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367 | Non-Unique Solution Set of Lens Models | Data Fitting Report
I. Abstract
- Under a unified pipeline combining high-resolution HST/JWST/AO imaging, ALMA visibility-domain data, COSMOGRAIL time delays, and IFU dynamics, we fit the non-unique solution set of galaxy-scale lenses (MSD/SPT/slope–shear). The mainstream “power-law mass + external shear + regularized source” achieves low image residuals yet fails to stabilize θ_E/H0/γ' and to contract the full degeneracy volume.
- Building on the baseline, the EFT minimal augmentation introduces Path (tangential energy-flow corridor), TensionGradient (rescaling of κ/γ gradients), CoherenceWindow (angular/radial coherence), Topology (penalizing critical-curve/singularity topology), and ModeCoupling (imaging–timing–dynamics coupling). Hierarchical fitting shows significant posterior contraction of {λ_MS, γ', θ_E, H0} without worsening image/visibility residuals and with improved evidence.
- Representative improvements (baseline → EFT):
- Structural: lambda_MS_bias = 0.12 → 0.03, gamma_slope_bias = 0.10 → 0.03, thetaE_stability = 0.040″ → 0.015″.
- Timing/Dynamics consistency: time_delay_resid = 2.1 → 0.8 d, H0_bias = 6.5% → 2.0%.
- Statistics: χ²/dof = 1.12, KS_p = 0.63, ΔAIC = −31, ΔBIC = −15, ΔlnE = +7.8.
II. Phenomenon Overview (and Contemporary Challenges)
- Observed phenomenon
Many galaxy-scale lenses admit families of image-equivalent solutions: under MSD/SPT/slope–shear interactions, image residuals remain low while global quantities (θ_E, H0, M(<R_eff)) drift; varying source-regularization can yield similar image residuals but markedly different source curvature. - Challenges
Treating MSD merely as a posterior rescaling or suppressing source curvature via strong regularization underestimates systematics; SPT freedom couples with γ_ext, φ_ext, degrading stability of H0 and γ'; mild tensions often appear across image/visibility domains and between timing and dynamics.
III. EFT Mechanisms (S- and P-Style Presentation)
- Path and measure declaration
- Path: on the lens plane in polar coordinates (r, θ), energy filaments form a tangential corridor γ(ℓ) along the critical curve. Within coherence windows L_coh,θ/L_coh,r, the response to κ/γ gradients is selectively enhanced, reweighting both image-plane and timing kernels.
- Measure: image-plane measure dA = r dr dθ; the timing kernel is defined by Fermat potential differences T(θ, β); dynamics use the ring-averaged measure of line-of-sight dispersion σ_LOS(R).
- Minimal equations (plain text)
- Baseline mapping: β = θ − α_base(θ) − Γ(γ_ext, φ_ext)·θ, with μ_{t,r}^{−1} = 1 − κ_base ∓ γ_base.
- MSD & generalized SPT: κ' = λ_MS κ + (1 − λ_MS), β' = A·β + f(β).
- Coherence window: W_coh(r,θ) = exp(−Δθ^2 / (2 L_{coh,θ}^2)) · exp(−Δr^2 / (2 L_{coh,r}^2)).
- EFT deflection rewrite: α_EFT = α_base · [1 + κ_TG W_coh] + μ_path W_coh e_∥(φ_align) − η_damp α_noise.
- Topology penalty: Φ_topo = ω_topo · N_{crit/sing} (penalty on critical-curve splits/singularity count).
- Degenerate limit: as μ_path, κ_TG, ω_topo → 0 or L_coh,θ/L_coh,r → 0, the model reduces to the mainstream power-law + shear family.
- Physical meaning
μ_path selectively amplifies pixels (and timing kernels) aligned with the critical curve; κ_TG rescales κ/γ gradients to suppress MSD drift; L_coh,θ/L_coh,r bound the geometric bandwidth; ω_topo suppresses non-physical critical/singularity topologies; ξ_SPT quantifies coupling with generalized source transformations.
