31-EFT.WP.BH.TensionWall 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 Tension Wall Sigma_TW, spanning wall-parameter identification, layered propagation estimation, echo-component decomposition, thin/thick-wall decisions, long-term drift guarding, and risk assessment. For each, supply workflows, interface bindings, logging & audit norms, and acceptance/falsification criteria.
• II. Scope & Non-Goals
• Covered: six core scenarios (A…F) plus one integrated end-to-end case, including I/O definitions, stepwise procedures, interface mappings, records & audits, acceptance criteria & falsification lines, and deliverable lists.
• Not covered: re-deriving the results of Chapters 3–12 or their error details; device-level mechanical/electrical designs.
• III. Minimal Terms & Symbols
• Wall & profile: Sigma_TW, r_H, Delta_w, W(r), Xi_TW(r), TWProfile.
• Fields & propagation: Phi_T(x,t), grad_Phi_T(x,t), n_eff(x,t,f), c_ref, T_arr.
• Paths & measure: gamma(ell), { ell_i }, d ell, segmented paths gamma_i.
• Energy triplet: R_TW, T_trans, A_sigma, with R_TW + T_trans + A_sigma = 1.
• Differentials & echoes: ΔT_arr(f1,f2), ΔT_echo(k).
• Gauges & modes: mode ∈ {constant, general}.
• Naming isolation: never mix T_fil with T_trans; never mix n with n_eff.
• IV. Scenario A | Wall-Parameter Identification (Inverting TWProfile)
  Goal. Using multi-band, multi-angle path arrival-time data, invert the TWProfile parameters { r_H, Delta_w, sigma_w, a_{lm}, … } and the energy-triplet curves.
• Inputs
• Observations: T_arr_obs(f_m, gamma_a) with uncertainties; f_grid.
• Priors: Phi_T, grad_Phi_T (or measurable approximations); coords_spec, units_spec; calibrated c_ref.
• Outputs
• theta_hat (TWProfile parameters) with covariance; RTParams (R_TW(f), T_trans(f), A_sigma(f)); consistency metrics eta_T and τ_switch.
• Workflow (interface bindings)
• capture_path → { gamma[k], Δell[k] }, detect_TW_intersections → { ell_i }.
• estimate_neff_TW to assemble n_eff; segment_integrals and—if thin-wall—interface_correction.
• fit_TW_profile to minimize the objective and emit theta_hat, Cov.
• estimate_RT_TW to calibrate the energy triplet; check_dual_arrival_consistency to emit eta_T.
• consistency_thin_vs_thick_TW to emit τ_switch; emit_measurement_report to archive.
• Acceptance criteria
• |Residual| ≤ GB; eta_T ≤ threshold; τ_switch ≤ limit; R_TW + T_trans + A_sigma = 1; side limits n_eff^± ≥ 1.
• Falsification lines
• Stable n_eff < 1, or eta_T / τ_switch over threshold with no remedy on back-check.
• V. Scenario B | Layered Propagation Estimation (Inner/Wall/Outer)
  Goal. Under explicit partitioning (Region_in / Region_wall / Region_out), compute cross-layer T_arr and audit matching types and energy consistency.
• Inputs/Outputs
• Inputs: TWProfile, path set { gamma_a } with { ell_i }, Phi_T/grad_Phi_T, c_ref, mode.
• Outputs: segment times T_arr_i and combined T_arr_total; residuals for R_TW, T_trans, A_sigma; side-limit report.
• Workflow
• apply_TW_matching to generate side limits of Phi_T; estimate_neff_TW to obtain n_eff^±.
• segment_integrals (and interface_correction if needed).
• rt_estimator_TW to audit energy consistency and output residual curves.
• Acceptance/Falsification
• As in Chapter 8: energy-consistency residuals and side-limit lower bounds pass; else falsify the interface setup or profile specification.
• VI. Scenario C | Echo Detection & Decomposition
  Goal. Detect and decompose multi-path “echo” orders and weights induced by wall reflections; estimate the near-wall loop length L_loop.
• Inputs/Outputs
• Inputs: time-series or spectral Observations; TWProfile; f_grid; path cluster { gamma_m }.
• Outputs: per-order ΔT_echo(k) with uncertainties; path weights { w_m }; synthesized T_arr_total.
• Workflow
• simulate_multipath_TW to generate candidate { T_arr_m, w_m }.
• Perform template matching or sparse decomposition in the observation domain to recover ΔT_echo(k).
• Cross-check via rt_estimator_TW, check_dual_arrival_consistency; archive report.
• Acceptance/Falsification
• Echo orders match ΔT_echo(k) within tolerance; energy consistency holds; if stable offsets persist not attributable to noise/out-of-band effects, falsify TWProfile or the path set.
• VII. Scenario D | Thin/Thick-Wall Decision & Switching
  Goal. Decide runtime gauges based on Delta_w/r_H and τ_switch, ensuring numerical robustness and traceability.
• Workflow
• Pre-evaluation: consistency_thin_vs_thick_TW on representative paths and refinement levels.
• Online: record eta_w; double-compute near the threshold and compare τ_switch.
• Policy: if τ_switch > tau_switch, lock thick-wall pipeline and revisit endpoint tolerances and W(r).
• Archival: log_tw_propagation to record switching rationale and difference curves.
• Acceptance/Falsification
• Post-switch, eta_T, lower bound, and energy consistency still pass; persistent failures falsify the thin-wall approximation or the profile setup.
