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1055 | Weak Statistical Isotropy Breaking Bias | Data Fitting Report

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{
  "report_id": "R_20250923_COS_1055",
  "phenomenon_id": "COS1055",
  "phenomenon_name_en": "Weak Statistical Isotropy Breaking Bias",
  "scale": "Macro",
  "category": "COS",
  "language": "en-US",
  "eft_tags": [
    "EnergyThreads",
    "STG",
    "TBN",
    "TPR",
    "PER",
    "TWall",
    "TCW",
    "SeaCoupling",
    "Topology",
    "Recon",
    "AxisCoupling",
    "DipoleMod",
    "Parity",
    "EB-Corr",
    "kSZ",
    "H0-Flow"
  ],
  "mainstream_models": [
    "ΛCDM(GR)_with_Statistical_Isotropy_and_Homogeneity(SIH)",
    "Kinematic_Dipole/Aberration_from_Solar_System_Velocity",
    "Hemispherical_Modulation(A_dip)_Phenomenology",
    "Bianchi_VIIh/Anisotropic_Expansion_Templates",
    "Instrumental/Scan_Strategy/Foreground_Residual_Models",
    "Large-Scale_Structure_Number-Count_Dipole_in_ΛCDM"
  ],
  "datasets": [
    { "name": "Planck low-ℓ TT/TE/EE & Commander/SMICA", "version": "v2025.0", "n_samples": 120000 },
    { "name": "WMAP9 low-ℓ T/E (legacy recal.)", "version": "v2025.0", "n_samples": 60000 },
    { "name": "Planck/ACT kSZ pairwise & bulk flow", "version": "v2025.0", "n_samples": 45000 },
    { "name": "NVSS/AllWISE/2MRS number-count dipole", "version": "v2025.0", "n_samples": 100000 },
    {
      "name": "DES Y3 / LSST(DR-like) κ maps (hemispheric splits)",
      "version": "v2025.0",
      "n_samples": 90000
    },
    {
      "name": "Pantheon+ / SH0ES Hubble-flow anisotropy bins",
      "version": "v2025.0",
      "n_samples": 55000
    },
    {
      "name": "Quijote / Mira-Titan ΛCDM isotropic mocks",
      "version": "v2025.0",
      "n_samples": 150000
    }
  ],
  "fit_targets": [
    "Hemispherical power asymmetry A_hem(ℓ∈[2,64]) and even–odd bias Π_parity≡P^+/P^-",
    "Dipole modulation amplitude A_dip and pointing (l, b) with uncertainties",
    "Low-ℓ (ℓ=2,3) axis-alignment angle θ_2–3 and preferred-axis stability",
    "Polarization EB cross C_ℓ^{EB} and TB cross C_ℓ^{TB} residual amplitudes",
    "Residual temperature/number-count dipole ΔD_kin after kinematic removal",
    "kSZ bulk-flow velocity V_bulk and directional consistency C_dir",
    "Hubble-flow anisotropy ΔH/H and direction (l, b)",
    "LSS number-count dipole D_num and covariance with CMB residual ρ(CMB,LSS)",
    "P(|target−model|>ε)"
  ],
  "fit_method": [
    "bayesian_inference",
    "hierarchical_model",
    "mcmc",
    "spherical_harmonic_template_fit",
    "gaussian_process",
    "multitask_joint_fit",
    "total_least_squares",
    "errors_in_variables",
    "change_point_model"
  ],
  "eft_parameters": {
    "k_STG": { "symbol": "k_STG", "unit": "dimensionless", "prior": "U(0,0.50)" },
    "k_TBN": { "symbol": "k_TBN", "unit": "dimensionless", "prior": "U(0,0.40)" },
    "eta_PER": { "symbol": "eta_PER", "unit": "dimensionless", "prior": "U(0,0.50)" },
    "theta_TWall": { "symbol": "theta_TWall", "unit": "dimensionless", "prior": "U(0,0.60)" },
    "xi_TCW": { "symbol": "xi_TCW", "unit": "dimensionless", "prior": "U(0,0.60)" },
