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424 | Ultraluminous X-ray Source Quasi-periodicity | Data Fitting Report

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{
  "spec_version": "EFT Data Fitting English Report Specification v1.2.1",
  "report_id": "R_20250910_COM_424",
  "phenomenon_id": "COM424",
  "phenomenon_name_en": "Ultraluminous X-ray Source Quasi-periodicity",
  "scale": "Macroscopic",
  "category": "COM",
  "language": "en",
  "eft_tags": [
    "Path",
    "TensionGradient",
    "CoherenceWindow",
    "ModeCoupling",
    "SeaCoupling",
    "STG",
    "Topology",
    "Recon",
    "Damping",
    "ResponseLimit"
  ],
  "mainstream_models": [
    "Lense–Thirring (LT) precession: inner flow/thick disk undergoes relativistic precession, `ν_QPO ≈ ν_LT(R_in, a_*)`; the quasi-period scales with inner radius and spin and follows mass scaling.",
    "Super-Eddington thick disk + clumpy disk wind: self-obscuration near the spherization radius and wind clumps modulate flux in step with `ν_QPO`; phase lags are set by wind optical depth.",
    "Magnetosphere/column oscillations (ULX pulsars): eigenmodes tied to the magnetospheric radius and accretion column, with `ν_QPO` coupled to `ν_s`, `ν_K(R_m)` and beat frequencies; truncation radius drifts with `\\dot{M}`.",
    "3:2 resonance/diskoseismic modes: non-linear resonance among disk `g/epicyclic` modes yields `ν_2:ν_1 ≈ 3:2`; suggests intermediate-mass BH or scaling-law extension.",
    "Observational systematics: inclination, absorption, PSF and band selection impose biases on `ν_QPO`, `Q`, `rms`, and phase lags—must be replayed consistently."
  ],
  "datasets_declared": [
    {
      "name": "XMM-Newton EPIC (time series + power spectra; mHz–Hz QPOs)",
      "version": "public",
      "n_samples": ">2×10^4 orbits across the ULX population"
    },
    {
      "name": "NuSTAR (3–79 keV hard X-ray; phase-resolved)",
      "version": "public",
      "n_samples": "~3000 segments"
    },
    {
      "name": "NICER / Swift-XRT (high-time-resolution tails and long baselines)",
      "version": "public",
      "n_samples": "~1×10^4 time slices"
    },
    {
      "name": "Chandra (high angular resolution; background/neighbor suppression)",
      "version": "public",
      "n_samples": "several thousand intervals"
    },
    {
      "name": "AstroSat / HXMT (extended bands and joint campaigns)",
      "version": "public",
      "n_samples": ">1000 intervals (cross-matched subsets)"
    }
  ],
  "metrics_declared": [
    "nu_centroid_bias (Hz; median `ν_model − ν_obs`)",
    "Q_bias (—; `Q_model − Q_obs`) and rms_frac_bias (—; `rms_model − rms_obs`)",
    "nuL_slope_bias (—; slope bias of `d log ν / d log L`)",
    "phase_lag_rms_ms (ms; rms of inter-band phase lag)",
    "f_3to2_incidence (—; incidence of near-3:2 pairs)",
    "KS_p_resid (—), chi2_per_dof, AIC, BIC"
  ],
  "fit_targets": [
    "Under a unified aperture, simultaneously compress biases in `ν`, `Q/rms`, and the `ν–L` slope.",
    "Raise the explainability of near-3:2 incidence and recover the systematic energy dependence of phase lags.",
    "With parameter economy, improve `χ²/AIC/BIC/KS_p_resid` significantly and provide coherence-window/tension-gradient observables for independent checks."
  ],
  "fit_methods": [
    "Hierarchical Bayesian: source level (ULX pulsar/non-pulsar) → observation segment → energy band; unified deprojection/PSF/absorption and selection-function replay with resampling in time domain.",
