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%.