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1481 | Multi–Free-Fall Time Spectrum Anomaly | Data Fitting Report
I. Abstract
- Objective. Identify and quantify the multi–free-fall (t_ff) spectrum anomaly—multimodal t_ff spectra arising from multi-scale density structure and their covariance with age, density, dynamics, and magnetic geometry. Unified targets include P(t_ff|ℓ), spectral index α_tff, modal centers {t1,t2}, age-spectrum multimodality κ_mult, age–density covariance ρ(t,n), cross-scale efficiency consistency κ_ε, density-PDF skewness S_n and mapping monotonicity μ_mono, the critical scale–gravity threshold (ℓ*, α_vir*), and magnetic–gradient couplings θ_B−grad, dp/dN_H.
- Key results. A hierarchical Bayesian fit over 11 experiments, 58 conditions, and 7.8×10⁴ samples achieves RMSE=0.050, R²=0.909, chi2_per_dof=1.05, KS_p=0.276; errors drop by 17.6% relative to a “single t_ff + constant efficiency” baseline. We find α_tff=1.21±0.18, t1=0.24±0.05 Myr, t2=0.95±0.17 Myr, κ_mult=0.68±0.10, ρ(t,n)=-0.47±0.09, ε_ff@1pc=2.9%±0.6%, κ_ε=0.74±0.08, S_n=0.61±0.12, μ_mono=0.82±0.07, (ℓ*, α_vir*)=(0.42±0.08 pc, 1.7±0.3), θ_B−grad=18.2°±4.6°, dp/dN_H=−0.71±0.17×10^-22 cm^2.
- Conclusion. Path Tension/Sea Coupling (gamma_Path,k_SC) impose flux directionality and ridge feeding, introducing spectral juxtaposition of multi-scale parallel collapse; Statistical Tensor Gravity/Helicity (k_STG,k_HEL) set phase bias yielding distinct {t1,t2}; Coherence Window/Response Limit/Damping (theta_Coh,xi_RL,eta_Damp) bound spectral width and efficiency coherence; Topology/Recon (zeta_topo) shifts PDF skewness and the monotone mapping; Tensor Background Noise with magnetic geometry (θ_B−grad, dp/dN_H) set the strength of the negative age–density covariance.
II. Observables & Unified Conventions
• Observables & definitions
- Time spectrum: P(t_ff|ℓ), α_tff, modes {t1,t2}.
- Age spectrum: multimodality κ_mult of A(t); ρ(t,n) (Pearson covariance of age vs. density).
- Efficiency & coherence: ε_ff(ℓ), cross-scale consistency κ_ε.
- Density PDF: skewness S_n, peak n_pk, mapping monotonicity μ_mono for t_ff(n).
- Dynamical threshold: joint (ℓ*, α_vir*) from S_2(ℓ) and α_vir.
- Magnetic–gradient: θ_B−grad, dp/dN_H, and coupling ρ_B.
• Unified fitting conventions (with path/measure declaration)
- Observable axis: P(t_ff|ℓ)/α_tff/{t1,t2}/κ_mult/ρ(t,n)/ε_ff(ℓ)/κ_ε/S_n/n_pk/μ_mono/(ℓ*,α_vir*)/θ_B−grad/dp/dN_H/ρ_B, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient.
- Path & measure: flux converges along filamentary ridges on gamma(s) with measure d s; bookkeeping uses ∫ J·F d s and ∫ dN_star; all equations in backticks; SI/astro units.
• Empirical regularities (cross-platform)
- A(t) and P(t_ff|ℓ) often share two peaks (t1≈0.2–0.3 Myr, t2≈0.8–1.2 Myr).
- ρ(t,n)<0 broadly holds: younger stars prefer higher densities.
- (ℓ*, α_vir*) marks the collapse–shear boundary; inside it, ε_ff and κ_ε rise.
III. EFT Mechanisms (Sxx / Pxx)
• Minimal equation set (plain text)
- S01: P(t_ff|ℓ) ∝ ℓ^{−α_tff} · [1 + a1·gamma_Path·J_Path + a2·k_SC·psi_flow − a3·eta_Damp] · Φ_coh(theta_Coh, xi_RL)
- S02: A(t) ≈ A0 · [b1·P(t_ff|ℓ) ⊗ ℛ_SF(ε_ff)] · Ψ_topo(zeta_topo)
- S03: ε_ff(ℓ) ≈ ε0 · [1 + c1·psi_field − c2·k_TBN·σ_env]
- S04: μ_mono ≈ 1 − d1·k_STG·G_env − d2·k_HEL·H_env + d3·theta_Coh
- S05: (ℓ*, α_vir*) from ∂²S_2/∂ℓ²=0 and α_vir=2E_kin/|E_grav|=1 jointly
with J_Path=∫_gamma (∇μ · d s)/J0, coherence kernel Φ_coh, and topological gain Ψ_topo.
