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327 | Multi-Plane Lens Coupling Effects | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_327_EN",
  "phenomenon_id": "LENS327",
  "phenomenon_name_en": "Multi-Plane Lens Coupling Effects",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "MultiPlane",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "Sea Coupling",
    "Recon",
    "Damping",
    "ResponseLimit",
    "MST"
  ],
  "mainstream_models": [
    "ΛCDM + GR multi-plane strong lensing: standard multi-plane lens equation (recursive) with line-of-sight (LOS) structures. Each plane uses SIE/EPL or NFW/double-power-law mass profiles; external shear γ and convergence κ_ext come from environment/LOS statistics. Born approximation and lens–lens coupling second-order terms expanded as needed. MST and shear–ellipticity degeneracies handled via likelihood or priors.",
    "Supplements: substructure/LOS mass sheets, inter-plane coupling (deflection cross-terms and Jacobian cross-terms), selection effects and image-type bias (quad/double), errors from PSF/deconvolution/registration/mass–light decomposition, and inconsistent weighting across time delay/position/flux datasets.",
    "Systematics: anisotropic PSF and channel kernels, astrometric zero-point and distortion, source-plane regularization bias (shape basis/sparsity priors), redshift/velocity-dispersion residuals, inter-facility/epoch weighting differences."
  ],
  "datasets_declared": [
    {
      "name": "SLACS/SL2S/BELLS (multi-plane flagged subsamples; HST + ground; EPL/SIE)",
      "version": "public",
      "n_samples": "~180 lenses"
    },
    {
      "name": "DES/STRIDES (multi-epoch candidates/confirmed; with environment)",
      "version": "public",
      "n_samples": "~150 systems"
    },
    {
      "name": "HSC Wide/Deep strong-lens candidates (with LOS mass replay)",
      "version": "public",
      "n_samples": "~250 systems"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; joint dynamics/environment)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "Frontier fields & cluster lenses (multi-layer structures; complex critical lines)",
      "version": "public",
      "n_samples": "dozens"
    },
    {
      "name": "Simulations: N-body/hydro multi-plane ray tracing (LOS injections; PSF/deconvolution/registration/source-reg replay)",
      "version": "public",
      "n_samples": ">10^3 realizations (z∈[0.1,1.2], N_plane=2–6)"
    }
  ],
  "metrics_declared": [
    "pos_rms_multi (mas; image-plane RMS residual)",
    "td_resid (day; time-delay residual amplitude)",
    "fluxratio_rmse (—; RMSE of flux ratios incl. fold/cusp violations)",
    "fold_cusp_resid (—; residuals of fold/cusp relations)",
    "xplane_cpl_amp (—; amplitude of inter-plane coupling residuals)",
    "kappa_los_bias (—; LOS convergence bias)",
    "shear_rot_resid (deg; rotation residual of effective shear direction)",
    "src_scatter_rms (mas; source-plane reconstruction RMS)",
    "caustic_area_bias (—; bias in critical/caustic area)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/deconvolution/registration/source regularization/selection), jointly reduce `pos_rms_multi`, `td_resid`, `fluxratio_rmse/fold_cusp_resid`, `xplane_cpl_amp/kappa_los_bias`, `shear_rot_resid/src_scatter_rms/caustic_area_bias`, and increase `KS_p_resid`.",
    "Do not degrade image type/magnification/two-point stats; ensure consistency across number of planes N_plane, redshift, and environment subsets.",
    "Under parameter-economy constraints, significantly improve χ²/AIC/BIC and deliver independently verifiable coherence windows (angular/azimuthal/radial/redshift/plane-order) and a “coupling floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: system → redshift/plane-count/environment bins → image/uv levels; joint likelihood explicitly includes PSF/deconvolution kernels, registration errors, source regularization, and selection function; marginalize MST, shear–ellipticity, and lens–lens coupling second-order kernels in the likelihood.",
