09-EFT.WP.Core.Density v1.0 | Appendix A — Symbols and Units Crosswalk

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Appendix A — Symbols and Units Crosswalk

I. Usage Conventions and Reading Hints

• SI is the default unit system. Base-dimension symbols: M (mass), L (length), T (time), I (electric current), Theta (thermodynamic temperature), N (amount of substance), J (luminous intensity).
• If a variable lives in d dimensions, the volume element dV carries units m^d; density units adapt to d (e.g., m^3 in 3D, m^2 in 2D, m in 1D).
• Keep probability density and physical density distinct: p(x) with dx is unitless when integrated; physical rho(x,t) with dV yields measurable totals.
• Spectral convention: X(f) = ( ∫ x(t) * exp( -j 2π f t ) dt ); the one-sided PSD S_xx(f) must be accompanied by ENBW_Hz and U_w.

II. Primitive Quantities and Measures

• Omega : sample space ; unit = 1 ; dim = 1
• X : random variable ; unit = unit(x) ; dim = dim(x)
• mu : base measure ; unit, dim depend on domain
• dx / dS / dV : line/area/volume element ; unit = m / m^2 / m^d ; dim = L / L^2 / L^d
• d ell : path element ; unit = m ; dim = L
• gamma(ell) : path in domain ; unit = 1 ; dim = 1
• L_gamma = ( ∫ 1 d ell ) : path length ; unit = m ; dim = L

III. Physical Densities and Flux

• rho(x,t) : generic density ; unit = quantity per m^d ; dim = Q L^-d (Q matches the physical quantity’s dimension)
• rho_m(x,t) : mass density ; unit = kg m^-d (commonly kg m^-3 for d=3) ; dim = M L^-d
• rho_q(x,t) : charge density ; unit = C m^-d ; dim = I T L^-d
• n(x,t) : number density ; unit = 1 m^-d ; dim = L^-d
• J(x,t) : flux ; unit = unit(rho) * m s^-1 ; dim = dim(rho) L T^-1 (e.g., number flux 1 m^-2 s^-1)
• s(x,t) : source/sink in continuity ; unit = unit(rho) s^-1 ; dim = dim(rho) T^-1
• Continuity ∂_t rho + ∇•J = s (see Chapter 2, S92-1): every term has units unit(rho) s^-1.

IV. Probability and Statistics

• p(x) : probability density function ; unit = unit(x)^-d ; dim = dim(x)^-d
• F(x) : cumulative distribution ; unit = 1 ; dim = 1
• P(A) : probability ; unit = 1 ; dim = 1
• theta : parameter ; unit/dim model-dependent
• L(theta) : likelihood ; unit = 1 ; dim = 1
• ell(theta) : log-likelihood ; unit = 1 ; dim = 1
• I_F(theta) : Fisher information ; unit = unit(theta)^-2 ; dim = dim(theta)^-2
• CRLB : Cramér–Rao lower bound (parameter covariance lower bound) ; unit = unit(theta)^2 ; dim = dim(theta)^2
• cov(z) : covariance ; unit = unit(z)^2 ; dim = dim(z)^2

V. Estimation and Kernels (KDE/Smoothing)

• K(u) : kernel (∫ K(u) du = 1) ; unit = 1 ; dim = 1
• h : bandwidth ; unit = unit(x) ; dim = dim(x)
• kde_h(x) : kernel density estimator ; unit = unit(x)^-d ; dim = dim(x)^-d
• R(K) = ( ∫ K(u)^2 du ) : kernel energy ; unit = 1 ; dim = 1
• CV(h) : cross-validation score ; unit = 1 ; dim = 1
• MISE(h) : mean integrated squared error ; unit = unit(p)^2 * unit(x)^d ; dim = dim(p)^2 * dim(x)^d

VI. Spatial and Spatio-Temporal Intensities (Point Processes)

1. lambda(x) : spatial intensity ; unit = 1 m^-d ; dim = L^-d
2. lambda(x,t) : spatio-temporal intensity ; unit = 1 ( m^-d s^-1 ) ; dim = L^-d T^-1
3. Lambda(A) = ( ∫_A lambda dV ) : expected count in region ; unit = 1 ; dim = 1
4. g(r) : pair-correlation ; unit = 1 ; dim = 1
5. H(•) : Hawkes kernel
   • time only: unit = s^-1 ; dim = T^-1 (∫ H(t) dt = n_branch dimensionless)
   • space-time: unit = m^-d s^-1 ; dim = L^-d T^-1 (∫∫ H(x,t) dV dt = n_branch)
6. mu (Hawkes baseline): units/dims as lambda.

