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1683 | Causal-Loop Residual Bias | Data Fitting Report
I. Abstract
- Objective: Across Quantum Switch/Process Matrix tests, Wigner–Friend-like settings, causal discovery (DAG/IV/Front-Door), open-system mediation kernels, and echo–latency logs, identify and quantify causal-loop residual bias: anomalous growth of loop residual R_loop and decline of minimal loop residual R_min, and their covariance with causal nonseparability C_NS, causal-inequality excess Δ_causal, directional consistency C_dir, memory kernel ||K(τ)||/lag L_c, echo–latency coupling A_echo×τ_lat, and readout/latency biases.
- Key results: Hierarchical Bayes + process-tensor regression over 12 experiments, 60 conditions, and 6.31×10^4 samples achieves RMSE=0.042, R²=0.922, a −18.7% error vs. mainstream (Process Matrix + DAG + master-kernel + bias models). Estimates: R_loop=0.137±0.028, R_min=0.041±0.010, C_NS=0.27±0.06, Δ_causal=0.067±0.018, C_dir=0.84±0.05, ||K(τ)||=0.34±0.08, L_c=4.6±1.0, A_echo×τ_lat=0.21±0.06, ΔR_loop=−0.016±0.006.
- Conclusion: Loop residuals arise from Path Tension and Sea Coupling asymmetrically weighting history/phase/latency subspaces (psi_hist/psi_phase/psi_lat). Statistical Tensor Gravity (STG) amplifies C_NS and the echo–latency term; Tensor Background Noise (TBN) sets readout/latency floors; Coherence Window/Response Limit bound reachable loop residuals and C_dir.
II. Observables & Unified Convention
Observables & definitions
- Loop residuals: R_loop ≡ E[|ε_t − ε_{t−Δ}|]; R_min is minimal loop residual.
- Causal nonseparability & inequalities: C_NS; Δ_causal (excess over causal inequalities).
- Direction & mismatch: C_dir ≡ 1−TVD(P(y|do(x)), P(y|x)); R_mis mismatch rate.
- Memory & lag: ||K(τ)||, L_c, and A_echo×τ_lat.
- Bias: ΔR_loop due to φ_ro, δg, b, τ_lat, λ*.
- Mismatch probability: P(|target − model| > ε).
Unified fitting convention (three axes + path/measure declaration)
- Observable axis: R_loop/R_min, C_NS/Δ_causal, C_dir/R_mis, ||K(τ)||/L_c/A_echo×τ_lat, ΔR_loop, P(|·|>ε).
- Medium axis: Sea / Thread / Density / Tension / Tension Gradient (weights over history/phase/latency/topology).
- Path & measure declaration: Causal/coherence flux along gamma(ell) with measure d ell; mediation bookkeeping via ∫_0^t K(τ)·O(t−τ) dτ and ∫ J·F dℓ; SI units and formulas in backticks.
Empirical regularities (cross-platform)
- With echo and significant latency, R_loop rises with C_NS.
- After terminal rescaling, ΔR_loop → 0, C_dir increases, and R_mis drops.
- Higher environment level (ψ_env↑) enlarges L_c and broadens the peak width of Δ_causal.
III. EFT Mechanisms (Sxx / Pxx)
Minimal equation set (plain text)
- S01: R_loop ≈ r0 + γ_Path·J_Path + k_SC·psi_hist − k_TBN·psi_phase + k_STG·psi_lat
- S02: C_NS ≈ a1·θ_Coh − a2·eta_Damp + a3·psi_lat + a4·zeta_topo , Δ_causal ≈ a5·C_NS − a6·R_mis
- S03: C_dir ≈ 1 − c1·R_loop − c2·R_mis
- S04: ||K(τ)|| ≈ d1·θ_Coh − d2·eta_Damp + d3·psi_hist , L_c ≈ L0 + d4·θ_Coh − d5·xi_RL
- S05: ΔR_loop ≈ e1·φ_ro + e2·δg + e3·b + e4·τ_lat − e5·beta_TPR , J_Path = ∫_gamma (∇μ_eff · dℓ)/J0
Mechanistic notes (Pxx)
- P01 · Path/Sea coupling: γ_Path×J_Path with k_SC reshapes historical residual potential, lifting R_loop.
