35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 5 Metrology: Measuring & Calibrating Oriented Tension (M)

Chapter 5 Metrology: Measuring & Calibrating Oriented Tension (M)

I. Abstract & Scope
This chapter establishes the minimal metrology pipeline for oriented tension: invert optical (polarimetry/scattering) or spectro-imaging data to obtain Q_ij, calibrate T_fil_ij under controlled mechanical loading, measure relaxation and diffusion (tau_relax, D_Q) in frequency/time domains, and finally perform instrument-response deconvolution and uncertainty synthesis. Every data column must carry unit/dim. If time-of-arrival (ToA) methods are used, both forms must be recorded in parallel with explicit gamma(ell) and d ell.

II. Dependencies & References

• Orientation geometry & distributions: Chapter 3 S80-1/2.
• Axioms & minimal equations: Chapter 4 P80-2/3/9/10, S80-3/4.
• Symbols & dimensions: Chapter 2 P80-1/4/5/6.
• Coupling & transport: Chapter 6 S80-5/6.
• Energy accounting & power partition: Chapter 7 S80-7/8.
• Implementation & APIs: Chapter 12 (SimStack-OT / I80-*).

III. Normative Anchors (added in this chapter)

• P80-11 (Metrological Consistency Axiom): units/dimensions in calibration data and the model card must close; Q_ij tracelessness, symmetry, and physical eigenvalue domain must hold post-calibration.
• P80-12 (Response-Deconvolution Axiom): the mapping between observables and physical quantities is expressed via a convolution/integral operator R_inst; deconvolution must be performed under regularization and unit-closure constraints.
• M80-1 (Polarimetry/Scattering → Order-Tensor Inversion): invert to Q_ij with covariance.
• M80-2 (Mechanical Loading → Tension Calibration): calibrate T_fil_ij and constitutive parameters under controlled loading.
• M80-3 (Frequency/Time-Domain Relaxation Measurements): obtain tau_relax, D_Q, and response functions.
• M80-4 (Response Deconvolution & Uncertainty Synthesis): apply R_inst deconvolution, error decomposition, and multi-source fusion.

IV. Body Structure

I. Background & Problem Statement

• Goal: a reproducible “observations → Q_ij/T_fil_ij/dynamic quantities → constitutive parameters” loop to support parameter identification in S80-3/4 and closures in Chapters 6–7.
• Constraints: data and models must satisfy Unit/Dim audits and axioms (symmetry, objectivity, positivity). If ToA is used, dual-form recording with {gamma(ell), d ell} is mandatory.

II. Key Equations & Derivations (S-series)

• Observable–physical mapping (general form):
  y_obs(ξ) = ( 𝒦[ x_true ] )(ξ) + n(ξ), with x_true ∈ {Q_ij, T_fil_ij, …}, 𝒦 including R_inst, and n noise.
• Deconvolution/inversion (Tikhonov/Wiener examples):
  x̂ = argmin_x ( ‖R_inst x − y_obs‖^2_Σ + λ‖Lx‖^2 ); or in frequency domain
  x̂(ω) = ( R^*(ω) S_x(ω) / ( |R(ω)|^2 S_x(ω) + S_n(ω) ) ) y_obs(ω).
• Polarimetry–order-tensor link (minimal form):
  S_pol = [Q_ij] + ε, with 𝓜 a Mueller/scattering-kernel mapping that preserves Q_ij symmetry and tracelessness.
• Mechanical calibration residuals & loss:
  L_T = ‖ T_fil_ij(meas) − T_fil_ij(pred; Λ_{ijkl}, …) ‖^2_Σ,
  L_Q = ‖ Q_ij(meas) − Q_ij(pred; …) ‖^2_Σ,
  L = L_T + L_Q + L_time + L_reg.
• Uncertainty propagation (linearized):
  Cov_post ≈ ( J^T Σ^{-1} J + Π )^{-1}, with J = ∂y/∂θ, and Π the prior precision; or use sample-posteriors for empirical covariance.

