Appendix III — Emergent Closure Constants

III.1 Logical Role of This Appendix

This appendix introduces a new class of constants that are neither assumed nor identified with known physical quantities.

They arise exclusively from the closure thresholds extracted in Appendix HHH and serve as the internal language of the theory.

At this stage, these constants are:

• purely emergent,
• dimensionless or operationally scaled,
• uninterpreted physically.

No connection to $c$, $\hbar$, $G$, or $\Lambda$ is permitted here.

III.2 Principle of Construction

All constants introduced below are constructed from the minimal set of numerically extracted quantities:

No additional parameters are introduced. No rescaling by external units is allowed.

Each constant encodes a distinct structural aspect of closure dynamics.

III.3 Definition of the Closure Strength Constant $\mathcalC _*$

We define the closure strength constant as:

This quantity measures the balance between dissipative loss and historical accumulation at the closure threshold.

Numerically, $\mathcal{C}_*$ is found to be:

• $\mathcal{O}(1)$ across all validated runs,
• insensitive to numerical resolution,
• invariant under rescaling of simulation time.

At this stage, $\mathcal{C}_*$ is interpreted only as an internal consistency parameter of closure.

III.4 Definition of the Memory Saturation Constant $\mathcalM _*$

Define the dimensionless memory saturation constant:

where $\langle | \mathbf{L} | \rangle_{\mathrm{sat}}$ denotes the typical angular-momentum magnitude observed deep within the ORBIT phase.

Empirically:

with tight clustering across runs.

This constant quantifies how close the system operates to the minimal closure condition once stable existence is established.

No thermodynamic or quantum interpretation is assigned here.

III.5 Definition of the Existential Gap Constant $\mathcalG _*$

We define the existential gap constant as:

This quantity measures the separation between:

• dissipation scale,
• minimal angular-memory scale.

It governs how sharply the system transitions between closure and non-closure.

Numerical analysis shows that $\mathcal{G}_*$ controls the steepness of the phase boundary identified in Appendix JJJ.

III.6 Independence and Completeness

The three constants $\mathcal{C}_*$, $\mathcal{M}_*$, and $\mathcal{G}_*$ are algebraically independent.

No constant can be eliminated in favor of the others without loss of information.

Together, they form a complete basis for describing closure behavior within the validator framework.

III.7 Absence of Physical Interpretation

It is essential to emphasize:

\beginquote At this stage, the constants $\mathcal{C}_*$, $\mathcal{M}_*$, and $\mathcal{G}_*$ are not identified with any known physical constants, nor are they assumed to correspond to measurable quantities. \endquote

They constitute the internal grammar of the theory and nothing more.

Any attempt to interpret them prematurely would constitute circular reasoning.

III.8 Preparatory Role for Subsequent Appendices

These constants will serve as the sole inputs for:

• the phase analysis in Appendix JJJ,
• the causal bound in Appendix KKK,
• the emergence of known constants in Appendix PPP.

No new independent quantities will be introduced beyond this point.

\paragraph*Concluding Statement.