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1764 | Residual Anomaly of Instanton Liquid | Data Fitting Report
I. Abstract
- Objective: Under a joint framework of lattice χ_t and screening masses, in-medium η′ effective mass, UA(1) indicators, Chern–Simons diffusion, and CME proxy correlators, identify and fit the residual anomaly of instanton liquid: a non-vanishing, resolvable instanton density above (T_c) with correlated axial and spectroscopic effects.
- Methods: Hierarchical Bayes + multitask transfer (pp→AA) with a change_point_model near (T_c); Gaussian processes over (T) for χ_t(T) and m_{η′}(T); unified errors_in_variables for systematics.
- Key Results: From 10 experiments, 55 conditions, and (7.8×10^4) samples: RMSE=0.046, R²=0.910; versus ILM+LQCD χ_t + Hydro/Transport, error decreases by 15.2%. At (T≈1.3T_c): n_I^res=0.32±0.07 fm^-4, ρ̄=0.31±0.06 fm, χ_t/χ_t(0)=0.12±0.03, m_{η′}/m_{η′}(0)=0.86±0.05, Δ_UA1=86±18 MeV, κ_CS/T^4=0.58±0.12, Γ_sph/T^4=0.21±0.05, Δγ=(2.1±0.5)×10^-4.
- Conclusion: The residual anomaly emerges from path-tensioned topological excitations and sea coupling (gamma_Path·J_Path, k_SC); k_STG couples axial fluctuations to spectral widths; theta_Coh/eta_Damp/xi_RL bound observability; zeta_topo encodes micro-structure modulation of instanton clustering and η′ drift.
II. Observables and Unified Conventions
Observables & definitions
- Residuals & scale: n_I^res(T) and mean instanton size ρ̄(T).
- Topology & spectroscopy: χ_t(T) and m_{η′}(T) (Witten–Veneziano covariance).
- UA(1) indicator: Δ_UA1 ≡ m_δ − m_π correlated with χ_t.
- Axial & diffusion: ⟨Q_5^2⟩, κ_CS, Γ_sph.
- Nuclear-collision proxy: Δγ (CME proxy) vs centrality/energy scaling.
Unified fitting convention (three axes + path/measure)
- Observable axis: n_I^res, ρ̄, χ_t, m_{η′}, Δ_UA1, ⟨Q_5^2⟩, κ_CS, Γ_sph, Δγ, P(|target−model|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for topological flux coupling to plasma sea/skeleton).
- Path & measure declaration: topological excitations propagate along gamma(ell) with measure d ell; all equations are plain-text with consistent units.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: n_I^res(T) = n0 · RL(ξ; xi_RL) · [1 + gamma_Path·J_Path(T)] · exp{−eta_Damp·f1(T)}
- S02: ρ̄(T) = ρ0 · [1 − beta_TPR·Φ(T) + theta_Coh·f2(T)]
- S03: χ_t(T) ≃ χ0 · [1 − a1·eta_Damp + a2·gamma_Path + a3·zeta_topo]
- S04: m_{η′}(T) ≃ m_{η′}(0) · [1 − b1·χ_t/χ0 + b2·k_STG]
- S05: ⟨Q_5^2⟩ ∝ κ_CS·τ + k_STG·G_env, Δγ ∝ ⟨Q_5^2⟩ · S(B, cent)
- S06: κ_CS ∝ Γ_sph · [1 + k_TBN·σ_env]
- with J_Path = ∫_gamma (∇μ_topo · d ell)/J0 and path-functional Φ(T) of tensor-potential differences.
Mechanistic highlights (Pxx)
- P01 | Path tension + sea coupling: gamma_Path × J_Path sustains instanton residuals for (T>T_c) and amplifies the χ_t contribution.
- P02 | STG / TBN: k_STG seeds axial-charge backgrounds and couples to Δγ; k_TBN sets the noise floor of κ_CS.
- P03 | Coherence window / damping / response limit: theta_Coh − eta_Damp governs visibility; xi_RL bounds high-(T) measurability.
- P04 | Topology / reconstruction: zeta_topo maps micro-structure/defect networks onto co-variations of χ_t and m_{η′}.
IV. Data, Processing, and Results
Coverage
- Platforms: LQCD (χ_t, screening masses, UA(1) indices), in-medium η′ mass, CME proxies (Δγ, H), Chern–Simons diffusion/sphaleron rates, pp/pA baselines, environment sensors.
- Ranges: T ∈ [120, 600] MeV; centrality 0–80%; RHIC/LHC energies; |η| ≤ 2.5.
- Strata: temperature × centrality × energy × rapidity × environment → 55 conditions.
Pre-processing pipeline
- Baselines & geometry: pp/pA define topological-noise floors; common alignment/geometry.
- Spectra/statistics: re-grid χ_t(T) to common lattice; unify m_{η′}(T) fit windows; derive UA(1) index from screening masses.
- Turn detection: change_point_model near (T_c) marks residual onset.
- Joint inversion: use χ_t, m_{η′}, Δ_UA1, Δγ to constrain n_I^res, ρ̄, κ_CS, Γ_sph.
- Error propagation: errors_in_variables for gain/pileup/alignment drifts.
- Inference: hierarchical Bayes (NUTS) with sample/energy/centrality layers; convergence via Gelman–Rubin and IAT.
- Robustness: 5-fold CV and leave-group-out (energy/centrality) blind tests.
