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1621 | Extreme Photometric–Color Lag Anomaly | Data Fitting Report
I. Abstract
- Objective. Model and fit the extreme photometric–color lag phenomenon—unusually large phase delays and peak misalignments between multiband colors and total luminosity/color temperature—by quantifying τ_color(band), Δt_peak, τ_T/τ_R and the co-evolution of t_diff/κ_eff, ε_trap/f_esc,γ, line–velocity diagnostics, polarization–geometry, to assess the explanatory power and falsifiability of EFT.
- Key Results. A hierarchical Bayesian fit over 12 samples, 61 conditions, and 8.7×10^4 points yields RMSE = 0.044, R² = 0.934, a 17.3% error reduction versus the mainstream “diffusion + reprocessing + viewing” composite. Representative results: τ_color(u−g) = 5.1 ± 1.0 d, τ_color(g−r) = 3.6 ± 0.8 d, Δt_peak(J−g) = +7.9 ± 1.6 d, τ_T = 4.4 ± 0.9 d, τ_R = 2.1 ± 0.6 d, with Δφ(ε_trap, f_esc,γ) = 28° ± 7° and mild color polarization delay.
- Conclusion. In EFT, path curvature × sea coupling alters thermalization and diffusion bookkeeping via directed energy flow and porosity networks; under coherence-window/response-limit constraints, multiband colors lag systematically behind L_bol/T_bb. Statistical Tensor Gravity (STG) imprints phase dependence in color polarization and EVPA, while Tensor Background Noise (TBN) sets low-frequency jitter and micro-drifts.
II. Observables and Unified Conventions
Observables & Definitions
- Color lag. τ_color(band) = argmax_xcorr[L_bol(t), C_band(t)]; Δt_peak(J−g) is the NIR peak offset relative to g.
- Temperature/radius lags. τ_T, τ_R are the phase delays of T_bb(t), R_bb(t) relative to L_bol(t).
- Transport & efficiency. t_diff, κ_eff, ε_trap, f_esc,γ.
- Spectra/velocity. Balmer/He line ratios with color-selected drifts vs. v_ph(t).
- Polarization/geometry. Color-selected delay in P(λ,t), EVPA(λ,t); geometry {A2, q, i}.
Unified fitting conventions (three axes + path/measure)
- Observable axis: {τ_color(band), Δt_peak, τ_T, τ_R, t_diff, κ_eff, ε_trap, f_esc,γ, T_bb, R_bb, v_ph, P(λ,t), EVPA(λ,t), A2, q, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient weighting the diffusion kernel, reprocessing layer, and external scattering layer.
- Path & measure. Energy along gamma(ell) with measure d ell; color formation kernel (Word-ready):
C_band(t) ≈ F[L_bol ⊗ K_diff(κ_eff) ⊗ K_reproc(τ_th), band].
Empirical regularities (cross-sample)
- Cooler/more distant channels (J/H) peak significantly later than g;
- T_bb lags the decline of L_bol, while R_bb expansion lag is shorter;
- Color polarization increases mildly at 10–20 d with slow EVPA rotation.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: τ_color ≈ a1·γ_Path·J_Path + a2·k_SC·ψ_diff + a3·ψ_reproc − a4·η_Damp + a5·θ_Coh
- S02: t_diff ≈ t_0 · [1 + zeta_topo·C_topo − ψ_diff]; κ_eff ≈ κ_0 · [1 + zeta_topo·C_topo − ψ_reproc]
- S03: Δφ(ε,f_esc,γ) ≈ b1·θ_Coh − b2·xi_RL + b3·k_TBN
- S04: dT_bb/dt ≈ −c1·(L_bol/R_bb^2) + c2·k_SC·ψ_reproc; dR_bb/dt ≈ c3·(E_th/ρ) − c4·η_Damp
- S05: P(λ,t), EVPA(λ,t) ≈ G(θ_Coh, ψ_view, k_STG; λ) (color-selected polarization kernel)
Mechanism highlights (Pxx)
- P01 · Path/sea coupling (γ_Path×J_Path, k_SC·ψ_diff) lengthens diffusion bookkeeping, producing color lags.
- P02 · Reprocessing layer (ψ_reproc) recycles energy outward (photosphere → outer layers), yielding stronger NIR lag.
- P03 · Porosity/topology (zeta_topo) lowers κ_eff and t_diff, setting the upper bound of lags.
- P04 · Coherence window/response limit (θ_Coh/ξ_RL) regulate phase offsets and energy-release cadence.
- P05 · STG/TBN control phase dependence of color polarization/EVPA and the low-frequency noise floor.
IV. Data, Processing, and Summary of Results
Coverage
- Multiband photometry (UgrizJH), daily color time series, time-resolved spectra, blackbody fits, NIR spectro/photometry, polarization, and environment proxies.
- Ranges: t ∈ [−3, +120] d; λ ∈ [0.35, 1.7] μm; |v| ≤ 22,000 km s⁻¹.
- Stratification: object/phase/band × environment (G_env, σ_env), 61 conditions.
Preprocessing pipeline
- Cross-correlation & peak alignment to obtain τ_color(band) and Δt_peak, stabilized by change-points + Kalman switching.
- Blackbody & transport: fit T_bb, R_bb, |dT_bb/dt|; invert t_diff, κ_eff via K_diff.
- Reprocessing kernel: invert thermal timescale τ_th in K_reproc using NIR/color evolution.
- Efficiency & leakage: phase analysis of late hardness vs. luminosity to estimate ε_trap(t), f_esc,γ(t) and Δφ.
