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Structural Selection
Part VIAppendix3 min read·609 words

Appendix UUU — Robustness & Uncertainty Quantification

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Appendix UUU — Robustness & Uncertainty Quantification

\labelapp:UUU

UUU.1 Purpose and role in the validation chain

This appendix evaluates the robustness of the ranking produced in Appendix ‘§app:RRR‘ under controlled uncertainty. Two effects are tested:

  • explicit measurement noise injected via Monte Carlo perturbations,
  • finite-sample uncertainty assessed via bootstrap resampling.

The goal is strictly methodological:

\beginquote To determine whether the identified winner is a fragile artifact of small metric perturbations, or a statistically preferred configuration under a fixed, auditable uncertainty model. \endquote

No physical assumptions or model equations are modified in this appendix.

UUU.2 Dataset slice and provenance

All diagnostics are computed using the same NPZ-derived scoring table as Appendix ‘§app:RRR‘, restricted to the terminal stability slice

steps=640000.\text{steps} = 640000.

A reproducible snapshot of the dataset slice is stored as:

UUU_dataset_used.csv.\texttt{UUU\_dataset\_used.csv}.

All outputs for this appendix were generated in the following directory:

/Users/fcp/Desktop/STRUCTURAL_STABILITY_SERIES/validation_results_big/
out_A_npz_REDO2_20260117_180544

UUU.3 Baseline score and configuration

The unified score analyzed here is identical to Appendix ‘§app:RRR‘. Define:

F=1CVband+ε,A=μv,L=log⁡ ⁣(1+max⁡(Σ0,0)).F = \frac{1}{\mathrm{CV}_{\mathrm{band}}+\varepsilon}, \qquad A = \mu_v, \qquad L = \log\!\left(1+\max(\Sigma_0,0)\right).

The baseline score is

<a id="eq-eq-uuu-score" />

S=F A L=μv log⁡(1+max⁡(Σ0,0))CVband+ε.S = F\,A\,L = \frac{\mu_v\,\log(1+\max(\Sigma_0,0))} {\mathrm{CV}_{\mathrm{band}}+\varepsilon}.

Configuration:

ε=10−6,lens_norm = False,use_sigma_max = False.\varepsilon = 10^{-6}, \qquad \texttt{lens\_norm = False}, \qquad \texttt{use\_sigma\_max = False}.

UUU.4 Monte Carlo noise model

Uncertainty is injected independently into the three measured components μv\mu_v, Σ0\Sigma_0, and CVband\mathrm{CV}_{\mathrm{band}}:

  • Amplitude noise:
μv′=μv⋅ηA,ηA∼LogNormal(σA=0.02).\mu_v' = \mu_v \cdot \eta_A, \qquad \eta_A \sim \mathrm{LogNormal}(\sigma_A = 0.02).
  • Lensing proxy noise:
Σ0′=Σ0⋅ηΣ,ηΣ∼LogNormal(σΣ=0.05).\Sigma_0' = \Sigma_0 \cdot \eta_\Sigma, \qquad \eta_\Sigma \sim \mathrm{LogNormal}(\sigma_\Sigma = 0.05).
  • Flatness noise:
CVband′=clip ⁣(CVband+δCV,0,10),δCV∼N(0,0.012).\mathrm{CV}_{\mathrm{band}}' = \mathrm{clip}\!\left( \mathrm{CV}_{\mathrm{band}} + \delta_{\mathrm{CV}}, 0, 10 \right), \quad \delta_{\mathrm{CV}} \sim \mathcal{N}(0,0.01^2).

Each Monte Carlo draw recomputes the score using Eq. ‘(eq:UUU_score)‘ and re-ranks all candidates.

\paragraph*Run parameters.

mc=2000,topk=15,seed=1337.\texttt{mc}=2000, \quad \texttt{topk}=15, \quad \texttt{seed}=1337.

UUU.5 Monte Carlo winner stability

Let w⋆w^\star denote the baseline winner at steps =640000=640000. Define:

P(win)=Pr⁡[w⋆ ranks first],P(top-K)=Pr⁡[w⋆∈top-K].P(\text{win}) = \Pr[w^\star \text{ ranks first}], \qquad P(\text{top-}K) = \Pr[w^\star \in \text{top-}K].