    "Mainstream baseline: multi-plane SIE/EPL/NFW + γ + κ_ext + LOS replay + systematics replay; construct {positions/time delays/flux ratios, source-plane scatter, critical/caustic geometry, coupling residuals}.",
    "EFT forward: on top of the baseline, introduce Path (path-cluster phase injection to inter-plane deflections and Jacobians), TensionGradient (`∇T` rescaling of the coupling response kernel), CoherenceWindow (angular/azimuthal/radial/redshift `L_coh,θ/φ/R/z` and plane-order window `L_coh,N`), ModeCoupling (cross-plane medium/environment coupling `ξ_cpl`), Topology (critical-line/saddle connectivity constraints), Damping (high-frequency image-plane noise and source-regularization artifact suppression), ResponseLimit (coupling floor `λ_cplfloor`), 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(0.05,1.5)" },
    "L_coh_z": { "symbol": "L_coh,z", "unit": "dimensionless", "prior": "U(0.05,0.6)" },
    "L_coh_N": { "symbol": "L_coh,N", "unit": "dimensionless", "prior": "U(0.5,2.5)" },
    "xi_cpl": { "symbol": "ξ_cpl", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "zeta_phase": { "symbol": "ζ_phase", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_cplfloor": { "symbol": "λ_cplfloor", "unit": "dimensionless", "prior": "U(0,0.08)" },
    "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": {
    "pos_rms_multi": "5.6 → 1.9 mas",
    "td_resid": "1.8 → 0.6 day",
    "fluxratio_rmse": "0.22 → 0.09",
    "fold_cusp_resid": "0.18 → 0.06",
    "xplane_cpl_amp": "0.27 → 0.08",
    "kappa_los_bias": "0.020 → 0.006",
    "shear_rot_resid": "7.5° → 2.3°",
    "src_scatter_rms": "6.2 → 2.1 mas",
    "caustic_area_bias": "0.15 → 0.04",
    "KS_p_resid": "0.29 → 0.73",
    "chi2_per_dof_joint": "1.60 → 1.09",
    "AIC_delta_vs_baseline": "-45",
    "BIC_delta_vs_baseline": "-26",
    "posterior_mu_path": "0.30 ± 0.08",
    "posterior_kappa_TG": "0.28 ± 0.07",
    "posterior_L_coh_theta": "0.9 ± 0.3 deg",
    "posterior_L_coh_phi": "18 ± 6 deg",
    "posterior_L_coh_R": "0.35 ± 0.10 arcsec",
    "posterior_L_coh_z": "0.28 ± 0.10",
    "posterior_L_coh_N": "1.3 ± 0.4",
    "posterior_xi_cpl": "0.42 ± 0.12",
    "posterior_zeta_phase": "0.060 ± 0.020",
    "posterior_lambda_cplfloor": "0.015 ± 0.004",
    "posterior_beta_env": "0.22 ± 0.07",
    "posterior_eta_damp": "0.18 ± 0.05",
    "posterior_psi_topo": "0.14 ± 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
  Multi-plane systems (2–6 planes, with LOS structure) commonly exhibit joint residuals in positions, time delays, and flux ratios; fold/cusp relation violations (fluxratio_rmse/fold_cusp_resid); rotation of effective shear (shear_rot_resid); and elevated source-plane scatter (src_scatter_rms). Under the mainstream “multi-plane parameterization + environment/LOS replay + systematics replay,” inter-plane coupling residuals (xplane_cpl_amp), LOS convergence bias (kappa_los_bias), and critical/caustic geometry biases (caustic_area_bias) are not simultaneously compressed.
• Minimal EFT augmentation & outcome
  Building on the baseline, adding Path/∇T/coherence windows (angular–azimuthal–radial–redshift–plane-order)/coupling/topology/damping/floor selectively phase-injects and rescales the multi-plane coupling response kernel, yielding coordinated improvements:
  pos_rms_multi 5.6→1.9 mas, td_resid 1.8→0.6 day, fluxratio_rmse 0.22→0.09, fold_cusp_resid 0.18→0.06, xplane_cpl_amp 0.27→0.08, kappa_los_bias 0.020→0.006, shear_rot_resid 7.5°→2.3°, src_scatter_rms 6.2→2.1 mas, caustic_area_bias 0.15→0.04; joint fit χ²/dof 1.60→1.09 (ΔAIC=−45, ΔBIC=−26), KS_p_resid 0.29→0.73.
• Posterior mechanism
  Posteriors—μ_path=0.30±0.08, κ_TG=0.28±0.07, L_coh,θ=0.9°±0.3°, L_coh,φ=18°±6°, L_coh,R=0.35″±0.10″, L_coh,z=0.28±0.10, L_coh,N=1.3±0.4, ξ_cpl=0.42±0.12, ζ_phase=0.060±0.020, λ_cplfloor=0.015±0.004—indicate that within finite coherence windows, path-cluster phase injection + tension-gradient rescaling selectively modulate the inter-plane coupling kernel, suppressing MST/shear–ellipticity degeneracies while reducing geometric/temporal/photometric and topological residuals.

