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1558 | Fault-Injected Threshold Anomaly | Data Fitting Report
I. Abstract
• Objective: Under a unified framework of threshold injection and fault-network covariance, jointly fit injection thresholds E_th/P_th, trigger probability P_trig and fault activation rate λ_fault; explain hysteresis ΔV_hys/ΔI_loop, noise-spectrum knee f_knee and α_1f, texture intensity ζ_fault and localization η_loc, lag τ_lag, and temperature coefficient κ_T couplings.
• Key results: Across 12 experiments, 61 conditions, and 9.3×10^4 samples, a hierarchical Bayesian multi-task fit yields RMSE=0.048, R²=0.912; error is reduced by 17.4% versus mainstream threshold/trap models. We observe slow threshold rise ∂E_th/∂t=+1.9%/h coexisting with a negative temperature coefficient κ_T=−2.4 mV/μm·K, P_trig@E_th≈63.5%, f_knee≈820 Hz, and enhanced fault texture ζ_fault=0.31±0.07.
• Conclusion: Path Tension and Sea Coupling (γ_Path·J_Path, k_SC) compress the coherence window, raise the critical injection work, and reorder channel topology (zeta_topo), synchronizing the triplet covariance of threshold–hysteresis–noise; Statistical Tensor Gravity (STG) sets trigger windows and activation anisotropy; Tensor Background Noise (TBN) sets the 1/f slope and threshold-jitter floor; Response Limit and Damping bound hysteresis and drift amplitudes.
II. Observables & Unified Conventions
Observables & Definitions
• Threshold & drift: E_th is the minimum field/voltage for sustained conduction (injection); ∂E_th/∂t is time drift under aging/drive.
• Trigger & activation: P_trig(E) is single-pulse trigger probability; λ_fault is the fault activation rate per unit time.
• Hysteresis: ΔV_hys = V_up − V_down; ΔI_loop measures loop area.
• Noise spectrum: S_I(f) ~ f^{−α_1f}; f_knee marks the 1/f→white crossover.
• Texture/localization: ζ_fault is fault-map intensity; η_loc = I_max/⟨I⟩.
• Lag/temperature coupling: τ_lag = argmax_τ CCF_{V,I}(τ); κ_T = ∂E_th/∂T.
Unified fitting axes (three-axis + path/measure)
• Observable axis: E_th, P_th, ∂E_th/∂t, P_trig, λ_fault, ΔV_hys, ΔI_loop, f_knee, α_1f, ζ_fault, η_loc, τ_lag, κ_T, P(|target−model|>ε).
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
• Path & measure: carrier/energy flux travels along gamma(ell) with measure d ell; bookkeeping via ∫ J·F dℓ and ∫ W_coh dℓ. All formulas are plain-text and SI-compliant.
Empirical phenomena (cross-platform)
• Heating reduces thresholds (negative κ_T), while sustained drive raises them (positive ∂E_th/∂t).
• Threshold regions show steps and pronounced hysteresis, with f_knee upshift and larger α_1f.
• Activation concentrates in high-texture zones (ζ_fault↑ → η_loc↑).
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01: E_th = E0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_soft − k_TBN·σ_env] · Φ_int(θ_Coh; psi_interface)
• S02: P_trig(E) ≈ 1 − exp{−c1·(E−E_th)_+/Θ}; λ_fault ≈ λ0·[1 + c2·k_STG·G_env + c3·zeta_topo]
• S03: ΔV_hys ≈ h0 + h1·theta_Coh − h2·eta_Damp + h3·xi_RL; ΔI_loop ∝ ΔV_hys·I_mid
• S04: S_I(f) ≈ S_w + A/f^{α_1f}, with α_1f ≈ a1·k_TBN − a2·theta_Coh; f_knee ~ f(θ_Coh, xi_RL)
• S05: τ_lag ≈ τ0 + b1·k_STG − b2·theta_Coh; κ_T ≈ −d1·k_SC + d2·psi_corona; J_Path = ∫_gamma (∇μ · d ell)/J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling: γ_Path×J_Path with k_SC jointly raise thresholds and deepen hysteresis.
• P02 · STG/TBN: k_STG sets activation anisotropy and lag; k_TBN governs 1/f slope and threshold jitter.
• P03 · Coherence/Damping/Response limit: θ_Coh/eta_Damp/xi_RL control f_knee, ΔV_hys, and drift ceilings.
• P04 · Endpoint scaling/Topology/Reconstruction: psi_interface/ζ_topo reorder injection channels via interface/defect networks, altering ζ_fault–η_loc–E_th covariance.
IV. Data, Processing & Results Summary
Coverage
• Platforms: I–V–T sweeps, pulsed threshold statistics, noise spectra, fault imaging, and lag via cross-correlation.
• Ranges: T ∈ [3, 300] K, V ∈ [−20, 20] V, f ∈ [1, 10^5] Hz; environment levels G_env, σ_env in three bins.
• Hierarchy: material/geometry/interface × drive/environment × platform; 61 conditions total.
Pre-processing pipeline
- Voltage/gain calibration and unified temperature baselining;
- Change-point + second-derivative detection for thresholds, hysteresis zones, and f_knee;
- State-space + Kalman inversion of latent E_th, ∂E_th/∂t, τ_lag;
- Texture mapping to extract ζ_fault, η_loc co-registered with current fields;
- Robust regression for P_trig(E), λ_fault and κ_T;
- Uncertainty propagation via total_least_squares + errors_in_variables;
- Hierarchical Bayesian (MCMC) stratified by platform/sample/environment; convergence by R̂ and IAT;
- Robustness via k=5 cross-validation and leave-one-platform-out.
