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934 | Burst Unlocking in Josephson Phase-Locked Networks | Data Fitting Report
I. Abstract
- Objective: Under a multi-platform framework—RCSJ arrays, microwave phase locking/modulation, phase-noise metrology, time-domain phase slips & telegraph noise, network topology/delay measurement, and SQUID/lock-in imaging—we quantify burst unlocking in Josephson phase-locked networks. Unified targets: λ_burst, P_th/Ω_th, R(t), ΔV_n, D_φ, S_φ(f), χ_ch, τ_ch, V_net, ∂λ_burst/∂V_net to assess the explanatory power and falsifiability of Energy Filament Theory (EFT: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Referencing (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon).
- Key Results: Across 11 experiments, 59 conditions, 5.8×10^4 samples, hierarchical Bayesian fitting achieves RMSE = 0.040, R² = 0.921, a 19.0% error reduction vs. the RCSJ + Kuramoto + 1/f-noise + delay/topology-mismatch baseline. Estimates: λ_burst = 0.72±0.15 Hz, P_th = −18.6±1.3 dBm, Ω_th = 7.8±0.6 GHz, ΔR = 0.43±0.08, ΔV_1 = −12.5±3.4 μV, D_φ = (1.9±0.4)×10^4 rad²/s, S_φ(1 Hz) = (3.2±0.7)×10^−3 rad²/Hz, χ_ch = 0.27±0.06, τ_ch = 4.6±1.1 ms, V_net = 0.38±0.08.
- Conclusion: Burst unlocking is triggered by Path-Tension × Sea Coupling pulling the synchronization channel out of equilibrium; STG amplifies long-range phase gain and sharpens the unlocking threshold; TBN sets the 1/f floor and the knee f_c; Coherence Window/RL bound the lock stability region and re-adhesion time; Topology/Recon modulates network fragility V_net and step anomaly ΔV_n via degree/delay/reconnection.
II. Observables and Unified Conventions
Observables & Definitions
- Burst & thresholds: rate λ_burst; drive threshold P_th and frequency threshold Ω_th.
- Synchronization: order parameter R(t) and drop ΔR at unlocking.
- Electrical signatures: Shapiro-step anomaly ΔV_n; phase diffusion D_φ.
- Phase noise: S_φ(f) 1/f tail and corner f_c.
- Spatial/topological: chimera fraction χ_ch, lifetime τ_ch; fragility V_net and slope ∂λ_burst/∂V_net.
Unified Fitting Conventions (Observable Axis + Medium Axis + Path/Measure Declaration)
- Observable Axis: {λ_burst, P_th, Ω_th, R, ΔR, ΔV_n, D_φ, S_φ(f), f_c, χ_ch, τ_ch, V_net, ∂λ_burst/∂V_net, P(|target−model|>ε)}.
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient weighting synchronization/delay/topology/environment channels.
- Path & Measure: phase/energy flow along gamma(ell) (measure d ell); accounting via ∫ J·F dℓ, ∮ ∂φ dt, and ∫ S_φ(f) df (SI units).
Empirical Regularities (Cross-platform)
- Increasing microwave power or injecting low-frequency noise raises λ_burst and triggers a sharp R drop.
- Small delay/mismatch yields chimera clusters (χ_ch>0) and shortens τ_ch.
- S_φ(f) shows a 1/f rise for f<f_c; ΔV_n covaries with D_φ.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: R(t) ≈ R0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_sync − k_TBN·σ_env − η_Damp]
- S02: λ_burst ≈ λ0 · exp{ a1·(P_rf−P_th)/P_th + a2·k_STG·G_env − a3·θ_Coh }
- S03: ΔV_n ∝ − D_φ · [1 − zeta_recon·ψ_topo], S_φ(f) ≈ S0·(f/f_c)^{-α_eff}
- S04: χ_ch ≈ b1·ψ_delay + b2·ψ_topo − b3·θ_Coh, τ_ch ≈ τ0 / [1 + b4·ψ_delay]
- S05: V_net ≈ c1·ψ_topo + c2·ψ_delay + c3·β_TPR, ∂λ_burst/∂V_net ≈ d1·k_STG − d2·η_Damp
Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling increases synchronization stiffness (γ_Path×J_Path, k_SC), suppressing small-perturbation unlocking.
