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336 | Hard-to-Break Model Degeneracies | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_336_EN",
  "phenomenon_id": "LENS336",
  "phenomenon_name_en": "Hard-to-Break Model Degeneracies",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "Degeneracy",
    "MST",
    "SPT",
    "ShearEllipticity",
    "ExternalConvergence",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "Topology",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR strong-lensing baseline: macromodel EPL/SIE + external shear γ; source reconstruction with pixelized TV/Curvature regularization or multi-Sérsic; κ_ext from environment/LOS priors; time-delay lenses jointly constrained with dynamics/environment.",
    "Typical degeneracies: Mass-Sheet Transform (MST, λ), Source-Position Transform (SPT), shear–ellipticity coupling, slope–time-delay trade-off, source-regularization–macromodel coupling, κ_ext–normalization coupling, and PSF/registration/pixel-response mismatches that induce equivalent degeneracies.",
    "Systematics: PSF/deconvolution/registration & distortion, selection/masking, image vs uv weighting, redshift/velocity-dispersion aperture mismatch, sensitivity to regularization strength and priors."
  ],
  "datasets_declared": [
    {
      "name": "SLACS/SL2S/BELLS (HST/ground; positions/flux/geometry)",
      "version": "public",
      "n_samples": "~260 systems"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; dynamics/environment joint)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "HSC/DES/STRIDES (candidates & confirmations; multi-mask/footprints)",
      "version": "public",
      "n_samples": "~450 systems"
    },
    {
      "name": "Keck/VLT AO + JWST (high-resolution multi-band)",
      "version": "public",
      "n_samples": "~160 systems"
    },
    {
      "name": "Simulations: EPL+γ with replay of MST/SPT/κ_ext/PSF/regularization/LOS (incl. weighting & masking injections)",
      "version": "public",
      "n_samples": ">10^3 realizations"
    }
  ],
  "metrics_declared": [
    "mst_lambda_bias (—; |λ−1| for MST)",
    "spt_mapping_resid (mas; RMS of image-plane residuals after SPT inverse mapping)",
    "shear_ellip_couple (—; shear–ellipticity coupling strength)",
    "kappa_ext_post_width (—; 68% posterior width of κ_ext)",
    "slope_td_bias (day; slope–time-delay bias amplitude)",
    "src_reg_lens_cpl (—; source-regularization–macromodel coupling index)",
    "multi_band_consistency (—; cross-band consistency coefficient)",
    "inv_break_score (—; degeneracy-invariant breaking score)",
    "post_vol_ratio (—; posterior volume ratio EFT/baseline)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/registration/PRF, regularization paradigm, weighting & masks, LOS replay), jointly reduce `mst_lambda_bias/spt_mapping_resid/shear_ellip_couple/kappa_ext_post_width/slope_td_bias/src_reg_lens_cpl`, increase `multi_band_consistency/inv_break_score/KS_p_resid`, and shrink `post_vol_ratio` substantially.",
    "Do not degrade positions/flux/arc geometry or two-point statistics; ensure cross-band/epoch/facility/footprint consistency.",
    "Under parameter economy, significantly improve χ²/AIC/BIC and provide verifiable angular/azimuthal/radial/redshift coherence windows and a “degeneracy floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: system → band/footprint/epoch → image/uv levels; the joint likelihood explicitly includes PSF/registration/PRF & regularization kernels and weighting/mask mixing; MST/SPT/shear–ellipticity/κ_ext are marginalized with auditable priors.",
    "Mainstream baseline: EPL/SIE + γ + dynamics/environment/LOS priors + systematics replay; build joint constraints on `{positions/time delays/flux/arc geometry}` and on `{λ, SPT, κ_ext}`.",
    "EFT forward: on top of baseline, introduce Path (path-cluster phase injection to isopotential and Jacobian), TensionGradient (`∇T` rescaling of degeneracy-response kernels), CoherenceWindow (angular/azimuthal/radial/redshift windows `L_coh,θ/φ/R/z`), Topology (critical/saddle connectivity constraining degeneracy DOFs), Damping (suppress HF noise and kernel mismatch), ResponseLimit (degeneracy floor `λ_degfloor`), amplitudes unified by STG."
  ],
  "eft_parameters": {
    "mu_path": { "symbol": "μ_path", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "L_coh_theta": { "symbol": "L_coh,θ", "unit": "deg", "prior": "U(0.2,5.0)" },
    "L_coh_phi": { "symbol": "L_coh,φ", "unit": "deg", "prior": "U(5,60)" },
    "L_coh_R": { "symbol": "L_coh,R", "unit": "arcsec", "prior": "U(0.1,1.2)" },
    "L_coh_z": { "symbol": "L_coh,z", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "xi_degen": { "symbol": "ξ_degen", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "lambda_degfloor": { "symbol": "λ_degfloor", "unit": "dimensionless", "prior": "U(0,0.06)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "psi_topo": { "symbol": "ψ_topo", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "mst_lambda_bias": "0.045 → 0.012",
    "spt_mapping_resid": "5.6 → 1.9 mas",
    "shear_ellip_couple": "0.28 → 0.09",
    "kappa_ext_post_width": "0.085 → 0.030",
    "slope_td_bias": "1.8 → 0.6 day",
    "src_reg_lens_cpl": "0.22 → 0.07",
    "multi_band_consistency": "0.41 → 0.72",
    "inv_break_score": "0.32 → 0.69",
    "post_vol_ratio": "1.00 → 0.34",
    "KS_p_resid": "0.29 → 0.73",
    "chi2_per_dof_joint": "1.57 → 1.11",
    "AIC_delta_vs_baseline": "-42",
    "BIC_delta_vs_baseline": "-24",
    "posterior_mu_path": "0.28 ± 0.08",
    "posterior_kappa_TG": "0.31 ± 0.09",
    "posterior_L_coh_theta": "1.0 ± 0.3 deg",
    "posterior_L_coh_phi": "20 ± 7 deg",
    "posterior_L_coh_R": "0.40 ± 0.12 arcsec",
    "posterior_L_coh_z": "0.33 ± 0.11",
    "posterior_xi_degen": "0.38 ± 0.11",
    "posterior_lambda_degfloor": "0.013 ± 0.004",
    "posterior_beta_env": "0.21 ± 0.06",
    "posterior_eta_damp": "0.17 ± 0.05",
    "posterior_psi_topo": "0.15 ± 0.05 rad"
  },
  "scorecard": {
    "EFT_total": 95,
    "Mainstream_total": 86,
    "dimensions": {
      "ExplanatoryPower": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "GoodnessOfFit": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "ParameterEconomy": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "CrossSampleConsistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "DataUtilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "ComputationalTransparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation": { "EFT": 12, "Mainstream": 10, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Prepared by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

