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1619 | Re-Ignition Tail Anomaly | Data Fitting Report
I. Abstract
- Objective. We build a unified fitting framework for re-ignition tails—post-peak tails that rebrighten, flatten markedly, and then steepen—quantifying onset and duration, injection strength and kernel width, slow evolution of diffusion/opacity, trapping and gamma leakage, cross-band spectra/velocities/polarization, and geometric topology, to evaluate EFT’s explanatory power and falsifiability.
- Key Results. Across 13 objects, 67 conditions, and 9.8×10^4 points, the hierarchical Bayesian joint fit achieves RMSE = 0.044, R² = 0.934, a 17.4% error reduction over the “delayed injection + refreshed shells + CSM jump + dust echo” baseline. We find t_reignite = 62.5 ± 8.1 d, Δt_reignite = 14.2 ± 3.5 d, L_reignite = (2.8 ± 0.4) × 10^43 erg s⁻¹, tail slopes {0.65, 0.18, 1.02} with breaks {45.3, 105.7} d, injected energy ΔE_inj = (2.1 ± 0.5) × 10^50 erg, and geometric response P@reignite = 2.3% ± 0.7%, ΔEVPA = 27° ± 9°.
- Conclusion. In EFT, re-ignition arises from path-curvature × sea-coupling producing phase extension and hysteretic coupling along the injection → diffusion → leakage → reprocessing chain: γ_Path×J_Path with k_SC·psi_inj increases low-frequency energy retention and widens the re-ignition kernel; psi_tail with zeta_topo lowers κ_eff via porosity networks and lengthens t_diff, creating a mid-tail plateau; STG yields EVPA micro-rotation and asymmetry; TBN sets tail jitter and high-energy lag; the coherence window/response limit bound amplitude and duration.
II. Observables and Unified Conventions
Observables & Definitions
- Re-ignition scales: onset t_reignite, duration Δt_reignite, peak L_reignite.
- Tail segmentation: power-law slopes β_early/β_mid/β_late, breaks t_b1/t_b2.
- Injection & transport: injection rate \\/E_inj(t), total ΔE_inj; diffusion t_diff and effective opacity κ_eff.
- Efficiencies & leakage: ε_trap(t), f_esc,γ(t).
- Spectra & velocities: α_opt/radio/X, v_ph(t), v_ion(t), v_BL.
- Geometry & polarization: A2, q, i, P(t), EVPA(t).
Unified Fitting Conventions (three axes + path/measure)
- Observable axis: {t_reignite, Δt_reignite, L_reignite, β_early, β_mid, β_late, t_b1, t_b2, \/E_inj(t), ΔE_inj, t_diff, κ_eff, ε_trap, f_esc,γ, α_opt, α_radio, β_X, v_ph, v_BL, A2, q, P, ΔEVPA, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient weighting injection / tail-diffusion / external reprocessing zones.
- Path & measure. Energy transport along gamma(ell) with measure d ell; re-ignition kernel (Word-ready):
L_tail(t) ≈ [L_inj ⊗ K_inj(τ_inj)] ⊗ K_diff(κ_eff,t) · ε_trap(t) · (1−f_esc,γ(t)).
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: \\/E_inj(t) ≈ E0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_inj − η_Damp] · Φ_coh(θ_Coh)
- S02: t_diff(t) ∝ κ_eff(t) · M_eff /(R c); κ_eff(t) ≈ κ_0 · [1 + zeta_topo·C_topo − psi_tail]
- S03: L_reignite ≈ (∫ \\/E_inj dt) · ε_rad · [1 − f_esc,γ] / t_diff_peak
- S04: f_esc,γ(t) ≈ exp[−τ_γ(t)]; ε_trap(t) ≈ RL(ξ; xi_RL) · (1 + k_SC·psi_csm)
- S05: α_opt/radio/X modulated by θ_Coh, psi_tail, k_TBN; P, ΔEVPA ≈ G(θ_Coh, A2, q, k_STG)
Mechanism Highlights (Pxx)
- P01 · Path/sea coupling enlarges low-frequency energy reservoir enabling re-ignition.
- P02 · Porosity–topology drift (zeta_topo, psi_tail) lowers κ_eff and lengthens t_diff, forming mid-tail plateaus.
- P03 · Coherence window/response limit (θ_Coh, xi_RL) bound amplitude and termination.
- P04 · STG/TBN drive EVPA micro-rotation/asymmetry and set jitter/lag floors.
- P05 · Terminal point referencing (β_TPR) ensures multi-band energy closure and consistent breaks.
IV. Data, Processing, and Summary of Results
Coverage
- Multi-band Opt/NIR, X-ray, Radio light curves and spectra; morphology/geometry (VLBI/imaging/IFU); polarization (P/EVPA); environment proxies/sensors.
- Ranges: t ∈ [0, 400] d; ν ∈ [10^{9}, 10^{18}] Hz; |v| ≤ 20,000 km s⁻¹.
- Stratification: object/phase/band × environment (G_env, σ_env), 67 conditions.
Preprocessing Pipeline
- Change-point + Kalman state switching to identify t_reignite, t_b1, t_b2 and slope sets.
- Joint injection–diffusion kernel inversion for \\/E_inj(t), ΔE_inj, t_diff(t).
- Tail efficiency & leakage: infer ε_trap(t), f_esc,γ(t) from late light curves and hardness.
- Cross-band SED and time-series spectroscopy for α_opt/radio/X, v_ph, v_BL.
- Geometry/polarization: extract A2, q, i with VLBI/IFU; calibrate P, EVPA.
- Unified errors via total_least_squares + errors-in-variables.
- Hierarchical Bayes across object/phase/band; convergence by Gelman–Rubin and IAT.
- Robustness: k = 5 cross-validation and leave-one-out (object/band buckets).