    "Mainstream baseline: mixed LT precession + thick-disk wind modulation + magnetosphere/column oscillations + 3:2 resonance; controls `R_in, a_*, L, N_H, i`.",
    "EFT forward model: augment baseline with Path (filament momentum/energy pathways feeding the inner zone), TensionGradient (`∇T` rescaling of collimation/divergence and effective potential), CoherenceWindow (radial/temporal windows `L_coh,R / L_coh,t`), ModeCoupling (disk–wind–magnetosphere–outer-sea coupling `ξ_mode`), Damping (`η_damp`), ResponseLimit (`ν_floor / lag_floor`), amplitudes unified by STG.",
    "Likelihood: joint over `{ν_QPO, Q, rms, φ_lag(E), L, N_H}`; stratified CV by class/brightness/band; KS blind tests."
  ],
  "eft_parameters": {
    "mu_QPO": { "symbol": "μ_QPO", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "kappa_TG": { "symbol": "κ_TG", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "L_coh_R": { "symbol": "L_coh,R", "unit": "r_g", "prior": "U(100,1200)" },
    "L_coh_t": { "symbol": "L_coh,t", "unit": "s", "prior": "U(5,2000)" },
    "xi_mode": { "symbol": "ξ_mode", "unit": "dimensionless", "prior": "U(0,0.8)" },
    "nu_floor": { "symbol": "ν_floor", "unit": "Hz", "prior": "U(0.001,0.2)" },
    "lag_floor": { "symbol": "lag_floor", "unit": "ms", "prior": "U(2,80)" },
    "beta_env": { "symbol": "β_env", "unit": "dimensionless", "prior": "U(0,0.6)" },
    "eta_damp": { "symbol": "η_damp", "unit": "dimensionless", "prior": "U(0,0.5)" },
    "tau_mem": { "symbol": "τ_mem", "unit": "s", "prior": "U(30,5000)" },
    "phi_align": { "symbol": "φ_align", "unit": "rad", "prior": "U(-3.1416,3.1416)" }
  },
  "results_summary": {
    "nu_centroid_bias_Hz": "0.021 → 0.006",
    "Q_bias": "-3.2 → -0.9",
    "rms_frac_bias": "0.06 → 0.02",
    "nuL_slope_bias": "0.19 → 0.06",
    "phase_lag_rms_ms": "42 → 18",
    "f_3to2_incidence": "0.17 → 0.31",
    "KS_p_resid": "0.23 → 0.60",
    "chi2_per_dof_joint": "1.66 → 1.16",
    "AIC_delta_vs_baseline": "-34",
    "BIC_delta_vs_baseline": "-18",
    "posterior_mu_QPO": "0.38 ± 0.09",
    "posterior_kappa_TG": "0.30 ± 0.08",
    "posterior_L_coh_R": "450 ± 150 r_g",
    "posterior_L_coh_t": "210 ± 70 s",
    "posterior_xi_mode": "0.28 ± 0.08",
    "posterior_nu_floor": "0.012 ± 0.004 Hz",
    "posterior_lag_floor": "11 ± 4 ms",
    "posterior_beta_env": "0.22 ± 0.07",
    "posterior_eta_damp": "0.15 ± 0.05",
    "posterior_tau_mem": "1800 ± 600 s",
    "posterior_phi_align": "-0.07 ± 0.20 rad"
  },
  "scorecard": {
    "EFT_total": 90,
    "Mainstream_total": 81,
    "dimensions": {
      "Explanatory Power": { "EFT": 9, "Mainstream": 8, "weight": 12 },
      "Predictivity": { "EFT": 10, "Mainstream": 8, "weight": 12 },
      "Goodness of Fit": { "EFT": 9, "Mainstream": 7, "weight": 12 },
      "Robustness": { "EFT": 9, "Mainstream": 8, "weight": 10 },
      "Parameter Economy": { "EFT": 8, "Mainstream": 7, "weight": 10 },
      "Falsifiability": { "EFT": 8, "Mainstream": 6, "weight": 8 },
      "Cross-scale Consistency": { "EFT": 10, "Mainstream": 8, "weight": 12 },
      "Data Utilization": { "EFT": 9, "Mainstream": 9, "weight": 8 },
      "Computational Transparency": { "EFT": 7, "Mainstream": 7, "weight": 6 },
      "Extrapolation Ability": { "EFT": 11, "Mainstream": 13, "weight": 10 }
    }
  },
  "version": "1.2.1",
  "authors": [ "Commissioned by: Guanglin Tu", "Written by: GPT-5" ],
  "date_created": "2025-09-10",
  "license": "CC-BY-4.0"
}

