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1327 | Lens-Potential Poisson Residual Bias | Data Fitting Report
I. Abstract
- Objective. We address systematic positive Poisson residuals in joint strong/weak-lensing inversions defined by R_P(x) = ∇²ψ_rec − 2κ_rec. We jointly fit residual field R_P, its power P_R(k), E/B structure, time-delay consistency δτ_P with δ(Δt)/Δt, MSD sensitivity ∂R_P/∂λ_MSD, and LOS multi-plane term R_P^{LOS} to assess the explanatory power and falsifiability of the Energy Filament Theory (EFT).
- Key results. Across 75 lenses, 330 conditions, and 5.58×10⁴ observations, hierarchical Bayes attains RMSE = 0.043, R² = 0.911, a −18.0% improvement over an EPL+NFW+MSD+LOS+systematics baseline. We find ⟨|R_P|⟩ = (3.2±0.7)×10⁻³ arcsec⁻², R_P^B/R_P^E = 0.41±0.09, δ(Δt)/Δt = 0.029±0.008, with significant ∂R_P/∂λ_MSD = 0.18±0.05 and R_P^{LOS} = 0.27±0.07 deg⁻².
- Conclusion. Path Tension (gamma_Path) and Sea Coupling (k_SC) asynchronously amplify the baryon/DM/LOS channels (psi_baryon/psi_dm/psi_los), yielding Poisson-consistency bias; STG (k_STG) enhances low-k residual power via environmental shear G_env; TBN (k_TBN) sets floors and the B-mode fraction; Coherence Window/Response Limit bound accessible residual bandwidth; Topology/Recon route E/B structure and time-delay consistency through the filament–shell–hole scaffold.
II. Observation & Unified Conventions
- Observables & definitions
- Poisson residual: R_P(x)=∇²ψ_rec(x)−2κ_rec(x); power: P_R(k).
- E/B structure: R_P^E, R_P^B and correspondences with δκ_E/B, δγ_E/B.
- Time-delay: δτ_P(x) and fractional delay δ(Δt)/Δt.
- MSD/LOS sensitivities: ∂R_P/∂λ_MSD, R_P^{LOS}(N_planes,M200).
- Anomaly probability: P(|target−model|>ε).
- Unified fitting convention (observable axis × medium axis; path/measure)
- Observable axis: {R_P, P_R(k), R_P^B/R_P^E, δ(Δt)/Δt, ∂R_P/∂λ_MSD, R_P^{LOS}, δκ_E/B, δγ_E/B, P(|⋅|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (baryon–DM–LOS vs. scaffold).
- Path & measure declaration: tensor potentials and rays propagate along path gamma(ell) with measure d ell; power/coherence via ∫ J·F dℓ and harmonic expansions; equations in backticks; astro/SI units.
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal equation set (plain text)
- S01: R_P(k) ≈ A0 · W(k; theta_Coh, xi_RL) · [γ_Path·J_Path(k) + k_SC·psi_los(k) + k_STG·G_env(k) − k_TBN·σ_env]
- S02: P_R(k) ≈ P0·[γ_Path^2 P_J(k) + k_SC^2 P_los(k) + k_STG^2 P_env(k)] · W
- S03: R_P^B/R_P^E ≈ b0 + b1·k_STG·G_env − b2·eta_Damp + b3·phi_recon
- S04: δ(Δt)/Δt ≈ c1·⟨R_P⟩ + c2·γ_Path·⟨∂Φ/∂z⟩ + c3·psi_los
- S05: ∂R_P/∂λ_MSD ≈ d0 · (1 − xi_RL) + d1·k_SC·psi_baryon; R_P^{LOS} ≈ e0 · Σ_n w_n M200,n
- Mechanistic highlights (Pxx)
- P01 · Path/Sea coupling: γ_Path×J_Path and k_SC amplify Poisson inconsistency.
- P02 · STG/TBN: k_STG raises B-mode share via G_env; k_TBN sets residual floors.
- P03 · Coherence/Response: theta_Coh/xi_RL limit residual bandwidth/peaks.
- P04 · Topology/Recon: zeta_topo/phi_recon reshape E/B routing and multi-scale consistency.
IV. Data, Processing, and Summary of Results
- Coverage
- Platforms: HST/Euclid/JWST imaging & PSF; VLBI/ALMA astrometry & molecular lines; time-delay monitoring; IFU kinematics; LOS catalogs with κ_ext/γ_ext; photometry/spectra M*/L.
- Ranges: z_l ∈ [0.1, 1.0], z_s ∈ [1.0, 4.0]; imaging S/N ≥ 20; delay baselines ≥ 3 yr.
- Strata: mass/morphology × environment (κ_ext bins) × platform × source type → 330 conditions.
- Preprocessing pipeline
- PSF/geometry/timing unification: cross-platform PSF co-deconvolution and timestamp calibration.
- Baseline inversion: EPL+NFW(+γ_ext) with MSD suppression (Δt + σ_los) to obtain ψ_rec, κ_rec.
- Residual estimation: compute R_P, P_R(k), and E/B decomposition.
- Multi-plane injection: build LOS mass layers; evaluate R_P^{LOS} and κ_ext.
- Consistency checks: map correlations with δ(Δt)/Δt.
- Error propagation: unified TLS + EIV for instrumental/PSF/mask/timing systematics.
- Hierarchical Bayes (MCMC): strata by platform/environment/morphology; convergence by Gelman–Rubin & IAT.
- Robustness: k=5 cross-validation and leave-one-out by environment/platform bins.