\begintable[h!]

| Candidate | (γ,rep)(\gamma,\mathrm{rep}) | steps | P(win)P(\text{win}) | P(top-15)P(\text{top-}15) | | — | — | — | — | — | | Baseline winner | (0.10, 0) | 640000 | 0.641 | 0.9995 | | Runner-up | (0.02, 0) | 640000 | 0.090 | 0.9800 | | |

Figure: Monte Carlo stability under the specified noise model. \endtable

Pairwise separation probability:

Pr⁡ ⁣(Swinner′>Srunner-up′)=1.000.\Pr\!\left(S'_{\text{winner}} > S'_{\text{runner-up}}\right) = 1.000.

UUU.6 Ranking stability metrics

Global ordering stability is assessed using three metrics:

  • Spearman rank correlation,
  • Kendall τ\tau correlation,
  • Top-15 Jaccard set overlap.

Monte Carlo statistics (mean ±\pm std):

Spearman=−0.0209±0.0857,Kendall=−0.0143±0.0590,J(Top-15)=0.7471±0.0920.\begin{aligned} \text{Spearman} &= -0.0209 \pm 0.0857, \\ \text{Kendall} &= -0.0143 \pm 0.0590, \\ J(\text{Top-15}) &= 0.7471 \pm 0.0920. \end{aligned}

\paragraph*Direction convention. Near-zero or negative correlations may arise from opposite rank-order conventions. Top-KK overlap and winner probabilities are direction-invariant.

UUU.7 Bootstrap uncertainty

Bootstrap resampling assesses finite-sample sensitivity.

\paragraph*Protocol. For each replicate (NN rows sampled with replacement):

  1. recompute the baseline score,
  2. re-rank all candidates,
  3. record winner identity and stability metrics.

Parameters:

boots=500.\texttt{boots} = 500.

Results:

P(same winner)=0.606,J(Top-15)=0.578.P(\text{same winner}) = 0.606, \qquad J(\text{Top-15}) = 0.578.

UUU.8 Interpretation

Under the stated uncertainty model:

  • the winner remains in the elite set with very high probability,
  • score separation from the runner-up is strong,
  • bootstrap variability is moderate and consistent with finite-sample effects.

UUU.9 Artifacts written

  • ‘UUU_summary_640000.txt‘
  • ‘UUU_dataset_used.csv‘
  • ‘UUU_baseline_rank_640000.csv‘
  • ‘UUU_mc_winner_stability_640000.csv‘
  • ‘UUU_mc_topk_membership_640000.csv‘
  • ‘UUU_mc_rank_distributions_640000.csv‘
  • ‘UUU_mc_pairwise_winner_vs_runnerup_640000.csv‘
  • ‘UUU_bootstrap_rank_stability_640000.csv‘

\paragraph*Appendix UUU status. Monte Carlo robustness and bootstrap uncertainty quantification completed for the terminal stability slice under a fixed, auditable uncertainty model.

Source: Gravity as a Temporally Closed Dynamical Phase/69_Appendix UUU — Robustness & Uncertainty Quantification.TEX in the verified v2 revision. Found an issue with this section? Submit a criticism.
Cite this section

Plain text

Hassan, A. (2026). Appendix UUU — Robustness &amp; Uncertainty Quantification. In Gravity as a Temporally Closed Dynamical Phase, The Complete Structural Selection Corpus. Nuronova Genix Corp. https://structuralselection.org/book/appendix/appendix-uuu-robustness-and-amp-uncertainty-quantification

BibTeX

@incollection{hassan2026appendixuuurobustnes,
  author    = {Hassan, Akram},
  title     = {Appendix UUU — Robustness &amp; Uncertainty Quantification},
  booktitle = {The Complete Structural Selection Corpus},
  publisher = {Nuronova Genix Corp},
  year      = {2026},
  url       = {https://structuralselection.org/book/appendix/appendix-uuu-robustness-and-amp-uncertainty-quantification}
}

RIS

TY  - CHAP
AU  - Hassan, Akram
TI  - Appendix UUU — Robustness &amp; Uncertainty Quantification
T2  - The Complete Structural Selection Corpus
PB  - Nuronova Genix Corp
PY  - 2026
UR  - https://structuralselection.org/book/appendix/appendix-uuu-robustness-and-amp-uncertainty-quantification
ER  -
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