04-EFT.WP.Core.Metrology v1.0 | Appendix B — Metrology Process Templates and Examples

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Appendix B — Metrology Process Templates and Examples

I. Process Overview

• Goal: provide a minimal, end-to-end template from registration to reporting, together with three plug-and-play examples that cover register_measurement, check_dim, uncertainty budgeting and round_by_unc, and guard_band.
• Input domain: units/dimensions (see Appendix A), reference conditions RefCond, measurement model y = f(x), samples and specifications.
• Output domain: metrological result y_hat, combined standard uncertainty u_c(y), expanded uncertainty U = k * u_c(y), conformity decision, and a traceability chain.

II. Template B1: Measurement Model Registration and Dimensional Check (builds on Mx-1)

1. Applicability: first-time modeling and registry of any measurand.
2. Steps
   • Define the measurand and unit: measurand = "y", make unit(y) explicit; declare the input list inputs = [x_1, ..., x_p] and their units.
   • Modeling: provide y = f(x); if a path integral is involved, declare gamma(ell) and the measure d ell explicitly.
   • Dimensional check: call check_dim( y - f(x) ); it must return [] or the target dimension string identically.
   • Register:
     1. register_measurement(code, model="explicit", measurand="y", inputs, unit=unit(y), trace=[...]) -> MRef
     2. Record traceability_chain(MRef) and ensure traceability to SI.
3. Constraints
   • If using nondimensionalization: declare L0, t0 (default t0 def= L0 / c_ref) and provide the nondim / re_dim mapping.
   • Conflict-name review: T_fil (N) and T_trans (1) must not be mixed.

III. Template B2: Uncertainty Budgeting and Combination (builds on Chapter 5)

1. Applicability: linear or linearizable measurement models.
2. Steps
   • Type A: u_A(x_i) = unc_typeA(samples_i) (sample-based standard uncertainty).
   • Type B: u_B(x_i) = unc_typeB(spec_i) (datasheet, calibration certificate, or inter-comparison).
   • Combine inputs: u(x_i) = sqrt( u_A(x_i)^2 + u_B(x_i)^2 ).
   • Form input covariance: Cov[x], including correlations; if approximately independent, state the approximation explicitly and set off-diagonals to zero.
   • Linearization & Jacobian: J = ∂f/∂x |_{x = x_hat}.
   • Combined standard uncertainty: u_c(y) = sqrt( J * Cov[x] * J^T ).
   • Expanded uncertainty: U = k * u_c(y), with k chosen by coverage probability and effective degrees of freedom.
3. Checks
   check_dim(J_i * x_i) and check_dim(u_c(y)) must both match dim(y).

IV. Template B3: Reporting and Conformity Decision

1. Applicability: scenarios requiring a published result and a pass/fail decision against tolerance tol.
2. Steps
   • Rounding: (y_disp, digits) = round_by_unc(y_hat, U); follow “combine first, then round.”
   • Guard band: guard_band(y_hat, U, tol, rule="shared-risk") -> decision.
   • Report fields
     1. result = y_disp (unit(y))
     2. U = k * u_c(y) and coverage probability
     3. RefCond and corr_env (if applicable)
     4. traceability_chain
     5. unit_policy and a statement that check_dim passed

V. Example S-B1: Arrival-Time T_arr Measurement Loop (two conventions included)

1. Model and conventions
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )
   • General form: T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell )
   • Piecewise approximation: ∫_gamma n_eff d ell approx n_eff_avg * L_gamma
   • Substitute to obtain: T_arr approx ( n_eff_avg * L_gamma ) / c_ref
2. Units and dimensions
   unit(T_arr) = s, dim(T_arr) = "[T]"; check_dim passes: "[L]" * "[]" / "[L][T]^-1" -> "[T]".
3. Inputs and uncertainties
   • x = [ n_eff_avg, L_gamma, c_ref ]
   • u(n_eff_avg) from Type-B refractive-index model; u(L_gamma) from path traceability; u(c_ref) from the reference light speed or calibrated phase velocity in medium.
   • Jacobian
     1. ∂T_arr/∂n_eff_avg = L_gamma / c_ref
     2. ∂T_arr/∂L_gamma = n_eff_avg / c_ref
     3. ∂T_arr/∂c_ref = - ( n_eff_avg * L_gamma ) / c_ref^2
   • If Cov[x] is approximately diagonal:
     u_c(T_arr) = sqrt( Σ_i (∂T_arr/∂x_i)^2 * u(x_i)^2 ).
4. Reporting and decision
   • U = k * u_c(T_arr); (T_arr_disp, digits) = round_by_unc( T_arr_hat, U )
   • With tolerance tol_T: guard_band(T_arr_hat, U, tol_T, "shared-risk")
5. Traceability chain (example)
   • ∫_gamma d ell → gauge block / interferometer → m (SI)
   • time base → frequency standard → s (SI)
   • refractive index n_eff → published model with operating RefCond