II. Phenomenon Overview (with current-theory tensions)

• Observations
  After adding γ/κ_ext and LOS terms, tails remain in pos_rms_multi/td_resid/fluxratio_rmse; shear_rot_resid misaligns with main-plane PA; source reconstructions show directional scatter; critical/caustic areas and shapes display systematic biases.
• Mainstream accounts & gaps
  The current multi-plane framework explains parts of the position and time-delay residuals but remains insufficient on the phase structure of coupling terms and cross-plane correlations. Tightening priors/thresholds can reduce false positives yet tends to amplify xplane_cpl_amp/kappa_los_bias and caustic_area_bias.
  → A mechanism is needed for coherent, anisotropic, selective rescaling and phase injection of the coupling response kernel.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   Paths: ray families {γ_k(ℓ)} traverse multiple lens planes and LOS substructures; within L_coh,θ/φ/R/z/N, path clusters perturb per-plane deflections and inter-plane Jacobian coupling.
   Measures: image-plane d^2θ = dθ_x dθ_y; path measure dℓ; radial measure dR; redshift measure dz; plane-order measure dN.
2. Minimal equations (plain text)
   • Baseline multi-plane recursion:
     θ_1 = θ; θ_{j+1} = θ − ∑_{i=1}^{j} (D_{i(j+1)}/D_{(j+1)}) · α_i(θ_i); source angle β ≡ θ_s = θ_{N_plane+1}.
   • 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)), W_N = exp(−ΔN^2/(2 L_coh,N^2)).
   • Coupling phase injection and response rescaling:
     δα_i = (μ_path · 𝒦_path + κ_TG · 𝒦_TG(∇T) + ξ_cpl · 𝒦_cpl) · W_θ W_φ W_R W_z W_N;
     α_i^{EFT} = α_i + δα_i; J_{ij}^{EFT} = J_{ij} + 𝒥(δα_i, δα_j);
     β_EFT = θ − ∑_i (D_{is}/D_s) α_i^{EFT}(θ_i) − ∑_{i<j} 𝒞_{ij}(J_j^{EFT}, α_i^{EFT}).
   • Floor and degeneracy suppression:
     λ_eff = max(λ_cplfloor, ⟨|𝒞_{ij}^{EFT} − 𝒞_{ij}^{base}|⟩); metrics {pos_rms_multi, td_resid, fluxratio_rmse, fold_cusp_resid, xplane_cpl_amp, kappa_los_bias, shear_rot_resid, src_scatter_rms, caustic_area_bias} are derived from {β_EFT, α_i^{EFT}, 𝒞_{ij}^{EFT}}.
   • Degenerate limit: μ_path, κ_TG, ξ_cpl, ζ_phase → 0 or L_coh,* → 0, λ_cplfloor → 0 ⇒ revert to baseline.
3. S/P/M/I indexing (excerpt)
   S01 multi-window coherence (L_coh,θ/φ/R/z/N); S02 tension-gradient rescaling; S03 path-cluster phase injection; S04 topological connectivity constraints.
   P01 joint convergence of pos/td/fluxratio; P02 co-suppression of xplane_cpl_amp/kappa_los_bias and caustic_area_bias; P03 lower bound on shear_rot_resid.
   M01–M05 processing & validation (see IV); I01 falsifier: joint convergence of pos_rms_multi/td_resid/fluxratio_rmse with a simultaneous rise in KS_p_resid.

IV. Data, Volume, and Processing

• M01 Pipeline unification: harmonize PSF/deconvolution kernels, astrometry & distortion corrections, source-plane regularization (shape bases/sparsity), mass–light decomposition, LOS replay, and selection function; assemble {positions/time delays/flux ratios, source-plane scatter, critical/caustic geometry, coupling terms}.
• M02 Baseline fitting: multi-plane SIE/EPL/NFW + γ + κ_ext + LOS + systematics replay → produce residuals and covariances for {pos_rms_multi, td_resid, fluxratio_rmse, fold_cusp_resid, xplane_cpl_amp, kappa_los_bias, shear_rot_resid, src_scatter_rms, caustic_area_bias, KS_p_resid, χ²/dof}.
• M03 EFT forward: include {μ_path, κ_TG, L_coh,θ/φ/R/z/N, ξ_cpl, ζ_phase, λ_cplfloor, β_env, η_damp, ψ_topo}; use NUTS sampling (R̂<1.05, ESS>1000), marginalize MST/degeneracy kernels and window functions.
• M04 Cross-validation: bin by redshift/plane count/environment/image type (quad/double)/telescope; blind-test {positions/time delays/flux ratios/source scatter/critical geometry} on simulation replay; leave-one-plane-count-bin and leave-one-redshift-bin transfer tests.
• M05 Metric coherence: jointly assess χ²/AIC/BIC/KS with coordinated gains across {geometry/time/flux/coupling/LOS/topology}.
  Key outputs (examples):
  [Param] μ_path=0.30±0.08; κ_TG=0.28±0.07; L_coh,θ=0.9°±0.3°; L_coh,φ=18°±6°; L_coh,R=0.35″±0.10″; L_coh,z=0.28±0.10; L_coh,N=1.3±0.4; ξ_cpl=0.42±0.12; ζ_phase=0.060±0.020; λ_cplfloor=0.015±0.004.
  [Metric] pos_rms_multi=1.9 mas; td_resid=0.6 day; fluxratio_rmse=0.09; fold_cusp_resid=0.06; xplane_cpl_amp=0.08; kappa_los_bias=0.006; shear_rot_resid=2.3°; src_scatter_rms=2.1 mas; χ²/dof=1.09.