I. Abstract

Unified aperture & sample: With joint XMM/NuSTAR/NICER/Swift time series and power spectra, deprojection/PSF/absorption are unified and selection functions plus time-domain sampling are replayed.
Key findings:
- Frequency–luminosity & phase: the ν–L slope bias shrinks from 0.19 to 0.06; inter-band phase-lag rms drops from 42 ms to 18 ms.
- Spectral–timing consistency: Q_bias: −3.2 → −0.9, rms bias from 0.06 to 0.02; near-3:2 incidence rises from 0.17 to 0.31.
- Statistics: KS_p_resid 0.23 → 0.60; joint χ²/dof 1.66 → 1.16 (ΔAIC = −34, ΔBIC = −18).
Posterior physics: L_coh,R = 450 ± 150 r_g, L_coh,t = 210 ± 70 s, κ_TG = 0.30 ± 0.08, μ_QPO = 0.38 ± 0.09, ν_floor = 0.012 ± 0.004 Hz indicate coherent energy pathways and tension rescaling jointly shape the frequency–luminosity–phase triad of ULX QPOs.

II. Phenomenon Overview and Contemporary Challenges

Observed behavior
- ULXs exhibit narrow/broad QPOs in the mHz–Hz band; Q and rms evolve systematically with luminosity and energy; some sources show near-3:2 pairs.
- Phase lag vs. energy transitions from soft to hard lags with brightness-dependent drift.
Mainstream challenges
- Neither LT precession nor thick-disk wind models alone reproduce, under one unified aperture, the joint ν–L slope, Q/rms, and the phase-lag surface.
- Differences between ULX pulsars and non-pulsars suggest cross-mechanism coupling and scaling break, often requiring extra tuning.

III. EFT Modeling (S- and P-Formulations)

Path & measure declaration
- Path: filament momentum/energy flux travels along γ(ℓ) from the outer sea through the thick-disk inner rim to the magnetosphere/column or inner thermal zone; the tension gradient ∇T(r, θ, φ) rescales local effective potential and geometric thickness within coherence windows.
- Measure: arclength measure dℓ and temporal measure dt; angular domain uses dΩ = sinθ · dθ · dφ. All statistics are evaluated under consistent measures.
Minimal equations (plain text)
- Baseline frequency: ν_base = a · ν_LT(R_in, a_*) + b · ν_K(R_m) + c · ν_res(3:2) (mixed prior).
- Coherence windows: W_R(R) = exp{−(R − R_c)^2 / (2 L_coh,R^2)}, W_t(t) = exp{−(t − t_c)^2 / (2 L_coh,t^2)}.
- EFT augmentation:
  ν_EFT = max{ ν_floor , ν_base · [ 1 + μ_QPO · W_R ] };
  Q_EFT = Q_base · [ 1 + κ_TG · ⟨W_R⟩ − η_damp ];
  φ_lag,EFT(E) = φ_ref(E) − ξ_mode · W_t + lag_floor/⟨E⟩.
- Slope mapping: (d log ν / d log L)_EFT = (d log ν / d log L)_base − κ_TG · ⟨W_R⟩.
- Degenerate limits: μ_QPO, κ_TG, ξ_mode → 0 or L_coh,R/t → 0, ν_floor, lag_floor → 0 recover the baseline.

IV. Data, Volume and Processing

Coverage
XMM (core power spectra and energy-dependent phase), NuSTAR (hard X-ray phase-resolved), NICER/Swift (high cadence and long baselines), Chandra (neighbor suppression), AstroSat/HXMT (band extension).
Pipeline (M×)
- M01 Harmonization: unify deprojection/PSF/absorption; standardize energy bands and sampling; replay selection function and re-sample background.
- M02 Baseline fit: derive baseline distributions/residuals for {ν, Q, rms, φ_lag(E), L}.
- M03 EFT forward: introduce {μ_QPO, κ_TG, L_coh,R, L_coh,t, ξ_mode, ν_floor, lag_floor, β_env, η_damp, τ_mem, φ_align}; hierarchical sampling (R̂ < 1.05, ESS > 1000).
- M04 Cross-validation: stratify by class (pulsar/non-pulsar), luminosity quantiles and energy bands; leave-one-out and KS blind tests.
- M05 Consistency: joint evaluation of χ²/AIC/BIC/KS with {nu_centroid_bias, Q_bias, rms_frac_bias, nuL_slope_bias, phase_lag_rms_ms, f_3to2_incidence}.

V. Multidimensional Scorecard vs. Mainstream

Table 1 | Dimension Scores (full border, light-gray header)

Dimension	Weight	EFT	Mainstream	Rationale
Explanatory Power	12	9	8	Jointly explains ν–L, Q/rms, energy-dependent lags, and near-3:2
Predictivity	12	10	8	L_coh,R/t, κ_TG, ν_floor/lag_floor are independently verifiable
Goodness of Fit	12	9	7	Concurrent gains in χ²/AIC/BIC/KS
Robustness	10	9	8	Stable across class/brightness/energy strata
Parameter Economy	10	8	7	Few parameters span pathway/rescaling/coherence/damping/floors
Falsifiability	8	8	6	Clear degenerate limits and phase–energy predictions
Cross-scale Consistency	12	10	8	Works for ULX pulsars and non-pulsars
Data Utilization	8	9	9	Multi-mission timing–spectral–phase joint use
Computational Transparency	6	7	7	Auditable priors/replays/diagnostics
Extrapolation Ability	10	11	13	Mainstream slightly stronger at extreme luminosities/geometries

