01-EFT.WP.Core.Terms v1.0 | Chapter 5, Operators & Statistics

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Chapter 5, Operators & Statistics

I. Chapter Objectives & Boundaries

• Fix the notations for space-, time-, and path-related operators and statistical operators so they are directly reusable across volumes and amenable to machine validation.
• This chapter defines only operator domains/ranges, composition rules, and statistical conventions. Physical equations appear in Core.Equations; metrology workflows in Core.Metrology; falsification and power in Methods.Falsification.

II. Whitelisted Operators and Domains/Ranges (Unified Interface)

1. Spatial operators
   • grad[f] def= spatial gradient; dom: f: R^3→R → rng: R^3
   • div[F] def= spatial divergence; dom: F: R^3→R^3 → rng: R
   • curl[F] def= spatial curl; dom: F: R^3→R^3 → rng: R^3
   • lap[f] def= Laplacian; dom: f: R^3→R → rng: R
2. Path operators
   • D_ell f def= directional derivative along gamma(ell); D_ell f def= d f(gamma(ell)) / d ell
   • avg_gamma[f] def= ( 1 / L_gamma ) * ( ∫_gamma f d ell )
3. Statistical operators
   • avg_t[f; Δt] def= time-windowed mean over window Δt
   • avg_V[f; V=Ω] def= volume mean over region Ω
   • var_t[f; Δt] def= time-windowed variance; var_t[f; Δt] def= avg_t[(f - avg_t[f; Δt])^2; Δt]
   • cov_t[f,g; Δt] def= time-windowed covariance; cov_t[f,g; Δt] def= avg_t[(f - avg_t[f; Δt]) (g - avg_t[g; Δt]); Δt]
   • xcorr_t[f,g; τ; Δt] def= time-windowed cross-correlation at lag τ
4. Notation & relations
   • Definition sign: def=; approximation: approx; dimensional similarity: sim.
   • Expectation/probability symbols are kept as alias mappings: add_alias(canonical="avg_t", alias="<•>_t") (use avg_t in main text).

III. Spatial Operators (Rules & Examples)

1. Linearity and product rules
   • grad[a f + b g] = a grad[f] + b grad[g]
   • div[a F + b G] = a div[F] + b div[G]
   • grad[f g] = g grad[f] + f grad[g]
   • div[f F] = grad[f] • F + f div[F]
2. Compositions and identities
   • curl[grad[f]] = 0
   • div[curl[F]] = 0
   • lap[f] = div[grad[f]]
3. Domain/regularity requirements
   f is at least piecewise C^1 to define grad; F is at least piecewise C^1 to define div, curl; lap[f] requires f ∈ C^2.
4. Interaction with an orientation field
   • If p(x,t) is a unit-norm orientation, define parallel/orthogonal components: F_parallel def= (F • p) p, F_perp def= F - F_parallel.
   • Use the parallel/orthogonal decomposition only when p(x,t) is declared and |p|=1 (see this volume, Chapter 2).

IV. Path Operators & Time-of-Arrival Coupling

1. Path derivative
   D_ell f def= d f(gamma(ell)) / d ell = gradf • d gamma / d ell
2. Path average and piecewise paths
   • avg_gamma[f] = ( 1 / L_gamma ) * ( ∫_gamma f d ell )
   • If gamma = ⋃_k gamma_k, then ∫gamma f d ell = Σ_k ( ∫{gamma_k} f d ell )
3. Unified time-of-arrival (reference)
   • T_arr(gamma) def= ( ∫_gamma ( n_eff / c_ref ) d ell )
   • Constant factored: T_arr(gamma) = ( 1 / c_ref ) * ( ∫_gamma n_eff d ell )
   • Constraint: only n_eff is allowed in the integrand for arrival-time expressions; n, rho, etc., must not substitute; make gamma(ell) and d ell explicit (see this volume, Chapter 3).