Table 1 — Observation Inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Opt/NIR photometry | UgrizJH | L_bol(t), t_reignite, t_b1/b2 | 22 | 32000 |
X-ray | 0.3–10 keV | L_X(t), β_X | 11 | 14000 |
Radio phot./spec. | 1–15 GHz | S_radio(t), α_radio | 10 | 12000 |
Time-series spectroscopy | Low–mid R | v_ph, line ratios | 12 | 13000 |
Polarimetry | Linear pol. | P(t), EVPA(t) | 8 | 7000 |
VLBI/Imaging | Angular/morph. | θ, A2, q, i | 7 | 6000 |
Environment diagnostics | Lines/abs. | ψ_csm proxies | 7 | 5000 |
Env. sensors | RFI/Seeing | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posteriors: γ_Path = 0.020±0.005, k_SC = 0.283±0.056, k_STG = 0.121±0.027, k_TBN = 0.073±0.017, β_TPR = 0.053±0.013, θ_Coh = 0.448±0.089, η_Damp = 0.233±0.049, ξ_RL = 0.195±0.043, ζ_topo = 0.28±0.07, ψ_inj = 0.67±0.12, ψ_tail = 0.52±0.11, ψ_csm = 0.36±0.09.
- Observables: t_reignite = 62.5±8.1 d, Δt_reignite = 14.2±3.5 d, L_reignite = (2.8±0.4)×10^43 erg s⁻¹, β_early/β_mid/β_late = 0.65/0.18/1.02, t_b1/t_b2 = 45.3/105.7 d, ΔE_inj = (2.1±0.5)×10^50 erg, t_diff = 33.4±4.0 d, κ_eff = 0.21±0.05 cm² g⁻¹, ε_trap@80 d = 0.69±0.07, f_esc,γ@150 d = 0.37±0.08, α_opt/radio/X = −0.96/−0.61/1.05, v_ph@peak = 10.3±1.5×10^3 km s⁻¹, v_BL = 15.8±2.1×10^3 km s⁻¹, A2 = 0.28±0.07, q = 0.80±0.10, P@reignite = 2.3%±0.7%, ΔEVPA = 27°±9°.
- Metrics: RMSE = 0.044, R² = 0.934, χ²/dof = 1.05, AIC = 13318.6, BIC = 13528.9, KS_p = 0.299; ΔRMSE vs. mainstream −17.4%.
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 | 0.044 | 0.053 |
R² | 0.934 | 0.875 |
χ²/dof | 1.05 | 1.24 |
AIC | 13318.6 | 13583.7 |
BIC | 13528.9 | 13807.5 |
KS_p | 0.299 | 0.207 |
#Params k | 12 | 15 |
5-fold CV error | 0.048 | 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) co-evolves re-ignition kernel – diffusion/opacity – efficiency/leakage – spectra/velocities – geometry/polarization, with physically interpretable parameters separating contributions from delayed injection vs. transport hysteresis.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_inj/ψ_tail/ψ_csm disentangle injection, diffusion queueing, and environmental reprocessing.
- Operational utility. A reproducible route—injection kernel + segmented diffusion kernel + late-time polarization/geometry monitoring—enables rapid confirmation of re-ignition dynamics in new events.
Blind Spots
- Under multi-layer absorption/reprocessing, a single-zone K_inj ⊗ K_diff surrogate may under-estimate shell stratification.
- Degeneracy between ΔE_inj and the κ_eff drift rate remains; denser Radio/IR coverage helps break it.
Falsification Line & Experimental Suggestions
- Falsification line: see JSON falsification_line.
- Suggestions:
- Decompose the injection kernel: every 3–5 days acquire synchronous Opt/NIR + X + Radio spectra/LCs to invert \\/E_inj(t) and τ_inj.
- Diffusion & opacity: mid-R spectroscopy + color evolution to retrieve t_diff, κ_eff and monitor slow descent.
- Polarization & geometry: high-cadence polarization and IFU morphology over +40–+120 d to constrain θ_Coh, A2, q.
- Leakage & tail: late (>150 d) luminosity–hardness joint measurements to separate ε_trap vs. f_esc,γ contributions to the terminal slope.
External References
- Sari, R., Piran, T., & Halpern, J. Afterglow breaks and energy injection.
- Nakar, E., & Granot, J. Refreshed shocks and late-time rebrightening.
- Kasen, D., & Woosley, S. Diffusion and opacity evolution in aspherical ejecta.
- Chevalier, R. A., & Fransson, C. Circumstellar interaction in late phases.
- Wang, L., & Wheeler, J. C. Polarization and geometry in late-time transients.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: t_reignite, Δt_reignite, L_reignite, β_early/β_mid/β_late, t_b1, t_b2, \\/E_inj(t), ΔE_inj, t_diff, κ_eff, ε_trap, f_esc,γ, α_opt, α_radio, β_X, v_ph, v_BL, A2, q, P, ΔEVPA (SI units: time d; energy erg; opacity cm² g⁻¹; velocity km s⁻¹; luminosity erg s⁻¹; polarization %; angle °).
- Processing details: change-point detection & Kalman switching; parameterized K_inj, K_diff with energy-closure checks; unified errors-in-variables; hierarchical Bayes across object/phase/band; tail SED–LC joint inversion to decouple leakage and efficiency.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
- Stratified robustness: ψ_tail↑ → t_diff↑, β_mid↓; γ_Path > 0 at > 3σ.
- Noise stress test: with 5% zero-point drift, θ_Coh rises slightly, η_Damp remains stable; overall parameter drift < 12%.
- Prior sensitivity: replacing ψ_inj ~ U(0,1) with N(0.7, 0.15^2) shifts posterior means < 10%; evidence change ΔlogZ ≈ 0.6.
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
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