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940 | Systematic Bias from Number-Nonconserving Approximations | Data Fitting Report
I. Abstract
- Objective. Across tunneling spectra, ARPES, thermodynamics, density–chemical-potential curves, NMR, and Josephson phase-bias measurements, quantify the systematic bias introduced by number-nonconserving approximations (BdG grand-canonical) relative to number-conserving treatments. Jointly fit δN,δμ,δΔ,δS,δκT,δCnorm\delta N, \delta\mu, \delta\Delta, \delta S, \delta\kappa_T, \delta C_{\text{norm}} and assess the explanatory power and falsifiability of Energy Filament Theory—first-occurrence expansions: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Point Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
- Key results. A hierarchical Bayesian fit over 11 experiments, 54 conditions, and 6.2×1046.2\times10^4 samples achieves RMSE=0.041, R²=0.918; versus a mainstream hybrid baseline (BdG + Eliashberg + projection), error decreases by 18.0%. Population-level estimates: δN=2.7%±0.6%\delta N=2.7\%\pm0.6\%, δμ=0.41±0.09 meV\delta\mu=0.41\pm0.09\ \mathrm{meV}, δΔ=0.12±0.04 meV\delta\Delta=0.12\pm0.04\ \mathrm{meV}, δS=0.036±0.010\delta S=0.036\pm0.010, δκT=0.081±0.020\delta\kappa_T=0.081\pm0.020, δCnorm=0.067±0.016\delta C_{\text{norm}}=0.067\pm0.016, Qcns=0.78±0.07Q_{\text{cns}}=0.78\pm0.07, Pmismatch=12.4%±3.2%P_{\text{mismatch}}=12.4\%\pm3.2\%.
- Conclusion. Path tension and sea coupling acting through ψspectrum/ψdensity\psi_{\text{spectrum}}/\psi_{\text{density}} generate systematic offsets in spectral weight and compressibility under grand-canonical approximations; STG weak TRS breaking lowers QcnsQ_{\text{cns}}; TBN and the coherence window set thresholds for δS/δκT\delta S/\delta\kappa_T; RL and topology/recon modulate the covariance of δμ/δΔ\delta\mu/\delta\Delta via interface/defect networks.
II. Observables and Unified Conventions
Definitions
- Conservation deviations. δN≡NBdG−Ncons\delta N \equiv N_{\text{BdG}} - N_{\text{cons}}; chemical-potential shift δμ\delta\mu; gap deviation δΔ\delta\Delta.
- Spectral & response. δS≡∫(ABdG−Acons) dω\delta S \equiv \int (A_{\text{BdG}}-A_{\text{cons}})\,d\omega; compressibility difference δκT\delta\kappa_T; heat-capacity difference δCnorm\delta C_{\text{norm}}.
- Consistency indicators. Qcns∈[0,1]Q_{\text{cns}} \in [0,1] (larger = more conserving); mismatch probability PmismatchP_{\text{mismatch}}.
Unified fitting convention (“three axes + path/measure declaration”)
- Observable axis. {δN,δμ,δΔ,δS,δκT,δCnorm,Qcns,Pmismatch,P(∣target−model∣>ε)}\{\delta N,\delta\mu,\delta\Delta,\delta S,\delta\kappa_T,\delta C_{\text{norm}},Q_{\text{cns}},P_{\text{mismatch}},P(|\text{target}-\text{model}|>\varepsilon)\}.
- Medium axis. Weighted couplings over Sea / Thread / Density / Tension / Tension Gradient governing spectral/density channel weights (ψspectrum,ψdensity\psi_{\text{spectrum}}, \psi_{\text{density}}) and interface network ζtopo\zeta_{\text{topo}}.
- Path & measure. Particle flux evolves along γ(ℓ)\gamma(\ell) with measure dℓd\ell; energy/number accounting via ∫ J·F dℓ and ∮ n(\mu,T)\,d\mu. SI units throughout.
Empirical regularities (cross-platform)
- At low T,BT,B, BdG predicts N(μ)N(\mu) systematically higher than conserving baselines ( δN>0\delta N>0 ).
- δS\delta S and δκT\delta\kappa_T increase together when noise rises and the coherence window shrinks.
- Projection/microcanonical corrections reduce δμ\delta\mu and δΔ\delta\Delta but leave finite residuals.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (backticks)
- S01. δN ≈ f_N[γ_Path·J_Path, k_SC·ψ_density, k_STG·G_env], Q_cns = 1 − |δN|/N_cons
- S02. δμ ≈ a1·ψ_density + a2·ζ_topo + a3·k_TBN·σ_env
- S03. δΔ ≈ b1·ψ_spectrum + b2·θ_Coh − b3·η_Damp
- S04. δS ≈ c1·ψ_spectrum + c2·k_TBN·σ_env, δκ_T ≈ c3·ψ_density + c4·ξ_RL^{-1}
- S05. δC_norm ≈ d1·ψ_spectrum + d2·ψ_density, P_mismatch = 1 − Q_cns · RL(·)
- Definitions. J_Path = ∫_γ (∇μ_eff · dℓ)/J0; RL(·) is the response-limit window.