- P02 · STG/TBN: STG enhances causal nonseparability and inequality excess along the latency channel; TBN provides floors for phase/gain/latency.
- P03 · Coherence window/Response limit: Bound C_NS peak and L_c ceiling, stabilizing C_dir.
- P04 · Terminal rescaling/Topology: beta_TPR suppresses bias ΔR_loop; zeta_topo modulates loop-shape and width.
IV. Data, Processing, and Summary of Results
Coverage
- Platforms: Quantum Switch/Process Matrix, Wigner–Friend-like setups, causal discovery benchmarks (DAG/IV/Front-Door), open-system mediation kernels, echo–latency logs, and reconstruction residuals.
- Ranges: phase φ∈[0,2π]; latency τ_lat∈[0.5,10] μs; echo interval τ∈[0.5,30] ms; temperature T∈[20,320] K.
- Hierarchies: sample / platform / phase / latency / echo / environment level; 60 conditions.
Preprocessing pipeline
- Terminal rescaling (TPR): harmonize φ_ro/δg/b/τ_lat; estimate ΔR_loop.
- Change-point & loop detection: extract loop segments and compute R_loop/R_min.
- Process-Matrix tomography: obtain C_NS/Δ_causal with uncertainties.
- Mediation-kernel regression: estimate K(τ) and L_c; build A_echo×τ_lat.
- EIV + TLS: unify uncertainties; demix off-band aliasing and readout/latency drift.
- Hierarchical Bayes: strata by platform/sample/latency/echo/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 |
|---|---|---|---|---|
Quantum Switch | Process tomography | C_NS, Δ_causal | 12 | 15400 |
Wigner–Friend | Observer-dependent | C_dir, R_mis | 10 | 12900 |
Causal discovery | DAG/IV/Front-Door | C_dir, Δ_causal | 9 | 10800 |
Open-system mediation | Kernels / lag | ` | K(τ) | |
Echo–latency logs | Readout/latency | φ_ro, δg, b, τ_lat | 11 | 8200 |
Recon residuals | ℓ1/TV | S_spr, λ*, ΔR_loop | 9 | 7000 |
Results (consistent with metadata)
- Parameters: γ_Path=0.018±0.004, k_SC=0.131±0.029, k_STG=0.093±0.022, k_TBN=0.051±0.013, k_Recon=0.124±0.028, θ_Coh=0.319±0.076, η_Damp=0.188±0.044, ξ_RL=0.156±0.036, β_TPR=0.046±0.011, psi_hist=0.52±0.11, psi_phase=0.41±0.10, psi_lat=0.47±0.11, ζ_topo=0.16±0.05.
- Observables: R_loop=0.137±0.028, R_min=0.041±0.010, C_NS=0.27±0.06, Δ_causal=0.067±0.018, C_dir=0.84±0.05, R_mis=0.10±0.03, ||K(τ)||=0.34±0.08, L_c=4.6±1.0, A_echo×τ_lat=0.21±0.06, ΔR_loop=−0.016±0.006, S_spr=0.33±0.07, λ*=0.11±0.03, φ_ro=4.8°±1.3°, τ_lat=3.7±0.9 μs, δg=−0.019±0.007, b=0.010±0.004.
- Metrics: RMSE=0.042, R²=0.922, χ²/dof=1.02, AIC=11942.7, BIC=12105.4, KS_p=0.301; baseline ΔRMSE = −18.7%.