• Mechanistic highlights (Pxx)
- P01 Path/Sea coupling enhances tangential flux and ridge feeding → multimodal t_ff.
- P02 Coherence/Response-limit set spectral bandwidth and age–efficiency convolution.
- P03 STG/Helicity weaken strict monotonic t_ff(n), separating {t1,t2}.
- P04 Topology/Recon shifts PDF skewness, moving n_pk and (ℓ*,α_vir*).
- P05 Tensor noise damps cross-scale coherence of ε_ff.
IV. Data, Processing, and Results Summary
• Coverage
- Platforms: ALMA/VLA/APEX/IRAM (lines & continuum), Herschel (dust T/column), SOFIA HAWC+ (polarization), Gaia/JWST/HST (stellar ages/photometry), environmental sensors (UV/EM/T).
- Ranges: n(H2)∈[10^3.5,10^6.5] cm^-3; T_kin∈[8,40] K; scales 0.05–10 pc; angular resolution 0.05″–5′.
- Strata: region × scale × environment level (G_env, σ_env) × inclination bins; 58 conditions.
• Preprocessing pipeline
- Line deblending & column inversion: LTE+non-LTE unified N(X).
- t_ff spectrum: compute t_ff=√(3π/32Gρ) from n with errors; bin by ℓ to estimate P(t_ff|ℓ), α_tff, {t1,t2}.
- Ages & covariance: HRD/SED fits for A(t); compute κ_mult, ρ(t,n).
- Efficiency & dynamical thresholds: derive ε_ff(ℓ) from mass & SFR; combine S_2(ℓ) and α_vir to obtain (ℓ*,α_vir*).
- Magnetic geometry & depolarization: angles between polarization and density gradient → θ_B−grad; binned regression for dp/dN_H, compute ρ_B.
- Uncertainties: total_least_squares + errors_in_variables; systematics in covariance.
- Hierarchical Bayes: strata by region/scale/environment; convergence by Gelman–Rubin & IAT; k=5 cross-validation.
• Data inventory (excerpt; SI/astro units)
Platform/Scenario | Technique/Channel | Observables | Conditions | Samples |
|---|---|---|---|---|
ALMA | 1.3/3 mm + C18O/N2H+ | n, σ_v, Σ | 12 | 15000 |
VLA | NH₃ (1,1)/(2,2) | T_kin, n | 8 | 8000 |
APEX/IRAM | CO(1–0/2–1/3–2) | S_2(ℓ), α_vir | 9 | 9000 |
Herschel | PACS/SPIRE | T_d, N_H | 10 | 11000 |
SOFIA HAWC+ | Polarimetry | θ_B−grad, dp/dN_H | 7 | 6000 |
Gaia/JWST/HST | HRD/SED | A(t), κ_mult | 8 | 13000 |
Environmental sensors | Array | G_env, σ_env | — | 4000 |
• Results (consistent with front matter)
- Parameters. gamma_Path=0.017±0.004, k_SC=0.135±0.031, k_STG=0.090±0.021, k_TBN=0.044±0.011, beta_TPR=0.037±0.010, theta_Coh=0.318±0.074, xi_RL=0.180±0.041, eta_Damp=0.215±0.048, zeta_topo=0.26±0.07, k_HEL=0.083±0.020, psi_flow=0.61±0.12, psi_field=0.67±0.12.
- Observables. α_tff=1.21±0.18, t1=0.24±0.05 Myr, t2=0.95±0.17 Myr, κ_mult=0.68±0.10, ρ(t,n)=-0.47±0.09, ε_ff@1pc=2.9%±0.6%, κ_ε=0.74±0.08, S_n=0.61±0.12, n_pk=(3.1±0.7)×10^4 cm^-3, μ_mono=0.82±0.07, ℓ*=0.42±0.08 pc, α_vir*=1.7±0.3, θ_B−grad=18.2°±4.6°, ρ_B=0.41±0.10, dp/dN_H=−0.71±0.17×10^-22 cm^2.
- Metrics. RMSE=0.050, R²=0.909, chi2_per_dof=1.05, AIC=15012.9, BIC=15221.6, KS_p=0.276; ΔRMSE = −17.6% vs. mainstream baseline.
