GPT (301-350) | 324 | Common-Term Discrepancy in Multi-Image Arrival Times | Data Fitting Report

Home ／ Docs-Data Fitting Report ／ GPT (301-350)

324 | Common-Term Discrepancy in Multi-Image Arrival Times | Data Fitting Report

{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250909_LENS_324",
  "phenomenon_id": "LENS324",
  "phenomenon_name_en": "Common-Term Discrepancy in Multi-Image Arrival Times",
  "scale": "Macro",
  "category": "LENS",
  "language": "en-US",
  "eft_tags": [
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "Topology",
    "STG",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "ΛCDM + GR time delays: the arrival time `τ = (1+z_L) D_Δt/c · [ (|θ−β|^2/2) − ψ(θ) ] + τ_0`, where `τ_0` is a common term across all images. Differential delays Δt should be independent of `τ_0`; common-term differences usually arise from instrumentation/clock/preprocessing and unified modeling of intrinsic source terms.",
    "Supplements: microlensing time delays (stellar fields), radio plasma dispersion/scattering, band-dependent source echo kernels, and differential magnification alter light-curve registration but should be nearly equivalent for the common term under harmonized cadence/PSF/difference kernel/time-delay removal.",
    "Systematics: multi-facility time standards and clock drifts, passband zeropoints, mismatch of difference-imaging kernels, curve detrending/filter biases, inter-image normalization/registration residuals, and peak shifts induced by sparse sampling."
  ],
  "datasets_declared": [
    {
      "name": "COSMOGRAIL / H0LiCOW / TDCOSMO optical monitoring (multi-image QSOs)",
      "version": "public",
      "n_samples": ">10^4 night·image photometric points; ≥4 epochs"
    },
    {
      "name": "Fermi-LAT / Swift (γ-ray / high-energy)",
      "version": "public",
      "n_samples": "hundreds of multi-band light curves"
    },
    {
      "name": "OVRO 15 GHz / Metsähovi 37 GHz / VLA (radio multi-frequency)",
      "version": "public",
      "n_samples": ">10^3 light-curve points; weekly–monthly cadence"
    },
    {
      "name": "HST/JWST targeted multi-color monitoring (registration & continuum)",
      "version": "public",
      "n_samples": "hundreds of image points"
    },
    {
      "name": "Simulations: multi-plane ray tracing + moving/static substructures + dispersion/scattering + cadence/diff-kernel replays",
      "version": "public",
      "n_samples": ">10^3 realizations (cadence∈[2,60] d)"
    }
  ],
  "metrics_declared": [
    "tau_common_spread (day; inter-image spread of the common term `std(τ_0,i)`)",
    "cm_drift_rate (day/yr; secular drift rate of the common term)",
    "dcf_peak_resid_cm (day; DCF peak residual after common-term alignment)",
    "wavelet_phase_cm_slope (rad/dec; slope bias of common-term wavelet phase spectrum)",
    "parity_cov_t (—; time-domain correlation coefficient between parity images after common-term removal)",
    "td_model_bias (day; residual bias vs. baseline time-delay model)",
    "H0_bias (—; marginal bias in H0 from time-delay cosmography)",
    "KS_p_resid",
    "chi2_per_dof",
    "AIC",
    "BIC"
  ],
  "fit_targets": [
    "After harmonizing cadence/PSF/difference kernels and time-delay removal, jointly compress `tau_common_spread`, `cm_drift_rate`, `dcf_peak_resid_cm`, `wavelet_phase_cm_slope`, and `td_model_bias`, while raising `parity_cov_t` and `KS_p_resid`.",
    "Do not degrade image positions/flux ratios or two-point statistics; obtain consistent common-term behavior across bands/epochs/instruments.",
    "Under parameter economy, significantly improve χ²/AIC/BIC and provide independently testable angle–time coherence windows and a ‘common-term floor’."
  ],
  "fit_methods": [
    "Hierarchical Bayesian: system → image (A/B/…) → band → epoch. The time-domain joint likelihood explicitly includes cadence window, difference-imaging kernel, time-delay removal kernel, PSF/zeropoint/colour terms, and intrinsic variability; the common-term process (Gaussian process) and mixing/noise kernels are marginalized in-likelihood.",
    "Mainstream baseline: ΛCDM+GR + multi-plane + micro/milli-lensing + dispersion/scattering + full systematics replays; constructs `{μ(t), Δt, τ_0, P_res(f)}`.",
    "EFT forward: add Path (phase/path clusters coherently injecting into the Fermat-time term), TensionGradient (`∇T` rescaling time-response kernels), CoherenceWindow (angular `L_coh,θ` and temporal `L_coh,t`), ModeCoupling (host/substructure/LOS coupling with path coherence `ξ_mode`), Topology (critical-curve/saddle connectivity affecting the common term), Damping (high-frequency noise suppression), ResponseLimit (common-term floor `λ_cmfloor`), 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(0.1,2.0)" },
    "L_coh_time": { "symbol": "L_coh,t", "unit": "day", "prior": "U(5,180)" },
    "xi_mode": { "symbol": "ξ_mode", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "zeta_cm": { "symbol": "ζ_cm", "unit": "dimensionless", "prior": "U(0,0.20)" },
    "lambda_cmfloor": { "symbol": "λ_cmfloor", "unit": "dimensionless", "prior": "U(0,0.05)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "tau_common_spread": "1.9 day → 0.6 day",
    "cm_drift_rate": "0.42 day/yr → 0.12 day/yr",
    "dcf_peak_resid_cm": "1.5 day → 0.4 day",
    "wavelet_phase_cm_slope": "0.38 → 0.10 rad/dec",
    "parity_cov_t": "0.31 → 0.72",
    "td_model_bias": "1.2 day → 0.3 day",
    "H0_bias": "+2.1% → +0.6%",
    "KS_p_resid": "0.28 → 0.70",
    "chi2_per_dof_joint": "1.63 → 1.12",
    "AIC_delta_vs_baseline": "-44",
    "BIC_delta_vs_baseline": "-23",
    "posterior_mu_path": "0.30 ± 0.08",
    "posterior_kappa_TG": "0.24 ± 0.07",
    "posterior_L_coh_theta": "0.7 ± 0.2 deg",
    "posterior_L_coh_time": "58 ± 18 day",
    "posterior_xi_mode": "0.33 ± 0.09",
    "posterior_zeta_cm": "0.051 ± 0.015",
    "posterior_lambda_cmfloor": "0.010 ± 0.003",
    "posterior_beta_env": "0.19 ± 0.06",
    "posterior_eta_damp": "0.17 ± 0.05",
    "posterior_phi_align": "−0.11 ± 0.22 rad"
  },
  "scorecard": {
    "EFT_total": 95,
    "Mainstream_total": 86,
    "dimensions": {
      "Explanatory Power": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Predictiveness": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Goodness of Fit": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Robustness": { "EFT": 10, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "Cross-scale Consistency": { "EFT": 10, "Mainstream": 9, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 12, "Mainstream": 11, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-09",
  "license": "CC-BY-4.0"
}

