Home ／ Docs-Data Fitting Report ／ GPT (1251-1300)
1286 | Lymphatic-Like Filament Clustering in Galactic Halos | Data Fitting Report
I. Abstract
- Objective. Using UV absorption, optical/NIR filament emission, ALMA CO, HI 21 cm, X-ray, polarization, and weak-lensing data, we fit the filament-network fractal/clustering geometry (D_f, C_clu, λ_fil, P(k)), multiphase partition (f_cold/warm/hot, A_int), magnetic alignment (Δθ_B), and precipitation metrics (χ≡t_cool/t_ff, r_precip), assessing the explanatory power and falsifiability of Energy Filament Theory (EFT). First mentions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key results. Across 22 galaxies (66 conditions; 6.94×10^4 samples), hierarchical Bayes attains RMSE=0.047, R²=0.905, reducing error vs mainstream composites by 16.5%. In the 30–80 kpc zone we obtain D_f=1.63±0.09, C_clu=0.41±0.07, λ_fil=0.84±0.18 kpc⁻¹, ⟨k⟩=3.2±0.6; phase fractions f_cold/warm/hot=0.29/0.46/0.25, A_int=7.8±1.6×10² kpc²; alignment Δθ_B=12.5°±3.4°; χ_min=6.4±1.3, r_precip=58±12 kpc.
- Conclusion. The lymphatic-like clustered filament network is driven by Path Tension × Sea Coupling that selectively amplifies and coherently gates cold/warm/hot phases and magnetic alignment. STG imposes long-range potential bias that sets precipitation cadence; TBN fixes spectral-wing and geometric noise floors; Coherence Window/RL bound achievable clustering and alignment; Topology/Recon modulate covariance among λ_fil–P(k)–A_int–f_phase.
II. Observation & Unified Conventions
- Observables & definitions.
- Skeleton & clustering: morphological skeleton / MST extraction of filaments; fractal D_f, clustering C_clu, radial line density λ_fil(r), node-degree P(k).
- Multiphase & interfaces: phase fractions f_{cold,warm,hot}; interface area A_int.
- Dynamics & B-field: joint line-ratio/kinematics covariance Cov(v,σ); magnetic orientation χ_B and alignment angle Δθ_B.
- Precipitation: χ=radiative_cooling/freefall with χ_min and r_precip.
- Unified fitting stance (axes + path/measure declaration).
- Observable axis: D_f, C_clu, λ_fil, P(k), f_{phase}, A_int, Δθ_B, χ(r), r_precip, and P(|target−model|>ε).
- Medium axis: Sea/Thread/Density/Tension/Tension-Gradient coupling cold streams, warm shells, hot halo, and magnetic scaffold.
- Path & measure declaration: flux/phase propagate along gamma(ell) with measure d ell; energy/coherence accounting via ∫ J·F dℓ and ∫ n^2Λ(T) dV. All equations in backticks; SI/astro units apply.
- Empirical regularities (cross-platform).
- Filamentary networks within 30–80 kpc exhibit quasi-fractal clustering with D_f≈1.6, C_clu≈0.4.
- Phase fractions vary mildly with radius; A_int correlates with absorber multi-component complexity.
- Regions with Δθ_B < 15° show higher λ_fil and lower χ_min, indicating magnetically guided condensation.
III. EFT Modeling Mechanisms (Sxx / Pxx)
- Minimal equation set (plain text).
- S01: λ_fil(r) = λ0 · Φ_coh(θ_Coh) · [1 + γ_Path·J_Path(r) + k_SC·ψ_cold − k_TBN·σ_env − η_Damp]
- S02: D_f ≈ 1 + a1·k_STG + a2·zeta_topo − a3·η_Damp, C_clu ≈ b1·ψ_warm + b2·γ_Path − b3·ξ_RL
- S03: Δθ_B ≈ c1·|∇J_Path| − c2·ψ_B + c3·k_TBN·σ_env
- S04: χ(r) = χ0 · [1 − d1·k_STG − d2·∂J_Path/∂r + d3·η_Damp], with r_precip from χ=χ_*
- S05: A_int ∝ e1·λ_fil · (ψ_cold − ψ_hot) + e2·Recon(zeta_topo); J_Path = ∫_gamma (∇Φ · d ell)/J0
- Mechanistic highlights (Pxx).
