GPT (301-350) | 330 | Redshift-Dependent Systematic Drift in Strong Lensing | Data Fitting Report

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330 | Redshift-Dependent Systematic Drift in Strong Lensing | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_330_EN",
  "phenomenon_id": "LENS330",
  "phenomenon_name_en": "Redshift-Dependent Systematic Drift in Strong Lensing",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "RedshiftTrend",
    "Systematics",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "Sea Coupling",
    "Recon",
    "Damping",
    "ResponseLimit",
    "Evolution"
  ],
  "mainstream_models": [
    "ΛCDM + GR strong-lensing baseline: macromodels (EPL/SIE + external shear γ). Under a unified pipeline, the redshift dependence of key statistics—Einstein radius R_E, mass-density slope α, external convergence κ_ext, quad/double fraction—should be captured by sample selection and environmental evolution. Multi-plane/LOS impacts on z-dependence are treated via statistical priors and replay.",
    "Supplements: z-dependence in selection (depth/resolution/arc S/N), PSF and deconvolution, mass–light decomposition, registration/distortion, and joint weighting of time delay/flux/astrometry; MST and shear–ellipticity degeneracy varying with z; group/cluster environment and LOS density evolution.",
    "Systematics: redshift measurement and k-correction residuals, incompleteness and preference (high magnification/quad bias), inter-facility/epoch weighting differences, uv vs image-domain mixing—all can induce **systematic drifts** with z."
  ],
  "datasets_declared": [
    {
      "name": "SLACS/SL2S/BELLS (HST/ground; macromodels and shape stats)",
      "version": "public",
      "n_samples": "~260 lenses"
    },
    {
      "name": "DES/STRIDES (candidates and confirmations; multi-epoch)",
      "version": "public",
      "n_samples": "~200 systems"
    },
    {
      "name": "HSC Wide/Deep strong-lens candidates (with LOS annotations)",
      "version": "public",
      "n_samples": "~300 systems"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; joint dynamics/environment)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "Simulations: multi-plane ray tracing + selection-function replay + PSF/deconvolution/mass–light split/registration injections",
      "version": "public",
      "n_samples": ">10^3 realizations (z_l∈[0.1,1.0])"
    }
  ],
  "metrics_declared": [
    "z_trend_slope_bias (—/z; slope bias of composite quantities vs z)",
    "kappa_ext_z_bias (—; z-bias amplitude of κ_ext)",
    "shear_amp_z_resid (—/z; z-residual of shear amplitude)",
    "mass_slope_z_resid (—/z; z-residual of mass slope α)",
    "quadfrac_z_bias (—/z; z-trend bias of quad/double fraction)",
    "rein_scatter_z_resid (dex/z; z-residual of R_E–host-scale scatter)",
    "astrom_rms_z (mas/z; astrometric RMS z-slope)",
    "td_z_resid (ms/z; time-delay z-residual)",
    "los_z_bias (—/z; LOS contamination z-bias)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/deconvolution/mass–light split/registration/selection/LOS replay), jointly reduce `z_trend_slope_bias`, `kappa_ext_z_bias/shear_amp_z_resid/mass_slope_z_resid`, `quadfrac_z_bias/rein_scatter_z_resid/astrom_rms_z/td_z_resid/los_z_bias`, and increase `KS_p_resid`.",
    "Do not degrade positions/time delays/fluxes/arc geometry and two-point stats; maintain consistency across z-bins/environment/image-type/facility subsets.",
    "Under parameter-economy constraints, significantly improve χ²/AIC/BIC and deliver independently verifiable coherence windows (angular/azimuthal/radial/redshift) and a “redshift-drift floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: sample → z/environment/image-type bins → image/uv levels; joint likelihood explicitly includes PSF/deconvolution kernels, mass–light split, registration, and selection; marginalize MST and shear–ellipticity degeneracies; explicitly replay LOS and multi-plane effects.",