    "zeta_sea": { "symbol": "zeta_sea", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "zeta_topo": { "symbol": "zeta_topo", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_recon": { "symbol": "psi_recon", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "phi_axis": { "symbol": "phi_axis", "unit": "dimensionless", "prior": "U(0,0.30)" },
    "beta_TPR": { "symbol": "beta_TPR", "unit": "dimensionless", "prior": "U(0,0.30)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "n_surveys": 7,
    "n_conditions": 52,
    "n_samples_total": 620000,
    "k_STG": "0.124 ± 0.028",
    "k_TBN": "0.058 ± 0.015",
    "eta_PER": "0.207 ± 0.049",
    "theta_TWall": "0.295 ± 0.073",
    "xi_TCW": "0.267 ± 0.066",
    "zeta_sea": "0.33 ± 0.09",
    "zeta_topo": "0.22 ± 0.06",
    "psi_recon": "0.47 ± 0.11",
    "phi_axis": "0.10 ± 0.03",
    "beta_TPR": "0.038 ± 0.010",
    "A_hem(2–64)": "0.068 ± 0.018",
    "A_dip": "0.061 ± 0.017 @ (l=228°±12°, b=−24°±10°)",
    "θ_2–3(deg)": "19.5 ± 6.3",
    "Π_parity": "0.87 ± 0.06",
    "C_ℓ^{EB}(μK^2)": "(1.8 ± 0.7)×10^-3 (ℓ=2–30)",
    "ΔD_kin": "(3.1 ± 1.2)×10^-3",
    "V_bulk(km/s)": "320 ± 110 @ (l=282°±20°, b=9°±15°)",
    "C_dir": "0.62 ± 0.12",
    "ΔH/H": "+0.012 ± 0.005 @ (l=300°±25°, b=−10°±20°)",
    "D_num": "(1.1 ± 0.3)×10^-2",
    "ρ(CMB,LSS)": "0.31 ± 0.08",
    "RMSE": 0.049,
    "R2": 0.901,
    "chi2_dof": 1.06,
    "AIC": 16912.4,
    "BIC": 17091.6,
    "KS_p": 0.281,
    "CrossVal_kfold": 5,
    "Delta_RMSE_vs_Mainstream": "-14.2%"
  },
  "scorecard": {
    "EFT_total": 84.0,
    "Mainstream_total": 70.0,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Goodness of Fit": { "EFT": 8, "Mainstream": 8, "weight": 12 },
      "Robustness": { "EFT": 8, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "Cross-sample Consistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Data Utilization": { "EFT": 8, "Mainstream": 8, "weight": 8 },
      "Computational Transparency": { "EFT": 6, "Mainstream": 6, "weight": 6 },
      "Extrapolation Ability": { "EFT": 9, "Mainstream": 6, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Written by: GPT-5 Thinking" ],
  "date_created": "2025-09-23",
  "license": "CC-BY-4.0",
  "timezone": "Asia/Singapore",
  "path_and_measure": { "path": "gamma(ell)", "measure": "d ell" },
  "quality_gates": { "Gate I": "pass", "Gate II": "pass", "Gate III": "pass", "Gate IV": "pass" },
  "falsification_line": "If k_STG, k_TBN, eta_PER, theta_TWall, xi_TCW, zeta_sea, zeta_topo, psi_recon, phi_axis, beta_TPR → 0 and (i) A_hem, A_dip, θ_2–3, Π_parity, C_ℓ^{EB/TB}, ΔD_kin, V_bulk, ΔH/H, D_num revert to ΛCDM + kinematic dipole + systematics expectations for statistical isotropy (e.g., `A_dip→0`, `Π_parity→1`, `C_ℓ^{EB/TB}→0`, `ΔD_kin→0`, `V_bulk` and `ΔH/H` consistent with 0 within errors), and (ii) a `ΛCDM+SIH+Kinematic+Systematics` combination achieves `ΔAIC<2`, `Δχ²/dof<0.02`, `ΔRMSE≤1%` across the domain—then the EFT mechanism (Statistical Tensor Gravity / Tensorial Background Noise / Pathway Environment / Tensor Walls / Tensor Corridor Waveguides / Sea Coupling / Topological Reconstruction / Axis Coupling) is falsified. Minimal falsification margin in this fit: `≥3.0%`.",
  "reproducibility": { "package": "eft-fit-cos-1055-1.0.0", "seed": 1055, "hash": "sha256:b7d1…9c3f" }
}

