03-EFT.WP.Core.Parameters v1.0 | Chapter 9 — Cross-Volume Binding and Use Cases

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Chapter 9 — Cross-Volume Binding and Use Cases

I. Chapter Goals and Binding Landscape

• Goal: define the standard way to bind this Parameters volume to Core.Equations, covering arrival time T_arr, the tension field T_fil(x,t), and continuity/flux J(x,t), and provide end-to-end use cases (register → calibrate → propagate → regress).
• Binding scope: S20-* (paths and arrival time), S40-* (minimal equations for the tension field), S50-* (continuity and transport), S70-* (variational and weak forms), S80-* (statistics and coarse-graining); implementation side binds to I30-* and I20-*.
• Invariants (cross-volume): never mix T_fil with T_trans; strictly distinguish n and n_eff; whenever a line integral or division appears, the expression must be parenthesized and must declare gamma(ell) with the measure d ell.

II. Cross-Volume Interfaces and Anchors

1. Equation anchors (see EFT.WP.Core.Equations v1.1)
   • S20-1…S20-3 arrival time: T_arr = ( ∫ ( n_eff / c_ref ) d ell ) and the constant-factored convention.
   • S40-* tension field: strong/weak forms of T_fil(x,t) with source S_src(x,t).
   • S50-* continuity: ∂_t rho + div[J] = S_src.
   • S70-* weak-form notation: inner_V[u,v], Lagr[•], delta[Lagr].
   • S80-* statistical windows: avg_t[f; Δt], avg_V[f; V=Ω], avg_gamma[f].
2. Implementation anchors (this volume)
   • Parameter side: I30 1…12 (registration, priors, transforms, inference, identifiability, propagation, import/export, scenario governance).
   • Equation side: I20-* (path discretization, assembly/solve, arrival-time interface propagate_time, regression and comparison).

III. Binding Postulates and Consistency Checks (P91-1…P91-4)

• P91-1 Path explicitness: whenever T_arr appears, write
  T_arr = ( ∫_gamma ( n_eff / c_ref ) d ell ), and declare gamma(ell), L_gamma = ∫_gamma 1 d ell.
• P91-2 Dimensional closure: validate binding expressions with check_dim(expr); the integrand ( n_eff / c_ref ) * d ell is dimensionless.
• P91-3 Constitutive-map conservation: for n_eff def= F_map(T_fil, TensionGrad, ...), declare the domain of validity and the approximation level approx.
• P91-4 Window alignment: any windowed observation must state its window/domain using avg_t / avg_V / avg_gamma.

IV. Use Case A: Path–Arrival-Time Parameter Chain (S20- / I30- / I20-4)

1. Scenario and objective
   • Given measured arrival times T_obs and a path family gamma = ⋃_k gamma_k with segment lengths L_k = ∫_{gamma_k} 1 d ell.
   • Objective: estimate segmentwise effective indices theta = { n_eff_k } and a shared constant c_ref, and quantify uncertainty.
2. Modeling and binding
   • Discrete form (Minimal Equation S90-1)
     T_arr(theta) = Σ_k ( ( n_eff_k / c_ref ) * L_k ), with n_eff(ell) ≈ n_eff_k, ell ∈ gamma_k.
   • Derivatives (sensitivities, S90-2)
     ∂ T_arr / ∂ n_eff_k = L_k / c_ref；∂ T_arr / ∂ c_ref = - ( Σ_k n_eff_k * L_k ) / ( c_ref^2 )。
3. Parameters and priors
   • Registration: register_param("ref speed","c_ref","scalar","[L][T]^-1","physical",...); register_param("segment n_eff_k",...) for each segment k.
   • Bounds: n_eff_k ≥ 1.0, c_ref > 0; transforms: n_eff_k via log, c_ref via softplus.
   • Priors: prior(n_eff_k) = LogNormal(mu_k, sigma_k); prior(c_ref) = LogNormal(mu_c, sigma_c).
4. Likelihood and inference
   • Noise model: T_obs = T_arr(theta) + ε, ε ~ Normal(0, σ_T).
   • Likelihood: L(data | theta) = ∏_j Normal( T_obs^j | T_arr^j(theta), σ_T ).
   • Inference flow:
     1. infer_mle for initialization;
     2. infer_map to incorporate priors;
     3. posterior_sample_mcmc(..., method="NUTS") for posterior samples.
5. Validation and propagation
   • Dimensionality: check_dim("( n_eff / c_ref ) * d ell") must pass before comparison.
   • Arrival-time propagation: propagate_uncertainty_mc(model=T_arr, prior_spec, n=10^4); report CI_{1-α}[T_arr].
   • Regression: compare_param_sets to audit theta updates; compare_solutions(..., metrics=["T_arr"]) against T_obs.
6. Interface checklist (script skeleton)
   • discretize_path(gamma, scheme="piecewise", h=...) → {L_k}
   • register_param / set_bounds / set_prior / set_transform
   • infer_mle / infer_map / posterior_sample_mcmc
   • compute_jacobian(eqn="S20-*", params=["n_eff_k","c_ref"])
   • propagate_uncertainty_mc / export_params

