23-EFT.WP.Metrology.PathCorrection v1.0 | Chapter 10 — Path Integration and Dual-Form Comparison (Numerical Implementation)

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Chapter 10 — Path Integration and Dual-Form Comparison (Numerical Implementation)

One-Sentence Goal
Provide a stable numerical scheme to compute T_arr on a discrete path gamma(ell) with gridded/analytic n_eff(f,x), ensuring parallel evaluation of the two formulations, estimable errors, controlled delta_form, and auditable persistence.

I. Scope and Objects

1. Inputs
   • Path: gamma(ell) (from geometry or the Chapter 9 ray solution), arclength domain [0, L_gamma].
   • Medium: n_eff(f,x) or its piecewise/grid representation with an interpolator Interp(x).
   • Constants & references: c_ref, RefCond, publication time bases tau_mono/ts.
   • Events: set of interface/refraction/reflection/material-switch points E = { ell_j }.
2. Outputs
   • T_form1 = ( 1 / c_ref ) * ( ∫_{gamma} n_eff d ell ),
     T_form2 = ( ∫_{gamma} ( n_eff / c_ref ) d ell ),
     delta_form and error estimates u_q, u_interp, u_geom.
   • Components & auxiliaries: ∫ 1 d ell, per-segment integration logs, sample set and weights.
3. Applicability and Boundaries
   Fiber/cable (n_g(f,T)), free-space/atmosphere (n_eff(f,x)), and hybrid paths; higher-order dispersion/absorption terms may be inserted into the integrand per Chapters 5–8.

II. Terms and Variables

• gamma: [0, L_gamma] → R^3, unit(ell) = "m".
• n_eff(f,x), unit = "1"; use n_g if it is a group-index model.
• w_i, quadrature weights; x_i = gamma(ell_i) sampling points.
• Q[p], a p-point quadrature operator (e.g., Simpson, Gauss–Legendre).
• eps_abs/eps_rel, absolute/relative error tolerances; Δell_max step-size cap.
• Error terms: u_q (quadrature truncation/round-off), u_interp (interpolation field error), u_geom (path-geometry error), u_c (combined).

III. Axioms P810-*

• P810-1 (Parallel Dual Forms) — Any T_arr computation must output T_form1 and T_form2 simultaneously, using identical samples and weights to minimize numerical discrepancy.
• P810-2 (Explicit Measure) — Express all integrals in the arclength measure d ell; for non-arclength parameterizations, multiply explicitly by the arclength factor | d gamma / d ξ | d ξ.
• P810-3 (Event Segmentation) — Discontinuities in the medium or its derivatives must be treated as event points; within each segment the integrand is continuous and differentiable.
• P810-4 (Adaptivity & Conservation) — Use adaptive subdivision with embedded error estimates; evaluate L_gamma = ( ∫_gamma 1 d ell ) in parallel to verify path-length conservation.
• P810-5 (Units & Time Base) — All quantities entering the quadrature must pass check_dim; publish all results on tau_mono.
• P810-6 (Traceability) — Sampling, weights, error estimates, event points, and parameters must be persisted as a reproducible manifest.

IV. Minimal Equations S810-*

• S810-1 (Dual-Form Definitions)
  T_form1 = ( 1 / c_ref ) * ( ∫_{0}^{L_gamma} n_eff( f, gamma(ell) ) d ell ),
  T_form2 = ( ∫_{0}^{L_gamma} ( n_eff( f, gamma(ell) ) / c_ref ) d ell ).
  delta_form = | T_form1 - T_form2 |, check_dim( T_form* ) = "[T]".
• S810-2 (Change of Parameterization)
  If the path uses parameter ξ with ell(ξ), then
  ( ∫_{gamma} g(ell) d ell ) = ( ∫_{ξ0}^{ξ1} g( ell(ξ) ) * | d gamma / d ξ | d ξ ).
• S810-3 (Piecewise Quadrature)
  With event points E = { 0 = ell_0 < ell_1 < … < ell_M = L_gamma },
  T = ∑_{m=0}^{M-1} ( ∫_{ell_m}^{ell_{m+1}} F(ell) d ell ), where F(ell) = n_eff( f, gamma(ell) ) / c_ref or n_eff( f, gamma(ell) ).
• S810-4 (Error Estimation and Combination)
  Given segmentwise embedded estimates e_m,
  u_q = sqrt( ∑ e_m^2 );
  u_interp from bounds of the interpolation error of the n_eff field or multi-resolution differencing;
  u_geom from numerical/discretization errors of gamma propagated via Chapter 9 sensitivities;
  u_c = sqrt( u_q^2 + u_interp^2 + u_geom^2 ).
• S810-5 (Numerical Stability)
  For long links or large sample counts, use compensated summation (Kahan or Neumaier):
  S = compensate_sum( w_i * f_i ), with error O(ε_machine).
• S810-6 (Sampling Criterion)
  Step size respects both medium variation and geometric curvature:
  Δell ≤ min( Δell_n , Δell_k , Δell_max ), with
  Δell_n ≈ η_n * ( | n_eff | / | d n_eff / d ell | ),
  Δell_k ≈ η_k * R_curv (curvature radius), η_n, η_k ∈ (0,1).
• S810-7 (Grid Interpolation)
  For 3-D grids n_eff[x]: Interp(x) may be trilinear/cubic-spline; an upper bound follows grid spacing h and derivative bounds:
  | n - n_I | ≤ C * h^p * max | ∂^p n | (p=1,3).
• S810-8 (Path-Length Check)
  L_gamma_est = ∑ || gamma(ell_{i+1}) - gamma(ell_i) ||; its difference from analytic L_gamma is the geometric consistency metric res_L.