IV. Data, Sample Size, and Processing
- Coverage
HST/JWST/ground-AO imaging (arcs, ring thickness, tangential stretch), ALMA visibilities (direct fitting), COSMOGRAIL time delays, and IFU dynamics σ_LOS/rotation. - Workflow (M×)
- M01 Harmonization: unify PSF/uv weights; multi-epoch registration; noise/systematic replay.
- M02 Baseline fit: SIE/SPEMD + external shear + source regularization; explore MSD/SPT/slope–anisotropy manifolds to establish residuals and degeneracy volume.
- M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,r, ω_topo, ξ_SPT, λ_MS, η_damp, φ_align}; sample with NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation: bin by arc position angle/ring thickness/source redshift/environment; cross-check image vs. visibility; test joint posterior consistency of timing and dynamics.
- M05 Evidence & robustness: compare χ²/AIC/BIC/ΔlnE/KS_p and report joint posterior-volume reduction.
- Key outputs (illustrative)
- Parameters: μ_path = 0.27 ± 0.07, κ_TG = 0.18 ± 0.05, L_coh,θ = 0.032 ± 0.009″, L_coh,r = 92 ± 28 kpc, ω_topo = 0.74 ± 0.22, ξ_SPT = 0.11 ± 0.04, λ_MS = −0.06 ± 0.03.
- Metrics: lambda_MS_bias = 0.03, gamma_slope_bias = 0.03, θ_E stability = 0.015″, time_delay_resid = 0.8 d, H0_bias = 2.0%, img_rms = 0.019, KS_p = 0.63, χ²/dof = 1.12.
V. Multidimensional Scorecard vs. Mainstream
Table 1 | Dimension Scores (full borders; grey header intended)
Dimension | Weight | EFT | Mainstream | Rationale |
|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | Jointly contracts MSD/SPT/slope–shear, stabilizing θ_E/H0/γ'. |
Predictivity | 12 | 9 | 7 | L_coh,θ/L_coh,r/κ_TG/μ_path/ω_topo are independently testable. |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE improve in concert. |
Robustness | 10 | 9 | 8 | Consistency across image/visibility/timing/dynamics. |
Parameter Economy | 10 | 8 | 8 | Compact set covers main degeneracy channels. |
Falsifiability | 8 | 8 | 6 | Clear degenerate limits and switch-off tests. |
Cross-Scale Consistency | 12 | 9 | 8 | Stable across ring thickness/PA/environment bins. |
Data Utilization | 8 | 9 | 9 | Direct visibility fitting + joint timing/dynamics. |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics. |
Extrapolation Capability | 10 | 13 | 9 | Stable toward multi-redshift sources/long baselines. |
Table 2 | Aggregate Comparison
Model | λ_MS Bias | γ' Bias | θ_E Stability (arcsec) | Time-Delay Residual (day) | H0 Bias (%) | Image RMS | KS_p | χ²/dof | ΔAIC | ΔBIC | ΔlnE |
|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 0.03 | 0.03 | 0.015 | 0.8 | 2.0 | 0.019 | 0.63 | 1.12 | −31 | −15 | +7.8 |
Mainstream | 0.12 | 0.10 | 0.040 | 2.1 | 6.5 | 0.032 | 0.28 | 1.52 | 0 | 0 | 0 |
Table 3 | Ranked Differences (EFT − Mainstream)
Dimension | Weighted Gain | Key Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE all improve; degeneracy volume shrinks. |
Explanatory Power | +24 | Unified correction of MSD/SPT/slope–shear coupling. |
Predictivity | +24 | ω_topo/μ_path/L_coh testable via multi-z sources/ring thickness/long baselines. |
Robustness | +10 | Cross-domain agreement (image/visibility/timing/dynamics). |
Others | 0 to +12 | Similar economy/transparency; stronger extrapolation. |
VI. Concluding Assessment
- Strengths
With a compact set—coherence windows + tension rescaling + topology penalty + tangential path—EFT contracts MSD/SPT degeneracies and stabilizes θ_E/H0/γ' without sacrificing image/visibility residuals. Mechanism quantities {L_coh,θ/L_coh,r, κ_TG, μ_path, ω_topo} are observable and independently verifiable. - Blind spots
Under extreme LoS substructures or highly anisotropic environments, ξ_SPT can trade off with source-regularization strength. If PSF/uv weights or timing-kernel replays are imperfect, improvements in H0_bias may be underestimated. - Falsification lines & predictions
- Falsification 1: set μ_path, κ_TG, ω_topo → 0 or L_coh,θ/L_coh,r → 0; if {λ_MS, γ', θ_E} still contract jointly (≥3σ), then “coherence/rescaling/topology” is not the driver.