V. Time and Volume Statistics (Windows, Weights, and Missingness)

1. Windows and normalization
   • Define a weight w_t(τ) with ∫ w_t(τ) d τ = 1; then avg_t[f; w_t] def= ∫ f(t-τ) w_t(τ) d τ
   • Uniform-window special case: w_t(τ) = 1/Δt for τ ∈ [0, Δt] else 0
2. Discretization mappings
   • Sampling {t_k}: avg_t[f; Δt] approx ( Σ_k f(t_k) Δt ) / ( Σ_k Δt )
   • Volume integral discretization: avg_V[f; Ω] approx ( Σ_j f(x_j) ΔV_j ) / ( Σ_j ΔV_j )
3. Missing data and masks
   • Define a mask m(t) ∈ {0,1}; avg_t[f; Δt, m] def= ( Σ_k m_k f(t_k) Δt ) / ( Σ_k m_k Δt )
   • Variance is computed analogously with mask weights.
4. Cross statistics
   • cov_t[f,g; Δt] = avg_t[f g; Δt] - avg_t[f; Δt] avg_t[g; Δt]
   • xcorr_t[f,g; τ; Δt] def= avg_t[f(t) g(t+τ); Δt]

VI. Commutativity & Exchange Conditions (Use with Care)

• With time averaging
  If grad[f] is approximately stationary over the window Δt, then
  avg_t[grad[f]; Δt] approx grad[avg_t[f; Δt]]; otherwise, the exchange is forbidden.
• With volume averaging
  If boundary fluxes vanish and f is sufficiently smooth inside Ω, then
  avg_V[div[F]; Ω] = 0 (the averaged form of the divergence theorem).
• With path averaging
  In general this does not hold: avg_gamma[grad[f]] is not equal to grad[avg_gamma[f]]; it only approximates when grad[f] is nearly constant along gamma.

VII. Dimensions & Units (Consistency Requirements)

1. Required checks
   • Use check_dim(expr) to verify, e.g., dim[( n_eff / c_ref ) * d ell] = [T^0]; after integration along the path, dim[T_arr] = [T].
   • Statistics preserve the dimension of the measured quantity: dim[avg_t[f]] = dim[f], dim[var_t[f]] = dim[f]^2.
2. Prohibitions
   Do not substitute n(x,t) into ( n_eff / c_ref ); and never overload T_fil to mean temperature or time in the same context (see P10-4).

VIII. Implementation Binding (I10-4 Validation & Whitelist)

1. Expression-validation API
   • validate_expr(expr:str, allowed:set[str]) -> bool
   • Recommended whitelist: { "+","-","*","/","^","(",")", "grad","div","curl","lap", "avg_t","avg_V","avg_gamma","D_ell" }
   • Permit only the operators above and the symbols defined in Chapters 2–4 of this volume: { T_fil, rho, n, n_eff, c_ref, p, gamma, d ell, L_gamma }.
2. Rendering API
   render_expr(expr:str, style="text") -> str outputs a plain-text expression for reports and cards.
3. Consistency examples
   • validate_expr("T_arr = ( ∫ ( n_eff / c_ref ) d ell )", allowed=...) -> True
   • validate_expr("T_arr = ∫ n d ell / c", allowed=...) -> False（uses forbidden symbols and lacks parentheses）

IX. Common Misuses & Corrections

• Misuse: ∫ n_eff dl (unnamed path and improper measure)
  Correction: ∫_gamma n_eff d ell, and define gamma(ell) and L_gamma.
• Misuse: avg_t[grad[f]] = grad[avg_t[f]] (unconditional exchange)
  Correction: only permissible as approx when grad[f] is approximately stationary.
• Misuse: T_arr = ( ∫ ( n / c_ref ) d ell )
  Correction: T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
• Misuse: using T_fil to stand in for transmission, e.g., T_fil ~ T_trans
  Correction: strictly distinguish T_fil and T_trans.

X. Quick Checklist

• Have you used only operators from this chapter’s whitelist and symbols already defined?
• Are all line integrals explicit in gamma(ell) and d ell, and compliant with parentheses and constant-factoring rules?
• Do means and variances specify windows and measures, and do discrete implementations provide weights or masks?
• Do all expressions pass check_dim, and do statistical outputs have correct dimensions?
• Have you avoided mixing T_fil/T_trans and n/n_eff, and any improper exchanges?

Chapter Summary

This chapter establishes unified notation, domains/ranges, and composition rules for spatial, path, and statistical operators; provides executable conventions for windowing and discretization; and aligns these with the time-of-arrival T_arr, dimensional checks, and the implementation-validation interfaces. Chapter 6 (Dimensions, Non-Dimensionalization & Units) builds on these results to deliver canonical scale mappings and the full unit-registration workflow.