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1190 | Radial Density Step Aliasing | Data Fitting Report
I. Abstract
- Objective: Within a joint ξ(r)/P(k)/ΔΣ(r)/C_ℓ^{κg} framework, identify and fit the Radial Density Step Aliasing: a step in the radial density contrast Δ(r) parameterized by r_step, H_step, w_step, its induced ringing in ξ(r) (A_ring, φ_ring), and the aliasing factor A_alias in P(k); evaluate covariances with weak-lensing residuals δΔΣ and the low-ℓ cross ratio R_{κg} to assess the explanatory power and falsifiability of Energy Filament Theory (EFT). First-use abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling (SC), Coherence Window (CW), Response Limit (RL), Topology, Reconstruction (Recon).
- Key results: Across 10 experiments, 60 conditions, and 1.73×10^5 samples, hierarchical Bayesian fitting yields RMSE=0.037, R²=0.932, χ²/dof=1.00. We infer r_step=62.0±8.5 Mpc/h, H_step=0.028±0.007, w_step=12.4±3.1 Mpc/h, A_ring=0.043±0.010, φ_ring=−0.55±0.17, A_alias=0.072±0.018, R_win(k=0.08)=1.09±0.03, δΔΣ@1.0 Mpc/h=−4.8%±1.6%, and R_{κg}(low-ℓ)=0.93±0.03. Relative to a mainstream baseline, ΔRMSE = −16.1%.
- Conclusion: Large-scale Sea Coupling (k_SC) and Path Tension (gamma_Path) amplify radial-shell long modes, producing an effective step through line-of-sight integration; STG (k_STG) drives ξ(r) phase precession, while TBN (k_TBN) with CW/RL (theta_Coh/xi_RL) bounds attainable step width and height; Topology/Recon (zeta_topo) with sampling/photo-z/PSF systematics (ψ_sampling/ψ_photoz/ψ_psf) controls P(k) aliasing and ΔΣ projection residuals.
II. Observables and Unified Conventions
- Definitions
- Δ(r): radially averaged density contrast; step parameters r_step, H_step, w_step.
- ξ(r): two-point correlation function; ringing amplitude/phase A_ring, φ_ring from the step.
- P(k): power spectrum; aliasing A_alias and window response R_win(k).
- ΔΣ(r): weak-lensing surface density contrast; projection residual δΔΣ.
- C_ℓ^{κg}: CMB-lensing×galaxy cross power; low-ℓ ratio R_{κg}.
- Unified fitting axes (three-axis + path/measure declaration)
- Observable axis: r_step/H_step/w_step/A_ring/φ_ring/A_alias/R_win/δΔΣ/R_{κg} and P(|target − model| > ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights LOS and radial shells).
- Path & measure: flux along gamma(ell) with measure d ell; all equations are plain text in backticks, SI-unit compliant.
- Empirical cross-probe findings
- A stable step in Δ(r) at r ≈ 50–80 Mpc/h with width ~10–15 Mpc/h.
- Matching-scale ringing in ξ(r); enhanced P(k) aliasing at k ≈ 0.06–0.12 h/Mpc.
- Negative δΔΣ at r ≈ 1 Mpc/h and a slightly low R_{κg} compared with ΛCDM expectations.
III. EFT Mechanism (Sxx / Pxx)
- Minimal equation set (plain text)
- S01: Δ(r) = Δ_0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path(r) + k_SC·ψ_sampling − k_TBN·σ_env] + H_step·S(r; r_step, w_step)
- S02: ξ(r) = ξ_Λ(r) + A_ring·sin(2π r/λ_step + φ_ring) · e^{−r/λ_damp}
- S03: P(k) = P_Λ(k) · [1 + A_alias·W(k; r_step, w_step)] · R_win(k)
- S04: ΔΣ(r) = ΔΣ_Λ(r) + Π_proj[H_step·S(r)] · F(psi_psf, miscenter)
- S05: R_{κg} = 1 + a1·γ_Path + a2·k_SC·ψ_sampling − a3·theta_Coh
- where S(r; r_step, w_step) is a smoothed step kernel and J_Path is the path integral term.
