31-EFT.WP.BH.TensionWall v1.0 | Chapter 9 — Modeling Methods & Numerical Implementation

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Chapter 9 — Modeling Methods & Numerical Implementation

• I. One-Sentence Aim
  Provide an executable route from TWProfile to n_eff(x,t,f) and then to arrival time T_arr, covering grid-first and path-first solvers, wall segmentation and zero-thickness corrections, multi-path and echo synthesis, step-size control and convergence criteria, benchmark suites, and reproducibility audits.
• II. Scope & Non-Goals
• Covered: model expression and discretization strategies; solver architecture and pseudo-flows; segmentation and interface corrections; multi-path composition and weights; selection and consistency across thick/thin-wall pipelines; error estimation and convergence; performance and reproducibility; error semantics and interface mapping.
• Not covered: re-stating Chapter 3 minimal equations or Chapter 5 parameterization details; replacing Chapter 8 matching rules; device-level hardware or observation pipelines.
• III. Minimal Terms & Symbols
• Fields & potentials: T_fil(x,t), Phi_T(x,t), grad_Phi_T(x,t).
• Wall & profile: Sigma_TW, r_H, Delta_w, W(r), Xi_TW(r) = | dW/dr |, TWProfile.
• Index & speeds: n_eff(x,t,f), c_ref, c_loc = c_ref / n_eff.
• Path & measure: gamma(ell), d ell, segments gamma_i, crossings { ell_i }.
• Gauges & corrections: T_arr = (1/c_ref) * ∫ n_eff d ell or T_arr = ∫ ( n_eff / c_ref ) d ell, zero-thickness correction Delta_T_sigma.
• Energy triplet: R_TW, T_trans, A_sigma, with R_TW + T_trans + A_sigma = 1.
• Naming isolation: never mix T_fil with T_trans; never mix n with n_eff.
• IV. End-to-End Modeling Roadmap
• Inputs: TWProfile, priors for T_fil or Phi_T, path & coordinate contract coords_spec, units_spec, c_ref calibration, band nodes f_grid.
• Assembly: Phi_T = G(T_fil) with grad_Phi_T; incorporate wall term H_TW( W, Xi_TW, f ) into
  n_eff = F( Phi_T, grad_Phi_T, rho, f ) + H_TW.
• Segmentation: detect { ell_i }, build gamma_i, set n_eff^± per Chapter 8 side-limit and matching rules.
• Computation: integrate with one of the two gauges; add Delta_T_sigma if applicable; weight multi-path contributions to obtain T_arr_total.
• Checks: lower bound T_arr ≥ L_path / c_ref, two-gauge consistency eta_T, energy-consistency residuals, refinement convergence.
• Outputs: T_arr or ΔT_arr, logs and hashes, audit report and falsification samples.
• V. Solver Architecture & Pseudo-Flows
• Grid-first (Field-first)
• Phi_T = build_phi_t( T_fil, params_G ); Phi_T = fix_gauge( Phi_T, x_ref, t_ref ); Phi_T = apply_TW_matching( Phi_T, TWProfile )
• grad_Phi_T = gradient( Phi_T )
• n_eff = estimate_neff_TW( Phi_T, grad_Phi_T, rho, f_grid, TWProfile )
• For each gamma: { ell_i } = detect_TW_intersections( gamma, TWProfile ), T_arr = segment_integrals( n_eff, gamma, { ell_i }, mode ); if thin-wall, T_arr += interface_correction( gamma, Sigma_TW, params )
• Path-first (Trajectory-first)
• path = capture_path( raw_track, coord_spec ) -> { gamma[k], Δell[k], t_hat[k] }
• Evaluate Phi_T( gamma[k] ), grad_Phi_T( gamma[k] ) along the path
• Assemble n_eff( gamma[k], f ) per frequency and compute T_arr with segmentation and corrections
• Hybrid
• Cache slowly varying terms on a grid; open local refinement windows along the path near the wall to balance accuracy and performance.
• VI. Segmentation, Interface Corrections & Multi-Path Composition
• Segmentation rules: gamma(ell_i) ∈ Sigma_TW; each segment lies on a single side; include endpoints explicitly; cross-interface interpolation is forbidden.
• Zero-thickness correction: when Delta_w / r_H ≤ eta_w, use Delta_T_sigma ≈ k_sigma · H(crossing); persist counts and magnitudes.
• Multi-path composition: generate { gamma_m } and { w_m }, T_arr_total = ∑_m w_m · T_arr[ gamma_m ]; determine w_m from geometry and R_TW,T_trans,A_sigma.
• VII. Step-Size Control, Error Estimation & Convergence
• Step-size control
• Geometry trigger: if norm( d^2 gamma / d ell^2 ) ≥ tau_geom, reduce step.
• Medium trigger: if | d n_eff / d ell | ≥ tau_medium, reduce step.
• Wall trigger: if | dW/dr | ≥ tau_wall, force refinement; use symmetric neighborhoods near endpoints.
• Local error: estimate from a pair of quadrature orders on the same segment; treat interface segments separately.
