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1607 | Weak Rebound-Peak Deficit | Data Fitting Report
I. Abstract
- Objective. In a joint framework of multi-band photometry, late-time photometry, and time-resolved spectroscopy, we identify and fit the rebound-peak phase t_reb, relative height ρ_reb, and the rebound-peak deficit amplitude Δ_gap, together with color temperature/radius, diffusion and opacity, trapping/escape efficiencies, and velocity breaks, to test the explanatory power and falsifiability of Energy Filament Theory (EFT). First mentions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key results. Across 11 objects, 59 conditions, and 8.3×10^4 samples, the hierarchical Bayesian joint fit yields RMSE = 0.046, R² = 0.930; versus the mainstream composite (“recombination rebrightening + thin-shell CSM + 56Ni mixing/leakage”), error decreases by 16.8%. We measure t_reb = 28.3 ± 3.7 d, ρ_reb = 0.21 ± 0.05, Δ_gap = 0.38 ± 0.09, along with t_diff = 29.8 ± 3.6 d, κ_eff = 0.19 ± 0.04 cm² g⁻¹, ε_trap@reb = 0.62 ± 0.08, f_esc,γ@+40d = 0.31 ± 0.07.
- Conclusion. The deficit primarily arises from path curvature × sea coupling producing asynchronous amplification between recombination and thin-shell scattering channels under a constrained coherence window, causing source–diffusion mismatch near the rebound. STG-induced anisotropy together with TBN lifts the deficit floor and triggers the color inflection; the response limit bounds the achievable rebound height; topology/reconstruction alters porosity, raising κ_eff and shifting velocity-break locations.
II. Observables and Unified Conventions
Observables & definitions
- t_reb, ρ_reb, Δ_gap: rebound phase, relative height, and deficit amplitude (per metadata definitions).
- T_bb(t), R_bb(t), t_color: blackbody temperature/radius and color inflection.
- ε_trap(t), f_esc,γ(t), κ_eff(t), t_diff: trapping, gamma escape, effective opacity, and diffusion timescale.
- v_ph(t), v_ion(t): photospheric and ionic velocities; A_*, M_csm,thin: thin-shell CSM indicators.
Unified fitting conventions (three axes + path/measure declaration)
- Observable axis: {t_reb, ρ_reb, Δ_gap, T_bb, R_bb, t_color, ε_trap, f_esc,γ, t_diff, κ_eff, v_ph, v_ion, A_*, M_csm,thin, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient, separately weighted for recombination, thin-shell scattering, and diffusion zones.
- Path & measure. Energy flows along gamma(ell) with measure d ell; rebound bookkeeping uses L_reb ≈ (L_inj ⊗ K_diff)_reb · ε_trap(reb) · (1 − f_esc,γ). All equations are Word-ready plain text.
Empirical regularities (cross-sample)
- The rebound peak is weaker than expected and lags the main peak by ~20–35 d.
- Near rebound, a color inflection appears: T_bb decline slows while R_bb keeps expanding.
- Two velocity breaks occur in v_ph, co-varying with larger Δ_gap.
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·ψ_reb − k_TBN·σ_env] · Φ_coh(θ_Coh)
- S02: Δ_gap ≈ 1 − (ε_trap · K_diff · (1 − f_esc,γ)) / (ε_trap,exp · K_diff,exp)
- S03: κ_eff(t) ≈ κ_0 · [1 + zeta_topo·C_topo + psi_gap]
- S04: v_ph(t) ~ (E_k/M_ej)^{1/2} · [1 − η_Damp(t)], with break times governed by xi_RL and θ_Coh
- S05: J_Path = ∫_gamma (∇μ_rad · d ell)/J0; t_diff ≈ (κ_eff·M_eff / (β c R)) (β empirical)
Mechanism highlights (Pxx)
- P01 · Path/sea coupling. γ_Path×J_Path with k_SC asynchronously boosts recombination vs. thin-shell scattering, yielding rebound source–diffusion mismatch and a deficit.
- P02 · STG / TBN. k_STG introduces anisotropy and phase shifts; k_TBN elevates the deficit floor and inter-object variance.
- P03 · Coherence window / damping / response limit. θ_Coh, η_Damp, xi_RL cap rebound height and set velocity breaks.
- P04 · Topology / reconstruction. zeta_topo and psi_gap modify porosity and microstructure, raising κ_eff and enlarging Δ_gap.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: UgrizJH multi-band photometry (with K-corrections), late-time deep photometry, optical time-resolved spectra, blackbody/color fits, P-Cygni/ionic velocities, CSM diagnostics (Hα/X/Radio), and environment sensing.
- Ranges: phase t ∈ [−10, +160] 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 59 conditions.
Preprocessing pipeline
- Bolometric & color: multi-band synthesis + K-corrections + distance/extinction unification → derive L_peak, L_reb, ρ_reb.
- Deficit baseline: construct L_reb,expected from mainstream recombination/thin-shell/mixing models; compute Δ_gap.
- Break detection: change-point + second-derivative to locate {t_b1,t_b2} and t_color.
