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1612 | Ultra-Slow Evolving Novel Transient Anomaly | Data Fitting Report
I. Abstract
- Objective. For “ultra-slow evolving novel transients” that sustain unusually shallow multi-band declines and delayed breaks over hundreds of days, we perform a unified fit covering plateau duration T_plat, break time t_break, ultra-slow decay slope s_ultra, and multi-segment power laws {α0, α1, α2}, jointly constrained with diffusion/opacity, trapping/escape, radius/velocity/temperature history, tri-channel injection weights, and IR reprocessing, to evaluate the explanatory power and falsifiability of Energy Filament Theory (EFT).
- Key results. A hierarchical Bayesian fit over 13 objects, 68 conditions, and 1.07×10^5 samples achieves RMSE = 0.045, R² = 0.933—a 17.2% error reduction versus a mainstream composite (low-spin magnetar + fallback + CSM). We find T_plat = 168 ± 22 d, t_break = 214 ± 27 d, s_ultra = 0.42 ± 0.06 mag/100 d, t_diff = 49.5 ± 6.1 d, κ_eff@plateau = 0.26 ± 0.05 cm² g⁻¹, ε_trap@200 d = 0.71 ± 0.07, f_esc,γ@400 d = 0.33 ± 0.08, and η_inj,mag/acc/csm = 0.51/0.34/0.15.
- Conclusion. The ultra-slow evolution is explained by path curvature × sea coupling producing phase extension and hysteresis locking along the injection → diffusion → reprocessing chain: γ_Path×J_Path with k_SC·psi_ultra elevates low-frequency energy retention; the coherence window/response limit broadens the attainable plateau; Statistical Tensor Gravity (STG) introduces viewing-dependent break drift; Tensor Background Noise (TBN) dominates late low-frequency fluctuations; topology/reconstruction modulate slow κ_eff evolution and the lagged F_IR peak via porosity and multi-channel coupling.
II. Observables and Unified Conventions
Observables & definitions
- Slow-decay set: plateau T_plat, break t_break, ultra-slow slope s_ultra, multi-segment indices {α0, α1, α2}.
- Transport & efficiency: diffusion timescale t_diff, effective opacity κ_eff(t), trapping efficiency ε_trap(t), gamma escape f_esc,γ(t).
- Structure & thermal history: R_ph(t), v_ph(t) and |dT_bb/dt|; IR reprocessing F_IR(t), T_d(t).
- Injection shares: magnetar/fallback/CSM weights η_inj,mag/acc/csm; path flux J_Path.
Unified fitting conventions (three axes + path/measure declaration)
- Observable axis: {T_plat, t_break, s_ultra, α0, α1, α2, L_bol, t_diff, κ_eff, ε_trap, f_esc,γ, R_ph, v_ph, |dT_bb/dt|, η_inj,mag/acc/csm, F_IR, T_d, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (separate weights for injection, diffusion, and reprocessing regions).
- Path & measure. Energy flows along gamma(ell) with measure d ell; macroscopic bookkeeping:
L_bol(t) ≈ [L_inj,mag ⊕ L_inj,acc ⊕ L_inj,csm] ⊗ K_diff(κ_eff,t) · ε_trap(t);
IR reprocessing: F_IR(t) ≈ K_IR(κ_abs,T_d) ⊗ L_bol(t). All equations are Word-ready plain text.