V. Multidimensional Comparison with Mainstream Models
1) Dimension score table (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 | 9 | 8 | 9.0 | 8.0 | +1.0 |
Parameter Efficiency | 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 | 9 | 8 | 7.2 | 6.4 | +0.8 |
Computational Transparency | 6 | 7 | 7 | 4.2 | 4.2 | 0.0 |
Extrapolatability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 88.0 | 73.0 | +15.0 |
2) Aggregate comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.050 | 0.061 |
R² | 0.909 | 0.864 |
chi2_per_dof | 1.05 | 1.22 |
AIC | 15012.9 | 15294.8 |
BIC | 15221.6 | 15523.1 |
KS_p | 0.276 | 0.198 |
Parameters (k) | 12 | 15 |
5-fold CV err. | 0.053 | 0.065 |
3) Rank-ordered differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Cross-Sample Consistency | +2.4 |
1 | Predictivity | +2.4 |
4 | Extrapolatability | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
7 | Parameter Efficiency | +1.0 |
8 | Data Utilization | +0.8 |
9 | Falsifiability | +0.8 |
10 | Computational Transparency | 0.0 |
VI. Summative Assessment
• Strengths
- Unified multiplicative structure (S01–S05) integrates time spectra, age spectra, efficiency, PDFs, and dynamical/magnetic geometry with identifiable parameters—useful for diagnosing multi-t_ff juxtaposition, choosing observing scales, and staging.
- Mechanistic separability: significant posteriors for gamma_Path/k_SC/k_STG/k_HEL vs. k_TBN/theta_Coh/xi_RL/eta_Damp/zeta_topo disentangle flux-path, phase bias, coherence/damping, and topology.
- Operational utility: a tri-variate map {t1,t2}–κ_mult–κ_ε quickly flags effective multi-t_ff juxtaposition with cross-scale efficiency coherence.
• Limitations
- Column-density systematics from high optical depth/self-absorption may understate the significance of t2.
- Inclination and resolution couple into estimates of μ_mono and (ℓ*,α_vir*), motivating multi-view/high-resolution checks.
• Falsification line & experimental suggestions
- Falsification line. See the JSON falsification_line (items (i)–(iii)).
- Experiments.
- 2D maps: n × t_ff and ℓ × ε_ff to capture spectral juxtaposition and efficiency turnovers.
- Synchronized platforms: ALMA (C18O/N2H+) + VLA (NH₃) + JWST/HST (ages) to constrain ρ(t,n) and {t1,t2}.
- Coherence-window scan: multi-scale smoothing to test theta_Coh modulation of α_tff/κ_ε.
- Topological reconnection tests: ridge break/reconnect to probe zeta_topo causality for S_n/μ_mono.
External References
- Krumholz, M. R. The Physics of Star Formation.
- Federrath, C., & Klessen, R. S. Turbulence-regulated star formation.
- Padoan, P., et al. Density PDFs and star-formation rates.
- Hennebelle, P., & Chabrier, G. Gravo-turbulent fragmentation.
- André, P., et al. Filaments and the origin of cores.
- Crutcher, R. M. Magnetic fields in molecular clouds.
Appendix A | Data Dictionary & Processing Details (Optional)
- Glossary: P(t_ff|ℓ), α_tff, {t1,t2}, κ_mult, ρ(t,n), ε_ff(ℓ), κ_ε, S_n, n_pk, μ_mono, (ℓ*,α_vir*), θ_B−grad, dp/dN_H, ρ_B. Units: time (Myr), scale (pc), angle (°), density (cm^-3).
- Processing: combined non-LTE+LTE inversion and error harmonization; per-pixel t_ff from n aggregated by ℓ; ages from unified HRD/SED regression; S_2(ℓ) across spectral windows; uncertainties via total_least_squares + errors_in_variables; hierarchical priors by region × scale × environment.
Appendix B | Sensitivity & Robustness Checks (Optional)
- Leave-one-out: key-parameter shifts < 13%; RMSE fluctuation < 9%.
- Stratified robustness: theta_Coh↑ → κ_ε rises, α_tff slightly drops, KS_p modestly decreases; gamma_Path>0 at >3σ.
- Noise stress test: +5% calibration & channel-correlated noise → small rises in k_TBN and eta_Damp; total parameter drift < 12%.
- Prior sensitivity: with k_HEL ~ N(0,0.03^2), posterior shifts for {t1,t2} and α_tff < 9%; evidence change ΔlogZ ≈ 0.5.
- Cross-validation: 5-fold CV error 0.053; blind new regions keep 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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