40-EFT.WP.Materials.Superconductivity v1.0 | Chapter 8: Metrology Chain & Measurement Matrix

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Chapter 8: Metrology Chain & Measurement Matrix

I. Positioning & Objectives
This chapter defines the measurement matrix mapping from material parameters to observables, y = M(θ). It specifies traceability, construction and linearization, sensitivities and identifiability, error propagation, and power-optimized experiment design. Interfaces align with Chapter 6 (coherence windows) and Chapter 7 (arrival-time data contracts). All formulas/symbols/definitions are in English; citations and interfaces use “volume + version + anchor (P/S/M/I)”.

II. Metrology Chain & Traceability

• S80-1 (Traceability elements)
  Timebase/frequency, field/geometry, temperature/pressure, phase unwrapping, and path de-embedding each trace to SI or the project reference set. Constants use c_ref, Phi0_ref, k_B, h, e.
• S80-2 (Instrument segmentation & baselines)
  Path gamma = ⊕_k gamma_k and differential baselines (blank/substrate-only) freeze non-sample contributions in dataset cards; record delta_form, gamma(ell), d ell.
• M8-1 (Metrology calibration workflow)
  Input: standards/reference samples, environmental/drift monitors, instrument simulations and geometry maps.
  Steps: (i) multi-point calibration in frequency/time/field/geometry; (ii) drift modeling and online correction; (iii) build cal_bundle reusable by M7-*/M8-*.
  Output: cal_bundle = {timebase, freq_ref, field_ref, geom_ref, drift_model}.

III. Measurement Matrix Construction (Definition & Linearization)

• S80-3 (Observable and parameter vectors)
  y = [ T_arr(ω), Δf/f, 1/Q, |S21|, arg S21, M(H), R(H,T,θ,φ), … ]^T
  θ = [ lambda_L, xi, kappa_i, κ_ij, sigma1(ω), sigma2(ω), d, H_c1, H_c2, … ]^T
  ν = instrument/environment nuisance parameters (modeled inside cal_bundle).
• S80-4 (Mapping & linearization)
  y = M(θ, ν; x), with x experiment design variables (band/geometry/temperature/field/orientation/strain). Around a nominal θ0:
  Δy ≈ J(θ0) · Δθ + B · Δν, where J = ∂y/∂θ, B = ∂y/∂ν.
• S80-5 (Multi-band/multi-geometry stacking)
  Form J_stack = blkstack{ J(ω_k, geom_m) }. Within W_coh(ω), weight by w(ω) ∝ L_coh(ω) (or information density).
• M8-2 (Matrix generation)
  Input: instrument model, cal_bundle, data contracts (Ch. 7), W_coh (Ch. 6).
  Output: J, B, Σ_y (measurement noise covariance, including piecewise terms from unwrapping).

IV. Observation Geometry & Sensitivity

• S80-6 (Geometry parameterization)
  geom = { mode ∈ {TE, TM}, θ, φ, pol, incidence, L, aperture }. For anisotropy κ_ij, define directional derivatives ∂y/∂ê and principal-axis rotation sensitivities.
• S80-7 (Channel standardization)
  Normalize observable scales: ỹ = W_y · y, J̃ = W_y · J, preventing ill-conditioning from heterogeneous channel magnitudes.
• M8-3 (Sensitivity spectra & geometry optimization)
  Output: per-channel sensitivity magnitudes |J̃| and directions (singular vectors); recommend “most informative” geom* sets.

V. Error Propagation & Uncertainty Budget

• S80-8 (Linear error propagation)
  cov(Δθ) ≈ (J^T Σ_y^{-1} J)^{-1} (no priors). With calibration perturbations, use Σ_y ← Σ_y + B Σ_ν B^T.
• S80-9 (Joint channels & coherence weighting)
  Σ_y = blkdiag{ Σ_y(ω_k, geom_m) / w_k }; data outside W_coh receive low weights or enter as systematics.
• M8-4 (Uncertainty synthesis)
  Output: uncertainties and confidence intervals for derived quantities; generate report-ready error-budget tables (consistent with Chapter 2 u(x)/U).

