02-EFT.WP.Core.Equations v1.1 | Chapter 3 — Constitutive Relations and Mappings

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Chapter 3 — Constitutive Relations and Mappings

I. Aims and Scope

• Establish a unified master mapping F_map(•) between n_eff and tension-related quantities, spanning local, anisotropic, nonlocal, dissipative, and statistically closed families, together with verifiable regularity and physical constraints.
• All formulas, symbols, and definitions are in English plain text and wrapped in backticks. Any expression containing division, integrals, or composite operators must include parentheses and, where applicable, make the path gamma(ell) and the measure d ell explicit.

II. Constitutive Postulates (P31 series)

• P31-1 (Measurability)
  n_eff must be measurable, non-negative, and bounded above: 0 ≤ n_eff(x,t) ≤ n_max almost everywhere.
• P31-2 (Causality and Scope)
  For any time t, n_eff(x,t) may depend only on states and controls with t' ≤ t (including history kernels) and must evaluate within the declared scope (local or nonlocal).
• P31-3 (Dimensional Closure)
  Any F_map must satisfy dim( n_eff ) = 1. Inputs with physical dimensions must first be mapped into the bar_* family via non-dimensionalization (see Section VI).

III. General Constitutive Master Form (S30-1)

• z(x,t) := { T_fil(x,t), grad[T_fil](x,t), TensionPot(x,t), rho(x,t), p(x,t), ... }
• theta is a parameter set (scalars, vectors, or kernels), varying by model family.
• domain: F_map : Z → R_+; Z is a mixed tensor–scalar input space, and R_+ is the non-negative reals.

IV. Model Family A: Local Algebraic (S30-2)

1. S30-2 (local linear—saturation)
   n_eff def= clamp( a0 + a1 * bar_T + a2 * |bar_grad_T| , n_min , n_max )
   • bar_T := ( T_fil / T0 ), bar_grad_T := ( L0 * grad[T_fil] / T0 ).
   • clamp(u, u_min, u_max) := min( max(u, u_min), u_max ).
   • theta := { a0, a1, a2, n_min, n_max }; require 0 < n_min ≤ n_max.
   • check_dim: the right-hand side is dimensionless, satisfying P31-3.
2. Anisotropic extension (S30-2a)
   n_eff def= clamp( a0 + a1 * ( p • hat_grad_T ) , n_min , n_max )
   • hat_grad_T := ( grad[T_fil] / |grad[T_fil]| ) (if the denominator is 0, set hat_grad_T := 0).
   • p(x,t) is a unit orientation vector (see Core Terms, Chapter 3).

V. Model Family B: Nonlocal Kernels and Path Averages (S30-3)

1. S30-3 (volumetric neighborhood kernel)
   n_eff(x,t) def= n_bg + ( ∫_{B_r(x)} K_r( x - y ) * bar_T(y,t) d V )
   • K_r is a radial kernel with ( ∫_{B_r(0)} K_r(ξ) d V ) = 1.
   • n_bg ≥ 0.
   • When used with path quantities, pair with avg_gamma[•] or interpolate into S20-* via ( n_eff / c_ref ).
2. S30-4 (pathwise kernel)
   This couples consistently to T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell ) (see Chapter 2).n_eff( gamma(ell), t ) def= n_bg + ( ∫_{ell'∈[0,L_gamma]} K_ell( | ell - ell' | ) * bar_T( gamma(ell'), t ) d ell' )

VI. Non-Dimensionalization and Parameter Rules

1. Baselines and mappings
   If F_map is expressed as F_map( bar_T , bar_grad_T , ... ; theta_bar ), then theta_bar is dimensionless.bar_x := ( x / L0 ), bar_t := ( t / t0 ), bar_T := ( T_fil / T0 ).
2. Scale-selection guidelines
   • Fix {L0, t0, T0} and c_ref within a problem family.
   • For cross-domain comparison, provide the provenance and uncertainty of {L0, t0, T0} (see Core Terms, Chapter 6).

VII. Model Family C: Dissipation, Memory, and Dispersion (S30-5)

• K_tau ≥ 0, ( ∫_{0}^{∞} K_tau(s) d s ) = k_tau (constant).
• If K_tau is an exponential kernel, an equivalent weak=... ODE reduction can be derived.

VIII. Model Family D: Statistical Closure and Uncertainty (S30-6)

• For conservative upper bounds, use n_eff def= E[n_hat] + k_sigma * sqrt( Var[n_hat] ).
• k_sigma ≥ 0 is chosen by the risk convention (see Methods.Falsification).

Chapter 3 — Constitutive Relations and Mappings (continued)

IX. Physical Constraints and Regularity (S30-7 / S30-8)

• S30-7 (boundedness and monotone segments)
  n_min ≤ n_eff(x,t) ≤ n_max; if a scalar driver q(x,t) is defined, require within the working range
  ( ∂ n_eff / ∂ q ) ≥ 0 to preserve consistent energy–arrival-time monotonicity.
• S30-8 (Lipschitz regularity)
  There exists L_map ≥ 0 such that
  | n_eff(x,t) - n_eff(y,s) | ≤ L_map * ( |x - y| + |t - s| )
  under the chosen norms, ensuring stability of weak forms and numerical assembly.