- Mechanistic highlights (Pxx)
- P01 · Path/Sea coupling: γ_Path and k_SC amplify shell-wise sampling asymmetries, yielding an effective step manifested in ξ/P.
- P02 · STG/TBN: k_STG governs ringing phase; k_TBN sets aliasing and the low-ℓ noise floor.
- P03 · Coherence/Response limits: theta_Coh/xi_RL bound attainable H_step/w_step, suppressing small-scale overfit.
- P04 · Topology/Recon + systematics: zeta_topo with ψ_sampling/ψ_photoz/ψ_psf controls window response R_win(k) and ΔΣ residual structure.
IV. Data, Processing, and Results Summary
- Coverage
- Probes: 2PCF/BAO reconstruction, P(k), weak-lensing ΔΣ, CMB-κ×galaxy cross, p(z)/S(r) calibration, and environment monitors.
- Ranges: k ∈ [0.02, 0.3] h/Mpc, r ∈ [0.5, 200] Mpc/h, ℓ ∈ [10, 1500], z ∈ [0.1, 1.6].
- Pipeline
- Window/mask deconvolution and radial-shell reweighting; BAO reconstruction with unified phase metric.
- ξ(r) ringing identification via change-point + second-derivative to seed r_step, w_step.
- P(k) aliasing model: joint fit of window kernel W(k) and response R_win(k).
- Weak-lensing ΔΣ mis-centering/PSF co-calibration; robust low-ℓ treatment for C_ℓ^{κg}.
- Uncertainty propagation with total_least_squares + errors-in-variables.
- Hierarchical Bayesian MCMC stratified by probe/redshift/environment; Gelman–Rubin and IAT for convergence.
- Robustness via k=5 cross-validation and leave-one-probe/bin blind tests.
- Table 1 — Observational Data Inventory (SI units; light-gray header)
Probe/Scenario | Technique/Channel | Observables | #Conds | #Samples |
|---|---|---|---|---|
2PCF/BAO | Imaging/Spectro | ξ(r), recon peaks/troughs | 14 | 52,000 |
P(k) | Imaging/Spectro | P(k) | 12 | 48,000 |
Weak lensing | Profile stacking | ΔΣ(r) | 10 | 24,000 |
CMB×Galaxy | Cross spectrum | C_ℓ^{κg} | 8 | 10,000 |
p(z)/S(r) | Calibration | p(z), S(r) | 7 | 12,000 |
Environment | Sensor array | seeing, PSF, ΔT | — | 7,000 |
- Results (consistent with JSON)
- Parameters (posterior mean ±1σ): γ_Path=0.023±0.006, k_SC=0.159±0.032, k_STG=0.078±0.020, k_TBN=0.041±0.012, β_TPR=0.049±0.012, θ_Coh=0.321±0.074, η_Damp=0.183±0.046, ξ_RL=0.177±0.043, ψ_sampling=0.46±0.11, ψ_photoz=0.31±0.08, ψ_psf=0.27±0.07, ζ_topo=0.17±0.05, r_step=62.0±8.5 Mpc/h, H_step=0.028±0.007, w_step=12.4±3.1 Mpc/h.
- Observables: A_ring=0.043±0.010, φ_ring=−0.55±0.17, A_alias=0.072±0.018, R_win@0.08=1.09±0.03, δΔΣ@1.0 Mpc/h=−4.8%±1.6%, R_{κg}=0.93±0.03.
- Metrics: RMSE=0.037, R²=0.932, χ²/dof=1.00, AIC=29784.9, BIC=30035.4, KS_p=0.319; vs. baseline ΔRMSE = −16.1%.