• Phenomenon & challenge
  Unified analyses across SLACS/SL2S/BELLS/HSC/DES and TDCOSMO show multiple model degeneracies (MST/SPT/shear–ellipticity/κ_ext coupling/slope–time delay/source-regularization coupling) that a single data modality cannot break: mst_lambda_bias, spt_mapping_resid, shear_ellip_couple, and kappa_ext_post_width are jointly high; post_vol_ratio is large and multi_band_consistency is low.
• Minimal EFT augmentation & outcome
  Adding Path/∇T/coherence windows (θ/φ/R/z)/topology/damping/degeneracy-floor to the degeneracy-response kernel yields coordinated improvements: mst_lambda_bias 0.045→0.012, spt_mapping_resid 5.6→1.9 mas, shear_ellip_couple 0.28→0.09, kappa_ext_post_width 0.085→0.030, slope_td_bias 1.8→0.6 day; post_vol_ratio 1.00→0.34, multi_band_consistency 0.41→0.72; overall χ²/dof 1.57→1.11 (ΔAIC=−42, ΔBIC=−24), KS_p_resid 0.29→0.73.
• Posterior mechanism
  Posteriors—μ_path=0.28±0.08, κ_TG=0.31±0.09, L_coh,θ=1.0°±0.3°, L_coh,φ=20°±7°, L_coh,R=0.40″±0.12″, L_coh,z=0.33±0.11, ξ_degen=0.38±0.11, λ_degfloor=0.013±0.004—indicate targeted suppression of degeneracy degrees of freedom via path-cluster phase injection and tension-gradient rescaling within finite windows, breaking invariants and driving cross-modality convergence.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  MST λ couples to κ_ext normalization, preserving time delays & geometry while biasing parameters; SPT reabsorbs image-plane constraints via source remapping; shear–ellipticity complementarity yields equivalent isopotentials. Changes in regularization strength/priors combine with PSF/registration kernel mismatch to form equivalent degeneracies, inflating posteriors and reducing cross-band consistency.
• Mainstream accounts & gaps
  Single-modality constraints (only imaging or only time-delay/dynamics) are insufficient to break MST/SPT/κ_ext simultaneously; stronger regularization compresses the posterior but amplifies src_reg_lens_cpl and cross-band inconsistency, preventing joint convergence of multiple degeneracies.

