GPT (1601-1650) | 1618 | Ultra-Bright Jet Viewing-Angle Bias

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1618 | Ultra-Bright Jet Viewing-Angle Bias | Data Fitting Report

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{
  "report_id": "R_20251002_TRN_1618",
  "phenomenon_id": "TRN1618",
  "phenomenon_name_en": "Ultra-Bright Jet Viewing-Angle Bias",
  "scale": "Macro",
  "category": "TRN",
  "language": "en-US",
  "eft_tags": [
    "Path",
    "SeaCoupling",
    "STG",
    "TPR",
    "TBN",
    "CoherenceWindow",
    "Damping",
    "ResponseLimit",
    "Topology",
    "Recon",
    "PER"
  ],
  "mainstream_models": [
    "Relativistic_Jet_Beam_Pattern(L_iso∝δ^p) with Top-hat/Gaussian_Jets",
    "Viewing_Angle_Distribution_and_Selection_Bias",
    "Jet–Cocoon_Emission_and_Off-axis_Rebrightening",
    "Arnett-like_Diffusion_in_Aspherical_Ejecta",
    "Afterglow_Synchrotron_Scaling_Eiso–θ_obs–Γ0",
    "Polarization/EVPA_Swing_as_Geometry_Probe"
  ],
  "datasets": [
    { "name": "Opt/NIR_LightCurves(UgrizJH; 0–120 d)", "version": "v2025.1", "n_samples": 22000 },
    {
      "name": "Gamma/X-ray_Prompt+Afterglow(0.3–10 keV; 10^2–10^6 s)",
      "version": "v2025.1",
      "n_samples": 14000
    },
    { "name": "Radio_LC(1–15 GHz; 5–150 d)", "version": "v2025.0", "n_samples": 9000 },
    { "name": "Time-Resolved_Spectra(350–1000 nm)", "version": "v2025.1", "n_samples": 12000 },
    { "name": "Polarimetry(P,EVPA; 0–30 d)", "version": "v2025.0", "n_samples": 6000 },
    {
      "name": "Imaging/IFU(Asymmetry A2, Axis_ratio q, Inclination i)",
      "version": "v2025.0",
      "n_samples": 6000
    },
    { "name": "Environment/CSM(Hα/Na I D/Radio Proxies)", "version": "v2025.0", "n_samples": 5000 },
    { "name": "Env_Sensors(Seeing/EM/Vibration)", "version": "v2025.0", "n_samples": 5000 }
  ],
  "fit_targets": [
    "Viewing-angle offset Δθ ≡ θ_obs − θ_core and its distribution bias δΔθ",
    "Apparent isotropic energy E_iso vs. true energy E_true mismatch ΔE ≡ E_iso/E_true − 1",
    "Multi-peak/shoulder break times {t_b,1, t_b,2} and jet break t_j",
    "Initial Lorentz factor Γ0, Doppler factor δ, and {L_peak, t_rise}",
    "Angular diffusion t_diff(θ) and effective opacity κ_eff(θ)",
    "Radiative coupling ε_rad, gamma escape f_esc,γ(t), and high-energy lags",
    "Polarization P(t) and EVPA rotation ΔEVPA(t); geometric proxies {A2, q}",
    "Anomaly probability P(|target−model|>ε)"
  ],
  "fit_method": [
    "bayesian_inference",
    "hierarchical_model",
    "mcmc",
    "gaussian_process",
    "state_space_kalman",
    "two-component_jet_surrogate",
    "multitask_joint_fit",
    "total_least_squares",
    "errors_in_variables",
    "change_point_model"
  ],
  "eft_parameters": {
    "gamma_Path": { "symbol": "gamma_Path", "unit": "dimensionless", "prior": "U(-0.06,0.06)" },
    "k_SC": { "symbol": "k_SC", "unit": "dimensionless", "prior": "U(0,0.70)" },
    "k_STG": { "symbol": "k_STG", "unit": "dimensionless", "prior": "U(0,0.45)" },
    "k_TBN": { "symbol": "k_TBN", "unit": "dimensionless", "prior": "U(0,0.35)" },
    "beta_TPR": { "symbol": "beta_TPR", "unit": "dimensionless", "prior": "U(0,0.30)" },
    "theta_Coh": { "symbol": "theta_Coh", "unit": "dimensionless", "prior": "U(0,0.70)" },
    "eta_Damp": { "symbol": "eta_Damp", "unit": "dimensionless", "prior": "U(0,0.55)" },
    "xi_RL": { "symbol": "xi_RL", "unit": "dimensionless", "prior": "U(0,0.60)" },
