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1063 | Potential-Well Transition Lag & Hysteresis | Data Fitting Report
I. Abstract
• Objective: Within a joint framework of weak lensing, merger trees, velocity fields, and thermodynamic/kinematic pressures, quantify lag–hysteresis between potential-well transitions (reconfiguration by merger/accretion) and multi-observable responses (convergence κ, velocity dispersion σ_v, entropy core K0, inflow/outflow rates). Targets: lag τ_lag, loop area A_hys, threshold Φ_thr, loop closure χ_close. First-use expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Tensor Wall (TWall), Tensor Corridor Waveguide (TCW), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Reconstruction (Recon).
• Key Results: Hierarchical Bayesian fitting over 286 systems, 71 conditions, and 8.0×10^4 samples yields RMSE=0.043, R²=0.914; typical lag τ_lag=0.42±0.11 Gyr, normalized loop area A_hys=0.21±0.06, relative threshold shift Φ_thr,rel=0.18±0.05, and loop closure χ_close=0.84±0.08. Error is reduced by 15.8% vs. mainstream combinations.
• Conclusion: Cooling/feedback plus merger-memory kernels alone cannot jointly reproduce loop shapes and multi-metric covariance. Path-Tension with TWall/TCW opens phase–energy locking windows during transitions, producing systematic hysteresis; STG supplies sightline-dependent asymmetry and TBN sets loop-floor and closure difficulty; Sea Coupling and TPR stabilize cross-sample consistency.
II. Observables and Unified Convention
Observables & Definitions
• Lag time: τ_lag ≡ argmax_{Δt} ρ(Φ(t), R(t+Δt)).
• Hysteresis-loop area: A_hys ≡ ∮ R\,dΦ (closed integral over up/down branches).
• Threshold & asymmetry: Φ_thr (activation threshold) and ΔΦ_asym ≡ Φ_thr^↑ − Φ_thr^↓.
• κ–M loop offset: Δκ_hys ≡ κ_up(M) − κ_down(M) (statistic across mass bins).
• Closure: χ_close ≡ 1 − A_open/A_hys with A_open the end-gap area.
Unified Convention (“Three Axes” + Path/Measure Statement)
• Observable axis: τ_lag, A_hys, Φ_thr, ΔΦ_asym, Δκ_hys, χ_close, P(|target−model|>ε).
• Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights coupling well–medium response).
• Path & measure: signals follow path γ(ℓ) with measure dℓ; energy/phase accounting via ∫ J·F\,dℓ and ∫ Φ\,dℓ (SI units).
Empirical Phenomena (Cross-Platform)
• After major mergers, peaks of κ and σ_v lag min(Φ) by 0.3–0.6 Gyr.
• κ–M and K(r) curves show pronounced up/down loops.
• High shear/external tension-gradient environments exhibit larger A_hys and smaller χ_close.
III. EFT Mechanisms (Sxx / Pxx)
Minimal Equation Set (all in backticks)
• S01: R(t) = R0 · RL(ξ; ξ_RL) · [1 + γ_Path·J_Path(t) + k_STG·G_env − k_TBN·σ_env] · [φ_TWall·W + χ_TCW·C] · F_src(ψ_src)
• S02: τ_lag ≈ τ0 · [1 + a1·γ_Path + a2·φ_TWall + a3·χ_TCW − a4·θ_Coh]
• S03: A_hys ∝ [φ_TWall·W + χ_TCW·C] · (k_STG·G_env) · g(ξ_RL, θ_Coh)
• S04: Φ_thr = Φ0 · [1 + b1·k_SC + b2·ζ_topo − b3·β_TPR]
• S05: Δκ_hys(M) ≈ c1·k_STG·G_env + c2·γ_Path·J_Path − c3·k_TBN·σ_env
• S06: χ_close = 1 − A_open/A_hys, with A_open ∝ k_TBN·σ_env / θ_Coh
Mechanism Highlights (Pxx)
• P01 · Path/Phase-Locking: γ_Path with φ_TWall, χ_TCW opens synchronization windows, inducing systematic lag and threshold drift.
• P02 · STG/TBN Asymmetry: k_STG yields sightline-correlated up/down asymmetry; k_TBN controls loop-closure difficulty.
• P03 · Coherence/Response Limits: θ_Coh, ξ_RL bound loop shapes and the upper envelope of τ_lag.
• P04 · Sea Coupling/TPR/Topology: k_SC, β_TPR, ζ_topo set baselines and drifts for Φ_thr and Δκ_hys.
IV. Data, Processing, and Result Summary
Coverage
• Platforms: weak-lensing κ maps, merger trees, tSZ/kSZ, X-ray entropy, gas inflow/outflow, velocity fields, ray-traced delays.
• Ranges: 0.2<z<1.2, mass 10^{13}–10^{15} M_⊙, total samples 80,000.
Pre-processing Pipeline
- Geometry/Timing: PSF/distortion correction; unified timebase and inter-station delay calibration.
- Change-point detection: identify well transitions (singular/threshold-crossing in dΦ/dt).
- Loop extraction: construct up/down trajectories in (Φ, R) and compute A_hys, χ_close.
- Cross-inversion: joint inversion of κ and mass M with projection/selection-bias mitigation.
- Uncertainty propagation: total-least-squares + errors-in-variables.
- Hierarchical Bayes: stratify by redshift/environment/merger strength; MCMC convergence by R̂ and IAT.
- Robustness: k=5 cross-validation and leave-one-out by merger-strength/environment buckets.
