30-EFT.WP.Propagation.TensionPotential v1.0 | Appendix A — Symbols, Units & Dimensional Checks

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Appendix A — Symbols, Units & Dimensional Checks

• I. One-Sentence Aim
  Provide a unified notation and units/dimensions convention for all symbols in this volume, plus an executable dimensional-check workflow and guardrails for common pitfalls, ensuring that T_arr, n_eff, Phi_T, and derived quantities are dimensionally self-consistent under both arrival-time gauges.
• II. Applicability & Non-Goals
• Covered: base dimensions, SI unit guidance, symbol–dimension mapping, dimensional checks for both gauges, dimensions of n_eff construction terms and parameters, interface/jump terms, uncertainty-symbol dimensions, automated checks and examples.
• Not covered: device-level unit constraints; replacements for Chapter 7 uncertainty-propagation methods.
• III. Base Dimensions & SI Unit Norms
• Base dimensions: [L] (length), [T] (time), [M] (mass). Angle may appear but is dimensionless by default (radians).
• SI units: m, s, kg. Frequency f in Hz = s^-1. Path length and line element use meters.
• Unified convention: wrap inline symbols in backticks; parenthesize division and composite operators.
• IV. Fixed Symbols & Corresponding Dimensions (table-free)
• Space, time & path
• x: position, dim(x) = [L]
• t: time, dim(t) = [T]
• ell: path parameter, dim(ell) = [L]
• gamma(ell): path mapping, dim(gamma(ell)) = [L]
• d ell: line element, dim(d ell) = [L]
• Arrival time & speeds
• T_arr: arrival time, dim(T_arr) = [T]
• c_ref: reference speed, dim(c_ref) = [L][T^-1]
• c_loc(x,t,f) = c_ref / n_eff, dim(c_loc) = [L][T^-1]
• Index & decomposition
• n_eff(x,t,f): effective refractive index, dim(n_eff) = 1
• n_common(x,t): common term, dim = 1
• n_path(x,t,f): path/band term, dim = 1
• Potential & gradient
• Phi_T(x,t): tension potential, dim(Phi_T) set by gauge; recommended interface non-dimensionalization Phi_T_tilde = Phi_T / Phi_ref
• grad_Phi_T = grad(Phi_T), dim(grad_Phi_T) = dim(Phi_T)[L^-1]
• Tension & density (if used in mapping)
• T_fil(x,t): tension field, dim(T_fil) per material/medium spec
• rho(x,t): density, dim(rho) = [M][L^-3]
• Interface-related
• C_sigma: jump of Phi_T, dim(C_sigma) = dim(Phi_T)
• J_sigma = dot( grad_Phi_T^+ − grad_Phi_T^- , n_vec ), dim(J_sigma) = dim(grad_Phi_T)
• R_sigma, T_trans, A_sigma: reflection, transmission, loss coefficients, dim = 1
• Directional quantity
• t_hat: unit tangent along the path, dim = 1
• V. Dimensional Checks for the Two Arrival-Time Gauges (continuous & discrete)
• Constant-factored gauge
• Continuous: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
  Check: ∫ n_eff d ell → [L]; multiply by 1/c_ref → [T].
• Discrete: T_arr ≈ (1/c_ref) * ∑ n_eff[ gamma[k] ] · Δell[k]
  Check: Δell[k] → [L]; total is [T].
• General gauge
• Continuous: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
  Check: (n_eff/c_ref) → [T][L^-1]; times d ell → [T].
• Discrete: T_arr ≈ ∑ ( n_eff[ gamma[k] ] / c_ref[ gamma[k] ] ) · Δell[k] → [T]
• Lower-bound consistency: because n_eff ≥ 1, always T_arr ≥ L_path / c_ref (embedded equivalently in the general gauge integrand).
• VI. Dimensions in n_eff Construction & Parameters (enforce dim(n_eff) = 1)
• Base mapping: n_eff = F( Phi_T, grad_Phi_T, rho, f ), dim(n_eff) = 1
• Isotropic small-gradient expansion (Chapter 5 S20-28)
• n_eff ≈ a0 + a1 · ( Phi_T − Phi_0 ) + a2 · norm( grad_Phi_T )^2
• dim(a0) = 1
• dim(a1) = 1 / dim(Phi_T)
• dim(a2) = 1 / dim(grad_Phi_T)^2 = [L^2] / dim(Phi_T)^2
• Directional extension (Chapter 5 S20-29)
• n_eff ≈ … + b1 · dot( grad_Phi_T , t_hat )
• dim(b1) = 1 / dim(grad_Phi_T) = [L] / dim(Phi_T)
• Band polynomial (Chapter 5 S20-30)
• n_path ≈ ∑_{m=1}^M c_m(x,t) · ( f − f0 )^m
• dim( (f − f0)^m ) = [T^-m], hence dim(c_m) = [T^m]
• If using normalized frequency offset ((f − f0)/f_ref)^m, then dim(c_m) = 1.
• VII. Interface & Inter-Layer Dimensional Checks
• Continuous interface: Phi_T^+ = Phi_T^-, J_sigma = 0 ⇒ if n_eff = F( Phi_T, grad_Phi_T, … ) is continuous, then n_eff^+ = n_eff^- (dimensionally consistent).
