32-EFT.WP.Cosmo.LayeredSea v1.0 | Chapter 6 — Propagation & Arrival Time: Near-Layer Formulations

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Chapter 6 — Propagation & Arrival Time: Near-Layer Formulations

• I. One-Sentence Objective
  One-sentence goal: Under the piecewise framework induced by layer interfaces Sigma_sea, unify the two computational formulations of arrival time T_arr, define consistency criteria between zero-thickness corrections and thick-layer volumetrics, provide an executable band-differencing workflow to isolate the path term, and specify multi-path/“echo” approximations together with numerical convergence, auditing, and logging conventions.
• II. Scope & Non-Goals
• Coverage: control equations and segmentation rules for the two formulations; zero-thickness correction Delta_T_sigma; thick-layer consistency tau_switch; band differencing Delta_T_arr; multi-path weighting and echo delays; discrete realizations and convergence criteria; formulation selection and logging fields; metrology steps M60-*; interface anchors.
• Non-Goals: do not repeat the construction of n_eff (Chapter 3) or the geometry/segmentation derivations (Chapter 4); do not replace interface matching (Chapter 8) or solver implementation details (Chapter 9).
• III. Minimal Terms & Symbols
• Geometry & Layers: SeaProfile, layer profile W_k(chi), layer strength Xi_k(chi), interface set Sigma_sea, intersection sequence { ell_i }.
• Path & Measure: gamma(ell) (arclength parameterized), ell ∈ [0, L], line element d ell, piecewise segments gamma_i.
• Propagation & Speed: n_eff(x,t,f) (dimensionless, with n_eff ≥ 1), c_ref, local speed c_loc = c_ref / n_eff.
• Events & Energy: R_sea, T_trans, A_sigma, with R_sea + T_trans + A_sigma = 1.
• Corrections & Consistency: zero-thickness Delta_T_sigma, thin/thick consistency tau_switch; two-formulation consistency eta_T.
• Name isolation: T_fil ≠ T_trans; n ≠ n_eff (strictly no conflation).
• IV. Segmentation Rules & Two-Formulation Control Equations (inheriting S60-5)
• Segmentation rules
• Partition the path gamma(ell) at Sigma_sea via { ell_i }, yielding gamma_i = gamma|_[ell_i, ell_{i+1}].
• Include segment endpoints explicitly in integrals; interpolation across an interface is forbidden. Record endpoints and tolerances in interface_marks.
• Control equations
• Constant factored out: T_arr = ( 1 / c_ref ) * ( ∑_i ∫_{gamma_i} n_eff d ell )
• General form: T_arr = ∑_i ∫_{gamma_i} ( n_eff / c_ref ) d ell
• Lower bound
• From n_eff ≥ 1, obtain T_arr ≥ L_path / c_ref (equivalently embedded in the general integrand).
• V. Zero-Thickness Correction vs. Thick-Layer Volumetrics: Consistency
• Zero-thickness correction (thin-layer condition)
• When a layer meets Delta_k / L_char ≤ eta_w, use an event-type correction: Delta_T_sigma ≈ k_sigma · H(crossing) where H indicates a crossing event.
• Thick-layer volumetrics (non-thin layer)
• Explicit in-layer integral: T_arr^{layer_k} = ∫_{layer_k} ( n_eff / c_ref ) d ell.
• Quantifying consistency
• Define tau_switch = | T_arr^{thick} − ( T_arr^{thin} + Delta_T_sigma ) | and require tau_switch ≤ a preset limit (recorded in the Contract).
• Enforce a dual-run near eta_w; if the limit is not met, fix the computation to thick-layer volumetrics and backtrack endpoint tolerances and SeaProfile parameters.
• VI. Band Differencing & Path-Term Isolation (inheriting S60-4)
• Differencing on the same path
• Constant factored out: Delta_T_arr(f1,f2) = ( 1 / c_ref ) * ∫ ( n_path(f1) − n_path(f2) ) d ell
• General form: Delta_T_arr(f1,f2) = ∫ ( ( n_path(f1) − n_path(f2) ) / c_ref ) d ell
• Execution notes
• Reuse the same { gamma[k], Δell[k] }, the same segmentation, and the same Delta_T_sigma configuration. Aggregate out-of-band residuals into u_sys and log the leakage ratio.
• The linearity region / chosen polynomial order follow the band model of Chapter 3 and the metrology workflow of Chapter 7.
• VII. Multi-Path Weighting & “Echo” Approximation
• Multi-path weighting
• For an observable path family { gamma_m } with weights { w_m },
  T_arr_total = ∑_m w_m · T_arr[ gamma_m ], with ∑_m w_m = 1 (or an amplitude-normalization strategy).
• Weight sources: geometry, layer-event R_sea, T_trans, A_sigma, and attenuation/threshold policies (frozen in the Contract).
• Echo-delay approximation
• If a near-layer closed loop of length L_loop exists, the k-th echo delay is approximated by
  Delta_T_echo(k) ≈ k · ∫_{loop} ( n_eff / c_ref ) d ell (general form).
