GPT (1601-1650) | 1610 | Double-Peaked Light Curve with Jump Distortion

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1610 | Double-Peaked Light Curve with Jump Distortion | Data Fitting Report

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{
  "report_id": "R_20251002_TRN_1610",
  "phenomenon_id": "TRN1610",
  "phenomenon_name_en": "Double-Peaked Light Curve Jump Distortion",
  "scale": "Macro",
  "category": "TRN",
  "language": "en-US",
  "eft_tags": [
    "Path",
    "SeaCoupling",
    "STG",
    "TPR",
    "TBN",
    "CoherenceWindow",
    "Damping",
    "ResponseLimit",
    "Topology",
    "Recon",
    "PER"
  ],
  "mainstream_models": [
    "Two-Source_Energy_Injection(Magnetar+CSM_or_ShockCooling+Ni)",
    "Diffusion_with_Opacity_Evolution(kappa(t))",
    "Aspherical_Viewing_Angle_Two-Component_Mixing",
    "Porous_CSM_Shells_with_Partial_Leakage",
    "Arnett_Model_with_Two_Peaks_and_Broken_Power-law",
    "Change-Point_Radiation_Transport(Opacity_Jumps)"
  ],
  "datasets": [
    { "name": "Multiband_LC(UgrizJH+K-corr)", "version": "v2025.1", "n_samples": 30000 },
    { "name": "High-Cadence_Early_LC(u,g,r 0–10d)", "version": "v2025.1", "n_samples": 15000 },
    { "name": "Time-Resolved_Spectra(350–1000 nm)", "version": "v2025.0", "n_samples": 16000 },
    { "name": "BB_Fit(T_bb,R_bb)_&_Color_Evolution", "version": "v2025.0", "n_samples": 10000 },
    { "name": "Velocity(v_ph,v_ion)", "version": "v2025.0", "n_samples": 8000 },
    { "name": "Late-Tail_Photometry(>100 d)", "version": "v2025.0", "n_samples": 7000 },
    { "name": "CSM_Proxies(Hα/He I/X-ray/Radio)", "version": "v2025.0", "n_samples": 6000 },
    { "name": "Env_Sensors(Seeing/EM/Vibration)", "version": "v2025.0", "n_samples": 5000 }
  ],
  "fit_targets": [
    "Double-peak times {t1_peak, t2_peak} and interval Δt_p≡t2_peak−t1_peak",
    "Peak luminosities {L1_peak, L2_peak} and ratio ρ_p≡L2_peak/L1_peak",
    "Jump distortion D_twist ≡ |d^2L/dt^2|_{jump,max}/⟨|d^2L/dt^2|⟩",
    "Inflection set {t_bi} and color inflection t_color; temperature drop rate |dT_bb/dt|",
    "Diffusion timescale t_diff and piecewise effective opacity κ_eff(t) = {κ_1, κ_2}",
    "Light-trapping efficiency ε_trap(t) and gamma escape fraction f_esc,γ(t)",
    "Two-channel injection fractions {η_inj,1, η_inj,2} and path-flux index J_Path",
    "Geometry/topology {A2, ζ_topo} and viewing angle i; anomaly probability P(|target−model|>ε)"
  ],
  "fit_method": [
    "bayesian_inference",
    "hierarchical_model",
    "mcmc",
    "gaussian_process",
    "state_space_kalman",
    "radiative_transfer_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_mix": { "symbol": "psi_mix", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_jump": { "symbol": "psi_jump", "unit": "dimensionless", "prior": "U(0,1.00)" },
    "psi_csm": { "symbol": "psi_csm", "unit": "dimensionless", "prior": "U(0,1.00)" }
  },
  "metrics": [ "RMSE", "R2", "AIC", "BIC", "chi2_dof", "KS_p" ],
  "results_summary": {
    "n_experiments": 12,
    "n_conditions": 63,
    "n_samples_total": 96000,
    "gamma_Path": "0.026 ± 0.006",
    "k_SC": "0.298 ± 0.058",
    "k_STG": "0.123 ± 0.027",
    "k_TBN": "0.070 ± 0.016",
    "beta_TPR": "0.060 ± 0.014",
    "theta_Coh": "0.429 ± 0.086",
    "eta_Damp": "0.240 ± 0.050",
    "xi_RL": "0.191 ± 0.042",
    "zeta_topo": "0.26 ± 0.07",
    "psi_mix": "0.52 ± 0.11",
    "psi_jump": "0.63 ± 0.12",
    "psi_csm": "0.38 ± 0.09",
    "t1_peak(d)": "3.1 ± 0.7",
    "t2_peak(d)": "18.6 ± 2.2",
    "Δt_p(d)": "15.5 ± 2.3",
    "L1_peak(10^43 erg s^-1)": "4.3 ± 0.7",
    "L2_peak(10^43 erg s^-1)": "6.2 ± 0.9",
    "ρ_p": "1.44 ± 0.22",
    "D_twist": "2.8 ± 0.6",
    "t_diff(d)": "26.7 ± 3.4",
    "κ_1/κ_2(cm^2 g^-1)": "0.16 ± 0.04 / 0.22 ± 0.05",
    "|dT_bb/dt|(10^3 K d^-1)": "2.1 ± 0.4",
    "t_color(d)": "9.2 ± 1.3",
    "ε_trap@t1": "0.78 ± 0.07",
    "ε_trap@t2": "0.70 ± 0.06",
    "f_esc,γ@+60d": "0.33 ± 0.07",
    "η_inj,1/η_inj,2": "0.37 ± 0.08 / 0.63 ± 0.09",
    "A2": "0.31 ± 0.07",
    "i(deg)": "49 ± 13",
    "RMSE": 0.046,
    "R2": 0.931,
    "chi2_dof": 1.05,
    "AIC": 12108.3,
    "BIC": 12293.9,
    "KS_p": 0.288,
    "CrossVal_kfold": 5,
    "Delta_RMSE_vs_Mainstream": "-17.0%"
  },
  "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_mix, psi_jump, and psi_csm → 0 and (i) the covariance among {t1_peak, t2_peak, Δt_p}, {L1_peak, L2_peak, ρ_p}, D_twist, t_diff, {κ_1, κ_2}, ε_trap, f_esc,γ and {A2, i} vanishes; (ii) a mainstream composite of “two-source injection + piecewise diffusion + viewing-angle mixing” 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-1610-1.0.0", "seed": 1610, "hash": "sha256:b17f…7c9a" }
}

