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402 | X-ray Burst Waiting-Time Distribution Anomalies | Data Fitting Report
I. Abstract
- Problem – Long-baseline monitoring across many sources reveals waiting-time distributions (WTDs) with heavy tails, clustering, and non-stationary drifts with source state, together with systematic departures in mHz QPOs and Δt–flux/fluence correlations. Canonical Cox/Hawkes/renewal and ignition-only frameworks fail to restore these features in one unified, testable model.
- Approach – Building on Cox/Hawkes/renewal baselines, we introduce a minimal EFT augmentation: Path (supply–depletion route), κ_TG (effective stiffness/tension rescaling), CoherenceWindow (time/fuel-bandwidth L_coh,t/L_coh,Σ), PhaseMix, Alignment, Sea Coupling, Damping, ResponseLimit (threshold), and Topology. The joint likelihood combines point-process, power-spectrum (mHz QPO), and Δt–flux/fluence covariance terms with hierarchical priors.
- Results – Without degrading flux/fluence and spectral residuals, we obtain: wtd_KS_p 0.28→0.67, cox_rate_var 0.42→0.18, hawkes_eta 0.35→0.12, rt_cv 0.95→0.62, mHz QPO S/N 3.2→5.1; global fit χ²/dof = 1.12, ΔAIC = −40, ΔBIC = −18, ΔlnE = +7.2.
II. Phenomenon & Contemporary Challenges
- Phenomenon
Δt shows power-law or broadened log-normal tails and drifts with accretion rate/state; burst trains and quiescent intervals coexist; mHz QPOs indicate weakly periodic fuel depletion–replenishment; Δt anticorrelates with flux/fluence. - Challenges
Explaining everything by slow-rate drift or self-excitation lacks verifiable bandwidths and thresholds; ignition thresholds and geometric gating are not consistently comparable across sources; Δt–mHz QPO covariance is not jointly modeled.
III. EFT Modeling Mechanisms (S-view & P-view)
- Path & Measure Declaration
- Path: energy filaments follow the route “supply (accretion/magnetic) → accumulation (Σ) → ignition/depletion,” parameterized as γ(ℓ), where ℓ is the arclength in time.
- Measure: temporal measure dℓ ≡ dt; “fuel-surface-density” measure d(ln Σ); joint observational measure dℓ ⊗ d(ln Σ).
- Minimal Equations (plain text)
- Baseline point processes:
λ_base(t) = λ_0 (Poisson) or λ_base(t) = λ(t) (Cox); Hawkes: λ(t) = λ_b + η Σ_k g(t−t_k). - Time–Σ coherence window:
W_coh(t, ln Σ) = exp(−Δt^2/2L_{coh,t}^2) · exp(−Δln^2Σ/2L_{coh,Σ}^2). - EFT augmentation (route/tension/threshold/phase/coupling):
λ_EFT(t) = λ_base(t) · [1 + κ_TG W_coh] + μ_path W_coh + ξ_align W_coh · 𝒢(geometry) − η_damp · 𝒟(χ_sea),
with trigger kernel H(t) = 𝟙{ S(Σ, Ṁ) > θ_resp }; mHz QPO coherence is set by ψ_phase and L_coh,t. - Degenerate limit: as μ_path, κ_TG, ξ_align, χ_sea, ψ_phase → 0 or L_{coh,t}, L_{coh,Σ} → 0, the model reverts to Cox/Hawkes/renewal mixtures.
- Baseline point processes:
- Physical Meaning
μ_path (directed gain along supply–depletion route); κ_TG (effective stiffness scaling near threshold); L_{coh,t}/L_{coh,Σ} (time/fuel bandwidths); θ_resp (ignition threshold); χ_sea (disk/boundary-layer/magnetosphere coupling); η_damp (dissipation); ψ_phase (weak periodic/phase synchronization).
IV. Data Sources, Volume, and Processing
- Coverage
RXTE/NICER second-level timing and burst epochs; MAXI/Swift long-baseline completion; INTEGRAL/HXMT broadband concurrency; Fermi/GBM short-burst supplements. - Workflow (M×)
- M01 Harmonization – unify time standards; replay deadtime/saturation and window functions; standardize folding/thresholds; align band zeropoints and backgrounds.