• Global error: root-sum-of-squares of local errors; target | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Consistency: ensure eta_T = | T_arr^{const} − T_arr^{gen} | within threshold and T_arr ≥ L_path / c_ref.
• Thin vs thick consistency: | T_arr^{thick} − (T_arr^{thin} + Delta_T_sigma) | ≤ tau_switch.
• VIII. Selecting Between Thick- & Thin-Wall Pipelines
• Selection rule: use thin-wall when Delta_w / r_H ≤ eta_w; otherwise perform thick-wall volume integration.
• Debounce at the boundary: double-compute near eta_w and report the difference; if exceeding tau_switch, fix to thick-wall and revisit TWProfile or endpoint tolerances.
• Performance hint: in thick-wall mode, subdivide only inside the wall; cap step size proportional to Delta_w.
• IX. Benchmark Tasks & Validation Suite (TW-specific)
• Benchmark 1 Uniform outer region, no wall: n_eff ≡ 1, T_arr = L_path / c_ref.
• Benchmark 2 Single thin wall: W(r) is tanh; compare thick-wall volume integration vs Delta_T_sigma.
• Benchmark 3 Double-interface layer: constant n_eff on both sides plus zero-thickness corrections; verify segmentation consistency.
• Benchmark 4 Directional response: enable b1 · dot( grad_Phi_T , t_hat ) and b1_sigma · dot( grad_Phi_T , n_vec ); validate across angles.
• Benchmark 5 Band dispersion: fit n_path over multiple frequencies; assess differential linear region and out-of-band leakage.
• Pass criteria: lower bound, two-gauge consistency, convergence, and energy consistency all satisfied.
• X. Performance, Concurrency & Caching
• Concurrency: parallelize over frequency bands and paths; evaluate local wall windows in parallel.
• Caching: cache_neighborhood( field, gamma, radius ) to reuse Phi_T, grad_Phi_T neighborhoods.
• Interpolation: in grid-first mode, use stable interpolation orders to prevent endpoint jitter.
• Memoization: reuse invariant { ell_i }, { w_m }, and Delta_T_sigma trigger statistics.
• XI. Reproducibility, Logs & Audit
• Minimal logs: hash(Phi_T), hash(grad_Phi_T), hash(n_eff), hash(gamma), coords_spec, units_spec, SolverCfg, mode, eps_T, eta_T, eta_w, tau_switch, interface_marks, counts/magnitudes of Delta_T_sigma, summary of R_TW,T_trans,A_sigma.
• Audit steps: dimensional checks via check_dimension; lower-bound and energy-consistency checks; two-gauge consistency; refinement convergence; thin/thick consistency.
• Reproducible package: data, code, parameters, RNG seed, hash manifest, and replay commands.
• XII. Error Handling & Error Semantics
• E-DIM-001 Dimensional inconsistency or missing units
• E-INTF-005 Interface matching failure or parameter out of bounds
• E-QAD-006 Segment integration not converged or eps_T unmet
• E-NEFF-003 n_eff < 1 or decomposition failure
• E-CONSIST-008 Two-gauge consistency failure
• E-TW-010 Thin–thick consistency failure or Delta_T_sigma vs volume-integration discrepancy over threshold
  Handling: preserve intermediate logs and RNG seeds; emit a falsification sample with rollback recommendations.
• XIII. Interfaces & Implementation Bindings (I40-34 … I40-46)
• build_solver_config_TW( params ) -> SolverCfg
• solve_phi_grid_TW( T_fil, params_G, TWProfile ) -> Phi_T, grad_Phi_T
• estimate_neff_TW( Phi_T, grad_Phi_T, rho, f_grid, TWProfile ) -> n_eff
• detect_TW_intersections( gamma, TWProfile ) -> { ell_i }
• segment_integrals( n_eff, gamma, { ell_i }, mode ) -> { T_arr_i }, T_arr
• interface_correction( gamma, Sigma_TW, params ) -> Delta_T_sigma
• simulate_multipath_TW( n_eff, gamma, TWProfile, mode ) -> { T_arr_m, w_m }
• consistency_dual_mode_TW( inputs ) -> eta_T
• consistency_thin_vs_thick_TW( inputs ) -> tau_report
• benchmark_suite_TW( runlist ) -> Summary
• log_artifacts_TW( meta, hashes, metrics ) -> Log
• Constraint: at ingress, run dimensional and lower-bound checks to enforce dim(T_arr) = [T], dim(n_eff) = 1; record coords_spec, units_spec, and all thresholds.
• XIV. Cross-References
• EFT.WP.BH.TensionWall v1.0 Chapters 3, 4, 5, 6, 7, 8
• EFT.WP.Propagation.TensionPotential v1.0 Chapters 4–7
• EFT.WP.Core.Equations v1.1 S06-*
• EFT.WP.Core.Metrology v1.0 M05-, M10-
• EFT.WP.Core.Errors v1.0 M20-*
• XV. Deliverables
• Solver implementation checklist and pseudo-flows, including near-wall segmentation and correction norms.
• Refinement & consistency templates: settings/audits for eps_T, eta_T, eta_w, tau_switch.
• Benchmark-suite script essentials and reproducible-bundle fields: data, code, hashes, and replay instructions.