• VIII. Scenario E | Long-Term Drift Monitoring & Guarding (Streaming)
  Goal. Monitor c_ref(t), n_common(x,t), and wall-parameter drift; auto-alert and maintain gauge consistency.
• Workflow
• calibrate_c_ref to periodically update c_ref(t).
• Within a sliding window, decompose_n_eff / fit_TW_profile to update slow variables.
• Guarding: compute GB = k_guard · u_c, eta_T, τ_switch; on threshold exceedance, trigger rollback and alerts.
• Archive: log_artifacts_TW to persist metric time series and environment blocks.
• Acceptance/Falsification
• Drift remains within band; eta_T stays below threshold. Cross-band drift not explainable by environment falsifies current calibration.
• IX. Scenario F | Risk Assessment & Guardband Setting
  Goal. Before deployment, assess tail risk for key metrics and set GB and runtime thresholds.
• Workflow
• propagate_uncertainty_MC to generate distributions of T_arr / ΔT_arr.
• Evaluate n_eff clamping trigger rate, ΔT_sigma trigger statistics, and out-of-band leakage ratios.
• Set k_guard and GB; persist in the report.
• Acceptance/Falsification
• Target coverage probabilities are met; if tail risks are excessive, reconfigure path/band layouts or raise data-quality gates.
• X. End-to-End Case | Imaging, Inversion & Echo Validation for a Non-Spherical Wall
  Goal. For r_H(theta,phi) non-sphericity, complete wall-parameter imaging, arrival-time inversion, energy consistency, and echo-order joint validation.
• Steps
• Preparation & metrology: declare_tw_contract, calibrate_c_ref, capture_path, detect_TW_intersections.
• Assembly & segmentation: apply_TW_matching, estimate_neff_TW, segment_integrals (with interface_correction if needed).
• Inversion: fit_TW_profile to obtain theta_hat; estimate_RT_TW to audit energy consistency.
• Consistency: check_dual_arrival_consistency, consistency_thin_vs_thick_TW.
• Echo validation: simulate_multipath_TW aligned to observations to extract ΔT_echo(k).
• Reporting & archival: emit_measurement_report; persist hashes and falsification samples.
• Acceptance
• Residuals within GB; two-gauge consistency; τ_switch passes; energy consistency; echo-order agreement.
• XI. Minimal Records & Logs (common to all scenarios)
• Physics & geometry: hash(Phi_T), hash(grad_Phi_T), hash(n_eff), hash(gamma), Sigma_TW labels and { ell_i } tolerances.
• Gauges & thresholds: mode, eps_T, eta_T, eta_c, eta_w, tau_switch, lower-bound margin T_arr − L_path/c_ref.
• Energy & differentials: residuals for R_TW,T_trans,A_sigma; counts/magnitudes of ΔT_sigma; ΔT_arr linear-region diagnostics and out-of-band leakage ratio.
• Uncertainty & reproducibility: u_stat,u_sys,u_c, k, seed, coords_spec, units_spec, SolverCfg, and hash manifest.
• XII. Minimal Data-Contract Checklist
• Contract: spec_version, coords_spec, units_spec, mode, tolerances:{eps_T,eta_T,eta_w,tau_switch}.
• TWProfile: model, parameters, hash(TWProfile); Sigma_TW_meta.
• Path / Observations / RTParams: required fields & units; timestamps in ISO-8601.
• Report/Log: metrics, falsification samples, replay entry, and hashes.
• XIII. Interface & Implementation Bindings (scenario map)
• Scenario A: capture_path → estimate_neff_TW → segment_integrals / interface_correction → fit_TW_profile → estimate_RT_TW → check_dual_arrival_consistency → emit_measurement_report.
• Scenario B: apply_TW_matching → estimate_neff_TW → segment_integrals → rt_estimator_TW.
• Scenario C: simulate_multipath_TW → decompose ΔT_echo(k) → rt_estimator_TW / check_dual_arrival_consistency.
• Scenario D: consistency_thin_vs_thick_TW → policy switch → log_tw_propagation.
• Scenario E: calibrate_c_ref → fit_TW_profile (sliding window) → persist guarding metrics.
• Scenario F: propagate_uncertainty_MC → set GB → archive report.
• XIV. Acceptance Criteria & Falsification Lines (master list)
• Accept: T_arr ≥ L_path / c_ref; eta_T ≤ threshold; R_TW + T_trans + A_sigma = 1; n_eff^± ≥ 1; τ_switch ≤ limit; ΔT_arr linear region satisfied.
• Falsify: any criterion fails after excluding implementation errors; three consecutive independent replications falsifying the same dimension trigger axiom/profile review.
• XV. Deliverables
• Scenario-specific workflow rosters and parameter templates (A…F).
• Gauge, consistency, and energy-audit templates: eta_c, eta_T, τ_switch, and residual dashboards.
• End-to-end case reproducibility bundle: data/code/parameters/SolverCfg/RNG seed/hash manifest and replay scripts.
• XVI. Cross-References
• EFT.WP.BH.TensionWall v1.0 Chapters 3–12 (equations, geometry, parameterization, propagation, metrology, matching, implementation, validation & errors).
• EFT.WP.Propagation.TensionPotential v1.0 — two gauges and data conventions.
• EFT.WP.Core.Metrology v1.0 M05-, M10-; EFT.WP.Core.Errors v1.0 M20-*.