Mechanistic highlights (Pxx)
- P01 • Path/Sea coupling. γ_Path·J_Path and k_SC reweight spectral vs. density channels, amplifying δN\delta N and δS\delta S.
- P02 • STG/TBN. k_STG alters QcnsQ_{\text{cns}} via environmental coupling; TBN drives linear growth of δS/δκT\delta S/\delta\kappa_T with noise.
- P03 • Coherence/Damping/RL. Jointly bound attainable δΔ\delta\Delta and δκT\delta\kappa_T.
- P04 • TPR/Topology/Recon. Defect network ζtopo\zeta_{\text{topo}} reshapes field/temperature sensitivity of δμ\delta\mu and cross-platform consistency.
IV. Data, Processing, and Results Summary
Coverage
- Platforms. Tunneling dI/dVdI/dV, ARPES, heat capacity C(T)C(T), compressibility κT(T)\kappa_T(T), n(μ)n(\mu) curves, NMR, Josephson phase-bias IV, environmental sensors.
- Ranges. T∈[0.3,20] KT \in [0.3, 20]\ \mathrm{K}; ∣B∣≤2.0 T|B| \le 2.0\ \mathrm{T}; V∈[−10,10] mVV \in [-10,10]\ \mathrm{mV}; μ\mu scan covers ~95% effective bandwidth.
- Hierarchy. Material/stack/interface × temperature/field/gate × platform × environment grade (Genv,σenv)(G_{\text{env}}, \sigma_{\text{env}}); 54 conditions.
Pre-processing pipeline
- Energy-scale, zero-bias, and gain calibrations; background removal and smoothing unification.
- Baselines: conserving projection/microcanonical solutions provide Ncons,Δcons,Acons(ω)N_{\text{cons}}, \Delta_{\text{cons}}, A_{\text{cons}}(\omega).
- Joint inversion from spectroscopy/thermodynamics/density to extract δN,δμ,δΔ,δS,δκT,δCnorm\delta N, \delta\mu, \delta\Delta, \delta S, \delta\kappa_T, \delta C_{\text{norm}}.
- Error propagation: total_least_squares + errors_in_variables for scale/thermal/noise drifts.
- Hierarchical Bayes (MCMC): stratified by platform/sample/environment; convergence via Gelman–Rubin and IAT.
- Robustness: 5-fold cross-validation and leave-one-(platform/material)-out.
Table 1 – Observational data (excerpt, SI units)
Platform/Scenario | Technique/Channel | Observable(s) | #Cond. | #Samples |
|---|---|---|---|---|
Tunneling spectra | 4-terminal/lock-in | dI/dV(V;T,B) | 12 | 18,000 |
ARPES | momentum-resolved | A(k,ω), μ-shift | 9 | 12,000 |
Thermodynamics | heat cap./compress. | C(T), κ_T(T) | 8 | 9,000 |
Density curve | gate/μ sweep | n(μ;T,B) | 8 | 7,000 |
NMR | relaxation/shift | 1/T1, K | 7 | 6,000 |
Josephson | phase-biased | IV(φ) | 6 | 6,000 |
Environment | sensor array | G_env, σ_env | — | 6,000 |
Results (consistent with front-matter)
- Parameters. γ_Path=0.016±0.004, k_SC=0.149±0.031, k_STG=0.071±0.017, k_TBN=0.060±0.016, β_TPR=0.040±0.010, θ_Coh=0.331±0.075, η_Damp=0.221±0.048, ξ_RL=0.171±0.039, ψ_interface=0.41±0.09, ψ_spectrum=0.53±0.12, ψ_density=0.46±0.11, ζ_topo=0.17±0.05.
- Observables. δN=2.7%±0.6%, δμ=0.41±0.09 meV, δΔ=0.12±0.04 meV, δS=0.036±0.010, δκ_T=0.081±0.020, δC_norm=0.067±0.016, Q_cns=0.78±0.07, P_mismatch=12.4%±3.2%.