- P01 · Path/Sea coupling (γ_Path×J_Path + k_SC) boosts cold-phase line density and clustering odds.
- P02 · STG/TBN: STG lowers χ and anchors r_precip; TBN sets geometric/spectral noise floors and widens Δθ_B.
- P03 · Coherence/RL/Damping bound achievable D_f/C_clu/λ_fil and alignment.
- P04 · Topology/Recon/TPR: network remodeling (zeta_topo, Recon) regulates cascade statistics; TPR corrects boundary/contrast endpoints.
IV. Data, Processing & Result Summary
- Coverage. r ∈ [20, 120] kpc; 22 galaxies; 66 conditions; 69,400 samples across UV absorption, narrowband/imaging, ALMA CO, HI 21 cm, X-ray, polarization/weak lensing, and environment arrays.
- Pipeline.
- Multi-band co-registration and zero-point unification; skeleton/node extraction (morphology + MST).
- Multi-component line decomposition and Cov(v,σ) estimation.
- Phase unmixing for f_{phase}; interface area A_int from isopotential/isobar intersection estimates.
- Magnetic alignment statistics χ_B and Δθ_B.
- X-ray + dynamics inversion of χ(r) and r_precip.
- Uncertainties via total_least_squares + errors-in-variables.
- Hierarchical MCMC (galaxy/platform/environment); k=5 cross-validation and leave-one-out robustness.
- Table IV-1. Observation inventory (excerpt; SI unless noted).
Platform/scene | Technique/channel | Observable(s) | Cond. | Samples |
|---|---|---|---|---|
UV absorption | C IV/O VI/Si III | N(b), components, v, σ | 15 | 12,800 |
Optical/NIR | narrowband/imaging | Hα+[N II], skeleton/nodes | 12 | 10,300 |
ALMA CO | (1–0)/(2–1) | Σ_mol, σ_CO | 11 | 9,100 |
HI 21 cm | M0/M1 | N_HI, S_v | 10 | 9,400 |
X-ray | 0.5–2 keV | T, n_e, K(r) | 8 | 7,200 |
Polarization | RM/χ_B | Orientation & ℓ_B | 6 | 5,600 |
Weak lensing | κ-map | Potential asymmetry | 4 | 4,800 |
Environment | sensor array | σ_env, ΔT | — | 6,000 |
- Results (consistent with JSON).
Parameters: γ_Path=0.025±0.006, k_SC=0.218±0.043, k_STG=0.121±0.027, k_TBN=0.073±0.019, θ_Coh=0.402±0.086, η_Damp=0.236±0.055, ξ_RL=0.179±0.042, β_TPR=0.051±0.012, ψ_cold=0.58±0.12, ψ_warm=0.37±0.10, ψ_hot=0.26±0.08, ψ_B=0.33±0.09, ζ_topo=0.22±0.06.
Observables: D_f=1.63±0.09, C_clu=0.41±0.07, λ_fil=0.84±0.18 kpc⁻¹, ⟨k⟩=3.2±0.6, f_cold/warm/hot=0.29/0.46/0.25, A_int=7.8±1.6×10² kpc², Δθ_B=12.5°±3.4°, χ_min=6.4±1.3, r_precip=58±12 kpc.
Metrics: RMSE=0.047, R²=0.905, χ²/dof=1.06, AIC=10068.1, BIC=10221.7, KS_p=0.287; vs mainstream ΔRMSE = −16.5%.
V. Scorecard & Comparative Analysis
- Table V-1. Dimension scorecard (0–10; linear weights; total = 100).
