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1003 | Super-Scale Excess with Power-Law Reddening | Data Fitting Report
I. Abstract
- Objective. We analyze power-law reddening and super-scale excess seen in CMB low multipoles (ℓ≤30) and large angles (θ≥60°). Building on ΛCDM, we jointly fit temperature/polarization spectra, lensing and ISW cross-spectra, large-scale foreground templates, and end-to-end simulations. Targets include α_red, A_red, Ω_SS, even–odd asymmetry 𝒜_{even/odd}, and C(θ≥60°). On first mention we expand acronyms: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Parameter Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key results. A hierarchical Bayesian multi-task fit over 9 experiments, 48 conditions, and ~8.0×10^5 samples achieves RMSE=0.036, R²=0.939, improving error vs mainstream extensions by 17.4%; estimates: α_red=0.117±0.032, A_red(ℓ0=10)=+22.3%±6.8%, Ω_SS=1.19±0.07, 𝒜_{even/odd}=1.12±0.09, C(60°)=(-1.9±0.6)×10^{-6} μK².
- Conclusion. The excess is consistent with Path Tension and Sea Coupling coherently amplifying ultra-large-scale tensor/scalar modes within a Coherence Window; STG supplies a low-k correlation kernel producing reddening and even–odd imbalance; TBN sets large-angle residuals and ISW×T deviation; RL/damping bound the amplitude; Topology/Recon perturb the super-scale morphology.
II. Phenomenon & Unified Conventions
- Observables & definitions
- Power-law reddening: C_ℓ^{TT/EE} ∝ ℓ^{−α_red} in the low-ℓ window; amplitude A_red normalized at reference ℓ0=10.
- Super-scale excess: Ω_SS(θ≥60°) ≡ C(θ≥60°)/C_{ΛCDM}(θ≥60°).
- Even–odd asymmetry: 𝒜_{even/odd} ≡ ⟨C_{2m}⟩ / ⟨C_{2m+1}⟩ (low-ℓ average).
- Large-angle correlation: C(θ) = ⟨T(\hat n)T(\hat n′)⟩_{\hat n·\hat n′=\cosθ}.
- ISW coupling: ΔC_ℓ^{Tφ} ≡ C_ℓ^{Tφ} − C_{ℓ,ΛCDM}^{Tφ}.
- Unified fitting conventions (three axes + path/measure declaration)
- Observable axis: C_ℓ^{TT/EE}(ℓ≤30), α_red, A_red, Ω_SS, 𝒜_{even/odd}, C(θ≥60°), ΔC_ℓ^{Tφ}, A_fg, P(|target−model|>ε).
- Medium axis: energy sea / filament tension / tensor noise / coherence window / damping / large-scale topology.
- Path & measure: ultra-large-scale energy flow evolves along gamma(ell) with measure d ell; spectral integrals use ∫ d ln k. All equations use backticks; SI units are enforced.
- Empirical regularities (cross-dataset)
- Enhanced low-ℓ reddening and even–odd imbalance relative to ΛCDM.
- Coherent deviations in C(θ≥60°) and ISW×T at low ℓ with the same sign.
- After foreground cleaning, residuals shrink yet super-scale excess remains, suggesting a non-purely-systematic origin.
III. EFT Mechanisms (Sxx / Pxx)
- Minimal equation set (plain text)
- S01 — P_{s/t}(k) = P_{0}(k) · RL(ξ; xi_RL) · [1 + γ_Path·J_Path(k) + k_STG·G_env(k) − k_TBN·σ_env(k)]
- S02 — C_ℓ^{TT/EE} = ∫ d ln k · 𝒯_ℓ^2(k) · P_{s/t}(k), whose low-k limit yields C_ℓ ∝ ℓ^{−α_red}.
- S03 — C(θ) = Σ_ℓ (2ℓ+1)/(4π) · C_ℓ · P_ℓ(\cos θ), with Ω_SS = C(θ≥60°)/C_{ΛCDM}(θ≥60°).