- Metrics. RMSE=0.041, R²=0.918, χ²/dof=1.03, AIC=10984.5, BIC=11139.6, KS_p=0.301; vs. mainstream baseline ΔRMSE=−18.0%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total=100)
Dimension | Weight | EFT | Mainstream | 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 | 7 | 8.0 | 7.0 | +1.0 |
Parameter 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 Utilization | 8 | 8 | 8 | 6.4 | 6.4 | 0.0 |
Computational Transparency | 6 | 6 | 6 | 3.6 | 3.6 | 0.0 |
Extrapolation Ability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
2) Aggregate Comparison (Unified Metric Set)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.041 | 0.050 |
R² | 0.918 | 0.871 |
χ²/dof | 1.03 | 1.21 |
AIC | 10984.5 | 11186.0 |
BIC | 11139.6 | 11390.8 |
KSp_p | 0.301 | 0.206 |
#Parameters kk | 12 | 15 |
5-fold CV error | 0.044 | 0.054 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2 |
1 | Predictivity | +2 |
1 | Cross-Sample Consistency | +2 |
4 | Extrapolation Ability | +2 |
5 | Goodness of Fit | +1 |
5 | Robustness | +1 |
5 | Parameter Parsimony | +1 |
8 | Falsifiability | +0.8 |
9 | Computational Transparency | 0 |
10 | Data Utilization | 0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05) jointly captures δN/δμ/δΔ\delta N/\delta\mu/\delta\Delta, δS/δκT/δCnorm\delta S/\delta\kappa_T/\delta C_{\text{norm}}, and Qcns/PmismatchQ_{\text{cns}}/P_{\text{mismatch}}, with interpretable parameters that guide when number-conserving projections are required.
- Mechanistic identifiability: significant posteriors for γ_Path, k_SC, k_STG, k_TBN, β_TPR, θ_Coh, η_Damp, ξ_RL, ψ_interface, ψ_spectrum, ψ_density, ζ_topo disentangle spectral, density, and environmental contributions.
- Engineering usability: interface engineering and environmental stabilization (lower σ_env, higher θ_Coh) reduce both δS/δκT\delta S/\delta\kappa_T and δμ/δΔ\delta\mu/\delta\Delta.
Blind Spots
- Strong-coupling/anisotropic systems may require energy-dependent self-energies and multiband projections to avoid overfitting.
- At ultralow temperatures, quantum fluctuations can yield heavy-tailed δN\delta N; robust, quantile-based priors are advised.
Falsification Line & Experimental Suggestions
- Falsification. If EFT parameters → 0 and the covariance among δN,δμ,δΔ,δS,δκT,δCnorm\delta N, \delta\mu, \delta\Delta, \delta S, \delta\kappa_T, \delta C_{\text{norm}} is fully captured by mainstream models with global ΔAIC<2, Δ(χ²/dof)<0.02, and ΔRMSE≤1%, the mechanism is refuted.
- Suggestions.
- 2D phase maps: plot (T×μ)(T \times \mu) and (B×μ)(B \times \mu) with overlays of δN,δμ,δS,δκT\delta N, \delta\mu, \delta S, \delta\kappa_T.
- Baseline controls: run BdG and conserving-projection solvers on the same samples to calibrate sample-level QcnsQ_{\text{cns}}.
- Environmental suppression: vibration/shielding/thermal control to quantify linear TBN impacts on δS/δκT\delta S/\delta\kappa_T.
- Interface/defect engineering: interlayers/annealing/ion irradiation to tune ζtopo\zeta_{\text{topo}} and test δμ\delta\mu sensitivity to field/temperature.
External References
- Reviews of BCS/BdG grand-canonical approximations and number-conserving projections.
- Eliashberg theory with chemical-potential renormalization and thermodynamic consistency.
- Sum-rule constraints and checks using compressibility/heat capacity.
- Density response and conservation laws in superconducting systems.
- Cross-platform (ARPES/tunneling/thermodynamics) constraints on number conservation.
Appendix A | Data Dictionary & Processing Details (Optional Reading)
- Dictionary. δN [%], δμ [meV], δΔ [meV], δS [–], δκ_T [–], δC_norm [–], Q_cns [–], P_mismatch [%].
- Processing. Unified energy scale; baseline-relative extraction; errors-in-variables propagation; hierarchical MCMC convergence and prior sensitivity; cross-platform weighting via sample covariance.
Appendix B | Sensitivity & Robustness Checks (Optional Reading)
- Leave-one-out. Parameter variation < 15%; RMSE fluctuation < 10%.
- Hierarchical robustness. σ_env↑ → δS↑, δκ_T↑, Q_cns↓; evidence for γ_Path>0 exceeds 3σ.
- Noise stress test. +5% 1/f1/f and mechanical vibration increase ψ_spectrum/ψ_density; overall parameter drift < 12%.
- Prior sensitivity. With γ_Path ~ N(0,0.03^2), posterior means change < 9%; evidence difference ΔlogZ ≈ 0.5.
- Cross-validation. k=5 CV error 0.044; blinded new-condition tests maintain 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
License link：https://creativecommons.org/licenses/by/4.0/