04-EFT.WP.Core.Metrology v1.0 | Chapter 7 — Calibration and Verification Interfaces (I40-*)

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Chapter 7 — Calibration and Verification Interfaces (I40-*)

I. Objectives and Scope

• Define interfaces and workflows for calibration, verification, and adjustment, ensuring consistency with the measurement chain and the traceability_chain, dimensional closure, and reproducible reporting.
• This chapter binds directly to I40 2/3/4/5/6/8/9; uncertainty follows Chapter 5, and data presentation follows Chapter 6.

II. Terms and Roles

• calibration: solve for the mapping from indication to true value g_cal and its parameters phi, and supply uncertainty and validity period.
• verification: under a fixed g_cal, evaluate pass|fail|inconclusive against the tolerance function MPE(x) and a stated decision rule.
• adjustment: change internal instrument parameters to reduce bias; after adjustment, perform calibration again or, at minimum, run verification.
• references and traceability: all reference values x_ref must be traceable through a traceability_chain and valid under RefCond.

III. Calibration Models and Minimal Equation (S97-1)

1. Mother form: y_true = g_cal( y_ind; phi ) + epsilon, where epsilon is a zero-mean error term and dim( y_true ) = dim( y_ind ).
2. Inverse mapping (for indication correction): y_corr = g_inv( y_ind; phi ), with g_inv( g_cal( y; phi ); phi ) identically equal to y over the domain.
3. Typical families (choose one or combine):
   • Affine: g_cal( y; phi ) = a * y + b, phi = { a, b }.
   • Polynomial: g_cal( y; phi ) = ∑_{k=0..K} c_k * y^k.
   • Piecewise linear: g_cal( y; phi ) = interp1( table ), table = { (y_k, v_k) }, monotonic.
   • Logarithmic or exponential: g_cal( y; phi ) = p * log( y ) + q or p * exp( q * y ), with domain declared explicitly.
   • With deadband: g_cal( y; phi ) = a * y + b + deadband( y; τ ).
4. Constraint: check_dim( g_cal( y_ind; phi ) - y_true ) = "[1]"; priors and bounds for phi may reuse Core.Parameters.

IV. Calibration Data and Fitting

1. Data pairs: D = { (x_ref_i, y_ind_i, u_ref_i) }, where each x_ref_i is provided by a reference standard under RefCond and unified by convert to a common unit.
2. Weighted least squares (including reference uncertainty):
   • Weights: w_i = 1 / u_ref_i^2 (or, to combine indication noise, w_i = 1 / ( u_ref_i^2 + u_ind_i^2 )).
   • Estimate: phi_hat = argmin_phi ∑_i w_i * ( g_cal( y_ind_i; phi ) - x_ref_i )^2.
   • Covariance: Cov[phi_hat] approx ( J^T W J )^-1, with J = ∂g_cal/∂phi |_{phi_hat}, W = diag( w_i ).
3. Output correction: y_corr = g_inv( y_ind; phi_hat ); propagate uncertainty as in the next section.

V. Uncertainty Propagation and Coverage

• Point-value uncertainty: u_c( y_corr ) = combine_uncertainty( J_y, u_inputs, Cov ), where J_y = ∂g_inv/∂(inputs).
• Affine-model special case:
  y_corr = ( y_ind - b ) / a,
  u_c^2( y_corr ) = ( (1/a)^2 * u^2(y_ind) ) + ( ((y_ind - b)/a^2)^2 * u^2(a) ) + ( (1/a)^2 * u^2(b) ) + cross_terms。
  Cross terms supplied by Cov[a,b].
• Coverage and reporting: U = k * u_c, with k and nu_eff per Chapter 5.

VI. Verification and Tolerance Decisions

1. Define MPE(x): maximum permissible error as a function of x, in the same unit as the measurand.
2. Residual: e = g_cal( y_ind; phi_hat ) - x_ref (equivalently, y_corr - x_ref).
3. Decision (shared-risk, one-sided upper example):
   • pass if |e| + u_c(e) ≤ MPE(x_ref);
   • fail if |e| - u_c(e) > MPE(x_ref);
   • inconclusive otherwise.
4. Interface: guard_band( result=|e|, U=u_c(e), tol=MPE(x_ref), rule="shared-risk" ).

VII. Drift Modeling and Validity

• Linear drift approximation: bias(t) = bias0 + drift_rate * ( t - t0 ).
• Future-uncertainty inflation: u_c_future^2 = u_c_now^2 + u_drift^2, with u_drift = | drift_rate | * Δt / sqrt(3) (uniform-bound assumption).
• Validity policy: choose the minimal duration T_valid such that U_future( T_valid ) ≤ MPE(x) at the lower bound of the operating interval.

