21-EFT.WP.Metrology.Sync v1.0 | Chapter 2 — Axioms & Minimal Equations (Synchronization Modeling Baseline)

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Chapter 2 — Axioms & Minimal Equations (Synchronization Modeling Baseline)

One-line objective: Codify non-negotiable axioms and minimal computable equations for network time–frequency synchronization so that offset/skew/J, TIE/MTIE/tdev, and the dual-form arrival gauges are consistent, reproducible, and auditable across implementations.

I. Scope & Objects

1. Unified modeling objects
   • Two-way packet timing (NTP/PTP/White Rabbit) and frequency/phase holdover (SyncE/PLL).
   • Port roles, Boundary/Transparent Clocks, and injection of correctionField and residence_time into the link-delay model.
   • Calibration and online estimation of delay asymmetry asym, including its slow drift.
2. Applicable inputs
   • Timestamps & metadata {t1,t2,t3,t4, correctionField, residence_time, port_role}.
   • Topology & master selection {gm_id, domain, topology}; target loop H(f).
   • Arrival-time parameters n_eff, c_ref, path gamma(ell) (compute both forms in parallel).
3. Outputs & publication
   Estimands with uncertainty {offset, skew, J, TIE, MTIE, tdev, U = k * u_c}; manifest.sync.* and contracts.*.

II. Terms & Variables

• Time bases & phases: tau_mono, ts, offset, skew, phi(t) (phase), f(t) (instantaneous frequency).
• Four timestamps: t1 (master send), t2 (slave recv), t3 (slave send), t4 (master recv).
• Link terms: correctionField, residence_time, asym, delay_rt.
• Stability & spectra: TIE(t), MTIE(tau), tdev(tau), input PSD S_in(f), loop transfer H(f).
• Dual-form arrival: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ); difference delta_form.
• Metrology: unit(x), dim(x), check_dim(expr).

*III. Axioms P602- **

• P602-1 (Timestamp trust tier): Prefer hardware timestamps; for software timestamps, publish u(ts) and the compensation model.
• P602-2 (Explicit asymmetry): Do not assume symmetry by default; asym must be calibrated/estimated with validity and U(asym) published.
• P602-3 (Explicit insertion delays): All correctionField and residence_time terms must enter equations and be persisted.
• P602-4 (Unified time base): Estimate on tau_mono, publish on ts with U and the window for offset/skew/J.
• P602-5 (Dual-form mandatory): Any delay/path computation must record both arrival forms and delta_form.
• P602-6 (Steady-state windows): PSD and stability metrics are evaluated on declared steady-state windows; Delta_t must be persisted.
• P602-7 (Loop budget): The target bandwidth and pole–zero scheme of H(f) must be explicit; online test alpha spending is budgeted.
• P602-8 (Dimensional integrity): Run check_dim(expr) before release; units must be explicit.
• P602-9 (Traceability): TraceID, hash_sha256(blob), and signature are mandatory.

*IV. Minimal Equations S602- **

• S602-1 (Two-way offset / round-trip delay)
  offset = ( (t2 - t1) - (t4 - t3) - asym ) / 2
  delay_rt = ( (t2 - t1) + (t4 - t3) - 2 * asym ) / 2 - correctionField - residence_time
• S602-2 (One-way delay & asymmetry split)
  delay_ms = delay_rt/2 + asym/2，delay_sm = delay_rt/2 - asym/2
• S602-3 (Frequency offset estimation)
  skew = d( offset ) / dt；discrete skew[k] = ( offset[k] - offset[k-1] ) / Delta_t
• S602-4 (First-order PLL discretization)
  e[k] = offset[k]
  freq_corr[k] = freq_corr[k-1] + Ki * e[k] * Delta_t
  phase_corr[k] = Kp * e[k]
  u[k] = phase_corr[k] + freq_corr[k]
• S602-5 (Jitter & steady-state variance)
  J = rms( offset - lowpass(offset, BW) )
  var_out = ( ∫ | H(f) |^2 * S_in(f) df )
• S602-6 (Stability family)
  TIE(t) = phi(t) - phi(0)
  MTIE(tau) = max_over_t ( sup_window | TIE(t+tau) - TIE(t) | )
  tdev(tau) = sqrt( 0.5 * E[ ( (phi(t+2tau) - 2phi(t+tau) + phi(t)) / (2*pi*f0*tau) )^2 ] )
• S602-7 (Asymmetry drift model)
  asym(t) = a0 + a1 * t + w_asym(t) (low-frequency w_asym); publish U(asym).
• S602-8 (Dual-form arrival difference)
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |
• S602-9 (Uncertainty composition)
  U^2(offset) = U^2(ts) + U^2(asym) + U^2(model) + U^2(env)；publish U = k * u_c and k.
• S602-10 (TC/BC aggregation)
  correctionField = ( ∑_h corr_h )，residence_time = ( ∑_h res_h ) accumulated along the path.

V. Metrology Flow M60-2 (Modeling Baseline)

• Ready — standardize inputs; run check_dim; align to tau_mono.
• Two-way estimation — use S602-1/2/3 to obtain offset/skew/delay_rt; record U(ts) and U(asym).
• Loop & steady state — configure H(f); use S602-4/5 to evaluate u[k] and var_out.
• Stability — via S602-6, produce TIE/MTIE/tdev; declare TauSet and windows.
• Arrival — compute both forms and delta_form (S602-8); assert thresholds.
• Persist — generate manifest.sync.* including U, window, H(f), TraceID, and signature.

VI. Contracts & Assertions C60-2*

• C60-21 (Dimensional checks): check_dim(all)=true; unit(offset)="s", unit(skew)="1" or "ppb".
• C60-22 (Dual-form delta): delta_form ≤ tol_Tarr.
• C60-23 (Steady-state variance): var_out ≤ var_budget.
• C60-24 (Stability bounds): MTIE(tau) ≤ bound_mtie(tau) for tau ∈ TauSet.
• C60-25 (Asymmetry validity): age(asym_cal) ≤ T_cal_max or trigger online estimation.
• C60-26 (Uncertainty release): U(offset), k, and window must appear in manifest.sync.*.

VII. Implementation Bindings I60- (Prototype)*

• two_way_estimator(frames) -> {offset, delay_rt, U}
• estimate_asymmetry(frames, scheme) -> {asym, U}
• pll_update(offset_series, cfg) -> loop_state, u_series
• stability_metrics(offset_series, TauSet) -> {TIE, MTIE, tdev}
• toa_two_forms(path_fields) -> {T_arr_form1, T_arr_form2, delta_form}
• propagate_uncertainty(components) -> U_total
• emit_sync_manifest(results, meta) -> manifest.sync
• Invariants: Delta_t > 0; unique(gm_id, domain); sum(loss_window) ≤ tol_loss; delta_form ≤ tol_Tarr.

VIII. Cross-References

• Allan family, time bases & noise: EFT.WP.Metrology.TimeBase v1.0 Chapter 7.
• Cleaning, contracts & manifests: Methods.Cleaning v1.0 Chapter 10 and Appendix C.
• Statistical uncertainty & SLOs: Methods.CrossStats v1.0 Appendix E, Chapter 14.

IX. Quality Gauges & Risk Control

• SLIs: offset_ns.{p50,p95,p99}, skew_ppb_p95, mtie(tau), tdev(tau), loss_rate, gm_switch_count.
• Risks & mitigations: asymmetry drift (periodic calibration/online estimation); routing changes (path rediscovery); virtualization noise (pin/isolated cores); GNSS anomalies (unlock detection & fallback).

Summary