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arxiv: 2604.13230 · v1 · submitted 2026-04-14 · 💻 cs.LG · cs.NE

Does Dimensionality Reduction via Random Projections Preserve Landscape Features?

Iv\'an Olarte Rodr\'iguez , Anja Jankovic , Thomas B\"ack , Elena Raponi This is my paper

Pith reviewed 2026-05-10 16:29 UTC · model grok-4.3

classification 💻 cs.LG cs.NE
keywords exploratory landscape analysisrandom projectionsdimensionality reductionblack-box optimizationlandscape featuresfeature robustnessgaussian embeddings
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The pith

Random projections change most exploratory landscape features so they no longer represent the original optimization problem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Exploratory Landscape Analysis extracts numerical features that describe the geometry and topology of black-box optimization problems. High-dimensional cases make these features costly, leading researchers to apply random projections first to reduce dimension. The paper recomputes the same ELA features on identical sample points before and after projection across budgets and embedding sizes. Most feature values shift substantially, which means the reduced-space versions fail to reflect the original landscape structure. A small group of features remains more stable, yet stability alone does not guarantee they still capture intrinsic problem properties rather than projection artifacts.

Core claim

Linear random projections via Gaussian embeddings frequently modify the geometric and topological structures that Exploratory Landscape Analysis depends on, so the resulting feature values cease to be representative of the original high-dimensional problem.

What carries the argument

Direct recomputation of ELA features on the exact same sampled points and objective values, once in the original space and once after Random Gaussian Embedding, followed by numerical comparison across sample budgets and target dimensions.

If this is right

  • Most ELA features become unreliable indicators after linear random projection.
  • Reduced-space feature values typically do not match the original problem's landscape characteristics.
  • Only a small subset of features exhibits comparative stability across projections.
  • Even stable features may still encode projection-induced artifacts instead of original landscape traits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • High-dimensional algorithm selection that uses ELA should validate feature stability on the concrete problem rather than assume projection preserves information.
  • The observed sensitivity points toward testing structure-preserving alternatives such as PCA or manifold learning for future ELA pipelines.
  • Practitioners applying ELA to high-dimensional tasks may need to accept higher computational cost or develop projection-aware feature normalizations.

Load-bearing premise

That agreement or disagreement between ELA feature values computed on the same points before and after projection is sufficient to decide whether the reduced features still describe intrinsic properties of the original landscape.

What would settle it

A follow-up experiment on standard benchmark functions that shows the majority of ELA feature classes produce statistically indistinguishable values in the projected space for every tested dimension and sample size.

Figures

Figures reproduced from arXiv: 2604.13230 by Anja Jankovic, Elena Raponi, Iv\'an Olarte Rodr\'iguez, Thomas B\"ack.