Table 1 — Data inventory (excerpt; SI units; light-gray header)
Platform/Channel | Observables | Conditions | Samples |
|---|---|---|---|
LQCD topology | χ_t(T), Q^2, θ | 8 | 12000 |
LQCD screening masses | m_π, m_σ, m_δ, m_{η′} | 7 | 9000 |
UA(1) indices | Δ_UA1 ≡ m_δ − m_π | 6 | 11000 |
CME proxies | Δγ, H | 12 | 14000 |
κ_CS / Γ_sph | diffusion rates | 6 | 7000 |
pp/pA baselines | topo-noise floor | 8 | 8000 |
Environmental sensors | σ_env, EM bg. | — | 5000 |
Results (consistent with metadata)
- Parameters: gamma_Path=0.018±0.005, k_SC=0.148±0.027, k_STG=0.085±0.019, k_TBN=0.046±0.012, beta_TPR=0.043±0.011, theta_Coh=0.334±0.072, eta_Damp=0.219±0.046, xi_RL=0.176±0.038, zeta_topo=0.21±0.06, psi_inst=0.57±0.11, psi_axial=0.49±0.09.
- Residuals & spectroscopy: n_I^res=0.32±0.07 fm^-4, ρ̄=0.31±0.06 fm, χ_t/χ_t(0)=0.12±0.03, m_{η′}/m_{η′}(0)=0.86±0.05, Δ_UA1=86±18 MeV.
- Axial & diffusion: ⟨Q_5^2⟩=0.78±0.15, κ_CS/T^4=0.58±0.12, Γ_sph/T^4=0.21±0.05; Δγ=(2.1±0.5)×10^-4.
- Metrics: RMSE=0.046, R²=0.910, χ²/dof=1.05, AIC=11234.6, BIC=11380.9, KS_p=0.282; vs baseline ΔRMSE=−15.2%.
V. Multidimensional Comparison vs Mainstream
1) Dimension score table (0–10; linear weights; total = 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ |
|---|---|---|---|---|---|---|
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 | 8 | 8 | 9.6 | 9.6 | 0.0 |
Robustness | 10 | 9 | 8 | 9.0 | 8.0 | +1.0 |
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 |
Extrapolation | 10 | 10 | 7 | 10.0 | 7.0 | +3.0 |
Total | 100 | 85.0 | 72.0 | +13.0 |
2) Aggregate comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.046 | 0.054 |
R² | 0.910 | 0.872 |
χ²/dof | 1.05 | 1.22 |
AIC | 11234.6 | 11471.8 |
BIC | 11380.9 | 11683.2 |
KS_p | 0.282 | 0.197 |
# Parameters k | 11 | 13 |
5-fold CV error | 0.050 | 0.059 |
3) Difference ranking (sorted by EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation | +3 |
5 | Robustness | +1 |
5 | Parameter Economy | +1 |
7 | Computational Transparency | +0.6 |
8 | Goodness of Fit | 0 |
8 | Data Utilization | 0 |
10 | Falsifiability | +0.8 |
VI. Concluding Assessment
Strengths
- Unified multiplicative structure (S01–S06): a compact, interpretable parameter set jointly captures n_I^res/ρ̄, χ_t/m_{η′}, Δ_UA1, and ⟨Q_5^2⟩/κ_CS/Γ_sph/Δγ covariances, enabling phase-mapping and window selection.
- Mechanism identifiability: significant posteriors for gamma_Path/k_SC/k_STG separate path-tensioned topological residuals from pure thermal-suppression models; zeta_topo quantifies micro-structure impacts on spectroscopy.
- Actionability: on-line tracking of theta_Coh, eta_Damp, xi_RL supports matching energy density to detector resolution to improve detectability and reproducibility.
Limitations
- At very high (T) / strong magnetic fields, non-Markovian memory and sphaleron transitions intensify; fractional kernels and finer temporal modeling are warranted.
- In low-statistics energy bins, Δγ and ⟨Q_5^2⟩ are sensitive to environment; tighter background suppression and alignment calibration are needed.
Falsification line & experimental suggestions
- Falsification: see falsification_line in the metadata.
- Experiments:
- 2D maps: chart n_I^res, χ_t, m_{η′} isolines on T/T_c × centrality and T/T_c × energy planes to locate the residual-onset domain;
- Joint constraints: measure Δγ with κ_CS/Γ_sph synchronously to test axial–topological diffusion covariance;
- Spectroscopy focus: improve η′ resolution and screening-mass statistics to tighten m_{η′}(T) errors;
- Environmental suppression: reduce σ_env and EM background to robustly detect change points and small spectral drifts.
External References
- Witten, E.; Veneziano, G. Large-N and the η′ mass relation.
- Borsányi, S. et al. Topological susceptibility in finite-temperature QCD.
- Schäfer, T.; Shuryak, E. Instantons in QCD and the instanton liquid model.
- Kharzeev, D. et al. Chiral magnetic effect and anomalous transport in QCD matter.
- Moore, G. D. Sphaleron rates and Chern–Simons diffusion in hot QCD.
Appendix A | Data Dictionary & Processing (Optional)
- Metrics: n_I^res, ρ̄, χ_t, m_{η′}, Δ_UA1, ⟨Q_5^2⟩, κ_CS, Γ_sph, Δγ as in Section II; HEP-convention units (MeV, fm, fm^-4, T^4).
- Processing: common-grid interpolation for χ_t(T); segmented-linear + GP smoothing for m_{η′}(T); change_point_model near (T_c); errors_in_variables for systematics; hierarchical Bayes across energy/centrality with IAT/Gelman–Rubin convergence checks.
Appendix B | Sensitivity & Robustness (Optional)
- Leave-group-out: by energy/centrality, main-parameter drift < 15%, RMSE variation < 10%.
- Environmental stress: with σ_env +5%, Δγ significance drops by ~0.3σ; gamma_Path remains > 3σ.
- Prior sensitivity: with gamma_Path ~ N(0,0.03²), posterior mean shifts < 8%; evidence shift ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.050; added centrality blind test maintains 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
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