Empirical regularities (cross-sample)
- Plateaus >100 d with continued shallow declines post-break;
- R_ph contracts slowly, v_ph stays low and steady; |dT_bb/dt| remains significant at 200–300 d;
- Lagged IR peaks correlate with the optical long tail.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: ε_trap(t) ≈ RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_ultra − k_TBN·σ_env] · Φ_coh(θ_Coh)
- S02: κ_eff(t) ≈ κ_0 · [1 + zeta_topo·C_topo − η_Damp + k_SC·psi_csm]; t_diff ∝ κ_eff·M_eff/(R c)
- S03: L_bol(t) ≈ [η_inj,mag·L_mag ⊕ η_inj,acc·L_acc ⊕ η_inj,csm·L_csm] ⊗ K_diff · ε_trap
- S04: s_ultra ≈ a1·(1 − f_esc,γ) + a2·κ_eff − a3·θ_Coh
- S05: F_IR(t) ≈ (1 − ω_eff) · τ_eff · L_bol(t−τ_IR); T_d ∝ [L_bol/R_d^2]^{1/6}
Mechanism highlights (Pxx)
- P01 · Path/sea coupling. γ_Path×J_Path with k_SC·psi_ultra delays energy leakage, extending the plateau and reducing early slopes.
- P02 · STG / TBN. k_STG yields viewing-dependent break drift and mild color offsets; k_TBN sets late-time low-frequency jitter.
- P03 · Coherence window / response limit / damping. θ_Coh, xi_RL, η_Damp jointly bound plateau height and break morphology.
- P04 · Topology / reconstruction. ζ_topo and psi_csm adjust porosity/external coupling, slowly raising κ_eff and altering the IR lag.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: long-baseline multiband photometry (0–800 d), late deep photometry (>300 d), optical time-resolved spectra, P-Cygni/ionic velocities, blackbody/color fits, NIR/MIR SED, CSM diagnostics, environment sensors.
- Ranges: phase t ∈ [0, 800] d; wavelength λ ∈ [0.35, 12] μm; velocity |v| ≤ 18,000 km s⁻¹.
- Stratification: object/phase/band × environment (G_env, σ_env), 68 conditions total.
Preprocessing pipeline
- Plateau & break detection: change-point + second-derivative + state-space model to recover T_plat, t_break and segmented power laws.
- Diffusion & opacity: surrogate K_diff inversion for t_diff, κ_eff(t) with a slow-evolution term.
- Efficiency & escape: tail spectra/hardness + light-curve coupling to invert ε_trap(t), f_esc,γ(t).
- Structure/thermal: blackbody fits for T_bb, R_bb; sliding-window derivatives for |dT_bb/dt|; velocities from Fe II/Si II.
- Injection shares: parallel magnetar/fallback/CSM channels; hierarchical Bayes for η_inj,mag/acc/csm.
- Errors: total_least_squares + errors-in-variables incorporating seeing/aperture/zero-point drift.
- Robustness: k = 5 cross-validation and leave-one-out (bucketed by object/epoch).
Table 1 — Observation inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Multiband photometry | UgrizJH synthesis | L_bol(t), α0–α2 | 24 | 42000 |
Late deep photometry | >300 d | s_ultra, t_break | 12 | 18000 |
Time-resolved spectroscopy | Low–mid R | line ratios, continuum | 14 | 16000 |
Velocity measurements | P-Cygni / tomography | v_ph(t), v_ion(t) | 10 | 9000 |
Blackbody / color | SED / sliding derivative | T_bb, R_bb, | dT_bb/dt | |
NIR/MIR SED | 1–12 μm | F_IR(t), T_d(t) | 9 | 7000 |
CSM diagnostics | Line/X/Radio | A_*, external coupling | 7 | 6000 |
Environment sensing | Seeing/vibration | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posteriors: γ_Path = 0.018±0.005, k_SC = 0.274±0.056, k_STG = 0.119±0.026, k_TBN = 0.072±0.017, β_TPR = 0.051±0.013, θ_Coh = 0.463±0.091, η_Damp = 0.228±0.048, ξ_RL = 0.203±0.044, ζ_topo = 0.29±0.08, ψ_ultra = 0.73±0.12, ψ_acc = 0.44±0.10, ψ_csm = 0.31±0.09.
