02-EFT.WP.Core.Equations v1.1 | Chapter 2 — Minimal Equations for Paths and Arrival Time

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Chapter 2 — Minimal Equations for Paths and Arrival Time

I. Chapter Goals and Scope

• Establish a unified minimal equation for arrival time, the criterion for factoring out constants, the additivity of concatenated paths, and reparameterization invariance; provide discretization practices and alignment with implementation-binding interfaces.
• All formulas, symbols, and definitions in this chapter are English plain text. Wherever division, integration, or composite operators appear, use parentheses and state gamma(ell) and d ell explicitly.

II. Symbols and Preliminaries

• Path and measure: gamma(ell) is the arc-length–parameterized integration path, ell ∈ [0, L_gamma], d ell is the path measure, and L_gamma = ( ∫_gamma 1 d ell ).
• Medium convention: n_eff(x,t) is the effective refractive index (dimensionless). c_ref > 0 is the reference upper bound of propagation with dimension [L/T], and may be a constant or a slowly varying function c_ref(x,t).
• Dimensional closure: dim( ( n_eff / c_ref ) * d ell ) = [T], hence dim( T_arr ) = [T].
• Conflict-name constraints: strictly distinguish n and n_eff; do not mix T_fil with T_trans; the bare symbols "c", "T", "n" are forbidden.

III. Minimal Equation Definitions (S20 Series)

1. S20-1 (General path–arrival form)
   T_arr def= ( ∫_gamma ( n_eff / c_ref ) d ell )
   • domain: gamma(ell) piecewise smooth; n_eff ≥ 0 almost everywhere; c_ref > 0.
   • notes: applicable when n_eff(x,t) and c_ref(x,t) vary in space and time.
2. S20-2 (Constant-factoring form)
   T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )
   • iff: c_ref is constant along gamma (e.g., depends only on a global reference, independent of x,t).
   • criterion: use when ∂_ell c_ref = 0 holds along gamma.
3. S20-3 (Additivity under path concatenation)
   If gamma = gamma_1 ⊕ gamma_2 (junction geometrically continuous, measures consistent), then
   T_arr[gamma] = T_arr[gamma_1] + T_arr[gamma_2]。
4. S20-4 (Reparameterization invariance)
   For any strictly monotone change of parameter ell' = phi(ell),
   T_arr = ( ∫gamma ( n_eff / c_ref ) d ell ) = ( ∫{gamma'} ( n_eff / c_ref ) d ell' )。
5. S20-5 (Discrete summation approximation)
   error: order O(h^p) according to the discretization scheme scheme ∈ {trap, simpson, ...}.On node sequence {ell_k} with segment lengths ds[k],
   T_arr approx ( Σ_k ( n_eff[k] / c_ref[k] ) * ds[k] )。
6. S20-6 (Non-negativity and zero condition)
   T_arr ≥ 0; moreover, T_arr = 0 iff L_gamma = 0 or n_eff = 0 almost everywhere.
7. S20-7 (Sensitivities: discrete form)
   ∂ T_arr / ∂ n_eff[k] = ( ds[k] / c_ref[k] )；∂ T_arr / ∂ c_ref[k] = ( - n_eff[k] * ds[k] / ( c_ref[k]^2 ) )。

IV. Proof and Verification Highlights (Mandatory Conventions)

1. Dimensional-closure check:
   check_dim( ( n_eff / c_ref ) * d ell ) -> [T]；check_dim( T_arr ) -> [T]。
2. Additivity:
   • gamma = gamma_1 ⊕ gamma_2；
   • Lebesgue integrals are linear, with the common measure d ell;
   • Therefore S20-3 follows.
3. Reparameterization invariance:
   • d ell' = ( d ell / d ell' )^{-1} d ell；
   • Substitute variables in the integral to obtain S20-4.
4. Necessity and sufficiency for constant factoring:
   • If ∂_ell c_ref = 0, factor out the constant to get S20-2;
   • Conversely, if S20-2 holds for arbitrary n_eff, then c_ref cannot depend on ell and is constant along gamma.

V. Path Segmentation and Reference Path

• Segmentation: gamma = ⋃_{k=1}^K gamma_k, each segment satisfying geometric continuity and measure consistency.
• Reference path: gamma_ref(ell) is used for baseline comparison and regression testing, with
  T_arr_ref def= ( ∫_{gamma_ref} ( n_eff / c_ref ) d ell )。
• Differential arrival time:
  Δ T_arr = T_arr[gamma] - T_arr[gamma_ref] = ( ∫ ( ( n_eff / c_ref )_gamma - ( n_eff / c_ref )_ref ) d ell )。

VI. Coupling and Boundary Alignment (with the Mother Form)

• Observation definition: J_T[u] def= T_arr[gamma; n_eff(u), c_ref]；
• In the weak form, treat its first variation with respect to u via inner_V[•,•];
• Statistical conventions must be explicit: avg_gamma[•] or avg_t[•; Δt].

VII. Lint Rules and Forbidden Patterns

• ∫ n d ell / c (missing parentheses and non-canonical symbols);
• Omitting path or measure (e.g., ∫ n_eff / c_ref);
• Bare "c", "T", "n";
• Interchanging n_eff and n.

VIII. Implementation Binding and Interface Alignment

1. With I20-4:
   • propagate_time(n_eff_path:list[float], ds:list[float], c_ref:float) -> float
   • Semantics aligned with S20-2; if S20-1 with spatially varying c_ref[k] is needed, extend the implementation to segment-wise c_ref[k].
2. With I20-2:
   • discretize_path(gamma, scheme, h) -> {nodes, ds} to produce {ds[k]}；
   • Record the error order O(h^p) consistent with scheme in the equation metadata notes.
3. With regression:
   • compare_solutions(x,y,metrics=["L2","L_inf","T_arr"]) -> dict；
   • The T_arr metric uses gamma_ref as the baseline.

IX. Typical Use Cases (Consistency and Edge Conditions)

• Case A (constant c_ref, piecewise-constant n_eff):
  T_arr = ( 1 / c_ref ) * ( Σ_k n_eff[k] * ds[k] )。
• Case B (varying c_ref with smooth n_eff):
  T_arr approx ( Σ_k ( n_eff[k] / c_ref[k] ) * ds[k] )；increase order via scheme="simpson"。
• Case C (zero path or transparent segment):
  If L_gamma = 0 or n_eff = 0 almost everywhere, then T_arr = 0 (see S20-6).

X. Equation Cards (Summary List)

• S20-1 — T_arr def= ( ∫_gamma ( n_eff / c_ref ) d ell )
• S20-2 — T_arr = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )（iff ∂_ell c_ref = 0）
• S20-3 — T_arr[gamma_1 ⊕ gamma_2] = T_arr[gamma_1] + T_arr[gamma_2]
• S20-4 — Reparameterization invariance
• S20-5 — Discrete summation approximation
• S20-6 — Non-negativity and zero condition
• S20-7 — Discrete sensitivity formulas

XI. Pre-Release Checklist

• Are gamma(ell), d ell, and L_gamma all explicit?
• Does check_dim confirm dim( T_arr ) = [T]?
• If using S20-2, does ∂_ell c_ref = 0 hold?
• For discretization, are scheme and the error order p documented?
• Have all forbidden patterns been avoided and validate_equation lint rules passed?
• Are interface fields and semantics fully aligned with I20-*?