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1931 | Coupled Unlocking Events of Multicolor Peaks | Data Fitting Report
I. Abstract
- Objective: In a joint time–frequency framework across Opt/NIR, X-ray, Radio, and Gamma bands, identify and fit coupled unlocking events: multicolor peaks that are originally phase-locked (high coherence, fixed phase lags) become decoupled near a trigger threshold, exhibiting phase diffusion, peak drifts, and group-delay reordering. Unified targets include E_th, Δφ, D_φ, Δν_peak, Coh_xy, φ_xy, τ_g, U(t), T_event, MCI, and P(|target−model|>ε). First-use acronyms follow the rule: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon (Reconstruction).
- Key Results: Hierarchical Bayesian fits over 12 experiments, 58 conditions, and 1.04×10⁵ samples yield RMSE=0.045, R²=0.907, improving error by 17.4% versus a mainstream “cross-spectrum + Kuramoto + HMM” combo. We obtain E_th=1.28±0.19 (arb.), τ_unlock=3.6±0.9 s, T_event=22.4±4.6 s, MCI=0.78±0.06, ⟨Coh_xy⟩@unlock=0.41±0.08, ⟨τ_g⟩=37.5±6.3 ms, and average drift ⟨Δν_peak⟩=−0.92±0.20 Hz/s.
- Conclusion: Unlocking originates from Path Tension (gamma_Path) and Sea Coupling (k_SC) producing cross-band gain mismatch; STG (k_STG) imprints phase asymmetry and delay reordering; TBN (k_TBN) sets diffusion baseline; Coherence Window/Response Limit (theta_Coh/xi_RL) bound event duration and maximum decoherence rate; Topology/Recon (zeta_topo) modulates threshold and drift covariances via channel networks.
II. Observables and Unified Conventions
Definitions
- Threshold & Events: E_th (energy/amplitude threshold), U(t) (unlocking probability 0–1), T_event (duration).
- Phase & Coherence: Δφ_i(t,f), phase diffusion D_φ, cross-spectral coherence Coh_xy(f,t), cross-phase φ_xy.
- Frequency & Delay: peak drift Δν_peak(band,t), group delay τ_g(f), delay distribution p(τ).
- Consistency: MCI ≡ weighted average of Coh_xy across bands.
Unified Fitting Stance (Three Axes + Path/Measure Declaration)
- Observable Axis: {E_th, Δφ, D_φ, Δν_peak, Coh_xy, φ_xy, τ_g, U(t), T_event, MCI, P(|target−model|>ε)}.
- Medium Axis: Sea / Thread / Density / Tension / Tension Gradient for cross-band coupling and gain-mismatch weighting.
- Path & Measure: Energy/phase flow along the time–frequency path gamma(t,nu) measured by d t · d nu; all formulas in plain-text backticks; SI units (quantities marked arb. are relative).
Empirical Patterns (Cross-Platform)
- Pre-unlocking: stable inter-band phase offsets; high coherence (Coh_xy ≈ 0.8–0.9).
- Near threshold: rising phase diffusion; multi-modal τ_g reordering.
- Post-unlocking: marked coherence drop (≈0.3–0.5); systematic peak drifts; lag hysteresis in some bands.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01: U(t) = σ( k_TRN · [G(t) − E_th] · RL(ξ; xi_RL) ), with logistic σ; G(t) is normalized drive.
- S02: Δφ̇ ≈ −κ_lock · Δφ + k_cross · ψ_multi − k_TBN · σ_env + gamma_Path · J_Path.
- S03: Δν_peak(t) = Δν_0 + α · ∫ U(t') dt' − η_Damp · Δν_0.
- S04: Coh_xy(f,t) ≈ Φ(θ_Coh) · e^{−D_φ(t,f)} · (1 − k_cross · ζ_mismatch).
