20-EFT.WP.Metrology.TimeBase v1.0 | Chapter 9 — Arrival-Time Metrology & Path Harmonization

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Chapter 9 — Arrival-Time Metrology & Path Harmonization

One-line objective: Centered on the explicit path gamma(ell) and the dual-form gauge T_arr, unify the measure, units, and uncertainty propagation for arrival time; compute on tau_mono, publish on ts, and enforce delta_form ≤ tol_Tarr alongside path monotonicity and consistency.

I. Scope & Targets

1. Covered
   • Arrival/flight-time metrology: T_arr (ToA/ToF/TDoA/GDoP related).
   • Path harmonization: parameterization gamma(ell), L_gamma, monotonicity and sampling consistency.
   • Time-base mapping: tau_mono → ts, with attached offset/skew/J and traceable warp(t).
   • Environmental corrections and uncertainty: RefCond, corr_env(x; RefCond), u(x), U = k * u_c.
2. Inputs
   Path spline or segment set { ell_j, n_eff(ell_j) } or { ell_j, v(ell_j) }; reference constant c_ref; synchronization traces hat{offset/skew/J}.
3. Outputs
   Dual-form T_arr, delta_form, L_gamma, contract assertions, and manifest.arrival.*; peer-to-peer TDoA/TDoF as needed.

II. Terms & Symbols

• Path & measure: gamma(ell), ell ∈ [0, L_gamma], d ell, L_gamma = ( ∫_gamma 1 d ell ).
• Medium & speed: n_eff(ell), c_ref, v(ell) = c_ref / n_eff(ell), with unit(n_eff)=1.
• Arrival time: T_arr, dual-form difference delta_form, threshold tol_Tarr.
• Time bases: tau_mono, ts, mapping warp(t), and companions offset/skew/J.
• Environment & uncertainty: RefCond, corr_env(x; RefCond), u(x), U = k * u_c.
• Related quantities: TDoA = T_arr(B) - T_arr(A), residual = T_meas - T_pred.

III. Axioms P509- **

• P509-1 (Explicit measure): Every integral must name the path and measure—gamma(ell), d ell; “effective length” may not substitute an explicit path.
• P509-2 (Dual forms in parallel): Compute both
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ), and record delta_form.
• P509-3 (Dimensional integrity): unit(T_arr)="s", dim(T_arr)="[T]"; run check_dim(expr) prior to publication.
• P509-4 (Path monotonicity): non_decreasing(ell) is a mandatory contract; broken paths must be segmented with explicit transition conditions.
• P509-5 (Unified time base): Perform all comparisons/windowing on tau_mono, publish on ts, and attach offset/skew/J.
• P509-6 (Environmental coherence): When the medium varies with RefCond, publish corr_env and the reference conditions.
• P509-7 (Traceability): Persist all T_arr computations, kernels/steps, warp(t), and TraceID for full replay.

IV. Minimal Equations S509- **

• S509-1 (Dual-form arrival time)
  T_arr = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell )；
  T_arr = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell )。
• S509-2 (Difference gauge)
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。
• S509-3 (Discrete approximations)
  T_arr ≈ ( 1 / c_ref ) * ( ∑_j n_eff(ell_j) * Δell_j )；
  T_arr ≈ ( ∑_j ( n_eff(ell_j) / c_ref ) * Δell_j )。
• S509-4 (Path length & monotonicity)
  L_gamma = ( ∫_gamma 1 d ell ) with constraints Δell_j ≥ 0 and ∑ Δell_j = L_gamma。
• S509-5 (TDoA/GDoP)
  TDoA(A,B) = T_arr(B) - T_arr(A)；u^2(TDoA) = u^2(T_arr(A)) ⊕ u^2(T_arr(B)) (independence approximation)。
• S509-6 (Uncertainty propagation)
  u^2(T_arr) ≈ ( ∑_j ( ∂T_arr / ∂n_eff(ell_j) * u(n_eff_j) )^2 ) ⊕ ( ∑_j ( ∂T_arr / ∂Δell_j * u(Δell_j) )^2 ) ⊕ u^2(corr_env)。
• S509-7 (Time-base mapping)
  t_pub = warp( t_calc ) = a + ∫_0^{t_calc} ( 1 + s_rel(u) ) d u，with offset/skew/J attached at publication。
• S509-8 (Environmental correction)
  T_arr' = T_arr + corr_env( T_arr ; RefCond )，and publish RefCond with u(corr_env)。
• S509-9 (Fit residuals)
  residual = T_meas - T_pred；use thresholds for alerts and calibration updates。

