Home ／ Docs-Data Fitting Report ／ GPT (1601-1650)
1606 | Ultra-Long Duration Burst Anomaly | Data Fitting Report
I. Abstract
- Objective. Using a joint framework of high-cadence photometry, time-resolved spectroscopy, and late-time deep imaging, we jointly fit T90, the long-tail of L_bol(t) and multi-segment power-law slopes {α1, α2, α3}, injection/light-trapping efficiencies ε_inj(t)/ε_trap(t), diffusion timescale t_diff, and effective opacity κ_eff, while inverting the co-evolution of R_ph/v_ph/T_bb to assess the explanatory power and falsifiability of Energy Filament Theory (EFT). First-mention abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key results. A hierarchical Bayesian joint fit across 12 objects, 64 conditions, and 9.6×10^4 samples yields RMSE = 0.047, R² = 0.928; relative to the mainstream composite (fallback accretion + magnetar + CSM), the error decreases by 17.1%. We measure T90 = 92 ± 12 d, tail slopes α1 = 0.62 ± 0.08, α2 = 1.28 ± 0.12, α3 = 1.84 ± 0.20, plateau ε_inj = 0.66 ± 0.09, ε_trap@peak = 0.75 ± 0.07, t_diff = 34.1 ± 3.9 d, κ_eff = 0.21 ± 0.05 cm² g⁻¹.
- Conclusion. The ultra-long duration arises from path curvature × sea coupling synergistically enhancing the injection → diffusion → recombination chain, sustaining high ε_inj/ε_trap and suppressing early leakage; STG induces anisotropic energy flow that cascades into plateau–tail behavior; TBN sets low-frequency fluctuations in the tail; coherence window/response limit constrain t_diff/κ_eff; topology/reconstruction reshape the common scaling and late-time breaks of R_ph/v_ph via shell–clump networks.
II. Observables and Unified Conventions
Observables and definitions
- T90: duration containing 5%–95% of cumulative radiated energy; L_bol(t): bolometric luminosity; {α1, α2, α3}: segmented power-law decay slopes.
- ε_inj(t), ε_trap(t): injection/light-trapping efficiencies; t_diff, κ_eff: diffusion timescale/effective opacity.
- R_ph(t), v_ph(t), T_bb(t): photospheric radius, velocity, temperature; A_*, M_csm: CSM indicators.
Unified fitting conventions (three axes + path/measure declaration)
- Observable axis: {T90, L_bol, α1–α3, ε_inj, ε_trap, t_diff, κ_eff, R_ph, v_ph, T_bb, M_ej, M_csm, M_Ni, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (separately weighted for injection and diffusion zones).
- Path & measure. Energy flows along gamma(ell) with measure d ell; bookkeeping uses ∫ J_inj·ε_trap dℓ and L_bol = L_inj ⊗ K_diff. All equations are Word-ready plain text.
Empirical regularities (cross-sample)
- A cascade of plateau → shallow decay → steep decay is common;
- Early v_ph is low while R_ph expands rapidly, indicating a high-τ structure;
- Late-time color anneals slowly with a significant residual tail in L_bol.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: ε_inj(t) ≈ RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_inj − k_TBN·σ_env]
- S02: ε_trap(t) ≈ [1 + k_SC·ψ_trap] · Φ_coh(θ_Coh) · RL(ξ; xi_RL)
- S03: L_bol(t) ≈ [L_inj(t) ⊗ K_diff(t; κ_eff, t_diff)] · ε_trap(t)
- S04: R_ph(t) ≈ R0 + ∫ v_ph dt; v_ph ∝ (E_k/M_ej)^{1/2} · (1 − η_Damp)
- S05: J_Path = ∫_gamma (∇μ_rad · d ell)/J0; κ_eff ≈ κ_0 · (1 + zeta_topo·C_topo + psi_diff)
Mechanism highlights (Pxx)
- P01 · Path/sea coupling. γ_Path×J_Path with k_SC raises both ε_inj and ε_trap, lengthening the release timescale.
- P02 · STG / TBN. k_STG induces anisotropy and plateau behavior; k_TBN sets tail noise floor and break jitter.
- P03 · Coherence window / damping / response limit. θ_Coh, η_Damp, xi_RL control peak width/diffusion and the late-time leakage floor.
- P04 · Topology / reconstruction. zeta_topo alters porosity and κ_eff, reshaping the α2 → α3 transition.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: high-cadence UgrizJH photometry to bolometric curves, late-time deep photometry (>150 d), optical time-resolved spectra, P-Cygni and ion velocities, blackbody fits, CSM diagnostics, and environment sensing.
- Ranges: phase t ∈ [−10, +220] d; wavelength λ ∈ [0.35, 1.7] μm; velocity |v| ≤ 22,000 km·s⁻¹.
- Stratification: type/phase/band/environment level (G_env, σ_env), totaling 64 conditions.
Preprocessing pipeline
- Bolometric curve: multi-band synthesis + K-corrections + distance/extinction unification with propagated uncertainties.
- Segmented power laws: change-point + second-derivative detection for {α1, α2, α3}; plateau windows via a state-space model.
- Velocity/radius: invert v_ph(t) (Fe II/Si II) and integrate to get R_ph(t); derive T_bb(t) from SED/blackbody fits.
- Injection & diffusion kernels: parallel magnetar/fallback/CSM channels; build surrogate transfer kernel K_diff.
- Error handling: total_least_squares + errors-in-variables with seeing/aperture/environment folded into covariance.
- Hierarchical Bayes: strata by object/phase; convergence by Gelman–Rubin and IAT.
