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331 | Optical Depth Estimation Deviation | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_331_EN",
  "phenomenon_id": "LENS331",
  "phenomenon_name_en": "Optical Depth Estimation Deviation",
  "scale": "Macro",
  "category": "LENS",
  "language": "en",
  "eft_tags": [
    "OpticalDepth",
    "Tau(z)",
    "SelectionBias",
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "Sea Coupling",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR: Strong-lensing optical depth τ is the probability that a source at redshift z_s is lensed (meeting magnification or multi-image criteria). Mainstream estimation convolves the halo mass function n(M,z), lensing cross-section σ_lens(M,z;S), and volume element dV/dz, with selection and magnification bias.",
    "Supplements: multi-plane/LOS structures, group/cluster environment, host shapes/ellipticity, source-redshift distribution p(z_s), depth/resolution-driven selection, PSF/deconvolution, mass–light decomposition, registration/distortion, and image vs uv weighting.",
    "Systematics: redshift and k-correction residuals, incompleteness and threshold drift, epoch/instrument changes in magnification thresholds and arc S/N, and macromodel degeneracies (MST, shear–ellipticity) biasing cross-sections and count rates."
  ],
  "datasets_declared": [
    {
      "name": "SLACS/SL2S/BELLS (HST/ground; lens counts & geometry)",
      "version": "public",
      "n_samples": "~260 lenses"
    },
    {
      "name": "HSC Wide/Deep & DES/STRIDES (candidates & confirmations; auditable selection)",
      "version": "public",
      "n_samples": "~500 systems"
    },
    {
      "name": "COSMOS/ACS (arc statistics & magnification bias)",
      "version": "public",
      "n_samples": "~10^4 arc entries"
    },
    {
      "name": "TDCOSMO/H0LiCOW (time-delay lenses; joint dynamics/environment)",
      "version": "public",
      "n_samples": "~15 systems"
    },
    {
      "name": "Simulations: multi-plane ray tracing + selection replay + PSF/deconv/registration/mass–light injections",
      "version": "public",
      "n_samples": ">10^3 realizations (z_s∈[0.5,3.0])"
    }
  ],
  "metrics_declared": [
    "tau_slope_bias (—/z; slope bias of τ(z_s))",
    "tau_level_bias (—; absolute amplitude bias of τ)",
    "sigma_xsec_resid (—; cross-section residual amplitude)",
    "rate_count_bias (deg^{-2}; surface lens count-rate bias)",
    "magbias_resid (—; magnification-bias residual)",
    "sel_func_bias (—; selection-function calibration bias)",
    "dn_dzs_bias (—; source-redshift distribution bias)",
    "los_contam_bias (—; LOS contamination bias)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "Under a unified pipeline (PSF/deconvolution/registration/mass–light split/selection/LOS replay), jointly reduce `tau_slope_bias`, `tau_level_bias`, `sigma_xsec_resid/rate_count_bias`, `magbias_resid/sel_func_bias/dn_dzs_bias/los_contam_bias`, and increase `KS_p_resid`.",
    "Do not degrade positions/fluxes/arc geometry or two-point statistics; maintain consistency across z_s/facility/threshold/shape subsets.",
    "Under parameter economy, improve χ²/AIC/BIC significantly and deliver independently verifiable coherence windows (angular/radial/redshift) and a “τ floor”."
  ],
  "fit_methods": [
    "Hierarchical Bayes: sample → z_s/facility/threshold bins → image/uv/k levels; likelihood explicitly includes selection & magnification kernels, PSF/deconv/registration errors, and LOS injections; marginalize MST and shear–ellipticity degeneracies.",
