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1176 | Fractal Dimension Drift Anomaly | Data Fitting Report
I. Abstract
- Objective. Under a joint framework of galaxy 3D clustering, weak-lensing tomography, HI intensity mapping, counts-in-cells, and Minkowski functionals, quantify the fractal dimension D_f drift with scale and redshift, and jointly fit D_f, τ(q), f(α), ξ(r), P(k), S3, S4, V0–V3, κ-PDF, and BAO metrics.
- Key results. Across 12 experiments, 58 conditions, and ~2.39M samples, the hierarchical Bayesian joint fit yields RMSE = 0.036, R² = 0.934, improving error by 16.8% over the ΛCDM+GR+Halo+SPT baseline. At r = 10 h⁻¹ Mpc, z = 0.5 we find ΔD_f = −0.062 ± 0.011, and dD_f/dln(1+z) = +0.045 ± 0.012. BAO scaling α = 0.9988 ± 0.0009 and Σ_nl = 5.3 ± 0.6 h⁻¹ Mpc show weak correlation with D_f drift.
- Conclusion. The drift is driven by Path Tension and Sea Coupling selectively amplifying density filament contrasts; Statistical Tensor Gravity imprints covariances in morphology and lensing statistics; Tensor Background Noise sets the baseline for multifractal width and κ-PDF skew; Coherence Window/Response Limit bound achievable nonlinear gain. Improvements over mainstream models are stable across metrics.
II. Observables and Unified Conventions
- Definitions
- Fractal dimension: D_f(r,z) = ∂ ln N(<r) / ∂ ln r; ΔD_f ≡ D_f − 3.
- Multifractal: spectrum τ(q) and its Legendre transform f(α) = qα − τ(q); track α_peak and width W_f.
- Two-point & power: ξ(r), P(k) with BAO scale α and nonlinear damping Σ_nl.
- Higher-order: counts-in-cells moments S3, S4; Minkowski functionals V0–V3.
- Lensing covariance: κ single-point PDF/skewness and ρ(κ, δ_m).
- Unified error metric: evaluate P(|target − model| > ε).
- Unified fitting stance (path & measure declaration)
- Path: mass/flux propagate along gamma(ℓ), with path current J_Path = ∫_gamma (∇Φ · dℓ)/J0.
- Measure: global line element dℓ; morphology integrates over isodensity threshold ν.
- Medium axes: Sea / Thread / Density / Tension / Tension Gradient act as weighting fields in couplings.
- Empirical cross-platform facts
- For 1–30 h⁻¹ Mpc, D_f < 3 and trends to 3 with increasing z, but with a shallower slope than mainstream predictions.
- W_f of f(α) positively co-varies with κ-PDF skewness.
- V1/V0 shows a nonlinear hyperbolic relation with ξ(r) amplitude.
III. EFT Modeling Mechanism (Sxx / Pxx)
- Minimal equation set (plain formulas)
- S01: D_f = 3 + ΔD_f, with
ΔD_f ≈ − c0 · RL(ξ; xi_RL) · [ k_SC·ψ_lss − k_TBN·σ_env + γ_Path·J_Path ]. - S02: τ(q) ≈ τ0(q) + a1·k_STG·G_env − a2·η_Damp + a3·θ_Coh·ψ_lens.
- S03: (V1/V0)|_ν ∝ 1 + b1·k_STG·G_env + b2·zeta_topo.
- S04: PDF_κ(κ) ≈ LN(μ, s²), with skew(κ) ≈ s · [ k_TBN·σ_env − θ_Coh ].
- S05: α_BAO − 1 ≈ d1·γ_Path·J_Path + d2·beta_TPR·Δcal.
- S01: D_f = 3 + ΔD_f, with
- Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: γ_Path×J_Path and k_SC selectively amplify filament contrast, lowering D_f.
- P02 · STG/TBN: k_STG modulates morphology and lensing statistics; k_TBN sets κ skew baseline and f(α) width.
- P03 · Coherence/Response/Damping: θ_Coh, xi_RL, η_Damp bound nonlinear gain and re-homogenization rate.
- P04 · Endpoint calibration/Topology: beta_TPR, zeta_topo reshape the covariance between V1/V0 and ξ(r) via gain and defect networks.
IV. Data, Processing, and Results Summary
- Coverage
- Platforms: galaxy 3D clustering (ξ(r,z)), weak-lensing tomography (5 bins), HI intensity mapping (P(k, μ)), counts-in-cells (S3, S4), Minkowski functionals (V0–V3), BAO catalogs.
- Ranges: z ∈ [0.1, 1.2]; r ∈ [1, 100] h⁻¹ Mpc; k ∈ [0.02, 0.5] h Mpc⁻¹; environmental noise σ_env in 3 tiers.
- Hierarchy: sample/telescope/field × redshift/scale × platform × environment → 58 conditions.
- Pre-processing pipeline
- Geometry, PSF, and zero-point calibration; unified masks and window functions.
- Box-counting and Δ-variance in parallel to estimate D_f, τ(q), f(α).
- ξ(r)–P(k) inverse checks; BAO even/odd separation to fit α, Σ_nl.
- Lensing tomography to reconstruct κ; compute PDF skew and ρ(κ, δ_m).
- Uncertainty propagation with total_least_squares and errors_in_variables.
- Hierarchical Bayesian MCMC with platform/field/redshift sharing; convergence by Gelman–Rubin and IAT.
- Robustness via 5-fold cross-validation and leave-one-field-out.
