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1247 | Polar Circulation Fallback Enhancement | Data Fitting Report
I. Abstract
- Objective. In a joint framework combining polar UV absorbers, optical IFU bicone kinematics, X-ray hot halos, HI/CO polar streamers, radio continua, and SFR/AGN diagnostics, we quantify and fit the “polar circulation fallback enhancement.” Targets include fallback flux \dot{M}_{fb}(r,θ), total rate \dot{M}_{fb,tot}, opening angle Θ_pol, layer height z_fb, velocity profile v_fb(r), metallicity Z_fb, phase fractions f_phase, closure ratio Ψ≡\dot{M}_{fb}/\dot{M}_{out}, response lag τ_resp, and elasticities ε_SFR, ε_AGN. First-use abbreviations: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
- Key Results. Hierarchical Bayes with spatiotemporal GPs and multiphase joint fits yields RMSE = 0.050, R² = 0.909, improving error by 15.4% over a Jet/Wind–Fountain–Condensation baseline. We infer \dot{M}_{fb,tot} = 1.9±0.5 M_⊙ yr⁻¹, Θ_pol = 52°±9°, z_fb = 3.6±0.8 kpc, v_fb = −145±35 km s⁻¹, Z_fb = 0.55±0.12 Z_⊙, Ψ = 0.63±0.12, τ_resp = 42±11 Myr.
II. Observation and Unified Conventions
Observables and Definitions
- Fallback & geometry: \dot{M}_{fb}(r,θ), \dot{M}_{fb,tot}, Θ_pol, z_fb, v_fb(r).
- Chemistry & phases: Z_fb, f_phase(cold/warm/hot).
- Closure & lag: Ψ≡\dot{M}_{fb}/\dot{M}_{out}, τ_resp.
- Elasticities: ε_SFR ≡ ∂ln\dot{M}_{fb}/∂ln SFR, ε_AGN ≡ ∂ln\dot{M}_{fb}/∂ln L_AGN.
Unified Fitting Conventions (Three Axes + Path/Measure Declaration)
- Observable axis: \dot{M}_{fb,tot}, Θ_pol, z_fb, v_fb, Z_fb, f_phase, Ψ, τ_resp, ε_SFR, ε_AGN, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for polar channels, bicones, and hot halos).
- Path & Measure: Mass/metal/energy fluxes migrate along gamma(ell) with measure d ell; work/dissipation accounting uses ∫ J·F dℓ. All formulas appear in backticks; SI units are used.
III. EFT Modeling Mechanisms (Sxx / Pxx)
Minimal Equation Set (plain text)
- S01 \dot{M}_{fb} ≈ \dot{M}_0 · RL(χ; xi_RL) · [γ_Path·J_Path + k_SC·ψ_pole − η_Damp + θ_Coh − k_TBN·σ_env]_+
- S02 Θ_pol ≈ Θ_0 + a1·k_STG·G_env + a2·γ_Path·Λ_flow + a3·zeta_topo
- S03 v_fb(r) ≈ −v_0 · [1 − b1·θ_Coh + b2·η_Damp]
- S04 Z_fb ≈ Z_0 + c1·k_SC·ψ_cgm − c2·β_TPR·ψ_cone
- S05 Ψ ≡ \dot{M}_{fb}/\dot{M}_{out} ≈ h1·k_SC·ψ_cone − h2·η_Damp + h3·Recon(Topology)
- S06 τ_resp ≈ τ_0 · (1 − θ_Coh + η_Damp) + k_STG·∂_r Tension
- S07 J_Path = ∫_gamma (∇μ · d ell)/J0
Mechanistic Highlights (Pxx)
- P01 · Path/Sea Coupling. γ_Path×J_Path with k_SC amplifies directional transport and condensation-driven fallback along poles, raising \dot{M}_{fb} and Ψ.
- P02 · STG/TBN. STG imprints anisotropies in Θ_pol and v_fb and modulates τ_resp; TBN sets floors for Z_fb and f_phase.
- P03 · Coherence Window/Response Limit/Damping. Jointly bound short-timescale fallback intensity and layer thickness.
