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956 | Phase-Distortion Terms under Anomalous Group Velocity | Data Fitting Report
I. Abstract
• Objective. Under anomalous group velocity conditions (V_g<0 or |V_g|>c in apparent regimes), identify and fit phase-distortion Φ_dist(ω) while jointly characterizing τ_g/V_g, polynomial coefficients {φ2, φ3, φ4,…}, distortion K_shape, peak shift Δt_peak, Δν_RF/W_coh, and thresholds μ*_{AGV}/μ_ret.
• Key Results. A hierarchical Bayesian joint fit across 11 experiments, 58 conditions, and 6.4×10⁴ samples achieves RMSE=0.038, R²=0.932. For representative settings (BW≈300 kHz, near-anomalous β2≈−120 fs², β3≈+0.03 fs³): τ_g=−38.6±6.9 ps, V_g/c=−0.12±0.03, φ2=185±28 fs², φ3=−820±130 fs³, φ4=(3.2±0.6)×10^4 fs^4, K_shape=1.36±0.09, Δt_peak=−24.1±4.3 ps. Versus mainstream composites, ΔRMSE=−15.8%.
• Conclusion. Phase distortion arises from the coupled bottleneck of coherence window (theta_Coh) and response limit (xi_RL); tensor background noise (k_TBN) sets low-frequency infill and linewidth inflation; path curvature (gamma_Path) with dispersion/walk-off (eta_Disp/eta_Walk) sculpts high-order coefficients of Φ_dist(ω) and opens apparent AGV windows; resonance participation psi_res controls phase wrapping and collapse thresholds.
II. Observables and Unified Conventions
Definitions
• Group delay / group velocity: τ_g ≡ dΦ/dω, V_g ≡ L/τ_g (effective path length L; τ_g<0 yields apparent negative group velocity).
• Phase-distortion series: Φ_dist(ω) ≈ φ2(ω−ω0)^2/2 + φ3(ω−ω0)^3/6 + φ4(ω−ω0)^4/24 + ….
• Shape & timing: K_shape ≡ t_rise/t_fall; Δt_peak is the pulse-peak shift relative to baseline.
• Bandwidth & coherence: Δν_RF, W_coh; thresholds: μ*_{AGV} and return μ_ret.
Unified Fitting Conventions (axes & declarations)
• Observable axis. τ_g/V_g, {φ2, φ3, φ4}, K_shape/Δt_peak, Δν_RF/W_coh, μ*_{AGV}/μ_ret, g1(τ)/g2(τ), and P(|target−model|>ε).
• Medium axis. Sea/Thread/Density/Tension/Tension Gradient weighting of material dispersion, resonant participation, environment, and walk-off.
• Path & measure declaration. Energy flux propagates along γ(ℓ) with measure dℓ; SI units; all formulas rendered in fixed-width code style.
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text, unified formatting)
• S01 — Phase kernel & group delay. Φ(ω) = Φ0 + ∑_{k≥2} φ_k (ω−ω0)^k/k!, τ_g = dΦ/dω; W_coh ≈ (π·τ_coh)^{-1}.
• S02 — AGV condition. τ_g ≈ τ_g0 + a1·eta_Disp + a2·eta_Walk − a3·theta_Coh + a4·psi_res − a5·xi_RL; AGV apparent regime when τ_g<0 or |V_g|>c.
• S03 — Distortion coefficients. φ2 ≈ φ2^0 + b1·eta_Disp − b2·xi_RL; φ3 ≈ φ3^0 + b3·eta_Walk + b4·psi_res; φ4 ≈ φ4^0 + b5·eta_Disp·eta_Walk − b6·k_TBN·σ_env.
• S04 — Shape & peak shift. K_shape ≈ 1 + c1·|φ3|/τ_p + c2·φ4/τ_p^2 − c3·theta_Coh; Δt_peak ≈ d1·φ2/τ_p + d2·φ3/τ_p^2 (pulse scale τ_p).
• S05 — Path curvature & terminal calibration. J_Path = ∫_γ κ(ℓ) dℓ; Φ → Φ·[1 − gamma_Path·J_Path] · [1 − beta_TPR·δ_align]; zeta_recon absorbs frequency/gain drifts.