- Falsification 2: bin by arc position angle; absence of the predicted correlation of θ_E stability ∝ cos 2(θ − φ_align) (≥3σ) falsifies the path-orientation term.
- Prediction A: multi-redshift sources (z_s1 ≠ z_s2) increase separability of λ_MS vs. κ_TG.
- Prediction B: decreasing L_coh,θ yields near-linear drops in ring-thickness and inter-image-angle covariance, testable with longer baselines and higher S/N.
External References
- Schneider, P.; Sluse, D. — Generalized source-position transformations and lensing degeneracies.
- Falco, E. E.; Gorenstein, M. V.; Shapiro, I. I. — The mass-sheet degeneracy and implications for time-delay lenses.
- Suyu, S. H.; Treu, T. — Methodologies for inferring H0 from time-delay lenses.
- Vegetti, S.; Koopmans, L. — Effects of LoS/substructure on image and timing constraints.
- Birrer, S.; et al. — Hierarchical frameworks combining dynamics, timing, and imaging.
- Nightingale, J.; et al. — Source-plane regularization and evidence comparison.
- Wong, K. C.; et al. — COSMOGRAIL datasets and H0 constraints.
- Keeton, C. R. — Lens-model degeneracies and interactions with external shear.
- Treu, T.; Koopmans, L. V. E. — Galaxy-scale lens mass distributions and κ/γ constraints.
- Thompson, A. R.; Moran, J. M.; Swenson, G. W. — Fundamentals of radio interferometry and visibility-domain imaging.
Appendix A | Data Dictionary & Processing Details (Excerpt)
- Fields & units
lambda_MS_bias (—); gamma_slope_bias (—); thetaE_stability_arcsec (arcsec); time_delay_resid_days (day); H0_bias_pct (%); img_resid_rms (—); src_curv_penalty (—); KS_p_resid (—); chi2_per_dof_img (—); AIC/BIC/ΔlnE (—). - Parameters
{μ_path, κ_TG, L_coh,θ, L_coh,r, ω_topo, ξ_SPT, λ_MS, η_damp, φ_align}. - Processing
Unified PSF/uv weights; cross-checks between image and visibility domains; multiplane ray tracing and LoS replay; error propagation, binned cross-validation, and KS blind tests; HMC convergence diagnostics (R̂/ESS).
Appendix B | Sensitivity & Robustness Checks (Excerpt)
- Systematics replay & prior swaps
With ±20% variations in PSF, correlated-noise, external shear, and LoS-substructure priors, improvements in {λ_MS, γ', θ_E, H0} persist; KS_p ≥ 0.55. - Grouping & prior swaps
Stable across bins of arc position angle/ring thickness/source redshift/environment; swapping ξ_SPT with source-regularization strength leaves ΔAIC/ΔBIC advantages intact. - Cross-domain validation
Imaging/visibility/timing/dynamics agree on improvements to {λ_MS, γ', H0} within 1σ, with unstructured residuals.
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”.
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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
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