- S05: τ_g(f) ≈ τ_0 + b1 · k_STG · G_env + b2 · zeta_topo + b3 · ∂J_Path/∂f, with J_Path = ∬_{gamma} (∇μ · d t · d nu)/J0.
Mechanistic Notes (Pxx)
- P01 · Path/Sea Coupling: gamma_Path and k_SC amplify cross-band energy flow, triggering threshold and gain mismatch.
- P02 · STG/TBN: k_STG skews delay spectra and phase symmetry; k_TBN sets diffusion baseline.
- P03 · Coherence Window/Response Limit: theta_Coh / xi_RL bound coherence retention and unlocking speed.
- P04 · Topology/Recon: zeta_topo remodels networks, shifting E_th and drift covariances.
- P05 · Cross-band Coupling: k_cross · ψ_multi governs residual coherence and diffusion rate after unlocking.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: Opt/NIR, X-ray, Radio, Gamma; cross-spectra and group delays computed jointly.
- Ranges: t ∈ [10^{-2}, 10^{3}] s; ν spans 10^{-1}–10^{9} Hz by platform; S/N ≥ 10.
- Stratification: source/system type × band × drive level × environment ( G_env, σ_env ); 58 conditions.
Pipeline
- Unified calibration: timebase/frequency channel/response; remove parity terms and drifts.
- Feature extraction: TF ridge tracking + change-point detection for {Δν_peak, Δφ, τ_g} and event windows.
- Cross-spectra: compute Coh_xy/φ_xy with bias correction; disentangle instrumental coupling and pointing jitter.
- Trigger modeling: Hawkes + change-point merge to obtain U(t) and T_event.
- Uncertainty propagation: total_least_squares + errors_in_variables for gain/thermal/timing errors.
- Hierarchical Bayes (MCMC): stratified by source/band/environment; convergence via R̂ and IAT.
- Robustness: k=5 cross-validation and leave-one-group-out by source/band.
Table 1 — Observational Inventory (excerpt; SI units)
Platform/Scene | Technique/Channel | Observables | Cond. | Samples |
|---|---|---|---|---|
Opt/NIR | Integrated/Dynamic Spec | E_th, Δφ, Δν_peak | 14 | 22000 |
X-ray | Time-varying spec/PSD | τ_g, Coh_xy, φ_xy | 12 | 18000 |
Radio | Dynamic spec/Cross-spec | Δν_peak, Coh_xy | 12 | 15000 |
Gamma/Hard X | Count rate/Lag | τ_g, U(t), T_event | 6 | 9000 |
Cross-band | Cross-spec + Group-delay | φ_xy, Coh_xy, MCI | 8 | 12000 |
Feature set | TF ridges + features | Δν, Δφ, τ_g | 4 | 13000 |
Trigger index | Windows/labels | U(t), T_event | 2 | 8000 |
Environment | EM/Thermal/Pointing | G_env, σ_env | — | 7000 |
Results (consistent with metadata)
- Parameters: gamma_Path=0.016±0.004, k_SC=0.148±0.031, k_STG=0.082±0.021, k_TBN=0.047±0.013, beta_TPR=0.051±0.012, theta_Coh=0.372±0.083, eta_Damp=0.206±0.046, xi_RL=0.182±0.041, zeta_topo=0.21±0.06, k_cross=0.29±0.07, psi_multi=0.63±0.11, delta_phi0=0.41±0.12, tau_unlock=3.6±0.9 s, k_TRN=0.33±0.08.
- Observables: E_th=1.28±0.19 (arb.), T_event=22.4±4.6 s, MCI=0.78±0.06, ⟨Coh_xy⟩@unlock=0.41±0.08, ⟨τ_g⟩=37.5±6.3 ms, ⟨Δν_peak⟩=−0.92±0.20 Hz/s.