V. Metrological Workflow M80-10

1. Ready — Ingest gamma, n_eff/Interp, RefCond; assemble event set E (medium/interface/turning points).
2. Segment Modeling — For each segment, form the local integrand F(ell) and an error policy { eps_abs, eps_rel, Δell_max, η_n, η_k }.
3. Adaptive Quadrature
   • Prefer Gauss–Kronrod (G7–K15) or adaptive Simpson.
   • Use the embedded pair to estimate e_m; subdivide segments until tolerances are met.
4. Parallel Dual Forms — Accumulate T_form1 and T_form2 on the same refined grid with shared weights and compensated summation.
5. Interpolation & Sampling — At each sample x_i, call Interp(x_i); near boundaries, use one-sided limits and align with E.
6. Error Combination — Aggregate u_q, u_interp, u_geom into u_c; produce delta_form and res_L.
7. Checks
   • check_dim( T_form* ) = "[T]".
   • delta_form ≤ tol_Tarr.
   • res_L ≤ tol_L and L_gamma > 0.
8. Persist — Output
   manifest.path.integral = { T_form1, T_form2, delta_form, parts:{ L_gamma }, qc:{ u_q, u_interp, u_geom, u_c, res_L }, policy:{ quad, eps_abs, eps_rel, Δell_max, η_n, η_k }, RefCond, samples:{ N_total, E } }.
9. Monitor — Track rolling delta_form_p99, failure rate, and subdivision-depth distributions; trigger policy cards and fallbacks on anomalies.

VI. Contracts and Assertions (C80-101x)

• C80-1011 Dual-Form Difference — delta_form ≤ tol_Tarr (default suggestion ≤ 0.02 ns).
• C80-1012 Error Tolerances — Ensure u_q ≤ eps_abs + eps_rel * |T_arr|; otherwise refine or raise quadrature order.
• C80-1013 Event Alignment — Discontinuities must be event points; sampling across discontinuities is forbidden.
• C80-1014 Sampling Density — Δell must not exceed min(Δell_n, Δell_k, Δell_max); otherwise tag undersampled.
• C80-1015 Geometric Conservation — res_L ≤ tol_L (suggest ≤ 1e-6 * L_gamma + 1e-4 m).
• C80-1016 Interpolation Domain — If x_i leaves the grid domain, extrapolation is forbidden; fall back or extend the field and tag out_of_domain.
• C80-1017 Dimensional Consistency — check_dim( n_eff ) = "1", check_dim( c_ref ) = "[L][T]^-1".
• C80-1018 Traceability — Policies, sampling statistics, and event lists must be persisted; otherwise tag trace_missing and refuse publication.

VII. Implementation Bindings I80-*

• I80-101 integrate_path(n_eff, gamma, c_ref) -> { T_form1, T_form2, delta_form, qc, parts }
  Invariants: non_decreasing(ell), L_gamma > 0, delta_form ≤ tol_Tarr.
• I80-102 build_quadrature(policy) -> Q (policy ∈ { GK15, Simpson, GL-8, GL-16 }).
• I80-103 sample_along(gamma, Δell_rule, events) -> { x_i, w_i }.
• I80-104 interpolate_n(Interp, x_i) -> n_i, u_interp_i.
• I80-105 combine_errors(e_m, u_interp, u_geom) -> u_c.
• I80-106 assert_integral_contracts(payload, rules) -> report.
• I80-107 emit_path_manifest_integral(payload, policy) -> manifest.path.integral.

VIII. Cross-References

• Ray/path geometry sources: Chapter 9.
• Medium-field construction and dispersion: Chapters 4–6 and 8.
• Dual-form criteria and cleaning strategies: EFT.WP.Methods.Cleaning v1.0.
• Time base and publication semantics: EFT.WP.Metrology.TimeBase v1.0, EFT.WP.Metrology.Sync v1.0.
• Instrument/processing components: EFT.WP.Metrology.Instrument v1.0.

IX. Quality and Risk Control

1. SLO — p95( delta_form ) ≤ 0.02 ns, p99 ≤ 0.05 ns; integrator_fail_rate ≤ 1e-5.
2. Stability — Guard maximum subdivision depth and sample counts; use compensated summation with 64-bit accumulation, optionally 80/128-bit when needed.
3. Fallback Strategy
   • Raise order (e.g., GL-8 → GL-16 or densify adaptive Simpson).
   • Strengthen event subdivision.
   • Straight-line/stratified approximations with inflated u_c.
   • Refuse publication and tag the cause (undersampled / out_of_domain / trace_missing).

Summary
This chapter specifies the adaptive numerical implementation of path integrals, the parallel dual-form evaluation, and the error-composition framework. Core output:
manifest.path.integral = { T_form1, T_form2, delta_form, parts:{ L_gamma }, qc:{ u_q, u_interp, u_geom, u_c, res_L }, policy:{ quad, eps_abs, eps_rel, Δell_max, η_n, η_k }, RefCond, samples:{ N_total, E } }.
Together with Chapters 4–9 and the Cleaning/TimeBase/Instrument volumes, it enables stable computation and auditable publication of arrival times across media.