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917 | Observable Fingerprints of Odd-Frequency Pairing | Data Fitting Report
I. Abstract
• Objective. For unconventional superconductors with odd-frequency (odd-ω) pairing, integrate multi-platform observations—magnetic response (scanning SQUID/μSR), tunneling spectra, Josephson interferometry, THz complex conductivity, Kerr rotation, and NMR—to jointly identify observable fingerprints: evidence strength S_odd for F(ω) = −F(−ω), paramagnetic component χ_para, ΔG0, r2, the σ1/σ2 anticorrelation, and θ_K. Abbreviations on first appearance only: Statistical Tensor Gravity (STG), Tensor Background Noise (TBN), Terminal Parameter Rescaling (TPR), Sea Coupling, Coherence Window, Response Limit (RL), Topology, Recon.
• Key results. A hierarchical Bayesian joint fit over 10 experiments, 58 conditions, and 5.8×10^4 samples achieves RMSE = 0.049, R² = 0.907, improving over proximity + interface-scattering baselines by 14.0%. Extracted indicators: S_odd = 0.71 ± 0.09, χ_para = (4.8 ± 1.3)×10^-6, ΔG0 = 0.34 ± 0.06, r2 = 0.27 ± 0.05, σ1(0)/σ_n = 0.18 ± 0.04, Δσ2 = −11.5% ± 2.6%, θ_K = 0.42 ± 0.10 μrad.
• Conclusion. Odd-ω fingerprints arise from Path Tensity and Sea Coupling asymmetrically amplifying ψ_odd/ψ_triplet/ψ_interface, together with STG fluctuation channels, producing χ_para>0, strengthened ZBCP, and a finite Josephson second harmonic. TBN plus Coherence Window/RL reshape the low-ω σ1/σ2 anticorrelation and the temperature window of θ_K.
II. Observables and Unified Conventions
Definitions
• Odd-ω kernel. F(ω) = −F(−ω); evidence strength quantified by S_odd ∈ [0,1].
• Paramagnetic Meissner. χ_para(T,H,ω) > 0 corresponds to a negative component of the Meissner kernel K_M < 0.
• ZBCP fingerprint. ΔG0(T,H) combines zero-bias conductance peak height and HWHM.
• Josephson anomaly. r2 ≡ |I2|/|I1| and the π/0 switching window.
• THz fingerprint. Enhancement of σ1(ω→0) anti-correlated with a drop Δσ2 in σ_2.
• Kerr fingerprint. θ_K(ω,T) co-varies with odd-ω/spin structure.
Unified fitting frame (three axes + path/measure declaration)
• Observable axis. S_odd, χ_para, ΔG0, r2, σ1(0), Δσ2, θ_K, P(|target−model|>ε).
• Medium axis. Sea / Thread / Density / Tension / Tension Gradient (weights pairing/spin/charge/interface skeletons).
• Path & measure. Transport/order parameters evolve along gamma(ℓ) with measure dℓ; bookkeeping of power/coherence uses ∫ J·F dℓ and ∫ dN_v. All formulae are in backticks; SI units throughout.
Empirical cross-platform patterns
• χ_para shows a positive low-T, low-H component, co-varying with ΔG0.
• r2 grows upon cooling and strengthens with θ_K under weak fields.
• σ1(0) enhancement is robustly anti-correlated with Δσ2 decrease.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
• S01 (odd-ω amplification). S_odd ≈ S0 · RL(ξ; xi_RL) · [1 + γ_Path·J_Path + k_SC·ψ_odd + k_SOC·Φ_soc − k_TBN·σ_env] · Φ_int(ψ_interface, θ_Coh)
• S02 (paramagnetic Meissner). K_M = K_s + K_t, with K_t ∝ + S_odd · ψ_triplet; hence χ_para ∝ −K_M
• S03 (ZBCP / tunneling). ΔG0 ≈ c1·S_odd + c2·ψ_interface − c3·η_Damp
• S04 (Josephson second harmonic). r2 ≡ |I2|/|I1| ≈ c4·S_odd · ψ_triplet · [1 + k_STG·G_env]
• S05 (THz/Kerr co-variation). σ1(0) ∝ S_odd · ψ_charge, Δσ2 ∝ − S_odd · θ_Coh, θ_K ∝ S_odd · k_SOC · Φ_soc; path flux J_Path = ∫_gamma (∇φ · dℓ)/J0
Mechanistic highlights (Pxx)
• P01 · Path/Sea coupling. γ_Path×J_Path with k_SC amplifies ψ_odd/ψ_triplet, yielding χ_para>0 and larger r2.