VI. Example S-B2: Length Measurement with Environmental Correction (Interferometer)

1. Model
   • Raw reading: L_raw (internal scale length of the optical system)
   • Air correction: L = corr_env( L_raw; RefCond ), where corr_env = L_raw * ( n_air / n_ref ) or an equivalent form
2. Dimensions and units
   unit(L) = m; dim(L) = "[L]"; corr_env returns with the same dimension.
3. Uncertainty budget
   • Inputs: x = [ L_raw, n_air ] (treat n_ref as a constant or include in Type B)
   • Jacobian: ∂L/∂L_raw = n_air / n_ref, ∂L/∂n_air = L_raw / n_ref
   • u_c(L) = sqrt( (∂L/∂L_raw)^2 u(L_raw)^2 + (∂L/∂n_air)^2 u(n_air)^2 + 2 Cov(L_raw, n_air) (∂...)(∂...) )
4. Traceability chain
   • L_raw → laser wavelength → frequency standard → s ↔ Hz → m
   • n_air → empirical model (p, Temp, humidity), with p, Temp traced to Pa, K

VII. Example S-B3: Temperature with Offset Units and Conversion

1. Model
   • Temp_K = Temp_C + 273.15 (affine conversion)
   • dim(Temp_K) = "[Temp]", check_dim passes (offset b is legitimate)
2. Uncertainty
   u(Temp_K)^2 = u(Temp_C)^2 + u(273.15)^2, typically u(273.15) approx 0
3. Reporting
   Use K as the primary reporting unit; if a parallel unit is shown, state the affine relation.

VIII. Traceability Chain Samples (three summaries)

• T_arr: length (m) ↔ frequency (Hz) → time (s); n_eff from a model and environmental parameters RefCond.
• L (environment-corrected): interferometer (λ) → frequency standard → m; p, Temp, humidity each traced to Pa, K, 1.
• Temp_K: thermometer calibration → fixed points / SPRT → K (affine to degC).

IX. Script Snippets (I40- binding)**

1. Registration and dimensional check
   • register_measurement("MEAS.T_ARR", "explicit", "T_arr", ["n_eff_avg","L_gamma","c_ref"], "s", trace=["m","s"])
   • check_dim("(1/c_ref) * ∫ n_eff d ell") -> "[T]"
2. Uncertainty budgeting
   • J = [ L_gamma/c_ref, n_eff_avg/c_ref, - (n_eff_avg*L_gamma)/c_ref^2 ]
   • u_c = combine_uncertainty(J, u_inputs, Cov=None)
   • U = expanded_uncertainty(u_c, k=2.0)
3. Reporting and decision
   • (val, digits) = round_by_unc(T_arr_hat, U)
   • guard_band(val, U, tol=tol_T, rule="shared-risk")

X. Regression Baseline and Checklist

• check_dim: in all examples, the dimensions of y and f(x) are identical.
• unit_policy = "SI-Strict": unit set aligns with Appendix A.
• Reference conditions: whenever corr_env appears, explicitly list RefCond = { p_ref, Temp_ref, humidity_ref }.
• Two-form equivalence: T_arr yields identical u_c(T_arr) and U under both conventions (by integrand equivalence); the dimensionless statement with t0 = L0 / c_ref can be used as a cross-check.
• Conflict-name review: T_fil (N) vs T_trans (1), n (m^-3) vs n_eff (1) are never mixed.

XI. User-Reusable Placeholders (substitute the braces)

• register_measurement("{CODE}", "{MODEL}", "{MEASURAND}", [{INPUTS}], "{UNIT}", trace=[{TRACE}])
• unc_typeA({samples}) -> u_A; unc_typeB({spec}) -> u_B
• combine_uncertainty({J}, {u_inputs}, {Cov}) -> u_c; expanded_uncertainty({u_c}, {k}) -> U
• round_by_unc({value}, {U}) -> (value_disp, digits); guard_band({value}, {U}, {tol}, "{rule}") -> decision

XII. Alignment with Other Volumes

• T_arr and path integrals: see the companion white paper Energy Threads, Chapter 20 S/M/I.
• Unit consistency for tension and transmission-related quantities: see Energy Threads, Chapter 40 S/P/M/I.
• Consistent propagation from parameter uncertainty to outputs: see EFT.WP.Core.Parameters v1.0, Chapter 7.