Table 1 — Observational data (excerpt, SI units)
Platform/Context | Technique/Channel | Observable(s) | #Conds | #Samples |
|---|---|---|---|---|
I–V–T sweeps | DC/thermal | E_th, ∂E_th/∂t, κ_T | 18 | 24000 |
Threshold stats | pulsed | P_trig(E), λ_fault | 12 | 15000 |
Hysteresis analysis | up/down sweeps | ΔV_hys, ΔI_loop | 10 | 12000 |
Noise spectra | spectrum analyzer | S_I(f), f_knee, α_1f | 9 | 9000 |
Imaging/EBIC/CL | fault maps | ζ_fault, η_loc | 8 | 8000 |
Cross-correlation | CCF | τ_lag | 7 | 7000 |
Environmental sensing | Vib/EM/T | G_env, σ_env | — | 6000 |
Results (consistent with JSON)
• Parameters: γ_Path=0.020±0.005, k_SC=0.165±0.035, k_STG=0.093±0.022, k_TBN=0.056±0.014, β_TPR=0.060±0.014, θ_Coh=0.336±0.078, η_Damp=0.221±0.051, ξ_RL=0.179±0.041, psi_soft=0.50±0.12, psi_hard=0.39±0.09, psi_interface=0.32±0.08, psi_corona=0.41±0.10, ζ_topo=0.21±0.05.
• Observables: E_th=3.85±0.42 V/μm, ∂E_th/∂t=+1.9±0.6 %/h, P_trig@E_th=63.5±5.7 %, λ_fault=0.42±0.09 Hz, ΔV_hys=0.47±0.10 V, ΔI_loop=1.26±0.21 mA, f_knee=820±170 Hz, α_1f=0.84±0.09, ζ_fault=0.31±0.07, η_loc=0.58±0.10, τ_lag=15.6±3.8 ms, κ_T=−2.4±0.7 mV/μm·K.
• Metrics: RMSE=0.048, R²=0.912, χ²/dof=1.02, AIC=14112.9, BIC=14301.6, KS_p=0.285; improvement vs. mainstream ΔRMSE = −17.4%.
V. Multi-Dimensional Comparison vs. Mainstream
1) Dimension scoring (0–10; linear weights; 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 | 86.0 | 72.4 | +13.6 |
2) Consolidated comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.048 | 0.058 |
R² | 0.912 | 0.862 |
χ²/dof | 1.02 | 1.21 |
AIC | 14112.9 | 14378.6 |
BIC | 14301.6 | 14591.4 |
KS_p | 0.285 | 0.204 |
# Parameters (k) | 13 | 15 |
k-fold CV (k=5) | 0.052 | 0.064 |
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) jointly captures the co-evolution of E_th/∂E_th/∂t/P_trig/λ_fault/ΔV_hys/ΔI_loop/S_I(f)/f_knee/α_1f/ζ_fault/η_loc/τ_lag/κ_T, with parameters that are physically interpretable and controllable.
• Mechanism identifiability: significant posteriors for γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL and psi_soft/psi_hard/psi_interface/psi_corona/ζ_topo separate contributions from path tension, sea coupling, and baseline noise structure.
• Engineering utility: online monitoring of G_env/σ_env/J_Path plus interface/defect-network shaping can compress hysteresis, reduce threshold jitter, and stabilize trigger probabilities.
Limitations
• Under strong non-equilibrium/self-heating, fractional-memory kernels and non-Gaussian noise are needed to capture long correlations and bursty mis-thresholding.
• Multi-physics coupling (thermal/electrical/stress) can bias κ_T; joint calibration across fields is required.
Falsification Line & Experimental Suggestions
• Falsification line: see the JSON falsification_line; require global ΔAIC/Δχ²/dof/ΔRMSE thresholds and the disappearance of key covariances.
• Suggestions:
- Threshold maps: dense scans in (T, E_th), (t, ∂E_th/∂t), and (ζ_fault, η_loc);
- Geometry/interface: tune ζ_topo/psi_interface via interlayers/annealing to test threshold–hysteresis covariance slopes;
- Synchronized acquisition: I–V–T + noise spectra + fault imaging to validate the hard link between f_knee and hysteresis/drift;
- Noise control: reduce σ_env and quantify linear effects of k_TBN on α_1f and threshold jitter.
External References
• Sze, S. M., & Ng, K. K. Physics of Semiconductor Devices (thresholds & interface states).
• Hooge, F. N. 1/f noise sources and mechanisms.
• Poole, H. H.; Frenkel, J.; Fowler–Nordheim. Classical field-injection and tunneling theories.
• Shklovskii, B. I., & Efros, A. L. Electronic Properties of Doped Semiconductors (percolation & defect networks).
• van der Ziel, A. Noise in Solid State Devices.
Appendix A | Data Dictionary & Processing Details (optional)
• Metric dictionary: E_th, ∂E_th/∂t, P_trig, λ_fault, ΔV_hys, ΔI_loop, S_I(f), f_knee, α_1f, ζ_fault, η_loc, τ_lag, κ_T as defined in Section II; SI units.
• Processing details: change-point & second-derivative detection for thresholds/hysteresis; Kalman estimation of E_th/τ_lag; robust regression/ML for P_trig/λ_fault; piecewise spectral fits for α_1f/f_knee; uncertainty propagation with TLS+EIV; hierarchical MCMC with shared priors and convergence by R̂/IAT.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out: parameter shifts < 15%, RMSE fluctuation < 10%.
• Stratified robustness: G_env↑ → f_knee rises and KS_p slightly drops; 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 0.052; 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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