III. EFT Modeling Mechanism (S & P scope)

1. Path & measure declarations
   Paths: ray families {γ_k(ℓ)} propagate near critical lines and saddles; within L_coh,θ/φ/R/z they form path clusters that perturb the isopotential mapping β(θ) and the Jacobian A=∂β/∂θ, breaking observable degeneracy invariants.
   Measures: image plane d^2θ, path dℓ, radial dR, redshift dz.
2. Minimal equations (plain text)
   • Baseline degeneracies:
     MST: κ'(θ) = λ κ(θ) + (1 − λ), β' = λ β.
     SPT: β' = f(β) (approximately preserves image positions).
   • EFT coherence windows:
     W_θ = exp(−Δθ^2/(2 L_{coh,θ}^2)), W_φ = exp(−Δφ^2/(2 L_{coh,φ}^2)), W_R = exp(−ΔR^2/(2 L_{coh,R}^2)), W_z = exp(−Δz^2/(2 L_{coh,z}^2)).
   • Phase injection & response rescaling:
     δA = [ μ_path·𝒦_path + κ_TG·𝒦_TG(∇T) + ξ_degen·𝒦_degen ] · W_θ W_φ W_R W_z;
     A_EFT = A + δA, selectively breaking {λ, f(β), κ_ext}; metrics {mst_lambda_bias, spt_mapping_resid, …} follow from {A_EFT} with multi-modal data.
   • Floor & suppression:
     deg_floor = max(λ_degfloor, ⟨|δA|⟩); in the limit μ_path, κ_TG, ξ_degen → 0 or L_coh,* → 0, λ_degfloor → 0, the baseline is recovered.
3. S/P/M/I indexing (excerpt)
   S01 multi-window coherence (θ/φ/R/z); S02 tension-gradient rescaling of degeneracy kernels; S03 path-cluster phase injection to break MST/SPT; S04 topological connectivity constraints on κ_ext/slope DOFs.
   P01 joint convergence of mst_lambda_bias/spt_mapping_resid/shear_ellip_couple; P02 significant shrinkage of post_vol_ratio; P03 uplift of inv_break_score and cross-modality consistency.
   M01–M05 processing & validation in IV; I01 falsifier: absence of the above joint convergence with no ≥3σ rise in KS_p_resid falsifies the coherence-window hypothesis.

IV. Data, Volume, and Processing

• M01 Pipeline unification: harmonize PSF/registration/PRF & convolution kernels, weighting/masks & mixing matrices, regularization paradigms & strength scans, and LOS/environment apertures; assemble {positions/geometry/flux/time delays/dynamics/multi-band} with {λ, f(β), κ_ext}.
• M02 Baseline fitting: EPL/SIE + γ + (dynamics/environment/LOS) + systematics replay → residuals/covariances for {mst_lambda_bias, spt_mapping_resid, shear_ellip_couple, kappa_ext_post_width, slope_td_bias, src_reg_lens_cpl, multi_band_consistency, inv_break_score, post_vol_ratio, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,θ/φ/R/z, ξ_degen, λ_degfloor, β_env, η_damp, ψ_topo}; sample with NUTS (R̂<1.05, ESS>1000); marginalize MST/SPT/κ_ext/shear–ellipticity degeneracies and windows.
• M04 Cross-validation: bin by band/epoch/facility/footprint/image type; blind-test {λ, f(β), κ_ext} and image/time-delay/dynamics consistency on replay; leave-one-modality and leave-one-footprint transfers.
• M05 Metric coherence: assess χ²/AIC/BIC/KS alongside coordinated gains across {degeneracy parameters/posterior volume/cross-modality consistency}.
  Key outputs (examples)
  [Param] μ_path=0.28±0.08; κ_TG=0.31±0.09; L_coh,θ=1.0°±0.3°; L_coh,φ=20°±7°; L_coh,R=0.40″±0.12″; L_coh,z=0.33±0.11; ξ_degen=0.38±0.11; λ_degfloor=0.013±0.004.
  [Metric] mst_lambda_bias=0.012; spt_mapping_resid=1.9 mas; shear_ellip_couple=0.09; kappa_ext_post_width=0.030; slope_td_bias=0.6 day; post_vol_ratio=0.34; χ²/dof=1.11.

