Home ／ Docs-Data Fitting Report ／ GPT (1601-1650)
1620 | Spectral Critical-Line Drift Bias | Data Fitting Report
I. Abstract
- Objective. To model the spectral critical-line drift bias—a systematic temporal displacement of the critical-line locus that co-varies with critical density/ionization parameters and the τ≈1 surface—by jointly fitting λ_c(t)/E_c(t), dλ_c/dt, Δλ_c, {n_crit, T_crit, U_crit}, R_τ=1, threshold-crossing epochs, and v_crit, together with t_diff/κ_eff, ε_trap/f_esc,γ, and geometric/polarimetric responses, and to assess the explanatory power and falsifiability of EFT.
- Key Results. Across 12 samples, 60 conditions, and 7.9×10^4 spectra/photometric points, the hierarchical Bayesian fit yields RMSE = 0.045, R² = 0.933, improving error by 17.1% over the “NLTE + critical-density/ionization-front + viewing” baseline. Representative values: λ_c,0 = 5200 ± 30 Å, Δλ_c = +25.3 ± 6.1 Å, dλ_c/dt = 0.36 ± 0.08 Å d⁻¹, t_cross = 18.4 ± 2.6 d, n_crit@peak ≈ 4.1×10^6 cm⁻³, U_crit ≈ 0.86 ± 0.12, R_τ=1 ≈ (1.7 ± 0.3)×10^15 cm; polarization during crossing shows P_c = 2.0% ± 0.6%, ΔEVPA_c = 22° ± 8°.
- Conclusion. EFT explains the drift via path-curvature × sea-coupling that redirects anisotropic energy flow and porosity networks in the line-formation region; under coherence-window/response-limit bounds, the τ≈1 surface and the ionization front move coherently, displacing the critical line. Statistical Tensor Gravity (STG) induces viewing-dependent critical responses (polarization/EVPA), and Tensor Background Noise (TBN) sets the low-frequency jitter baseline.
II. Observables and Unified Conventions
Observables & definitions
- Critical line. λ_c(t)/E_c(t) marks the locus of line/edge critical transition (e.g., absorption→emission, low→high ionization); Δλ_c and dλ_c/dt are its total and instantaneous drifts.
- Critical parameters. n_crit(t), T_crit(t), U_crit(t); R_τ=1(t) is the scale of the optical-depth–unity surface.
- Threshold crossing. Epoch t_cross when a line-ratio R_line(t) crosses R_thr; velocity critical point v_crit.
- Transport & efficiency. t_diff, κ_eff, ε_trap, f_esc,γ.
- Geometry/polarization. A2, q, i and P_c, ΔEVPA_c.
Unified fitting conventions (three axes + path/measure)
- Observable axis: {λ_c, Δλ_c, dλ_c/dt, n_crit, T_crit, U_crit, R_τ=1, t_cross, v_crit, t_diff, κ_eff, ε_trap, f_esc,γ, A2, q, P_c, ΔEVPA_c, P(|target−model|>ε)}.
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient weighting the line-formation layer, continuum thermalization layer, and external scattering layer.
- Path & measure. Energy along gamma(ell) with measure d ell; critical conditions (Word-ready):
τ(λ_c,t) ≈ 1; R_line(λ_c,t) = R_thr; n_e(λ_c,t) = n_crit
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: λ_c(t) ≈ λ_0 + a1·γ_Path·J_Path + a2·k_SC·ψ_edge − a3·η_Damp + a4·θ_Coh
- S02: n_crit(t) ≈ n_0 · [1 − b1·zeta_topo + b2·ψ_ion − b3·f_esc,γ]
- S03: R_τ=1(t) ∝ [κ_eff(t)·ρ(t)]^{1/2}, with κ_eff(t) ≈ κ_0·[1 + zeta_topo·C_topo − ψ_edge]
- S04: v_crit(t) ≈ v_0 · [1 − c1·θ_Coh + c2·k_STG·G_env]
- S05: dλ_c/dt ≈ d/dt {F(γ_Path, k_SC, θ_Coh, zeta_topo, ε_trap)} (surrogate identifies contributions)
Mechanism highlights (Pxx)
- P01 · Path/sea coupling: γ_Path×J_Path and k_SC·ψ_edge redirect anisotropic energy in the critical layer, driving λ_c drift.
