21-EFT.WP.Metrology.Sync v1.0 | Appendix E — Error & Uncertainty Propagation (Synchronization Edition)

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Appendix E — Error & Uncertainty Propagation (Synchronization Edition)

I. Purpose & Scope

• Provide a unified, computable framework for uncertainty across the synchronization pipeline—from measurement → modeling → servo → publication—covering sources, the combined standard uncertainty u_c, and coverage U = k * u_c, aligned with manifests, contracts, and panels.
• All statistics are computed on tau_mono and published on ts; any arrival-time metric must record both forms and delta_form.

II. Error Sources & Notation

1. Timestamp chain
   • Hardware timestamp quantization: q (LSB), u_q = q / sqrt(12).
   • Soft-interrupt jitter: u_sw; DMA/PHY latency drift: u_hw.
   • Routing/medium asymmetry: asym, calibration residual u(asym).
2. Protocol & packets
   Round-trip measurement noise: u_rtt; PDV-induced: u_pdv.
3. Servo & holdover
   Measurement-equivalent input noise: u_meas; servo output after filtering: u_servo; holdover (thermal/timekeeping) drift: u_ho.
4. References
   Reference stability mapping: u_ref.offset(tau), u_ref.skew(tau)—recommended via TDEV/ADEV.
5. Dual-form arrivals
   • T_arr.form1 = ( 1 / c_ref ) * ( ∫ n_eff d ell ), T_arr.form2 = ( ∫ ( n_eff / c_ref ) d ell ), delta_form = | T_arr.form1 - T_arr.form2 |.
   • Treat model-form mismatch as a bounded systematic: u_form (see S60E-7).

III. Axioms P60E- **

• P60E-1 (Dimensional integrity): Run check_dim(expr) before publication; declare unit(x) and dim(x) for every metric.
• P60E-2 (Statistical gauge): Use u(offset; tau) := TDEV(tau) for time error and u(skew; tau) := ADEV(tau) for fractional frequency unless a richer noise map is declared.
• P60E-3 (Explicit correlation): Correlated noise from shared clocks / common routes must appear in the covariance matrix Sigma.
• P60E-4 (Dual-form mandatory): Any T_arr publication must include both forms and delta_form, and include them in the budget.
• P60E-5 (Coverage factor): Report coverage as U = k * u_c, default k = 2 (~95%).

IV. Minimal Equations S60E- (Definitions & Propagation)*

• S60E-1 (Two-way offset estimator)
  offset_hat = 0.5 * ( ( t2 - t1 ) - ( t4 - t3 ) ) + asym
  Vector form y = f(x), x = [t1,t2,t3,t4,asym]^T, gradient g = ∂f/∂x = [ -0.5, 0.5, 0.5, -0.5, 1 ]^T.
  Propagation: u(offset_hat)^2 = g^T Sigma g, where Sigma is the covariance of x.
• S60E-2 (Quantization & timestamp uncertainty)
  u_ts^2 = u_q^2 + u_sw^2 + u_hw^2, with u_q = q / sqrt(12).
• S60E-3 (Servo filtering, frequency domain)
  Given input PSD S_in(f) and servo transfer H(f),
  u_servo^2 = ( ∫ | H(f ) |^2 * S_in(f) df )
  Discrete approximation: u_servo^2 ≈ ( ∑_k | H(e^{j ω_k}) |^2 * S_in(ω_k) Δω ).
• S60E-4 (Holdover drift)
  For holdover window tau_ho, approximate via reference stability:
  u_ho(offset; tau_ho) := TDEV_ref(tau_ho);
  u_ho(skew; tau_ho) := ADEV_ref(tau_ho).
• S60E-5 (Multi-reference fusion)
  With normalized weights w_i, offsets o_i, covariance Sigma_o:
  offset_fused = ( ∑ w_i o_i ) / ( ∑ w_i )
  u(offset_fused)^2 = ( w^T Sigma_o w ) / ( ( ∑ w_i )^2 ).
• S60E-6 (Link asymmetry calibration & residual)
  asym = asym_cal + epsilon_cal, with epsilon_cal ~ U[-a, a];
  u(asym)^2 = u(asym_cal)^2 + a^2 / 3.
• S60E-7 (Dual-form arrival delta → uncertainty)
  If model bias bias_arr ∈ [ -delta_form, +delta_form ] (rectangular),
  u_form = delta_form / sqrt(3);
  Combine: u(T_arr)^2 = u_form^2 + u_meas^2 + u_env^2 (u_env is residual of environmental corrections).
• S60E-8 (Combined & coverage)
  u_c^2 = u(offset_hat)^2 + u_servo^2 + u_ho^2 + u_ref^2 + u_form^2 ;
  U = k * sqrt( u_c^2 ), default k = 2.
• S60E-9 (Sequential fusion, information form)
  Independent updates: u_{k+1}^{-2} = u_k^{-2} + u_meas^{-2};
  With forgetting lambda ∈ (0,1]: u_{k+1}^2 = (1 - lambda)^2 * u_k^2 + lambda^2 * u_meas^2.

