30-EFT.WP.Propagation.TensionPotential v1.0 | Chapter 13 — Application Scenarios & Case Studies

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Chapter 13 — Application Scenarios & Case Studies

• I. One-Sentence Aim
  Provide deployable, end-to-end scenarios centered on the unified gauges for Phi_T(x,t), n_eff(x,t,f), and arrival time T_arr. Cover arrival-time imaging, path comparison & inversion, multi-layer propagation estimation, drift monitoring, and anisotropy diagnosis. For each, define workflows, interface bindings, and acceptance criteria.
• II. Scope & Non-Goals
• Covered: four core scenarios plus one end-to-end integrated case, including I/O definitions, steps & interfaces, gauge selection, logging & audit, acceptance metrics, and falsification lines.
• Non-goals: no theory re-derivation; no replacement of Chapter 7 metrology flows or Chapter 9 solver details; no device-level mechanical/electrical designs.
• III. Minimal Terms & Symbols
• Potential & gradient: Phi_T(x,t), grad_Phi_T(x,t).
• Effective index & speeds: n_eff(x,t,f), c_ref, c_loc(x,t,f) = c_ref / n_eff(x,t,f).
• Path & measure: gamma(ell), d ell, segmented paths gamma_i, interface set Sigma.
• Arrival time & differentials: T_arr, ΔT_arr(f1,f2).
• Decomposition: n_eff = n_common(x,t) + n_path(x,t,f).
• Gauges: mode ∈ {constant, general}.
• IV. Scenario A | Arrival-Time Imaging (T_arr Imaging)
  Goal. Reconstruct the spatial distribution of n_common(x,t)—and hence produce T_arr isochrone maps—using multi-path arrival-time measurements under known (or approximately known) geometry.
• Inputs
• Sensor layout and multi-path set { gamma_a }; band nodes f_grid.
• Metrology Contract (coordinates, units, gauge, thresholds, gauge fix).
• Optional priors or Chapter 4 outputs for Phi_T and grad_Phi_T.
• Outputs
• Reconstructed map of n_common(x,t) over the region; per-pixel u_c and pass/fail flags; T_arr isochrone map.
• Workflow (interface bindings)
• Path capture & interfaces: capture_path → { gamma[k], Δell[k] }; detect_interfaces → { ell_i }, Sigma.
• Arrival-time acquisition: obtain T_arr_obs(f_grid, gamma_a) with uncertainties.
• Low-band background: decompose_n_eff or fit n_common directly on low bands.
• Imaging inversion (linearized):
• Model: T_arr_obs ≈ (1/c_ref) · ∑_pixels n_common[p] · L_p, where L_p is the pixel chord length.
• Solve with least squares or sparse priors; produce n_common̂ and pixel covariance.
• Consistency checks: check_dual_arrival_consistency → eta_T; enforce T_arr_obs ≥ L_path / c_ref.
• Artifacts & logs: emit_report to seal the contract, thresholds, hashes, and metrics.
• Gauge selection
• Use the constant-factored gauge when max |δc_ref / c_ref| ≤ eta_c; otherwise use the general gauge and record c_ref(x,t,f) estimates and uncertainties.
• Acceptance & falsification
• Accept: imaging residuals |T_arr_obs − T_arr_mod| ≤ GB, and eta_T ≤ threshold.
• Falsify: any stable pixel implying n_eff < 1, or simultaneous conflict with the lower bound.
• V. Scenario B | Path Comparison & Inversion (Inverting Phi_T / n_eff)
  Goal. Use same-path multi-band ΔT_arr(f1,f2) to isolate the path term, then jointly invert over multiple paths for theta = {a0,a1,a2,b1,c_m(·)}, yielding a parameterized n_eff. If needed, recover a gauge-consistent representative of Phi_T.
• Inputs
• Same-path multi-band measurements T_arr_obs(f_m, gamma) and multi-path set { gamma_a }.
• Phi_T, grad_Phi_T from Chapter 4 (or measurable approximations).
• Outputs
• theta_hat with covariance; parameter sets for n_common(x,t) and n_path(x,t,f); consistency and lower-bound report.
• Workflow
• Band differencing: delta_arrival to compute ΔT_arr(f1,f2) and cancel n_common.
• Parameter fitting: fit_n_eff_params solves
  min_theta ∑ ((T_arr_obs − T_arr_mod(theta))/u_c)^2 + R(theta) → theta_hat.
• Gauge check: if the model depends only on grad_Phi_T, verify invariance under Phi_T → Phi_T + const.
• Cross-validation: evaluate generalization on independent paths; run check_dual_arrival_consistency.
• Acceptance & falsification
• Accept: differential correlation/slope within thresholds; T_arr residuals within GB.
• Falsify: any band combination that no n_common + n_path decomposition can fit within the error budget.
• VI. Scenario C | Multi-Layer Propagation Estimation (Layered Sea)
  Goal. In a stratified Sea with interfaces Sigma, compute cross-layer T_arr under matching rules, evaluate interface corrections, and check energy consistency.
