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Structural Selection
Part VIAppendix2 min read·403 words

Appendix QQQ — Numerical Extraction of Closure Invariants

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Appendix QQQ — Numerical Extraction of Closure Invariants

\labelapp:QQQ

QQQ.1 Purpose and non-negotiable scope

This appendix provides the unique numerical bridge of the manuscript:

data  →  filters  →  estimators  →  invariants  →  uncertainty.\text{data} \;\rightarrow\; \text{filters} \;\rightarrow\; \text{estimators} \;\rightarrow\; \text{invariants} \;\rightarrow\; \text{uncertainty}.

No external physical constants (c,G,ℏ,Λ,kB)(c,G,\hbar,\Lambda,k_B), no observational priors, and no unit identifications are introduced. All quantities remain in validator units.

QQQ.2 Data sources

\paragraph*Canonical raw table. The authoritative dataset is

orbit_runs.csv.\texttt{orbit\_runs.csv}.

\paragraph*Supplementary availability. The CSV file is provided as supplementary material to enable independent reproduction.

QQQ.3 Primitive observables and phase label

From each run (one row), we retain only directly reported validator quantities:

γ,L≡⟨∣L∣⟩  (mean_abs_L),Norb  (estimated_orbits),status∈{ORBIT,COLLAPSE},rep,steps,dt.\begin{aligned} &\gamma,\quad \mathcal{L} \equiv \langle |\mathbf{L}| \rangle \;(\texttt{mean\_abs\_L}),\quad N_{\rm orb}\;(\texttt{estimated\_orbits}),\\ &\texttt{status}\in\{\mathrm{ORBIT},\mathrm{COLLAPSE}\},\quad \texttt{rep},\quad \texttt{steps},\quad \texttt{dt}. \end{aligned}

The binary closure indicator is defined as

C≡{1,status = ORBIT,0,status = COLLAPSE.\mathsf{C} \equiv \begin{cases} 1, & \text{status = ORBIT},\\ 0, & \text{status = COLLAPSE}. \end{cases}

QQQ.4 Derived bookkeeping quantity

The operational integration horizon (validator time units) is

<a id="eq-eq-qqq-tsim" />

Tsim≡(steps)×(dt).T_{\mathrm{sim}} \equiv (\texttt{steps})\times(\texttt{dt}).

QQQ.5 Inclusion rules

Rows are included only if:

  1. γ,L,status,steps,dt\gamma,\mathcal{L},\texttt{status},\texttt{steps},\texttt{dt} are finite;
  2. ‘status‘ is exactly ORBIT or COLLAPSE;
  3. for fixed (γ,rep)(\gamma,\texttt{rep}), the longest available TsimT_{\mathrm{sim}} is used.

QQQ.6 Extracted closure invariants

Four dimensionless invariants are extracted:

Lℏ,VT,VΛ,Vc.\mathcal{L}_{\hbar},\quad \mathcal{V}_{T},\quad \mathcal{V}_{\Lambda},\quad \mathcal{V}_{c}.

They are not identified with physical constants here.

QQQ.6.1 Minimal stable closure scale

Tmax⁡≡max⁡(Tsim),T_{\max} \equiv \max(T_{\mathrm{sim}}), Lℏ≡min⁡{L:C=1,  Tsim=Tmax⁡}.\mathcal{L}_{\hbar} \equiv \min\Big\{\mathcal{L}:\mathsf{C}=1,\;T_{\mathrm{sim}}=T_{\max}\Big\}.

QQQ.6.2 Micro-closure fluctuation scale

σL(γ)≡Std({L}rep∣C=1,  Tmax⁡),\sigma_L(\gamma) \equiv \mathrm{Std}\big(\{\mathcal{L}\}_{\texttt{rep}} \mid \mathsf{C}=1,\;T_{\max}\big), VT≡Median({σL(γ)}γ).\mathcal{V}_{T} \equiv \mathrm{Median}\big(\{\sigma_L(\gamma)\}_{\gamma}\big).