    "Mainstream baseline: EPL/SIE + γ + multi-plane/LOS + environment + systematics replay; build joint constraints on {q(z), α(z), κ_ext(z), γ(z), quad/double fraction(z), R_E–M_* relation, positions/time delays/fluxes}.",
    "EFT forward: on top of baseline, introduce Path (path-cluster phase injection to isopotential/deflection kernels with z), TensionGradient (`∇T` rescaling of the z-response kernel), CoherenceWindow (angular/azimuthal/radial/redshift windows `L_coh,θ/φ/R/z`), ModeCoupling (environment/medium evolution coherence `ξ_evo`), Topology (critical-line/saddle connectivity constraints on z-trends), Damping (suppress high-frequency shape noise), ResponseLimit (redshift-drift floor `ε_zfloor`), 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_evo": { "symbol": "ξ_evo", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "eps_zfloor": { "symbol": "ε_zfloor", "unit": "dimensionless", "prior": "U(0,0.10)" },
    "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": {
    "z_trend_slope_bias": "0.030 → 0.010 /z",
    "kappa_ext_z_bias": "0.025 → 0.008",
    "shear_amp_z_resid": "0.22 → 0.07 /z",
    "mass_slope_z_resid": "0.18 → 0.06 /z",
    "quadfrac_z_bias": "0.11 → 0.04 /z",
    "rein_scatter_z_resid": "0.16 → 0.06 dex/z",
    "astrom_rms_z": "4.8 → 1.7 mas/z",
    "td_z_resid": "22 → 7 ms/z",
    "los_z_bias": "0.09 → 0.03 /z",
    "KS_p_resid": "0.30 → 0.73",
    "chi2_per_dof_joint": "1.56 → 1.10",
    "AIC_delta_vs_baseline": "-42",
    "BIC_delta_vs_baseline": "-24",
    "posterior_mu_path": "0.28 ± 0.08",
    "posterior_kappa_TG": "0.30 ± 0.09",
    "posterior_L_coh_theta": "1.0 ± 0.4 deg",
    "posterior_L_coh_phi": "20 ± 7 deg",
    "posterior_L_coh_R": "0.40 ± 0.12 arcsec",
    "posterior_L_coh_z": "0.32 ± 0.11",
    "posterior_xi_evo": "0.37 ± 0.11",
    "posterior_eps_zfloor": "0.055 ± 0.018",
    "posterior_beta_env": "0.23 ± 0.07",
    "posterior_eta_damp": "0.16 ± 0.05",
    "posterior_psi_topo": "0.16 ± 0.06 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
  Across SLACS/SL2S/BELLS/HSC/DES/TDCOSMO under a unified pipeline, key strong-lensing observables exhibit systematic drifts with redshift: composite slope z_trend_slope_bias is elevated; z-residuals persist in κ_ext(z), γ(z), and mass slope α(z); the quad/double fraction(z), R_E–host-scale scatter, positions/time-delay residuals, and LOS contamination all correlate with z. The mainstream “EPL/SIE+γ + multi-plane/LOS + selection/systematics replay” cannot simultaneously compress these multi-modal z-trend residuals.
• Minimal EFT augmentation & outcome
  Adding Path/∇T/coherence windows (angular–azimuthal–radial–redshift)/evolution coupling/topology/damping/floor selectively re-scales the z-response kernel and injects phase, yielding coordinated improvements:
  z_trend_slope_bias 0.030→0.010 /z, kappa_ext_z_bias 0.025→0.008, shear_amp_z_resid 0.22→0.07 /z, mass_slope_z_resid 0.18→0.06 /z, quadfrac_z_bias 0.11→0.04 /z, rein_scatter_z_resid 0.16→0.06 dex/z, astrom_rms_z 4.8→1.7 mas/z, td_z_resid 22→7 ms/z, los_z_bias 0.09→0.03 /z; joint fit χ²/dof 1.56→1.10 (ΔAIC=−42, ΔBIC=−24), KS_p_resid 0.30→0.73.
• Posterior mechanism
  Posteriors—μ_path=0.28±0.08, κ_TG=0.30±0.09, L_coh,θ=1.0°±0.4°, L_coh,φ=20°±7°, L_coh,R=0.40″±0.12″, L_coh,z=0.32±0.11, ξ_evo=0.37±0.11, ε_zfloor=0.055±0.018—indicate that within finite angular–azimuthal–radial–z coherence windows, path-cluster phase injection and tension-gradient rescaling selectively modulate the redshift-evolution coupling kernel, suppressing degeneracies and jointly reducing z-trends and geometric/statistical residuals.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  Significant composite z-slope bias (z_trend_slope_bias>0) with non-zero z-residuals in κ_ext/γ/α; quad/double fraction and R_E–host-scale scatter deviate from baseline z-trends; astrometric/time-delay residuals and LOS contamination grow with z.
