49-Error Budget Card Template v1.0 | Chapter 3 — Metrological Models & Observables (Control Equations)

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Chapter 3 — Metrological Models & Observables (Control Equations)

I. Purpose & Scope

• Provide unified conventions for the metrological model and control equations for observables used in uncertainty propagation, including continuous/discrete forms, dimensional checks, and applicability.
• Applicable to timing / imaging / spectral / mixed; paraxial/small-angle, slowly varying medium within coherence windows; publication requires p_dim = 1.0.

II. Prerequisites & Inputs

• Explicit path: the text must show both gamma(ell) and measure d ell; data/metadata record delta_form ∈ {general, factored}.
• Reference quantities & units: c_ref (m/s), lambda_ref (m), k_ref = 2π/λ_ref (1/m).
• Consistency: len(gamma_ell) = len(d_ell) = len(n_eff) ≥ 2; Δell satisfies sampling constraints; clock_state = locked, |ts_start − calib.timestamp| ≤ τ_calib.

III. Control Equations (continuous form)

1. Arrival time (two equivalent forms)
   • Constant factored:
     T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General form:
     T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Dimensional check: [1]/[m·s^-1]·[m] = [s]. Applicability: n_eff varies slowly within the coherence window.
2. Phase accumulation
   Phi = ( 2π / λ_ref ) * ( ∫ n_eff d ell )
   Dimensional check: [1/m]·[m] = [rad].
3. Paraxial propagation & conservation (indicative)
   • Paraxial propagation (scalar envelope):
     ∂_z A + (1/2k_ref) ∇_⊥^2 A + i k_ref ( n_eff − 1 ) A = 0 (paraxial, small-angle, slowly varying medium).
   • Conservation error: ε_flux → 0 @ O(θ^2).
4. Mass conservation (continuity)
   ∂_t ρ + ∇·( ρ v ) = 0; for protocol-level checks, report ΔM = |∫ ρ dV|_{t2} − |∫ ρ dV|_{t1}.

IV. Discrete & Numerical Forms

1. Arrival time (summation approximation)
   T_arr ≈ ∑_i ( n_i / c_ref ) · d ell_i
   where n_i = n_eff(ell_i), and d ell_i is the discretized path measure.
2. Phase (summation approximation)
   Phi ≈ ( 2π / λ_ref ) ∑_i n_i · d ell_i
3. Sampling bounds
   • Arrival-time path step: Δell ≤ ( c_ref / f_s ) / max(n_eff)
   • Phase time step: Δt ≤ λ_ref / ( 2 c_ref )
4. Boundaries & occlusion
   When occlusions/boundaries exist, explicitly provide valid path segments and masks, keeping gamma_ell/d_ell/n_eff aligned.

V. Applicability & Assumptions

• Slowly varying medium: |∇ n_eff|·L_coh ≪ 1; segment the path and apply bandwise approximations if needed.
• Paraxial small-angle: |θ| ≤ θ_max; outside domain, switch to full-wave/numerical propagation and tag [Restricted].
• Metrological closure: all expressions pass check_dim; publication requires p_dim = 1.0.
• Symbols & conflicts: forbid bare c, T, n; use c_ref, T_arr (or specific quantity), n_eff (or n with explicit unit); T_fil ≠ T_trans, n ≠ n_eff.

VI. Interfaces for Uncertainty Propagation

• Delta method (linearization)
  Target y = f(x); Jacobian J = ∂f/∂x |_{x̂}; u^2(y) ≈ J · Cov(x) · Jᵀ.
• Arrival-time first-order sensitivity
  ∂T_arr/∂c_ref = − (1 / c_ref^2) ∫ n_eff d ell; path-wise covariance may be modeled via a kernel with correlation length L_c.
• Phase sensitivity
  ∂Phi/∂λ_ref = − ( 2π / λ_ref^2 ) ∫ n_eff d ell.
• Suggested interfaces: I80-prop_delta(f, x_hat, Cov), I81-prop_mc(simulator, draws, seed, cov_model), I83-coverage(...), I70-dim_check(...).

VII. Exports & Compliance

• Required: check_dim_report.json (covering T_arr/Phi dimensional checks), uncertainty.md (documenting chosen control equations and conventions).
• Citations & versioning: text and see[]/references[] consistent, using “volume + version + anchor (P/S/M/I)”, anchor coverage ≥ 90%, public releases v1.* only.
• Restricted mode: if any core assumption (paraxial/coherence/metrological closure) fails, enter [Restricted] and publish qualitative results only.

VIII. Checklist

• gamma(ell) and d ell are explicit in text; delta_form recorded in data/metadata.
• T_arr and Phi are provided using the unified forms; discrete sums and sampling bounds stated.
• p_dim = 1.0 and check_dim_report.json generated; clock_state = locked, τ_calib compliant.
• len(gamma_ell)=len(d_ell)=len(n_eff)≥2, Δell within constraints; occlusion/boundaries handled.
• Citations compliant (volume+version+anchor, coverage ≥ 90%), no external links/aliases; public releases v1.* only.