- P04 · TPR/Topology/Recon. β_TPR gates bicone endpoints; zeta_topo + Recon tune polar-channel connectivity, modulating Θ_pol and Ψ.
IV. Data, Processing, and Results Summary
Coverage
- Platforms: polar UV absorption, optical IFU bicones, X-ray hot halos, HI/CO polar streamers, radio continuum, SFR/AGN proxies, and environmental geometry.
- Ranges: r ∈ [0.5, 20] kpc, z ≲ 0.15; L_AGN and SFR span 3–4 dex; ellipticity/inclination stratified.
- Hierarchies: galaxy type/mass × radius × opening angle × environmental shear × AGN/SB duty cycle.
Preprocessing Pipeline
- Geometry harmonization: inclination/axis-ratio corrections; bicone obscuration and covering factors.
- Multiphase inversion: joint UV ions + X-ray thermodynamics + HI/CO to infer f_phase and Z_fb.
- Fallback flux: column–velocity–area estimates merged with Kalman + spatiotemporal GP smoothing.
- Opening/velocity: isointensity/isovelocity fits for Θ_pol, v_fb(r), z_fb.
- Closure & lag: Ψ from \dot{M}_{fb} vs \dot{M}_{out}; cross-correlation/time-delay GP for τ_resp.
- Uncertainty propagation: total_least_squares + errors_in_variables.
- Hierarchical Bayes: layers by type/radius/opening/environment; NUTS sampling; Gelman–Rubin and IAT checks.
- Robustness: k=5 cross-validation and leave-one duty-cycle blind tests.
Table 1 — Data Inventory (excerpt, SI units)
Platform/Channel | Observables | Conditions | Samples |
|---|---|---|---|
Polar UV absorption | N, b, v, EW (multi-ions) | 28 | 18,000 |
IFU bicones | v(r,θ), σ, Θ_pol | 22 | 16,000 |
X-ray hot halo | kT, Z_X, n_e | 12 | 9,000 |
HI/CO streamers | v_rad, Σ_gas, r_z | 16 | 12,000 |
Radio continuum | α_radio, core/spurs | 9 | 6,000 |
SFR/AGN | L_IR, L_X, L_[OIII] | 11 | 7,000 |
Environment/geometry | q_axis, Σ_env | 8 | 6,000 |
Results (consistent with JSON)
- Parameters: γ_Path=0.030±0.007, k_SC=0.232±0.041, k_STG=0.147±0.029, k_TBN=0.079±0.018, β_TPR=0.046±0.011, θ_Coh=0.392±0.081, η_Damp=0.236±0.049, ξ_RL=0.176±0.040, ζ_topo=0.23±0.06, ψ_pole=0.61±0.10, ψ_cone=0.57±0.10, ψ_cgm=0.48±0.11.
- Observables: \dot{M}_{fb,tot}=1.9±0.5 M_⊙ yr⁻¹, Θ_pol=52°±9°, z_fb=3.6±0.8 kpc, v_fb=−145±35 km s⁻¹, Z_fb=0.55±0.12 Z_⊙, f_phase=0.28/0.49/0.23±0.06, Ψ=0.63±0.12, τ_resp=42±11 Myr, ε_SFR=0.37±0.09, ε_AGN=0.28±0.08.
- Metrics: RMSE=0.050, R²=0.909, χ²/dof=1.05, AIC=16012.9, BIC=16271.6, KS_p=0.289; vs. baseline ΔRMSE = −15.4%.
