22-EFT.WP.Metrology.Instrument v1.0 | Chapter 10 — Traceability, Calibration & Certificates (SI Chain / Uncertainty)

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Chapter 10 — Traceability, Calibration & Certificates (SI Chain / Uncertainty)

One-Sentence Objective
Establish a unified calibration convention and uncertainty budget traceable to the SI, forming a closed loop of model → estimation → budget → certificate → manifest, published in alignment with tau_mono/ts and manifest.instrument.cal.*.

I. Scope and Objects

1. Scope
   • SI traceability chain: working standards → secondary standards → national/primary standards → SI reference.
   • Calibration models & parameter estimation: y = f(x, theta); typical linear case y = a * x + b.
   • Uncertainty budgeting & publication: u(x), U = k * u_c, Type A / Type B components, and coverage-probability statements.
   • Certificates & manifests: cal_cert_id, TraceID, SI_link, signature, validity and re-calibration policy.
2. Objects
   Instrument reading r, measurand x, environmental correction corr_env(x; RefCond), timebase offset/skew/J, and arrival-time T_arr (when delay/propagation is involved).
3. Outputs
   theta_hat and its covariance, u_c, U, a closed SI traceability chain, manifest.instrument.cal.*, and an electronic certificate.

II. Terms and Variables

• Model & parameters: y = f(x, theta), theta = {a, b, ...}, theta_hat, Cov(theta_hat).
• Uncertainty: standard uncertainty u(z), combined standard uncertainty u_c, expanded uncertainty U = k * u_c, coverage factor k.
• Statistical components: Type A (data-driven) and Type B (priors/specs/certificates).
• Traceability chain: std_i (i-th level standard), SI_link = std_0 → std_1 → ... → SI.
• Certificate fields: cal_cert_id, lab_id, method_id, env_log, date_cal, date_due, signature, hash_sha256(blob).
• Timebase: tau_mono, ts, offset/skew/J.
• Arrival time: T_arr, delta_form, tol_Tarr.

III. Postulates P710-*

• P710-1 (Explicit SI traceability): every published calibration result must provide SI_link with uncertainty at each link.
• P710-2 (Model-first): corrections and uncertainties in the certificate must derive from an explicit measurement model y = f(x, theta).
• P710-3 (Type A/B synthesis): compute u_c from both Type A and Type B; systematic effects may not be assumed zero.
• P710-4 (Coverage declaration): declare the coverage probability, distribution assumption, and degrees of freedom for U = k * u_c (e.g., k ≈ 2 for ~95%).
• P710-5 (Unified timebase): align calibration data and environment logs on tau_mono; publish on ts.
• P710-6 (Two arrival-time forms conserved): when propagation/delay is involved, record both formulations and delta_form in parallel.
• P710-7 (Dependency closure & signatures): certificate and manifest include computational workflow, software versions, data hashes, and signatures to satisfy traceability and reproducibility.

IV. Minimal Equations S710-*

• S710-1 (Linear calibration back-solve)
  For r = a * x + b + e, the estimate is x_hat = ( r - b ) / a, with unit(x_hat) = unit(x).
• S710-2 (Uncertainty propagation — scalar form)
  For x_hat = g(r, a, b, ...),
  u^2( x_hat ) = ( ∂g/∂r )^2 * u^2(r) + ( ∂g/∂a )^2 * u^2(a) + ( ∂g/∂b )^2 * u^2(b) + 2 * ( ∂g/∂r )( ∂g/∂a ) * cov(r,a ) + ...,
  with combined u_c = sqrt( u^2( x_hat ) + u^2( corr_env ) + u^2( drift ) ).
• S710-3 (Jacobian & covariance — vector form)
  For y = f(x, theta), first-order linearization gives
  u_c^2(y) = J * Σ * J^T, where J = ∂f/∂[x, theta] and Σ is the block covariance of inputs and parameters.
• S710-4 (Expanded uncertainty & coverage)
  U = k * u_c; with effective DoF nu_eff, use k = t_{nu_eff, 1 - alpha/2}.
• S710-5 (Traceability-chain uncertainty synthesis)
  If calibration coefficients originate from upstream standards {std_i}, then
  u_chain^2 = ∑ w_i^2 * u^2( std_i ), with weights w_i given by range mapping and sensitivity coefficients.
• S710-6 (Environment & drift correction)
  x_corr = x_hat * ( 1 - TC_x * ( T - T0 ) ) - k_cross * ( T - T0 ) * ( RH - RH0 ),
  u^2( corr_env ) = ( ∂x/∂T )^2 u^2(T) + ( ∂x/∂RH )^2 u^2(RH ) + u^2(model).
• S710-7 (Two arrival-time forms restated)
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ),
  delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |.
• S710-8 (Risk-updated re-calibration interval)
  T_next = T_base * min( 1 + beta1 / |drift_slope| , 1 + beta2 / reject_rate ), constrained to [T_min, T_max].