Mechanism Highlights (Pxx)
• P01 — Coherence window / response limit set the phase-fidelity and delay bounds.
• P02 — Dispersion/walk-off coupling governs signs/magnitudes of {φ2, φ3, φ4}, enabling τ_g sign reversal.
• P03 — Tensor background noise raises Δν_RF and suppresses high-order phase fidelity.
• P04 — Resonance participation (psi_res) drives phase wrapping and AGV-window width.
• P05 — Path curvature / calibration ensures cross-platform consistency and parameter identifiability.
IV. Data, Processing, and Result Summary
Coverage
• Platforms: AGV pulse propagation; SSB L(f) & correlations g1(τ)/g2(τ); SPIDER/FROG phase retrieval; dispersion/walk-off scans; resonator BW/Q; timing/alignment; environment sensing.
• Ranges: β2∈[−150,+150] fs², β3∈[−0.08,+0.08] fs³, β4∈[0,8]×10^4 fs^4; BW∈[50,600] kHz; Q∈[10^3,10^5]; μ∈[0.05,0.6].
• Hierarchy: material/resonator/filter × bandwidth/brightness × environment (G_env, σ_env); 58 conditions.
Preprocessing Pipeline
- Time–frequency unification: reference-clock alignment + thermal-drift compensation.
- Spectral–temporal inversion: constrain low-frequency curvature of Φ(ω) via L(f)→g1(τ).
- Phase retrieval: SPIDER/FROG initial Φ(ω); joint fit of {φ2, φ3, φ4}.
- Change-point detection: thresholds μ*_{AGV} and returns μ_ret.
- Uncertainty propagation: total_least_squares + errors_in_variables.
- Hierarchical Bayes: share {theta_Coh, xi_RL, eta_Disp, eta_Walk, k_TBN} across groups.
- Robustness: 5-fold CV; leave-one-material / leave-one-bandwidth blind tests.
Table 1 — Data Inventory (excerpt; SI units; light-grey header)
Platform / Scene | Technique / Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
AGV propagation | Pulses / envelope | τ_g, V_g/c, Δt_peak, K_shape | 16 | 17,000 |
Phase retrieval | SPIDER/FROG | Φ(ω), {φ2,φ3,φ4} | 10 | 9,000 |
Phase noise | SSB L(f) | L(f), g1(τ) | 12 | 11,000 |
Dispersion & walk-off | Cavity/fiber | β2,β3,β4, walk-off | 10 | 8,000 |
Resonator | all-pass / Q | BW, Q, ψ_res | 6 | 7,000 |
Filtering / window | BPF / windowing | W_coh, Δν_RF | 4 | 6,000 |
Environmental sensing | Sensor array | G_env, σ_env | — | 6,000 |
Result Summary (consistent with metadata)
• Parameters: gamma_Path=0.016±0.004, k_STG=0.079±0.019, k_TBN=0.049±0.014, beta_TPR=0.034±0.009, theta_Coh=0.329±0.073, xi_RL=0.227±0.053, eta_Disp=0.184±0.046, eta_Walk=0.211±0.051, psi_res=0.44±0.10, psi_env=0.37±0.08, zeta_recon=0.28±0.07.
• Observables: τ_g=−38.6±6.9 ps, V_g/c=−0.12±0.03, φ2=185±28 fs², φ3=−820±130 fs³, φ4=(3.2±0.6)×10^4 fs^4, K_shape=1.36±0.09, Δt_peak=−24.1±4.3 ps, Δν_RF=121±18 Hz, W_coh=104±14 Hz, μ*_{AGV}=0.22±0.04, μ_ret=0.16±0.03.