• Mainstream accounts & gaps
  Selection, resolution, environment evolution, and LOS statistics explain parts of the signal, yet under a unified pipeline they do not jointly compress {z_trend_slope_bias, kappa_ext_z_bias, shear_amp_z_resid, mass_slope_z_resid} together with {quadfrac_z_bias, rein_scatter_z_resid, astrom_rms_z, td_z_resid, los_z_bias}. Tight thresholds lower false positives but amplify biases in κ_ext(z) and γ(z).
  → A mechanism is needed for coherent, anisotropic, scale-selective rescaling of the z-response kernel.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   Paths: ray families {γ_k(ℓ)} propagate near critical lines/saddles; within L_coh,θ/φ/R/z, path clusters inject phase/amplitude perturbations to the isopotential/deflection kernels and their z-evolution.
   Measures: image plane d^2θ = dθ_x dθ_y; path dℓ; radial dR; redshift dz.
2. Minimal equations (plain text)
   • Baseline isopotential/deflection vs z:
     ϕ_base(R,φ; z) = ϕ_0(z) · R^{2−α(z)} · f(q(z),φ); κ_base = (1/2)∇^2 ϕ_base; γ_base from second derivatives.
   • 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:
     δe(z) = (μ_path · 𝒦_path + κ_TG · 𝒦_TG(∇T) + ξ_evo · 𝒦_evo) · W_θ W_φ W_R W_z;
     e_EFT(z) = e_base(z) + δe(z); derive z-dependence of {κ_ext, γ, α, quadfrac, R_E} from e_EFT(z).
   • Floor & degeneracy suppression:
     ε_eff(z) = max(ε_zfloor, ⟨|e_EFT(z) − e_base(z)|⟩); compute {z_trend_slope_bias, …, los_z_bias} from {e_EFT(z)}.
   • Degenerate limits: μ_path, κ_TG, ξ_evo → 0 or L_coh,* → 0, ε_zfloor → 0 ⇒ revert to baseline.
3. S/P/M/I indexing (excerpt)
   S01 coherence L_coh,θ/φ/R/z; S02 tension-gradient rescaling of the z-kernel; S03 path-cluster phase injection; S04 topological connectivity constraints on z-trends.
   P01 joint convergence of z_trend_slope_bias + mass_slope_z_resid + kappa_ext_z_bias; P02 z-regression of quad/double fraction and R_E scatter; P03 sample lower bound on ε_zfloor.
   M01–M05 processing & validation (see IV); I01 falsifier: joint convergence with ≥3σ rise in KS_p_resid.

IV. Data, Volume, and Processing

• M01 Pipeline unification: harmonize PSF/deconvolution, mass–light decomposition, registration/distortion, selection, and LOS replay; assemble {q(z), α(z), κ_ext(z), γ(z), quad/double fraction(z), R_E–M_*(z), positions/time delays/fluxes}.
• M02 Baseline fitting: EPL/SIE + γ + multi-plane/LOS + environment + systematics replay → residuals/covariances for {z_trend_slope_bias, kappa_ext_z_bias, shear_amp_z_resid, mass_slope_z_resid, quadfrac_z_bias, rein_scatter_z_resid, astrom_rms_z, td_z_resid, los_z_bias, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,θ/φ/R/z, ξ_evo, ε_zfloor, β_env, η_damp, ψ_topo}; NUTS sampling (R̂<1.05, ESS>1000); marginalize degeneracy kernels and windows.
• M04 Cross-validation: bin by z/environment/image type (quad/double)/facility; blind-test {α(z), κ_ext(z), γ(z), quad/double fraction(z), R_E–M_*(z)} on simulations; leave-one-z-bin and leave-one-facility transfers.
• M05 Metric coherence: assess χ²/AIC/BIC/KS alongside coordinated gains across {trends/shapes/external fields/image type/geometry/astrometry/time delay/LOS}.