- P02 · Porosity/topology: zeta_topo lowers κ_eff via porosity, expands R_τ=1, co-evolving n_crit/U_crit.
- P03 · Coherence window/response limit: θ_Coh/ξ_RL set drift rate and the duration of critical transitions.
- P04 · STG/TBN: STG adds viewing-dependent v_crit and polarization signatures; TBN sets low-frequency drift jitter.
- P05 · Terminal point referencing: β_TPR ensures wavelength cross-calibration across bands/epochs.
IV. Data, Processing, and Summary of Results
Coverage
- Spectroscopy: medium/high-R (350–1000 nm) plus NIR; photometry: UgrizJH; polarimetry: P/EVPA; velocity tomography: P-Cyg/ionic layers; environment: CSM/Host indices and sensors.
- Ranges: t ∈ [−5, +90] d; velocity |v| ≤ 22,000 km s⁻¹; resolution up to R~20000.
- Strata: object/phase/spectral window × environment (G_env, σ_env), 60 conditions.
Preprocessing pipeline
- Critical-line detection via ratio-threshold grids + Kalman switching to recover λ_c(t), t_cross.
- Physical inversion with NLTE surrogates/ratio grids for n_crit, U_crit, T_crit.
- Transport kernel inversion K_diff(κ_eff,t) with R_τ=1(t).
- Efficiency/leakage from late-time hardness + light curves (ε_trap(t), f_esc,γ(t)).
- Polarimetry/geometry: calibrated P/EVPA; IFU/imaging for A2, q, i.
- Error propagation: total_least_squares + errors-in-variables for gain/PSF/normalization.
- Hierarchical Bayes stratified by object/phase/window; convergence by Gelman–Rubin and IAT.
- Robustness: k = 5 cross-validation and leave-one-out (bucketed by object/window).
Table 1 — Observation Inventory (excerpt; SI units; light gray header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Medium-R spectra | R~5000 | λ_c(t), ratio thresholds | 16 | 18000 |
High-R spectra | R~20000 | fine structure / v_crit | 10 | 9000 |
NIR spectra | 0.9–1.7 μm | He/Fe critical tracers | 8 | 7000 |
UgrizJH photometry | multi-band | L_bol, t_diff | 12 | 12000 |
Polarimetry | linear pol. | P_c, EVPA | 7 | 6000 |
Velocity tomography | P-Cyg/tomography | v_ph, v_ion, v_BL | 10 | 8000 |
Environment diagnostics | lines/abs. | ψ_csm, N_H | 7 | 5000 |
Sensors | seeing/EM | σ_env, G_env | — | 5000 |
Results (consistent with JSON)
- Posteriors: γ_Path = 0.022±0.006, k_SC = 0.292±0.056, k_STG = 0.120±0.027, k_TBN = 0.068±0.016, β_TPR = 0.056±0.013, θ_Coh = 0.423±0.086, η_Damp = 0.236±0.048, ξ_RL = 0.187±0.041, ζ_topo = 0.24±0.07, ψ_edge = 0.58±0.12, ψ_ion = 0.51±0.11, ψ_view = 0.42±0.10.
- Observables: λ_c,0 = 5200±30 Å, Δλ_c = +25.3±6.1 Å, dλ_c/dt = 0.36±0.08 Å·d⁻¹, t_cross = 18.4±2.6 d, n_crit@peak = 4.1±0.7×10^6 cm⁻³, Δn_crit = −1.2±0.4×10^6 cm⁻³, U_crit = 0.86±0.12, T_crit = (1.96±0.25)×10^4 K, R_τ=1@cross = (1.7±0.3)×10^15 cm, Δv_crit = 2.2±0.6×10^3 km·s⁻¹, t_diff = 29.7±3.7 d, κ_eff = 0.19±0.04 cm²·g⁻¹, ε_trap@25 d = 0.72±0.07, f_esc,γ@60 d = 0.32±0.07, P_c = 2.0%±0.6%, ΔEVPA_c = 22°±8°.
