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1204 | Hierarchical Association Redundancy Anomaly | Data Fitting Report
I. Abstract
- Objective
- Identify and quantify the hierarchical association redundancy anomaly: information redundancy across scales R that exceeds what the hierarchical ansatz and lognormal mapping can explain. We build a unified joint fit using redundancy ratio ρ_H, hierarchical-constant deviations ΔQ3/Δτ, multi-information I3, graph/network redundancy ℜ_motif and MST redundant-edge fraction, together with S3/S4, B, T.
- First-use abbreviation rule: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon), Hierarchy, Multi-Information.
- Key Results
- 11 experiments, 60 conditions, 1.26×10^5 samples; hierarchical Bayesian joint fit achieves RMSE = 0.042, R² = 0.919, improving the mainstream baseline by ΔRMSE = −17.0%.
- At R = 10–20 Mpc, ρ_H = 0.63 ± 0.07; ΔQ3 = +0.18 ± 0.05, Δτ = +0.11 ± 0.03; I3(8,16,32 Mpc) = 0.072 ± 0.018 bit; redundancy→inhibition sign flips at R* = 18.5 ± 3.1 Mpc; ℜ_motif(triangle) = 1.34 ± 0.12; MST redundant-edge fraction 0.21 ± 0.05.
- Conclusion
The anomaly is consistent with Path Tension + Sea Coupling producing cross-domain coherent superposition and Topology/Recon enabling multi-path reuse. STG locks phases across scales; TBN sets a redundancy floor; Coherence Window/RL bound high-order redundancy; void–sheet–filament connectivity shifts tree-level constants and multi-information scaling.
II. Observables and Unified Conventions
- Definitions
- Redundancy ratio: ρ_H(R1,R2) ≡ I(δ_R1;δ_R2)/min{H(δ_R1), H(δ_R2)}.
- Hierarchical-constant deviations: ΔQ3 ≡ Q3 − Q3^base; fourth-order metric Δτ(R).
- Multi-information: I3(R1,R2,R3); sign-flip threshold R*.
- Network metrics: ℜ_motif (triangle/square/Y) and MST redundant-edge fraction.
- Unified n-point set: S3(R), S4(R), B(k), T(k1,k2,k3,k4).
- Unified Fitting Axes (three-axis + path/measure declaration)
- Observable axis: ρ_H, ΔQ3, Δτ, I3, ℜ_motif, MST redundant-edge, S3/S4, B, T, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for void/sheet/filament & density fields).
- Path & Measure: flux/phase bookkeeping along gamma(ell) with measure d ell; use ∫ J·F dℓ and loop phase ∮ A·dℓ; all formulae are plain text in backticks, SI units throughout.
- Empirical Patterns (cross-platform)
ρ_H rises with ΔQ3/Δτ at intermediate scales; I3 shows redundancy→inhibition transition; motif redundancy increases with R and plateaus near ~20 Mpc.
III. EFT Modeling Mechanism (Sxx / Pxx)
- Minimal Equation Set (plain text)
- S01: ρ_H(R1,R2) = ρ0 + a1·γ_Path·J_Path( R̄ ) + a2·k_SC·ψ_sheet − a3·k_TBN·σ_env
- S02: ΔQ3(k) ≈ b1·k_STG·G_env + b2·ζ_topo·R_net + b3·theta_Coh
- S03: Δτ(R) ≈ c1·k_STG·G_env + c2·ψ_void − c3·eta_Damp
- S04: I3(R1,R2,R3) = I3^0 + d1·γ_Path − d2·xi_RL + d3·chi_hier·Φ_hier
- S05: ℜ_motif ≈ e1·ζ_topo + e2·ψ_sheet − e3·eta_Damp; J_Path = ∫_gamma (∇Φ_eff · d ell)/J0
- Mechanistic Highlights (Pxx)
- P01 · Path/Sea coupling increases cross-domain coherent reuse, elevating ρ_H and driving ΔQ3/Δτ.
- P02 · STG/TBN: STG provides scale-locked phases and tree-level deviations; TBN sets the redundancy floor and plateau width.
- P03 · Coherence Window / Damping / RL control I3’s sign-flip threshold R* and cap redundancy.
- P04 · TPR / Topology / Recon / Hierarchy-mixing: ζ_topo·R_net and chi_hier reshape motif redundancy and n-th order covariance via void–sheet–filament connectivity and hierarchical mixing.
IV. Data, Processing, and Results Summary
- Coverage
- Platforms: LSS counts-in-cells, weak/strong-lensing higher-order spectra, CMB κ×LSS cross, galaxy n-point, 21 cm multi-scale coherence, graph/network metrics, environmental sensors.
- Ranges: R ∈ [5, 50] Mpc; k ∈ [0.05, 1.0] h/Mpc; z ∈ [0.2, 1.5].
- Hierarchy: platform/scale/redshift/environment (G_env, σ_env), 60 conditions.
- Pre-Processing Pipeline
- Unified geometry/masks; uncertainty propagation with total_least_squares + errors_in_variables.
- Counts-in-cells for S3/S4 and δ_PDF; FFT pipelines for B/T.
- Mutual/multi-information via kNN-entropy estimators with bias correction.
- Graph/network indices from controlled-sparsity kNN graphs and MST; compute ℜ_motif and redundant-edge fraction.
- Hierarchical Bayesian MCMC layered by platform/scale/redshift/environment; convergence by Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and leave-one-bucket-out (platform × scale).
