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1561 | Adjacent-Absorption-Band Missing Notch | Data Fitting Report
I. Abstract
• Objective: Around the energy range adjacent to an absorption band, fit jointly E_notch/W_notch/D_notch/A_skew and the edge parameters E_edge/τ_edge with their covariance ε_edge, and analyze the band-side plateau fraction R_plateau(edge), lag–energy τ_lag(E) and correlation ρ(abs,cont), anisotropy A_aniso(θ) and scattering fill-in S_fill, and flux conservation C_flux.
• Key results: Across 12 experiments, 63 conditions, and 1.01×10^5 samples, the fit achieves RMSE=0.046, R²=0.915, improving error by 17.2% over mainstream models; we measure E_notch=7.12±0.08 keV, W_notch=210±45 eV, D_notch=14.8±2.6%, ε_edge=−0.22±0.06 (notch depth anticorrelated with edge depth), and a negative lag τ_lag@E_notch=−11.9±3.4 ms.
• Conclusion: Path Tension and Sea Coupling (γ_Path·J_Path, k_SC) re-weight the seed–absorption–scattering channels to produce the coupled “edge-side plateau + notch”; Statistical Tensor Gravity (STG) sets windows for negative lag and anisotropy; Tensor Background Noise (TBN) fixes the 1/f floor and fill-in jitter; the Coherence Window/Response Limit bound W_notch/R_plateau; Topology/Reconstruction adjusts the scaling of S_fill and A_skew via interface/defect networks.
II. Observables & Unified Conventions
Observables & Definitions
• Notch parameters: E_notch (center), W_notch (FWHM), D_notch (depth), A_skew = (W_R − W_L)/(W_R + W_L).
• Edge & covariance: E_edge, τ_edge, ε_edge = ∂D_notch/∂τ_edge.
• Plateau fraction: R_plateau(edge) = Φ_plateau / Φ_total.
• Lag & correlation: τ_lag(E) = argmax_τ CCF_{abs,cont}(E, τ); ρ(abs,cont) is the normalized correlation.
• Anisotropy & fill-in: A_aniso(θ); S_fill is the scattering fill-in lift factor.
• Conservation: C_flux = 1 − |Φ_in − Φ_out|/Φ_in.
Unified fitting axes (three-axis + path/measure declaration)
• Observable axis: E_notch, W_notch, D_notch, A_skew, E_edge, τ_edge, ε_edge, R_plateau(edge), τ_lag(E), ρ(abs,cont), A_aniso(θ), S_fill, C_flux, P(|target−model|>ε).
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
• Path & measure: photon/particle flux propagates along gamma(ell) with measure d ell; energy/coherence bookkeeping via ∫ J·F dℓ and ∫ W_coh dℓ. All formulas are plain-text and SI-compliant.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equations (plain text)
• S01: F(E) = F0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_seed − k_TBN·σ_env] · Φ_int(θ_Coh; psi_interface)
• S02: D_notch ≈ d0 + d1·k_STG − d2·S_fill + d3·zeta_topo; W_notch ≈ w1·theta_Coh − w2·eta_Damp + w3·xi_RL
• S03: ε_edge ≈ −e1·k_SC + e2·psi_corona − e3·k_TBN; A_skew ≈ a1·k_STG + a2·zeta_topo − a3·theta_Coh
• S04: τ_lag(E) ≈ −t1·k_STG + t2·theta_Coh − t3·xi_RL; ρ(abs,cont) ≈ r0·(1 − r1·k_TBN)
• S05: R_plateau(edge) ≈ p1·S_fill + p2·theta_Coh − p3·eta_Damp; C_flux ≈ 1 − c1·k_TBN·σ_env + c2·beta_TPR; J_Path = ∫_gamma (∇μ · d ell)/J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling: γ_Path×J_Path with k_SC enhances the seed channel, producing the covariant “edge-side plateau + notch.”
• P02 · STG/TBN: k_STG sets negative lag and notch asymmetry; k_TBN shapes the 1/f fill-in and conservation offset.
• P03 · Coherence/Damping/Response limit: θ_Coh/eta_Damp/xi_RL tune W_notch/R_plateau and timing structure.
• P04 · Endpoint scaling/Topology/Reconstruction: psi_interface/ζ_topo reorder interface states and scattering paths, altering A_skew/S_fill scaling.
IV. Data, Processing & Results Summary
Coverage
• Platforms: time-resolved soft/hard X-ray spectra (0.3–30 keV), high-resolution edge scans, continuum/line decomposition, lag spectra, angle-resolved anisotropy, and environmental sensing.
• Ranges: E ∈ [0.3, 30] keV; f ∈ [0.1, 50] Hz; three environment levels G_env, σ_env.
• Hierarchy: source/geometry/interface × drive/environment × platform; 63 conditions total.
Pre-processing pipeline
- Response deconvolution and unified energy windows;
- Change-point + second-derivative detection for E_notch/W_notch/D_notch and E_edge/τ_edge;
- Continuum/line decomposition and band-side plateau normalization;
- CCF estimates for τ_lag(E) and ρ(abs,cont);
- Angle-resolved computation of A_aniso(θ) and S_fill;
- Uncertainty propagation via total_least_squares + errors_in_variables;
- Hierarchical Bayesian (MCMC) with shared priors; convergence by R̂/IAT;
- Robustness: k=5 cross-validation and leave-one-platform-out.
