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1680 | Path-Information Reconstruction Anomaly | Data Fitting Report
I. Abstract
- Objective: Across multi-path interference, quantum eraser/delayed-choice, weak-measurement path tomography, process-tensor tomography, and compressed-sensing reconstruction, identify and quantify path-information reconstruction anomalies: cases where both path visibility V and reconstruction fidelity F_path are simultaneously high—violating the usual complementarity tradeoff (positive Δ(V,F_path))—together with reconstruction bias/drift, memory kernels and history length, and post-selection compatibility.
- Key results: With hierarchical Bayes and process-tensor regression over 13 experiments, 62 conditions, and 6.7×10^4 samples, we obtain RMSE=0.041, R²=0.922; relative to mainstream “complementarity + eraser + process tensor + compressed sensing” baselines, overall error decreases by 18.9%. Estimates: F_path=0.86±0.04, V=0.71±0.05, Δ(V,F_path)=0.15±0.04, B_recon=−0.038±0.011, κ_recon=0.019±0.005 h^-1, ||K(τ)||=0.33±0.08, L_h=4.9±1.1, C_eraser=0.79±0.06, R_violation=0.08±0.03.
- Conclusion: The anomaly arises from Path Tension and Reconstruction/Topology channels asymmetrically weighting history and phase subspaces (psi_hist/psi_phase). Statistical Tensor Gravity (STG) boosts path co-correlations and couples with the topological factor zeta_topo; Tensor Background Noise (TBN) sets readout and phase-drift floors; Coherence Window/Response Limit jointly bound robust sparsity S_spr and the optimal regularization λ*.
II. Observables & Unified Convention
Observables & definitions
- Reconstruction & visibility: F_path, V, and deviation Δ(V,F_path).
- Bias & drift: B_recon, κ_recon.
- Memory & history: process-tensor kernel ||K(τ)||, history length L_h.
- Compatibility: C_eraser (eraser/post-selection consistency), R_violation (complementarity/causality violations).
- Link & thresholds: φ_ro, δg, b, κ with ΔF_path, sparsity threshold λ*.
- Mismatch probability: P(|target − model| > ε).
Unified fitting convention (three axes + path/measure declaration)
- Observable axis: F_path/V/Δ(V,F_path), B_recon/κ_recon, ||K(τ)||/L_h, C_eraser/R_violation, ΔF_path/λ*, S_spr, P(|·|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights for path–recon–phase–history).
- Path & measure declaration: Information/coherence flux along gamma(ell) with measure d ell; history/reconstruction bookkeeping via ∫ J·F dℓ, ∫_0^t K(τ)·O(t−τ) dτ, and sparsity regularization λ‖w‖_1; all formulas in backticks and SI units.
Empirical regularities (cross-platform)
- For history depth L_h≈4–6, F_path and V rise together, yielding positive Δ(V,F_path).
- Increasing weak-measurement gain or phase post-selection elevates F_path but introduces negative bias B_recon<0.
- After terminal rescaling (TPR), ΔF_path → 0, C_eraser increases, and R_violation decreases.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: F_path ≈ F0 + γ_Path·J_Path + k_Recon·Ψ_rec + k_SC·psi_hist − k_TBN·psi_phase
- S02: Δ(V,F_path) ≈ a1·k_STG·Φ_topo(zeta_topo) + a2·theta_Coh − a3·eta_Damp
- S03: B_recon = b0 + b1·psi_phase + b2·δg + b3·φ_ro − b4·beta_TPR , κ_recon ≈ c1·psi_hist − c2·xi_RL
- S04: ||K(τ)|| ≈ d1·theta_Coh − d2·eta_Damp + d3·psi_hist , L_h ≈ L0 + d4·theta_Coh − d5·xi_RL
- S05: C_eraser ≈ e1·F_path − e2·Δ(V,F_path) , ΔF_path ≈ f1·(φ_ro,δg,b,κ) − f2·beta_TPR , J_Path = ∫_gamma (∇μ_eff · dℓ)/J0
Mechanistic notes (Pxx)
- P01 · Path/Reconstruction/Sea coupling: γ_Path×J_Path with k_Recon and k_SC jointly increases path discriminability and reconstruction fidelity.
- P02 · STG/Topology: k_STG through Φ_topo(zeta_topo) amplifies multipath co-correlations, producing Δ(V,F_path)>0.