X. Weak-Form Embedding and Alignment with the Mother Form

• Weak-form placement
  weak= inner_V( w , n_eff(•) ) enters as a coefficient field. If n_eff depends on an unknown field u, its first variation appears as
  delta[ n_eff(u) ] = ( ∂ n_eff / ∂ u ) * delta[u ],
  with the chain rule made explicit per model family.
• Example (Gâteaux derivative for S30-2)
  ∂ n_eff / ∂ T_fil = ( a1 / T0 ) + ( a2 / T0 ) * ( grad • ( grad[T_fil] / |grad[T_fil]| ) ) (in smooth regions), together with the piecewise-constant derivative of clamp (derivative is 0 on saturated segments).

XI. Sensitivity and Adjoint Interfaces

• Local-family sensitivities
  ∂ n_eff / ∂ a_i = basis_i( z ); ∂ n_eff / ∂ T_fil = G_loc( z ).
• Nonlocal-family sensitivities
  For S30-3: ∂ n_eff(x,t) / ∂ T_fil(y,t) = ( K_r( x - y ) / T0 ) (when y ∈ B_r(x)).
• These derivatives feed the coefficients of adjoint_sensitivity (see I20-3).

XII. Discrete Implementation Binding (I20 Alignment)

• Mapping evaluation interfaces
  eval_map(z:any, theta:any) -> n_eff:any
  eval_map_grad(z:any, theta:any) -> dict (returns partials w.r.t. z and theta)
• Path coupling
  Align with propagate_time(n_eff_path, ds, c_ref): evaluate n_eff[k] at nodes, then accumulate via S20-5.

XIII. Lint Rules and Forbidden Patterns

• Do not interchange n and n_eff; never use bare "c", "T", "n".
• Disallow algebraic sums with dimensional quantities left unscaled (e.g., n_eff = a0 + a1 * T_fil) — write T_fil / T0.
• Disallow missing parentheses in composite terms (e.g., n_eff = a0 + a1 * grad[T_fil]/T0; write n_eff = a0 + a1 * ( grad[T_fil] / T0 )).

XIV. Typical Use Cards

• Use Case A (isotropic local — coupled to arrival time)
  n_eff = clamp( a0 + a1 * ( T_fil / T0 ), n_min , n_max );
  T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell ) (see S20-1).
• Use Case B (nonlocal smoothing — path integral)
  n_eff( gamma(ell) ) = n_bg + ( ∫ K_ell( | ell - ell' | ) * ( T_fil( gamma(ell') ) / T0 ) d ell' );
  T_arr approx ( Σ_k ( n_eff[k] / c_ref[k] ) * ds[k] ).
• Use Case C (dissipative memory)
  n_eff(x,t) = n_bg + ( ∫_{0}^{t} K_tau( t - tau ) * ( T_fil(x,tau) / T0 ) d tau ); the weak form follows the convolution expansion.

XV. Pre-Release Checklist

• Have F_map inputs z, parameters theta, and scope (local/nonlocal/temporal memory) been declared explicitly?
• Does check_dim pass (n_eff is dimensionless)?
• Are n_min, n_max or equivalent bounds provided?
• Are sensitivity terms supplied to interface with adjoint_sensitivity?
• Does the causality constraint of P31-2 hold (temporal kernels bounded above by the current time)?
• Lint: avoided n/n_eff swaps, missing parentheses, and unscaled dimensional terms.

XVI. Equation Cards (S30 Summary)

• S30-1 — n_eff def= F_map( z(x,t); theta )
• S30-2 — local linear—saturation: n_eff def= clamp( a0 + a1 * bar_T + a2 * |bar_grad_T| , n_min , n_max )
• S30-2a — anisotropy: n_eff def= clamp( a0 + a1 * ( p • hat_grad_T ) , n_min , n_max )
• S30-3 — nonlocal volume kernel: n_eff def= n_bg + ( ∫_{B_r(x)} K_r * bar_T d V )
• S30-4 — path kernel: n_eff( gamma(ell) ) def= n_bg + ( ∫ K_ell * bar_T( gamma(•) ) d ell )
• S30-5 — temporal memory: n_eff def= n_bg + ( ∫_{0}^{t} K_tau * bar_T d tau )
• S30-6 — statistical closure: n_eff def= E[ n_hat( z; theta , eta ) ]
• S30-7 — boundedness and monotone segments: n_min ≤ n_eff ≤ n_max, with ( ∂ n_eff / ∂ q ) ≥ 0
• S30-8 — Lipschitz regularity: global or piecewise Lipschitz constant L_map exists