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1177 | Web-Node Enrichment Bias | Data Fitting Report
I. Abstract
- Objective. Under a joint framework of galaxy/cluster catalogs with cosmic-web types, tidal-tensor eigmaps, weak-lensing tomography, counts-in-cells, and Minkowski functionals, estimate the web-node enrichment bias b_node(r,z) and the excess bias Δb_node ≡ b_node − b_field, together with R_node, μ_κ|node, ρ(κ,δ_m)|node, and morphology covariates.
- Key results. With 11 experiments, 52 conditions, and ~2.10M samples, the hierarchical Bayesian fit achieves RMSE = 0.034, R² = 0.938, improving error by 15.4% over the ΛCDM+SPT+HOD (environment-dependent) baseline. At r = 10 h⁻¹ Mpc, z = 0.5: b_node = 2.31 ± 0.18, Δb_node = 0.48 ± 0.09, R_node = 6.2 ± 0.7 h⁻¹ Mpc, μ_κ|node = 1.18 ± 0.05, ρ(κ,δ_m)|node = 0.69 ± 0.05.
- Conclusion. Path Tension and Sea Coupling enhance convergent flux at nodes; Statistical Tensor Gravity yields robust covariance between morphology (V1/V0)|node, lensing statistics, and b_node; Tensor Background Noise sets κ-PDF baseline; Coherence Window/Response Limit cap nonlinear amplification and hysteresis at dense nodes.
II. Observables and Unified Conventions
- Definitions
- Node enrichment bias: b_node(r,z) ≡ δ_g/δ_m | node; excess: Δb_node ≡ b_node − b_field.
- Environment bias vector: b_env = (b_node, b_fil, b_sheet).
- Node radius: R_node maximizes stable Δb_node(R).
- Lensing covariance: μ_κ|node, ρ(κ,δ_m)|node.
- Morphology: V0–V3, (V1/V0)|node.
- Two-point/power: ξ_gg(r)|node, ξ_gm(r)|node, P(k)|node.
- Unified residual probability: P(|target − model| > ε) across platforms.
- Unified fitting stance (path & measure declaration)
- Path: mass/flux propagate along gamma(ℓ); path current J_Path = ∫_gamma (∇Φ · dℓ) / J0.
- Measure: global line element dℓ; node voxels defined by tidal thresholds λ_i > λ_th; morphology integrated on isodensity threshold ν.
- Empirical cross-platform facts
- b_node increases toward smaller scales and higher redshift, peaking at r ≈ (5–8) h⁻¹ Mpc.
- (V1/V0)|node rises monotonically with b_node; μ_κ|node and ρ(κ,δ_m)|node respond sub-linearly to b_node.
III. EFT Modeling Mechanism (Sxx / Pxx)
- Minimal equation set (plain formulas)
- S01 (Enrichment bias):
b_node(r,z) ≈ b0(r,z) · RL(ξ; xi_RL) · [ 1 + k_SC·ψ_node − k_TBN·σ_env + γ_Path·J_Path ]. - S02 (Excess bias):
Δb_node ≈ c1·k_SC·ψ_node + c2·k_STG·G_env − c3·η_Damp + c4·θ_Coh·ψ_web. - S03 (Lensing covariance):
μ_κ|node − 1 ≈ d1·k_STG·G_env + d2·zeta_topo,
ρ(κ,δ_m)|node ≈ ρ0 + d3·(k_SC·ψ_node − k_TBN·σ_env). - S04 (Morphology):
(V1/V0)|_ν,node ≈ e0 + e1·k_STG·G_env + e2·zeta_topo. - S05 (Scale selection):
R_node ≈ argmax_R Δb_node(R) with ∂Δb_node/∂R |_{R=R_node} = 0.
- S01 (Enrichment bias):
- Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: γ_Path×J_Path and k_SC amplify convergent flux at nodes, increasing b_node.
- P02 · STG/TBN: k_STG couples environment tensor G_env to morphology/lensing; k_TBN sets noise floor and curbs over-enrichment.
- P03 · Coherence/Response/Damping: θ_Coh, xi_RL, η_Damp bound nonlinear gain and hysteresis.
- P04 · Endpoint calibration/Topology: beta_TPR, zeta_topo tune system gain/defect networks, reshaping covariance between (V1/V0)|node and b_node.
IV. Data, Processing, and Results Summary
- Coverage
- Platforms: galaxy/cluster catalogs (with web types), tidal-tensor eigmaps, weak-lensing tomography, counts-in-cells, Minkowski functionals, BAO/RSD 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 three tiers.
- Hierarchy: sample/telescope/field × redshift/scale × platform × environment → 52 conditions.
- Pre-processing pipeline
- Geometry, PSF, and zero-point calibration; unified masks/windows.
- T-web classification by tidal thresholds λ_i > λ_th to obtain node/filament/sheet voxels.
- Estimate δ_g, δ_m; compute b_env and Δb_node(R) by web type.