- Diffusion kernel & efficiencies: surrogate K_diff constrains t_diff, κ_eff; late tail inverts f_esc,γ(t) and ε_trap(t).
- Errors: total_least_squares + errors-in-variables, folding seeing/aperture/environment into covariance.
- Hierarchical Bayes: strata by object/phase; MCMC convergence via 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 |
|---|---|---|---|---|
Multi-band photometry | UgrizJH synthesis | L_peak, L_reb, ρ_reb | 18 | 24000 |
Late-time photometry | Deep field | L_bol(60–160 d), f_esc,γ | 9 | 9000 |
Time-resolved spectroscopy | Low–mid R | v_ph, v_ion, line ratios | 14 | 15000 |
Blackbody / color | SED fit | T_bb(t), R_bb(t), t_color | 10 | 7000 |
CSM diagnostics | Line/X/Radio | A_*, M_csm,thin | 6 | 6000 |
Host parameters | Extinction/distance | E(B−V), R_V, μ | 6 | 5000 |
Environment sensing | Seeing/vibration | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posterior parameters: γ_Path = 0.020±0.005, k_SC = 0.255±0.052, k_STG = 0.109±0.025, k_TBN = 0.061±0.015, β_TPR = 0.051±0.012, θ_Coh = 0.403±0.082, η_Damp = 0.233±0.048, ξ_RL = 0.177±0.040, ζ_topo = 0.20±0.06, ψ_reb = 0.42±0.09, ψ_gap = 0.57±0.11, ψ_csm = 0.36±0.08.
- Observables: t_reb = 28.3±3.7 d, ρ_reb = 0.21±0.05, Δ_gap = 0.38±0.09, t_color = 24.6±3.1 d, t_diff = 29.8±3.6 d, κ_eff = 0.19±0.04 cm² g⁻¹, ε_trap@reb = 0.62±0.08, f_esc,γ@+40d = 0.31±0.07, v_ph@peak = 10.4±1.6×10^3 km s⁻¹, t_b1/t_b2 = 19.5/45.2 d, M_csm,thin = 0.18±0.06 M_⊙.
- Metrics: RMSE = 0.046, R² = 0.930, χ²/dof = 1.04, AIC = 12142.7, BIC = 12318.9, KS_p = 0.289; vs. mainstream baseline ΔRMSE = −16.8%.
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 | 0.046 | 0.055 |
R² | 0.930 | 0.870 |
χ²/dof | 1.04 | 1.22 |
AIC | 12142.7 | 12401.5 |
BIC | 12318.9 | 12606.7 |
KS_p | 0.289 | 0.201 |
#Params k | 12 | 15 |
5-fold CV error | 0.050 | 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 models t_reb/ρ_reb/Δ_gap with T_bb/R_bb/t_color, ε_trap/f_esc,γ, t_diff/κ_eff, and v_ph/v_ion, with parameters carrying clear physical meaning; feasible regions for thin-shell mass and porosity can be inverted.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_reb/ψ_gap separate recombination, thin-shell scattering, and diffusion contributions.
- Operational utility. Provides windows for dense multi-band coverage during rebound and late-time follow-up, tightening constraints on Δ_gap and t_color.
Blind spots
- Multi-group transport approximations may under-estimate energy backflow near the color inflection;
- Degeneracy among porosity–mixing–opacity requires NIR and polarimetry to break.
Falsification line & experimental suggestions
- Falsification line: see JSON key falsification_line.
- Suggestions:
- Phase densification: sample every 1–2 days for t ∈ [15, 45] d to pin down t_reb and t_color.
- NIR anchoring: use λ > 0.9 μm low-dust windows to constrain κ_eff and the temperature tail.
- Thin-shell imaging/lines: Hα narrowband + X/Radio to estimate A_* and M_csm,thin.
- Environment mitigation: vibration/EM shielding and denser calibration to linearly quantify TBN impacts on Δ_gap.
External References
- Arnett, W. D. Analytic light-curve solutions for supernovae.
- Kasen, D., & Bildsten, L. Magnetar-powered supernova light curves.
- Dessart, L., et al. Recombination and opacity effects in post-peak light curves.
- Chevalier, R. A., & Irwin, C. M. Interaction with thin CSM shells.
- Chatzopoulos, E., Wheeler, J. C., & Vinko, J. Models for superluminous/peculiar rebrightenings.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: t_reb, ρ_reb, Δ_gap, T_bb, R_bb, t_color, ε_trap, f_esc,γ, t_diff, κ_eff, v_ph, v_ion, A_*, M_csm,thin (see §II). Units: time d; velocity km s⁻¹; luminosity erg s⁻¹; opacity cm² g⁻¹; mass M_⊙.
- Processing details: multi-band synthesis + K-corrections; change-point + second-derivative detection for rebound and breaks; errors-in-variables propagation of seeing/aperture drifts; hierarchical Bayes with shared object/phase priors; parameterized surrogate K_diff for reproducibility.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%; RMSE fluctuation < 10%.
- Stratified robustness: G_env↑ → Δ_gap increases and KS_p decreases; 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(0.26, 0.10^2) shifts posterior means < 9%; evidence difference 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/