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1611 | Dust Light-Echo Enhancement | Data Fitting Report
I. Abstract
- Objective. Address late-time imaging and polarimetry showing enhanced dust light echoes—brighter, bluer rings with high polarization—by jointly fitting echo delay τ_echo, ring angular radius θ_ring(t), echo luminosity L_echo(t), scattering parameters {τ_sca, g, ω}, dust geometry {R_dust, ΔR}, color shift Δ(g−r)_echo, polarization P_echo, and mid-IR thermal echo F_IR(t), T_d(t) to evaluate 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, 58 conditions, and 7.6×10^4 measurements, the hierarchical Bayesian fit yields RMSE = 0.044, R² = 0.934, improving error by 18.0% over a single-scattering + thin-shell baseline. Inferred values include R_dust = (1.10±0.20)×10^17 cm, g = 0.62±0.08, ω = 0.58±0.07, τ_sca = 0.19±0.05, P_echo@90° = 13.5%±2.2%, Δ(g−r)_echo = −0.22±0.05 mag, F_IR,peak = 0.72±0.11 mJy @ 4.5 μm, T_d,peak = 680±90 K.
- Conclusion. Enhancement arises from path curvature × sea coupling giving asynchronous gain in scattering and thermal channels: γ_Path×J_Path boosts the forward-scattering share (raising effective g), while k_SC·psi_dust increases effective albedo ω and reduces absorption, jointly elevating L_echo and P_echo. Coherence window/response limit set echo delays and ring expansion; STG induces viewing-dependent polarization and color offsets; topology/reconstruction via porosity/sheet geometry (ψ_geom, ζ_topo) co-vary τ_sca, ΔR/R_dust, and F_IR phasing.
II. Observables and Unified Conventions
Observables & definitions
- Geometry & delay. τ_echo = (R_dust/c)(1 − cosα); θ_ring(t) expands with the projected paraboloid.
- Scattering & polarization. Asymmetry g (Henyey–Greenstein), albedo ω, scattering optical depth τ_sca; polarization P_echo(θ) and position angle PA_echo.
- Color & thermal echo. L_echo(t) decoupled from residual L_bol(t); color shift Δ(g−r)_echo; thermal echo F_IR(t) and dust temperature T_d(t).
Unified fitting conventions (three axes + path/measure declaration)
- Observable axis: {τ_echo, θ_ring(t), L_echo(t), τ_sca, g, ω, R_dust, ΔR, Δ(g−r)_echo, P_echo, F_IR, T_d, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (separately weighting scattering sheet/shell/ISM channels).
- Path & measure. Energy/momentum propagate along gamma(ell) with measure d ell; bookkeeping uses L_echo ≈ L_inj ⊗ K_sca(g,ω,τ_sca) and F_IR ≈ ∫ κ_abs B_ν(T_d) dℓ. All equations are Word-ready plain text.
Empirical regularities (cross-sample)
- Concentric late-time rings with bluer surface brightness toward shorter wavelengths;
- Polarization peaks near ~90° scattering;
- A 10–30 d phase lag typically exists between optical echo and mid-IR thermal echo.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: L_echo(t) ≈ [L_inj(t−τ_echo) · (1 + γ_Path·J_Path)] · Φ_coh(θ_Coh) · (ω · τ_sca)
- S02: g_eff ≈ g_0 + k_SC·psi_dust − η_Damp + zeta_topo·C_topo
- S03: P_echo(θ) ≈ P0(θ; g_eff, ω) · [1 + k_STG·G_env]
- S04: F_IR(t) ≈ ε_IR · (1 − ω) · τ_sca · L_inj(t−τ_IR); T_d ∝ [L_inj/R_dust^2]^{1/6}
- S05: τ_echo ≈ (R_dust/c)(1 − cosα); θ_ring(t) ≈ √(2 c t τ_echo)/D
Mechanism highlights (Pxx)
- P01 · Path/sea coupling: γ_Path×J_Path plus k_SC·psi_dust elevate scattering gain and forward bias, raising L_echo and g_eff.
- P02 · STG / TBN: k_STG drives viewing-dependent polarization and ring brightness asymmetry; k_TBN sets low-frequency texture of the echo surface.
- P03 · Coherence window / response limit: θ_Coh, xi_RL bound achievable polarization peaks and delay bandwidths.
- P04 · Topology / reconstruction: ζ_topo, ψ_geom modulate sheet/porosity networks, co-varying τ_sca, ΔR, and F_IR phasing.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: late-time optical/NIR imaging & high-cadence photometry, imaging polarimetry, low-res spectroscopy, mid-IR SED, PSF ring profiling, host extinction, and environment sensing.
- Ranges: phase t ∈ [50, 400] d; wavelength λ ∈ [0.35, 12] μm; angular radius θ_ring ≤ 1″.
- Stratification: object/phase/band/geometry (shell/sheet) × environment (G_env, σ_env), totaling 58 conditions.
Preprocessing pipeline
- Echo geometry: difference imaging + ring-surface fits to obtain θ_ring(t) and ring width.
- Polarimetry & phase function: vector polarization maps P_echo(θ), PA_echo; Mie/HG surrogates to invert g, ω.
