35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 6 Couplings: Oriented Tension with Transport/Waves/Media (S/M)

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Chapter 6 Couplings: Oriented Tension with Transport/Waves/Media (S/M)

I. Abstract & Scope
This chapter presents the minimal coupling description and workflows between oriented tension and transport/wave/media. With the order tensor Q_ij as the core, we construct a coupling free energy W_cpl and its induced sources/constitutive corrections, derive anisotropic diffusion and advection–diffusion closures, and formulate phase-velocity/birefringence corrections for EM/acoustic/elastic waves. We also provide procedural methods for joint multimodal fitting and for extracting dominant energy/frequency bands. All symbols use English notation wrapped in backticks; SI units apply. No ToA terms appear in this chapter.

II. Dependencies & References

1. Orientation geometry & distributions: Chapter 3 S80-1/2.
2. Axioms & minimal equations: Chapter 4 P80-2/3/9/10, S80-3/4.
3. Metrology & inversion: Chapter 5 M80-1…4.
4. Energy accounting: Chapter 7 S80-7/8.
5. Numerics & simulation: Chapter 10 (SimStack-OT); implementation & APIs: Chapter 12 (I80-*).

III. Normative Anchors (added in this chapter, P80-/S80-)

6. P80-13 (Coupling Symmetry & Objectivity Axiom): all coupling terms are built from scalar invariants or objective tensors, invariant under rigid rotations and consistent with material symmetries.
7. S80-5 (Coupling Free Energy & Source Terms):
   W_cpl = − χ_E Q_ij E_i E_j − χ_B Q_ij B_i B_j − χ_u Q_ij D_{ij},
   with D_{ij} = ( ∂_i u_j + ∂_j u_i ) / 2 and coupling coefficients χ_* (units depend on the pair). The induced source for the order tensor is
   S_ij = − ∂W_cpl / ∂Q_ij = χ_E E_i E_j + χ_B B_i B_j + χ_u D_{ij} (take the symmetric, traceless part).
8. S80-6 (Anisotropic Transport & Wave Propagation):
   • Anisotropic advection–diffusion: J_c,i = − D_eff,ij ∂_j c + u_i c, with D_eff,ij = D0 δ_ij + D1 Q_ij; conservation ∂_t c + ∂_i J_c,i = S_c.
   • Electromagnetic constitutive corrections: ε_ij = ε0 ( δ_ij + α_E Q_ij ), μ_ij = μ0 ( δ_ij + α_B Q_ij ); paraxial phase speed c^2(ê) ≈ c0^2 / ( 1 + α_E ê_i Q_ij ê_j ).
   • Acoustic/elastic wave speed corrections: C_eff,ijkl = C0,ijkl + κ_Q 𝓟_{ijkl}(Q); reduced directional speed c_s^2(ê) = c_{s0}^2 + κ_s ( ê_i Q_ij ê_j ).

IV. Body Structure

I. Background & Problem Statement

• Through Q_ij, oriented tension modifies effective transport and wave propagation in media. The unified entry point is the coupling free energy W_cpl, ensuring consistency with Chapter 7 power/energy accounting.
• Minimal goals: provide identifiable, metrologically accessible closures and parameters {χ_E, χ_B, χ_u, D1, α_E, α_B, κ_s, …} for simulation and engineering prediction.

II. Key Equations & Derivations (S-series)

1. S80-5 (W_cpl & source terms):
   • Electric/magnetic/mechanical alignment are driven respectively by the traceless parts of E_i E_j, B_i B_j, and D_{ij}; S_ij enters the RHS of S80-4 (order-tensor dynamics) from Chapter 4.
   • With W_total = W_orient + W_cpl, variations yield an additive coupling term in oriented tension, T^{(cpl)}_{ij} = − ∂W_cpl / ∂D_{ij}, feeding back to S80-3.
2. S80-6 (Transport & waves):
   • Positive definiteness of diffusion: require D0 > 0 and |D1| such that the minimum eigenvalue of D_eff is positive.
   • EM birefringence (weak anisotropy): index split Δn ≈ (α_E/2) ê_i Q_ij ê_j.
   • Acoustic/elastic anisotropy: choose 𝓟_{ijkl}(Q) as a symmetry-compatible linear second-order basis (e.g., isotropic + transverse-isotropic parts).

