01-EFT.WP.Core.Terms v1.0 | Chapter 3, Fields & Paths

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Chapter 3, Fields & Paths

I. Chapter Goals & Scope

• Specify the normative definitions, domains, and units for the field quantities T_fil(x,t), rho(x,t), n(x,t), n_eff(x,t), p(x,t), and enforce a consistent pairing with the path gamma(ell) and the measure d ell.
• Unify time-of-arrival expressions and path-related operators by providing reusable templates and validation checkpoints, in compliance with P10-1, P10-2, P10-3, P10-4 (see this volume, Chapter 1).

II. Fundamental Fields (Unified Entries)

1. Tension field — T_fil(x,t)
   • Definition (def=) T_fil(x,t) def= the intrinsic tension field of the medium-weave system.
   • Domain and regularity: x ∈ R^3, t ∈ R; assume piecewise continuously differentiable. TensionGrad = grad[T_fil] and TensionPot mappings are given in Core.Equations.
   • Units and dimensions: see Core.Metrology; consistency via check_dim(expr).
   • Constraints and conflicts: must not be confused with T_trans (see P10-4).
2. Density field — rho(x,t)
   • Definition (def=) rho(x,t) def= a field describing matter/energy distribution; its statistics act as sources or constraints for path- and gravity-like effects.
   • Domain and statistics: rho(x,t) ≥ 0; define avg_t[rho; Δt], avg_V[rho] as needed.
   • Units and dimensions: [M L^-3]; registration and checks per I10-3.
   • Constraint: do not conflate with n_eff(x,t).
3. Number density — n(x,t)
   • Definition (def=) n(x,t) def= the particle number density not to be confused with effective refractive index.
   • Constraint: in any time-of-arrival expression, n must never substitute for n_eff (see P10-4).
4. Effective refractive index — n_eff(x,t)
   • Definition (def=) n_eff(x,t) def= the effective refractive index governing propagation along a path, paired with reference limit c_ref.
   • Time-of-arrival conventions:
     1. Constant factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
     2. General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Constraint: any line integral must make gamma(ell) and d ell explicit (see P10-2).
5. Orientation — p(x,t)
   • Definition (def=) p(x,t) def= a unit-norm orientation vector field associated with local weave alignment.
   • Constraint: |p(x,t)| = 1; required field for Thread (see Chapter 2).

III. Path & Measure (Canonical Definitions)

1. Path — gamma(ell)
   • Definition (def=) gamma(ell) def= a piecewise C1 curve in R^3 parameterized by arc length ell ∈ [0, L_gamma].
   • Concatenation and reversal: gamma = gamma_1 ∘ gamma_2 denotes end-to-end concatenation; gamma^-1(ell) = gamma(L_gamma - ell).
   • Reparameterization invariance: if ell = s(u) is monotone, then ∫_gamma f d ell = ∫ f(gamma(u)) (d s/ d u) d u, preserving the value.
2. Measure — d ell
   • Definition (def=) d ell def= the line-element measure induced by arc length along gamma.
   • Piecewise expression: if gamma = ⋃_k gamma_k, then ∫_gamma f d ell = Σ_k ( ∫_{gamma_k} f d ell ).
3. Path length — L_gamma
   • Definition (def=) L_gamma def= ∫_gamma 1 d ell.
   • Path average: Definition (def=) avg_gamma[f] def= ( 1 / L_gamma ) * ( ∫_gamma f d ell ).

IV. Time-of-Arrival Expressions (Unified Templates)

1. Standard definition:
   • Definition (def=) T_arr(gamma) def= ( ∫_gamma ( n_eff / c_ref ) d ell ).
   • Constant-factored form: T_arr(gamma) = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell ).
2. Concatenation property: if gamma = gamma_1 ∘ gamma_2, then
   Identity T_arr(gamma) = T_arr(gamma_1) + T_arr(gamma_2).
3. Differentials and perturbations: for a small perturbation tilde_n_eff, first-order approximation
   Approximation tilde_T_arr(gamma) approx ( ∫_gamma ( tilde_n_eff / c_ref ) d ell ).
4. Applicability boundaries: the choice and drift calibration of c_ref are specified in Core.Metrology; mappings n_eff ↔ TensionPot/TensionGrad and their validity domains are specified in Core.Equations.

V. Field–Path Compatibility Rules (Mandatory)

• Explicit path: whenever ∫ (...) d ell appears, specify gamma(ell) and d ell (see P10-2).
• Symbol isolation: only n_eff may appear under the integrand for time-of-arrival; n, rho, etc., must not substitute.
• Dimensional conservation: [T_arr] = [T]; use check_dim(expr) to verify the dimension of n_eff/c_ref is [T L^-1], canceling with [L] from d ell.
• Statistical conventions: time or volume averages must specify window and domain, e.g., avg_t[n_eff; Δt], avg_V[rho]; along-path averages use avg_gamma[•].
• Conflict names: do not mix T_fil with T_trans; do not mix n with n_eff (see P10-4).

VI. Typical Expression Templates (Directly Reusable)

• Path-averaged effective refractive index:
  avg_gamma[n_eff] = ( 1 / L_gamma ) * ( ∫_gamma n_eff d ell )
• Arrival time over piecewise paths:
  T_arr = Σ_k ( ∫_{gamma_k} ( n_eff / c_ref ) d ell )
• Near-field tension–path coupling placeholder (mapping provided elsewhere):
  n_eff(x,t) def= F( T_fil(x,t), TensionGrad(x,t), ... ) (form of F specified in Core.Equations).

VII. Common Misuses & Corrections

• Misuse: T_arr = ∫ n_eff d ell / c
  Correction: T_arr = ( ∫ ( n_eff / c_ref ) d ell ) (parentheses complete, c_ref explicit).
• Misuse: ∫ n_eff dl (unnamed path, improper measure)
  Correction: ∫_gamma n_eff d ell, and define gamma(ell) and L_gamma in context.
• Misuse: T_arr = ( ∫ ( n / c_ref ) d ell )
  Correction: T_arr = ( ∫ ( n_eff / c_ref ) d ell ) (n must not replace n_eff).

VIII. Interfaces for Implementation Binding

• Expression validation: use validate_expr(expr:str, allowed:set) -> bool to ensure only permitted identifiers and operators are present (see I10-4).
• Time-of-arrival computation: arrival_time(n_eff_path:list, ds:list, c_ref:float) -> float implements discrete evaluation of T_arr(gamma) (see I10-5).
• Unit registration: register_unit(...) and check_dim(expr) enforce dimensional consistency (see I10-3).

IX. Quick Checklist

• Are all line integrals explicit in gamma(ell) and d ell, with parentheses used correctly?
• Is n_eff—rather than n—used under the time-of-arrival integrand, with c_ref stated explicitly?
• Is L_gamma provided, together with any necessary piecewise or concatenation statements?
• Has check_dim verified dimensional consistency?
• Are T_fil/T_trans and n/n_eff never interchanged, in full compliance with P10-4?

Chapter Summary

This chapter, using unified terminology and path conventions, standardizes the definitions of T_fil(x,t), rho(x,t), n_eff(x,t) and the path/measure pair gamma(ell), d ell; it provides the canonical expressions and templates for T_arr(gamma), and clarifies dimensional checks, statistical treatments, and implementation interfaces. Subsequent chapters build on this foundation to specify constants and baselines, operators and statistics, and the full rules for units and non-dimensionalization.