V. Metrology Flow M50-9 (Ready → Compute → Harmonize → Audit → Publish)

• Readiness
  Validate unit/dim; check non_decreasing(ell) and L_gamma; load RefCond and synchronization traces.
• Dual-form computation
  Choose discrete steps and kernels; compute both T_arr forms; obtain delta_form and u(T_arr).
• Environmental harmonization
  Apply corr_env and combine uncertainties to produce T_arr'.
• Time-base mapping
  Perform comparisons and windowing on tau_mono; map to ts via warp(t).
• Path consistency
  For broken paths, segment integration and record splice points and continuity conditions; mask anomalous segments with m ∈ {0,1}.
• Audit & assertions
  Evaluate contracts (next section); produce a report and remediation plan (backtest, recompute, rollback).
• Persistence & publication
  Emit manifest.arrival.*: both T_arr forms, delta_form, L_gamma, RefCond, warp, uncertainties, and signature.

VI. Contracts & Assertions (Arrival & Path)

• C50-91 (Monotone path): non_decreasing(ell)=true; if violated, segment and document.
• C50-92 (Dual-form threshold): delta_form ≤ tol_Tarr.
• C50-93 (Dimensional integrity): check_dim( T_arr )=pass and unit(T_arr)="s".
• C50-94 (Length consistency): L_gamma > 0 and | L_gamma - L_expected | ≤ tol_L (when a baseline exists).
• C50-95 (Uncertainty bound): U(T_arr') ≤ U_Tarr_max or u(T_arr') / T_arr' ≤ r_Tarr_max.
• C50-96 (Traceable publication): warp, kernels, step size, and TraceID are complete; omissions are violations.
• C50-97 (Residual gate): | residual | ≤ tol_res; on breach, trigger recalibration or rollback.

VII. Implementation Bindings I50-9*

• integrate_arrival_two_forms(path, n_eff, c_ref, kernel) -> {T1, T2, delta_form, L_gamma}
• apply_env_correction(T_arr, RefCond, model) -> {T_arr', u_corr}
• map_to_ts(T_arr_series, warp) -> {T_pub_series, offset/skew/J}
• compute_tdoa(T_arr_A, T_arr_B) -> {TDoA, u}
• audit_arrival_contracts(meta, thresholds) -> report
• emit_arrival_manifest(meta, results, signature) -> manifest.arrival
• Invariants: unique(TraceID); delta_form ≤ tol_Tarr; unit/dimension checks pass; warp monotone; L_gamma matches the segmented path.

VIII. Cross-References

• Synchronization & servos: Chapters 5–6 of this volume (offset/skew/J estimation and robustification).
• Buffering & time-base conversion: Chapter 8 (warp(t) and VRS implementation constraints).
• Cleaning arrival-time conventions: Methods.Cleaning v1.0 Chapters 6 and 12 (dual forms and environmental corrections).
• Imaging time/path gating: Methods.Imaging v1.0 Chapter 13 (gating & T_arr alignment).
• Data specification: EFT.WP.Core.DataSpec v1.0 (path & metrology field lexicon).

IX. Quality Gauges & Risk Control

1. Key metrics
   delta_form, U(T_arr'), psi(T_arr) (stability), residual_p95/p99, latency_ms_p99 (computation chain).
2. Risk policies
   • For threshold breaches: recompute both forms → switch to backup kernel/step-size → roll back to last qualified warp and RefCond → trigger field calibration.
   • Persistent breaches trigger audit: freeze publication and emit a detailed, TraceID-stamped report with replay scripts.

Summary

• This chapter grounds arrival-time metrology in an explicit-path, dual-form, auditable framework—defining axioms P509-*, equations S509-*, flow M50-9, contracts C50-9*, and interfaces I50-9*.
• Unified computation on tau_mono, publication on ts, monotone and traceable warp(t), environmental coherence via RefCond, and persistent manifest.arrival.* together ensure that T_arr is reproducible and cross-system consistent.