16-EFT.WP.Methods.Cleaning v1.0 | Chapter 6 Path and Arrival-Time Cleaning

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Chapter 6 Path and Arrival-Time Cleaning

One-Sentence Goal
Define paths explicitly via gamma(ell) with the line-element measure d ell, compute the two concurrent expressions of T_arr and their difference delta_form, and complete monotonicity, dimensional, and uncertainty audits to produce a publishable arrival.* manifest.

I. Scope & Objects

1. Applicable targets
   • All path-related fields in D_time from Chapter 5: path.ell, path.param = gamma(ell), path.measure = d ell, path.L_gamma.
   • Arrival-time medium fields: arrival.c_ref, arrival.n_eff, together with their samples or modeled values along the path.
2. Target artifacts
   Produce D_arrival and manifest.arrival_forms = { gamma(ell), d ell, L_gamma, c_ref, n_eff, T_arr_form1, T_arr_form2, delta_form, tol_Tarr, method }, plus local-difference diagnostics and an uncertainty breakdown.

II. Terms & Variables (Memory Anchors)

• Path parametrization & measure: gamma(ell), d ell, ell ∈ [0, L_gamma], L_gamma = ( ∫_gamma 1 d ell ).
• Medium & constants: n_eff(ell), c_ref(ell) or constant c_ref.
• Two arrival-time forms:
  T_arr_form1 = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  T_arr_form2 = ( ∫ ( n_eff / c_ref ) d ell )
• Difference & threshold: delta_form = | T_arr_form1 - T_arr_form2 |, tol_Tarr.
• Uncertainty: u(x), U = k * u_c, discretization error u_disc.
• Constraints: non_decreasing(ell), check_dim( T_arr_form1 - T_arr_form2 ) = true.

III. Axioms (P106-*)

• P106-01 Explicit path & measure
  All path computations must use gamma(ell) with d ell as the measure, ell as the arc-length parameter, and domain [0, L_gamma].
• P106-02 Monotone path
  non_decreasing(ell) = true, with each step Delta_ell_k = ell_{k+1} - ell_k > 0; zero or negative steps indicate data errors or the need for reconstruction.
• P106-03 Two-form concurrency
  Any arrival-time estimate must compute both T_arr_form1 and T_arr_form2 and persist delta_form.
• P106-04 Explicit medium
  The origin, units, and dimensions of n_eff(ell) and c_ref(ell) must be declared in the manifest; if c_ref is treated as constant, say so explicitly.
• P106-05 Dimension conservation
  dim(T_arr_form1) = dim(T_arr_form2) = [T], and check_dim( T_arr_form1 - T_arr_form2 ) must pass.
• P106-06 Discretization method
  Integrals use a declared quadrature (e.g., trapezoidal/Simpson); method and order are written to the manifest, and u_disc must be estimated and published.
• P106-07 Deferred environmental corrections
  First compute and audit both forms under the raw RefCond; apply corr_env(•; RefCond) in Chapter 12 and write back.
• P106-08 Topological consistency
  Self-intersections or loops must record loop_count and be segmented to avoid arc-length parametrization distortion.
• P106-09 Contracts before publication
  If any of delta_form ≤ tol_Tarr, monotonicity, dimensional, or unit contracts fail, the dataset must not be published.

IV. Minimal Equations (S106-*)

• S106-01 Path length
  L_gamma = ( ∫_gamma 1 d ell )
• S106-02 Two arrival-time forms
  T_arr_form1 = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  T_arr_form2 = ( ∫ ( n_eff / c_ref ) d ell )
• S106-03 Two-form difference
  delta_form = | T_arr_form1 - T_arr_form2 |
• S106-04 Discrete quadrature (trapezoidal rule)
  Let Delta_ell_k = ell_{k+1} - ell_k:
  T_arr_form1_hat = ( 1 / c_ref_hat ) * sum_k ( 0.5 * ( n_eff_k + n_eff_{k+1} ) * Delta_ell_k )
  T_arr_form2_hat = sum_k ( 0.5 * ( n_eff_k / c_ref_k + n_eff_{k+1} / c_ref_{k+1} ) * Delta_ell_k )
• S106-05 Discretization-error upper bound (trapezoid)
  Let f(ell) = n_eff(ell) or f(ell) = n_eff(ell)/c_ref(ell), and if sup_ell | f''(ell) | ≤ M2, then
  u_disc ≤ ( M2 / 12 ) * sum_k ( Delta_ell_k )^3
• S106-06 Composite threshold convention
  tol_Tarr = k * sqrt( u^2_form1 + u^2_form2 + u^2_disc + u^2_env )
  where u_env is provided by the Chapter 12 environmental correction.
• S106-07 Arc-length from coordinates
  If only coordinates gamma_k ∈ R^d are available,
  ell_0 = 0, ell_{k+1} = ell_k + || gamma_{k+1} - gamma_k ||
• S106-08 Local-difference localization
  delta_local_k = | ( 1 / c_ref_bar_k ) * ( 0.5 * ( n_eff_k + n_eff_{k+1} ) * Delta_ell_k ) - ( 0.5 * ( n_eff_k / c_ref_k + n_eff_{k+1} / c_ref_{k+1} ) * Delta_ell_k ) |

