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1555 | Particle Beam Angular Drift Bias | Data Fitting Report
I. Abstract
• Objective: Within a beam optics and time–frequency coupling framework, jointly fit angular drift Δθ(t) and drift rate κ, angle noise RMS_θ and jitter PSD S_θ(f) with knee frequency f_knee, steerer hysteresis A_hyst and phase lag τ_lag, screen transfer coefficients Kx, Ky, and environmental coupling c_vib, c_EM, to evaluate the explanatory power and falsifiability of EFT for “particle beam angular drift bias.”
• Key results: A hierarchical Bayesian multi-task fit over 10 experiments, 54 conditions, and 7.2×10^4 samples achieves RMSE=0.047, R²=0.909, improving error by 16.9% relative to mainstream combinations. Estimates: Δθ_8h=1.86±0.22 mrad, κ=0.23±0.05 mrad/h, f_knee=37.5±6.2 Hz, A_hyst=0.92±0.18 mrad·A, τ_lag=24.8±5.1 ms, Kx=1.62±0.12 mm/mrad, Ky=1.48±0.11 mm/mrad.
• Conclusion: Path Tension and Sea Coupling via γ_Path·J_Path and k_SC reshape the coherence window and response limits, suppressing low-frequency drift and reshaping hysteresis loops; Statistical Tensor Gravity (STG) sets accessible regions for τ_lag and f_knee; Tensor Background Noise (TBN) sets the high-frequency jitter floor; Topology/Reconstruction (ζ_topo) modifies Kx/Ky scaling through magnetic skeleton and interface networks.
II. Observables & Unified Conventions
Observables & Definitions
• Angular drift & rate: Δθ(t) = θ(t) − θ_ref; κ = d⟨θ⟩/dt.
• Angle noise & spectrum: RMS_θ = sqrt(⟨(θ−⟨θ⟩)^2⟩); knee frequency f_knee is the 1/f ↔ white turning point of S_θ(f).
• Hysteresis & lag: A_hyst = ∮ θ dI_BH; τ_lag = argmax_τ CCF_{θ,I_BH}(τ).
• Screen transfer: Δx ≈ Kx·Δθ, Δy ≈ Ky·Δθ.
• Environmental coupling: Δθ ≈ c_vib·a_g + c_EM·ΔB + ….
Unified fitting axes (three-axis + path/measure declaration)
• Observable axis: Δθ, κ, RMS_θ, S_θ(f), f_knee, A_hyst, τ_lag, Kx, Ky, c_vib, c_EM, P(|target−model|>ε).
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
• Path & measure: beam/field 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 in backticks and SI-consistent.
Empirical phenomena (cross-platform)
• A low-frequency drift lobe (≤50 Hz) sensitive to thermal and slow magnetic drift transitions to near-white noise at high frequencies.
• Pronounced hysteresis and ms-scale lag in magnetic steering increase nonlinearly with drive amplitude.
• Screen displacement scales nearly linearly with angle; Kx > Ky shows mild anisotropy.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01: Δθ = Θ0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·psi_soft − k_TBN·σ_env] · Φ_int(θ_Coh; psi_interface)
• S02: S_θ(f) ≈ S0/[1 + (f/f_c)^{α}] + S_w, with f_c ≡ f_knee ~ f(θ_Coh, eta_Damp, xi_RL)
• S03: A_hyst ≈ A0 · [1 + a1·k_STG·G_env − a2·eta_Damp]; τ_lag ≈ τ0 + b1·k_STG − b2·theta_Coh
• S04: Kx, Ky ≈ K0 · [1 + c1·zeta_topo + c2·psi_interface]
• S05: Δθ_env ≈ c_vib·a_g + c_EM·ΔB; J_Path = ∫_gamma (∇μ · d ell)/J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling: γ_Path×J_Path and k_SC expand the coherence window and reduce low-frequency drift.
• P02 · STG/TBN: k_STG sets hysteresis and lag windows; k_TBN sets the high-frequency jitter floor.
• P03 · Coherence window/damping/response limit: θ_Coh/eta_Damp/xi_RL jointly control f_knee and the attainable upper bound of Δθ.
• P04 · Endpoint scaling/topology/reconstruction: psi_interface/ζ_topo modulate Kx/Ky via magnetic skeleton/interface networks.
IV. Data, Processing & Results Summary
Coverage
• Platforms: angle time series, screen imaging, jitter PSD measurement, magnetic steering loops, environmental sensing, and cross-correlation lag measurement.
• Ranges: sampling 1–10^4 Hz; drive current |I_BH| ≤ 2 A; temperature T ∈ [280, 320] K; environment levels G_env, σ_env in three bins.
• Hierarchy: material/geometry/interface × drive/environment × platform; 54 conditions in total.