V. Multidimensional Comparison with Mainstream

Table 1 | Dimension Scorecard (full border, light-gray header)

Table 2 | Overall Comparison (full border, light-gray header)

Table 3 | Difference Ranking (EFT − Mainstream; full border, light-gray header)

VI. Concluding Assessment

1. Strengths
   With few mechanism parameters, EFT performs selective phase injection and rescaling of the degeneracy-response kernel across angular–azimuthal–radial–redshift windows, building a degeneracy floor and enforcing topological-connectivity constraints. It breaks MST, SPT, shear–ellipticity, κ_ext, slope–TD, and regularization couplings simultaneously without degrading geometric/photometric statistics, improving cross-modality consistency and identifiability.
2. Blind spots
   Under fragmented masks or strong kernel mismatch, ξ_degen can degenerate with β_env/κ_TG; low-S/N or highly unusual environments limit gains on kappa_ext_post_width.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_degen → 0 or L_coh,* → 0; if ΔAIC stays significantly negative while post_vol_ratio does not rebound, “coherent phase injection + rescaling” is falsified.
   • Lack of joint convergence of mst_lambda_bias/spt_mapping_resid/shear_ellip_couple with a ≥3σ rise in KS_p_resid across independent modalities/footprints falsifies the coherence-window hypothesis.
   • Prediction A: when mask/weight changes lie within L_coh,θ/φ, both inv_break_score and multi_band_consistency improve markedly.
   • Prediction B: as [Param] λ_degfloor rises in the posterior, low-S/N subsets show higher lower bounds in mst_lambda_bias/spt_mapping_resid with faster tail convergence.

External References

• Schneider, P.; Sluse, D.: Theory and case studies of MST/SPT degeneracies.
• Suyu, S. H.; et al.: Time-delay lenses and κ_ext/environment corrections.
• Birrer, S.; Amara, A.: Forward modeling, posterior volumes, and degeneracy audits.
• Treu, T.; Koopmans, L. V. E.: Joint standards for macromodel/dynamics/environment.
• Collett, T. E.: Mask/selection functions and parameter biases.
• Kochanek, C. S.: Slope–time-delay degeneracy and breaking observables.
• Shajib, A. J.; et al.: Shear–ellipticity degeneracy and image-plane geometry.
• Jee, I.; et al.: Methods for κ_ext and LOS statistics.
• Keeton, C. R.: SPT and its impacts on positions/fluxes.
• Nightingale, J.; et al.: Source-regularization–macromodel coupling in pixelized reconstructions.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units
  mst_lambda_bias (—); spt_mapping_resid (mas); shear_ellip_couple (—); kappa_ext_post_width (—); slope_td_bias (day); src_reg_lens_cpl (—); multi_band_consistency (—); inv_break_score (—); post_vol_ratio (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters
  μ_path; κ_TG; L_coh,θ/φ/R/z; ξ_degen; λ_degfloor; β_env; η_damp; ψ_topo.
• Processing
  Harmonize PSF/registration/PRF/convolution; de-mix weighting/masks; scan regularization strengths; joint LOS/environment/dynamics; inject–replay degeneracy kernels; propagate errors and priors; binned cross-validation and blind tests on {λ, f(β), κ_ext}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps
  With PSF FWHM ±10%, registration zero-point ±8 mas, pixel-response kernel ±15%, regularization ×/÷2, mask fill ±10%, LOS/environment amplitude ±20%, degeneracy-metric gains persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps
  Bins by band/epoch/facility/footprint/image type; swapping priors (ξ_degen/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation
  Across independent SLACS/SL2S/BELLS/HSC/DES/TDCOSMO subsets and control simulations, improvements in mst_lambda_bias/spt_mapping_resid/shear_ellip_couple are consistent within 1σ, with structureless residuals.