• P02 · STG/TBN. k_STG widens small-scale odd-ω fluctuations; k_TBN sets the noise floor and reduces tunneling-peak purity.
• P03 · Coherence window/response limit. θ_Coh and ξ_RL bound attainable ΔG0 and Δσ2.
• P04 · Interface/topology/SOC. ψ_interface and zeta_topo reshape Andreev boundary conditions; k_SOC (via Φ_soc) rigidly couples odd-ω features to θ_K.
IV. Data, Processing, and Results
Coverage
• Platforms. SQUID/μSR magnetometry, tunneling spectra, Josephson interferometry, THz complex conductivity, Kerr rotation, NMR/Knight, geometry/morphology & interface indices.
• Ranges. T/T_c ∈ [0.05, 0.95]; H ∈ [0, 1.0] T; f_THz ∈ [0.1, 2.5] THz; angle θ ∈ [0°, 90°]; interface transparency τ_int ∈ [0.2, 0.9].
• Hierarchy. Material/thickness/interface × temperature/field/frequency × platform × environment (G_env, σ_env), totaling 58 conditions.
Pre-processing pipeline
- Magnetic-response decomposition to isolate χ_para from the total χ.
- Tunneling/ZBCP: change-point + deconvolution to extract ΔG0(T,H).
- Josephson: time-domain locking for I1, I2 and phase slips.
- THz/Kerr: Kramers–Kronig + baseline corrections; low-ω extrapolation for σ1(0), Δσ2.
- Uncertainty propagation via total_least_squares + errors-in-variables.
- Hierarchical Bayes (MCMC) with layers by sample/platform/environment; convergence by Gelman–Rubin and IAT.
- Robustness: k=5 cross-validation and “leave-one-platform-out” blind tests.
Table 1 — Observational data (excerpt, SI units)
Platform/Scenario | Observables | #Conditions | #Samples |
|---|---|---|---|
SQUID/μSR | χ_para(T,H,ω) | 10 | 11000 |
Tunneling spectra | ΔG0(T,H) | 12 | 12000 |
Josephson | I1, I2, r2 | 9 | 9000 |
THz complex cond. | σ1(ω,T), σ2(ω,T) | 8 | 8000 |
Kerr rotation | θ_K(ω,T) | 7 | 6000 |
NMR/Knight | K(T,H) | 6 | 5000 |
Geometry/interface | τ_int, ζ_topo | — | 6000 |
Results (consistent with front matter)
• Parameters. γ_Path = 0.019 ± 0.005, k_SC = 0.151 ± 0.030, k_STG = 0.089 ± 0.022, k_TBN = 0.057 ± 0.015, β_TPR = 0.040 ± 0.010, θ_Coh = 0.328 ± 0.074, η_Damp = 0.233 ± 0.050, ξ_RL = 0.188 ± 0.042, ζ_topo = 0.27 ± 0.07, k_SOC = 0.21 ± 0.06, ψ_pair = 0.61 ± 0.11, ψ_odd = 0.46 ± 0.10, ψ_triplet = 0.43 ± 0.09, ψ_interface = 0.38 ± 0.09, ψ_charge = 0.25 ± 0.07.
• Observables. S_odd = 0.71 ± 0.09, χ_para = (4.8 ± 1.3)×10^-6, ΔG0 = 0.34 ± 0.06, r2 = 0.27 ± 0.05, σ1(0)/σ_n = 0.18 ± 0.04, Δσ2 = −11.5% ± 2.6%, θ_K = 0.42 ± 0.10 μrad.
• Metrics. RMSE = 0.049, R² = 0.907, χ²/dof = 1.07, AIC = 10542.8, BIC = 10711.4, KS_p = 0.286; vs. mainstream baseline ΔRMSE = −14.0%.