QQQ.6.3 Non-closure drift scale

For COLLAPSE runs:

ln⁡L≈α−k Tsim,k≥0,\ln\mathcal{L} \approx \alpha - k\,T_{\mathrm{sim}}, \qquad k\ge0, VΛ≡Median({k}).\mathcal{V}_{\Lambda} \equiv \mathrm{Median}\big(\{k\}\big).

QQQ.6.4 Effective propagation-speed scale

∣v∣(x,t)=vx2+vy2,ceffval=Q0.999(∣v∣),|v|(x,t)=\sqrt{v_x^2+v_y^2},\qquad c_{\mathrm{eff}}^{\mathrm{val}} = Q_{0.999}(|v|), Vc≡(ceffval)2.\mathcal{V}_{c} \equiv \big(c_{\mathrm{eff}}^{\mathrm{val}}\big)^2.

QQQ.7 Numerical results

Tmax⁡=320.0,T_{\max}=320.0, Lℏ=1.017373×10−2,VT=1.994383×10−2,VΛ=7.869909×10−4.\mathcal{L}_{\hbar}=1.017373\times10^{-2},\quad \mathcal{V}_{T}=1.994383\times10^{-2},\quad \mathcal{V}_{\Lambda}=7.869909\times10^{-4}. ceffval=1.6719949745,Vc=2.7955671948.c_{\mathrm{eff}}^{\mathrm{val}}=1.6719949745,\quad \mathcal{V}_{c}=2.7955671948.

QQQ.8 Summary table

\begintable[h]

\begintabularx\linewidthlXcc

Invariant & Operational definition & Value & 95%95\% CI \

Lℏ\mathcal{L}_{\hbar} & Minimum sustained L\mathcal{L} in ORBIT at Tmax⁡T_{\max} & 1.02×10−21.02\times10^{-2} & fixed \ VT\mathcal{V}_{T} & Median replicate dispersion of L\mathcal{L} in ORBIT & 1.99×10−21.99\times10^{-2} & [1.16,3.40]×10−2[1.16,3.40]\times10^{-2} \ VΛ\mathcal{V}_{\Lambda} & Median decay rate from ln⁡L=α−kT\ln\mathcal{L}=\alpha-kT (COLLAPSE) & 7.87×10−47.87\times10^{-4} & [5.96,9.78]×10−4[5.96,9.78]\times10^{-4} \ Vc\mathcal{V}_{c} & Median (Q0.999(∣v∣))2(Q_{0.999}(|v|))^2 from velocity fields & 2.802.80 & pending \

\endtabularx Figure: Closure invariants extracted purely from validator outputs. \endtable

QQQ.9 Logical handoff

All numerical invariants are now fixed. Appendix PPP may apply exactly one physical calibration.

Source: Gravity as a Temporally Closed Dynamical Phase/58.5_Appendix QQQ — Numerical Extraction of Closure Invariants.tex in the verified v2 revision. Found an issue with this section? Submit a criticism.
Cite this section

Plain text

Hassan, A. (2026). Appendix QQQ — Numerical Extraction of Closure Invariants. In Gravity as a Temporally Closed Dynamical Phase, The Complete Structural Selection Corpus. Nuronova Genix Corp. https://structuralselection.org/book/appendix/appendix-qqq-numerical-extraction-of-closure-invariants

BibTeX

@incollection{hassan2026appendixqqqnumerical,
  author    = {Hassan, Akram},
  title     = {Appendix QQQ — Numerical Extraction of Closure Invariants},
  booktitle = {The Complete Structural Selection Corpus},
  publisher = {Nuronova Genix Corp},
  year      = {2026},
  url       = {https://structuralselection.org/book/appendix/appendix-qqq-numerical-extraction-of-closure-invariants}
}

RIS

TY  - CHAP
AU  - Hassan, Akram
TI  - Appendix QQQ — Numerical Extraction of Closure Invariants
T2  - The Complete Structural Selection Corpus
PB  - Nuronova Genix Corp
PY  - 2026
UR  - https://structuralselection.org/book/appendix/appendix-qqq-numerical-extraction-of-closure-invariants
ER  -
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Appendix HHH — Critical Damping and Closure Thresholds
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Appendix III — Emergent Closure Constants