VIII. Multi-Point Linearization and Table Lookups

• Lookup table table = { (y_k, v_k, u_k) }, where v_k is the reference true value; interp1 uses segment-wise linear interpolation.
• Monotonicity constraint: v_{k+1} - v_k ≥ 0 (or ≤ 0); non-monotonic datasets should be adjusted or fit with a regression model.
• Uncertainty interpolation: u_c( y_corr ) = linear_interp( u_k ) ⊕ u_interp, where u_interp budgets interpolation-model error.

IX. Environmental and Unit Consistency

• Apply convert( x_ref, U_ref, U_target ) before fitting; record corr_env( y; RefCond ) alongside the reference corrections.
• Perform all computations under U_target; mixed-unit fitting is forbidden.
• check_dim must pass; calibration for arrival-time integrals is detailed in §XI.

X. I40- Interface Mapping and Minimal Implementation**

• register_measurement( code="cal_model_X", model="g_cal_family", measurand="Y", inputs=["y_ind"], unit="U_target", trace=[...] )。
• Use convert and corr_env to unify x_ref and y_ind to common RefCond and units.
• Estimate phi_hat and Cov[phi_hat]; produce u_c with combine_uncertainty.
• Generate guard_band decisions and populate ReportV1 fields (see Chapter 6).
• Fix formats via export_units, export_refcond; use compare_reports for regression monitoring.
• When binding to equations, call bind_to_equation( Sref, unit_policy ), and if arrival time is involved, invoke enforce_arrival_time_convention().

XI. Calibration Use Case for Arrival-Time Sensing Chains

1. Objective: calibrate the scale factor of an n_eff sensor for use in T_arr.
2. Two conventions consistent:
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )。
   • General form: T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell )。
3. Assume sensor reading n_meas(ell) and true value n_true(ell) follow n_true(ell) = a * n_meas(ell) + b.
4. Calibration data: with known path gamma(ell) and L_gamma = ( ∫_gamma 1 d ell ), and a reference arrival time T_ref:
   • Residual function:
     r(a,b) = ( ( 1 / c_ref ) * ( ∫_gamma ( a * n_meas(ell) + b ) d ell ) ) - T_ref。
   • Closed-form (when b=0 holds):
     a_hat = ( c_ref * T_ref ) / ( ∫_gamma n_meas d ell )。
   • In the general case, fit a,b via weighted least squares, with weights from u(T_ref) and the path-integral uncertainty.
5. Reporting: bind S20-* (arrival time) and list { c_ref(cert), path(gamma), integrator(spec) } in trace; propagate U per Chapter 5.

XII. Minimal Certificate Fields (CalibrationCertificateV1)

• instrument, serial, measurand, unit, RefCond_name, date, location.
• model_family, phi_hat, Cov[phi_hat], valid_range, residual_stats.
• traceability_chain, standards (with validity and U), software_version, version_semver.
• environment (actual), adjustment (if any), drift_assessment, T_valid.
• decision_rule, source of MPE, verification_result.

XIII. Quality Control and Reproducibility

• Before/after comparison: compare_reports( a, b, metrics=["mean","U","pass_rate"] ); threshold breaches trigger recalibration.
• Recomputation policy: any change to RefCond, unit, model, or phi requires recomputing u_c and U and updating the certificate.
• Randomness record: if MC is used, store seed and sample size n; report nu_eff to align across batches.

XIV. Calibration and Verification Workflow (Mx-5)

• Plan: determine measurand, U_target, MPE(x), RefCond, and the reference’s traceability_chain.
• Acquire: collect D = { (x_ref_i, y_ind_i, u_ref_i) }; run convert and corr_env.
• Fit: choose g_cal; estimate phi_hat, Cov[phi_hat]; form g_inv.
• Propagate: assess u_c, U, and nu_eff at representative points or intervals; compute residuals and coverage intervals.
• Verify: decide with guard_band and MPE(x); produce verification_result.
• Document: generate CalibrationCertificateV1; export via export_units, export_refcond; archive the traceability_chain.
• Maintain: estimate drift_rate and T_valid; set up triggers and schedules for recalibration.

XV. Interface Anchors with Other Volumes

• Core.Equations: when g_cal is embedded in Sxx-*, use bind_to_equation and maintain check_dim.
• Core.Parameters: register priors and bounds for phi via I30 1/3/4; for Fisher or sensitivity analysis, see that volume’s Chapter 5.
• Arrival-time parameters: adhere to the fixed conventions and path declarations gamma(ell), d ell, c_ref.