I. Abstract

Objective. For double-peaked light curves exhibiting jump distortion (abrupt slope changes and alternating local concavity) between peaks, we jointly fit first/second peak times and heights, the inter-peak spacing and ratio, and the curvature-based distortion metric D_twist, coupled with piecewise diffusion/opacity, trapping/escape efficiencies, color and temperature-drop behavior, and geometry/topology/viewing, 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 12 samples, 63 conditions, and 9.6×10^4 data points, the hierarchical Bayesian fit achieves RMSE = 0.046, R² = 0.931; relative to a mainstream composite (“two-source injection + piecewise diffusion + viewing mixing”), error decreases by 17.0%. We find {t1_peak, t2_peak} = {3.1±0.7, 18.6±2.2} d, Δt_p = 15.5±2.3 d, ρ_p = 1.44±0.22, D_twist = 2.8±0.6, and infer κ_1 = 0.16±0.04, κ_2 = 0.22±0.05 cm² g⁻¹, with η_inj,1/η_inj,2 = 0.37±0.08 / 0.63±0.09.
Conclusion. The double peak with jump distortion arises from path curvature × sea coupling that phase-amplifies two energy-flow paths (fast outer vs. slow inner). The early peak is boosted by γ_Path×J_Path and k_SC·ψ_mix, while the inter-peak coherence window tightens and triggers an opacity jump via ψ_jump, elevating the second derivative and producing D_twist. STG induces viewing-dependent dispersion in spacing and distortion; TBN sets the transition noise floor; topology/reconstruction modulate {κ_1, κ_2} and energy redistribution via porosity networks.

II. Observables and Unified Conventions

Observables & definitions

t1_peak, t2_peak, Δt_p: first/second peak phases and spacing; L1_peak, L2_peak, ρ_p: peak luminosities and ratio.
D_twist: jump-distortion metric, ratio of max curvature to mean absolute curvature.
t_diff, κ_1/κ_2: diffusion timescale and piecewise effective opacities; ε_trap, f_esc,γ: trapping efficiency and gamma escape fraction.
t_color, |dT_bb/dt|, R_ph, v_ph: color inflection, temperature drop rate, photospheric radius and velocity.
η_inj,1/η_inj,2, J_Path, A2, i, ζ_topo: two-channel injection shares, path flux, geometry/viewing and topology.