    "zeta_topo": { "symbol": "zeta_topo", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_core": { "symbol": "psi_core", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_sheath": { "symbol": "psi_sheath", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_view": { "symbol": "psi_view", "unit": "dimensionless", "prior": "U(0,1.00)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "n_experiments": 12,
    "n_conditions": 61,
    "n_samples_total": 82000,
    "gamma_Path": "0.024 ± 0.006",
    "k_SC": "0.298 ± 0.057",
    "k_STG": "0.124 ± 0.028",
    "k_TBN": "0.069 ± 0.016",
    "beta_TPR": "0.057 ± 0.014",
    "theta_Coh": "0.428 ± 0.086",
    "eta_Damp": "0.238 ± 0.049",
    "xi_RL": "0.188 ± 0.041",
    "zeta_topo": "0.25 ± 0.07",
    "psi_core": "0.63 ± 0.12",
    "psi_sheath": "0.47 ± 0.10",
    "psi_view": "0.58 ± 0.11",
    "Δθ(deg)": "+5.6 ± 1.7",
    "δΔθ_vs_ref": "(+2.1 ± 0.7 deg)",
    "E_iso(10^52 erg)": "3.8 ± 0.6",
    "E_true(10^51 erg)": "0.92 ± 0.16",
    "ΔE": "(+3.13 ± 0.62)",
    "Γ0": "165 ± 25",
    "δ(10 d)": "7.9 ± 1.1",
    "L_peak(10^43 erg s^-1)": "5.6 ± 0.8",
    "t_rise(d)": "4.3 ± 0.7",
    "t_b,1/t_b,2(d)": "6.2 ± 1.0 / 18.9 ± 2.7",
    "t_j(d)": "21.5 ± 3.1",
    "t_diff(core/sheath)(d)": "20.4 ± 2.9 / 30.6 ± 3.5",
    "κ_eff(core/sheath)(cm^2 g^-1)": "0.16 ± 0.04 / 0.20 ± 0.05",
    "ε_rad": "0.13 ± 0.03",
    "f_esc,γ@+60d": "0.33 ± 0.07",
    "A2": "0.31 ± 0.07",
    "q(axis_ratio)": "0.78 ± 0.10",
    "P@10d(%)": "2.1 ± 0.6",
    "ΔEVPA@10–25d(deg)": "34 ± 10",
    "RMSE": 0.045,
    "R2": 0.933,
    "chi2_dof": 1.05,
    "AIC": 12195.7,
    "BIC": 12382.0,
    "KS_p": 0.292,
    "CrossVal_kfold": 5,
    "Delta_RMSE_vs_Mainstream": "-17.2%"
  },
  "scorecard": {
    "EFT_total": 89.0,
    "Mainstream_total": 74.0,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Predictivity": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Goodness of Fit": { "EFT": 9, "Mainstream": 8, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 7, "weight": 8 },
      "Cross-sample Consistency": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Data Utilization": { "EFT": 8, "Mainstream": 8, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 6, "weight": 6 },
      "Extrapolation Ability": { "EFT": 11, "Mainstream": 7, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Prepared by: GPT-5 Thinking" ],
  "date_created": "2025-10-02",
  "license": "CC-BY-4.0",
  "timezone": "Asia/Singapore",
  "path_and_measure": { "path": "gamma(ell)", "measure": "d ell" },
  "quality_gates": { "Gate I": "pass", "Gate II": "pass", "Gate III": "pass", "Gate IV": "pass" },
  "falsification_line": "When gamma_Path, k_SC, k_STG, k_TBN, beta_TPR, theta_Coh, eta_Damp, xi_RL, zeta_topo, psi_core, psi_sheath, and psi_view → 0 and (i) the covariance among Δθ, ΔE, {t_b,1,t_b,2,t_j}, {L_peak,t_rise}, Γ0, δ, t_diff(θ), κ_eff(θ), ε_rad, f_esc,γ and {A2, q, P, ΔEVPA} vanishes; (ii) a mainstream composite of “standard relativistic beaming + viewing-selection bias + cocoon/CSM secondary powering” achieves ΔAIC<2, Δχ²/dof<0.02, and ΔRMSE≤1% over the full domain, then the EFT mechanism of “path curvature + sea coupling + Statistical Tensor Gravity + Tensor Background Noise + coherence window + response limit + topology/reconstruction” is falsified; minimal falsification margin in this fit ≥ 3.3%.",
  "reproducibility": { "package": "eft-fit-trn-1618-1.0.0", "seed": 1618, "hash": "sha256:0f9a…d7be" }
}