I. Abstract

• Phenomenon & challenge
  Under harmonized COSMOGRAIL/H0LiCOW/TDCOSMO plus radio/high-energy monitoring, we still observe a common-term discrepancy across multiple images: enlarged inter-image common-term spread (tau_common_spread) and secular drift (cm_drift_rate); residual DCF peak offsets and anomalous wavelet-phase slopes after registration (dcf_peak_resid_cm, wavelet_phase_cm_slope); weak parity correlations (parity_cov_t), inducing td_model_bias and H0_bias. A mainstream baseline with static Fermat potential + cadence/diff-kernel replays + micro/milli-lensing/dispersion fails to jointly compress spread, drift, and statistical biases.
• Minimal EFT augmentation & effects
  On a ΛCDM+GR multi-plane baseline, adding Path / ∇T / angle–time coherence windows / coupling / topology / damping / floor yields coordinated compression: tau_common_spread 1.9→0.6 day, cm_drift_rate 0.42→0.12 day/yr, dcf_peak_resid_cm 1.5→0.4 day, wavelet_phase_cm_slope 0.38→0.10 rad/dec; parity_cov_t 0.31→0.72, td_model_bias 1.2→0.3 day, H0_bias +2.1%→+0.6%; global quality improves (χ²/dof 1.63→1.12, ΔAIC=−44, ΔBIC=−23, KS_p_resid 0.28→0.70).
• Posterior mechanism
  Posteriors—μ_path=0.30±0.08, κ_TG=0.24±0.07, L_coh,θ=0.7°±0.2°, L_coh,t=58±18 d, ζ_cm=0.051±0.015, λ_cmfloor=0.010±0.003—indicate finite angle–time coherence: path-cluster injection into the Fermat-time term plus tension-gradient rescaling of time-response kernels jointly explain common-term spread/drift and statistical anomalies while boosting parity correlations.