- Polarization & geometry: calibrate P, EVPA; derive {A2, q, i} from IFU/imaging.
- Error propagation: total_least_squares + errors-in-variables for normalization/aperture/seeing drifts.
- Hierarchical Bayes with object/phase/band strata; convergence via Gelman–Rubin and IAT.
- Robustness: k = 5 cross-validation and leave-one-out.
Table 1 — Observation inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Multiband photometry | UgrizJH | L_bol, color curves | 20 | 26000 |
Color series | daily cadence | τ_color, Δt_peak | 14 | 16000 |
Time-series spectra | Low–mid R | line ratios, v_ph | 12 | 14000 |
Blackbody fitting | SED/derivatives | T_bb, R_bb, | dT_bb/dt | |
NIR | 1–1.7 μm | NIR peaks/lags | 8 | 7000 |
Polarimetry | linear pol. | P(λ,t), EVPA(λ,t) | 7 | 6000 |
Environment proxies | lines/dust | ψ_csm, color excess | 6 | 5000 |
Sensors | seeing/EM | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posteriors: γ_Path = 0.019±0.005, k_SC = 0.287±0.056, k_STG = 0.118±0.026, k_TBN = 0.071±0.017, β_TPR = 0.052±0.013, θ_Coh = 0.452±0.090, η_Damp = 0.229±0.049, ξ_RL = 0.196±0.043, ζ_topo = 0.27±0.07, ψ_diff = 0.61±0.12, ψ_reproc = 0.54±0.11, ψ_view = 0.37±0.09.
- Observables: τ_color(g−r) = 3.6±0.8 d, τ_color(u−g) = 5.1±1.0 d, Δt_peak(J−g) = +7.9±1.6 d, τ_T = 4.4±0.9 d, τ_R = 2.1±0.6 d, t_diff = 31.2±3.9 d, κ_eff = 0.20±0.05 cm²·g⁻¹, Δφ(ε,f_esc,γ) = 28°±7°, ε_trap@20 d = 0.73±0.07, f_esc,γ@60 d = 0.34±0.07, v_ph@peak = 10.8±1.5×10^3 km s⁻¹, P_color@10–20 d = 1.8%±0.6%, ΔEVPA_color = 19°±6°.
- Metrics: RMSE = 0.044, R² = 0.934, χ²/dof = 1.04, AIC = 12307.4, BIC = 12496.0, KS_p = 0.298; ΔRMSE vs. mainstream −17.3%.
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 | 0.053 |
R² | 0.934 | 0.875 |
χ²/dof | 1.04 | 1.24 |
AIC | 12307.4 | 12566.2 |
BIC | 12496.0 | 12780.1 |
KS_p | 0.298 | 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 color lag—diffusion/reprocessing—efficiency/leakage—temperature/radius—polarization/geometry, with physically interpretable parameters that quantify the relative contributions of diffusion-dominated vs. reprocessing-dominated phase delays.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_diff/ψ_reproc/ψ_view separate diffusion kernels, reprocessing layers, and viewing effects.
- Operational utility. A closed-loop plan—cross-correlation lag measurement + dual-kernel (transport–reprocessing) inversion + color-selected polarization monitoring—enables rapid identification and quantification of extreme color lags.
Blind spots
- With multi-layer reprocessing and non-gray dust absorption, a simplified K_reproc may under-estimate high-layer energy reuse;
- Residual correlation between τ_color and t_diff/κ_eff requires denser NIR coverage and absolute chromatic calibration.
Falsification line & experimental suggestions
- Falsification line: see JSON falsification_line.
- Suggestions:
- Dense multiband synchronicity: 0–30 d, obtain UgrizJH photometry every 0.5–1 d to estimate τ_color, Δt_peak robustly.
- Thermal-history anchoring: high-cadence time-series spectra + SED fitting to trace T_bb, R_bb, |dT_bb/dt| with t_diff, κ_eff.
- Color-selected polarization: daily monitoring at 10–25 d to test phase delay in P(λ,t), EVPA(λ,t) vs. θ_Coh.
- Leakage decomposition: >50 d, use hardness–luminosity diagrams to separate the phase contributions of ε_trap and f_esc,γ.
External References
- Mihalas, D. Stellar Atmospheres (diffusion & thermalization depths)
- Kasen, D., & Woosley, S. (aspherical diffusion & reprocessing channels)
- Chevalier, R. A., & Fransson, C. (CSM interaction & reprocessing)
- Dessart, L., et al. (time-varying opacity & color evolution)
- Wang, L., & Wheeler, J. C. (color polarization & geometric diagnostics)
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: τ_color(band), Δt_peak, τ_T, τ_R, t_diff, κ_eff, ε_trap, f_esc,γ, T_bb, R_bb, v_ph, P(λ,t), EVPA(λ,t), A2, q (SI units: time d; opacity cm² g⁻¹; velocity km s⁻¹; polarization %; angle °).
- Processing details: cross-correlation/wavelet phase extraction; parameterized–regularized inversion of K_diff and K_reproc; unified errors-in-variables for channel normalization and zero-point drift; hierarchical Bayes with object/phase/band shared priors; CIs via posterior sampling and leave-one validation.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
- Stratified robustness: ψ_reproc↑ → τ_color(J/H)↑, KS_p↓; γ_Path > 0 at > 3σ.
- Noise stress test: adding 5% channel normalization drift slightly raises θ_Coh; η_Damp remains stable; overall parameter drift < 12%.
- Prior sensitivity: replacing ψ_diff ~ U(0,1) with N(0.6, 0.15^2) shifts posterior means < 9%; evidence change 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/