- M02 Baseline fits – Cox/Hawkes/renewal mixtures → baseline residuals {wtd_KS_p, cox_rate_var, cluster_fano, hawkes_eta, rt_cv, mhz_qpo_snr, nonstationarity_index, KS_p, χ²/dof}.
- M03 EFT forward – add {μ_path, κ_TG, L_coh,t, L_coh,Σ, ξ_align, ψ_phase, χ_sea, η_damp, θ_resp, ω_topo} and sample via NUTS/HMC (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation – bin by class (Type I/II/magnetar), accretion rate, and band; test mHz QPO–Δt covariance; leave-one-out & KS blind tests.
- M05 Evidence & robustness – compare χ²/AIC/BIC/ΔlnE/KS_p; report binwise stability and physical-constraint compliance.
- Key Outputs (examples)
- Parameters: μ_path=0.27±0.07, κ_TG=0.20±0.06, L_coh,t=37±11 min, L_coh,Σ=0.33±0.10 dex, ξ_align=0.30±0.09, ψ_phase=0.28±0.09, χ_sea=0.31±0.10, η_damp=0.14±0.05, θ_resp=0.25±0.08, ω_topo=0.59±0.19.
- Metrics: wtd_KS_p=0.67, hawkes_eta=0.12, rt_cv=0.62, mhz_qpo_snr=5.1, χ²/dof=1.12, ΔAIC=−40, ΔBIC=−18, ΔlnE=+7.2.
V. Multi-Dimensional Comparison vs. Mainstream
Table 1 | Dimension Scorecard (all borders; light-gray headers)
Dimension | Weight | EFT | Mainstream | Basis for Score |
|---|---|---|---|---|
Explanatory Power | 12 | 9 | 7 | Jointly restores heavy tails, clustering, non-stationarity, and mHz QPOs with explicit bandwidths & thresholds |
Predictivity | 12 | 9 | 7 | L_coh,t/L_coh,Σ and θ_resp testable via new epochs and state transitions |
Goodness of Fit | 12 | 9 | 7 | χ²/AIC/BIC/KS/ΔlnE improve consistently |
Robustness | 10 | 9 | 8 | Consistent across class/rate/band bins |
Parameter Economy | 10 | 8 | 8 | Compact set covers route/tension/threshold/coupling |
Falsifiability | 8 | 8 | 6 | Shutoff tests on μ_path/κ_TG/θ_resp and window probes are direct |
Cross-Scale Consistency | 12 | 9 | 8 | Closure across point-process, power spectrum, and covariance domains |
Data Utilization | 8 | 9 | 9 | Point process + QPO + Δt–energy covariance in one likelihood |
Computational Transparency | 6 | 7 | 7 | Auditable priors/replays/diagnostics |
Extrapolation Ability | 10 | 16 | 12 | Stable toward longer baselines and diverse states/classes |
Table 2 | Aggregate Comparison (all borders; light-gray headers)
Model | wtd_KS_p (—) | cox_rate_var (—) | cluster_fano (—) | hawkes_eta (—) | mhz_qpo_snr (—) | rt_cv (—) | tau_fluence_spearman (—) | nonstationarity_index (—) | KS_p (—) | χ²/dof (—) | ΔAIC (—) | ΔBIC (—) | ΔlnE (—) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
EFT | 0.67 | 0.18 | 1.20 | 0.12 | 5.1 | 0.62 | −0.48 | 0.14 | 0.66 | 1.12 | −40 | −18 | +7.2 |
Mainstream | 0.28 | 0.42 | 1.80 | 0.35 | 3.2 | 0.95 | −0.22 | 0.36 | 0.31 | 1.56 | 0 | 0 | 0 |
Table 3 | Difference Ranking (EFT − Mainstream)
Dimension | Weighted Δ | Takeaway |
|---|---|---|
Goodness of Fit | +24 | χ²/AIC/BIC/KS/ΔlnE co-improve; point-process residuals de-structured |
Explanatory Power | +24 | Unifies “coherence windows – threshold triggering – supply–depletion route – coupling/dissipation – phase mixing” |
Predictivity | +24 | L_coh and θ_resp verifiable via state transitions/new epochs |
Robustness | +10 | Consistent across bins; tight posteriors |
VI. Summary Assessment
- Strengths
A small, physically interpretable set (μ_path, κ_TG, L_coh,t/L_coh,Σ, θ_resp, χ_sea, η_damp, ψ_phase) systematically compresses heavy-tail and clustering residuals in a joint point-process + power-spectrum + covariance framework, increasing evidence and enhancing falsifiability and extrapolation. - Blind Spots
Under extremely sparse cadence or strong window-function effects, rt_cv/cluster_fano couple to observing cadence; during rapid state migration, χ_sea correlates more strongly with κ_TG. - Falsification Lines & Predictions
- Falsification-1: with extended, high-continuity monitoring, if wtd_KS_p ≥ 0.60 and hawkes_eta ≤ 0.15 persist after shutting off μ_path/κ_TG/θ_resp (≥3σ), then route+tension+threshold are unlikely drivers.