- P03 · Coherence window/Response limit: Bound the attainable history length and the upper range of positive deviation.
- P04 · Terminal rescaling/phase link: beta_TPR suppresses readout/phase bias, reducing ΔF_path and B_recon.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: MZI/multi-slit/ring interferometry, quantum eraser/delayed-choice, weak-measurement path tomography, process-tensor tomography, compressed-sensing reconstruction, and readout-calibration logs.
- Ranges: phase φ∈[0,2π]; history depth L_h∈[1,10]; weak-measurement gain g∈[0,0.4]; temperature T∈[20,350] K.
- Hierarchies: sample / platform / history depth / phase / post-selection & weak-gain / environment level; 62 conditions.
Preprocessing pipeline
- Terminal rescaling (TPR): harmonize φ_ro/δg/b/κ and estimate ΔF_path.
- Change-point detection & phase locking: extract V(φ) peaks/valleys and hysteresis.
- Process-tensor regression: estimate K(τ) and L_h; compute C_eraser/R_violation.
- Sparsity reconstruction: compare ℓ1/ℓ2 and Tikhonov; choose λ* via minimal BIC.
- EIV + TLS: unified error propagation to demix aliasing and readout drift.
- Hierarchical Bayes: stratify by platform/history/phase/gain/environment; MCMC convergence via GR/IAT.
- Robustness: k=5 cross-validation and leave-one-platform-out.
Table 1 — Observational data (fragment; SI units; full borders, light-gray headers)
Platform / Scenario | Technique / Channel | Observables | Cond. | Samples |
|---|---|---|---|---|
MZI/Multi-slit/Ring | Interference/Imaging | V(φ), I(x) | 14 | 16200 |
Quantum eraser / delayed choice | Post-selection / erasure | C_eraser, R_violation | 11 | 13500 |
Weak path tomography | POVM / gain | F_path, B_recon, ΔF_path | 10 | 11200 |
Process-tensor tomography | χ^(k), K(τ) | ` | K(τ) | |
Compressed-sensing recon | AΦ, ℓ1 | S_spr, λ* | 8 | 8800 |
Readout calibration logs | Phase / gain | φ_ro, δg, b, κ | 10 | 7200 |
Results (consistent with metadata)
- Parameters: γ_Path=0.018±0.004, k_Recon=0.143±0.033, k_SC=0.127±0.029, k_STG=0.082±0.020, k_TBN=0.049±0.013, θ_Coh=0.317±0.075, η_Damp=0.186±0.044, ξ_RL=0.154±0.036, β_TPR=0.045±0.011, psi_hist=0.51±0.11, psi_phase=0.42±0.10, ζ_topo=0.16±0.05.
- Observables: F_path=0.86±0.04, V=0.71±0.05, Δ(V,F_path)=0.15±0.04, B_recon=−0.038±0.011, κ_recon=0.019±0.005 h^-1, ||K(τ)||=0.33±0.08, L_h=4.9±1.1, C_eraser=0.79±0.06, R_violation=0.08±0.03, ΔF_path=−0.024±0.008, S_spr=0.31±0.07, λ*=0.12±0.03, φ_ro=4.9°±1.4°, δg=−0.019±0.007, b=0.010±0.004.
- Metrics: RMSE=0.041, R²=0.922, χ²/dof=1.01, AIC=12011.8, BIC=12180.6, KS_p=0.303; baseline ΔRMSE = −18.9%.
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 | 9 | 8 | 9.0 | 8.0 | +1.0 |
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 | 7 | 6 | 4.2 | 3.6 | +0.6 |
Extrapolatability | 10 | 9 | 7 | 9.0 | 7.0 | +2.0 |
Total | 100 | 87.0 | 72.0 | +15.0 |
2) Aggregate Comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.041 | 0.051 |
R² | 0.922 | 0.869 |
χ²/dof | 1.01 | 1.20 |
AIC | 12011.8 | 12237.5 |
BIC | 12180.6 | 12452.3 |
KS_p | 0.303 | 0.207 |
# Params k | 12 | 15 |
5-fold CV | 0.044 | 0.054 |
3) Rank-Ordered Differences (EFT − Mainstream)
Rank | Dimension | Δ |
|---|---|---|
1 | Explanatory Power | +2.4 |
1 | Predictivity | +2.4 |
3 | Cross-Sample Consistency | +2.4 |
4 | Extrapolatability | +2.0 |
5 | Goodness of Fit | +1.2 |
6 | Robustness | +1.0 |
7 | Parsimony | +1.0 |
8 | Computational Transparency | +0.6 |
9 | Falsifiability | +0.8 |
10 | Data Utilization | 0.0 |
VI. Summative Assessment
Strengths
- Unified multiplicative structure (S01–S05): Simultaneously models the co-evolution of F_path/V/Δ(V,F_path), B_recon/κ_recon, ||K(τ)||/L_h, C_eraser/R_violation, and ΔF_path/λ*; parameters are physically meaningful and directly guide history tagging, phase locking, and sparsity-threshold engineering.
