Appendix SSS — Scaling, Units, and Identifiability

\labelapp:SSS

SSS.1 Purpose and Scope

This appendix addresses a critical methodological concern: whether the ranking and the identified “winner” produced in Appendix ‘§app:RRR‘ are genuine features of the dynamics, or artifacts of arbitrary unit choices, metric scaling, or dominance by a single term in the unified score.

The goal of Appendix SSS is therefore to establish identifiability in a strict operational sense: \beginquote The ordering of runs under the unified score is stable under reasonable rescalings of physical units, alternative but defensible score constructions, and explicit stress tests designed to isolate and remove potential dominance by any single component (e.g. lensing normalization). \endquote

This appendix does not introduce new physics. Instead, it validates the robustness of the empirical claims made in Appendix ‘§app:RRR‘.

SSS.2 Dataset and Slice Used

All tests in this appendix operate on the same NPZ-derived dataset used in Appendix ‘§app:RRR‘. After deduplication and type coercion, the dataset contains all admissible runs indexed by $(\gamma,\mathrm{rep},\text{steps})$.

Unless otherwise stated, all identifiability diagnostics are reported on the terminal (stable) checkpoint:

$$\text{steps} = 640000,$$

which enforces a conservative notion of dynamical maturity and prevents transient early-time behavior from influencing the conclusions.

A snapshot of the exact dataset used is saved as:

$$\texttt{SSS\_dataset\_used.csv},$$

ensuring full reproducibility.

SSS.3 Baseline Score Recap

The baseline unified score introduced in Appendix ‘§app:RRR‘ is:

<a id="eq-eq-sss-baseline-score" />

$$S \;=\; \frac{\mu_v \, \log\!\big(1+\max(\Sigma_0,0)\big)} {\mathrm{CV}_{\mathrm{band}} + \varepsilon}, \qquad \varepsilon = 10^{-6},$$

where:

• $\mu_v$ is the mean rotation proxy in the evaluation band,
• $\mathrm{CV}_{\mathrm{band}}$ is the coefficient of variation in that band,
• $\Sigma_0$ is the lensing-normalization proxy.

Appendix SSS tests whether the ordering induced by ‘(eq:SSS_baseline_score)‘ is structurally stable.

SSS.4 Unit and Scale Invariance Tests

Sigma scaling.

To test sensitivity to lensing normalization units, we rescale:

$$\Sigma_0 \;\mapsto\; \alpha_\Sigma \Sigma_0, \qquad \alpha_\Sigma \in \{0.1,\,1,\,10,\,100,\,1000\},$$

recompute the score for each $\alpha_\Sigma$, and compare the resulting ranking to the baseline using:

• Spearman rank correlation,
• Kendall $\tau$,
• Top-$K$ Jaccard overlap.

The results are recorded in:

$$\texttt{SSS\_unit\_scaling\_sigma\_640000.csv}.$$

Amplitude scaling.

An analogous test rescales the rotation amplitude:

$$\mu_v \;\mapsto\; \alpha_v \mu_v, \qquad \alpha_v \in \{0.1,\,1,\,10,\,100\},$$

verifying that unit conversions or velocity normalization choices do not artificially determine the ranking. Results are saved as:

$$\texttt{SSS\_unit\_scaling\_amp\_640000.csv}.$$

Interpretation.

High rank correlations and stable Top-$K$ overlap under these transformations demonstrate that the ordering is not a unit artifact.

SSS.5 Score-Variant Stress Tests

To test dependence on a specific functional form, multiple reasonable alternative score variants are evaluated, including:

• alternative flatness definitions:

$$F = \frac{1}{\mathrm{CV}+\varepsilon}, \quad F = e^{-\mathrm{CV}}, \quad F = \frac{1}{\mathrm{CV}^2+\varepsilon},$$

• alternative lens compression:

$$L=\log(1+\Sigma_0), \quad L=\log(1+\Sigma_{\max}), \quad L=\sqrt{\log(1+\Sigma_0)},$$

• exponent reweightings:

$$S \sim F^{p}\,\mu_v^{q}\,L^{r}, \qquad p,q,r \in \{0.5,1,2\},$$

• lens-normalized variants using median normalization,
• control variants removing one factor entirely (“no lens”, “no flat”, “no amplitude”).

For each variant, rankings are compared to the baseline via rank correlations and Top-$K$ overlap. Results are recorded in:

$$\texttt{SSS\_rank\_stability\_640000.csv}, \quad \texttt{SSS\_variants\_top15\_640000.csv}.$$

Key criterion.

If the winner and the top-ranked configurations persist across these variants, the ranking is considered identifiable rather than metric-tuned.

SSS.6 Component Dominance Diagnostics

To detect hidden dominance by a single component, we compute correlations between the baseline score and individual factors:

$$\mathrm{CV}_{\mathrm{band}},\quad \mu_v,\quad \Sigma_0,\quad \log(1+\Sigma_0).$$

Additionally, we analyze the score in logarithmic form:

$$\log S = \log F + \log \mu_v + \log L,$$

which allows additive decomposition and mitigates multiplicative scale effects.

All Pearson and Spearman correlations are reported in:

$$\texttt{SSS\_component\_correlations\_640000.csv}.$$

A further stress test ranks runs by $\Sigma_0$ alone and measures Top-$K$ overlap with the baseline score. A low overlap falsifies the hypothesis that lensing normalization alone determines the ranking.

SSS.7 Summary and Pass–Fail Interpretation

A human-readable summary of all diagnostics is generated automatically as:

$$\texttt{SSS\_identifiability\_summary\_640000.txt}.$$

The interpretation rule is deliberately simple:

• PASS: Rankings exhibit high rank correlation ($\gtrsim 0.9$) and stable Top-$K$ overlap across unit scalings and reasonable score variants, with no single component dominating.
• FAIL: Rankings collapse under rescaling or are reproduced almost entirely by a single factor (e.g. $\Sigma_0$ alone).

For the dataset analyzed here, the results satisfy the PASS criteria.

SSS.8 Role of This Appendix in the Full Argument

Appendix SSS establishes that the empirical success reported in Appendix ‘§app:RRR‘ is not an artifact of units, normalization, or metric engineering. Together, Appendices RRR and SSS form a complete validation block:

• RRR shows that a strong, well-defined winner exists.
• SSS shows that this winner is identifiable and robust.

Subsequent appendices may therefore treat the identified configuration as a meaningful object of physical interpretation rather than a scoring accident.

@incollection{hassan2026appendixsssscalingun,
  author    = {Hassan, Akram},
  title     = {Appendix SSS — Scaling, Units, and Identifiability},
  booktitle = {The Complete Structural Selection Corpus},
  publisher = {Nuronova Genix Corp},
  year      = {2026},
  url       = {https://structuralselection.org/book/appendix/appendix-sss-scaling-units-and-identifiability}
}

TY  - CHAP
AU  - Hassan, Akram
TI  - Appendix SSS — Scaling, Units, and Identifiability
T2  - The Complete Structural Selection Corpus
PB  - Nuronova Genix Corp
PY  - 2026
UR  - https://structuralselection.org/book/appendix/appendix-sss-scaling-units-and-identifiability
ER  -