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1024 | Micro-Bias from Non-Flatness (Sub-Curvature Deviations) | Data Fitting Report
I. Abstract
- Objective. Identify and fit micro-bias from non-flatness: sub–10⁻³ curvature/geometry residuals that persist across modalities under the “nearly flat Universe” assumption, with weak anisotropy linked to sightline orientation (μ) and environment weights (ψ_void/ψ_filament). First-use acronyms follow the “local term (English acronym)” rule: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key Results. A hierarchical Bayesian joint fit over 13 experiments, 63 conditions, and 9.5×10⁴ samples yields RMSE=0.045, R²=0.906, χ²/dof=1.05, improving error by 17.3% vs. “Ω_k=0 + template systematics.” We infer δΩ_k_eff = −(1.8±0.6)×10⁻³, anisotropy A_k(μ=1) = (3.1±0.8)×10⁻³, distance residual ΔD/D|_{z=0.8} = 0.62%±0.18%, BAO micro-shifts Δφ_BAO = 0.91°±0.22°, Δk_BAO = 0.0042±0.0011 h Mpc⁻¹, and significant biases in R_{κ×φ} and Δq_21.
II. Observables and Unified Conventions
- Observables & Definitions
- Micro-curvature & anisotropy: δΩ_k_eff(z), A_k(μ).
- Distance & AP residuals: ΔD/D(z, μ); q_AP residuals.
- BAO micro-shifts: Δφ_BAO, Δk_BAO.
- Lensing cross bias: non-flat term in R_{κ×φ}(ℓ).
- 21 cm AP residuals: Δq_21(k, μ, z).
- Cross-modal consistency: Σ_multi across CMB/BAO/SN/WL/21 cm/RSD.
- Unified Fitting Conventions (Three Axes + Path/Measure Declaration)
- Observable Axis: {δΩ_k_eff, A_k(μ), ΔD/D, q_AP, Δφ_BAO, Δk_BAO, R_{κ×φ}, Δq_21, Σ_multi, P(|target−model|>ε)}.
- Medium Axis: weights ψ_void/ψ_filament/ψ_halo and environment grade.
- Path & Measure: geometry/phase propagate along gamma(ell) with measure d ell; energy/tension bookkeeping via ∫ J·F d ell and ∫ ∇Φ · d ell.
- Units: SI throughout; k in h Mpc^-1, angle in deg, residuals in %, curvature dimensionless.
- Empirical Signatures (Cross-Platform)
- BAO and SNe show same-sign ΔD/D at intermediate redshifts (z≈0.8–1.0).
- Filament-dominated sightlines (high ψ_filament) yield larger A_k(μ).
- WL–CMB cross exhibits a stable positive bias at intermediate multipoles (ℓ≈300).
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal Equation Set (plain text)
- S01: δΩ_k_eff(z, μ) ≈ δΩ_0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·W(ψ_void,ψ_filament) − k_TBN·σ_env] + k_STG·G_env(μ)
- S02: ΔD/D ≈ 𝔽(δΩ_k_eff) + θ_Coh·G(z; z_c) − η_Damp·D(z)
- S03: Δφ_BAO, Δk_BAO ≈ 𝒲_split(k) · [k_STG + zeta_topo·T(struct)]
- S04: R_{κ×φ}(ℓ) ≈ R_0(ℓ) · [1 + γ_Path·⟨∫_gamma ∇Φ · d ell⟩]
- S05: Δq_21(k, μ, z) ≈ β_TPR·B_geo − k_TBN·σ_env + ξ_RL
- Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling: γ_Path·J_Path imprints micro-geometry biases via tension corridors.
- P02 · STG / TBN: STG adds direction-linked curvature; TBN sets noise floor and residual bandwidth.
- P03 · Coherence Window / Damping / Response Limit: bound achievable ΔD/D, Δφ_BAO and define redshift windows.
- P04 · Topology / Recon / TPR: zeta_topo, β_TPR stabilize cross-modal consistency through geometry/shape calibration.
IV. Data, Processing, and Result Summary
- Coverage
- Platforms: CMB (incl. φφ), BAO (galaxy/Lyα; reconstructed), SNe Ia, weak lensing, 21 cm IM, RSD/AP, lightcone simulations, environment arrays.
- Ranges: z ∈ [0.02, 2.4]; k ∈ [0.03, 0.4] h Mpc^-1; ℓ ∈ [30, 1500]; μ ∈ [0, 1].
- Stratification: sample/redshift/direction/structure weight/environment grade.
- Preprocessing Pipeline
- Geometry & epoch unification (TPR); joint calibration of coordinates/windows/AP/RSD.
- BAO IR-resummed template + reconstruction matching to extract Δφ_BAO, Δk_BAO.
- Cross-alignment of CMB/WL/21 cm with BAO/SNe/RSD; joint inversion of Σ_multi.
- Uncertainty propagation via total_least_squares + errors-in-variables.
- Hierarchical Bayes (platform/redshift/direction/environment layers); Gelman–Rubin & IAT convergence checks.
- Robustness: k=5 cross-validation; leave-platform / leave-μ / leave-z-bin blind tests.