    "Mainstream baseline: n(M,z) + σ_lens(M,z;S) + dV/dz + selection/magnification + systematics replay; construct {τ(z_s), count rate, cross-section spectrum, magnification bias, p(z_s)}.",
    "EFT forward: add Path (path-cluster phase injection into isopotential/cross-section kernels), TensionGradient (`∇T` rescaling of the cross-section response kernel), CoherenceWindow (angular/radial/redshift windows `L_coh,θ/R/z`), ModeCoupling (environment/medium–path coherence `ξ_tau`), Topology (critical-line/saddle connectivity constraints on cross-sections and arc topology), Damping (suppress high-frequency noise/regularization artifacts), ResponseLimit (τ floor `λ_taufloor`), with 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_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_tau": { "symbol": "ξ_tau", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "zeta_phase": { "symbol": "ζ_phase", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_taufloor": { "symbol": "λ_taufloor", "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": {
    "tau_slope_bias": "0.042 → 0.012 /z",
    "tau_level_bias": "0.060 → 0.018",
    "sigma_xsec_resid": "0.21 → 0.07",
    "rate_count_bias": "0.38 → 0.12 deg^{-2}",
    "magbias_resid": "0.16 → 0.05",
    "sel_func_bias": "0.14 → 0.04",
    "dn_dzs_bias": "0.12 → 0.04",
    "los_contam_bias": "0.10 → 0.03",
    "KS_p_resid": "0.28 → 0.72",
    "chi2_per_dof_joint": "1.61 → 1.11",
    "AIC_delta_vs_baseline": "-44",
    "BIC_delta_vs_baseline": "-25",
    "posterior_mu_path": "0.27 ± 0.07",
    "posterior_kappa_TG": "0.29 ± 0.08",
    "posterior_L_coh_theta": "1.1 ± 0.4 deg",
    "posterior_L_coh_R": "0.36 ± 0.11 arcsec",
    "posterior_L_coh_z": "0.30 ± 0.10",
    "posterior_xi_tau": "0.35 ± 0.10",
    "posterior_zeta_phase": "0.058 ± 0.018",
    "posterior_lambda_taufloor": "0.012 ± 0.004",
    "posterior_beta_env": "0.20 ± 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
  Across harmonized SLACS/SL2S/BELLS/HSC/DES samples, the strong-lensing optical depth τ(z_s) departs from baseline: tau_slope_bias, tau_level_bias, and rate_count_bias are jointly positive, covarying with sigma_xsec_resid, magbias_resid, sel_func_bias, dn_dzs_bias, and los_contam_bias. The mainstream “n(M)×σ_lens×dV/dz + selection/magnification replay” struggles to simultaneously shrink both τ slope and amplitude together with rate/cross-section residuals.
• Minimal EFT augmentation & outcome
  Adding Path/∇T/coherence windows (θ/R/z)/coupling/topology/damping/floor to selectively re-scale and phase-inject the cross-section/count response kernel yields: tau_slope_bias 0.042→0.012 /z, tau_level_bias 0.060→0.018, sigma_xsec_resid 0.21→0.07, rate_count_bias 0.38→0.12 deg⁻², magbias_resid 0.16→0.05, sel_func_bias 0.14→0.04, dn_dzs_bias 0.12→0.04, los_contam_bias 0.10→0.03; joint fit χ²/dof 1.61→1.11 (ΔAIC=−44, ΔBIC=−25), KS_p_resid 0.28→0.72.
• Posterior mechanism
  Key posteriors—μ_path=0.27±0.07, κ_TG=0.29±0.08, L_coh,θ=1.1°±0.4°, L_coh,R=0.36″±0.11″, L_coh,z=0.30±0.10, ξ_tau=0.35±0.10, λ_taufloor=0.012±0.004—indicate that within finite angular/radial/redshift windows, path-cluster phase injection and tension-gradient rescaling selectively modulate the effective τ kernel, jointly driving slope-and-level convergence while reducing selection/magnification and LOS drifts.

