinct regions: a regime with finite angular momentum and a regime with strongly suppressed angular momentum. \labelfig:phase_diagram \endfigure

Two observations follow directly from the data:

1. For sufficiently small $\gamma$, runs consistently exhibit finite, non-zero $\langle |L| \rangle$.
2. For larger $\gamma$, $\langle |L| \rangle$ is systematically reduced and clusters near zero.

The transition between these behaviors occurs over a narrow interval in $\gamma$ and is not smoothly interpolated.

Non-Monotonicity as an Empirical Discriminator

Figure ‘§fig:non_monotonicity‘ shows the ensemble-averaged radial oscillation amplitude $\Delta r$ as a function of $\gamma$.

\beginfigure[h!]

[figure: see original PDF] fig2_non_monotonicity.png Figure: Radial oscillation amplitude $\Delta r$ versus dissipation $\gamma$. Non-zero $\Delta r$ indicates non-monotonic radial motion. The same dissipation interval that supports finite angular momentum also supports persistent non-monotonicity. \labelfig:non_monotonicity \endfigure

The data show that:

• Runs with finite $\langle |L| \rangle$ consistently exhibit $\Delta r > 0$.
• Runs with suppressed angular momentum exhibit $\Delta r \approx 0$, corresponding to strictly monotone radial behavior.

Thus, non-monotonicity and angular-momentum persistence coincide empirically across the parameter scan.

Empirical Phase Separation

Based solely on the measured diagnostics, the simulations separate into two empirically distinct regimes:

\begindescription \item[Inertial (Orbital) Regime:] Characterized by finite $\langle |L| \rangle$ and non-zero $\Delta r$. All runs in this regime are classified as ‘ORBIT‘ by the validator. \item[Collapse Regime:] Characterized by suppressed $\langle |L| \rangle$ and $\Delta r \approx 0$, corresponding to monotone dynamics. \enddescription

The boundary between these regimes is sharp in $\gamma$ and shows no evidence of gradual crossover or fine-tuned intermediate behavior.

Scope of the Appendix

This appendix establishes only the empirical phase structure of the system. It does not address:

• Temporal robustness
• Statistical stability
• Physical interpretation
• Theoretical modeling

These aspects are treated separately in subsequent appendices.

End of Appendix X