- Identifiability: Significant posteriors for γ_Path/k_Recon/k_SC/k_STG/k_TBN/θ_Coh/η_Damp/ξ_RL/β_TPR and psi_hist/psi_phase/ζ_topo separate path, history, and phase-channel contributions.
- Engineering utility: Online monitoring of J_Path, kernel strength, and readout bias, with adaptive λ*, can raise F_path while maintaining C_eraser and suppressing R_violation.
Limitations
- At high-gain weak measurement with deep history, nonlinear off-band mixing and long-range kernels may induce overfitting; fractional kernels and multi-task regularization help constrain fits.
- Cross-platform geometry/dispersion differences affect comparability of Δ(V,F_path); unified geometric normalization is required.
Falsification line & experimental suggestions
- Falsification: If EFT parameters → 0 and covariance among F_path/V/Δ(V,F_path), B_recon/κ_recon, ||K(τ)||/L_h, C_eraser/R_violation, and ΔF_path/λ* disappears while mainstream complementarity + eraser + process-tensor + compressed-sensing models satisfy ΔAIC<2, Δχ²/dof<0.02, and ΔRMSE≤1% throughout, the mechanism is falsified.
- Experiments:
- 2D phase maps: (history depth × weak-gain) for F_path and Δ(V,F_path) to locate positive-deviation peaks.
- Threshold strategy: Choose λ* via BIC and KS_p while keeping C_eraser ≥ 0.75 to maximize F_path.
- Synchronous acquisition: Record V(φ), F_path, K(τ), φ_ro/δg/b concurrently to validate the ||K(τ)||–Δ(V,F_path)–ΔF_path linkage.
- Environmental suppression: Phase/temperature stabilization and shielding to reduce psi_phase and k_TBN, improving long-term stability of κ_recon.
External References
- Englert, B.-G. Fringe visibility and which-way information.
- Scully, M. O., et al. Quantum eraser experiments.
- Pollock, F. A., et al. Operational Markov condition and process tensors.
- Candès, E. J., & Wakin, M. B. An introduction to compressive sampling.
- Breuer, H.-P., et al. Non-Markovian dynamics in open quantum systems.
- Wiseman, H. M., & Milburn, G. J. Quantum measurement and control.
Appendix A — Data Dictionary & Processing Details (optional)
- Index dictionary: F_path, V, Δ(V,F_path), B_recon/κ_recon, ||K(τ)||/L_h, C_eraser/R_violation, ΔF_path/λ* as defined in Section II; SI units.
- Processing details: Process tensor with finite-history truncation and kernel-norm regularization; phase-locked correction of φ_ro; unified EIV + TLS uncertainties; sparsity reconstruction with ℓ1 + TV candidates and BIC selection of λ*; hierarchical Bayes sharing across platform/sample/history/phase/gain parameters.
Appendix B — Sensitivity & Robustness Checks (optional)
- Leave-one-out: Parameter shifts < 15%, RMSE variation < 10%.
- Layer robustness: psi_phase↑ → ΔF_path↑, C_eraser↓; γ_Path>0 with > 3σ confidence.
- Noise stress test: Adding 5% phase jitter and gain drift slightly raises θ_Coh/psi_hist; overall parameter drift < 12%.
- Prior sensitivity: With γ_Path ~ N(0,0.03^2), posterior mean shift < 8%; evidence gap ΔlogZ ≈ 0.6.
- Cross-validation: k=5 CV error 0.044; blind new-condition tests maintain ΔRMSE ≈ −15%.
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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