V. Typical Propagation Scenarios & Flows

1. M60-E1 (PTP two-way → servo → publish)
   • Acquire t1..t4 and asym; estimate Sigma.
   • Compute offset_hat via S60E-1 and u(offset_hat).
   • Obtain u_servo through H(f) (S60E-3).
   • Combine to u_c and U; persist under manifest.sync.unc.*.
2. M60-E2 (GNSS reference holdover/fallback)
   • Enter holdover, select tau_ho;
   • Evaluate u_ho from reference TDEV/ADEV (S60E-4);
   • Combine into u_c, adapt k or SLO thresholds dynamically.
3. M60-E3 (Multi-reference fusion & consistency)
   • Form o_i and Sigma_o;
   • Compute offset_fused and u(offset_fused) (S60E-5);
   • Publish consistency residuals and q_score; trigger mismatch alerts as needed.
4. M60-E4 (Arrival harmonization)
   • Compute T_arr.form1/form2 in parallel to get delta_form;
   • Map u_form (S60E-7) and fold into u_c;
   • Assert thresholds under *C60-arr- ** and execute runbooks.

VI. Reporting & Decomposition (Publication Discipline)

1. Fields
   • manifest.sync.unc.offset.u_c, manifest.sync.unc.offset.U_k2
   • manifest.sync.unc.skew.u_c
   • manifest.sync.unc.decomp = {u_ts, u_asym, u_servo, u_ho, u_ref, u_form}
   • manifest.sync.arrival = {T_arr.form1, T_arr.form2, delta_form, u_form}
2. Decomposition rules
   • Publish ranked variance shares u_i^2 / u_c^2 (“top contributors”).
   • Merge same-origin correlated terms before second-level breakdown to avoid misleading attribution.

VII. Contracts & Thresholds C60E- (Examples)*

• C60E-01: U(offset; k=2) ≤ U_max (tiered by site/profile).
• C60E-02: u_servo ≤ beta * u_meas (filter must not amplify noise, beta ≤ 1).
• C60E-03: u(asym) ≤ asym_budget; when calibration is stale, degrade to soft.
• C60E-04: delta_form ≤ tol_Tarr ∧ u_form ≤ tol_Tarr / sqrt(3).
• C60E-05: in multi-reference fusion, cond(Sigma_o) ≤ kappa_max else reject or reweight.
• C60E-06: in holdover, U(offset; k=2, tau_ho) must satisfy SLO_holdover.

VIII. Implementation Bindings I60- (Synchronization Uncertainty APIs)*

• unc_ptp_offset(t1,t2,t3,t4, asym, Sigma) -> {offset_hat, u_offset} (S60E-1)
• unc_servo_psd(S_in, H) -> u_servo (S60E-3)
• unc_holdover(tau_ho, ref_stab) -> {u_offset, u_skew} (S60E-4)
• fuse_offsets(o_list, Sigma_o, weights) -> {offset_fused, u_fused} (S60E-5)
• arrival_uncertainty(T_arr_form1, T_arr_form2) -> {delta_form, u_form} (S60E-7)
• emit_unc_manifest(bundle, policy) -> manifest.sync.unc
• Invariants: check_dim(all); window coherence; Sigma positive semidefinite; delta_form ≤ tol_Tarr; k and confidence discipline declared.

IX. Cross-References

• Allan family & metric definitions: Chapter 9 and Appendix D of this volume.
• Dual-form arrivals & path harmonization: EFT.WP.Metrology.TimeBase v1.0 Chapter 9.
• Statistical propagation & coverage: Methods.CrossStats v1.0 Appendix E.
• Cleaning & contract publication: Methods.Cleaning v1.0 Chapter 10 & allied appendices.

Summary