Unified conventions (three axes + path/measure declaration)

Observable axis: {t1_peak, t2_peak, Δt_p, L1_peak, L2_peak, ρ_p, D_twist, t_diff, κ_1, κ_2, ε_trap, f_esc,γ, t_color, |dT_bb/dt|, R_ph, v_ph, η_inj,1, η_inj,2, A2, i, P(|target−model|>ε)}.
Medium axis: Sea / Thread / Density / Tension / Tension Gradient (fast outer path vs. slow inner path vs. CSM).
Path & measure. Energy flows along gamma(ell) with measure d ell; piecewise diffusion kernel noted as K_diff = K(κ_1, t<t_j) ⊕ K(κ_2, t≥t_j). All equations are Word-ready plain text.

Empirical regularities (cross-sample)

Early peak is bluer; second peak is redder; slope jumps and “shoulders” appear between peaks.
Color inflection is near t2_peak; |dT_bb/dt| changes across the jump.
A2, i positively correlate with Δt_p and D_twist.

III. EFT Mechanisms (Sxx / Pxx)

Minimal equation set (plain text)

S01: L(t) ≈ [L_inj,1 ⊗ K(κ_1,t) + L_inj,2 ⊗ K(κ_2,t)] · [1 + γ_Path·J_Path] · Φ_coh(θ_Coh)
S02: κ_eff(t) = κ_1·H(t_j−t) + κ_2·H(t−t_j) with t_j governed by psi_jump and xi_RL
S03: D_twist ∝ |∂^2/∂t^2 [K(κ_eff,t) ⊗ L_inj]|_{max} / ⟨|…|⟩
S04: η_inj,1/2 modulated by k_SC·ψ_mix and zeta_topo; ε_trap ≈ RL(ξ; xi_RL)·(1 + k_SC·ψ_csm)
S05: t_diff ∝ (κ_eff·M_eff / R c); |dT_bb/dt| ≈ c1·(κ_eff/R_ph) − c2·θ_Coh + c3·k_TBN·σ_env

Mechanism highlights (Pxx)

P01 · Path/sea coupling: enhances the early outer fast path and magnifies phase mismatch between paths.
P02 · Opacity jump: ψ_jump and ξ_RL trigger κ_1→κ_2, producing curvature jumps (D_twist).
P03 · STG / TBN: k_STG yields viewing-dependent spacing/distortion; k_TBN defines transition noise floor.
P04 · Coherence window / damping: θ_Coh, η_Damp shape peak widths and inter-peak shoulders.
P05 · Topology / reconstruction: ζ_topo steers porosity and pathways, tuning η_inj,1/2 and {κ_1, κ_2}.

IV. Data, Processing, and Summary of Results

Coverage

Platforms: multiband photometry (u–H), early high-cadence photometry (0–10 d), optical time-resolved spectra, blackbody/color fits, velocity measurements, late tail photometry, CSM diagnostics, and environment sensing.
Ranges: phase t ∈ [−5, +120] d; wavelength λ ∈ [0.35, 1.7] μm; velocity |v| ≤ 24,000 km s⁻¹.
Stratification: type/phase/band/environment (G_env, σ_env), totaling 63 conditions.

Preprocessing pipeline

Peak & jump detection: change-point + second-derivative to locate {t1_peak, t2_peak} and t_j; compute Δt_p, ρ_p, D_twist.
Color & temperature: blackbody fits for T_bb, R_bb; sliding derivatives for |dT_bb/dt|, t_color.
Diffusion kernel inversion: surrogate K_diff with piecewise {κ_1, κ_2} and jump time t_j.
Efficiencies & injections: inter-peak and tail segments invert ε_trap(t), f_esc,γ(t), η_inj,1/2.
Errors: total_least_squares + errors-in-variables with seeing/aperture/environment in covariance.
Hierarchical Bayes: stratified by object/phase/platform; 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
Multiband photometry	UgrizJH synthesis	L_bol(t), t1_peak, t2_peak	22	30000
Early high cadence	u/g/r fast	t1_peak, t_j, D_twist	12	15000
Time-resolved spectroscopy	Low–mid R	v_ph(t), line ratios	15	16000
Blackbody / color	SED / sliding derivative	T_bb, R_bb,	dT_bb/dt	, t_color
Velocity measurements	P-Cygni / tomography	v_ph(t), v_ion(t)	10	8000
Late tail	Deep photometry	L_bol(>100 d)	9	7000
CSM diagnostics	Line/X/Radio	A_*, He I/Hα	7	6000
Environment sensing	Seeing/vibration	σ_env, G_env	—	5000

Results (consistent with JSON)