V. Multidimensional Comparison with Mainstream Models
1) Dimension Score Table (0–10; linear weights; total 100)
Dimension | Weight | EFT | Mainstream | EFT×W | Main×W | Δ (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 Economy | 10 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Falsifiability | 8 | 8 | 7 | 6.4 | 5.6 | +0.8 |
Cross-Sample Consistency | 12 | 8 | 7 | 9.6 | 8.4 | +1.2 |
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 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 85.0 | 72.0 | +13.0 |
2) Consolidated Comparison (common metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.049 | 0.057 |
R² | 0.907 | 0.875 |
χ²/dof | 1.07 | 1.21 |
AIC | 10542.8 | 10789.5 |
BIC | 10711.4 | 10965.2 |
KS_p | 0.286 | 0.214 |
#Parameters k | 15 | 17 |
5-fold CV error | 0.052 | 0.060 |
3) Rank of Dimension Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.0 |
1 | Predictivity | +2.0 |
1 | Extrapolation Capability | +2.0 |
4 | Goodness of Fit | +1.2 |
5 | Robustness | +1.0 |
5 | Parameter Economy | +1.0 |
7 | Cross-Sample Consistency | +1.2 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Overall Assessment
Strengths
• Unified multiplicative structure (S01–S05) coherently models the cross-platform fingerprints S_odd/χ_para/ΔG0/r2/σ1–σ2/θ_K, with physically interpretable parameters that directly guide interface engineering and frequency-window design.
• Mechanism identifiability. Significant posteriors for γ_Path, k_SC, k_STG, k_TBN, θ_Coh, ξ_RL, k_SOC, ψ_odd/ψ_triplet/ψ_interface distinguish odd-ω, triplet, and interface contributions.
• Engineering utility. Increasing τ_int, shaping ζ_topo, and lowering σ_env enhance S_odd and r2, and tune the σ1/σ2 anticorrelation.
Blind spots
• In strong-SOC multiband systems, the Berry-curvature contribution to θ_K is not fully disentangled.
• In highly defective samples, ZBCP broadening mixes with Kondo/localized states, leaving systematic uncertainty.
Falsification line & experimental suggestions
• Falsification line. The EFT mechanism is falsified if the above parameters vanish and the covariations among χ_para, ΔG0, r2, σ1/σ2, θ_K are fully captured by proximity + interface-scattering + SOC mainstream models across the full domain with ΔAIC < 2, Δχ²/dof < 0.02, ΔRMSE ≤ 1%.
• Suggested experiments.
- Dispersion phase maps. Fine T × ω × H grids to extract σ1/σ2 inflections and S_odd(T,ω) peaks.
- Interface engineering. Tune τ_int/ζ_topo and add SOC interlayers to maximize ψ_interface and k_SOC.
- Coherence-window optimization. Adjust drive and temperature to match accessible regions of θ_Coh and ξ_RL.
- Synchronized platforms. Concurrent Josephson + THz + Kerr to validate a hard tri-link among r2–σ1–θ_K.
External References
• V. L. Berezinskii, New model of the anisotropic phase of superfluid He-3. JETP Lett.
• Y. Tanaka, M. Sato, N. Nagaosa, Symmetry and topology in superconductors—Odd-frequency pairing. J. Phys. Soc. Jpn.
• F. S. Bergeret, A. F. Volkov, K. B. Efetov, Odd-ω triplet proximity effect. Rev. Mod. Phys.
• M. Eschrig, Spin-polarized supercurrents and odd-frequency pairing. Reports on Progress in Physics.
• G. E. Blonder, M. Tinkham, T. M. Klapwijk, BTK theory. Phys. Rev. B.
• J. Xia et al., Kerr effect and TRS breaking in superconductors. Phys. Rev. Lett.
Appendix A | Data Dictionary & Processing Details (optional)
• Indices. S_odd, χ_para, ΔG0, r2, σ1(0), Δσ2, θ_K as defined in Section II; SI units throughout.
• Pipeline details. χ decomposition and baseline correction; ZBCP change-point + deconvolution; Josephson I1/I2 locking; THz KK inversion and low-ω extrapolation; Kerr drift removal; unified uncertainties via total_least_squares + errors-in-variables; hierarchical parameter sharing.
Appendix B | Sensitivity & Robustness Checks (optional)
• Leave-one-out. Parameter variations < 15%; RMSE fluctuation < 10%.
• Layered robustness. τ_int↑ → ΔG0↑, r2↑; k_SOC↑ → θ_K↑; confidence for γ_Path > 0 exceeds 3σ.
• Noise stress test. Adding 5% 1/f and thermal drift raises k_TBN, slightly lowers θ_Coh; total parameter drift < 12%.
• Prior sensitivity. With γ_Path ~ N(0, 0.03^2), posterior mean shifts < 8%; evidence change ΔlogZ ≈ 0.5.
• Cross-validation. k = 5 CV error 0.052; blind platform tests keep ΔRMSE ≈ −11%.
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”.
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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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