Figure 1
Figure 1. Figure 1: Surface plot and pointwise projection onto a [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Heatmap of the median normalized absolute feature distribution shift for datasets of size 200 with a compression [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normalized aggregated feature distribution shift of Schwefel (f20) function with [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the ela_level.lda_qda_10 feature of instance 1 of the Sphere (f1) function. The blue distribution corresponds to the measured feature by using 𝑆 = 2000 and the orange corresponds to the measurement of the feature by using 𝑆 = 200. 0 20 40 Full (no reduction) 0 10 20 Density 𝑟 = 0.5 0.8 1.0 1.2 1.4 1.6 ic.eps_s 0 5 10 15 𝑟 = 0.25 ic.eps_s|fid=16, instance=1 𝑆 = 2000 𝑆 = 200 [PITH_FULL_IMAGE… view at source ↗
Figure 5
Figure 5. Figure 5: Distribution of the ic.eps_s feature of instance 1 of the Weierstrass (f16) function. The blue distribution corresponds to the measured feature by using 𝑆 = 2000 and the orange corresponds to the measurement of the feature by using 𝑆 = 200 sample sizes 𝑆 = 2000 and 𝑆 = 200 become increasingly aligned in terms of their mean values. This apparent agreement suggests that the feature estimator has stabilized, … view at source ↗
Figure 1
Figure 1. Figure 1: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 1
Figure 1. Figure 1: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Heatmap of the median normalized absolute feature distribution shift for [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 5
Figure 5. Figure 5: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p026_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p031_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p032_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p034_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
Figure 12
Figure 12. Figure 12: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p036_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 13
Figure 13. Figure 13: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p038_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p039_14.png] view at source ↗
Figure 14
Figure 14. Figure 14: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p040_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p041_15.png] view at source ↗
Figure 15
Figure 15. Figure 15: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p042_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p043_16.png] view at source ↗
Figure 16
Figure 16. Figure 16: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p044_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p045_17.png] view at source ↗
Figure 17
Figure 17. Figure 17: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p046_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p047_18.png] view at source ↗
Figure 18
Figure 18. Figure 18: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p048_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p049_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p050_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p051_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p052_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p053_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p054_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p055_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p056_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p057_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p057_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p058_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p059_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p060_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p061_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p062_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p063_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p064_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Same as above, for 𝑺 = 200. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p065_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p066_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: Same as above for 𝑺 = 2000 and 𝒓 = 0.25. ela_meta.lin_simple.adj_r2 ela_meta.lin_simple.intercept ela_meta.lin_simple.coef.min ela_meta.lin_simple.coef.max ela_meta.lin_simple.coef.max_by_min ela_meta.lin_w_interact.adj_r2 ela_meta.quad_simple.adj_r2 ela_meta.quad_simple.cond ela_meta.quad_w_interact.adj_r2 ela_distr.skewness ela_distr.kurtosis ela_distr.number_of_peaks ela_level.mmce_lda_10 ela_level.mmc… view at source ↗
Figure 39
Figure 39. Figure 39: Same as above for 𝑺 = 200 and 𝒓 = 0.5. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p066_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p067_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: Same as above for 𝑺 = 2000 and 𝒓 = 0.25. ela_meta.lin_simple.adj_r2 ela_meta.lin_simple.intercept ela_meta.lin_simple.coef.min ela_meta.lin_simple.coef.max ela_meta.lin_simple.coef.max_by_min ela_meta.lin_w_interact.adj_r2 ela_meta.quad_simple.adj_r2 ela_meta.quad_simple.cond ela_meta.quad_w_interact.adj_r2 ela_distr.skewness ela_distr.kurtosis ela_distr.number_of_peaks ela_level.mmce_lda_10 ela_level.mmc… view at source ↗
Figure 42
Figure 42. Figure 42: Same as above for 𝑺 = 200 and 𝒓 = 0.5. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p067_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p068_43.png] view at source ↗
Figure 44
Figure 44. Figure 44: Same as above for 𝑺 = 2000 and 𝒓 = 0.25. ela_meta.lin_simple.adj_r2 ela_meta.lin_simple.intercept ela_meta.lin_simple.coef.min ela_meta.lin_simple.coef.max ela_meta.lin_simple.coef.max_by_min ela_meta.lin_w_interact.adj_r2 ela_meta.quad_simple.adj_r2 ela_meta.quad_simple.cond ela_meta.quad_w_interact.adj_r2 ela_distr.skewness ela_distr.kurtosis ela_distr.number_of_peaks ela_level.mmce_lda_10 ela_level.mmc… view at source ↗
Figure 45
Figure 45. Figure 45: Same as above for 𝑺 = 200 and 𝒓 = 0.5. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p068_45.png] view at source ↗
Figure 46
Figure 46. Figure 46: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p069_46.png] view at source ↗
Figure 47
Figure 47. Figure 47: Same as above for 𝑺 = 2000 and 𝒓 = 0.25. ela_meta.lin_simple.adj_r2 ela_meta.lin_simple.intercept ela_meta.lin_simple.coef.min ela_meta.lin_simple.coef.max ela_meta.lin_simple.coef.max_by_min ela_meta.lin_w_interact.adj_r2 ela_meta.quad_simple.adj_r2 ela_meta.quad_simple.cond ela_meta.quad_w_interact.adj_r2 ela_distr.skewness ela_distr.kurtosis ela_distr.number_of_peaks ela_level.mmce_lda_10 ela_level.mmc… view at source ↗
Figure 48
Figure 48. Figure 48: Same as above for 𝑺 = 200 and 𝒓 = 0.5. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p069_48.png] view at source ↗
Figure 49
Figure 49. Figure 49: Normalized aggregated feature distribution shift of [PITH_FULL_IMAGE:figures/full_fig_p070_49.png] view at source ↗
Figure 50
Figure 50. Figure 50: Same as above for 𝑺 = 2000 and 𝒓 = 0.25. ela_meta.lin_simple.adj_r2 ela_meta.lin_simple.intercept ela_meta.lin_simple.coef.min ela_meta.lin_simple.coef.max ela_meta.lin_simple.coef.max_by_min ela_meta.lin_w_interact.adj_r2 ela_meta.quad_simple.adj_r2 ela_meta.quad_simple.cond ela_meta.quad_w_interact.adj_r2 ela_distr.skewness ela_distr.kurtosis ela_distr.number_of_peaks ela_level.mmce_lda_10 ela_level.mmc… view at source ↗
Figure 51
Figure 51. Figure 51: Same as above for 𝑺 = 200 and 𝒓 = 0.5. Manuscript submitted to ACM [PITH_FULL_IMAGE:figures/full_fig_p070_51.png] view at source ↗
read the original abstract

