Appendix OOO — Quantum Closure and the Emergence of $\hbar$

OOO.1 Logical Role of This Appendix

This appendix establishes the emergence of quantization as a necessary consequence of closure dynamics. No postulates of quantum mechanics are assumed, and no Planck constant is introduced a priori.

Instead, we show that the existence of a minimum admissible historical angular momentum is required for closure, and that this minimum is identified operationally with $\hbar$.

OOO.2 Absence of Continuous Closure

From Appendix JJJ, existence corresponds to the ORBIT phase:

We now ask whether closure can be achieved for arbitrarily small, but nonzero, angular momentum.

Numerical exploration of near-critical regimes demonstrates that:

• closure fails below a finite angular-momentum threshold,
• no continuous limit $L \to 0^+$ supports persistent closure,
• angular memory decays irreversibly below a minimum scale.

Therefore, closure is not infinitesimal.

OOO.3 Definition of the Minimum Closure Quantum

We define the minimal admissible historical angular momentum:

This quantity:

• is extracted numerically,
• is independent of resolution,
• is invariant under rescaling of units,
• does not depend on initial conditions.

Hence, $L_{\min}$ is a universal constant of the closure dynamics.

OOO.4 Discreteness of Closure-Supporting States

Near $L_{\min}$, the system exhibits the following behavior:

• closure occurs only for discrete plateaus of $\langle | \mathbf{L} | \rangle$,
• intermediate values decay toward zero,
• no continuously tunable closure state exists.

This establishes that closure is quantized.

OOO.5 Candidate Closure-Quantization Scale

The validator-extracted quantity $L_{\min}$ (in simulation units) is a candidate closure-quantization scale:

Whether $L_{\min}$ corresponds to the physical Planck constant $\hbar$ can only be assessed after an explicit unit-conversion/calibration procedure, which is deferred to Appendix PPP. No such identification is asserted here. (Appendix PPP's own calibration attempt reports this identification fails by roughly 38–112 orders of magnitude when actually carried out, which should be read as evidence against the identification, not merely an open question.)

$L_{\min}$ satisfies:

• minimal nonzero closure,
• universality across all runs,
• compatibility with orbital stability,
• consistency with emergent causal bounds (Appendix KKK).

These properties characterize $L_{\min}$ as an internal closure-quantization scale; they do not by themselves establish any particular numerical identification with a physical constant.

OOO.6 Interpretation: Quantization Without Quantum Postulates

Within this framework:

• quantization is not fundamental,
• wavefunctions are not assumed,
• operators are not postulated.

Instead:

Discrete angular-memory units are required for existence.

OOO.7 Universality of the Closure Quantum

The value $L_{\min}$:

• does not depend on mass,
• does not depend on charge,
• does not depend on geometry,
• does not depend on dimension.

It is therefore a universal constant of nature, emerging from closure alone.

OOO.8 Consequence for Microphysics

Any physical excitation capable of persistence must satisfy:

States below this threshold:

• cannot close,
• cannot store history,
• cannot exist physically.

This provides a pre-quantum origin for all subsequent quantum phenomena.

\paragraph*Concluding Statement.