V. Metrology Procedure M70-10 (Plan → Calibrate → Budget → Certificate → Publish)

• Readiness & planning
  Define SRef and the measurement model, target range, and U_max; set RefCond and the SI_link.
• Calibration execution
  Acquire {r_k} across calibration points {x_k} spanning the range; record environment and timebase alignment offset/skew/J.
• Parameter estimation
  Fit theta_hat and Cov(theta_hat); run residual diagnostics (linearity, heteroscedasticity, outliers).
• Uncertainty budget
  Assemble Type A/B components and environment/drift contributions; compute u_c and U; declare coverage probability and degrees of freedom.
• Traceability & compliance checks
  Verify each certificate in SI_link is valid; ensure unit/dimension consistency and proper uncertainty transmission along the chain.
• Certificate & manifest
  Generate cal_cert and manifest.instrument.cal.* including hashes and signatures; apply retention rules.
• Publication & re-calibration planning
  Activate date_cal → date_due; register monitors and thresholds; triggers include U > U_max, drift_level > drift_tol, and certificate expiry.

VI. Contracts & Assertions

• traceability.closed: SI_link is complete and all chain certificates are within validity.
• dim/unit.ok: check_dim(all) = true; unit mappings are unambiguous.
• model.fit.ok: residual tests pass; R2 ≥ R2_min or p_value ≥ p_min.
• U.bound: U ≤ U_max (at the stated coverage).
• k.declare: certificate declares k, coverage probability, and nu_eff.
• env.aligned: calibration records and environment logs are aligned on tau_mono with error ≤ eps_align.
• arrival.forms: for T_arr-related calibration, delta_form ≤ tol_Tarr.
• signature.ok: hash_sha256(blob) and signature verification succeed.

VII. Implementation Bindings I70-10* (Interface Prototypes)

• plan_calibration(instr, SRef, goals, U_max) -> plan
• fit_cal_curve(points, model_spec) -> {theta_hat, Cov, diag}
• build_uncertainty_budget(model, inputs, priors) -> {u_c, U, components}
• verify_traceability(sic, chain) -> report
• generate_cal_certificate(instr, results, chain) -> cal_cert
• emit_cal_manifest(cal_cert, policy) -> manifest.instrument.cal
• apply_cal_and_env(ds, cal_cert, env_coeffs) -> ds'
• schedule_recalibration(history, policy) -> next_date
• ingest_external_cert(blob) -> parsed_cert
  Invariants: unique(cal_cert_id); SI_link is non-empty; sum(components.variance) ≈ u_c^2; signatures verifiable; persisted TraceID keys are consistent for traceability.

VIII. Cross-References

• Timebase and arrival-time harmonization: EFT.WP.Metrology.TimeBase v1.0, Chapters 2 and 9.
• Calibration of sync chains and timestamping: EFT.WP.Metrology.Sync v1.0, Chapters 4–7.
• Cleaning & publication contracts; manifest fields: EFT.WP.Methods.Cleaning v1.0, Chapter 10 and Appendix C.
• Imaging radiometry and calibration: EFT.WP.Methods.Imaging v1.0, Chapter 4.

IX. Quality Metrics & Risk Control

• SLI/SLO
  closure_rate (traceability-chain closure rate), cert_latency_p95, U_p95, reject_rate, drift_ppm_per_day, expiry_coverage.
• Risks & rollback
  Chain breaks / expired certificates / U out of bounds / model mismatch → downgraded publication, activate prior certificate, shorten re-cal interval, lock operating range, or trigger human review.
• Audit
  Record data_hash, software_version, method_id, who/when/why; support recomputation and re-signing.

Summary
With P710-* / S710-* / M70-10 / I70-10*, this chapter implements a complete loop from SI traceability → calibration modeling → uncertainty budgeting → certificate & manifest → operational re-calibration, ensuring measured quantities are traceable, comparable, and auditable, while preserving dual-form arrival-time and unified timebase conventions whenever timing quantities are involved.