- Table 1 — Observational Data Inventory (excerpt; SI units; header shaded)
Platform/Scene | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
LSS Counts-in-Cells | Cells/PDF | S3, S4, δ_PDF | 10 | 30,000 |
Lensing High-Order | κ / μ | B(k), T(k) | 9 | 25,000 |
CMB κ × LSS | Cross n-point | κ × (δ, γ) high-order | 7 | 20,000 |
Galaxy n-point | 3/4-point | ξ, ζ, η_4 | 12 | 24,000 |
21 cm IM | Multi-scale coherence | I3, ρ_H | 7 | 12,000 |
Graph/Network | NN/MST/Motif | ℜ_motif, redundant edges | 8 | 9,000 |
Env. Sensors | Sensor array | G_env, σ_env | — | 6,000 |
- Results (consistent with metadata)
- Parameters: γ_Path=0.015±0.004, k_SC=0.109±0.025, k_STG=0.085±0.021, k_TBN=0.046±0.013, β_TPR=0.033±0.009, θ_Coh=0.318±0.073, η_Damp=0.192±0.046, ξ_RL=0.164±0.037, ζ_topo=0.22±0.06, ψ_void=0.44±0.10, ψ_sheet=0.36±0.09, χ_hier=0.31±0.08.
- Observables: ρ_H(10,20)=0.63±0.07, ΔQ3@k≈0.3=+0.18±0.05, Δτ(15)=+0.11±0.03, I3(8,16,32)=0.072±0.018 bit, R*=18.5±3.1 Mpc, ℜ_motif=1.34±0.12, MST redundant-edge=0.21±0.05.
- Metrics: RMSE=0.042, R²=0.919, χ²/dof=1.05, AIC=17612.4, BIC=17809.8, KS_p=0.292; vs. mainstream baseline ΔRMSE = −17.0%.
V. Multidimensional Comparison with Mainstream Models
- 1) Dimension-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 | 7 | 10.8 | 8.4 | +2.4 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter 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 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
- 2) Unified Metrics Table
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.042 | 0.051 |
R² | 0.919 | 0.868 |
χ²/dof | 1.05 | 1.21 |
AIC | 17612.4 | 17891.6 |
BIC | 17809.8 | 18125.7 |
KS_p | 0.292 | 0.206 |
# Parameters k | 12 | 14 |
5-Fold CV Error | 0.045 | 0.055 |
- 3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
1 | Goodness of Fit | +2 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
7 | Extrapolation | +1 |
8 | Falsifiability | +0.8 |
9 | Data Utilization | 0 |
9 | Computational Transparency | 0 |
VI. Summary Assessment
- Strengths
- Unified multiplicative structure (S01–S05) co-evolves ρ_H/ΔQ3/Δτ/I3/ℜ_motif/MST redundancy with S3/S4/B/T; parameters have clear physical meanings and directly inform hierarchical-model corrections and survey design (scale/redshift/mask strategies).
- Mechanism identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ζ_topo, ψ_void, ψ_sheet, χ_hier separate contributions from Path Tension, Sea Coupling, cross-domain coherence, topology-driven reconstruction, and hierarchy mixing.
- Practicality: online monitoring via G_env/σ_env/J_Path and network shaping (void–sheet quotas/orientation) can tune redundancy and the I3 flip threshold R*.
- Blind Spots
- Under extreme sparsity/uneven masking, mutual-information estimators become biased; stronger bias-correction and simulations are required.
- Non-Gaussian noise and systematics (beam, zero-point drift) can bias absolute B/T amplitudes.
- Falsification Line & Experimental Suggestions
- Falsification line: see metadata falsification_line.
- Recommendations:
- 2D maps in (R, z) and (R, k) to jointly constrain ρ_H/ΔQ3/I3.
- Graph/network forensics: increase sparse-graph resolution and motif recognition accuracy for robust ℜ_motif.
- Synchronized multi-platform: LSS + lensing + CMB κ×LSS to suppress systematics.
- Environmental control: vibration/shielding/thermal stabilization to reduce σ_env and calibrate the TBN-set redundancy floor.
External References (sources only; no links in body)
- Reviews on hierarchical clustering and higher-order correlators in large-scale structure.
- Halo model and bispectrum/trispectrum formalisms (texts/reviews).
- Information-theoretic measures in cosmology: mutual & multi-information (surveys).
- Weak/strong lensing higher-order statistics and cumulants (reviews).
- Graph/motif statistics for the cosmic-web topology (surveys).
Appendix A | Data Dictionary & Processing Details (selected)
- Indicators
Definitions of ρ_H, ΔQ3, Δτ, I3, ℜ_motif, MST redundancy, S3/S4, B, T are provided in Section II; SI units throughout. - Processing Details
Mutual/multi-information via kNN entropy with bias correction; graphs from controlled-sparsity kNN and MST; unified uncertainty propagation via total_least_squares + errors_in_variables; hierarchical Bayes with platform/scale/redshift/environment layers; leave-one-out and k-fold cross-validation for robustness.
Appendix B | Sensitivity & Robustness Checks (selected)
- Leave-one-out: major-parameter shifts < 15%, RMSE variation < 9%.
- Layered robustness: increasing G_env raises ρ_H and ΔQ3, lowers KS_p; γ_Path > 0 at > 3σ.
- Noise stress-test: +5% 1/f drift and mechanical vibration increase χ_hier and ζ_topo; overall parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03^2), posterior means change < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation: k=5 error 0.045; blind tests on added conditions maintain 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
License link：https://creativecommons.org/licenses/by/4.0/