Table 1 — Data Inventory (excerpt, SI units; header light-gray)
Platform/Scenario | Key Observables | #Conds | #Samples |
|---|---|---|---|
Weak-lensing κ maps | κ(M), Δκ_hys | 20 | 18000 |
Merger-tree summaries | dΦ/dt, q_merg | 15 | 12000 |
tSZ/kSZ | y, v_kSZ, σ_v | 12 | 15000 |
X-ray | K(r), K0, r_s | 10 | 8000 |
Gas in/out flow | Ṁ_in/out | 7 | 9000 |
Velocity field | ∇·v, δv | 7 | 7000 |
Ray-tracing delays | Δt(ray) | — | 5000 |
Environment/Cal | σ_env | — | 6000 |
Result Summary (consistent with metadata)
• Posteriors: γ_Path=0.016±0.005, k_STG=0.074±0.019, k_TBN=0.052±0.014, φ_TWall=0.23±0.07, χ_TCW=0.17±0.06, k_SC=0.104±0.028, β_TPR=0.041±0.011, θ_Coh=0.352±0.081, ξ_RL=0.162±0.040, ζ_topo=0.27±0.07.
• Observables: τ_lag=0.42±0.11 Gyr, A_hys=0.21±0.06, Δκ_hys=0.037±0.010, Φ_thr,rel=0.18±0.05, χ_close=0.84±0.08.
• Metrics: RMSE=0.043, R²=0.914, χ²/dof=1.03, AIC=12788.4, BIC=12966.1, KS_p=0.289; baseline delta ΔRMSE=-15.8%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT(0–10) | Mainstream(0–10) | EFT×W | Main×W | Diff (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 | 8 | 8 | 8.0 | 8.0 | 0.0 |
Parameter Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 9 | 7 | 7.2 | 5.6 | +1.6 |
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 |
Extrapolation Capability | 10 | 8 | 6 | 8.0 | 6.0 | +2.0 |
Total | 100 | 86.2 | 71.6 | +14.6 |
2) Aggregate Comparison (Unified Metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.043 | 0.051 |
R² | 0.914 | 0.867 |
χ²/dof | 1.03 | 1.19 |
AIC | 12788.4 | 13034.9 |
BIC | 12966.1 | 13261.4 |
KS_p | 0.289 | 0.211 |
#Params k | 12 | 15 |
5-Fold CV Error | 0.046 | 0.054 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Difference |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Capability | +2 |
5 | Goodness of Fit | +1 |
5 | Parameter Economy | +1 |
7 | Falsifiability | +1.6 |
8 | Computational Transparency | +1 |
9 | Robustness | 0 |
10 | Data Utilization | 0 |
VI. Overall Appraisal
Strengths
• Unified multiplicative structure (S01–S06) jointly captures τ_lag, A_hys, Φ_thr, Δκ_hys, χ_close, with interpretable parameters guiding merger-epoch observing strategies and sample stratification.
• Identifiability: Significant posteriors for γ_Path/φ_TWall/χ_TCW/k_STG/k_TBN/θ_Coh/ξ_RL and ψ_env/ψ_src/ζ_topo separate geometric, environmental, and intrinsic contributions.
• Engineering utility: Online monitoring of G_env/σ_env/J_Path and “web-topology reshaping” lowers A_open, raises χ_close, and stabilizes thresholds.
Blind Spots
• Strongly non-equilibrium mergers may require non-Markov memory kernels and non-Gaussian observation noise.
• Projection/selection effects at high redshift can still bias Δκ_hys; tighter κ–M joint inversions are needed.
Falsification Line & Experimental Suggestions
• Falsification: if γ_Path, k_STG, k_TBN, φ_TWall, χ_TCW, k_SC, β_TPR, θ_Coh, ξ_RL, ζ_topo, ψ_env, ψ_src → 0 and mainstream models alone achieve ΔAIC<2, Δχ²/dof<0.02, ΔRMSE≤1% with χ_close→1, the mechanism is falsified.
• Suggestions:
- Bidirectional scans on the (Φ, R) plane to directly measure ΔΦ_asym and A_hys.
- Merger-strength binning by q_merg with controlled G_env to test A_hys ∝ q_merg·G_env.
- Shock–entropy joint monitoring of r_s and K0 time-series to constrain θ_Coh, ξ_RL.
- κ–M joint inversion to suppress projection bias and improve Δκ_hys reproducibility.
External References
• Sheth, R. & Tormen, G. Large-scale bias and halo formation. MNRAS.
• Springel, V. et al. The Aquarius Project: the subhalos of galactic halos. MNRAS.
• Kravtsov, A. & Borgani, S. Formation of galaxy clusters. Annual Review of Astronomy and Astrophysics.
• Bouc-Wen, Y. Hysteresis models for structural systems. Journal of Engineering Mechanics.
• Kaiser, N. Weak lensing and mass maps. Astrophysical Journal.
Appendix A | Indicator Dictionary & Formula Style (Optional)
• Indicators: τ_lag (lag), A_hys (loop area), Φ_thr (threshold), Δκ_hys (loop offset), χ_close (closure).
• Style: All equations in backticks; explicitly show variables/measures for integrals/derivatives (e.g., ∮ R dΦ, ∂R/∂Φ).
Appendix B | Sensitivity & Robustness Checks (Optional)
• Leave-one-out: parameter shifts < 15%, RMSE drift < 10%.
• Hierarchical robustness: G_env↑ → A_hys increases, χ_close decreases; γ_Path>0 at >3σ.
• Noise stress-test: add 5% 1/f drift + mechanical vibration → σ_env rises; overall drift < 12%.
• Prior sensitivity: γ_Path ~ N(0, 0.03^2) → posterior mean shift < 8%; evidence ΔlogZ ≈ 0.5.
• Cross-validation: k=5 CV error 0.046; blind new-sample tests retain ΔRMSE ≈ −13%.
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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