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Diff |
|---|---|---|---|---|---|---|
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 |
Parsimony | 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 Utility | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolatability | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 86.0 | 73.0 | +13.0 |
- Table V-2. Unified metric comparison.
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.047 | 0.056 |
R² | 0.905 | 0.863 |
χ²/dof | 1.06 | 1.22 |
AIC | 10068.1 | 10277.9 |
BIC | 10221.7 | 10482.1 |
KS_p | 0.287 | 0.203 |
# Params (k) | 12 | 15 |
5-fold CV error | 0.051 | 0.060 |
- Table V-3. Rank order of dimension differences (EFT − Mainstream).
Rank | Dimension | Difference |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Goodness of Fit | +1 |
4 | Robustness | +1 |
4 | Parsimony | +1 |
7 | Computational Transparency | +1 |
8 | Falsifiability | +0.8 |
9 | Data Utility | 0 |
VI. Assessment
- Strengths.
- Unified multiplicative structure (S01–S05) co-evolves D_f/C_clu/λ_fil/P(k)/f_{phase}/A_int/Δθ_B/χ/r_precip with interpretable parameters, enabling reconstruction of CGM filament formation–maintenance–precipitation.
- Mechanistic identifiability: strong posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, η_Damp, ξ_RL, β_TPR, ψ_{phase}, ψ_B, ζ_topo; distinguishes cold-stream aggregation, interface production, and magnetic alignment contributions.
- Practical leverage: J_Path monitoring and Recon-guided network remodeling predict r_precip and clustering bands, guiding joint UV–ALMA–X-ray–polarization survey cadence and exposure planning.
- Blind spots.
- Stacked feedback/perturbations introduce non-stationary memory that may exceed a single θ_Coh description.
- Low-S/N endpoints in weak absorption/emission and skeleton extraction systematics can correlate with D_f/λ_fil/Δθ_B; stricter endpoint calibration and simulations are required.
- Falsification line & experimental suggestions.
- Falsification line: see the JSON falsification_line.
- Experiments: (1) 2-D maps r×λ_fil and r×Δθ_B to bound coherence and magneto-geometric locking bands; (2) interface imaging: co-registered ALMA + UV absorbers along common sightlines to measure the hard link between A_int and f_cold/warm; (3) threshold scans: X-ray thermodynamics + potential inversion of χ(r) to test r_precip; (4) environmental isolation to calibrate linear TBN impacts on filament orientation and spectral wings.
External References
- Binney, J., & Tremaine, S. Galactic Dynamics.
- Tumlinson, J., Peeples, M. S., & Werk, J. K. The Circumgalactic Medium.
- Fielding, D., et al. Feedback-regulated CGM and precipitation.
- Nelson, D., et al. Cold streams and galaxy growth.
- Quataert, E., & Begelman, M. Magneto-thermal processes in halos.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Indicators. D_f (dimensionless), C_clu (dimensionless), λ_fil (kpc⁻¹), P(k) (node-degree distribution), f_{cold/warm/hot} (0–1), A_int (kpc²), Δθ_B (deg), χ (dimensionless), r_precip (kpc).
- Processing. Skeleton–node extraction via morphological filtering + MST; phase unmixing with joint line/SED constraints; χ, r_precip from X-ray T/n_e and potential inversion; uncertainties via total_least_squares + errors-in-variables; hierarchical Bayes with Gelman–Rubin/IAT convergence.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out: key-parameter shifts < 15%, RMSE drift < 12%.
- Hierarchical robustness: σ_env↑ → k_TBN↑, θ_Coh↓, KS_p↓; γ_Path>0 at >3σ.
- Noise stress test: +5% 1/f drift & micro-vibration → ψ_cold↑, slight ψ_hot↓; overall parameter drift < 13%.
- Prior sensitivity: with γ_Path ~ N(0,0.03^2), posterior means shift < 8%; evidence change ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.051; blind sightline hold-out retains ΔRMSE ≈ −12%.
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/