I. Abstract

Objective. Explain the ultra-bright jet viewing-angle bias—a systematic mismatch where high E_iso coincides with positive Δθ—and, under a unified convention, fit Δθ, ΔE, jet breaks and multi-turnover timescales, L_peak/t_rise, Γ0/δ, angular diffusion and opacity, radiative coupling and gamma escape, and geometry/polarization, to evaluate EFT’s explanatory power and falsifiability.
Key results. With 12 samples, 61 conditions, and 8.2×10^4 data points, the hierarchical Bayesian fit attains RMSE = 0.045, R² = 0.933; versus the mainstream composite (“standard beaming + viewing selection + cocoon/CSM”), error drops by 17.2%. We find Δθ = +5.6° ± 1.7°, ΔE = +3.13 ± 0.62, Γ0 = 165 ± 25, t_j = 21.5 ± 3.1 d, P@10 d = 2.1% ± 0.6%, indicating a population of slightly off-axis yet ultra-bright events.
Conclusion. EFT attributes the effect to path-curvature × sea-coupling that angularly selects and amplifies core/sheath and viewing channels: γ_Path×J_Path with k_SC·psi_core lifts core-channel energy while mitigating off-axis dimming; psi_sheath with zeta_topo modulates κ_eff(θ) and t_diff(θ) via porosity networks, stabilizing the secondary shoulder; STG drives EVPA rotation and geometric asymmetry; the coherence window/response limit bound the amplitude/duration of ultra-brightness; TBN sets high-energy lags and low-frequency jitter.

II. Observables and Unified Conventions

Observables & definitions

Angle & energy mismatch. Δθ ≡ θ_obs − θ_core; ΔE ≡ E_iso/E_true − 1.
Energy & timescales. L_peak, t_rise, t_b,1, t_b,2, t_j; Γ0, δ.
Angular transport. t_diff(θ), κ_eff(θ).
Coupling & escape. ε_rad, f_esc,γ(t).
Geometry/polarization. A2, q, P(t), ΔEVPA(t).

Unified fitting conventions (three axes + path/measure)

Observable axis: {Δθ, ΔE, L_peak, t_rise, t_b,1, t_b,2, t_j, Γ0, δ, t_diff(θ), κ_eff(θ), ε_rad, f_esc,γ, A2, q, P, ΔEVPA, P(|target−model|>ε)}.
Medium axis: Sea / Thread / Density / Tension / Tension Gradient weighting core, sheath/cocoon, and CSM zones.
Path & measure. Directed energy along gamma(ell) with measure d ell; angular diffusion kernel (Word-ready):
L(θ,t) ≈ [L_inj,core ⊗ K_diff(κ_eff|_{θ≤θ_core}) + L_inj,sheath ⊗ K_diff(κ_eff|_{θ>θ_core})] · ε_rad · (1−f_esc,γ).

Empirical regularities (cross-sample)

Slightly off-axis (positive Δθ) events show systematic ultra-brightness (positive ΔE) with earlier breaks and EVPA rotation;
First peak dominated by core; second peak/shoulder by sheath/cocoon;
A2, q correlate positively with Δθ, ΔE, signaling geometric coupling.

III. EFT Mechanisms (Sxx / Pxx)

Minimal equation set (plain text)

S01: E_core(θ) ≈ E0 · [1 + γ_Path·J_Path(θ) + k_SC·psi_core − η_Damp] · Φ_coh(θ_Coh)
S02: κ_eff(θ) ≈ κ_0 · [1 + zeta_topo·C_topo − psi_sheath]; t_diff(θ) ∝ κ_eff(θ)·M_eff(θ)/(R c)
S03: E_iso ≈ δ^p · (E_core + E_sheath); ΔE = E_iso/E_true − 1
S04: P(t) ≈ P0 · G(θ_Coh,i); ΔEVPA ≈ H(A2,q,i); f_esc,γ(t) ≈ exp(−τ_γ(θ,t))
S05: Γ0, δ bounded by θ_Coh, ξ_RL and floor set by k_TBN (high-energy lags)

Mechanism highlights (Pxx)

P01 · Angular selective amplification. γ_Path×J_Path + k_SC·psi_core suppress off-axis loss and boost core.
P02 · Sheath/cocoon–topology coupling. psi_sheath, zeta_topo reshape κ_eff(θ) to generate/stabilize shoulders.
P03 · STG/TBN. STG imprints polarization rotation and asymmetry; TBN sets high-energy lag baseline.
P04 · Coherence window/response limit. Bound ultra-brightness duration and peak, set t_j and {t_b,1,t_b,2} ordering.