• Inputs
• Layer metadata and Sigma; path gamma(ell) with crossings { ell_i }.
• Jump parameters C_sigma, J_sigma; reflection R_sigma and transmission T_trans (distinct from T_fil).
• Outputs
• Segment times T_arr_i and composite T_arr; interface correction ΔT_sigma; energy-consistency and one-sided feasibility checks.
• Workflow
• Matching: apply_matching(Phi_T, Sigma, params) to produce one-sided-consistent Phi_T.
• Index construction: estimate_n_eff, clamping n_eff ∈ [1, n_max] consistently on both sides.
• Segmentation & correction: segment_integrals + interface_correction; compose full-path T_arr.
• Audit: rt_estimator to verify R_sigma + T_trans + A_sigma = 1; enforce n_eff ≥ 1.
• Acceptance & falsification
• Accept: difference between segmented and corrected totals below threshold; energy consistency holds.
• Falsify: any one-sided limit implying n_eff < 1, or energy-conservation violations.
• VII. Scenario D | Streaming Monitoring & Drift Tracking
  Goal. Monitor long-run drift of c_ref, n_common, and interface parameters, maintaining gauge consistency and acceptance criteria.
• I/O
• Streaming T_arr_obs in a sliding window; periodic benchmark path gamma_ref; incremental reports and alerts.
• Workflow
• Periodic calibration: rolling calibrate_c_ref to estimate c_ref(t) and produce drift curves.
• Incremental inversion: within the window, run decompose_n_eff and fit_n_eff_params.
• Guarding: compute GB = k_guard · u_c and eta_T; exceedances trigger rollback and alerts.
• Archival: log_artifacts to persist hashes and metric time series.
• Acceptance & falsification
• Accept: drift stays within band; eta_T remains below threshold.
• Falsify: sustained threshold violations not fixed by re-calibration.
• VIII. Scenario E | Anisotropic-Channel Diagnosis
  Goal. Detect and quantify the directional term b1 · dot( grad_Phi_T , t_hat ), guiding model upgrades from isotropic to anisotropic.
• I/O
• Path sectors { gamma_a } spanning incidence angles; significance report and updated NeffParams.
• Workflow
• Geometric design: cover azimuths; emit t_hat[k] along each path.
• Regression & model selection: compare BIC/AIC with and without the directional term; use leave-one-out for robustness.
• Re-validation: test directional ΔT_arr response on independent paths; update b1.
• Acceptance & falsification
• Accept: statistical significance met; generalization error controlled; two-gauge consistency holds.
• Falsify: unstable directionality metrics or negligible T_arr improvement relative to complexity penalty.
• IX. End-to-End Case | Arrival-Time Imaging & Inversion in a Layered Medium
  Goal. Integrate Scenarios A, B, and C to deliver background imaging, band-differential inversion, and interface corrections in a single pipeline for a stratified Sea.
• Steps
• Preparation & metrology: capture_path, detect_interfaces, calibrate_c_ref; finalize Contract and gauge.
• Imaging (A): reconstruct n_common on low bands; generate T_arr isochrones.
• Differential inversion (B): estimate n_path and coefficients c_m(·) via multi-band ΔT_arr; include b1 if warranted.
• Cross-layer composition (C): apply_matching and segment_integrals; add interface corrections.
• Consistency & uncertainty: check_dual_arrival_consistency; propagate_uncertainty_GUM/MC to report mean ± k·u_c.
• Acceptance & audit: check lower bound, energy consistency, two-gauge agreement, and differential linear region; emit_report for archival.
• Pass criteria
• T_arr_obs − L_path/c_ref ≥ −k·u_c; eta_T ≤ threshold; R_sigma + T_trans + A_sigma = 1; n_eff ≥ 1; differential linear-region criteria met.
• X. Minimal Logging Set (Common to All Scenarios)
• Physics & geometry: hash(Phi_T), hash(grad_Phi_T), hash(n_eff), hash(gamma), Sigma and interface labels.
• Gauges & thresholds: mode, eps_T, eta_T, GB, u_c, n_eff clamping trigger rate.
• Metrology & calibration: details of c_ref calibration; f_grid; path generation & weighting rules.
• Errors & audit: GUM/MC configs, sample size and seed, out-of-band leakage assessment, falsification samples, and replay handle.
• XI. Cross-References
• EFT.WP.Propagation.TensionPotential v1.0 Chapters 4 (tension-potential construction), 5 (n_eff & propagation limits), 6 (paths & two gauges), 7 (metrology & calibration), 8 (boundaries & interfaces), 9 (modeling implementation), 11 (validation & benchmarks), 12 (error budget).
• EFT.WP.Core.Equations v1.1 S06-*; EFT.WP.Core.Metrology v1.0 M05-, M10-; EFT.WP.Core.Errors v1.0 M20-*.
• XII. Deliverables
• Scenario-specific workflow checklists (A/B/C/D/E) and reproducible parameter templates.
• Acceptance & falsification handbook, including differential linear-region and two-gauge consistency thresholds.
• Imaging & inversion script essentials, plus interface-correction and energy-consistency audit fields.