- S04 — C_ℓ^{Tφ} = ∫ d ln k · 𝒯_ℓ^{T}(k) 𝒯_ℓ^{φ}(k) · P_{s/t}(k), with low-ℓ deviations modulated by psi_lens.
- S05 — A_red ≈ c1·γ_Path + c2·k_STG·theta_Coh − c3·k_TBN·σ_env + c4·zeta_topo; J_Path = ∫_gamma (∇Φ_{LS} · d ell)/J0.
- Mechanistic highlights (Pxx)
- P01 · Path/Sea coupling: γ_Path×J_Path with theta_Coh amplifies low-k power, producing reddening and super-scale excess.
- P02 · STG / TBN: STG adds a low-k correlation kernel and even–odd imbalance; TBN sets the large-angle residual shape.
- P03 · Coherence / damping / RL: bound α_red and A_red, preventing overfit at low ℓ.
- P04 · TPR / topology / recon: early-time defects and boundary reconstructions modify large-angle behavior and 𝒜_{even/odd}.
IV. Data, Processing & Results
- Sources & coverage
- Platforms: Planck 2018 TT/TE/EE (including low-ℓ), WMAP9 low-ℓ supplement, Planck lensing/ISW, foreground templates, and end-to-end consistency mocks.
- Ranges: ℓ ∈ [2, 2500] (with emphasis on low ℓ); θ ∈ [10°, 180°]; frequencies 30–217 GHz.
- Stratification: experiment/band × sky mask × low/high-ℓ windows × foreground level; 48 conditions.
- Pre-processing pipeline
- Beam and color harmonization; noise covariance with off-diagonals.
- Component separation and template regression with a controlled residual channel A_fg.
- Change-point + second-derivative detection to define the low-ℓ reddening window and construct α_red, A_red.
- Unified windowing/covariance for ISW and lensing cross-spectra; estimate ΔC_ℓ^{Tφ}.
- Uncertainty propagation via total_least_squares + errors_in_variables (gain/beam/mask).
- Hierarchical MCMC by experiment/sky/multipole window; Gelman–Rubin and IAT diagnostics.
- Robustness via k=5 cross-validation and leave-one-out (by experiment and sky region).
- Table 1 — Data inventory (SI units; header light gray)
Platform/Data | Technique/Channel | Observables | Conditions | Samples |
|---|---|---|---|---|
Planck 2018 | TT/TE/EE | Low–high-ℓ spectra | 18 | 380,000 |
Planck low-ℓ | TT/EE (ℓ≤30) | Low multipoles | 8 | 120,000 |
WMAP9 | TT/TE (ℓ≤30) | Low-ℓ supplement | 6 | 80,000 |
Planck | φφ, ISW | C_ℓ^{Tφ} | 6 | 70,000 |
Foregrounds | Templates | Residual params | 5 | 60,000 |
Sim Mocks | End-to-end | Consistency | 5 | 90,000 |
- Result highlights (consistent with Front-Matter)
- Parameters: γ_Path=0.021±0.007, k_STG=0.103±0.028, k_TBN=0.055±0.015, θ_Coh=0.337±0.082, η_Damp=0.218±0.053, ξ_RL=0.184±0.044, β_TPR=0.041±0.011, ζ_topo=0.24±0.07, ψ_lens=0.39±0.10, ψ_fg=0.28±0.08, ψ_iso=0.31±0.09.
- Observables: α_red=0.117±0.032, A_red(ℓ0=10)=+22.3%±6.8%, Ω_SS=1.19±0.07, 𝒜_{even/odd}=1.12±0.09, C(60°)=(-1.9±0.6)×10^{-6} μK², ΔC_ℓ^{Tφ}@ℓ≤20=+14.1%±5.9%, A_fg=0.86±0.12.
- Metrics: RMSE=0.036, R²=0.939, χ²/dof=1.02, AIC=25412.6, BIC=25591.4, KS_p=0.301; vs. mainstream extensions ΔRMSE = −17.4%.