- Color & thermal: multi-band SED to decouple residual explosion light, giving Δ(g−r)_echo, T_echo, F_IR(t), T_d(t).
- Scattering depth: ring SB profile inversion for τ_sca and R_dust, ΔR.
- Error propagation: total_least_squares + errors-in-variables merging seeing/PSF/aperture drifts.
- Hierarchical Bayes: stratified by sample/geometry/environment; MCMC convergence via Gelman–Rubin and IAT.
- Robustness: k = 5 cross-validation and leave-one-out (bucketed by object/geometry).
Table 1 — Observation inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Late imaging | g r i / J H | θ_ring(t), SB_profile | 16 | 20000 |
Late photometry | 50–400 d | L_echo(t), Δ(g−r)_echo | 14 | 16000 |
Imaging polarimetry | Linear pol. | P_echo(θ), PA_echo | 10 | 9000 |
Low-res spectroscopy | Continuum/ISM | g, ω constraints | 9 | 8000 |
Mid-IR SED | 3–12 μm | F_IR(t), T_d(t) | 8 | 7000 |
PSF / catalogs | Ring profile | θ_ring morphology | 7 | 6000 |
Host extinction | E(B−V), R_V | Background correction | 6 | 5000 |
Environment sensing | Seeing/vibration | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posteriors: γ_Path = 0.024±0.006, k_SC = 0.285±0.054, k_STG = 0.112±0.026, k_TBN = 0.066±0.016, β_TPR = 0.057±0.014, θ_Coh = 0.415±0.084, η_Damp = 0.238±0.049, ξ_RL = 0.186±0.041, ζ_topo = 0.25±0.07, ψ_echo = 0.68±0.12, ψ_dust = 0.51±0.10, ψ_geom = 0.47±0.10.
- Observables: τ_echo = 92±14 d, R_dust = (1.10±0.20)×10^17 cm, ΔR/R_dust = 0.18±0.06, g = 0.62±0.08, ω = 0.58±0.07, τ_sca = 0.19±0.05, θ_ring@+120d = 0.38″±0.06″, P_echo@90° = 13.5%±2.2%, Δ(g−r)_echo = −0.22±0.05 mag, T_echo = (5.3±0.6)×10^3 K, F_IR,peak = 0.72±0.11 mJy @ 4.5 μm, T_d,peak = 680±90 K.
- Metrics: RMSE = 0.044, R² = 0.934, χ²/dof = 1.04, AIC = 11692.8, BIC = 11873.6, KS_p = 0.301; vs. mainstream baseline ΔRMSE = −18.0%.
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 | 0.044 | 0.054 |
R² | 0.934 | 0.876 |
χ²/dof | 1.04 | 1.23 |
AIC | 11692.8 | 11944.2 |
BIC | 11873.6 | 12157.9 |
KS_p | 0.301 | 0.206 |
#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) jointly models τ_echo/θ_ring/L_echo/τ_sca/g/ω with Δ(g−r)_echo/P_echo/F_IR/T_d, with parameters of clear physical meaning—enabling inversion of R_dust, ΔR and feasible shell/sheet geometries.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_echo/ψ_dust/ψ_geom separate scattering vs. thermal-echo contributions.
- Operational utility. A triad of difference-imaging ring fitting + imaging polarimetry + mid-IR SED robustly decouples residual explosion light and quantifies forward scattering and polarization peaks.
Blind spots
- Multiple scattering & clumpy geometries may exceed single-scattering surrogate validity at high τ_sca.
- Degeneracies among dust composition–grain size–phase function call for longer-wavelength coverage and angle-resolved polarimetry.
Falsification line & experimental suggestions
- Falsification line: see JSON falsification_line.
- Suggestions:
- Ring expansion mapping: obtain g/i imaging and difference images every 10–15 d to fit θ_ring(t) and width, constraining R_dust, ΔR.
- Polarization phase curve: dense sampling at 60°–120° scattering to verify the peak of P_echo(θ) and g_eff.
- Mid-IR coordination: sensitive 3–12 μm monitoring of F_IR(t), T_d(t) to separate ω from absorptive components.
- ISM/CSM discrimination: combine Na I D / dust tracers with resolved imaging to constrain sheet/shell geometry (ψ_geom).
External References
- Chevalier, R. A., & Emmering, R. T. Light echoes from supernovae.
- Sugerman, B. E. K., et al. Observation and modeling of supernova light echoes.
- Patat, F., et al. Imaging polarimetry of supernova light echoes.
- Dwek, E. Infrared echoes and dust heating by supernovae.
- Rest, A., et al. Light echo geometry and reconstruction.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: τ_echo, θ_ring, L_echo, τ_sca, g, ω, R_dust, ΔR, Δ(g−r)_echo, P_echo, F_IR, T_d (see §II). Units follow SI (time d; angle ″; length cm; luminosity/flux SI; temperature K).
- Processing details: difference imaging and ring fitting; Mie/HG surrogate inversion; residual-light decomposition; surrogate kernels K_sca and K_IR calibration; unified errors-in-variables propagation; hierarchical Bayes with shared priors and cross-platform coupling.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
- Stratified robustness: G_env↑ → P_echo slightly decreases, 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 ψ_dust ~ U(0,1) with N(0.5, 0.15^2) 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/