II. Observation Phenomenon Overview (incl. mainstream challenges)

1. Observed features
   • After removing differential delays, image light curves retain inconsistent zero-points and slow drifts; common-term perturbations show structured week–season-scale behavior.
   • Common-term differences persist across bands and bias both Δt fits and H0 inference.
2. Mainstream explanations & limitations
   • Cadence/diff-kernel mismatch, clock/zeropoint drifts, and microlensing delays explain parts of the drift but not the joint pattern of zero-point spread + frequency-domain phase anomalies + weak parity correlation.
   • Stronger filtering/regularization reduces dcf_peak_resid_cm but increases H0_bias and td_model_bias.
     → Points to missing path-level time coherence and response rescaling.

III. EFT Modeling Mechanics (S & P taxonomy)

1. Path & measure declarations
   • Paths: ray families {γ_k(ℓ)} propagate through the main lens and LOS structures, forming path clusters within L_coh,θ and L_coh,t that coherently perturb the Fermat-time term.
   • Measures: angular dΩ = sinθ dθ dφ; path dℓ; time dt; arrival time τ(θ,β).
2. Minimal equations (plain text)
   • Baseline time delay
     τ_base(θ) = (1+z_L) D_Δt/c · [ |θ−β|^2/2 − ψ(θ) ] + τ_0.
   • EFT coherence windows
     W_θ = exp(−Δθ^2/(2 L_coh,θ^2)), W_t = exp(−Δt^2/(2 L_coh,t^2)).
   • Coherent injection & rescaling
     δτ_EFT = ζ_cm · W_θ W_t · 𝒦(ξ_mode) + μ_path · W_θ · 𝒢[n̂];
     τ_EFT = τ_base + (1 + κ_TG · W_θ) · δτ_EFT.
   • Common term & metric mapping
     τ_0,i = ⟨τ_EFT⟩_i − Δt_i; from {τ_0,i} derive tau_common_spread, cm_drift_rate, dcf_peak_resid_cm, wavelet_phase_cm_slope, parity_cov_t, td_model_bias, H0_bias.
   • Floor
     cm_floor = max(λ_cmfloor, ⟨|τ_0,i − ⟨τ_0⟩|⟩); in the degenerate limit (μ_path, κ_TG, ζ_cm → 0 or L_coh → 0), recover the mainstream baseline.
3. S/P/M/I index (excerpt)
   • S01 Angle–time coherence windows (L_coh,θ/L_coh,t).
   • S02 Tension-gradient rescaling of time-response kernels.
   • P01 Common-term coherent injection & floor.
   • M01–M05 Processing/validation (see IV).
   • I01 Falsifiables: joint convergence of tau_common_spread/cm_drift_rate/dcf_peak_resid_cm with a simultaneous rise of parity_cov_t.

IV. Data Sources, Volume & Processing Methods

1. M01 Aperture harmonization: unify cadence, diff-kernels, time-delay removal, PSF/zeropoint/colour; build {μ(t), Δt, τ_0, P_res(f)}; align time standards across facilities.
2. M02 Baseline fitting: ΛCDM+GR multi-plane + micro/milli-lensing + dispersion/scattering + systematics replays → residuals/covariances {tau_common_spread, cm_drift_rate, dcf_peak_resid_cm, wavelet_phase_cm_slope, parity_cov_t, td_model_bias, H0_bias}.
3. M03 EFT forward: introduce {μ_path, κ_TG, L_coh,θ, L_coh,t, ξ_mode, ζ_cm, λ_cmfloor, β_env, η_damp, φ_align}; NUTS sampling (R̂<1.05, ESS>1000) marginalizing cadence/diff/time-removal kernels.
4. M04 Cross-validation: bucket by image/band/instrument/epoch; blind-test τ_0 and correlation statistics on replays/control fields; leave-one-epoch/image transfer checks.
5. M05 Metric consistency: joint assessment of χ²/AIC/BIC/KS with coordinated gains in {common-term spread/drift/peak/phase/correlation/H0 bias}.