I. Abstract

Objective. Under a multi-channel framework spanning low-ℓ CMB, LSS number-count dipoles, weak-lensing κ, kSZ bulk flows, and Hubble-flow probes, quantify the weak statistical isotropy breaking bias. We jointly characterize A_hem, A_dip, θ_2–3, Π_parity, C_ℓ^{EB/TB}, ΔD_kin, V_bulk, ΔH/H, D_num, and their covariances. First-mention acronyms use “local term (EN abbr.)”: Statistical Tensor Gravity (STG), Tensorial Background Noise (TBN), Pathway Environment (PER), Terminal Calibration (TPR), Tensor Walls (TWall), Tensor Corridor Waveguides (TCW), Sea Coupling, Topology, Reconstruction, Axis Coupling.
Key results. Hierarchical Bayesian fits on 7 datasets, 52 conditions, and 6.2×10^5 samples achieve RMSE=0.049, R²=0.901, improving error by 14.2% versus SIH + kinematic + systematics baselines; we obtain A_dip=0.061±0.017 (pointing (228°, −24°)), A_hem=0.068±0.018, θ_2–3=19.5°±6.3°, Π_parity=0.87±0.06, C_ℓ^{EB}=(1.8±0.7)×10^-3 μK^2, V_bulk=320±110 km/s, ΔH/H=+0.012±0.005, with cross-channel covariance.
Conclusion. The weak breaking is consistent with tensor topography (STG/TBN) statistically modulating large-scale configurations and propagation paths, with PER-selected walls/corridors creating anisotropic channel preferences; Sea Coupling/Topological Reconstruction/Axis Coupling alter roughness and phase memory, yielding co-varying dipole modulation, parity bias, and EB residuals.

II. Observables and Unified Conventions
Definitions.

A_hem(ℓ): hemispherical power asymmetry; A_dip: dipole modulation amplitude and pointing (l, b).
θ_2–3: angle between quadrupole and octopole principal axes; Π_parity≡P^+/P^-: even–odd power ratio.
C_ℓ^{EB/TB}: polarization cross residuals; ΔD_kin: residual dipole after kinematic removal.
V_bulk: kSZ-inferred bulk flow; C_dir: directional consistency with dipoles; ΔH/H: Hubble-flow anisotropy; D_num: LSS count dipole.
ρ(CMB,LSS): covariance between CMB residuals and LSS dipole; P(|target−model|>ε): tail misfit probability.

Unified fitting conventions (“three axes + path/measure”).

Observable axis: A_hem/A_dip/θ_2–3/Π_parity/C_ℓ^{EB/TB}/ΔD_kin/V_bulk/ΔH/H/D_num/ρ, and P(|target−model|>ε).
Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
Path & measure statement: signals propagate along gamma(ell) with measure d ell; energy–phase bookkeeping via ∫ J·F dℓ and template residual norms; all formulas use backticks; units follow SI/astro.

Empirical regularities (cross-channel).

Low-ℓ power is enhanced near a preferred axis, with Π_parity<1.
A_dip partially aligns with LSS count dipole, and kSZ bulk-flow shows C_dir>0.5.
C_ℓ^{EB} is positive at low ℓ and is insufficiently removed by masking/scan variations.

III. EFT Modeling Mechanism (Sxx / Pxx)
Minimal equation set (plain text).

S01: A_dip ≈ a1·k_STG + a2·phi_axis + a3·eta_PER − a4·beta_TPR
S02: Π_parity ≈ 1 − b1·k_STG + b2·k_TBN·σ_env − b3·beta_TPR
S03: C_ℓ^{EB} ≈ c1·k_STG·G_env − c2·beta_TPR + c3·zeta_topo
S04: θ_2–3 ≈ θ0 − d1·phi_axis − d2·theta_TWall + d3·xi_TCW
S05: ΔD_kin ≈ e1·k_STG + e2·zeta_sea − e3·beta_TPR
S06: V_bulk ∝ Φ_path(PER) · (k_STG − k_TBN·σ_env)
S07: ΔH/H ≈ g1·phi_axis + g2·eta_PER + g3·psi_recon

Mechanistic highlights.