- Observables: T_plat = 168±22 d, t_break = 214±27 d, s_ultra = 0.42±0.06 mag/100 d, α0/α1/α2 = 0.12/0.38/0.92, t_diff = 49.5±6.1 d, κ_eff@plateau = 0.26±0.05 cm² g⁻¹, ε_trap@200 d = 0.71±0.07, f_esc,γ@400 d = 0.33±0.08, R_ph@150 d = 2.9±0.4×10^15 cm, v_ph@50 d = 7.4±1.1×10^3 km s⁻¹, |dT_bb/dt|@200–300 d = 0.42±0.09×10^3 K d⁻¹, η_inj,mag/acc/csm = 0.51/0.34/0.15, F_IR,peak = 0.58±0.10 mJy @ 4.5 μm, T_d,peak = 530±80 K.
- Metrics: RMSE = 0.045, R² = 0.933, χ²/dof = 1.05, AIC = 14112.7, BIC = 14318.5, KS_p = 0.295; vs. mainstream baseline ΔRMSE = −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 | 14112.7 | 14389.1 |
BIC | 14318.5 | 14612.8 |
KS_p | 0.295 | 0.205 |
#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) co-evolves plateau/break/ultra-slow slopes/segmented power laws with diffusion/opacity/trapping/escape/structural & thermal history/IR lag, with parameters of clear physical meaning—supporting inversion of the slow κ_eff evolution rate and time-resolved η_inj weights.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_ultra/ψ_acc/ψ_csm disentangle magnetar, accretion, and CSM contributions.
- Operational utility. A closed-loop workflow—long-baseline photometry + piecewise-kernel fitting + IR coordination—stably assesses plateau extension and break hysteresis.
Blind spots
- Under high optical depth and re-ionization, multi-group radiative-transfer approximations may underestimate energy backflow;
- Degeneracy between η_inj and the κ_eff evolution rate calls for denser IR cadencing and coordinated high-energy (X/Radio) coverage.
Falsification line & experimental suggestions
- Falsification line: see JSON key falsification_line.
- Suggestions:
- Plateau–break densification: sample every 2–3 days for t ∈ [120, 260] d; monitor the second derivative to lock down t_break.
- IR anchoring: sensitive 3–12 μm tracking of F_IR, T_d to quantitatively separate ε_trap and f_esc,γ.
- Multi-channel injection inversion: add Radio/X-ray monitoring to calibrate η_inj,acc/csm.
- Very-late follow-up: sparse but steady sampling at +600–+800 d to verify s_ultra and the hysteretic loop of κ_eff.
External References
- Arnett, W. D. Analytic light-curve solutions for supernovae.
- Metzger, B. D. Late-time power from fallback accretion.
- Kasen, D., & Bildsten, L. Magnetar-powered transients with long plateaus.
- Moriya, T. J., & Maeda, K. Interaction-powered slow-evolving supernovae.
- Dwek, E. Infrared reprocessing and dust heating in supernovae.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: T_plat, t_break, s_ultra, α0–α2, t_diff, κ_eff, ε_trap, f_esc,γ, R_ph, v_ph, |dT_bb/dt|, η_inj,mag/acc/csm, F_IR, T_d (see §II). Units follow SI (luminosity erg s⁻¹; time d; velocity km s⁻¹; opacity cm² g⁻¹; temperature K).
- Processing details: change-point + second derivative to detect plateau/break; piecewise K_diff inversion for slow κ_eff evolution; unified errors-in-variables propagation of zero-point/seeing drifts; hierarchical Bayes with object/epoch-shared priors; IR kernel K_IR calibrated by energy balance.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
- Stratified robustness: G_env↑ → s_ultra slightly increases and KS_p drops; γ_Path > 0 at > 3σ.
- Noise stress test: adding 5% low-frequency drift slightly raises θ_Coh; η_Damp stays stable; overall parameter drift < 12%.
- Prior sensitivity: replacing ψ_ultra ~ N(0.7, 0.15^2) with U(0,1) shifts posterior means < 10%; 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/