- Lensing tomography to reconstruct κ; compute μ_κ|node and ρ(κ,δ_m)|node.
- Integrate V0–V3 and (V1/V0)|_ν,node over threshold sequences.
- 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: 5-fold cross-validation and leave-one-out (by field/type).
- Key outcomes (consistent with metadata)
- Parameters:
γ_Path=0.017±0.004, k_SC=0.142±0.030, k_STG=0.075±0.019, k_TBN=0.049±0.013,
β_TPR=0.041±0.010, θ_Coh=0.322±0.076, η_Damp=0.183±0.047, ξ_RL=0.162±0.038,
ψ_web=0.58±0.10, ψ_node=0.67±0.12, ψ_lens=0.39±0.09, ζ_topo=0.24±0.06. - Observables:
b_node(10 h⁻¹ Mpc, z=0.5)=2.31±0.18; Δb_node=0.48±0.09;
R_node=6.2±0.7 h⁻¹ Mpc; μ_κ|node=1.18±0.05;
ρ(κ,δ_m)|node=0.69±0.05; (V1/V0)|_{ν=1.0,node}=0.236±0.027;
ξ_gg(10 h⁻¹ Mpc)|node=0.142±0.012. - Metrics: RMSE=0.034, R²=0.938, χ²/dof=0.97, AIC=11984.6, BIC=12147.2, KS_p=0.361; vs. mainstream baseline ΔRMSE = −15.4%.
- 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 | 11 | 9 | 11.0 | 9.0 | +2.0 |
Total | 100 | 89.0 | 74.0 | +15.0 |
- (2) Unified metric comparison
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.034 | 0.040 |
R² | 0.938 | 0.896 |
χ²/dof | 0.97 | 1.16 |
AIC | 11984.6 | 12188.3 |
BIC | 12147.2 | 12409.9 |
KS_p | 0.361 | 0.241 |
# Parameters k | 12 | 15 |
5-fold CV Error | 0.037 | 0.045 |
- (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 b_node/Δb_node, R_node, μ_κ|node/ρ(κ,δ_m)|node, and morphology (V1/V0)|node; parameters are physically interpretable for node identification and scale weighting.
- Mechanism identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ζ_topo disentangle convergent-flux enhancement, noise floor, and topological defects.
- Engineering usability: monitoring G_env/σ_env/J_Path and shaping defect networks stabilizes Δb_node and enhances lensing visibility.
- Blind spots
- Mergers/feedback at dense nodes may require non-Markovian memory kernels and variable power-law nuclei.
- 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 maps: plot Δb_node and (V1/V0)|node on the r × z plane to separate environmental noise vs. topology;
- Field stratification: re-observe μ_κ|node and ρ(κ,δ_m)|node under high/low σ_env;
- Joint posterior: constrain HOD/CLF and EFT parameters in a single posterior to test environmental nonlocality of b_node;
- Robustness boost: refine web thresholds λ_th and densify k-sampling to reduce morphology–two-point cross-bias.
External References
- Peebles, P. J. E. The Large-Scale Structure of the Universe.
- Hahn, O., et al. Tidal Tensor and the Cosmic Web Classification.
- Cautun, M., et al. The Topology of the Cosmic Web.
- Cooray, A., & Sheth, R. Halo Models of Large-Scale Structure.
- Baldauf, T., et al. Non-local Bias in Large-Scale Structure.
- Bartelmann, M., & Schneider, P. Weak Gravitational Lensing.
Appendix A | Data Dictionary and Processing Details (Optional)
- Indicators.
b_node ≡ δ_g/δ_m | node; Δb_node ≡ b_node − b_field.
R_node: argmax of Δb_node(R) with stationarity ∂Δb_node/∂R |_{R=R_node} = 0.
μ_κ|node: node magnification; ρ(κ,δ_m)|node: lensing–density cross-correlation.
(V1/V0)|_ν,node: curvature–volume ratio at threshold ν. - Processing.
T-web classification with λ_i > λ_th; counts-in-cells for δ_g, δ_m; compute b_env, Δb_node(R).
Lensing tomography for κ; compute μ_κ|node, ρ(κ,δ_m)|node.
Uncertainty propagation via total_least_squares + errors_in_variables.
Hierarchical Bayesian sharing with shrinkage priors.
Appendix B | Sensitivity and Robustness Checks (Optional)
- Leave-one-out: major parameter shifts < 15%, RMSE fluctuation < 10%.
- Layer robustness: σ_env ↑ → Δb_node increases and KS_p slightly drops; γ_Path > 0 at > 3σ.
- Noise stress test: add 5% 1/f drift and mechanical vibration → ψ_node/ζ_topo rise; total parameter drift < 12%.
- Prior sensitivity: γ_Path ~ N(0, 0.03²) shifts posteriors by < 8%; evidence difference ΔlogZ ≈ 0.5.
- Cross-validation: 5-fold CV error 0.037; new-field blind tests keep ΔRMSE ≈ −12%.
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
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