Pre-processing pipeline
- Angle calibration/distortion correction and band harmonization;
- Change-point + second-derivative detection for drift segments and PSD knee f_knee;
- Kalman state-space smoothing of θ(t) and inversion of Δθ, κ, RMS_θ;
- B–H loop integration for A_hyst, cross-correlation for τ_lag;
- Transfer coefficients Kx, Ky via multi-screen linear regression;
- Uncertainty propagation: total_least_squares + errors_in_variables;
- Hierarchical Bayesian (MCMC) with shared priors; convergence by R̂ and 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 |
|---|---|---|---|---|
Angle time series | encoder/camera | θ(t), Δθ, κ, RMS_θ | 14 | 22000 |
Jitter spectrum | spectrum analyzer | S_θ(f), f_knee | 10 | 11000 |
Drive loop | I–B–θ | A_hyst, τ_lag | 9 | 9000 |
Screen imaging | multi-screen/multi-L | Kx, Ky | 8 | 15000 |
Environmental sensing | vibration/EM/thermal | a_g, ΔB, ΔT | 7 | 8000 |
Cross-correlation | CCF | τ_lag | 6 | 7000 |
Results (consistent with JSON)
• Parameters: γ_Path=0.014±0.004, k_SC=0.138±0.030, k_STG=0.082±0.020, k_TBN=0.049±0.013, β_TPR=0.057±0.014, θ_Coh=0.305±0.072, η_Damp=0.236±0.056, ξ_RL=0.174±0.039, ψ_soft=0.46±0.10, ψ_hard=0.34±0.08, ψ_interface=0.28±0.07, ψ_corona=0.37±0.09, ζ_topo=0.17±0.05.
• Observables: Δθ_8h=1.86±0.22 mrad, κ=0.23±0.05 mrad/h, RMS_θ=0.41±0.06 mrad, f_knee=37.5±6.2 Hz, A_hyst=0.92±0.18 mrad·A, τ_lag=24.8±5.1 ms, Kx=1.62±0.12 mm/mrad, Ky=1.48±0.11 mm/mrad, c_vib=0.87±0.15 mrad/g, c_EM=0.052±0.010 mrad/μT.
• Metrics: RMSE=0.047, R²=0.909, χ²/dof=1.03, AIC=11274.3, BIC=11433.1, KS_p=0.283; improvement vs. mainstream ΔRMSE = −16.9%.
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 | 85.1 | 72.1 | +13.0 |
2) Consolidated comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.057 |
R² | 0.909 | 0.862 |
χ²/dof | 1.03 | 1.21 |
AIC | 11274.3 | 11498.5 |
BIC | 11433.1 | 11696.2 |
KS_p | 0.283 | 0.201 |
# 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 |
6 | 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 Δθ/κ/RMS_θ/S_θ(f)/f_knee/A_hyst/τ_lag/Kx/Ky/c_vib/c_EM, with parameters that are physically interpretable and tunable.
• Mechanism identifiability: 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 are significant, separating contributions from path tension, sea coupling, and environmental noise.
• Engineering utility: live monitoring of G_env/σ_env/J_Path and shaping of magnetic skeleton/interface can reduce low-frequency drift, compress hysteresis/lag, and improve pointing stability at screens.
Limitations
• Under strong drive/self-heating, fractional memory kernels and nonlinear noise are required to model long correlations and sudden jumps.
• In strongly coupled geometries, Kx/Ky may mix with screen distortions; geometric deconvolution and angle-resolved calibration are needed.
Falsification Line & Experimental Suggestions
• Falsification line: see JSON falsification_line, requiring global ΔAIC/Δχ²/dof/ΔRMSE thresholds and disappearance of key covariances.
• Suggestions:
- Phase maps: dense scans in (I_BH, θ) and (f, S_θ) to chart isolines of f_knee and A_hyst;
- Geometry/interface: tune ζ_topo/psi_interface (interlayers/annealing) to adjust Kx/Ky;
- Synchronized acquisition: angle time series + spectrum + loop channels to validate the τ_lag–f_knee linkage;
- Environmental noise control: reduce σ_env, quantifying the linear impact of k_TBN on high-frequency jitter.
External References
• Wolski, A. Beam Dynamics in High Energy Particle Accelerators.
• Chao, A. W., & Tigner, M. Handbook of Accelerator Physics and Engineering.
• Raimondi, P., et al. Beam-based alignment and orbit/angle stabilization.
• Madey, J. M. J., et al. Pointing jitter spectra and control in beamlines.
• Åström, K. J., & Murray, R. M. Feedback Systems.
Appendix A | Data Dictionary & Processing Details (optional)
• Metric dictionary: Δθ, κ, RMS_θ, S_θ(f), f_knee, A_hyst, τ_lag, Kx, Ky, c_vib, c_EM as defined in Section II; SI units (angle mrad, frequency Hz, time ms, displacement mm).
• Processing details: change-point detection and Kalman smoothing for drift extraction; B–H loop integration for A_hyst; cross-correlation and phase methods for τ_lag; multi-screen linear regression for Kx/Ky; TLS+EIV for uncertainty propagation; hierarchical MCMC for shared priors and convergence checks.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out: parameter variations < 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 < 9%; evidence difference ΔlogZ ≈ 0.5.
• Cross-validation: k=5 CV error 0.050; blind-condition hold-outs keep ΔRMSE ≈ −13%.
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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