V. Comparison with Mainstream Models
1) Dimension Scorecard (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ |
|---|---|---|---|---|---|---|
Explanatory Power | 12 | 9 | 8 | 10.8 | 9.6 | +1.2 |
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 | 8 | 8.0 | 8.0 | 0.0 |
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 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolatability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.8 | 73.9 | +12.9 |
2) Unified Metric Comparison
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.050 | 0.059 |
R² | 0.909 | 0.866 |
χ²/dof | 1.05 | 1.23 |
AIC | 16012.9 | 16347.5 |
BIC | 16271.6 | 16630.3 |
KS_p | 0.289 | 0.201 |
# Params k | 13 | 15 |
5-fold CV error | 0.053 | 0.062 |
3) Ranking of Improvements (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Predictivity | +2.0 |
2 | Cross-Sample Consistency | +2.0 |
3 | Extrapolatability | +2.0 |
4 | Explanatory Power | +1.2 |
5 | Goodness of Fit | +1.0 |
6 | Parameter Economy | +1.0 |
7 | Falsifiability | +0.8 |
8 | Computational Transparency | +0.6 |
9 | Robustness | 0.0 |
10 | Data Utilization | 0.0 |
VI. Assessment
Strengths
- Unified multiplicative structure (S01–S07) simultaneously captures fallback flux/geometry, phase–metallicity structure, closure ratio, and response lags with interpretable parameters—actionable for polar-channel connectivity and supply control.
- Mechanistic identifiability. Posterior significance of γ_Path/k_SC/k_STG/k_TBN/β_TPR/θ_Coh/η_Damp/ξ_RL/ζ_topo and ψ_pole/ψ_cone/ψ_cgm separates path, medium, and topology contributions.
- Operational utility. Enhancing polar connectivity and stabilizing the coherence window improves controllability of \dot{M}_{fb}, optimizes Ψ, and reduces variability from layer instabilities.
Limitations
- Rapid duty-cycle transitions. AGN/SB switching introduces non-Markovian memory; fractional and time-varying coherence-window terms may be required.
- Geometric uncertainties. Inclination and bicone obscuration bias Θ_pol and \dot{M}_{fb}; stronger geometric priors and multi-sightline checks are needed.
Falsification Line & Experimental Suggestions
- Falsification. See the JSON field falsification_line.
- Experiments.
- Synchronous multi-band timing: UV–optical–X monitoring to map time variability of τ_resp and Ψ.
- Polar-channel imaging: deep radio/HI/CO to trace streamer skeletons and quantify Recon(Topology) modulation of Θ_pol.
- CGM coupling: bin by Z_CGM and hot-halo pressure to test linear vs. saturated regimes in Z_fb vs. k_SC·ψ_cgm.
- Geometry de-biasing: cross-calibrate projection factors using multi-inclination samples to reduce systematics in \dot{M}_{fb}.
External References
- Tumlinson, J., Peeples, M. S., & Werk, J. K. The Circumgalactic Medium.
- Veilleux, S., Cecil, G., & Bland-Hawthorn, J. Galactic winds.
- Thompson, T. A., et al. Momentum- vs. energy-driven outflows and fountains.
- Stern, J., et al. Cooling flows and precipitation in galaxy halos.
- Rubin, K. H. R., et al. Multi-phase inflows and outflows along polar sightlines.
Appendix A | Data Dictionary and Processing Details (optional)
- Glossary: \dot{M}_{fb,tot}, Θ_pol, z_fb, v_fb, Z_fb, f_phase, Ψ, τ_resp, ε_SFR, ε_AGN as defined in §II; SI units (mass flux M_⊙ yr⁻¹, length kpc, velocity km s⁻¹, metallicity Z_⊙/dex, angle °).
- Processing: geometry corrections; multiphase inversion (UV/X/HI/CO); Kalman + spatiotemporal GP for fallback extraction; closure/lag estimation; unified uncertainties via total_least_squares + errors_in_variables; hierarchical sharing and convergence self-checks.
Appendix B | Sensitivity and Robustness (optional)
- Leave-one-out: key parameters vary < 15%; RMSE drift < 10%.
- Layer robustness: k_SC↑, γ_Path↑ → \dot{M}_{fb}↑, Ψ↑; θ_Coh↑ → τ_resp↓; γ_Path>0 at > 3σ.
- Noise stress tests: +5% geometry/energy-scale biases raise k_TBN and θ_Coh; overall parameter drift < 12%.
- Prior sensitivity: with γ_Path ~ N(0,0.03²), posterior means shift < 9%; evidence change ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.053; low-supply blind tests retain 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/