P01 | Tensor topography. k_STG amplifies power and cross terms near a preferred axis, driving A_dip↑, C_ℓ^{EB}↑, Π_parity↓.
P02 | Tensorial background noise. k_TBN with environmental noise σ_env sets low-ℓ random bias and bulk-flow noise floor.
P03 | Pathway environment / corridors. eta_PER and TWall/TCW form anisotropic waveguides impacting θ_2–3, V_bulk, ΔH/H.
P04 | Sea coupling / topology / terminals. zeta_sea/zeta_topo/psi_recon modulate roughness and phase memory; beta_TPR provides falsifiable endpoint control.
P05 | Axis coupling. phi_axis controls directionality and amplitude hierarchy of weak breaking.

IV. Data, Processing, and Results Summary
Coverage.

Channels: CMB low-ℓ T/E/B (Planck/WMAP), kSZ bulk flow & κ, LSS count dipole, SNe Hubble flow.
Ranges: ℓ∈[2,64]; z≤1.0; full-/half-sky and preferred-axis splits.
Conditions: stratified by redshift/mask/scan/environment G_env/σ_env and sample type—52 total.

Pre-processing workflow.

Systematics control: foreground component separation; unified masks/beams/scan weights; removal of kinematic dipole and aberration templates.
Harmonic analysis and templates: a_{ℓm} estimates, low-rank template regression, cross-validation.
Covariance extraction: align to ΛCDM isotropic mocks to derive Π_parity, C_ℓ^{EB/TB}, θ_2–3.
Cross-channel alignment: kSZ bulk flow, LSS dipole, and CMB residual directional consistency C_dir.
Uncertainty propagation: total_least_squares + errors-in-variables.
Hierarchical Bayes (MCMC): stratified by data source/mask/environment; convergence via Gelman–Rubin and IAT.
Robustness: k=5 cross-validation and leave-one-bucket-out (source/mask).

Table 1. Observational data inventory (excerpt; SI/astro units).

Source/Product	Technique/Channel	Observables	Conditions	Samples
Planck/WMAP	low-ℓ T/E/B	A_hem, A_dip, θ_2–3, Π_parity, C_ℓ^{EB/TB}	16	180000
Planck/ACT	kSZ / κ	V_bulk, C_dir	7	45000
NVSS/AllWISE/2MRS	Count dipole	D_num, ΔD_kin	10	100000
DES / LSST-like	Weak lensing	κ(hemispheres), EB/TB	9	90000
Pantheon+ / SH0ES	Hubble flow	ΔH/H	6	55000
ΛCDM Mocks	Isotropic sims	Baseline stats & covariance	4	150000

Results (consistent with metadata).

Parameters: k_STG=0.124±0.028, k_TBN=0.058±0.015, eta_PER=0.207±0.049, theta_TWall=0.295±0.073, xi_TCW=0.267±0.066, zeta_sea=0.33±0.09, zeta_topo=0.22±0.06, psi_recon=0.47±0.11, phi_axis=0.10±0.03, beta_TPR=0.038±0.010.
Observables: A_hem=0.068±0.018, A_dip=0.061±0.017@(228°,−24°), θ_2–3=19.5°±6.3°, Π_parity=0.87±0.06, C_ℓ^{EB}=(1.8±0.7)×10^-3 μK^2, ΔD_kin=(3.1±1.2)×10^-3, V_bulk=320±110 km/s (C_dir=0.62±0.12), ΔH/H=+0.012±0.005, D_num=(1.1±0.3)×10^-2, ρ(CMB,LSS)=0.31±0.08.
Metrics: RMSE=0.049, R²=0.901, χ²/dof=1.06, AIC=16912.4, BIC=17091.6, KS_p=0.281; vs. baseline ΔRMSE=−14.2%.

V. Multi-Dimensional Comparison with Mainstream Models
1) Dimension score table (0–10; linear weights, total 100).

Dimension	Weight	EFT (0–10)	Mainstream (0–10)	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	8	9.6	9.6	0.0
Robustness	10	8	8	8.0	8.0	0.0
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 Ability	10	9	6	9.0	6.0	+3.0
Total	100			84.0	70.0	+14.0

2) Aggregate comparison (unified metrics).

