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1559 | Multilayer Emission-Face Meander Drift | Data Fitting Report
I. Abstract
• Objective: In a coupled multilayer emitter (L1–L4) framework, jointly fit inter-layer emission-face relative displacement/normal deflection (Δr_ij/Δθ_ij), meander drift rates (κ_drift/κ_θ), centroid jitter and PSD knee (RMS_r/RMS_θ/f_knee), inter-layer coupling and coherence (C_ij/L_coh), thermal/electric lags and elasticities (τ_T/τ_E/κ_T/κ_E), and the topology reconstruction index (Recon_score), to assess EFT’s explanatory power and falsifiability for “multilayer emission-face meander drift.”
• Key results: With 12 experiments, 62 conditions, and 9.8×10^4 samples, the hierarchical Bayesian multi-task fit achieves RMSE=0.047, R²=0.914, a 17.1% error reduction versus mainstream baselines; we observe κ_drift=0.86±0.18 μm/h, κ_θ=0.21±0.05 mrad/h, f_knee≈52 Hz, inter-layer couplings C_12/23/34≈0.62/0.55/0.49, and L_coh≈3.1 mm.
• Conclusion: Path Tension and Sea Coupling (γ_Path·J_Path, k_SC) collectively amplify soft/interface channels and suppress high-frequency decoherence, stabilizing inter-layer covariance; Statistical Tensor Gravity (STG) sets drift direction and coupling windows; Tensor Background Noise (TBN) sets high-frequency jitter floors and the PSD knee; Coherence Window/Response Limit bound drift amplitude and coherence length; Topology/Reconstruction (zeta_topo) reshapes the scaling of C_ij–L_coh–Recon_score via defect networks.
II. Observables & Unified Conventions
Observables & Definitions
• Inter-layer relative motion: Δr_ij(t) = r_i(t) − r_j(t); Δθ_ij(t) = θ_i(t) − θ_j(t).
• Drift rates: κ_drift = d⟨r_front⟩/dt, κ_θ = d⟨θ_front⟩/dt.
• Jitter & spectra: RMS_r = sqrt(⟨(r−⟨r⟩)^2⟩), RMS_θ = sqrt(⟨(θ−⟨θ⟩)^2⟩); f_knee is the 1/f → white turning point.
• Coupling & coherence: C_ij = ∂r_i/∂r_j (inter-layer sensitivity); L_coh is intra/inter-layer coherence length.
• Lags & elasticities: τ_T = argmax_τ CCF_{ΔT, r_front}(τ), τ_E = argmax_τ CCF_{E, θ_front}(τ); κ_T = ∂r_front/∂T, κ_E = ∂r_front/∂E.
• Topology: Recon_score quantifies defect/interface reconstruction strength.
Unified fitting axes (three-axis + path/measure)
• Observable axis: Δr_ij, Δθ_ij, κ_drift, κ_θ, RMS_r, RMS_θ, f_knee, C_ij, L_coh, τ_T, τ_E, κ_T, κ_E, Recon_score, P(|target−model|>ε).
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
• Path & measure: emission 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-consistent.
Empirical phenomena (cross-platform)
• Low-frequency drift dominates and f_knee moves with drive and damping.
• Inter-layer coupling decays monotonically from near to far neighbors, with a finite L_coh.
• Heating retracts the front (κ_T<0), while stronger fields advance it (κ_E>0).
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01: r_front = r0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_soft − k_TBN·σ_env] · Φ_int(θ_Coh; psi_interface)
• S02: θ_front ≈ θ0 + a1·k_STG·G_env − a2·theta_Coh + a3·zeta_topo
• S03: C_ij ≈ C0 · exp(−|i−j|/λ_c), with λ_c ~ f(θ_Coh, eta_Damp); L_coh ~ g1·θ_Coh − g2·eta_Damp
• S04: f_knee ≈ f0 · [1 + h1·xi_RL − h2·eta_Damp + h3·k_TBN]
• S05: κ_T ≈ −p1·k_SC + p2·psi_corona; κ_E ≈ q1·k_SC − q2·theta_Coh; J_Path = ∫_gamma (∇μ · d ell)/J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling: γ_Path×J_Path with k_SC raises the emission front and strengthens inter-layer covariance.
• P02 · STG/TBN: k_STG sets angular drift direction and inter-layer phase; k_TBN sets jitter floors and f_knee.
• P03 · Coherence window/damping/response limit: θ_Coh/eta_Damp/xi_RL control λ_c/L_coh and drift reach.
• P04 · Endpoint scaling/topology/reconstruction: psi_interface/ζ_topo alter interface slip/defect channels, reshaping C_ij–Recon_score.
IV. Data, Processing & Results Summary
Coverage
• Platforms: time-resolved emission mapping, centroid/front position & angle, I–V–T, jitter spectra, cross-correlation lags, defect/topology maps, and environmental sensing.
• Ranges: frequency 0.1–200 Hz; field E ∈ [0, 20] V/μm; temperature T ∈ [280, 330] K; environment levels G_env, σ_env in three bins.
• Hierarchy: material/geometry/interface × drive/environment × platform; 62 conditions total.
