Appendix GGG — Numerical Validator Framework

GGG.1 Scope and Logical Status

This appendix establishes the numerical foundation of the entire work. All subsequent appendices rely exclusively on results extracted here.

No physical constants, dimensional assumptions, continuum limits, or theoretical principles are introduced in this appendix. The analysis is strictly operational and data-driven.

The sole purpose of Appendix GGG is to define:

• the numerical validator,
• the operational variables,
• and the empirical criteria distinguishing closure from non-closure.

GGG.2 Source of Numerical Data

All numerical results are extracted from the dataset

where each row corresponds to an independent numerical evolution of the governing dynamical equations.

No preprocessing, rescaling, or filtering is applied beyond standard numerical consistency checks.

GGG.3 Structure of the Validator Dataset

Each run in the dataset is characterized by the following columns:

• $\gamma$ — numerical damping parameter (dimensionless),
• $dt$ — discrete time step,
• $\text{steps}$ — total number of integration steps,
• $\langle |\mathbf{L}| \rangle$ — time-averaged magnitude of the historical angular-momentum proxy,
• $\text{estimated\_orbits}$ — count of completed or quasi-stable orbital cycles,
• $\text{status}$ — categorical classification of the run outcome.

The allowed status labels are:

GGG.4 Explicit Absence of Physical Inputs

Crucially, the dataset contains no entries corresponding to:

nor any dimensional quantities associated with length, mass, energy, or physical time.

All quantities appearing in this appendix are purely numerical and internal to the validator.

GGG.5 Operational Definition of Closure

Closure is defined empirically, not theoretically.

A run is classified as ‘ORBIT‘ if and only if:

A run is classified as \textttNON\text-CLOSURE if:

No threshold value is assumed at this stage. All critical values are extracted statistically in subsequent appendices.

GGG.6 Numerical Time Normalization

For each run, the total simulation duration is defined as:

This quantity is used exclusively for relative comparisons between runs and carries no physical interpretation.

GGG.7 Stability and Consistency Filters

Runs exhibiting:

• numerical divergence,
• integration failure,
• sensitivity to machine precision

are labeled ‘UNSTABLE‘ and excluded from phase classification.

No physical meaning is attributed to such exclusions; they serve solely to protect numerical consistency.

GGG.8 Empirical Phase Separation

When aggregated over all runs, the dataset exhibits a clear bifurcation:

with no continuous interpolation or mixed states.

This separation emerges directly from the data without parameter tuning or imposed criteria.

GGG.9 Foundational Result

The central result of Appendix GGG is:

This establishes closure as an empirical property of the dynamical system, providing the raw numerical substrate for all subsequent derivations.

GGG.10 Logical Transition

Having defined closure operationally, the next appendix (Appendix HHH) extracts:

directly from the statistical structure of the validator data.

No new assumptions are introduced at any stage.

GGG.11 Data Provenance and Auditability

The file ‘orbit_runs.csv‘ is treated as the immutable raw record. All enriched tables (e.g. ‘orbit_runs_enriched_fixed.csv‘) are deterministic, script-generated transforms of this file.

No row is added, removed, or modified except by explicitly stated, reproducible rules.

\paragraph*Negative statement (critical for refereeing). The raw dataset contains no inserted values for $c$, $G$, $\hbar$, $\Lambda$, or $k_B$. All subsequent constants are claimed to emerge only after (i) defining closure operationally and (ii) extracting invariants from the validator outputs.