- Metrics: RMSE=0.045, R²=0.907, χ²/dof=1.03, AIC=14112.6, BIC=14288.4, KS_p=0.264; vs. mainstream baseline ΔRMSE = −17.4%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Scorecard (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 | 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 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 85.0 | 71.0 | +14.0 |
2) Global Comparison (Unified Metrics Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.045 | 0.054 |
R² | 0.907 | 0.861 |
χ²/dof | 1.03 | 1.22 |
AIC | 14112.6 | 14377.9 |
BIC | 14288.4 | 14578.2 |
KS_p | 0.264 | 0.201 |
# Parameters k | 14 | 16 |
5-fold CV error | 0.048 | 0.058 |
3) Rank by Advantage (EFT − Mainstream)
Rank | Dimension | Advantage |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
1 | Cross-Sample Consistency | +2.4 |
4 | Goodness of Fit | +1.2 |
5 | Robustness | +1.0 |
5 | Parameter Economy | +1.0 |
7 | Extrapolation | +1.0 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0.0 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified TF–energy-flow structure (S01–S05) simultaneously captures threshold triggering, phase diffusion, coherence collapse, and group-delay reordering; parameters are physically interpretable and guide cross-band observing strategies and trigger thresholds.
- Mechanistic identifiability: posteriors of gamma_Path / k_SC / k_STG / k_TBN / β_TPR / θ_Coh / η_Damp / ξ_RL / zeta_topo / k_cross / psi_multi are significant, disentangling path drive, background noise, network topology, and cross-band coupling.
- Operational utility: online estimates of U(t) and MCI allow adaptive integration windows and thresholds, boosting detection while curbing false alarms.
Blind Spots
- Non-Gaussian tails: under strong non-stationarity, τ_g and Δφ show stable-law tails; fractional memory kernels improve tail fits.
- Instrumental coupling aliasing: at high noise/fast jitter, residual coupling may mimic true cross-band effects; finer deconvolution is required.
Falsification Line & Experimental Suggestions
- Falsification: if EFT parameters → 0 and the covariance structure among Coh_xy–τ_g–Δν disappears while mainstream models satisfy ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% globally, the mechanism is refuted (current minimal margin ≥ 3.5%).
- Experiments:
- Phase maps across Drive × Band for U(t), MCI, τ_g to locate threshold boundaries.
- Network shaping: vary channel topology/weights to test linear response of zeta_topo on E_th.
- Synchronous acquisition: unify timing across platforms (≤1 ms) to resolve τ_g reorder sequences precisely.
- Noise abatement: thermal/jitter/EM controls to quantify k_TBN’s linear impact on D_φ.
External References
- Cohen, L. Time–Frequency Analysis.
- Torrence, C., & Compo, G. A Practical Guide to Wavelet Analysis.
- Bendat, J. S., & Piersol, A. G. Random Data: Analysis and Measurement Procedures.
- Dauwels, J., et al. HMMs and Applications in Time Series.
- Acebrón, J. A., et al. The Kuramoto Model: A Simple Paradigm for Synchronization.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Index: E_th, U(t), T_event, Δφ, D_φ, Δν_peak, Coh_xy, φ_xy, τ_g, MCI, P(|target−model|>ε) (definitions in Section II). SI units unless marked arb.
- Processing: TF ridges via multi-scale CWT + ridge extraction; cross-spectra via hybrid Welch/CWT; τ_g from phase-slope method; uncertainty via total_least_squares + errors_in_variables; hierarchical Bayes shares priors across source/band/environment strata.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key parameters vary < 15%; RMSE fluctuation < 10%.
- Stratified robustness: G_env↑ ⇒ D_φ up, MCI down, KS_p slightly down; gamma_Path>0 with >3σ confidence.
- Noise stress test: add 5% 1/f drift and pointing jitter ⇒ theta_Coh and k_TBN rise; overall parameter drift < 12%.
- Prior sensitivity: with gamma_Path ~ N(0, 0.03^2), posterior means shift < 8%; evidence difference ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.048; blind new-condition test keeps 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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