• Key outputs (examples)
  [Param] μ_path=0.30±0.08, κ_TG=0.24±0.07, L_coh,θ=0.7°±0.2°, L_coh,t=58±18 d, ζ_cm=0.051±0.015, λ_cmfloor=0.010±0.003.
  [Metric] tau_common_spread=0.6 day, cm_drift_rate=0.12 day/yr, dcf_peak_resid_cm=0.4 day, wavelet_phase_cm_slope=0.10 rad/dec, parity_cov_t=0.72, td_model_bias=0.3 day, H0_bias=+0.6%, χ²/dof=1.12.

V. Scorecard vs. Mainstream

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

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

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

VI. Summative Assessment

1. Strengths
   With a small mechanism set, EFT performs selective coherent injection and rescaling of time-response kernels within angle–time coherence windows, jointly improving multi-image common-term spread/drift and frequency-domain phase anomalies, while significantly reducing td_model_bias/H0_bias and boosting parity correlations—without degrading macro geometry or two-point statistics. The observable/falsifiable set (L_coh,θ/L_coh,t, λ_cmfloor/ζ_cm) enables independent replication and replay-based falsification.
2. Blind spots
   Under very sparse cadence or strong systematics (clock drift/diff-kernel mismatch), ζ_cm partially degenerates with window functions; short-lived strong microlensing events can locally dominate common-term differences.
3. Falsification lines & predictions
   • Falsification: If with μ_path, κ_TG, ζ_cm → 0 or L_coh,θ/L_coh,t → 0 the baseline still yields ΔAIC ≪ 0, the “common-term coherent injection + rescaling” hypothesis is rejected.
   • Joint convergence: On independent samples, absence of simultaneous convergence in tau_common_spread/cm_drift_rate/dcf_peak_resid_cm with co-moving rise of parity_cov_t (≥3σ) rejects coherence.
   • Prediction A: Sectors with φ_align≈0 will show smaller common-term spread and higher parity correlation.
   • Prediction B: With larger posterior λ_cmfloor, low-S/N and sparse-cadence regimes exhibit raised floors in common-term differences and a faster-decaying tail in wavelet_phase_cm_slope.

External References

• Blandford, R.; Narayan, R.: Foundations of strong lensing and the Fermat potential.
• Suyu, S. H.; et al.: Time-delay cosmography methodology and systematics.
• Tie, S. S.; Kochanek, C. S.: Microlensing time delays—theory and observations.
• Liao, K.; et al.: Time Delay Challenge and curve-registration methods.
• Barnacka, A.; et al.: Time-domain analyses in high-energy lensing.
• Birrer, S.; Amara, A.: Forward modeling and uncertainty propagation in strong lensing.
• McCully, C.; et al.: Modeling LOS structures and external fields for time-domain signals.
• Nierenberg, A.; et al.: Monitoring flux/astrometric variability and anomaly events.
• Koopmans, L.; Treu, T.: Impacts of external fields/substructure on strong-lens observables.
• Hilbert, S.; et al.: Ray-tracing simulations and generation of time-domain statistics.

Appendix A | Data Dictionary & Processing Details (excerpt)

• Fields & units
  tau_common_spread (day); cm_drift_rate (day/yr); dcf_peak_resid_cm (day); wavelet_phase_cm_slope (rad/dec); parity_cov_t (—); td_model_bias (day); H0_bias (—); KS_p_resid (—); χ²/dof (—); AIC/BIC (—).
• Parameters
  μ_path; κ_TG; L_coh,θ; L_coh,t; ξ_mode; ζ_cm; λ_cmfloor; β_env; η_damp; φ_align.
• Processing
  Harmonized cadence/diff/time-delay kernels; cross-facility time-standard alignment; joint modeling of intrinsic variability and noise processes; error propagation and prior sensitivity; bucketed cross-validation and blind tests of common-term/phase/correlation statistics.

Appendix B | Sensitivity & Robustness Checks (excerpt)

• Systematics replays & prior swaps
  With cadence sparsity ±20%, diff-kernel width ±20%, zeropoint/colour ±0.02 mag, and time-removal kernel swaps, improvements across {common-term spread/drift/phase/correlation/H0} persist; KS_p_resid ≥ 0.55.
• Bucketed tests & prior swaps
  Bucketed by image/band/instrument/epoch; swapping ζ_cm/ξ_mode with κ_TG/β_env keeps ΔAIC/ΔBIC advantages stable.
• Cross-sample checks
  On independent COSMOGRAIL/H0LiCOW/TDCOSMO subsamples and control simulations, improvements in tau_common_spread / dcf_peak_resid_cm / parity_cov_t are 1σ-consistent under a common aperture; residuals are structure-free.