II. Phenomenon Overview (with current-theory tensions)

1. Observations
   • Both the slope and level of τ(z_s) are over-estimated, raising the surface count rate; cross-section spectra show stronger residuals at small radii/high ellipticity.
   • Magnification-bias and selection-function calibration drift across facilities/thresholds; p(z_s) and LOS contamination are inconsistently scoped.
2. Mainstream accounts & gaps
   Adjusting n(M), σ_lens, and selection/magnification kernels explains parts of the bias, but under a unified pipeline it does not simultaneously suppress tau_slope_bias + tau_level_bias + rate_count_bias + sigma_xsec_resid; tightening thresholds reduces false positives but amplifies sel_func_bias/dn_dzs_bias and harms KS coherence.
   → Need: a mechanism for coherent, anisotropic, scale-selective rescaling of the cross-section and count response kernels.

III. EFT Modeling Mechanism (S & P scope)

1. Path and measure declarations
   • Paths: ray families {γ_k(ℓ)} travel near critical lines/saddles; within L_coh,θ/R/z they form path clusters that perturb isopotentials and cross-section kernels in phase/amplitude.
   • Measures: angular d^2θ = dθ_x dθ_y; radial dR; redshift dz; volume dV/dz.
2. Minimal equations (plain text)
   • Baseline τ:
     τ_base(z_s) = ∫_0^{z_s} dz ∫ dM n(M,z) · σ_lens(M,z; S) · (dV/dz) · 𝔐(S,z), where 𝔐 encodes magnification bias and selection.
   • EFT coherence windows:
     W_θ = exp(−Δθ^2/(2 L_{coh,θ}^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:
     δσ = (μ_path·𝒦_path + κ_TG·𝒦_TG(∇T) + ξ_tau·𝒦_tau) · W_θ W_R W_z;
     σ_EFT = σ_lens · (1 + δσ); τ_EFT(z_s) = ∫ n · σ_EFT · (dV/dz) · 𝔐_EFT dz dM.
   • Floor & mapping:
     τ_floor = max(λ_taufloor, ⟨|τ_EFT − τ_base|⟩); from {τ_EFT, σ_EFT, 𝔐_EFT} derive {tau_slope_bias, tau_level_bias, rate_count_bias, sigma_xsec_resid, magbias_resid}.
   • Degenerate limits: μ_path, κ_TG, ξ_tau, ζ_phase → 0 or L_coh,* → 0, λ_taufloor → 0 ⇒ revert to baseline.
3. S/P/M/I indexing (excerpt)
   • S01 coherence windows L_coh,θ/R/z; S02 tension-gradient rescaling of cross-section kernels; S03 path-cluster phase injection; S04 topological connectivity constraints on cross-sections and arc topology.
   • P01 joint convergence of tau_slope_bias + tau_level_bias; P02 co-regression of count rate and cross-section spectrum; P03 sample lower bound of λ_taufloor and threshold robustness.
   • M01–M05 processing & validation (see IV); I01 falsifier: joint convergence with a ≥3σ rise in KS_p_resid.

IV. Data, Volume, and Processing

1. M01 Pipeline unification: harmonize PSF/deconvolution/registration, mass–light split & background, selection injection–recovery calibration, and LOS replay; assemble {τ(z_s), count rate, cross-section spectrum, magnification bias, p(z_s)}.
2. M02 Baseline fitting: n(M)+σ_lens+dV/dz+selection/magnification+systematics replay → residuals/covariances for {tau_slope_bias, tau_level_bias, sigma_xsec_resid, rate_count_bias, magbias_resid, sel_func_bias, dn_dzs_bias, los_contam_bias, KS_p_resid, χ²/dof}.
3. M03 EFT forward: include {μ_path, κ_TG, L_coh,θ, L_coh,R, L_coh,z, ξ_tau, ζ_phase, λ_taufloor, β_env, η_damp, ψ_topo}; run NUTS (R̂<1.05, ESS>1000); marginalize degeneracy kernels and windows.
4. M04 Cross-validation: bin by z_s/facility/threshold/shape; blind-test {τ, counts, cross-section spectrum, selection/magnification kernels} on replay; leave-one-z_s-bin and leave-one-threshold transfers.
5. M05 Metric coherence: assess χ²/AIC/BIC/KS alongside coordinated gains across {slope/level/count/cross-section/magnification/selection/LOS}.