Exploratory Landscape Analysis (ELA) provides numerical features for characterizing black-box optimization problems. In high-dimensional settings, however, ELA suffers from sparsity effects, high estimator variance, and the prohibitive cost of computing several feature classes. Dimensionality reduction has therefore been proposed as a way to make ELA applicable in such settings, but it remains unclear whether features computed in reduced spaces still reflect intrinsic properties of the original landscape. In this work, we investigate the robustness of ELA features under dimensionality reduction via Random Gaussian Embeddings (RGEs). Starting from the same sampled points and objective values, we compute ELA features in projected spaces and compare them to those obtained in the original search space across multiple sample budgets and embedding dimensions. Our results show that linear random projections often alter the geometric and topological structure relevant to ELA, yielding feature values that are no longer representative of the original problem. While a small subset of features remains comparatively stable, most are highly sensitive to the embedding. Moreover, robustness under projection does not necessarily imply informativeness, as apparently robust features may still reflect projection-induced artifacts rather than intrinsic landscape characteristics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript investigates whether Exploratory Landscape Analysis (ELA) features remain representative of intrinsic landscape properties when computed on points projected from high-dimensional spaces to lower dimensions via Random Gaussian Embeddings. Starting from identical sampled points and objective values, the authors compare ELA feature values across original and projected spaces for varying sample budgets and embedding dimensions, concluding that most features are sensitive to projection (altering relevant geometric and topological structure) while a small subset appears stable but may reflect projection-induced artifacts rather than intrinsic characteristics.

Significance. If the central empirical findings hold after addressing methodological confounds, the work would be significant for black-box optimization and landscape analysis, as it provides evidence against naive use of dimensionality reduction to enable ELA in high dimensions and highlights the need for careful validation of reduced-space features.

major comments (1)
  1. [Methodology and Results] The experimental comparison (described in the methodology and results sections) fixes the point set and objective values before applying linear projection and recomputing ELA features. This necessarily increases relative point density in the lower-dimensional space by a factor scaling with the volume ratio, which directly affects density- and neighborhood-dependent ELA feature classes (dispersion, information content, convexity, etc.). No controls are reported to isolate projection-induced density bias from genuine structural alteration, such as density-matched resampling in the target space or validation against analytically known low-dimensional landscapes. This confound is load-bearing for the claim that observed feature shifts indicate loss of intrinsic structure rather than estimator bias.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from explicit enumeration of the test functions, exact original and target dimensions, sample sizes per budget, number of random projections per instance, and any statistical significance testing used to support comparative claims.
  2. [Results] Figure captions and tables reporting feature stability should include error bars or variability measures across random embeddings to allow readers to assess the reliability of 'stable' versus 'sensitive' classifications.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed and insightful comments on our manuscript. We address the major comment point by point below.

read point-by-point responses

  1. Referee: [Methodology and Results] The experimental comparison (described in the methodology and results sections) fixes the point set and objective values before applying linear projection and recomputing ELA features. This necessarily increases relative point density in the lower-dimensional space by a factor scaling with the volume ratio, which directly affects density- and neighborhood-dependent ELA feature classes (dispersion, information content, convexity, etc.). No controls are reported to isolate projection-induced density bias from genuine structural alteration, such as density-matched resampling in the target space or validation against analytically known low-dimensional landscapes. This confound is load-bearing for the claim that observed feature shifts indicate loss of intrinsic structure rather than estimator bias.

    Authors: We thank the referee for identifying this important methodological issue. Our experimental setup uses the same set of sampled points and their corresponding objective values in both the original and projected spaces to directly assess the impact of the random projection on the computed ELA features. This design choice simulates the practical application where evaluations obtained in the high-dimensional space are projected for landscape analysis in reduced dimensions. We agree, however, that the increased relative density of points in the lower-dimensional space could influence features that rely on dispersion, neighborhood information, or local convexity, and that this may confound the interpretation of feature changes as purely due to loss of intrinsic structure. To address this concern, we will add control experiments in the revised manuscript, including density-matched resampling in the target space and comparisons with analytically known low-dimensional landscapes. These revisions will help isolate the effects of projection from density bias and strengthen the validity of our conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; purely empirical feature comparison

full rationale

The manuscript conducts a direct numerical comparison of ELA features computed on identical sampled points and objective values before versus after random Gaussian projection. No derivations, first-principles results, fitted parameters, or predictions are claimed. The central claim rests on observed differences in feature values across dimensions and sample budgets, which does not reduce to any self-referential definition or input by construction. External citations (if present) concern standard ELA definitions and are not load-bearing for any uniqueness or ansatz. Methodological questions about projection-induced density changes are separate from circularity and do not involve any quoted reduction of the reported result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The study is empirical and introduces no new mathematical parameters, axioms, or postulated entities; all quantities are standard ELA features and standard random projection matrices.

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