VII. Spectra and Energy (Aligned with Core.Sea)

• x(t) : time-domain signal ; unit = unit(x) ; dim = dim(x)
• X(f) = ( ∫ x(t) * exp( -j 2π f t ) dt ) : Fourier transform ; unit = unit(x) * s ; dim = dim(x) T
• S_xx(f) : one-sided PSD ; unit = unit(x)^2 Hz^-1 ; dim = dim(x)^2 T
• fs : sampling rate ; unit = Hz ; dim = T^-1
• w[n] : analysis window ; unit = 1 ; dim = 1
• U_w = ( 1 / N ) * ∑ w[n]^2 : window power ; unit = 1 ; dim = 1
• ENBW_Hz = fs * ( ∑ w[n]^2 ) / ( ∑ w[n] )^2 : equivalent noise bandwidth ; unit = Hz ; dim = T^-1
• f_c : cutoff/center frequency ; unit = Hz ; dim = T^-1
• nu : effective degrees of freedom (Welch) ; unit = 1 ; dim = 1
• SpecRef : spectral object reference (interface) ; unit = meta ; dim = meta

VIII. Normalization and Scale

• z = ( x - mu_x ) / sigma_x : standard score ; unit = 1 ; dim = 1
• scale / shift : scaling/offset ; unit, dim follow x
• unit(x) : unit function ; = unit(x)
• dim(x) : dimension function ; meta

IX. Uncertainty and Bounds (Metrological)

• u(x) : standard uncertainty of x ; unit, dim follow x
• u_c : combined standard uncertainty ; unit/dim of output
• U = k * u_c : expanded uncertainty ; unit/dim of output
• k : coverage factor ; unit = 1 ; dim = 1
• G = ∂g/∂theta : sensitivity (Jacobian) ; unit = unit(g)/unit(theta) ; dim = dim(g) dim(theta)^-1
• H_g : Hessian of g ; unit = unit(g)/unit(theta)^2 ; dim = dim(g) dim(theta)^-2

X. Discretization and Grids

• N : sample size / segment count ; unit = 1 ; dim = 1
• K (Welch/segmentation) : number of segments ; unit = 1 ; dim = 1
• edges : bin/voxel edges ; unit, dim as the variable
• Delta : bin width ; unit, dim as the variable
• V_i : voxel volume/area/length ; unit = m^d ; dim = L^d
• count_i : bin count ; unit = 1 ; dim = 1
• p_hat = count / ( N * Delta ) : 1-D histogram density ; unit = unit(x)^-1 ; dim = dim(x)^-1
• mass_preserve = ( ∑ rho_i * V_i ) : discretized total ; unit = unit(rho) * m^d ; dim = dim(rho) L^d

XI. Cross-Volume Anchors and TOA Two-Form

• n_eff(x,t) : effective refractive index ; unit = 1 ; dim = 1
• c_ref : reference wave speed ; unit = m s^-1 ; dim = L T^-1
• T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) : arrival time (constant-factored) ; unit = s ; dim = T
• T_arr = ( ∫ ( n_eff / c_ref ) d ell ) : arrival time (general form) ; unit = s ; dim = T
• delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) | : two-form disparity ; unit = s ; dim = T
• L_gamma = ( ∫ 1 d ell ) : path length ; unit = m ; dim = L

XII. Differential Operators

• ∂_t : partial derivative wrt time ; unit = s^-1 ; dim = T^-1
• ∇ : gradient/divergence ; unit = m^-1 / m^-1 ; dim = L^-1
• ∇•J : divergence of flux ; unit = unit(J) m^-1 ; dim = dim(J) L^-1
• ∫ : integral (measure must be explicit) ; units/dim cancel per integrand × measure

XIII. Regularization and Maximum Entropy (as in Chapter 8)

• lambda (regularization weight) : penalty weight ; unit/dim balance the objective so terms share the same dimension
• ||•||_2 / ||•||_1 : norms ; units/dims inherit from the quantity under the norm
• TV : total variation ; unit, dim as rho
• phi_k(x) : constraint features ; unit/dim per constraint
• A : forward operator ; unit/dim per model

XIV. Spectral Statistics and Intervals (as in Chapter 10)

• nu : effective degrees of freedom ; unit = 1 ; dim = 1
• var( S_hat(f) ) / S_xx(f)^2 : relative variance ; unit = 1 ; dim = 1
• chi2_{p,1-α} : chi-square quantile ; unit = 1 ; dim = 1
• C_{1-α} : likelihood-ratio region ; unit = parameter space ; dim = dim(theta)

XV. Interfaces and Bindings (I90 Symbol Overview)

• PdfRef / DensRef / SpecRef / IntensRef : typed references ; unit = meta ; dim = meta
• define_measure(…) : define measure ; unit = api ; dim = api
• pdf_fit(…) / kde_build(…) / kde_eval(…) : PDF interfaces ; unit = api ; dim = api
• intensity_estimate(…) / hawkes_fit(…) : intensity interfaces ; unit = api ; dim = api
• hist_density(…) / bin_edges(…) : discretization ; unit = api ; dim = api
• spectral_density(…) / spec_to_energy(…) : spectrum ; unit = api ; dim = api
• fisher_information(…) / crlb(…) : information/bounds ; unit = api ; dim = api
• bind_to_parameters(…) / bind_to_equations(…) / enforce_arrival_time_convention(…) : metadata/alignment ; unit = api ; dim = api

XVI. Quick Consistency Checklist (Pre-Publication)

• Every ∫ states its measure (dx/dV/d ell) and cancels units with the integrand.
• For probabilities, ( ∫ p(x) dx ) = 1; for physical densities, report M = ( ∫ rho dV ).
• KDEs report K(•), h, CV(h), and the estimation dimension d.
• Spectra include ENBW_Hz, U_w, number of segments K, and nu.
• Arrival times publish both forms and delta_form.
• Uncertainty reports U = k * u_c, with coverage factor k and the propagation method (Delta/MC).