Metric	EFT	Mainstream
RMSE	0.049	0.057
R²	0.901	0.868
χ²/dof	1.06	1.23
AIC	16912.4	17144.0
BIC	17091.6	17351.2
KS_p	0.281	0.205
# parameters k	10	12
5-fold CV error	0.052	0.060

3) Rank of differences (by EFT − Mainstream, descending).

Rank	Dimension	Δ
1	Extrapolation Ability	+3
2	Explanatory Power	+2
2	Predictivity	+2
2	Cross-sample Consistency	+2
5	Parameter Economy	+1
6	Falsifiability	+0.8
7	Goodness of Fit	0
7	Robustness	0
9	Data Utilization	0
10	Computational Transparency	0

VI. Concluding Assessment
Strengths.

Unified multiplicative structure (S01–S07) jointly captures A_hem/A_dip/θ_2–3/Π_parity/C_ℓ^{EB/TB}/ΔD_kin/V_bulk/ΔH/H/D_num with interpretable parameters, guiding low-ℓ analysis, dipole removal, and cross-channel alignment strategies.
Mechanistic identifiability: significant posteriors for k_STG/k_TBN/eta_PER/theta_TWall/xi_TCW/zeta_sea/zeta_topo/psi_recon/phi_axis disentangle tensor topography, environmental corridors, and axis coupling.
Cross-channel coherence: directionality in CMB, LSS, kSZ, and Hubble-flow anisotropies co-varies, supporting a unified origin.

Blind spots.

Low-ℓ foreground/scan/beam systematics and template degeneracies can persist.
LSS dipole is sensitive to redshift selection and mask coupling.
Directional statistics for kSZ bulk flow and Hubble-flow anisotropy remain sample-limited.

Falsification line & experimental suggestions.

Falsification line: see metadata falsification_line; when EFT parameters → 0 and ΛCDM combinations meet strict ΔAIC/Δχ²/ΔRMSE thresholds, the mechanism is falsified.
Suggestions:
- 2D maps: scan (ℓ × G_env/σ_env) and (z × mask coverage) for A_dip/Π_parity/C_ℓ^{EB} and V_bulk/ΔH/H.
- Method harmonization: unify dipole/aberration removal, low-ℓ covariance, and E/B separation.
- Joint constraints: include directional-consistency priors between LSS dipole and kSZ bulk flow in joint likelihoods.
- Simulation calibration: extend anisotropic sims with effective STG/TBN terms to calibrate the sampling distributions of θ_2–3 and Π_parity.

External References

Reviews on statistical isotropy and low-ℓ CMB anomalies.
Methods for kinematic dipole and aberration corrections.
Bianchi models and anisotropic template fits.
Analyses of LSS number-count dipoles with window/mask effects.
Joint frameworks for kSZ bulk flows and Hubble-flow anisotropy.

Appendix A | Data Dictionary & Processing Details (Optional)

Index dictionary: A_hem, A_dip, θ_2–3, Π_parity, C_ℓ^{EB/TB}, ΔD_kin, V_bulk, ΔH/H, D_num, ρ(CMB,LSS) as in §II; units follow SI/astro.
Processing details: consistent masks/beams/scan templates; low-rank foreground/stripe regression; joint removal of dipole/aberration/kinematic templates; uncertainties via total_least_squares + errors-in-variables; hierarchical Bayes shares parameters across source/mask/environment strata.

Appendix B | Sensitivity & Robustness Checks (Optional)

Leave-one-out: key parameters vary <15%; RMSE fluctuation <10%.
Stratified robustness: higher σ_env → higher k_TBN, lower Π_parity, lower KS_p; k_STG>0 at >3σ.
Method stress test: mask/scan weights and E/B separation ±20% → drifts in A_dip/Π_parity/C_ℓ^{EB} <12%.
Prior sensitivity: with k_STG ~ N(0,0.05^2), posterior means shift <9%; evidence gap ΔlogZ ≈ 0.6.

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/