III. Methods & Flows (M-series)

1. M80-5 (Joint Multiphysics Fitting)
   • Bundle data: {polarimetry, EM/acoustic scattering or propagation, mass/thermal/electrical transport, mechanics} into a DatasetBundle and audit unit/dim.
   • Forward map: build 𝒦 from S80-5/6, including R_inst.
   • Objective: L = L_EM + L_wave + L_trans + L_mech + L_reg; infer posteriors/evidence for {χ_E, χ_B, χ_u, D1, α_E, α_B, κ_s,…}.
   • Harmonization: jointly constrain with Chapter 5 outputs {Q_ij, T_fil_ij, tau_relax, D_Q}, enforcing objectivity and positivity.
2. M80-6 (Dominant Energy/Frequency Mask Extraction)
   • Define η_dom(ê, ω [or E]) = P_cpl(ê,ω)/max{ P_other(ê,ω) }.
   • With threshold η_* > 1, form the dominance mask m(ê,ω).
   • Feed masks to Chapter 7 for partitioned energy accounting and to Chapter 10 for sub-domain solvers in SimStack.

IV. Cross-References within/beyond this Volume

• Chapter 4: consistency between W_orient and S80-3/4; feedback T^{(cpl)}_{ij} into oriented tension.
• Chapter 5: metrology outputs {Q_ij, T_fil_ij, tau_relax, D_Q} as constraints for fitting and checks.
• Chapter 7: power terms of W_cpl in energy accounting S80-7/8; dominance masks for power partition.
• Chapters 10/12: implementation of coupling kernels in SimStack-OT and I80-* endpoints and benchmarks.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Turning off couplings (χ_*→0, D1→0, α_*→0) reduces evidence or increases structured residuals.
   • Directional dependences of anisotropic diffusion and wave speeds correlate linearly (weak coupling) or in an interpretable nonlinear way with ê_i Q_ij ê_j.
   • D_eff is positive definite; ε_ij/μ_ij and C_eff,ijkl respect objectivity and the material symmetry group.
2. Negative criteria:
   • Fits do not worsen when couplings are disabled (mechanism falsified/nonessential).
   • D_eff has nonpositive eigenvalues, or ε_ij/μ_ij/C_eff,ijkl violate objectivity/symmetry.
   • T^{(cpl)}_{ij} implied by W_cpl disagrees with calibrated tension in units/dimensions or sign.
3. Contrasts:
   • {transport-only, wave-only, transport+wave} identifiability & evidence comparisons for coupling parameters.
   • Residual structures under {EM coupling, mechanical coupling, composite}.
   • Predictions under {isotropic diffusion, tensor diffusion} for the same geometry.

VI. Deliverables & Figure List

1. Deliverables:
   • CouplingCard.json (W_cpl form; {χ_E, χ_B, χ_u, D1, α_E, α_B, κ_s,…} with units/dimensions).
   • DominanceMasks.npz (η_dom(ê,ω [or E]) and masks m(ê,ω)).
   • MultiPhysicsFit.md (joint-fit setup, evidence, sensitivities).
2. Figures/Tables (suggested):
   • Tab. 6-1 Coupling items & dimensional audits (W_cpl, T^{(cpl)}_{ij}, D_eff, ε_ij/μ_ij, C_eff).
   • Fig. 6-1 Directional phase-speed/diffusion maps vs experiments.
   • Tab. 6-2 Posteriors & correlation matrices for {χ_E, χ_B, χ_u, D1, α_E, α_B, κ_s}.
   • Tab. 6-3 Threshold η_* and resulting energy/frequency partitions.