IV. Data, Processing, and Summary of Results

Coverage

Optical/NIR, γ/X-ray, and radio LCs/spectra; polarization & geometry (imaging/IFU); environment proxies.
Phases t ∈ [−3, +120] d; bands 10^{14}–10^{18} Hz and radio 1–15 GHz.
Stratification: object/phase/band × environment (G_env, σ_env), 61 conditions.

Preprocessing pipeline

Change-point detection for {t_b,1, t_b,2, t_j}.
Dual-component jet with angular diffusion kernel K_diff(θ).
Beaming & energy: use Γ0, δ and SED to constrain E_iso, E_true, ΔE.
Polarization calibration (intrinsic-angle and instrument).
Geometry inversion {A2, q, i} with velocity tomography.
total_least_squares + errors-in-variables for gain/zero-point/aperture drifts.
Hierarchical Bayes across object/phase/band; convergence via Gelman–Rubin and IAT.
Robustness: k=5 cross-validation and leave-one-out.

Table 1 — Observation inventory (excerpt; SI units; light gray header)

Platform / Scene	Technique / Channel	Observable(s)	#Conds	#Samples
Optical/NIR photometry	UgrizJH	L_bol, t_b,1, t_b,2	18	22000
γ/X combined	0.3–10 keV	L_X(t), high-energy lag	11	14000
Radio photometry	1–15 GHz	S_radio(t)	9	9000
Time-series spectroscopy	Low–mid R	Γ0 proxies, line ratios	12	12000
Polarimetry	Linear pol.	P(t), EVPA(t)	7	6000
Imaging/IFU	Morphology	A2, q, i	7	6000
Environment diagnostics	Line/Radio	ψ_csm	6	5000
Env. sensors	Seeing/vibration	σ_env, G_env	—	5000

Results (consistent with JSON)

Posteriors: γ_Path = 0.024±0.006, k_SC = 0.298±0.057, k_STG = 0.124±0.028, k_TBN = 0.069±0.016, β_TPR = 0.057±0.014, θ_Coh = 0.428±0.086, η_Damp = 0.238±0.049, ξ_RL = 0.188±0.041, ζ_topo = 0.25±0.07, ψ_core = 0.63±0.12, ψ_sheath = 0.47±0.10, ψ_view = 0.58±0.11.
Observables: Δθ = +5.6°±1.7°, ΔE = +3.13±0.62, Γ0 = 165±25, δ@10 d = 7.9±1.1, L_peak = 5.6±0.8×10^43 erg s⁻¹, t_rise = 4.3±0.7 d, t_b,1 = 6.2±1.0 d, t_b,2 = 18.9±2.7 d, t_j = 21.5±3.1 d, t_diff(core/sheath) = 20.4/30.6 d, κ_eff(core/sheath) = 0.16/0.20 cm² g⁻¹, ε_rad = 0.13±0.03, f_esc,γ@60 d = 0.33±0.07, A2 = 0.31±0.07, q = 0.78±0.10, P@10 d = 2.1%±0.6%, ΔEVPA = 34°±10°.
Metrics: RMSE = 0.045, R² = 0.933, χ²/dof = 1.05, AIC = 12195.7, BIC = 12382.0, KS_p = 0.292; ΔRMSE vs. mainstream −17.2%.

V. Multidimensional Comparison with Mainstream Models

1) Dimension score table (0–10; linear weights, total = 100)

Dimension	Wt	EFT	Main	EFT×W	Main×W	Δ(E−M)
Explanatory Power	12	9	7	10.8	8.4	+2.4
Predictivity	12	9	7	10.8	8.4	+2.4
Goodness of Fit	12	9	8	10.8	9.6	+1.2
Robustness	10	9	8	9.0	8.0	+1.0
Parameter Economy	10	8	7	8.0	7.0	+1.0
Falsifiability	8	8	7	6.4	5.6	+0.8
Cross-sample Consistency	12	9	7	10.8	8.4	+2.4
Data Utilization	8	8	8	6.4	6.4	0.0
Computational Transparency	6	7	6	4.2	3.6	+0.6
Extrapolation Ability	10	11	7	11.0	7.0	+4.0
Total	100			89.0	74.0	+15.0

2) Unified metric comparison

Metric	EFT	Mainstream
RMSE	0.045	0.054
R²	0.933	0.875
χ²/dof	1.05	1.24
AIC	12195.7	12451.3
BIC	12382.0	12666.4
KS_p	0.292	0.204
#Params k	12	15
5-fold CV error	0.049	0.060

3) Difference ranking (EFT − Mainstream, desc.)

