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%.