Appendix AA — Robustness and Dimensionless Controls

This appendix addresses reviewer-facing robustness concerns using strictly numerical and diagnostic checks. No new physical interpretation is introduced. All statements are to be read as validation protocol definitions and acceptance criteria.

Dimensionless Controls

To separate numerical choices from dynamical classification, we define dimensionless control groups constructed from the dissipation parameter $\gamma$, screening parameter $\mu$, and the discretization scales $(\Delta t,\Delta x)$.

We use the following groups (reported per run; see Fig. ‘§fig:AA_dimless‘):

• Damping-per-step:

This measures whether damping acts weakly or strongly within a single integration step.

• Screening-per-cell:

This measures how many grid cells resolve the screening length $\mu^-1$ (small $\Pi_\mu$ is better-resolved).

• Time-step stiffness proxy (diagnostic):

This provides a compact indicator of step size relative to spatial resolution (used diagnostically; not a claim of diffusion).

The purpose of $\{\Pi_{\gamma},\Pi_{\mu},\Pi_{s}\}$ is to ensure that phase classification and summary diagnostics (e.g. $\langle|L|\rangle$, non-monotonicity measures, and classifier labels) do not change under coordinated variation of $(\Delta t,\Delta x)$ that preserves these groups within tolerance.

\beginfigure[H]

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\vspace1em Missing figure: figAA_1_dimensionless_groups.png \vspace1em

Figure: Dimensionless control groups computed from run metadata. This figure is used to document the numerical control regime and to support invariance claims under coordinated discretization changes. \labelfig:AA_dimless \endfigure

Grid-Refinement Robustness

We test whether phase classification and key diagnostics are stable under grid refinement. Representative cases are selected from the validated dataset: (i) a clearly orbital case, (ii) a clearly collapsing case, and (iii) a borderline case near the transition region.

For each representative case, we rerun the simulation at multiple resolutions (e.g. $n_x \times n_y$ increased by factors of 2), with $\Delta x$ reduced accordingly, while holding all non-grid parameters fixed and coordinating $\Delta t$ as required by the numerical integrator. Each refinement run is processed by the same validator.

Acceptance criteria.

A case is considered grid-robust if:

1. the qualitative phase label (e.g. ‘ORBIT‘ vs. ‘COLLAPSE‘) is unchanged across refinements;
2. $\langle|L|\rangle$ remains within a specified tolerance band across refinements (reported in the validator tables);
3. the non-monotonicity diagnostic (e.g. radial oscillation amplitude or turning-point count, as used in the paper) is preserved up to tolerance.

Figure ‘§fig:AA_grid‘ summarizes the refinement sweep (values and tolerances to be read from validator outputs; no new quantities are introduced here).

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\vspace1em Missing figure: figAA_2_grid_refinement.png \vspace1em

Figure: Grid-refinement robustness summary for representative cases. The intent is to confirm phase-label stability and diagnostic stability under increased spatial resolution. \labelfig:AA_grid \endfigure

Domain Scaling and Boundary Robustness

We test sensitivity to domain size and boundary treatment by repeating representative runs under: (i) increased domain extents (e.g. $L_x, L_y$ scaled upward) and/or (ii) alternative boundary implementations available in the solver. All other parameters and initial-condition construction are held fixed.

Acceptance criteria.

A case is considered domain/boundary robust if:

1. the qualitative phase label is unchanged under domain scaling and boundary variations;
2. the diagnostics used for phase separation (e.g. $\langle|L|\rangle$ and non-monotonicity) remain within tolerance;
3. no new long-horizon instabilities appear solely due to boundary proximity (as evidenced by horizon-extension checks already reported elsewhere).

Figure ‘§fig:AA_domain‘ summarizes the domain-scaling/boundary sweep (values and tolerances to be read from validator outputs).

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\vspace1em Missing figure: figAA_3_domain_scaling.png \vspace1em

Figure: Domain scaling and boundary robustness summary. The purpose is to verify that classification and diagnostics do not change when boundary influence is reduced by enlarging the domain or by varying boundary treatment. \labelfig:AA_domain \endfigure

Continuous Closure Metric (Optional but Included)

To complement binary classifier labels (e.g. ‘ORBIT‘), we define a continuous diagnostic metric $M \in [0,1]$ as the time-fraction during which two independently measured conditions hold simultaneously over the horizon $T$: (i) angular-momentum proxy above a threshold $L_{\mathrm{crit}}$ and (ii) non-monotonic radial motion.

Let $\mathbf{1}(\cdot)$ denote the indicator function. Define:

This metric does not change any earlier results: it is a re-expression of already-used diagnostics in a single continuous score. It is included to (i) support robustness discussions and (ii) provide a smooth measure for borderline cases without modifying phase definitions.

Scope

This appendix documents numerical robustness with respect to:

• dimensionless discretization controls,
• grid refinement,
• domain size and boundary sensitivity,
• and an optional continuous diagnostic closure score.

No new physical interpretation is introduced, and no experimental numbers are asserted here beyond what is produced by validator outputs.

End of Appendix AA.