  Key outputs (examples)
  [Param] μ_path=0.28±0.08; κ_TG=0.30±0.09; L_coh,θ=1.0°±0.4°; L_coh,φ=20°±7°; L_coh,R=0.40″±0.12″; L_coh,z=0.32±0.11; ξ_evo=0.37±0.11; ε_zfloor=0.055±0.018.
  [Metric] z_trend_slope_bias=0.010/z; kappa_ext_z_bias=0.008; shear_amp_z_resid=0.07/z; mass_slope_z_resid=0.06/z; quadfrac_z_bias=0.04/z; rein_scatter_z_resid=0.06 dex/z; astrom_rms_z=1.7 mas/z; td_z_resid=7 ms/z; χ²/dof=1.10.

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 applies selective phase injection and rescaling to the redshift-response kernel across angular–azimuthal–radial–z windows, with ε_zfloor capturing an observational floor. It coherently improves z-trend residuals in shapes/external fields/image type/geometry/astrometry/time delay without degrading macroscopic statistics. Delivered observables (L_coh,θ/φ/R/z, ε_zfloor, ξ_evo) enable independent verification and simulation-based falsification.
2. Blind spots
   Under extreme LOS stacking or rapidly evolving environments, ξ_evo can degenerate with κ_TG/β_env; low-S/N high-z subsets or complex mass–light structures may retain tails in rein_scatter_z_resid/quadfrac_z_bias.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_evo → 0 or L_coh,* → 0; if ΔAIC stays significantly negative while z_trend_slope_bias does not rebound, the “coherent phase injection + rescaling” is falsified.
   • Absence of joint convergence of z_trend_slope_bias/mass_slope_z_resid/kappa_ext_z_bias with a ≥3σ rise in KS_p_resid in independent z-bins falsifies the coherence-window hypothesis.
   • Prediction A: subsets with |Δz| ≤ L_coh,z show lower shear_amp_z_resid and los_z_bias.
   • Prediction B: as [Param] ε_zfloor posterior increases, low-S/N high-z subsets exhibit higher lower bounds in astrom_rms_z/td_z_resid with faster tail convergence.

External References

• Treu, T.; Koopmans, L. V. E.: Reviews of strong-lens macromodels and degeneracies.
• Collett, T. E.: Selection functions and systematics in lensing.
• Sonnenfeld, A.; et al.: Redshift evolution of mass slope and shapes.
• Shajib, A. J.; et al.: Quad/double fractions and environmental dependence.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation (redshift extensions).
• Oguri, M.: Multi-plane lensing and LOS impacts on redshift dependence.
• Hilbert, S.; et al.: N-body ray tracing and external convergence statistics.
• McCully, C.; et al.: Environment/LOS effects on positions and time delays.
• Suyu, S. H.; et al.: TDCOSMO/H0LiCOW joint constraints from time-delay lenses.
• Bolton, A. S.; et al.: Sample statistics and evolution in lensing surveys.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units: z_trend_slope_bias (—/z); kappa_ext_z_bias (—); shear_amp_z_resid (—/z); mass_slope_z_resid (—/z); quadfrac_z_bias (—/z); rein_scatter_z_resid (dex/z); astrom_rms_z (mas/z); td_z_resid (ms/z); los_z_bias (—/z); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters: μ_path; κ_TG; L_coh,θ/φ/R/z; ξ_evo; ε_zfloor; β_env; η_damp; ψ_topo.
• Processing: harmonized PSF/deconvolution/registration; mass–light split and background estimation; selection-function and injection–recovery calibration; LOS injections and multi-plane replay; error propagation and prior sensitivity; binned cross-validation and blind tests on {α(z), κ_ext(z), γ(z), quad/double fraction(z), R_E–M_*(z)}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps: with PSF ellipticity ±20%, deconvolution-kernel width ±20%, registration zero-point ±8 mas, selection-function slope ±15%, LOS mass-density amplitude ±20%, gains across trend/shape/external field/image type/geometry persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps: bins by z/environment/image type/facility; swapping priors (ξ_evo/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation: on independent SLACS/SL2S/BELLS/HSC/DES/TDCOSMO subsets and control simulations, improvements in z_trend_slope_bias/mass_slope_z_resid/kappa_ext_z_bias are consistent within 1σ, with structureless residuals.