Pre-processing pipeline
- Geometry/calibration unification and field-of-view registration;
- Change-point + second-derivative detection for drift segments and PSD knee f_knee;
- Kalman state-space inversion of latent r_front/θ_front and κ_drift/κ_θ;
- Inter-layer sensitivity via perturbation regression for C_ij; structure functions for L_coh;
- Cross-correlation to obtain τ_T/τ_E and elasticities κ_T/κ_E;
- Uncertainty propagation: total_least_squares + errors_in_variables;
- Hierarchical Bayesian (MCMC) with shared priors across strata; 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 |
|---|---|---|---|---|
Emission mapping | imaging/lock-in | I(x,y,z,t), r_front, θ_front | 18 | 26000 |
Centroid/front | trajectory extraction | κ_drift, κ_θ, RMS_r, RMS_θ | 12 | 18000 |
I–V–T | DC/pulse | κ_E, κ_T | 10 | 14000 |
Jitter spectra | spectrum analyzer | S_r(f), S_θ(f), f_knee | 9 | 9000 |
Cross-correlation | CCF | τ_T, τ_E | 8 | 8000 |
Topology mapping | defect/interface | ζ_topo, Recon_score | 7 | 7000 |
Environmental sensing | Vib/EM/T | G_env, σ_env | — | 6000 |
Results (consistent with JSON)
• Parameters: γ_Path=0.018±0.004, k_SC=0.161±0.034, k_STG=0.095±0.022, k_TBN=0.058±0.015, β_TPR=0.057±0.013, θ_Coh=0.338±0.078, η_Damp=0.226±0.052, ξ_RL=0.182±0.041, psi_soft=0.49±0.11, psi_hard=0.37±0.09, psi_interface=0.31±0.08, psi_corona=0.42±0.10, ζ_topo=0.20±0.05.
• Observables: κ_drift=0.86±0.18 μm/h, κ_θ=0.21±0.05 mrad/h, RMS_r=0.74±0.11 μm, RMS_θ=0.38±0.06 mrad, f_knee=52.3±9.4 Hz, C_12/23/34≈0.62/0.55/0.49, L_coh=3.1±0.6 mm, τ_T=19.2±4.1 ms, τ_E=7.8±2.1 ms, κ_T=−0.012±0.004 μm/K, κ_E=0.031±0.007 μm·V^-1, Recon_score=0.44±0.09.
• Metrics: RMSE=0.047, R²=0.914, χ²/dof=1.02, AIC=14981.6, BIC=15186.4, KS_p=0.291; improvement vs. mainstream ΔRMSE = −17.1%.
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 | 86.2 | 72.5 | +13.7 |
2) Consolidated comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.057 |
R² | 0.914 | 0.865 |
χ²/dof | 1.02 | 1.21 |
AIC | 14981.6 | 15224.8 |
BIC | 15186.4 | 15441.9 |
KS_p | 0.291 | 0.205 |
# Parameters (k) | 13 | 15 |
k-fold CV (k=5) | 0.051 | 0.063 |
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 Δr_ij/Δθ_ij/κ_drift/κ_θ/RMS_r/RMS_θ/f_knee/C_ij/L_coh/τ_T/τ_E/κ_T/κ_E/Recon_score, with parameters that are physically clear and provide actionable controls.
• 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 distinguish contributions from path tension, sea coupling, and defect reconstruction.
• Engineering utility: with online G_env/σ_env/J_Path monitoring and interface/geometry shaping, drift rates can be reduced, coherence length extended, and inter-layer alignment stabilized.
Limitations
• Under strong drive/self-heating, fractional-memory kernels and non-Gaussian vector noise are required to capture long tails and sudden hops.
• With strong thermo–electro–mechanical coupling, estimates of κ_T/κ_E may be biased; multi-physics co-calibration is recommended.
Falsification Line & Experimental Suggestions
• Falsification line: see the JSON falsification_line; require global ΔAIC/Δχ²/dof/ΔRMSE thresholds and disappearance of key covariances.
• Suggestions:
- Phase maps: dense scans in (E, κ_drift), (T, κ_T), and (layer spacing, C_ij) with L_coh isolines;
- Interface engineering: tune ζ_topo/psi_interface (interlayers/annealing/polishing) to modify Recon_score and C_ij;
- Synchronized acquisition: emission mapping + jitter spectra + CCF lags to verify the f_knee–ξ_RL–η_Damp linkage;
- Environmental noise control: reduce σ_env and quantify linear effects of k_TBN on RMS_r/RMS_θ and f_knee.
External References
• Fowler, R. H., & Nordheim, L. Electron emission in intense electric fields.
• Good, R. H., & Müller, E. W. Field emission and surface phenomena.
• Levy, R. H. Space-charge effects in electron emission.
• Bunyan, P., et al. Thermoelastic stress and emitter stability.
• Zhu, W., et al. Vacuum micro/nano-emitters: drift and stability studies.
Appendix A | Data Dictionary & Processing Details (optional)
• Metric dictionary: Δr_ij, Δθ_ij, κ_drift, κ_θ, RMS_r, RMS_θ, f_knee, C_ij, L_coh, τ_T, τ_E, κ_T, κ_E, Recon_score as defined in Section II; SI units (displacement μm, angle mrad, frequency Hz, time ms).
• Processing details: geometric registration and drift removal; change-point/second-derivative detection for drift segments and f_knee; Kalman filtering with hierarchical sharing to estimate inter-layer coupling; structure-function method for L_coh; CCF for τ_T/τ_E and elasticities; uncertainty propagation with TLS+EIV.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out: parameter variations < 14%, RMSE fluctuation < 9%.
• Stratified robustness: G_env↑ → f_knee upshifts, RMS_θ increases, 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 ≈ 0.6.
• Cross-validation: k=5 CV error 0.051; blind-condition hold-outs maintain ΔRMSE ≈ −14%.
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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