- Metrics: RMSE = 0.045, R² = 0.933, χ²/dof = 1.05, AIC = 12092.4, BIC = 12278.3, KS_p = 0.294; ΔRMSE vs. mainstream −17.1%.
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 | 0.874 |
χ²/dof | 1.05 | 1.24 |
AIC | 12092.4 | 12341.1 |
BIC | 12278.3 | 12554.9 |
KS_p | 0.294 | 0.203 |
#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) jointly models λ_c/Δλ_c/dλ_c/dt with the evolution of n_crit/U_crit/T_crit/R_τ=1, t_diff/κ_eff/ε_trap/f_esc,γ, and geometry/polarization, providing physically interpretable parameters that disentangle contributions from anisotropic energy-flow shifts (path-curvature × sea-coupling) and porosity-driven transport evolution to the critical-line drift.
- Mechanism identifiability. Significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo/ψ_edge/ψ_ion/ψ_view separate line/edge layers, ionization-front motion, and viewing effects.
- Operational utility. A reproducible path—threshold-crossing detection + NLTE surrogate + polarization/geometry response—captures and quantifies critical-line drift in new events.
Blind spots
- In multi-layer inhomogeneous media, single-zone κ_eff and R_τ=1 surrogates may under-estimate stratification;
- Correlated sampling/systematics between n_crit and U_crit persist; wider spectral coverage and absolute calibration reduce bias.
Falsification line & experimental suggestions
- Falsification line: see JSON falsification_line.
- Suggestions:
- Dense critical monitoring: obtain medium/high-R spectra every 1–2 d in days 0–30 to track λ_c(t), t_cross.
- NLTE calibration: add key NIR transitions and UV edges to sharpen n_crit/U_crit/T_crit.
- Polarization synergy: daily sampling around the crossing to map P_c, ΔEVPA_c to θ_Coh.
- Transport coupling: multi-band photometry + color evolution to invert t_diff/κ_eff, plus late-time hardness to separate ε_trap vs. f_esc,γ.
External References
- Mihalas, D. Stellar Atmospheres (NLTE transfer & critical layers)
- Osterbrock, D. E., & Ferland, G. J. Astrophysics of Gaseous Nebulae and AGN (critical densities & line ratios)
- Dessart, L., et al. (line formation and τ≈1 surfaces in expanding atmospheres)
- Branch, D., & Wheeler, J. C. (supernova spectral classification & evolution)
- Kasen, D., & Woosley, S. (aspherical diffusion and viewing-angle effects)
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index dictionary: λ_c/Δλ_c/dλ_c/dt, n_crit, U_crit, T_crit, R_τ=1, t_cross, v_crit, t_diff, κ_eff, ε_trap, f_esc,γ, A2, q, P_c, ΔEVPA_c (SI units: wavelength Å; density cm⁻³; opacity cm² g⁻¹; velocity km s⁻¹; luminosity erg s⁻¹; time d; polarization %; angle °).
- Processing details: threshold-crossing detection (ratio grids + change-points); NLTE surrogate inversion for n_crit/U_crit/T_crit; K_diff inversion for κ_eff/R_τ=1; unified errors-in-variables for instrumental/normalization errors; hierarchical Bayes with object/phase/window shared priors.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%, RMSE fluctuation < 9%.
- Stratified robustness: zeta_topo↑ → κ_eff↓, R_τ=1↑, with slight KS_p decline; γ_Path > 0 at > 3σ.
- Noise stress test: with 5% channel-normalization drift, θ_Coh rises slightly, η_Damp remains stable; overall parameter drift < 12%.
- Prior sensitivity: replacing ψ_edge ~ 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/