VI. Identifiability & Condition Numbers

• S80-10 (Fisher information & conditioning)
  F = J^T Σ_y^{-1} J; evaluate cond(F), rank(F), and the correlation matrix C = diag(F)^{-1/2} F diag(F)^{-1/2} for parameter correlations and ill-conditioning.
• S80-11 (Priors & penalties)
  Add priors/physical constraints F ← F + F_prior (e.g., thin-film d prior window, kappa ≥ 1/√2); report posterior approximation cov(Δθ) ≈ F^{-1}.
• M8-5 (Identifiability assessment)
  Output: identifiability scores, weakest directions (principal components), and recommendations for new channels/geometries.

VII. Experiment Design & Power Optimization

• S80-12 (Design criteria)
  Support D/A/E-optimal objectives:
  Φ_D = -log det(F), Φ_A = tr(F^{-1}), Φ_E = -λ_min(F), jointly with a cost function cost(x).
• S80-13 (Scan grids & quotas)
  With discrete design sets X = {x_j} over temperature/field/band/geometry, solve
  min_x Φ(F(x)) + λ · cost(x) under overall time/cooling budgets.
• M8-6 (Design-optimization workflow)
  Output: recommended x* (e.g., sets of {ω, geom, θ, φ, T, H}), expected cov(Δθ), and per-channel quotas (samples per band/geometry).

VIII. Data Contract & Recording (Measurement-Matrix Section)

• S80-14 (Data structure)
  Extend dataset/pipeline cards with a measurement_matrix section with fixed fields:
• measurement_matrix:
• y_channels: ["T_arr(ω)","Δf/f","1/Q","|S21|","arg S21"]
• params: ["lambda_L","xi","kappa_i","kappa_tensor","sigma1(ω)","sigma2(ω)","d","H_c1","H_c2"]
• design_vars: ["ω","mode","θ","φ","T_temp","H","ε","p","geom","L"]
• J_shape: [Ny, Nθ]
• weighting: {rule: "coherence", L_coh_model: "..."}
• priors: {d: "N(μ,σ)", kappa_floor: "1/sqrt(2)"}
• covariance:
• Σ_y: "block-diagonal or sparse"
• Σ_ν: "calibration/systematic covariance"
• references:
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.6 W_coh"
• - "EFT.WP.Materials.Superconductivity v1.0:Ch.7 T_arr"
• - "EFT.WP.Core.Metrology v1.0:Ch.1–3,5"
• M8-7 (Consistency & reproducibility)
  Auto-validate shapes/units/anchors for J and y; emit a reproducibility bundle (scripts/parameters/environment digest), aligned with Methods.Repro v1.0.

IX. Implementation Bindings (I8- Prototypes)*

• I8-1 build_measurement_matrix(config, cal_bundle, contracts) -> {J, B, Σ_y, meta}
• I8-2 compute_fisher(J, Σ_y, priors=None) -> {F, cond, rank, C}
• I8-3 optimize_design(design_space, budget, criterion) -> {x*, F*, covθ*}
• I8-4 propagate_uncertainty(J, Σ_y, Σ_ν=None, priors=None) -> {covθ, ci}
• I8-5 assess_identifiability(J, Σ_y) -> {scores, weak_dirs, suggestions}
• I8-6 fuse_multi_band_matrices(J_list, Σy_list, weights) -> {J_stack, Σy_stack}

X. Interfaces to Adjacent Chapters

• With Chapter 4: parameterization and equations (κ_ij, xi, lambda_L, H_c1/H_c2) supply the forward model.
• With Chapter 6: W_c/W_coh and M6-* extractions produce observables that are linearized into J.
• With Chapter 7: the two T_arr conventions and de-embedding/unwrap outputs enter y and Σ_y.
• With Chapter 10: {J, Σ_y, priors} serve as the entry point for inversion and model comparison.

XI. Cross-Volume References & Anchors (This Chapter)

• Cross-volume (fixed style): EFT.WP.Core.Metrology v1.0 Ch.1–3,5; This volume Ch. 6 (coherence windows & extractions), Ch. 7 (arrival time & data contracts); EFT.WP.Methods.Repro v1.0; EFT.WP.Methods.SynthData v1.0.
• Anchors (S/M/I):
  S80-1—S80-14; M8-1—M8-7; I8-1—I8-6.

XII. Summary
This chapter establishes a unified path from metrological traceability to the measurement matrix y = M(θ), covering construction, sensitivity/identifiability analysis, error propagation, and optimal experiment design. With data contracts and implementation bindings, it ensures cross-instrument/cross-band consistency and reproducibility. The outputs {J, Σ_y, priors, cal_bundle} feed directly into Chapter 10 for inversion and model comparison.