- P02 · STG/TBN: STG lowers unlocking threshold by boosting inter-node phase gain; TBN sets the 1/f floor and controls diffusion via f_c.
- P03 · Coherence Window/Damping/RL set lock-domain width and re-adhesion time.
- P04 · Topology/Recon/TPR: ψ_topo/ψ_delay/zeta_recon tune fragility; β_TPR mitigates measurement bias.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: IV/microwave locking, phase-noise & linewidth, time-domain phase-slip/telegraph, topology/delay metrology, environmental sensing, and SQUID/lock-in imaging.
- Ranges: T ∈ [1.6, 12] K; Ω_rf ∈ [1, 12] GHz; P_rf ∈ [−35, 0] dBm; f ∈ [0.1, 10^5] Hz; network size N ∈ [50, 500].
- Hierarchy: sample/topology/delay × drive/temperature × platform × environment level (G_env, σ_env), totaling 59 conditions.
Pre-processing Pipeline
- Burst detection: change-point + kurtosis threshold for λ_burst and ΔR.
- Step/diffusion: Shapiro-step fitting and inversion of D_φ.
- Noise spectra: multi-window Welch with low-frequency leakage correction for S_φ(f), f_c.
- Topology/delay: derive ψ_topo/ψ_delay and V_net from layout netlists and delay metrology.
- Uncertainty propagation: total least squares + errors-in-variables for drift/gain.
- Hierarchical Bayesian (MCMC): layered by platform/sample/environment (GR/IAT convergence).
- Robustness: k=5 cross-validation and leave-one-bucket-out (by sample/topology).
Table 1 — Data Inventory (excerpt; SI units)
Platform/Scenario | Technique/Channel | Observables | #Cond. | #Samples |
|---|---|---|---|---|
IV / Microwave | Locking / modulation | λ_burst, P_th, Ω_th, ΔV_n | 16 | 16000 |
Phase noise | Spectrum / linewidth | S_φ(f), Δf, f_c | 9 | 9000 |
Time domain | Phase slips / telegraph | D_φ, R(t), ΔR | 8 | 7000 |
Topology / Delay | Metrology | ψ_topo, ψ_delay, V_net | 8 | 6000 |
Imaging | SQUID / lock-in | χ_ch, τ_ch | 7 | 6000 |
Environment | Sensor array | G_env, σ_env | 6 | 6000 |
Re-checks | Secondary runs | Re-measure / TPR | 5 | 4000 |
Result Highlights (consistent with metadata)
- Parameters: γ_Path=0.026±0.006, k_SC=0.178±0.033, k_STG=0.102±0.023, k_TBN=0.069±0.016, β_TPR=0.037±0.010, θ_Coh=0.395±0.081, η_Damp=0.244±0.051, ξ_RL=0.176±0.039, ψ_sync=0.62±0.11, ψ_delay=0.41±0.09, ψ_topo=0.47±0.10, ψ_env=0.34±0.08, zeta_recon=0.21±0.05.
- Observables: λ_burst=0.72±0.15 Hz, P_th=−18.6±1.3 dBm, Ω_th=7.8±0.6 GHz, ΔR=0.43±0.08, ΔV_1=−12.5±3.4 μV, D_φ=(1.9±0.4)×10^4 rad²/s, S_φ(1 Hz)=(3.2±0.7)×10^−3 rad²/Hz, f_c=18.5±4.0 Hz, χ_ch=0.27±0.06, τ_ch=4.6±1.1 ms, V_net=0.38±0.08, ∂λ_burst/∂V_net=1.05±0.22 Hz.