V. Use Case B: Constitutive Coupling from Tension Field → Effective Index (S40- / S70- / S20-*)

1. Scenario and objective
   With known boundaries and source S_src(x,t), solve T_fil(x,t) via S40-*; define n_eff def= F_map(T_fil, TensionGrad) and jointly calibrate constitutive parameters with arrival-time data.
2. Constitutive family and parameters
   • Choose a family (example; declare approximation level approx=1)
     n_eff(x) approx= n0 * ( 1 + a_T * T_fil(x) + a_G * |grad[T_fil](x)| )。
   • Parameters: theta = { n0, a_T, a_G }; bounds n0 ≥ 1, a_T, a_G free in sign.
3. Weak solve and chain-rule sensitivity
   • Weak form (notation inner_V[•,•])
     find T_fil ∈ V : inner_V( grad[T_fil], grad[v] ) = inner_V( S_src, v ) + BC terms, ∀ v ∈ V0。
   • Chain rule
     ∂ T_arr / ∂ theta_i = ∫_gamma ( ( ∂ n_eff / ∂ theta_i ) / c_ref ) d ell，
     where ∂ n_eff / ∂ a_T = n0 * T_fil, ∂ n_eff / ∂ a_G = n0 * |grad[T_fil]|, ∂ n_eff / ∂ n0 = 1 + a_T T_fil + a_G |grad[T_fil]|。
     Implementation: compute_jacobian(eqn="S20-* ∘ S40-*", params=["n0","a_T","a_G"]).
4. Observations and likelihood
   • Option A: use T_arr data (same noise model as Use Case A).
   • Option B: if local n_eff observations exist, add L_local = ∏ Normal( n_eff_obs | n_eff(theta), σ_n ).
   • Joint likelihood: L = L_Tarr * L_local (conditionally independent).
5. Steps (end-to-end)
   • register_param for {n0,a_T,a_G,c_ref}; set priors and transforms.
   • assemble_operator / solve_* for T_fil; use S70-* weak assembly if needed.
   • Line integral: avg_gamma[n_eff / c_ref] to obtain T_arr(theta).
   • infer_map or posterior_sample_mcmc to calibrate theta.
   • propagate_uncertainty_mc to report CI_{1-α}[n_eff(x)] and path CI_{1-α}[T_arr].
6. Consistency checks
   check_dim("n_eff / c_ref"), path declaration gamma(ell), and window alignment (if T_fil is time-averaged, synchronize with avg_t[•; Δt]).

VI. Use Case C: Continuity–Flux with Windowed Observations (S50- / S80-)

1. Scenario and objective
   Observations are time–volume windowed flux J or rate of density change. The goal is to identify transport parameters and co-constrain with T_arr.
2. Observation model (S90-3)
   Y_obs def= avg_t[ avg_V[ g(rho,J); V=Ω ] ; Δt ] + ε, where g(•) is the measurement operator (e.g., div[J] or boundary average of J•n_hat), and ε is observation noise.
3. Binding steps
   • On the I30 side, register transport parameters (e.g., diffusivity D, driving coefficient k_T); set bounds/prior/transform.
   • On the equation side, assemble/solve rho,J via S50-*, then form the model output Y(theta) using S80-* (avg_t / avg_V).
   • Build the likelihood L(Y_obs | theta) and run infer_map or sampling.
   • If co-constraining with T_arr, use the joint likelihood and, via compute_jacobian, check the condition number of coupled parameters; apply decorrelation strategies (e.g., regularize_cov) as suggested in Chapter 5 if needed.
4. Validation essentials
   Declare the window Δt and volume V; verify ( n_eff / c_ref ) * d ell is dimensionless and g(rho,J) is dimensionally closed.

VII. Executable Script Checklists (Minimal Workflows Mx-5A/B/C)

• Mx-5A (arrival-time chain)
  discretize_path → register_param → set_bounds/prior/transform → infer_mle → infer_map → posterior_sample_mcmc → propagate_uncertainty_mc → export_params
• Mx-5B (constitutive–tension field)
  register_param{n0,a_T,a_G,c_ref} → assemble_operator/solve_* (S40-*) → line integral (S20-*) → compute_jacobian → infer_map/posterior_sample_mcmc → propagate_uncertainty_mc
• Mx-5C (continuity–window)
  register_param{transport} → solve_* (S50-*) → avg_V/avg_t (S80-*) → infer_map → regularize_cov → export_params

VIII. Binding Quality Gates and Regression

• Quality gates
  validate_param_set passes; all check_dim evaluations pass; compare_solutions(...,"T_arr") error below the scenario threshold; Corr meets the baseline requirement.
• Regression
  export_params("yaml") to create the baseline artifact; monitor parameter drift with compare_param_sets; trigger bump_version and scenario audit as needed (see Chapter 8).

IX. Misuse and Lint Rules

• Do not write ∫ n d ell / c (missing parentheses and mixing n with n_eff); write ∫ ( n_eff / c_ref ) d ell.
• Do not use alias in artifacts in place of the canonical code; exports must always write back the canonical.
• Any windowed/path expression without explicit gamma(ell), d ell, Δt, or V is not admitted to the registry.
• Entries lacking the constitutive mapping’s declared validity range and approx level must not be released as MINOR/PATCH.

X. Output Anchors and Numbering

• Binding postulates: P91-1…P91-4.
• Minimal equations: S90-1 (piecewise discrete arrival time), S90-2 (arrival-time gradient), S90-3 (windowed observation model).
• Workflow IDs: Mx-5A/B/C (three end-to-end use cases).
• Interfaces: relevant function prototypes in I30 1…12 and I20 1…5.