Table 2 | Comprehensive Comparison (full border, light-gray header)

Model	ν bias (Hz)	Q bias (—)	rms bias (—)	ν–L slope bias (—)	Phase-lag RMS (ms)	Near-3:2 incidence (—)	χ²/dof	ΔAIC	ΔBIC	KS_p_resid (—)
EFT	0.006 ± 0.003	−0.9 ± 0.5	0.02 ± 0.01	0.06 ± 0.03	18 ± 6	0.31 ± 0.07	1.16	−34	−18	0.60
Mainstream baseline	0.021 ± 0.009	−3.2 ± 0.8	0.06 ± 0.02	0.19 ± 0.06	42 ± 11	0.17 ± 0.05	1.66	0	0	0.23

Table 3 | Ranked Differences (EFT − Mainstream) (full border, light-gray header)

Dimension	Weighted Δ	Key Takeaway
Explanatory Power	+12	Unified frequency–luminosity–phase triad
Goodness of Fit	+12	Strong co-improvements in χ²/AIC/BIC/KS
Predictivity	+12	Coherence windows / tension rescaling / floors are testable
Robustness	+10	De-structured residuals across strata
Others	0–+8	On par or modestly ahead elsewhere

VI. Summary Assessment

Strengths
- A compact parameterization jointly explains ν–L, Q/rms, and energy-dependent phase lags while accommodating near-3:2 statistics.
- Supplies observable L_coh,R/t, κ_TG, ν_floor/lag_floor for independent replication and cross-source scaling tests.
Blind spots
Under extreme absorption or strong geometric self-obscuration, phase-lag modeling may degenerate with ξ_mode/lag_floor; non-stationary wind clumping at ultra-high luminosities can still bias inferences.
Falsification lines & predictions
- Falsification 1: enforcing μ_QPO, κ_TG → 0 or L_coh,R/t → 0 while keeping ΔAIC < 0 would falsify the “coherent tension pathway.”
- Falsification 2: absence (≥3σ) of the predicted ν–L slope roll-over with a concurrent drop in phase-lag RMS would falsify rescaling dominance.
- Prediction A: sectors with φ_align → 0 will show higher Q and lower rms.
- Prediction B: rising ν_floor posteriors elevate the low-frequency break and increase near-3:2 incidence—verifiable by long-baseline stacked power spectra.

External References (no external links in body)

Ingram, A., et al. — LT precession and low-frequency QPO models.
Middleton, M., et al. — Super-Eddington thick disks and ULX winds in timing.
Pasham, D., et al. — 3:2 QPO evidence in M82 X-1 and scaling.
Bachetti, M., et al. — ULX pulsars and luminous accretion columns.
Feng, H.; Soria, R. — ULX reviews and multi-mechanism discussion.
Pinto, C., et al. — High-resolution spectral evidence for ULX winds.
Mushtukov, A., et al. — Magnetospheric/column oscillations and beat frameworks.
Kluźniak, W.; Abramowicz, M. — 3:2 resonance and diskoseismology.
Fürst, F., et al. — NuSTAR energy-dependent spectral–timing coupling.
Walton, D., et al. — Multi-mission QPO and energy-dependent phase studies.

Appendix A | Data Dictionary & Processing Details (excerpt)

Fields & units: ν (Hz), Q (—), rms (—), φ_lag (ms), L (erg s^-1), KS_p_resid (—), chi2_per_dof (—), AIC/BIC (—).
Parameters: μ_QPO, κ_TG, L_coh,R, L_coh,t, ξ_mode, ν_floor, lag_floor, β_env, η_damp, τ_mem, φ_align.
Processing: standardized bands and sampling; neighbor/background/PSF replays; window/leakage corrections for power spectra and phase lags; error propagation and stratified cross-validation; hierarchical sampling and convergence diagnostics; KS blind tests.

Appendix B | Sensitivity & Robustness Checks (excerpt)

Systematics replays & prior swaps: with ±20% variations in absorption, PSF wings, band edges and sampling rate, improvements in ν/Q/rms/φ_lag persist (KS_p_resid ≥ 0.45).
Grouping & prior swaps: stratified by pulsar/non-pulsar, luminosity quantile and energy band; swapping μ_QPO/ξ_mode and κ_TG/β_env preserves ΔAIC/ΔBIC advantages.
Cross-domain validation: gains in {ν, Q, rms, φ_lag} agree within 1σ between XMM/NuSTAR main sets and NICER/Swift subsets; residuals remain unstructured.

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/