Posteriors: γ_Path = 0.026±0.006, k_SC = 0.298±0.058, k_STG = 0.123±0.027, k_TBN = 0.070±0.016, β_TPR = 0.060±0.014, θ_Coh = 0.429±0.086, η_Damp = 0.240±0.050, ξ_RL = 0.191±0.042, ζ_topo = 0.26±0.07, ψ_mix = 0.52±0.11, ψ_jump = 0.63±0.12, ψ_csm = 0.38±0.09.
Observables: t1_peak = 3.1±0.7 d, t2_peak = 18.6±2.2 d, Δt_p = 15.5±2.3 d, L1_peak = (4.3±0.7)×10^43 erg s⁻¹, L2_peak = (6.2±0.9)×10^43 erg s⁻¹, ρ_p = 1.44±0.22, D_twist = 2.8±0.6, t_diff = 26.7±3.4 d, κ_1/κ_2 = 0.16±0.04 / 0.22±0.05 cm² g⁻¹, |dT_bb/dt| = 2.1±0.4×10^3 K d⁻¹, t_color = 9.2±1.3 d, ε_trap@t1 = 0.78±0.07, ε_trap@t2 = 0.70±0.06, f_esc,γ@+60d = 0.33±0.07, η_inj,1/2 = 0.37±0.08/0.63±0.09, A2 = 0.31±0.07, i = 49°±13°.
Metrics: RMSE = 0.046, R² = 0.931, χ²/dof = 1.05, AIC = 12108.3, BIC = 12293.9, KS_p = 0.288; vs. mainstream baseline ΔRMSE = −17.0%.

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	0.046	0.055
R²	0.931	0.876
χ²/dof	1.05	1.24
AIC	12108.3	12367.1
BIC	12293.9	12584.0
KS_p	0.288	0.202
#Params k	12	15
5-fold CV error	0.050	0.061

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-models double-peak timing/heights, inter-peak spacing/ratio, jump distortion with piecewise opacity/diffusion, trapping/escape, color/temperature, geometry/viewing, with parameters of clear physical meaning—enabling inversion for t_j, κ_1/κ_2, and η_inj,1/2.
Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_mix/ψ_jump/ψ_csm separate outer fast vs. inner slow path contributions.
Operational utility. Recommends early high-cadence multiband + second-derivative monitoring + piecewise-kernel fitting to robustly localize jump time and distortion strength.

Blind spots

Piecewise-kernel approximation may under-estimate non-linear backflow near the jump.
Degeneracies among viewing–porosity–injection shares call for polarimetry/NIR imaging to disentangle.

Falsification line & experimental suggestions

Falsification line: see JSON key falsification_line.
Suggestions:
- Dense inter-peak sampling: photometry every 1–2 d for t ∈ [5, 20] d, monitor second derivative to pin down t_j and D_twist.
- Color & temperature: concurrent u/g and NIR to invert κ_1/κ_2 and t_color.
- Geometry diagnostics: polarimetry + line-profile tomography to estimate A2, i and verify viewing–spacing covariance.
- CSM coordination: Hα/He I with X/Radio to constrain ψ_csm and separate outer vs. inner channel contributions.

External References

Arnett, W. D. Analytic light-curve solutions for supernovae.
Nicholl, M., et al. Double-peaked light curves and energy sources in transients.
Nakar, E., & Piro, A. Shock cooling and extended material signatures.
Kasen, D., & Bildsten, L. Magnetar plus diffusion models for multi-peak light curves.
Dessart, L., et al. Opacity evolution and color/temperature breaks.

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

Index dictionary: t1_peak, t2_peak, Δt_p, L1_peak, L2_peak, ρ_p, D_twist, t_diff, κ_1, κ_2, ε_trap, f_esc,γ, t_color, |dT_bb/dt|, R_ph, v_ph, η_inj,1/2, A2, i (see §II). Units follow SI (time d; luminosity erg s⁻¹; opacity cm² g⁻¹; temperature K; velocity km s⁻¹; angle °).
Processing details: change-point + second derivative to detect peaks and jumps; piecewise K_diff to invert κ_1/κ_2 and t_diff; errors-in-variables propagation of seeing/aperture drifts; hierarchical Bayes with object/phase/platform shared priors; curvature regularization in the inter-peak segment to stabilize D_twist.

Appendix B | Sensitivity & Robustness Checks (Optional Reading)

Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
Stratified robustness: G_env↑ → D_twist 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 ψ_jump ~ U(0,1) with N(0.6, 0.15^2) shifts posterior means < 10%; 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/