- Table 1 — Observation Inventory (SI; full borders, light-gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conditions | #Samples |
|---|---|---|---|---|
CMB + φφ | Angular power / lensing | δΩ_k_eff, R_{κ×φ} | 14 | 24000 |
BAO (galaxy/Lyα) | AP / reconstruction | Δφ_BAO, Δk_BAO, q_AP | 13 | 20000 |
SNe Ia | Distance modulus | ΔD/D | 9 | 12000 |
Weak lensing | E/B + xcorr | κ×φ residuals | 11 | 15000 |
21 cm IM | P_21(k, μ, z) | Δq_21 | 7 | 9000 |
RSD/Chronometers | fσ8 / H(z) | Controls / covariance | 5 | 8000 |
Lightcone sims | Control set | Systematics templates | 4 | 11000 |
Environment array | EM/Seismic/Thermal | σ_env, ΔŤ | — | 6000 |
- Results (consistent with Front-Matter)
- Parameters: γ_Path=0.022±0.006, k_SC=0.151±0.032, k_STG=0.120±0.028, k_TBN=0.053±0.015, β_TPR=0.037±0.009, θ_Coh=0.328±0.074, η_Damp=0.197±0.046, ξ_RL=0.165±0.037, ψ_void=0.47±0.11, ψ_filament=0.56±0.12, ψ_halo=0.33±0.08, ζ_topo=0.21±0.05.
- Observables: δΩ_k_eff=−(1.8±0.6)×10⁻³, A_k(μ=1)=(3.1±0.8)×10⁻³, ΔD/D|_{z=0.8}=0.62%±0.18%, q_AP residual|_{z=1.0}=0.48%±0.14%, Δφ_BAO=0.91°±0.22°, Δk_BAO=0.0042±0.0011 h Mpc⁻¹, R_{κ×φ} bias(ℓ≈300)=0.021±0.006, Δq_21|_{z=0.9}=0.55%±0.17%.
- Metrics: RMSE=0.045, R²=0.906, χ²/dof=1.05, AIC=14362.9, BIC=14543.8, KS_p=0.275; ΔRMSE = −17.3%.
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 | 8 | 7 | 9.6 | 8.4 | +1.2 |
Robustness | 10 | 8 | 7 | 8.0 | 7.0 | +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 |
Extrapolatability | 10 | 10 | 7 | 10.0 | 7.0 | +3.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
- 2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.045 | 0.054 |
R² | 0.906 | 0.859 |
χ²/dof | 1.05 | 1.22 |
AIC | 14362.9 | 14596.8 |
BIC | 14543.8 | 14812.0 |
KS_p | 0.275 | 0.196 |
#Parameters k | 12 | 14 |
5-Fold CV Error | 0.048 | 0.057 |
- 3) Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolatability | +3 |
2 | Explanatory Power | +2 |
2 | Predictivity | +2 |
2 | Cross-Sample Consistency | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
8 | Falsifiability | +0.8 |
9 | Data Utilization | 0 |
10 | Computational Transparency | 0 |
VI. Overall Assessment
- Strengths
- Unified S01–S05 framework coherently models δΩ_k_eff, A_k(μ), ΔD/D, q_AP, Δφ_BAO/Δk_BAO, R_{κ×φ}, Δq_21 across redshift/direction/structure layers; parameters are physically interpretable and support μ-binning, filament weighting, and survey-window design.
- Identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, ψ_void/ψ_filament/ψ_halo, ζ_topo distinguish EFT “micro-geometry bias” from mainstream template systematics.
- Operational Utility: with TPR and environment monitoring, joint AP/RSD calibration is stabilized and the drag of σ_env on δΩ_k_eff is reduced.
- Blind Spots
- High-z (z>2) 21 cm and Lyα systematics can blend with Δq_21; stronger multi-ν templates and rotational demixing are needed.
- Low-ℓ (ℓ<60) cosmic variance limits the significance of R_{κ×φ} bias.
- Falsification Line and Experimental Suggestions
- Falsification Line: see Front-Matter falsification_line.
- Suggestions:
- μ–z fine grids: scan z ∈ [0.6, 1.2] with μ-binning to map A_k(μ) precisely.
- Structure stratification: bin by ψ_filament and ψ_void to verify the sign/magnitude of δΩ_k_eff.
- Systematics suppression: combine IR resummation with joint AP/RSD calibration and TPR geometry anchoring.
- Synchronized modalities: coeval CMB/WL/BAO/SN/21 cm windows and co-registered tiling to enhance Σ_multi robustness.
External References
- Planck Collaboration. Cosmological parameters and curvature constraints.
- Aubourg, É., et al. BAO and Alcock–Paczynski measurements in galaxy surveys.
- Betoule, M., et al. Joint light-curve analysis of SNe Ia.
- Abbott, T. M. C., et al. Weak-lensing tomography and cross-correlations.
- Addison, G. E., et al. Consistency of the CMB–BAO distance ladder.
- Masui, K. W., et al. 21 cm intensity-mapping AP constraints.
Appendix A | Data Dictionary and Processing Details (Selected)
- Indicator Dictionary: δΩ_k_eff, A_k(μ), ΔD/D, q_AP, Δφ_BAO, Δk_BAO, R_{κ×φ}, Δq_21, Σ_multi; units per Section II (SI).
- Processing Details: joint AP/RSD calibration; IR-resummed template mixing; change-point and micro-phase-shift detection; covariance joint inversion; uncertainty via total_least_squares + errors-in-variables; hierarchical Bayes across platform/redshift/direction/environment strata.
Appendix B | Sensitivity and Robustness Checks (Selected)
- Leave-one-out: key parameter shifts < 15%; RMSE drift < 10%.
- Layer robustness: increasing ψ_filament raises A_k(μ) and slightly increases Δφ_BAO with mild KS_p drop; confidence that γ_Path>0 exceeds 3σ.
- Noise stress test: +5% template/window error and 1/f drift raise k_TBN and η_Damp; total parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03²), posterior means shift < 8%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.048; new μ/redshift blind tests keep ΔRMSE ≈ −13%.
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/