13-EFT.WP.Methods.SimStack v1.0 | Chapter 2: Continuous-Domain Physical Kernel (Sea, Density, Tension)

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Chapter 2: Continuous-Domain Physical Kernel (Sea, Density, Tension)

I. Scope & Objectives

• Define the continuous-domain physical core (ContinuousKernel) with its minimal inputs/outputs, dimensional conventions, and validation criteria. Provide the two equivalent formulations for the path arrival time T_arr, lock conservation and normalization quality gates, and offer S62-* discretization guidance.
• The goal is interchangeability: any implementation can be swapped or composed without changing symbols or units, and will couple seamlessly to the discrete braiding layer and the coupler layer via stable interfaces and manifests.

II. Minimal I/O and Dimensions of the Kernel

1. Minimal inputs
   • n_eff(x,t), unitless.
   • c_ref, units m/s, the reference propagation ceiling.
   • Optional fields: rho(x,t) (mass density, kg/m^3), S_xx(f) (power spectral density, units per measurement convention).
2. Derived quantities & conventions
   • Local propagation ceiling: c_loc(x,t) = c_ref / n_eff(x,t).
   • Tension field: T_fil(x,t) is a medium property. When it contributes to n_eff, its coefficient and unit mapping must be explicit.
   • Paths and measures: path written as gamma(ell), line measure d ell, volume measure dV, surface measure dS.
3. Dimensional checks
   • dim( n_eff ) = 1.
   • dim( ( n_eff / c_ref ) * d ell ) = [T].
   • Any flux term entering a volume integral must explicitly specify the measure via dS or dV.

III. Two Formulations of T_arr and Path Dependence

1. Definitions and the two forms
   • General form: T_arr(gamma) = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ).
   • “Pull-out constant” form: T_arr(gamma) = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ).
2. Discrepancy metric and reporting
   • Unified discrepancy:
     delta_form = | ( ∫ ( n_eff / c_ref ) d ell ) - ( 1 / c_ref ) * ( ∫ n_eff d ell ) |.
   • Minimum manifest fields: {gamma, d ell, c_ref, n_eff, T_arr, delta_form}.
3. Path dependence and segmentation
   • Segmented paths: if gamma = ⋃_k gamma_k, then T_arr = Σ_k T_arr(gamma_k).
   • When n_eff varies with frequency or environment, declare n_eff(x,t; context) with its context parameters, and keep the ingest fields consistent.

IV. Conservation & Validation: eps_norm, eps_mass

1. Probability-density normalization (when modeled as a probability density)
   • Target: ( ∫_V p(x,t) dV ) = 1.
   • Normalization error: eps_norm(t) = | 1 - ( ∫_V p(x,t) dV ) |.
2. Physical-density conservation (when modeled as mass or energy density)
   • Continuity equation: ∂_t rho + ∇•J = Q - Λ.
   • Conservation residual (discrete time step Δt):
     eps_mass = | ( ∫_V rho(t+Δt) dV ) - ( ∫_V rho(t) dV ) - ( ∫_V ( Q - Λ ) dV ) * Δt + ( ∮_{∂V} J•n dS ) * Δt |.
3. Quality gates and release conditions
   • eps_norm ≤ ε_norm_gate, eps_mass ≤ ε_mass_gate.
   • If gates are not met, artifacts must not be released, nor advanced into coupling with the discrete layer.

V. S62-* Discretization Guidance

1. S62-1 Method of Lines
   • Procedure: discretize space first to obtain an ODE system, then advance in time with a stable stepper.
   • Requirements: boundary conditions explicit; measures and units traceable term-by-term on the discrete operators.
2. S62-2 Finite Volume (conservative form)
   • Procedure: control volumes and interface fluxes as the core, prioritizing conservation.
   • Requirements: approximate J via interface fluxes; time marching and flux evaluation must share the same time base.
3. S62-3 Numerical quadrature for path integrals
   • Procedure: write T_arr as a weighted sum Σ w_i * n_eff(ell_i) / c_ref.
   • Requirements: declare nodes ell_i and weights w_i explicitly, and record them in the manifest.
4. S62-4 Stable step size and CFL criterion
   • Condition: dt ≤ CFL * min( Δx / c_loc(x,t) ).
   • Requirements: when c_loc is time-varying, the step controller and monitor points must share the same estimate.

VI. Data & Manifest Conventions

• Path and time base
  Parameterization and support of gamma(ell), definition of d ell, mapping parameters between tau_mono and ts.
• Kernel and environment
  Source of n_eff, interpolation or extrapolation strategy, c_ref selection notes, and the data conventions for T_fil and rho.
• Quality and uncertainty
  eps_norm, eps_mass, delta_form, along with their estimation windows and confidence conventions.
• Release and traceability
  Every artifact must carry measure/units/support/norm, and provide a replayable audit trail.

VII. Verification & Test Matrix

1. Minimum required tests
   • Homogeneous medium: n_eff = n0; verify agreement of the two T_arr forms.
   • Segmented path: gamma = gamma_1 ⋃ gamma_2; verify additivity across segments.
   • Conservation case: zero source and zero sink; verify eps_mass ≈ 0.
2. Boundary and extreme conditions
   • Stability and order-of-accuracy tests under steep gradients.
   • Step-size contraction and alerting strategy where c_loc is minimal.
3. Regression and thresholds
   Fix reference datasets and a baseline version; any change must report Δeps_norm, Δeps_mass, Δdelta_form.

VIII. Cross-References & Dependencies

• Time bases and arrival time: align with Core.Sea, with explicit paths and measures.
• Measures and conservation: align units and conservation checks with Core.Density.
• Concurrency and observation: share TS.* indices and time-alignment procedures with Core.Threads for end-to-end monitoring under coupling.

IX. Risks, Limitations & Open Questions

1. Risks
   • Confusing n with n_eff yields dimensional errors in T_arr; all expressions must pass check_dim(expr).
   • Over-large time steps in strongly inhomogeneous media can break both conservation and stability.
2. Limitations
   This chapter specifies numerical conventions and manifests only, not the physical provenance. Physical composition is defined in the corresponding core volumes.
3. Open questions
   Nonlinear coupling between T_fil and n_eff under strong events: identifiability and robust solvers.

X. Deliverables & Versioning

1. Deliverables
   • Kernel prototype and reference implementation with switchable S62-* options.
   • Input conditions and expected outputs for benchmark cases, plus the quality-gate threshold file.
2. Versioning
   Freeze symbols and manifest fields from v1.0 onward; add new fields with backward compatibility and record migration guidance in an appendix.

XI. New Terms & Symbols (to memorize)

• Core quantities: n_eff(x,t), c_ref, c_loc(x,t), T_fil(x,t), rho(x,t), S_xx(f).
• Paths & arrival time: gamma(ell), d ell, T_arr, delta_form.
• Conservation & quality gates: J(x,t), Q(x,t), Λ(x,t), eps_norm, eps_mass.
• Discretization & step size: CFL, Δx, dt, method labels S62-1..S62-4.