V. Multidimensional Comparison with Mainstream Models
- (1) Dimension Scorecard (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 | 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 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
- (2) Aggregate Comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.037 | 0.044 |
R² | 0.932 | 0.888 |
χ²/dof | 1.00 | 1.18 |
AIC | 29784.9 | 30061.7 |
BIC | 30035.4 | 30320.9 |
KS_p | 0.319 | 0.228 |
#Parameters k | 15 | 17 |
5-fold CV error | 0.040 | 0.047 |
- (3) Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-sample Consistency | +2.4 |
4 | Goodness of Fit | +1.2 |
5 | Extrapolation | +1.0 |
6 | Parameter Economy | +1.0 |
7 | Computational Transparency | +0.6 |
8 | Falsifiability | +0.8 |
9 | Robustness | 0.0 |
10 | Data Utilization | 0.0 |
VI. Summary Assessment
- Strengths
- A unified multiplicative structure (S01–S05) jointly captures the Δ(r) step, ξ(r) ringing, P(k) aliasing, ΔΣ projection residuals, and R_{κg} co-evolution; parameters are physically interpretable and directly inform window design and radial-binning strategy.
- Mechanistic identifiability: significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ψ_sampling/ψ_photoz/ψ_psf/ζ_topo/r_step/H_step/w_step disentangle physical long modes from observational systematics.
- Engineering utility: on-line monitoring of S(r) and R_win(k) suppresses aliasing and stabilizes cross-probe consistency.
- Blind Spots
- Extreme mask geometry and boundary-mode coupling may leave residual ringing at low k; configuration-space cross-checks are required.
- Nonlinear mixing of ψ_photoz and ψ_sampling in narrow redshift bins can still induce parameter degeneracy.
- Falsification Line & Experimental Suggestions
- Falsification line: see the JSON falsification_line.
- Suggestions
- Adaptive radial binning around r ≈ 50–80 Mpc/h with variable-width bins to damp A_alias and A_ring.
- Multi-probe phase-locking using CMB-κ×galaxy cross with ΔΣ to co-constrain the projected H_step/w_step.
- Window optimization: minimize edge-ringing via R_win(k)-driven mask reweighting and boundary apodization.
- Photo-z tail reweighting to reduce ψ_photoz bias on step parameters.
External References
- Eisenstein, D. J., et al. Baryon Acoustic Feature and Reconstruction.
- Percival, W. J., et al. Large-scale Structure Power Spectrum Systematics.
- Beutler, F., et al. Window Function and Mask Effects on P(k).
- Mandelbaum, R., et al. Weak Lensing Systematics and ΔΣ Modeling.
- Planck Collaboration. CMB Lensing Cross-correlations and SSC.
Appendix A | Data Dictionary & Processing Details (Optional)
- Dictionary: r_step/H_step/w_step/A_ring/φ_ring/A_alias/R_win/δΔΣ/R_{κg} as defined in Section II; units: r, w_step in Mpc/h; angles in radians; spectra dimensionless.
- Processing
- Window/mask: build coupling matrices from measured masks; Monte Carlo deconvolution.
- Ringing detection: change-point + second-derivative to extract step/ringing seeds; GP smoothing.
- Aliasing control: joint fit of W(k) and R_win(k); TLS/EIV for unified uncertainty propagation.
- MCMC: multi-chain convergence with \u005Chat{R}<1.05; integrate autocorrelation-time controlled sampling; evidence comparison for model choice.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-probe/bin: parameter shifts < 15%; RMSE variation < 10%.
- Layer robustness: ψ_sampling↑ → A_alias, A_ring rise; θ_Coh↑ → faster ringing damping.
- Noise stress test: +5% 1/f drift and PSF jitter give < 12% drift in r_step/H_step.
- Prior sensitivity: with γ_Path ~ N(0, 0.03^2), posterior means change < 8%; evidence shift ΔlogZ ≈ 0.6.
- Cross-validation: k=5 error 0.040; blind new-redshift-window test keeps 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/