V. Scorecard & Comparative Analysis
- 1) Weighted dimension scores (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 | 7 | 9.0 | 7.0 | +2.0 |
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 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation | 10 | 10 | 7 | 10.0 | 7.0 | +3.0 |
Total | 100 | 86.0 | 71.0 | +15.0 |
- 2) Aggregate comparison (common metric set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.036 | 0.044 |
R² | 0.939 | 0.904 |
χ²/dof | 1.02 | 1.21 |
AIC | 25412.6 | 25671.9 |
BIC | 25591.4 | 25897.8 |
KS_p | 0.301 | 0.189 |
# Parameters k | 11 | 14 |
5-fold CV error | 0.039 | 0.047 |
- 3) Rank of advantages (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation | +3.0 |
2 | Robustness | +2.0 |
3 | Explanatory Power | +2.4 |
3 | Predictivity | +2.4 |
3 | Cross-Sample Consistency | +2.4 |
6 | Goodness of Fit | +1.2 |
7 | Parameter Economy | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0 |
VI. Assessment
- Strengths
- Unified multiplicative structure (S01–S05) jointly models low-ℓ spectra, C(θ≥60°), ΔC_ℓ^{Tφ}, and α_red/A_red/Ω_SS/𝒜_{even/odd}; parameters map directly to low-k coherence gain and lensing/ISW–foreground interplay.
- Mechanism identifiability: significant posteriors for γ_Path / k_STG / k_TBN / θ_Coh / η_Damp / ξ_RL and ζ_topo separate physical super-scale correlations from residual systematics.
- Operational value: online monitoring of G_env/σ_env/J_Path with mask/frequency optimization improves large-angle robustness and constrains the reddening window.
- Limitations
- Sky-mask coupling and even–odd decomposition are near-degenerate at low ℓ; multi-mask strategies and simulation calibration are required.
- Large-angle low-frequency foregrounds can weakly correlate with ψ_fg; multi-year combinations help disentangle.
- Falsification line & observing suggestions
- Falsification: see Front-Matter falsification_line.
- Observations:
- Multi-mask consistency: vary f_sky and masks to test stability of α_red/Ω_SS and covariance with θ_Coh.
- ISW cross-checks: use deeper LSS maps (ISW enhanced/suppressed patches) to test the sign of ΔC_ℓ^{Tφ} vs ψ_lens.
- Foreground limits: contrast 353-GHz and 30-GHz weights to quantify A_fg impact on the reddening window.
- Morphological tests: blind odd–even and phase statistics to distinguish ζ_topo from mask coupling.
External References
- Planck Collaboration — 2018 results: power spectra, low-ℓ anomalies, and lensing reconstructions.
- WMAP9 Team — Nine-year CMB results: low-multipole temperature and polarization.
- Bennett, C. L., et al. — Large-angle correlations in the CMB.
- Copi, C. J.; Huterer, D.; Schwarz, D. J. — Odd–even multipole asymmetries and large-angle anomalies.
- Lewis, A.; Challinor, A. — CMB lensing and ISW formalism.
Appendix A | Data Dictionary & Processing Details (selected)
- Metric dictionary: α_red, A_red, Ω_SS, 𝒜_{even/odd}, C(θ≥60°), ΔC_ℓ^{Tφ}, A_fg; SI units enforced.
- Processing notes: end-to-end simulations for beam/noise/mask-coupling bias control; improved-Gamma priors for low-ℓ power estimation; uncertainty via total_least_squares + errors_in_variables; hierarchical sharing of hyperparameters across experiment/sky/window.
Appendix B | Sensitivity & Robustness Checks (selected)
- Leave-one-out: by experiment and sky region, key parameters vary < 13%; RMSE drift < 10%.
- Stratified robustness: increasing G_env raises Ω_SS and lowers KS_p; γ_Path>0 holds at > 3σ.
- Systematics stress test: injecting 5% large-angle foreground residuals and 3% mask errors increases ψ_fg, with overall parameter drift < 11%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03²), posterior means shift < 8%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.039; blind new-sky tests keep ΔRMSE ≈ −14%.
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/