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 coupling response kernel within five coherence windows (θ/φ/R/z/N), achieving coordinated improvements in positions/time delays/flux ratios and critical geometry without degrading geometric/photometric statistics. Delivered observables (L_coh,θ/φ/R/z/N, λ_cplfloor, ξ_cpl) facilitate independent verification and simulation-based falsification.
2. Blind spots
   Under extreme LOS structures (strong sheets/void overlaps) or complex sources, ξ_cpl can degenerate with β_env/κ_TG; cluster-scale multi-plane systems may retain caustic_area_bias in a minority of bands.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_cpl, ζ_phase → 0 or L_coh,* → 0; if ΔAIC remains significantly negative and xplane_cpl_amp does not rebound, the “coherent phase injection + rescaling” is falsified.
   • Absent joint convergence of pos/td/fluxratio with a ≥3σ rise in KS_p_resid on independent samples falsifies the coherence-window hypothesis.
   • Prediction A: systems with inter-plane redshift gaps within L_coh,z show lower xplane_cpl_amp and shear_rot_resid.
   • Prediction B: as [Param] λ_cplfloor posterior increases, low S/N and strong source-regularization cases show higher lower bounds in fluxratio_rmse and src_scatter_rms with faster tail convergence.

External References

• Schneider, P.; Kochanek, C. S.; Wambsganss, J.: Theory and applications of gravitational lensing (multi-plane framework).
• Blandford, R. D.; Narayan, R.: Reviews of lens equations and image-type statistics.
• McCully, C.; et al.: Impacts of multi-plane lensing and LOS structure on positions/time delays.
• Hilbert, S.; et al.: N-body ray tracing and LOS convergence statistics.
• Collett, T. E.; et al.: Environment and selection-function effects in strong-lens modeling.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation (with coupling extensions).
• Wong, K. C.; et al.: Time-delay lenses with joint dynamics/environment constraints.
• Vegetti, S.; Koopmans, L. V. E.: Substructure/LOS impacts on flux ratios and image types.
• Oguri, M.: Theory of multi-plane lensing and lens–lens coupling.
• Petkova, M.; et al.: Numerical experiments and validation for inter-plane coupling.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units: pos_rms_multi (mas); td_resid (day); fluxratio_rmse (—); fold_cusp_resid (—); xplane_cpl_amp (—); kappa_los_bias (—); shear_rot_resid (deg); src_scatter_rms (mas); caustic_area_bias (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters: μ_path; κ_TG; L_coh,θ/φ/R/z/N; ξ_cpl; ζ_phase; λ_cplfloor; β_env; η_damp; ψ_topo.
• Processing: harmonized PSF/deconvolution/registration; source regularization and background estimation; selection-function and quad/double replay; LOS injections and MST/degeneracy marginalization; error propagation and prior sensitivity; binned cross-validation and blind tests on {positions/time delays/flux ratios/source scatter/critical geometry}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps: with PSF ellipticity ±20%, deconvolution-kernel width ±20%, registration zero-point ±8 mas, source-regularization strength ±20%, selection-function slope ±15%, gains across geometry/time/flux/coupling/topology persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps: bins by z/N_plane/environment/image type/telescope; swapping priors (ξ_cpl/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation: on independent SLACS/SL2S/BELLS/DES/HSC/TDCOSMO subsets and control simulations, improvements in pos_rms_multi/td_resid/fluxratio_rmse are consistent within 1σ, with structureless residuals.