V. Cleaning Process (M10-6 Path & Arrival Time)

1. Path reconstruction & normalization
   If ell is missing, reconstruct from gamma_k via S106-07; remove duplicate points and zero steps; enforce non_decreasing(ell).
2. Measure & domain confirmation
   Make d ell and domain [0, L_gamma] explicit; compute and persist L_gamma; validate check_dim( L_gamma - ( ∫_gamma 1 d ell ) ).
3. Medium materialization
   Map n_eff(ell) and c_ref(ell) onto segments; record sources, interpolation mode, and coverage_med.
4. Two-form discrete integration
   Using the declared quadrature, compute T_arr_form1_hat and T_arr_form2_hat; also produce segment-level contributions and delta_local_k.
5. Uncertainty & threshold computation
   Estimate u_form1, u_form2, and u_disc; compute tol_Tarr via S106-06, recording k and modeling assumptions.
6. Contracts & diagnostics
   Assert check_dim( T_arr_form1_hat - T_arr_form2_hat ), delta_form ≤ tol_Tarr, and non_decreasing(ell); on failure, report the set of offending segments k and their share.
7. Manifest persistence & pre-signature checks
   • Write manifest.arrival_forms: { integrator, order, step_stats, M2_est, coverage_med, T_arr_form1, T_arr_form2, delta_form, tol_Tarr }.
   • Update qc.flags[] (e.g., PATH_LOOP, DELTA_EXCEED, MEDIUM_GAP).
8. Artifacts
   Output D_arrival, then proceed to Chapter 10’s contract gate and Chapter 12’s environmental correction.

VI. Contracts & Assertions (Chapter Must-Pass Items)

• Path monotonicity: non_decreasing(ell) = true, min(Delta_ell_k) > 0.
• Path length: L_gamma = ell_last - ell_first (coordinate-reconstructed paths must agree within tolerance).
• Dimension conservation: check_dim( T_arr_form1 - T_arr_form2 ) = true.
• Two-form bound: delta_form ≤ tol_Tarr.
• Medium coverage: coverage_med ≥ coverage_min, n_eff > 0, c_ref > 0.
• Discretization control: max(Delta_ell_k) ≤ step_max or u_disc ≤ tol_disc.
• Manifest completeness: exists(manifest.arrival_forms) with all fields present.

VII. Implementation Binding (I10-6)

1. Interface prototypes
   • build_path(ds, mode) -> path
   • materialize_medium(path, sources) -> { n_eff(ell), c_ref(ell), coverage_med }
   • integrate_path(path, f, method) -> ∫ f d ell
   • arrival_two_forms(path) -> { T_arr_form1, T_arr_form2, delta_form, delta_local[] }
   • estimate_disc_error(path, f, method) -> u_disc
   • enforce_arrival_time_convention(ds) -> ds'
2. Invariants & postconditions
   • path.ell is monotone and L_gamma consistent; both forms and delta_form are computed; manifest.arrival_forms is reproducible.
   • Any interpolation/extrapolation carries provenance and uncertainty annotations; silent inference is forbidden.
3. Failure semantics
   E_PATH_NON_MONOTONIC, E_MEDIUM_MISSING, E_INT_METHOD_UNDECLARED, E_DELTA_EXCEED, E_DISC_EXCESS; each must localize the affected segment set and proportion.

VIII. Quality Metrics & Risk Control

1. Indicators
   • Shape quality: loop_count, step_ratio = max(Delta_ell_k) / median(Delta_ell_k).
   • Form coherence: r_form = delta_form / max( T_arr_form1, T_arr_form2, eps ), p95(delta_local_k).
   • Coverage: coverage_med, proportion of medium-gap segments.
   • Discretization: u_disc / T_arr_form2 or absolute value.
2. Alerts & actions
   • If r_form > r_form_max → down-weight or roll back.
   • If coverage_med < coverage_min → resample the medium or quarantine.
   • If step_ratio > step_ratio_max → refine resampling or segment and re-integrate.

IX. Boundaries & Common Situations

• Constant c_ref
  Expect delta_form ≈ 0; exceedances usually stem from n_eff interpolation or overly large steps.
• Piecewise medium (c_ref(ell) piecewise constant)
  Align segment boundaries at breakpoints; avoid cross-segment averaging that induces bias.
• Path loops or noisy kinks
  Record loop_count and segment; if needed, merge micro-segments by a length threshold.
• Missing coordinates or positioning leaps
  Use robust interpolation and flag qc.flags += PATH_JUMP; do not reorder or fabricate data.

X. Audit & Panel Fields

• Minimal panel
  T_arr_form1, T_arr_form2, delta_form, tol_Tarr, r_form, coverage_med, u_disc, step_ratio, loop_count, p95(delta_local)
• Alert suggestions
  r_form > r_form_max, coverage_med < coverage_min, u_disc > tol_disc, loop_count > 0

XI. Cross-References

• Units & dimensions (check_dim, unit conventions): Chapter 4.
• Time axis & synchronization (segment windows and publication time base): Chapter 5.
• Environmental correction (corr_env and u_env): Chapter 12.
• Density & measures (n_eff conventions and normalization): Chapter 13 and EFT.WP.Core.Density v1.0.
• Acquisition & arrival-time semantics: EFT.WP.Core.Sea v1.0.
• Contracts & release freeze: Chapter 10.

Summary
This chapter fixes the path-computation convention via gamma(ell) and d ell, provides two equivalent expressions and discrete implementations for T_arr, and enforces a consistency gate through delta_form and tol_Tarr. With systematic audits of monotonicity, dimensions, and uncertainty, together with segment-level localization and remedial playbooks, the arrival.* fields can be standardized and persisted, establishing a reproducible, auditable foundation for Chapter 10’s contracts and Chapter 12’s environmental corrections.