• Echo identification and checks are detailed in Chapter 11 (validation) and Chapter 7 (metrology).
• VIII. Discrete Realization & Convergence Criteria
• Discrete formulas
• Constant factored out: T_arr ≈ ( 1 / c_ref ) * ∑_{k=0}^{N-1} n_eff( gamma[k] ) · Δell[k]
• General form: T_arr ≈ ∑_{k=0}^{N-1} ( n_eff( gamma[k] ) / c_ref( gamma[k] ) ) · Δell[k]
• Step-size strategy (three triggers)
• Geometric curvature: ‖ d^2 gamma / d ell^2 ‖ ≥ tau_geom.
• Medium variation: | d n_eff / d ell | ≥ tau_medium.
• Layer strength: Xi_k(chi) ≥ tau_sea (steep interfaces).
• Symmetric refinement near { ell_i }; include endpoints in local-error accounting.
• Convergence & consistency
• Refinement test: | T_arr^{(fine)} − T_arr^{(coarse)} | ≤ eps_T.
• Two-formulation consistency: eta_T = | T_arr^{const} − T_arr^{gen} | ≤ threshold.
• Thin/thick consistency: tau_switch ≤ limit; failing this, backtrack segmentation and SeaProfile.
• IX. Formulation Selection & Logging Standards
• Selection criterion
• If max_ell | δc_ref / c_ref | ≤ eta_c → use the constant-factored form; otherwise → use the general form, and record the c_ref(x,t,f) model and its uncertainty in the Contract.
• Minimal logging set
• hash(SeaProfile), hash(gamma), mode, eps_T, eta_T, eta_w, tau_switch, interface_marks:{ ell_i }, statistics of Delta_T_sigma triggers, summary of R_sea/T_trans/A_sigma, and DimReport.
• X. Metrology & Calibration Flows (M60-7 … M60-12)
• M60-7 Reference-speed review: update c_ref from gamma_ref, T_arr_ref; produce u_stat, u_sys.
• M60-8 Segmentation & interface logging: detect { ell_i }, freeze endpoint tolerance and method (bisection/secant/Newton); record interface_marks.
• M60-9 Two-formulation consistency: compute both forms in parallel and output eta_T; if exceeded, backtrack c_ref calibration and n_eff decomposition.
• M60-10 Band differencing: on the same path, compute multi-frequency Delta_T_arr, estimate n_path parameters, and output the out-of-band leakage ratio.
• M60-11 Multi-path weights: estimate R_sea, T_trans, A_sigma and synthesize { w_m }; audit energy consistency.
• M60-12 Archiving & falsification: compile the report and reproducible package; register falsification samples for lower-bound violations, formulation mismatch, and tau_switch exceeding threshold.
• XI. Interfaces & Implementation Bindings (aligned with the template family; examples I60-*)
• arrival_time_in_sea( n_eff, gamma, mode, c_ref ) -> T_arr (I60-6)
• detect_sea_intersections( gamma, SeaProfile ) -> { ell_i, layer_id } (I60-4)
• interface_correction_sea( gamma, SeaProfile, params ) -> Delta_T_sigma (I60-5)
• delta_arrival_in_sea( n_path_params, f1, f2, gamma, mode, c_ref ) -> Delta_T_arr (I60-7)
• estimate_RT_sea( data, SeaProfile ) -> R_sea, T_trans, A_sigma (I60-8)
  Constraints: enforce dimensional checks, lower-bound checks, and energy-consistency audits at entry; name isolation is mandatory.
• XII. Pass Criteria & Falsification Lines
• Pass: T_arr ≥ L_path / c_ref; eta_T meets target; tau_switch meets target; R_sea + T_trans + A_sigma = 1; segment endpoints included with no cross-interface interpolation; differencing lies within the intended linear region/order.
• Falsification: stable occurrences of n_eff < 1; persistent two-formulation inconsistency with no successful backtracking; tau_switch over threshold; broken energy consistency; missing segmentation or interpolation across interfaces.
• XIII. Cross-References
• EFT.WP.Cosmo.LayeredSea v1.0: Ch. 3 (Minimal Equations & Layered Representation), Ch. 4 (Geometry & Coordinate Choice), Ch. 5 (Layer Structure & Parameterization), Ch. 8 (Interface Matching), Ch. 9 (Modeling Methods & Numerical Realization), Ch. 11 (Validation & Benchmarks).
• EFT.WP.Propagation.TensionPotential v1.0: the two formulations and differencing workflow.
• EFT.WP.Core.Metrology v1.0: metrology steps and traceability.
• XIV. Deliverables
• Near-layer propagation implementation checklist: segmentation, corrections, differencing, multi-path handling, and logging templates.
• Formulation selection & audit template: fields for eta_c, eta_T, tau_switch, lower-bound margins, energy consistency, and falsification samples.
• Convergence & stability script notes: refinement tests, step-size strategy, endpoint error handling, and trigger statistics.