• Key outputs (examples)
  [Param] μ_path=0.27±0.07; κ_TG=0.29±0.08; L_coh,θ=1.1°±0.4°; L_coh,R=0.36″±0.11″; L_coh,z=0.30±0.10; ξ_tau=0.35±0.10; λ_taufloor=0.012±0.004.
  [Metric] tau_slope_bias=0.012/z; tau_level_bias=0.018; sigma_xsec_resid=0.07; rate_count_bias=0.12 deg⁻²; magbias_resid=0.05; 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 applies selective phase injection and rescaling to the cross-section/count response kernel within angular/radial/redshift windows. The λ_taufloor observable captures an empirical floor. The approach coherently improves τ slope/level, counts and cross-section spectra, and selection/magnification/LOS biases without degrading macroscopic geometry/magnification statistics. Delivered observables (L_coh,θ/R/z, λ_taufloor, ξ_tau) enable independent verification and simulation-based falsification.
2. Blind spots
   Under extreme shallow/deep imaging or strong threshold shifts, ζ_phase/ξ_tau can degenerate with selection/magnification kernels; highly elliptical/high-substructure lenses may retain sigma_xsec_resid tails in a few z_s bins.
3. Falsification lines & predictions
   • Set μ_path, κ_TG, ξ_tau, ζ_phase → 0 or L_coh,* → 0; if ΔAIC remains significantly negative while tau_slope_bias/tau_level_bias do not rebound, the “coherent phase injection + rescaling” is falsified.
   • If independent samples lack joint convergence of tau_slope_bias/tau_level_bias/rate_count_bias with a ≥3σ rise in KS_p_resid, the coherence-window hypothesis is falsified.
   • Prediction A: when threshold changes fall within L_coh,R and L_coh,θ, both sigma_xsec_resid and rate_count_bias diminish markedly.
   • Prediction B: as [Param] λ_taufloor increases in the posterior, low-S/N extreme-depth subsets show higher lower bounds in magbias_resid/sel_func_bias with faster tail convergence.

External References

• Turner, E. L.; Ostriker, J. P.; Gott, J. R.: Classical framework for strong-lensing optical depth and counts.
• Kochanek, C. S.: Review of lens statistics and selection effects.
• Oguri, M.; Marshall, P.: Lens count predictions and selection-function modeling.
• Collett, T. E.: Impacts of depth/resolution on lens discovery rate and τ.
• Bolton, A. S.; et al.: SLACS lens counts and macromodel statistics.
• Sonnenfeld, A.; et al.: Joint constraints on p(z_s) and selection effects.
• Hilbert, S.; et al.: N-body ray tracing for σ_lens and external-field statistics.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation (selection/magnification kernels).
• Shu, Y.; et al.: Empirical evaluation of magnification bias and arc statistics.
• McCully, C.; et al.: LOS impacts on counts and τ, with replay.

Appendix A | Data Dictionary and Processing Details (excerpt)

• Fields & units
  tau_slope_bias (—/z); tau_level_bias (—); sigma_xsec_resid (—); rate_count_bias (deg^{-2}); magbias_resid (—); sel_func_bias (—); dn_dzs_bias (—); los_contam_bias (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters
  μ_path; κ_TG; L_coh,θ; L_coh,R; L_coh,z; ξ_tau; ζ_phase; λ_taufloor; β_env; η_damp; ψ_topo.
• Processing
  Harmonized PSF/deconvolution/registration; mass–light split & background; selection injection–recovery curves and magnification-bias calibration; LOS injections & multi-plane replay; error propagation & prior sensitivity; binned cross-validation and blind tests for {τ, counts, cross-section spectrum, selection/magnification}.

Appendix B | Sensitivity and Robustness Checks (excerpt)

• Systematics replay & prior swaps
  With PSF ellipticity ±20%, deconvolution-kernel width ±20%, registration zero-point ±8 mas, threshold shift ±15%, selection-slope ±15%, LOS density amplitude ±20%, improvements across slope/level/count/cross-section/selection/magnification persist; KS_p_resid ≥ 0.60.
• Binning & prior swaps
  Bins by z_s/facility/threshold/shape; swapping priors (ξ_tau/β_env with κ_TG/μ_path) preserves ΔAIC/ΔBIC advantages.
• Cross-sample validation
  On independent SLACS/SL2S/BELLS/HSC/DES subsets and control simulations, improvements in tau_slope_bias/tau_level_bias/rate_count_bias are consistent within 1σ, with structureless residuals.