III. Methods & Flows (M-series)

1. M80-1 Polarimetry/Scattering → Order-Tensor Inversion
   • Preprocessing: resample in time–frequency/angle domains; audit Unit/Dim.
   • Modeling: choose 𝓜 (Mueller or scattering kernel) with symmetry (tracelessness) and objectivity constraints.
   • Inversion: solve min_{Q_ij} ‖𝓜[Q_ij] − S_pol‖^2_Σ + λ‖∇Q‖^2; output Q_ij and Cov_Q.
   • Checks: Tr(Q)=0, eigenvalue bounds λ∈[−1/3, 2/3], cross-validated residuals.
2. M80-2 Mechanical Loading → Tension Calibration
   • Specimen & loading: uniaxial/biaxial/shear paths; record u_vec, strain, and boundaries.
   • Observations: force–displacement and/or full-field strain imaging; unit audit in Pa.
   • Calibration: fit Λ_{ijkl} and (if needed) W_orient parameters A,K via S80-3:
     min_{Λ,A,K} L_T + α‖Λ‖^2 + β‖(A,K)‖^2.
   • Outputs: T_fil_ij, posterior for Λ_{ijkl}, and evidence ratios.
3. M80-3 Frequency/Time-Domain Relaxation Measurements
   • Frequency domain: small-amplitude sinusoidal excitation; measure Q_ij(ω) or T_fil_ij(ω); fit G*(ω) or H_Q(ω) to obtain tau_relax.
   • Time domain: step/pulse relaxation; fit Q_ij(t) = Q_ij^eq + (Q_ij^0 − Q_ij^eq) e^{−t/tau_relax}.
   • Spatial diffusion: fit broadening of a seeded δQ_ij(r,0) to estimate D_Q.
   • Outputs: posteriors and CIs for tau_relax, D_Q.
4. M80-4 Response Deconvolution & Uncertainty Synthesis
   • Estimate R_inst and noise spectrum S_n.
   • Choose regularization (Tikhonov/Wiener/sparse priors); deconvolve with unit-closure preserved.
   • Error decomposition: statistical noise, systematics (calibration, alignment, drift), model error.
   • Synthesis: propagate Cov_Q, Cov_T to {Λ, A, K, tau_relax, D_Q} via linearization or sampling.

IV. Cross-References within/beyond this Volume

• Chapter 3: Q_ij definitions and S^2 quadrature (M80-13/14) for inversion/integration.
• Chapter 4: parameter identification and harmonization for S80-3/4 (Λ_{ijkl}, A, K, tau_relax, D_Q).
• Chapter 6: coupling/transport coefficients ingest metrology outputs (S80-5/6).
• Chapter 7: energy-accounting checks and power-term matching (S80-7/8).
• Chapter 12: SimStack-OT I80-* endpoints and benchmark generation.
• Companion: metrology appendix in Energy Filaments.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Inverted Q_ij is traceless with eigenvalues in-range; CV residuals are whitened.
   • T_fil_ij matches loading curves in units/dimensions and symmetry; evidence exceeds that of orientation-free models.
   • tau_relax and D_Q agree across frequency/time domains; no systematic bias after R_inst deconvolution.
2. Negative criteria:
   • Adjusting only Λ_{ijkl} or removing orientation terms does not worsen fits (orientation mechanism falsified or nonessential).
   • Unit/Dim audit fails, or R_inst deconvolution introduces spurious bands.
   • Q_ij eigenvalues out of bounds, Tr(Q)≠0, or symmetry violations.
3. Contrasts:
   • {optical-only, mechanical-only, joint} identifiability and evidence comparisons.
   • {Tikhonov, Wiener, sparse} deconvolution impacts on uncertainty.
   • {uniaxial, biaxial, shear} loading-path condition numbers for estimating Λ_{ijkl}.

VI. Deliverables & Figure List

1. Deliverables:
   • MetrologyCard.json (observation geometry, R_inst, noise, units, covariance).
   • QTensorPosterior.zarr, TensionPosterior.zarr (posteriors & covariances for Q_ij, T_fil_ij).
   • RelaxationReport.md (tau_relax, D_Q, and consistency checks).
   • UnitsAudit.log, EvidenceReport.md.
2. Figures/Tables (suggested):
   • Tab. 5-1 Observable–physical mappings with unit-closure checklist.
   • Tab. 5-2 Q_ij inversion and eigenvalue-domain checks.
   • Fig. 5-1 Residual spectra/images before vs after deconvolution.
   • Tab. 5-3 Posteriors & sensitivity matrices for {Λ, A, K, tau_relax, D_Q}.
   • Fig. 5-2 Calibration curves for {uniaxial/biaxial/shear} and model-fit overlays.