Appendix HHH — Critical Damping and Closure Thresholds

HHH.1 Logical Position of This Appendix

This appendix constitutes the first point at which numerical regularities extracted in Appendix GGG are promoted to physically meaningful thresholds.

No new equations of motion are introduced. No physical constants are assumed. All results follow exclusively from statistical structure present in the validator dataset.

Here, the notion of closure ceases to be a numerical label and becomes a sharply defined dynamical state.

HHH.2 Ordering of Runs by Damping Parameter

Let the dataset be indexed by runs $i$, each characterized by a damping parameter $\gamma_i$ and a corresponding outcome classification.

We order all runs such that:

For each run, we retain:

This ordering reveals a sharp transition in behavior as $\gamma$ is increased.

HHH.3 Empirical Definition of the Critical Damping

Define the set of closed runs:

Define the complementary set of non-closed runs:

The critical damping $\gamma_{\mathrm{crit}}$ is defined purely operationally as:

This value marks the highest damping for which closure is still observed.

No averaging or fitting procedure is required.

HHH.4 Emergence of a Critical Angular-Momentum Threshold

For all runs with $\gamma < \gamma_{\mathrm{crit}}$, the quantity $\langle | \mathbf{L} | \rangle$ clusters around nonzero values.

For all runs with $\gamma \ge \gamma_{\mathrm{crit}}$, the same quantity collapses toward zero.

Define:

This value is not imposed. It is the smallest numerically observed angular-momentum magnitude compatible with sustained closure.

HHH.5 Sharpness of the Transition

To test whether the transition is gradual or sharp, we examine the conditional variance:

The data show:

• low variance well below $\gamma_{\mathrm{crit}}$,
• rapid variance increase near $\gamma_{\mathrm{crit}}$,
• collapse to zero above $\gamma_{\mathrm{crit}}$.

No intermediate plateau or mixed regime is observed.

This confirms that the closure transition is not a numerical artifact, but a genuine threshold phenomenon.

HHH.6 Closure Timescale

For each closed run, define the closure timescale $\tau_{\mathrm{cl}}$ as the earliest simulation time at which:

Extracting $\tau_{\mathrm{cl}}$ across runs yields:

Near the critical point, the divergence is described by:

No fitting exponent is imposed; the inverse scaling follows directly from numerical collapse.

HHH.7 Physical Status of the Extracted Quantities

At this stage:

• $\gamma_{\mathrm{crit}}$ is a numerically defined dissipation threshold,
• $L_{\mathrm{crit}}$ is the minimal angular-momentum content required for persistence,
• $\tau_{\mathrm{cl}}$ is the characteristic formation time of closure.

These quantities are not yet identified with any known physical constants. They are purely emergent scales of the validator.

HHH.8 Birth of Closure as a Physical State

This appendix establishes, without interpretation, that the system admits two and only two long-lived regimes:

The boundary between them is defined by $(\gamma_{\mathrm{crit}}, L_{\mathrm{crit}})$ and is dynamically sharp.

This is the first point in the work where existence becomes a state rather than a label.

HHH.9 Transition to Emergent Constants

The quantities extracted here serve as the raw ingredients for defining new dimensionless constants in Appendix III.

No reinterpretation is permitted before that step.