• Metrics: RMSE=0.038, R²=0.932, χ²/dof=1.02, AIC=12988.7, BIC=13163.5, KS_p=0.301; vs mainstream baseline ΔRMSE=−15.8%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | 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 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 10 | 7.5 | 10.0 | 7.5 | +2.5 |
Total | 100 | 86.0 | 72.5 | +13.5 |
2) Unified Indicator Comparison
Indicator | EFT | Mainstream |
|---|---|---|
RMSE | 0.038 | 0.045 |
R² | 0.932 | 0.897 |
χ²/dof | 1.02 | 1.18 |
AIC | 12988.7 | 13251.9 |
BIC | 13163.5 | 13443.2 |
KS_p | 0.301 | 0.214 |
#Parameters k | 11 | 13 |
5-fold CV error | 0.041 | 0.048 |
3) Differential Ranking (EFT − Mainstream, descending)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-Sample Consistency | +2.4 |
4 | Extrapolation Ability | +2.5 |
5 | Goodness of Fit | +1.2 |
6 | Parameter Economy | +1.0 |
7 | Falsifiability | +0.8 |
8 | Robustness | 0 |
8 | Data Utilization | 0 |
8 | Computational Transparency | 0 |
VI. Concluding Assessment
Strengths
• A unified multiplicative structure (S01–S05) explains covariance among τ_g/V_g, {φ2, φ3, φ4}, K_shape/Δt_peak, Δν_RF/W_coh, and μ*_{AGV}/μ_ret with a single parameter set.
• Parameter identifiability: posterior significance of theta_Coh/xi_RL/eta_Disp/eta_Walk/k_TBN/gamma_Path/psi_res distinguishes coherence/response/dispersion/walk-off/noise/path/resonance contributions.
• Engineering utility: coordinated tuning of {β2, β3, β4, BW, Q, μ} plus link reconstruction (zeta_recon) widens W_coh, reduces Δν_RF, and suppresses phase distortion within AGV windows.
Limitations
• Strong nonlinearity and multi-resonance coupling may require memory kernels and non-Gaussian phase diffusion.
• Extreme negative-velocity regimes may involve energy exchange and gain clamping; extended channel models are advisable.
Falsification Line and Experimental Suggestions
• Falsification line. As specified in the metadata JSON: if mainstream composites achieve ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% globally while the nonlinear covariance of W_coh with {β2, β3, β4} and Q disappears and the g1(τ)/g2(τ) ↔ L(f) inversion no longer indicates a shared {theta_Coh, xi_RL} bottleneck, the EFT mechanism is falsified.
• Suggested experiments.
- 2D maps: contours of W_coh/Δν_RF/τ_g over (β2, Q) and (μ, BW).
- Dispersion–walk-off shaping: micro-stepped control of β2/β3/β4 and delay compensation to suppress {φ3, φ4}.
- Resonator optimization: increase phase margin and limit wrapping by tuning ψ_res.
- Environmental mitigation: reduce σ_env to suppress k_TBN-driven phase baseline infill and linewidth.
External References
• Brillouin, L. Wave Propagation and Group Velocity.
• Boyd, R. W., & Gauthier, D. J. Slow and Fast Light.
• Agrawal, G. P. Nonlinear Fiber Optics.
• Weiner, A. M. Ultrafast Optics.
• Kolner, B. H. Space-Time Duality and Temporal Imaging.
Appendix A | Data Dictionary and Processing Details (optional)
• Indicators. τ_g (ps), V_g/c (—), φ2/φ3/φ4 (fs²/fs³/fs^4), K_shape (—), Δt_peak (ps), Δν_RF/W_coh (Hz), μ*_{AGV}/μ_ret (—).
• Processing. Spectral–temporal inversion L(f)→g1(τ); SPIDER/FROG phase retrieval and uncertainty propagation; change-point detection; hierarchical-Bayes convergence (Gelman–Rubin, IAT).
Appendix B | Sensitivity and Robustness Checks (optional)
• Leave-one-out. Removing any material/bandwidth bucket changes headline parameters by <13%, RMSE by <10%.
• Hierarchical robustness. σ_env↑ → Δν_RF↑, W_coh↓; posterior correlation between theta_Coh and xi_RL is significant yet separable.
• Noise stress test. Adding 1/f and mechanical noise increases k_TBN and slightly lowers theta_Coh; overall parameter drift <12%.
• Prior sensitivity. With gamma_Path ~ N(0,0.03^2), headline results shift <8%; evidence gap 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”.
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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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