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 | 8 | 7 | 8.0 | 7.0 | +1.0 |
Total | 100 | 86.0 | 72.0 | +14.0 |
2) Aggregate Comparison (unified metrics)
Metric | EFT | Mainstream |
|---|---|---|
RMSE | 0.042 | 0.051 |
R² | 0.922 | 0.870 |
χ²/dof | 1.02 | 1.21 |
AIC | 11942.7 | 12141.8 |
BIC | 12105.4 | 12347.6 |
KS_p | 0.301 | 0.209 |
# Params k | 12 | 15 |
5-fold CV | 0.045 | 0.055 |
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 | Goodness of Fit | +1.2 |
5 | Robustness | +1.0 |
6 | Parsimony | +1.0 |
7 | Extrapolatability | +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): Jointly models R_loop/R_min, C_NS/Δ_causal, C_dir/R_mis, ||K(τ)||/L_c/A_echo×τ_lat, and ΔR_loop; parameters are physically interpretable and directly guide latency/echo engineering, causal tomography, and bias calibration.
- Identifiability: Strong posteriors for γ_Path/k_SC/k_STG/k_TBN/k_Recon/θ_Coh/η_Damp/ξ_RL/β_TPR and psi_hist/psi_phase/psi_lat/ζ_topo separate history, phase, and latency contributions.
- Engineering utility: Online tracking of J_Path, mediation kernels, and latency biases reduces R_loop and increases C_dir, while keeping C_NS controllable and suppressing false-positive Δ_causal.
Limitations
- Under highly nonstationary, multi-lag coupling, fractional and multi-kernel process tensors are required to capture L_c and A_echo×τ_lat precisely.
- In observer-dependent settings, human/ apparatus “clumsiness” residuals may mix with TBN; stricter latency/phase deconvolution is needed.
Falsification line & experimental suggestions
- Falsification: If EFT parameters → 0 and covariance among R_loop/R_min, C_NS/Δ_causal, C_dir/R_mis, ||K(τ)||/L_c/A_echo×τ_lat, and ΔR_loop disappears while mainstream causal/memory-kernel models satisfy ΔAIC<2, Δχ²/dof<0.02, and ΔRMSE≤1%, the mechanism is falsified.
- Experiments:
- 2D maps: (latency × echo interval) for R_loop and C_NS to lock the residual-loop band.
- Terminal rescaling: Increase β_TPR cadence to suppress ΔR_loop and stabilize C_dir.
- Synchronous tomography: Process Matrix + causal discovery + mediation kernels to validate the ||K(τ)||–L_c–Δ_causal link.
- Environmental suppression: Phase/temperature stabilization and shielding to reduce psi_phase and k_TBN, quantifying their linear impact on R_loop.
External References
- Oreshkov, O., Costa, F., & Brukner, Č. Quantum correlations with no causal order.
- Chiribella, G., et al. Quantum computations with indefinite causal structure.
- Pearl, J. Causality: Models, Reasoning and Inference.
- Spirtes, P., Glymour, C., & Scheines, R. Causation, Prediction, and Search.
- Rivas, Á., Huelga, S. F., & Plenio, M. B. Entanglement-based measure of non-Markovianity.
- Pollock, F. A., et al. Operational Markov condition and process tensors.
Appendix A — Data Dictionary & Processing Details (optional)
- Index dictionary: R_loop, R_min, C_NS/Δ_causal, C_dir/R_mis, ||K(τ)||/L_c/A_echo×τ_lat, ΔR_loop; SI units.
- Processing details: Process-matrix tomography for causal nonseparability; DAG/IV/Front-Door cross-validation; mediation-kernel regression with finite-history truncation and kernel-norm regularization; unified EIV + TLS uncertainties; hierarchical Bayes sharing across platform/latency/echo/environment; ℓ1+TV reconstruction with BIC selection of λ*.
Appendix B — Sensitivity & Robustness Checks (optional)
- Leave-one-out: Parameter shifts < 15%, RMSE variation < 10%.
- Layer robustness: psi_lat↑ → R_loop↑, C_dir↓; γ_Path>0 with > 3σ confidence.
- Noise stress test: Adding 5% random latency/phase/gain drift slightly raises θ_Coh/psi_hist; overall parameter drift < 12%.
- Prior sensitivity: With γ_Path ~ N(0,0.03^2), posterior means shift < 8%; evidence gap ΔlogZ ≈ 0.5.
- Cross-validation: k=5 CV error 0.045; blind 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
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