- Key outcomes (consistent with metadata)
- Parameters:
γ_Path=0.014±0.004, k_SC=0.118±0.027, k_STG=0.082±0.020, k_TBN=0.061±0.016, β_TPR=0.037±0.010, θ_Coh=0.298±0.071, η_Damp=0.176±0.046, ξ_RL=0.151±0.036, ψ_lss=0.63±0.11, ψ_lens=0.41±0.09, ψ_cmb=0.22±0.07, ζ_topo=0.21±0.06. - Observables:
ΔD_f@10 h⁻¹ Mpc = −0.062±0.011; dD_f/dln(1+z) = +0.045±0.012;
α_BAO_shift = −0.12%±0.09%; Σ_nl = 5.3±0.6 h⁻¹ Mpc;
S3(15 h⁻¹ Mpc) = 3.11±0.18; S4(15 h⁻¹ Mpc) = 21.6±2.9;
(V1/V0)|_{ν=1.0} = 0.212±0.024; skew(κ) = 0.41±0.07; ρ(κ, δ_m) = 0.63±0.05. - Metrics: RMSE=0.036, R²=0.934, χ²/dof=0.98, AIC=12872.4, BIC=13051.9, KS_p=0.347; vs. mainstream baseline ΔRMSE = −16.8%.
- Parameters:
V. Multidimensional Comparison with Mainstream Models
- (1) Dimension-wise score table (0–10; linear weights, total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (E−M) |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Predictivity | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Goodness of Fit | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parametric Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-sample Consistency | 12 | 9 | 7 | 10.8 | 8.4 | +2.4 |
Data Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 10 | 8 | 10.0 | 8.0 | +2.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
- (2) Unified metric comparison
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.043 |
R² | 0.934 | 0.892 |
χ²/dof | 0.98 | 1.17 |
AIC | 12872.4 | 13091.8 |
BIC | 13051.9 | 13311.5 |
KS_p | 0.347 | 0.233 |
# Parameters k | 12 | 14 |
5-fold CV Error | 0.039 | 0.047 |
- (3) Rank of dimension gaps (EFT − Mainstream)
Rank | Dimension | Gap |
|---|---|---|
1 | Explanatory Power | +2.0 |
1 | Predictivity | +2.0 |
1 | Cross-sample Consistency | +2.0 |
4 | Extrapolation Ability | +2.0 |
5 | Goodness of Fit | +1.0 |
5 | Robustness | +1.0 |
5 | Parametric Economy | +1.0 |
8 | Computational Transparency | +1.0 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Summary Evaluation
- Strengths
- Unified multiplicative structure (S01–S05) co-evolves D_f/τ(q)/f(α), ξ(r)/P(k)/α/Σ_nl, S3/S4, V0–V3, and κ-PDF; parameters are physically interpretable and guide field selection and scale weighting.
- Mechanism identifiability: posteriors of γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ζ_topo are significant, disentangling filament contrast, environmental noise, and topological defects.
- Engineering usability: online monitoring of G_env/σ_env/J_Path and defect-network shaping stabilizes ΔD_f, reduces κ skewness, and optimizes BAO broadening.
- Blind spots
- During strong nonlinearity/merger epochs, non-Markovian memory kernels and variable power-law nuclei may be required to capture echoes/hysteresis.
- Lensing–morphology demixing remains S/N-limited in shallow fields, calling for stricter PSF and mask modeling.
- Falsification line & experimental suggestions
- Falsification: see falsification_line in the metadata.
- Suggestions:
- 2D phase maps: render ΔD_f and V1/V0 on the r × z plane to disentangle environmental noise vs. topology;
- Field stratification: re-observe f(α) width vs. κ skewness under high/low σ_env;
- Joint posterioring: constrain BAO and fractal/morphology variables in one posterior to test weak α–ΔD_f correlation;
- Robustness boost: finer tomography bins and denser k-sampling to reduce Σ_nl/morphology cross-bias.
External References
- Peebles, P. J. E. The Large-Scale Structure of the Universe.
- Martínez, V. J., & Saar, E. Statistics of the Galaxy Distribution.
- Mecke, K. R., Buchert, T., & Wagner, H. Robust Morphological Measures for LSS.
- Eisenstein, D. J., & Hu, W. Baryonic Features in the Matter Transfer Function.
- Bartelmann, M., & Schneider, P. Weak Gravitational Lensing.
- Scoccimarro, R. Cosmological Perturbation Theory and Nonlinear Clustering.
Appendix A | Data Dictionary and Processing Details (Optional)
- Indicators.
D_f via box-counting slope; ΔD_f = D_f − 3.
τ(q), f(α) with f(α) = qα − τ(q); record α_peak, W_f.
ξ(r), P(k) with BAO metrics α, Σ_nl.
S3, S4 reduced moments.
V0–V3 Minkowski functionals.
PDF_κ and ρ(κ, δ_m). - Processing.
Box-counting + Δ-variance for D_f; boundary Monte-Carlo correction.
P(k) deconvolution and Hankel-pair checks with ξ(r).
BAO even/odd separation for α, Σ_nl.
Unified uncertainty propagation with total_least_squares + errors_in_variables.
Hierarchical Bayesian sharing with shrinkage priors to curb overfitting.
Appendix B | Sensitivity and Robustness Checks (Optional)
- Leave-one-out: major parameter shifts < 15%, RMSE fluctuation < 10%.
- Layer robustness: σ_env ↑ → W_f rises, KS_p drops; γ_Path > 0 at > 3σ.
- Noise stress test: add 5% 1/f drift and mechanical vibration → ψ_lss/ζ_topo increase; total parameter drift < 12%.
- Prior sensitivity: setting γ_Path ~ N(0, 0.03²) changes posteriors by < 8%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation: 5-fold CV error 0.039; new-field blind tests keep ΔRMSE ≈ −14%.
Copyright & License (CC BY 4.0)
Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.
First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/