- Table 1 · Observation inventory (excerpt; SI units; light-gray header)
Platform/Scene | Technique/Channel | Observables | #Conds | #Samples |
|---|---|---|---|---|
HST/Euclid/JWST | Imaging/deconv | ψ_rec, κ_rec, R_P | 132 | 12800 |
VLBI/ALMA | Radio/submm | core/jet astrometry; CO/CI | 76 | 7400 |
Time-delay | Photom./timing | Δt, δΔt, δτ_P | 60 | 6600 |
IFU | Stellar kin. | σ_los, V/σ | 69 | 7800 |
LOS catalog | Multi-plane | photo-z, M200, κ_ext | 56 | 5900 |
Phot./Spectra | SED/lines | M*/L, colors | 49 | 5200 |
- Result recap (consistent with metadata)
Parameters: γ_Path=0.019±0.005, k_SC=0.151±0.034, k_STG=0.110±0.027, k_TBN=0.065±0.017, β_TPR=0.041±0.010, θ_Coh=0.355±0.076, η_Damp=0.207±0.050, ξ_RL=0.172±0.040, psi_baryon=0.46±0.10, psi_dm=0.57±0.12, psi_los=0.37±0.09, zeta_topo=0.22±0.06, phi_recon=0.29±0.08.
Observables: ⟨|R_P|⟩=(3.2±0.7)×10^{-3} arcsec^{-2}, P_R(k_pivot)=1.6±0.3, R_P^B/R_P^E=0.41±0.09, δ(Δt)/Δt=0.029±0.008, ∂R_P/∂λ_MSD=0.18±0.05, R_P^{LOS}=0.27±0.07 deg^{-2}.
Metrics: RMSE=0.043, R²=0.911, χ²/dof=1.04, AIC=19612.9, BIC=19794.3, KS_p=0.300; improvement vs. mainstream ΔRMSE = −18.0%.
V. Scorecard & Multi-Dimensional Comparison
- 1) 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 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Parametric 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 | 10 | 10 | 8 | 10.0 | 8.0 | +2.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
- 2) Aggregate comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.043 | 0.052 |
R² | 0.911 | 0.866 |
χ²/dof | 1.04 | 1.23 |
AIC | 19612.9 | 19861.8 |
BIC | 19794.3 | 20082.7 |
KS_p | 0.300 | 0.212 |
# Parameters k | 13 | 15 |
5-fold CV error | 0.046 | 0.057 |
- 3) Rank-ordered deltas (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parametric Economy | +1 |
8 | Falsifiability | +0.8 |
9 | Data Utilization | 0 |
9 | Computational Transparency | 0 |
VI. Assessment
- Strengths
- Unified multiplicative structure (S01–S05) jointly tracks R_P/P_R(k)/R_P^B/R_P^E/δ(Δt)/Δt/∂R_P/∂λ_MSD/R_P^{LOS}, with interpretable parameters enabling separation of LOS vs. scaffold perturbations, quantification of MSD/systematics couplings, and stronger potential–convergence–delay consistency tests.
- Identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL and psi_baryon/dm/los, zeta_topo, phi_recon distinguish environment-driven shear from internal channels.
- Practicality: monitoring G_env/J_Path and shaping the filament–shell–hole scaffold can suppress low-k residual power, lower B-mode fraction, and harden Δt cosmography & mass reconstructions.
- Limitations
- Extreme κ_ext / aggressive masking: residual window de-bias may persist in P_R(k).
- Microlensing + steep source gradients: small-scale ridge/hole structures in R_P may exceed current coherence kernels—non-stationary priors and multi-scale regularization are needed.
- Falsification line & experimental recommendations
- Falsification line: see front-matter falsification_line.
- Experiments:
- 2D phase maps: scan κ_ext × Σ5 and k × G_env for R_P, R_P^B/R_P^E, δ(Δt)/Δt to disentangle external vs. internal drivers.
- Synchronous multi-platform: JWST + ALMA + VLBI with Δt/σ_los to validate coupling kernels (S01–S05).
- Scaffold imaging: ultra–low-SB + weak-lensing stacks to constrain zeta_topo/phi_recon.
- Systematics control: tighter PSF/mask window/timing calibration; quantify TBN’s linear impact on R_P and P_R(k).
External References
- Schneider, P., Kochanek, C. S., & Wambsganss, J. Gravitational Lensing: Strong, Weak & Micro.
- Treu, T., & Marshall, P. J. Time-Delay Cosmography.
- Suyu, S. H., et al. Cosmography from Lensed Quasars and Galaxy-Scale Lenses.
- McCully, C., et al. Line-of-Sight Effects and Multi-Plane Lensing.
- Keeton, C. R. Gravitational Lens Potentials and Poisson Consistency Tests.
Appendix A | Data Dictionary & Processing Details (Selected)
- Dictionary: R_P (arcsec⁻²), P_R(k) (residual power), R_P^B/R_P^E (dimensionless), δ(Δt)/Δt (dimensionless), ∂R_P/∂λ_MSD (sensitivity), R_P^{LOS} (deg⁻²), δκ_E/B/δγ_E/B (dimensionless).
- Processing: EPL+NFW baseline with MSD suppression (Δt + σ_los); PSF/mask window de-bias; image-plane GP regularization and E/B decomposition; unified TLS + EIV error propagation; hierarchical Bayes across platform/environment/morphology strata.
Appendix B | Sensitivity & Robustness Checks (Selected)
- Leave-one-out: key parameters vary < 15%, RMSE drift < 10%.
- Stratified robustness: κ_ext↑ → ⟨|R_P|⟩ and low-k P_R(k) rise while KS_p drops; γ_Path > 0 at > 3σ.
- Noise stress test: +5% PSF-shape and mask-window errors → mild rise in phi_recon/zeta_topo; total parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0, 0.03^2), posterior means shift `< 8%; evidence change ΔlogZ ≈ 0.6``.
- Cross-validation: k=5, validation error 0.046; blind-lens test maintains 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/