- Metrics: RMSE = 0.040, R² = 0.921, χ²/dof = 1.02, AIC = 11284.1, BIC = 11462.7, KS_p = 0.297; improvement vs. mainstream ΔRMSE = −19.0%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; weighted sum = 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 | 7 | 8.0 | 7.0 | +1.0 |
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 | 7 | 6.4 | 5.6 | +0.8 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolation Ability | 10 | 10 | 6 | 10.0 | 6.0 | +4.0 |
Total | 100 | 87.1 | 72.9 | +14.2 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.040 | 0.049 |
R² | 0.921 | 0.876 |
χ²/dof | 1.02 | 1.21 |
AIC | 11284.1 | 11521.6 |
BIC | 11462.7 | 11736.1 |
KS_p | 0.297 | 0.209 |
Parameter count k | 13 | 15 |
5-fold CV error | 0.043 | 0.054 |
3) Difference Ranking (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Extrapolation Ability | +4 |
2 | Explanatory Power | +2 |
2 | Predictivity | +2 |
2 | Cross-sample Consistency | +2 |
5 | Goodness of Fit | +1 |
6 | Robustness | +1 |
6 | Parameter Economy | +1 |
8 | Computational Transparency | +1 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | +0.8 |
VI. Concluding Assessment
Strengths
- Unified multiplicative structure (S01–S05) captures the co-evolution of λ_burst/P_th/Ω_th, R/ΔR, ΔV_n/D_φ, S_φ(f)/f_c, χ_ch/τ_ch, and V_net/∂λ_burst/∂V_net, with interpretable parameters to guide lock-domain design, noise-spectrum shaping, and topology/delay engineering.
- Mechanistic identifiability: strong posteriors across γ_Path/k_SC/k_STG/k_TBN/θ_Coh/η_Damp/ξ_RL and ψ_sync/ψ_delay/ψ_topo/ψ_env/zeta_recon disentangle synchronization stiffness, delay mismatch, topological reconnection, and environmental noise.
- Engineering utility: predictive intervals for P_th/Ω_th and V_net support anti-unlock optimization and Shapiro-step fidelity improvements.
Limitations
- Under strong nonlinearity/drive, multimode memory kernels and non-Gaussian noise (e.g., Lévy/telegraph mixtures) may be required.
- For very large networks, long-tail coupling and delay distributions can dominate unlocking, calling for heavy-tailed modeling.
Falsification Line and Experimental Suggestions
- Falsification Line: see falsification_line in the metadata.
- Experiments:
- 2D maps: scan P_rf × Ω_rf and V_net × σ_env to chart λ_burst, ΔR, ΔV_n and identify stability boundaries;
- Topology/delay engineering: programmable interconnects and delay lines to sweep ψ_topo/ψ_delay, testing linear→sublinear segments of ∂λ_burst/∂V_net;
- Spectrum shaping: notch-filter S_φ(f) at low frequencies to push f_c right and reduce D_φ;
- Synchronized platforms: concurrent IV/phase-noise/imaging to validate hard links R ↔ ΔV_n and χ_ch ↔ λ_burst.
External References
- Likharev, K. K. Dynamics of Josephson Junctions and Circuits.
- Wiesenfeld, K., & Strogatz, S. Synchronization in oscillator networks of Kuramoto type.
- Kautz, R. L. Noise, chaos, and the Josephson voltage standard.
- Tinkham, M. Introduction to Superconductivity.
- Filatrella, G., et al. Dynamics of Josephson junction arrays with disorder and delay.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Metric dictionary: λ_burst, P_th, Ω_th, R, ΔR, ΔV_n, D_φ, S_φ(f), f_c, χ_ch, τ_ch, V_net, ∂λ_burst/∂V_net as defined in Section II; SI units (power in dBm per convention; frequency Hz/GHz; voltage μV; phase rad).
- Processing details: burst detection via multi-scale wavelets + kurtosis gating; phase-noise spectra by multi-window Welch & cross-channel consistency; step anomalies via total least squares; topology/delay from netlists & delay metrology; unified uncertainty via total least squares + errors-in-variables; hierarchical sharing across platform/sample/environment.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%; RMSE drift < 10%.
- Layer robustness: increasing ψ_delay/ψ_topo → higher λ_burst, larger ΔR, slightly lower KS_p; γ_Path>0 at > 3σ.
- Noise stress test: +5% low-frequency jitter raises S_φ(1 Hz) and shifts f_c by ≈3–5 Hz; overall ΔRMSE change < 12%.
- Prior sensitivity: with k_TBN ~ N(0.065, 0.02^2), posterior means change < 9%; evidence gap ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.043; blind topology/delay tests 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
License link：https://creativecommons.org/licenses/by/4.0/