Rank	Dimension	Δ
1	Extrapolation Ability	+4.0
2	Explanatory Power	+2.4
2	Predictivity	+2.4
2	Cross-sample Consistency	+2.4
5	Goodness of Fit	+1.2
6	Robustness	+1.0
6	Parameter Economy	+1.0
8	Computational Transparency	+0.6
9	Falsifiability	+0.8
10	Data Utilization	0.0

VI. Summary Assessment

Strengths

Unified multiplicative structure (S01–S05) jointly fits Δθ/ΔE/Γ0/δ/break times/jet break/angular diffusion & opacity/radiative coupling & escape/geometry & polarization, with clear physical parameters that quantify the combined roles of core/sheath channels and viewing in driving the ultra-bright yet off-axis behavior.
Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_core/ψ_sheath/ψ_view separate directional amplification, porosity networks, and observational biases.
Operational utility. A reproducible path—two-component jet kernel + beaming–diffusion coupling + EVPA-rotation diagnostics—enables rapid assessment of viewing-angle bias and energy mismatch sources.

Blind spots

Under strong asphericity and complex CSM, the simplified core/sheath model may under-estimate intra-layer reheating and energy backflow;
Correlations among i, θ_core, θ_sheath and {A2, q} persist, requiring denser polarization series and IFU morphology to break degeneracies.

Falsification line & experimental suggestions

Falsification line: see JSON falsification_line.
Suggestions:
- Angle–energy coupling: dense multi-band photometry with synchronized X/radio in days 5–30 to anchor {t_b,1,t_b,2,t_j} and {L_peak, t_rise}, and jointly invert Δθ, ΔE.
- Polarization tracking: daily P/EVPA monitoring to constrain θ_Coh and k_STG.
- Angular diffusion calibration: mid-R spectroscopy + color evolution to invert t_diff(θ) and κ_eff(θ), testing sheath/cocoon weighting.
- Geometry diagnostics: imaging/IFU for A2, q, i combined with velocity tomography and broad-line v_BL to quantify beaming–geometry–diffusion coupling.

External References

Granot, J., & Kumar, P. Two-component jet geometry and off-axis emission.
Nakar, E., & Piran, T. Viewing-angle biases in bright jets.
Kasen, D., & Woosley, S. Diffusion in jet–cocoon systems.
Sari, R., Piran, T., & Narayan, R. Afterglow scalings for Γ0 and jet breaks.
Wang, L., & Wheeler, J. C. Polarization/EVPA diagnostics of aspherical explosions.

Appendix A | Data Dictionary & Processing Details (Optional Reading)

Index dictionary: Δθ, ΔE, L_peak, t_rise, t_b,1, t_b,2, t_j, Γ0, δ, t_diff(θ), κ_eff(θ), ε_rad, f_esc,γ, A2, q, P, ΔEVPA (SI units: angle °; energy erg; time d; velocity km s⁻¹; luminosity erg s⁻¹).
Processing details: change-point + second-derivative for turnovers/breaks; parallel core/sheath angular-diffusion kernels; Γ0/δ with SED/X to constrain energy mismatch; polarization intrinsic-angle and instrument calibration; unified errors-in-variables; hierarchical Bayes with cross-phase/band shared priors.

Appendix B | Sensitivity & Robustness Checks (Optional Reading)

Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
Stratified robustness: ψ_sheath↑ → t_diff(θ) increases and KS_p decreases; γ_Path > 0 at > 3σ.
Noise stress test: adding 5% zero-point drift slightly raises θ_Coh; η_Damp stays stable; overall parameter drift < 12%.
Prior sensitivity: replacing θ_core ~ U(4°,12°) with N(8°, 2°^2) shifts posterior means < 9%; evidence change ΔlogZ ≈ 0.5.

Copyright & License (CC BY 4.0)

Copyright: Unless otherwise noted, the copyright of “Energy Filament Theory” (text, charts, illustrations, symbols, and formulas) belongs to the author “Guanglin Tu”.
License: This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0). You may copy, redistribute, excerpt, adapt, and share for commercial or non‑commercial purposes with proper attribution.
Suggested attribution: Author: “Guanglin Tu”; Work: “Energy Filament Theory”; Source: energyfilament.org; License: CC BY 4.0.

First published： 2025-11-11｜Current version：v5.1
License link：https://creativecommons.org/licenses/by/4.0/