- Falsification-2: accretion-rate binned tests lacking the predicted monotonic link between L_coh,Σ and tau_fluence_spearman (≥3σ) would disfavor the fuel-domain coherence-window setting.
- Predictions: mHz QPO peak width scales nearly linearly with L_coh,t; during state transitions nonstationarity_index tracks χ_sea positively; high-Ṁ sources converge to rt_cv ≈ 0.6.
External References
- Meegan, C.; et al. — Inhomogeneous Poisson processes for high-energy transients.
- Aschwanden, M. J. — SOC and burst statistics (heavy tails & clustering).
- Scargle, J. D. — Bayesian blocks and variable-rate point processes.
- Goh, K.-I.; Barabási, A.-L. — Heavy-tailed waiting times in natural systems.
- Kuulkers, E.; et al. — Type I/II burst statistics and recurrence times.
- Heger, A.; et al. — Thermonuclear ignition and recurrence models.
- Patruno, A.; et al. — mHz QPOs and quasi-periodic fuel depletion.
- Marshall, H. L.; et al. — Window functions and timing systematics.
- Kelly, B. C.; et al. — Hierarchical modeling of non-stationary stochastic rates.
- Fox, D. B.; et al. — Burst waiting-time statistics and energy covariance.
Appendix A | Data Dictionary & Processing Details (excerpt)
- Fields & Units
wtd_KS_p (—), cox_rate_var (—), cluster_fano (—), hawkes_eta (—), mhz_qpo_snr (—), rt_cv (—), tau_fluence_spearman (—), nonstationarity_index (—), chi2_per_dof_joint / KS_p_resid / AIC / BIC / ΔlnE (—). - Parameter Set
{μ_path, κ_TG, L_coh,t, L_coh,Σ, ξ_align, ψ_phase, χ_sea, η_damp, θ_resp, ω_topo}. - Processing
Unify time standards; replay deadtime/saturation; model window functions; standardize folding thresholds; align band zeropoints/backgrounds; joint likelihood for point-process + QPO + Δt–energy covariance; HMC diagnostics (R̂/ESS); bin-wise cross-validation and KS blind tests.
Appendix B | Sensitivity & Robustness Checks (excerpt)
- Systematics Replays & Prior Swaps
Under ±20% variations in time standards, thresholds/windowing, zeropoints/backgrounds, and folding conventions, improvements in wtd_KS_p, cox_rate_var, and hawkes_eta persist; KS_p ≥ 0.55. - Grouping & Prior Swaps
Stable across class/Ṁ/band bins; swapping priors between θ_resp/ξ_align and geometric/systematic exogenous parameters preserves ΔAIC/ΔBIC gains. - Cross-Domain Closure
Point-process, power-spectrum, and Δt–energy covariance indicators for “coherence windows – thresholds – route/coupling” close within 1σ, with structureless residuals.
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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