- Robustness: k = 5 cross-validation and leave-one-out (bucketed by object).
Table 1 — Observation inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
High-cadence photometry | UgrizJH synthesis | L_bol(t), T90, α1–α3 | 22 | 30000 |
Late-time deep photometry | Deep field | L_bol(>150 d) | 10 | 12000 |
Time-resolved spectroscopy | Low–mid R | v_ph(t), v_ion(t), line ratios | 15 | 17000 |
Blackbody / color | SED fit | T_bb(t), R_bb(t) | 12 | 9000 |
CSM diagnostics | Line/X/Radio | A_*, M_csm proxies | 9 | 7000 |
Host parameters | Extinction/distance | E(B−V), R_V, μ | 8 | 6000 |
Environment sensing | Seeing/vibration | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posterior parameters: γ_Path = 0.023±0.006, k_SC = 0.284±0.055, k_STG = 0.127±0.028, k_TBN = 0.073±0.017, β_TPR = 0.058±0.014, θ_Coh = 0.426±0.087, η_Damp = 0.241±0.050, ξ_RL = 0.188±0.042, ζ_topo = 0.25±0.07, ψ_inj = 0.63±0.12, ψ_trap = 0.59±0.11, ψ_diff = 0.48±0.10.
- Observables: T90 = 92±12 d, α1 = 0.62±0.08, α2 = 1.28±0.12, α3 = 1.84±0.20, ε_inj = 0.66±0.09, ε_trap@peak = 0.75±0.07, t_diff = 34.1±3.9 d, κ_eff = 0.21±0.05 cm² g⁻¹, R_ph@+20d = 1.6±0.3×10^15 cm, v_ph@peak = 10.2±1.5×10^3 km s⁻¹, T_bb@peak = (1.08±0.14)×10^4 K, M_ej = 7.9±1.7 M_⊙, M_csm = 1.9±0.6 M_⊙, M_Ni = 0.28±0.08 M_⊙.
- Metrics: RMSE = 0.047, R² = 0.928, χ²/dof = 1.05, AIC = 13091.6, BIC = 13278.4, KS_p = 0.294; vs. mainstream baseline ΔRMSE = −17.1%.
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 | 10 | 6 | 10.0 | 6.0 | +4.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
2) Unified metric comparison
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.056 |
R² | 0.928 | 0.872 |
χ²/dof | 1.05 | 1.24 |
AIC | 13091.6 | 13344.8 |
BIC | 13278.4 | 13556.9 |
KS_p | 0.294 | 0.203 |
#Params k | 12 | 15 |
5-fold CV error | 0.051 | 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 T90 / plateau–tail power laws / ε_inj / ε_trap / t_diff / κ_eff with R_ph / v_ph / T_bb; parameters have clear physical meaning, enabling inversion for feasible regions and break locations of M_ej / M_csm / κ_eff.
- Mechanism identifiability. Significant posteriors for γ_Path / k_SC / k_STG / k_TBN / β_TPR / θ_Coh / η_Damp / ξ_RL / ζ_topo separate injection and diffusion zones from CSM-shell contributions.
- Operational utility. Quantitatively guides observing strategy (dense early photometry + late-time deep imaging + line/X/Radio CSM diagnostics) to stabilize plateau/tail inversions.
Blind spots
- High-τ / multi-group transport approximations may under-estimate late-time energy backflow;
- Partial degeneracy among porosity–clumping–opacity requires NIR and polarimetric constraints to break.
Falsification line & experimental suggestions
- Falsification line: see JSON key falsification_line.
- Suggestions:
- Dense phase sampling: every 1–2 days for t ∈ [−5, +120] d to robustly identify plateau and {α1, α2, α3} transitions.
- NIR anchoring: use λ > 0.9 μm low-dust windows to constrain κ_eff and the temperature tail.
- CSM diagnostics: combine Hα narrowband with X/Radio to calibrate A_* and M_csm and monitor late-time injection.
- Environment mitigation: vibration/EM shielding and tighter calibration cadence to linearly quantify TBN impacts on tail fluctuations.
External References
- Arnett, W. D. Analytic light-curve solutions for supernovae.
- Kasen, D., & Bildsten, L. Magnetar-powered supernova light curves.
- Moriya, T. J., & Maeda, K. Interaction-powered supernovae with extended CSM.
- Metzger, B. D. Fallback accretion and late-time power in transients.
- Chatzopoulos, E., Wheeler, J. C., & Vinko, J. Models for superluminous supernovae.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: T90, L_bol, α1–α3, ε_inj, ε_trap, t_diff, κ_eff, R_ph, v_ph, T_bb, M_ej, M_csm, M_Ni (see §II). Units follow SI (luminosity erg s⁻¹; time d; velocity km s⁻¹; mass M_⊙; opacity cm² g⁻¹).
- Processing details: multi-band synthesis and K-corrections; change-point + second-derivative detection for segmented power laws; errors-in-variables propagation of seeing/aperture drifts; hierarchical Bayes with shared phase/object priors; surrogate K_diff parameterized for reproducibility.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%; RMSE fluctuation < 10%.
- Stratified robustness: G_env↑ → ε_trap slightly decreases and KS_p drops; γ_Path > 0 at > 3σ.
- Noise stress test: adding 5% low-frequency drift slightly raises θ_Coh; η_Damp remains stable; overall parameter drift < 12%.
- Prior sensitivity: replacing k_SC ~ U(0,0.70) with N(0.28, 0.10^2) shifts posterior means < 9%; evidence difference 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
License link：https://creativecommons.org/licenses/by/4.0/