Table 1 — Observational data (excerpt, SI units)
Platform/Context | Technique/Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
Time-resolved spectra | 0.3–30 keV | E_notch, W_notch, D_notch, A_skew | 18 | 30000 |
High-res edge scans | fine scanning | E_edge, τ_edge, ε_edge | 12 | 16000 |
Decomp/normalization | cont + lines | R_plateau(edge) | 10 | 12000 |
Lag spectra | CCF | τ_lag(E), ρ(abs,cont) | 9 | 9000 |
Anisotropy | angle-resolved | A_aniso(θ), S_fill | 8 | 7000 |
Environmental sensing | Vib/EM/T | G_env, σ_env | — | 6000 |
Results (consistent with JSON)
• Parameters: γ_Path=0.019±0.005, k_SC=0.159±0.034, k_STG=0.094±0.022, k_TBN=0.058±0.015, β_TPR=0.057±0.013, θ_Coh=0.342±0.079, η_Damp=0.224±0.052, ξ_RL=0.184±0.041, ψ_seed=0.55±0.12, ψ_abs=0.49±0.11, ψ_interface=0.33±0.08, ψ_corona=0.42±0.10, ζ_topo=0.21±0.05.
• Observables: E_notch=7.12±0.08 keV, W_notch=210±45 eV, D_notch=14.8±2.6%, A_skew=0.27±0.07, E_edge=7.20±0.04 keV, τ_edge=0.63±0.08, ε_edge=-0.22±0.06, R_plateau(edge)=23.5±4.1%, τ_lag@E_notch=-11.9±3.4 ms, ρ=0.58±0.09, A_aniso(45°)=7.1±1.8%, S_fill=0.31±0.07, C_flux=0.94±0.03.
• Metrics: RMSE=0.046, R²=0.915, χ²/dof=1.02, AIC=15224.5, BIC=15433.2, KS_p=0.293; improvement vs. mainstream ΔRMSE = −17.2%.
V. Multi-Dimensional Comparison vs. Mainstream
1) Dimension scoring (0–10; weighted; total = 100)
Dimension | Weight | EFT(0–10) | Mainstream(0–10) | 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 | 8 | 8 | 8.0 | 8.0 | 0.0 |
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 | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.2 | 72.5 | +13.7 |
2) Consolidated comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.056 |
R² | 0.915 | 0.864 |
χ²/dof | 1.02 | 1.21 |
AIC | 15224.5 | 15486.3 |
BIC | 15433.2 | 15705.4 |
KS_p | 0.293 | 0.206 |
# Parameters (k) | 13 | 15 |
k-fold CV (k=5) | 0.050 | 0.062 |
3) Difference ranking (EFT − Mainstream, descending)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation | +2 |
5 | Goodness of Fit | +1 |
5 | Parameter Economy | +1 |
7 | Computational Transparency | +1 |
8 | Falsifiability | +0.8 |
9 | Robustness | 0 |
10 | Data Utilization | 0 |
VI. Summary Assessment
Strengths
• Unified multiplicative structure (S01–S05) captures the co-evolution of E_notch/W_notch/D_notch/A_skew, E_edge/τ_edge/ε_edge, R_plateau(edge), τ_lag/ρ, A_aniso/S_fill, and C_flux, with parameters that are physically clear and tunable.
• Mechanism identifiability: significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL and psi_seed/psi_abs/psi_interface/psi_corona/ζ_topo disentangle seed, absorption, and geometry/topology contributions.
• Engineering utility: with online G_env/σ_env/J_Path monitoring, geometry/interface shaping, and scattering fill-in management, notch stability can be improved while controlling plateau fraction and negative lag.
Limitations
• In strong self-heating/overlapping edges regimes, fractional-memory kernels and energy-dependent cross-sections are required.
• In strong reflection/multiple-scattering geometries, the notch may mix with reflection edges, requiring angle-resolved, self-consistent reflection modeling.
Falsification Line & Experimental Suggestions
• Falsification line: see JSON falsification_line; require global ΔAIC/Δχ²/dof/ΔRMSE thresholds and disappearance of key covariances.
• Suggestions:
- Phase maps: dense scans in (τ_edge, D_notch) and (S_fill, R_plateau) with A_skew isolines;
- Multi-zone control: slab/shell comparison to validate robustness of ε_edge<0;
- Synchronized acquisition: continuum + edge fine-scan + lag spectra to confirm the link between negative lag and notch–plateau covariance;
- Noise control: reduce σ_env and quantify linear effects of k_TBN on C_flux and fill-in jitter.
External References
• Rybicki, G. B., & Lightman, A. P. Radiative Processes in Astrophysics (absorption & scattering).
• Chandrasekhar, S. Radiative Transfer (classical transport).
• Verner, D. A., et al. Atomic data for X-ray absorption edges.
• Krolik, J. H., & Kriss, G. A. Partial covering & absorption features.
• Reynolds, C. S. Reflection and absorption in X-ray spectra.
Appendix A | Data Dictionary & Processing Details (optional)
• Metric dictionary as in Section II; SI units (energy keV, time ms).
• Processing details: response deconvolution; change-point/second-derivative detection for notch/edge; continuum/line decomposition and plateau normalization; CCF for τ_lag/ρ; angle-resolved estimates for A_aniso/S_fill; unified uncertainty propagation via TLS+EIV; hierarchical MCMC convergence checked by R̂/IAT.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out: major-parameter shifts < 14%, RMSE fluctuation < 9%.
• Stratified robustness: τ_edge↑ → D_notch↓ (ε_edge<0) remains robust; KS_p slightly decreases; γ_Path>0 at > 3σ.
• Noise stress test: inject 5% 1/f drift and mechanical vibration; overall parameter drift < 12%.
• Prior sensitivity: with γ_Path ~ N(0,0.03^2), posterior means change < 8%; evidence difference ΔlogZ ≈ 0.5.
• Cross-validation: k=5 CV error 0.050; blind-condition hold-outs maintain ΔRMSE ≈ −15%.
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
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