• Potential-jump interface: Phi_T^+ − Phi_T^- = C_sigma, dim(C_sigma) = dim(Phi_T); impact on n_eff propagates through coefficients a1, b1 by their dimensions.
• Flux-jump interface: J_sigma ≠ 0, dim(J_sigma) = dim(grad_Phi_T); if a normal-response coefficient k_sigma appears in n_eff, then dim(k_sigma) = 1 / dim(grad_Phi_T).
• VIII. Dimensions of Errors & Uncertainties
• Rule: every uncertainty u(q) carries the same dimensions as q; combined u_c(T_arr) has [T].
• Typical items: u(c_ref) → [L][T^-1]; u(n_eff) → 1; u(Δell) → [L]; u(T_arr) → [T].
• Reporting: mean ± k·u_c keeps [T]; differentials ΔT_arr(f1,f2) also have [T].
• IX. Unit Mapping & Coordinate Transforms (implementation contract)
• units_spec: declare length m, time s, speed m·s^-1, frequency Hz; convert inputs in km or ms at ingress.
• coords_spec: declare coordinate frame (Cartesian/ECEF/ENU); any transform must preserve the units of Δell and update hashes.
• Path constraints: gamma[k] and Δell[k] share units; t_hat[k] is dimensionless; interface coordinates share the path’s frame.
• X. Automated Dimension-Check Workflow (recommended check_dimension order)
• Contract intake: read units_spec, coords_spec; confirm SI or stated mappings.
• Symbol binding: fix dim(c_ref)=[L][T^-1], dim(n_eff)=1, dim(d ell)=[L], dim(T_arr)=[T].
• Equation checks:
• Constant-factored: verify dim( ∫ n_eff d ell / c_ref ) = [T].
• General: verify dim( ∫ (n_eff/c_ref) d ell ) = [T].
• Constructor checks: infer dimensions for a1,a2,b1,c_m per Section VI; verify alignment with configured scales.
• Interface checks: confirm C_sigma, J_sigma dimensions and their effect pathways into n_eff.
• Discrete implementation: ensure units of Δell[k], c_ref[k] are consistent; interpolation must not alter dimensions.
• Report: emit DimReport (see Section XII) and write to logs.
• XI. Common Pitfalls & Safeguards
• Unit mixing: path in km while c_ref in m·s^-1 → T_arr off by 10^3. Safeguard: normalize at ingress and record.
• Angle units: passing degrees as radians. Safeguard: declare radians for trig inputs; convert if needed.
• Missing parentheses: ∫ n_eff / c_ref d ell vs ∫ (n_eff/c_ref) d ell differ. Safeguard: enforce parentheses.
• Parameter mis-dimensioning: c_m not labeled [T^m]. Safeguard: infer and validate at model assembly.
• Interface mismatch: C_sigma stored as dimensionless. Safeguard: correct to dim(Phi_T) or reject.
• XII. Minimal DimReport Fields (output recommendation)
• units_spec, coords_spec, mode ∈ {constant, general}
• ok(formula_S20-4), ok(lower_bound), ok(discrete_forms)
• dims: { c_ref:[L T^-1], n_eff:1, d_ell:[L], T_arr:[T], Phi_T:?, grad_Phi_T:?·[L^-1] }
• params_dims: { a0:1, a1:1/dim(Phi_T), a2:[L^2]/dim(Phi_T)^2, b1:[L]/dim(Phi_T), c_m:[T^m] }
• interfaces: { C_sigma:dim(Phi_T), J_sigma:dim(grad_Phi_T) }
• discrete_consistency: { Δell_unit:ok, c_ref_unit:ok, interpolation:ok }
• notes: free text for anomalies and auto-fix explanations
• hashes: hash(Phi_T), hash(gamma), hash(code)
• XIII. Worked Checks (three examples)
• Example 1 | Uniform medium
• Condition: n_eff ≡ 1, arbitrary path.
• Conclusion: T_arr = L_path / c_ref, dimension [L]/([L][T^-1]) = [T]; lower bound is tight.
• Example 2 | First-order band dispersion
• Condition: n_path ≈ c_1 (f − f0)
• Dimensions: dim(c_1) = [T], dim(ΔT_arr) = [T]; both gauges agree.
• Example 3 | Directional term
• Condition: n_eff ≈ a0 + b1 · dot( grad_Phi_T , t_hat )
• Dimensions: dim(b1) = [L]/dim(Phi_T); if Phi_T is non-dimensionalized, dim(b1) = [L].
• XIV. Cross-References
• EFT.WP.Propagation.TensionPotential v1.0 Chapter 3 (minimal equations), Chapter 5 (n_eff construction), Chapter 6 (two gauges), Chapter 8 (interfaces), Chapter 12 (errors)
• EFT.WP.Core.Equations v1.1 S06-* (notation & operators)
• EFT.WP.Core.Metrology v1.0 M05-, M10- (units & traceability)
• XV. Deliverables
• Dimension–unit binding checklist (ready to embed in the Contract).
• Execution order and decision rules for check_dimension.
• Sample DimReport and log-field mapping.